FRM - Financial Markets and Products

Financial Risk Manager
EXAM PART I
Financial Markets and Products
<$>GARP
2020
®
EXAM PART I
Financial Markets and Products
Pearson
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Chapter 1 Banks 1 Chapter 2 Insurance Companies
and Pension Plans 15
1.1 The Risks in Banking 2
Market Risks 2 2.1 Mortality Tables 16
Credit Risks 2
2.2 Life Insurance 16
Operational Risks 3
Whole Life Insurance 16
1.2 Bank Regulation 3 Term Life Insurance 19
Capital 3 Endowment Life Insurance 19
The Basel Committee 4 Group Life Insurance 19
Standardized Models versus Annuity Contracts 20
Internal Models 4 Longevity and Mortality Risk 20
Trading Book versus Banking Investments 21
Book 5
Liquidity Ratios 5 2.3 Pension Plans 21
1.3 Deposit Insurance 5 2.4 Property and Casualty Insurance 22
CAT Bonds 22
1.4 Investment Banking 6
Loss Ratios 23
IPOs 7
Dutch Auctions 7 2.5 Health Insurance 23
Advisory Services 8 2.6 Moral Hazard and Adverse
Trading 9 Selection 24
1.5 Conflicts of Interest 9 Moral Hazard 24
Adverse Selection 24
1.6 The Originate to Distribute
Model 10 2.7 Regulation 24
4.4 Options 50
Chapter 3 Fund Management 31
4.5 Market Participants 52
Hedgers 52
3.1 Mutual Funds 32 Speculators 52
Open-End Funds 32 Arbitrageurs 53
Closed-End Funds 33 4.6 Derivatives Risks 53
3.2 Exchange-Traded Funds 34
3.3 Undesirable TradingBehavior 34
3.4 Hedge Funds 35
Chapter 5 Exchanges and OTC
Prime Brokers 37 Markets 57
3.5 Types of Hedge Funds 38
Long-Short Equity 38 5.1 Exchanges 58
Dedicated Short 38 Central Counterparties 59
Distressed Debt 38
5.2 How CCPs Handle Credit Risk 60
Merger Arbitrage 38
Netting 60
Convertible Arbitrage 39
Variation Margin and Daily Settlement 60
Fixed-Income Arbitrage 39
Initial Margin 61
Emerging Markets 39
Default Fund Contributions 61
Global Macro 39
Managed Futures 39 5.3 Use of Margin Accounts
in Other Situations 61
3.6 Research on Returns 39
Options on Stocks 62
Mutual Fund Research 39
Short Sales 62
Hedge Fund Research 40
Buying on Margin 62
5.4 Over-the-Counter Markets 63
Bilateral Netting in OTC Markets 64
Chapter 4 Introduction Collateral 65
to Derivatives 45 Special Purpose Vehicles 65
Derivative Product Companies 65
Credit Default Swaps 65
4.1 The Markets 47
Exchange-Traded Markets 47
Over-the-Counter Markets 47
Market Size 48
4.2 Forward Contracts 48
4.3 Futures Contracts 50
iv ■ Contents
7.7 Placing Orders 88
Chapter 6 Central Clearing 71 Market Orders 89
Limit Orders 89
Stop-Loss Order 89
6.1 The Operation of CCPs 73
Stop-Limit Orders 89
6.2 Regulation of OTC Derivatives Market-if-Touched Orders 89
Market 73 Discretionary Orders 89
6.3 Standard versus Non-Standard Duration of Orders 89
Transactions 74 7.8 Regulation of Futures Markets 89
6.4 The Move to Central Clearing 75 7.9 Accounting 90
6.5 Advantages and Disadvantages 7.10 Forwards Compared
of CCPs 77 with Futures 90
6.6 CCP Risks 78
Model Risk 79
Liquidity Risk 79
Chapter 8 Using Futures
for Hedging 95
Chapter 7 Futures Markets 83
8.1 Long and Short Hedges 96
Short Hedge 96
7.1 Exchanges 84 Long Hedges 96
7.2 Operation of Exchanges 84 8.2 Pros and Cons of Hedging 98
7.3 Specification of Contracts 85 Shareholders May Prefer No Hedging 98
The Underlying Asset 85 There May Be Little or No Exposure 98
Contract Size 85 Hedging May Lose Money 99
Delivery Location 85 8.3 Basis Risk 99
Delivery Time 85
8.4 Optimal Hedge Ratios 100
Price Quotes 86
Tailing the Hedge 101
Price Limit 86
Position Limits 86 8.5 Hedging Equity Positions 102
Managing Beta 103
7.4 Delivery Mechanics 87
Cash Settlement 87 8.6 Creating Long-Term Hedges 104
7.5 Patterns of Futures Prices 87 8.7 Cash Flow Considerations 105
7.6 Market Participants 88
Contents ■ v
10.5 Valuing Forward Contracts 128
Chapter 9 Foreign Exchange
10.6 Forward versus Futures 129
Markets 109
10.7 Exchange Rates Revisited 130
10.8 Stock Indices 130
9.1 Quotes 110 Index Arbitrage 130
Outrights and Swaps 112
Indices Not Representing Tradable
Futures Quotes 113 Portfolios 131
9.2 Estimating FX Risk 113
Transaction Risk 113
Translation Risk 113 Chapter 11 Commodity
Economic Risk 114
Forwards
9.3 Multi-Currency Hedging and Futures 135
Using Options 114
9.4 Determination of Exchange
Rates 115 11.1 Why Commodities Are
Balance of Payments and Trade Flows 115 Different 136
Inflation 115 11.2 Types of Commodities 136
Monetary Policy 116 Agricultural Commodities 136
9.5 Real versus Nominal Interest Metals 137
Rates 116 Energy 137
Weather 138
9.6 Covered Interest Parity 116
Interpretation of Points 118 11.3 Commodities Held
for Investment 138
9.7 Uncovered Interest Parity 118
Lease Rates 139
11.4 Convenience Yields 139
Chapter 10 Pricing Financial 11.5 Cost of Carry 140

A Note on Compounding Frequencies
Forwards for Interest Rates 141
and Futures 123
11.6 Expected Future Spot Prices 141
Early Work 141
10.1 Short Selling 124 Modern Theory 141
Normal Backwardation and Contango 143
10.2 The No Income Case 124
Generalization 126
10.3 The Known Income Case 126
Generalization 127
10.4 The Known Yield Case 128
vi ■ Contents
Chapter 12 Options Markets 147 Chapter 14 Trading
Strategies 171
12.1 Calls and Puts 148
Moneyness 149 14.1 Strategies Involving
Profits from Call Options 149 a Single Option 172
Profits from Put Options 150 Principal Protected Notes 172
Payoffs 150
14.2 Spread Trading Strategies 173
12.2 Exchange-Traded Options Bull Spread 173
on Stocks 151 Bear Spread 174
Maturity 151 Box Spread 175
Strike Prices 151 Butterfly Spread 175
Dividends and Stock Splits 152 Calendar Spread 177
Index Options 152
14.3 Combinations 177
ETP Options 152
Straddle 177
Non-Standard Products 152
Strangle 178
12.3 Trading 152
14.4 Manufacturing Payoffs 178
12.4 Margin Requirements 153
12.5 Over-the-Counter Market 153
12.6 Warrants and Convertibles 153 Chapter 15 Exotic Options 183
12.7 Employee Stock Options 154
15.1 Exotics Involving a Single
Asset 184
Chapter 13 Properties Packages 184
of Options 159 Zero-Cost Products 184
Non-Standard American Options 184
Forward Start Options 185
13.1 Call Options 160 Gap Options 185
American versus European Options: No Cliquet Options 185
Dividends 160 Chooser Options 186
Employee Stock Options 161 Binary Options 186
Impact of Dividends 161 Asian Options 187
Lower Bound When There Are Dividends 162 Lookback Options 187
13.2 Put Options 162 Barrier Options 188
American versus European Options: No Compound Options 188
Dividends 163
13.3 Put-Call Parity 164
13.4 Use of Forward Prices 166
Contents ■ vii
15.2 Exotics Involving Multiple 16.9 Forward Rates 204
Assets 189 Forward Rate Agreement 205
Asset-Exchange Options 189
16.10 Determining Zero Rates 206
Basket Options 189
16.11 Theories of the Term
15.3 Exotics Dependent Structure 206
on Volatility 189
Volatility Swap 189
Variance Swap 190
Chapter 17 Corporate
15.4 Hedging Exotics 190
Bonds 213
Chapter 16 Properties 17.1 Bond Issuance 214

of Interest Rates 195 17.2 Bond Trading 214
17.3 Bond Indentures 216
16.1 Categories of Rates 196 Corporate Bond Trustee 216
Government Borrowing Rates 196 17.4 Credit Ratings 216
Libor 196 High-Yield Bonds 217
Repo Rate 196 17.5 Bond Risk 217
Overnight Interbank Borrowing 197 Event Risk 217
Swaps 197 Defaults 218
Risk-Free Rates 197
17.6 Classification of Bonds 218
16.2 Compounding Frequency 197 Issuer 218
Usual Conventions 199 Maturity 218
16.3 Continuous Compounding 199 Interest Rate 218
16.4 Zero Rates 199 Collateral 219
16.5 Discounting 200 17.7 Debt Retirement 220

Call Provisions 220
16.6 Bond Valuation 200
Bond Yield 201 17.8 Default Rate and Recovery
Par Yield 201
Rate 221
17.9 Expected Return from Bond
16.7 Duration 201
Investments 221
Modified Duration 202
Limitations of Duration 203
16.8 Convexity 203
viii ■ Contents
19.3 Treasury Bond Futures 244
Chapter 18 Mortgages and Quotes 245
Mortgage-Backed Cheapest-to-Deliver Bond Option 245
Securities 225 Calculating the Futures Price 246
19.4 Eurodollar Futures 247
Quotes 248
18.1 Calculating Monthly
Payments 226 Comparison with FRAs 248
Amortization Tables 227 The Libor Zero Curve 249
Hedging 250
18.2 Mortgage Pools 228 SOFR Futures 250
18.3 Agency Mortgage-Backed 19.5 Duration-Based Hedging 250
Securities (MBSs) 229
Trading of Pass-Throughs 229
Dollar Roll 230
Other Agency Products 230 Chapter 20 Swaps 257
Non-Agency MBSs 231
18.4 Modeling Prepayment 20.1 Mechanics of Interest Rate
Behavior 231 Swaps 258
Refinancing 231 Day Count Issues 258
Turnover 232 Confirmations 259
Defaults 232 Quotes 259
Curtailments 232 Swaps Based on Overnight Rates 260
18.5 Valuation of an MBS Pool 232 20.2 The Risk-Free Rate 260
18.6 Option Adjusted Spread 235 20.3 Reasons for Trading Interest
Rate Swaps 260
Comparative Advantage Arguments 262
Chapter 19 Interest Rate 20.4 Valuation of Interest Rate Swaps 263
Futures 241 Libor Forward Rates 264
Valuation Using Forward Rates 264
Libor Zero Rates 264
19.1 Day Count Conventions 242 20.5 Currency Swaps 265
Bonds 242 Valuation 265
Money Market Instruments 242 Other Currency Swaps 266
19.2 Price Quotes for Treasury 20.6 Other Swaps 266
Bonds and Bills 243
Bonds 243 20.7 Credit Risk 267
Treasury Bills 243 Index 273
Contents ■ ix
FRM
C O M M IT T EE
Chairman
Dr. Rene Stulz
Everett D. Reese Chair of Banking and M onetary Econom ics,
The Ohio State University
Members
Richard Apostolik Dr. Attilio Meucci, CFA
President and C E O , Global Association of Risk Professionals Founder, ARPM
Michelle McCarthy Beck, SMD Dr. Victor Ng, CFA, MD

C hief Risk Officer, T IA A Financial Solutions C hief Risk Architect, Market Risk M anagem ent and Analysis,
Goldm an Sachs
Richard Brandt, MD
Operational Risk M anagem ent, Citigroup Dr. Matthew Pritsker
Senior Financial Econom ist and Policy Advisor / Supervision,
Julian Chen, FRM, SVP
Regulation, and Credit, Federal Reserve Bank of Boston
FRM Program Manager, Global Association of Risk Professionals
Dr. Samantha Roberts, FRM, SVP
Dr. Christopher Donohue, MD
Balance Sheet Analytics & M odeling, PN C Bank
G A RP Benchmarking Initiative, Global Association of Risk
Professionals Dr. Til Schuermann
Partner, O liver Wyman
Donald Edgar, FRM, MD
Risk & Q uantitative Analysis, BlackRock Nick Strange, FCA
Director, Supervisory Risk Specialists, Prudential Regulation
Herve Geny
Authority, Bank of England
Group Head of Internal A udit, London Stock Exchange Group
Dr. Sverrir Porvaldsson, FRM
Keith Isaac, FRM, VP
Senior Q uant, SEB
Capital Markets Risk M anagem ent, TD Bank Group
William May, SVP

Global Head of Certifications and Educational Programs, Global
Association of Risk Professionals
x ■ FRM® Committee
Author
John C. Hull, PhD, University Professor, Maple Financial Group
Chair in Derivatives and Risk M anagem ent, University of Toronto
Reviewers
Erick W. Rengifo, PhD, Associate Professor of Econom ics, Charles Currat, PhD, FRM , SVP, Head of Investment Risk
Fordham University M ethodology, Wells Fargo Bank
David W. Wiley, M BA, C FA , President, W HW Investments, LLC Claus-Peter Mueller, PhD, FRM , BCS, Head of Group
Organization, Transformation, and IT, Heta A sset Resolution
Sam Wong, PhD, C E O and Chief Q uant, 5Lattice Securities
Limited Daniel Homolya, PhD, FRM , ERP, Treasury Risk and Control
Manager, Diageo
Tao Pang, PhD, FRM , C FA , Director of the Financial
M athematics Program, North Carolina State University Jeffrey Goodwin, FRM , Founder and President, Crescendo
Investments, LLC
Masao Matsuda, PhD, FRM , C A IA , President and C E O ,
Crossgates Investment and Risk M anagem ent Patrick Steiner, FRM , Large Institution Supervision Coordinating
Com m ittee, Federal Reserve Bank of New York
Colin White, PhD, FRM , Econom ist, Canadian Securities
Transition Office
Attributions ■ xi
Banks
Learning Objectives
A fter com pleting this reading you should be able to:
Identify the major risks faced by a bank, and explain ways Describe the potential conflicts of interest among com
in which these risks can arise. mercial banking, securities services, and investm ent bank
ing divisions of a bank and recommend solutions to the
Distinguish between econom ic capital and regulatory conflict of interest problems.
capital.
Describe the distinctions between the "banking book"
Summarize Basel Com m ittee regulations for regulatory and the "trading book" of a bank.
capital and their motivations.
Explain the originate-to-distribute model of a bank and
Explain how deposit insurance gives rise to a moral hazard discuss its benefits and drawbacks.
problem.
Describe investment banking financing arrangem ents

including private placem ent, public offering, best efforts,
firm com mitment, and Dutch auction approaches.
1
Banks are the cornerstone of the world's financial system . The Consider the exchange rate between the U.S. dollar (USD) and
activities of banks in many countries can be subdivided into the British pound (G BP). If the dem and to buy G B P using USD
com m ercial banking and investm ent banking. is greater than the dem and to sell G B P for USD, the value of
the exchange rate (USD per G BP) will increase. Sim ilarly, if the
Com m ercial banking involves the traditional activities of
dem and to sell G B P is greater than the dem and to buy GBP,
receiving deposits and making loans. These activities can be
the exchange rate will decrease.
categorized as either retail or wholesale. Retail banking involves
transacting with private individuals and small businesses. The values of market variables can be affected by many
W holesale banking involves transacting with large corporations. different events. For exam ple, the value of the British pound
Loans and deposits are much larger in wholesale banking than decreased in June 2016 after the United Kingdom voted to
in retail banking. As a result, the adm inistrative costs per dollar leave the European Union (an event that market participants
of deposits (or loans) are lower. The spread between the rates viewed as bad news for the British economy). Another exam ple
paid on deposits and the rates charged on loans is lower for can be seen with the reinstatem ent of sanctions by the U.S.
wholesale banking as well. governm ent on oil-producer Iran in May 2018. This event led
to an increase in the price of oil because market participants
M eanwhile, investment banking involves a variety of activities
thought that it could reduce the supply of oil in global markets.
such as:
A bank's exposure to movements in the values of market vari
• Raising debt or equity capital for com panies;
ables arises primarily from its trading operations. As previously
• Providing advice to com panies on m ergers, acquisitions, and
explained, proprietary trading by banks is not currently allowed
financing decisions; and
in the U.S. However, banks provide corporate clients and insti
• Acting as a broker-dealer for trading debt, equity, and other tutional investors with a wide range of products whose values
securities. depend on the prices of market variables. Consider again the
In some countries, the commercial banking and investment USD per G B P exchange rate. Among the transactions a corpo
banking sectors are strictly separated. The U.S., for example, rate client may request are as follows.
once limited the extent to which a single corporation could • S p o t transactions: where G B P is bought or sold for almost
engage in both commercial and investment banking. Until the immediate delivery.
repeal of the Glass-Steagall A ct in 1999, investment banks
• Forw ard contracts: where an exchange rate for the purchase
were not allowed to take deposits and make loans while
or sale of a certain amount of G B P on a future date is agreed.
commercial banks were not allowed to arrange equity issuances
• O ption s: where one side has the right (but not the obligation)
for other companies.
to buy or sell G B P at a pre-arranged price (i.e., the exercise
Following the financial crisis of 2007-2008, policym akers in price) at a certain future time.
some countries prohibited banks from putting depositors' funds
For many of these contracts, banks act as market makers by quot
at risk by engaging in proprietary trading (often referred to as
ing both a bid (i.e., the price at which they are prepared to buy)
prop trading). This is the speculative trading that an investment
and an ask (i.e., the price at which they are prepared to sell). Banks
bank does in the hope of increasing its profitability.
typically ensure that their exposures to market variables are kept
within certain limits, but they do not (usually) eliminate those
1.1 THE RISKS IN BANKING exposures entirely. As a result, banks are always exposed to some
market risk.
In this section, we explain three major risks that banks face.
In the next section, we will outline the way in which banks are
regulated to ensure that they can survive these risks.
Credit Risks
Credit risk arises from the possibility that borrowers will fail to
repay their debts. For banks, loans to corporations and individu
Market Risks
als are a major source of credit risk. If a borrower defaults, a loss
M arket risks are the risks arising from a bank's exposure to is usually incurred. In a bankruptcy, the size of the loss depends
movements in market variables (e.g ., exchange rates, interest on whether assets have been pledged as collateral and how the
rates, com m odity prices, and equity prices). These market vari bank's claims rank com pared with those of other cred itors.1
ables are often referred to as risk factors. The value of a market
variable is determ ined by trading in the financial markets. 1 This is discussed in more detail in C hap ter 17.
2 ■ Financial Risk Manager Exam Part I: Financial Markets and Products

A bank builds expected losses into the interest rate it charges This definition includes all risks that are not market or credit
on loans. For exam ple, suppose the bank's cost of funds (the risks (with the exception of strategic and reputational risks).
average interest rate paid on deposits and on the bank's debt) Operational risk is harder to quantify than either market risk or
is 1.5%. The average interest rate charged on loans might be credit risk. Exam ples from seven categories of operational risk
4%. The difference between the two interest rates (2.5% in identified by regulators include the following.
our exam ple) is referred to as the net interest margin. If a bank
• Internal fraud: Rogue traders intentionally misreporting posi
expects to lose 0.8% of what it lends, it will be left with 1.7% to
tions or em ployees stealing from the bank by creating loans
cover adm inistrative/operational costs and contribute to profits.
to fictitious com panies.
In this exam ple, 0.8% is the bank's expected (or average) loan • External fraud: Cyberattacks, bank robberies, forgery, and
losses. However, loan losses show significant variation from year check kiting.
to year. During stressed econom ic conditions, a bank might
• Em ploym ent practices and work place safety: W orker
experience loan losses as high as 4%, while during good
com pensation claims, em ployee discrimination claims, and
econom ic times these losses might be as low as 0 .2 % .2 Current
litigation arising from personal injury claims at bank branches.
regulations require banks to maintain enough capital to cover
losses estim ated to occur only once every thousand years.3 • Clients, p ro d u cts, and business p ractices: Money laundering
and other actions that are either unlawful or prohibited by
Other bank contracts also give rise to credit risk. For example, regulators.
banks trade a variety of derivatives (e.g., forward contracts and
• Dam age to physical assets: Terrorism, vandalism , earth
options). As already indicated, these give rise to market risk
quakes, fires, and floods.
because the value of a derivatives contract depends on the under
lying market variables. Derivatives also give rise to credit risk. This • Business disruption and system failures: Hardware and
comes from the possibility that the counterparty to a derivatives software failures, telecom m unication problem s, and utility
transaction will default when the transaction has a positive value outages.
to the bank (and therefore a negative value to the counterparty). • Execu tion, delivery, and p ro ce ss m anagem ent: Data entry
errors, collateral m anagem ent issues, and inadequate legal
Banks typically account for expected losses on transactions as
docum entation.
soon as they are initiated. An accounting rule known as IFRS 9
requires banks to estim ate expected losses on their loan port Operational risk is regarded by many to be a greater challenge
folios and to show the outstanding principal net of expected for banks than either market risk or credit risk. Between 2008
losses on their balance sheet. In the case of derivatives, and 2017, banks in North Am erica and Europe have been fined
banks calculate a credit value adjustm ent (CVA) reflecting the over USD 300 billion for operational risk violations such as
amount they expect to lose due to counterparty default. This money laundering, market manipulation, terrorist financing, and
is subtracted from the balance sheet value of the outstanding inappropriate activities in the mortgage market.
derivatives. In both cases, expected losses, even though they
Significant sources of operational risk in banking include cyber risk,
have not (yet) been incurred, are charged to income.
legal risk, and compliance risk (i.e., failure to comply with rules and
regulations, either accidentally or intentionally). These risks are dis
Operational Risks cussed further in Chapter 7 of Valuation and Risk M odels.
Operational risk is defined by bank regulators as:4
The risk o f loss resulting from inadequate or failed internal 1.2 BANK REGULATION
p ro cesses, p eo p le, and system s or from external events.
Banks are subject to regulations designed to protect depositors
as well as maintain confidence and stability in the financial sys
tem . In this section, we outline the developm ent of the global
2 Statistics published by Moody's show that between 1970 and 2016, banking regulatory environment.
the default rate for all rated com panies varied between 0.088% in 1979
to 4.996% in 2009.
3 This is explained in more detail in C hapter 6 of Valuation and Risk Capital

M o d els.
4 See Bank for International Settlem ents, "W orking Paper on the Regula It is im portant for banks to keep sufficient capital for the risks
tory Treatm ent of O perational Risk," Septem ber 2001. they are taking. The most im portant capital is equity capital.
Chapter 1 Banks ■ 3
Because losses have the effect of reducing equity capital, banks By the 1990s, however, bank trading activities had significantly
must try to maintain enough equity capital to cover potential increased. In response, the Basel Com m ittee agreed that banks
losses and remain solvent (i.e., have a positive amount of equity should keep capital for both market risk and credit risk. This
capital). Debt capital is the other main category of capital, and modification to Basel I, known as the M arket Risk A m endm ent,
it is usually subordinate to assets held for depositors (therefore was im plem ented in 1998.
providing an extra degree of protection for depositors).
In 1999, the Basel Com m ittee proposed what has become
Equity capital is som etim es referred to as goin g concern capital known as Basel II. This agreem ent revised the procedure for cal
because it absorbs losses while the bank is a going concern culating credit risk capital and introduced a capital requirement
(i.e., it remains in business). Debt capital is referred to as g o n e for operational risk. It took about eight years for the final Basel
concern capital because it is only affected by losses once a bank II rules to be worked out and im plem ented. The total capital
has failed. In theory, depositors are at risk only when losses are requirem ent was then the sum of amounts for (a) credit risk,
sufficiently large to wipe out both equity and debt capital. (b) market risk, and (c) operational risk.
We can distinguish between regulatory capital and econom ic The 2007-2008 crisis led to several bank failures and bailouts.
capital. Regulatory capital is the minimum capital that regulators Global bank regulators subsequently determ ined that the rules
require banks to keep. Economic capital is a bank's own estimate for calculating market risk capital were inadequate. Thus, the
of the capital it requires. In both cases, capital can be thought of rules were revised in what is referred to as Basel II.5
as funds that are available to absorb unexpected losses. A com
The Basel Com m ittee also decided that equity capital require
mon objective in calculating economic capital is to maintain a high
ments needed to be increased. This latest set of regulations,
credit rating (as will be described in later chapters). Economic
called Basel III, includes a large increase in the amount of equity
capital is allocated to a bank's business units so that they can be
capital that banks are required to keep and is expected to be
compared using a return on allocated econom ic capital metric.
fully im plem ented by 2027.7
The amount of capital that is necessary depends on the size of
M eanwhile, the rules for market risk have been revised yet again
possible losses. If a bank's equity capital is USD 4 billion and
with the Fundam ental R eview o f the Trading Book, which is due
there is a 1% chance that the bank will incur a loss higher than
to be im plem ented in 2022.
USD 4 billion over a year, the equity capital will be consid
ered insufficient by both regulators and the bank itself. This is
because even a 1% chance that the bank will becom e insolvent Standardized Models versus Internal
is unacceptable. As mentioned earlier, the regulatory capital Models
for credit risk is designed to be sufficient to cover a loss that is
expected to be exceeded only once every thousand years.5 Models are necessary to determ ine bank capital. Some models
are standardized tools developed by the Basel Com m ittee, while
others are internal models developed by the banks them selves.
The Basel Committee Generally, banks need approval from regulators before they can
use a specific internal model.
The Basel Com m ittee for Banking Supervision was established
in 1974 to provide a forum where the bank regulators from The models for credit risk that were introduced in Basel I were
different countries could exchange ideas.6 Prior to 1988, bank standardized models developed by the Basel Com m ittee. This
regulation and enforcem ent varied from country to country. means that two banks, when presented with the same portfolio,
In 1988, there was an international agreem ent (which became should calculate the same capital requirem ents. The M arket Risk
known as Basel I) that required regulators in all signatory Am endm ent included a standardized model approach and an
countries to calculate capital requirements in the same manner. internal model approach. Banks could determ ine market risk
Initially, these capital requirements were designed to cover capital using an internal model provided that the model satis
losses arising from defaults on loans and derivatives contracts. fied the requirements laid down by the Basel Com m ittee and
was approved by national regulators. Basel II allowed internal
models to be used to determ ine both credit risk capital and
operational risk capital.
5 See C hapter 6 of Valuation and Risk M o d els for further discussion.

7 Bank for International Settlem ents. (2019, March 20). Basel III m onitor
6 The Basel Com m ittee for Banking Supervision is based at the Bank for ing results p u b lish ed by the Basel C om m ittee [Press release]. Retrieved
International Settlem ents in Basel, Switzerland. from https://w w w .bis.org/press/p190320.htm

Since the crisis, the Basel Com m ittee has decided to reduce environments) is to issue a three-month commercial paper.
bank reliance on internal models. The com m ittee felt that it A t the end of the three months, a new three-month commercial
had given banks too much freedom to choose internal models paper is issued and used to repay the first issuance. A t the end
that would produce the lowest capital requirem ents. It now of a further three months, there is a third issuance of a three-
requires that banks use a standardized model for determining month commercial paper, which would be used to repay the
operational risk capital. For credit risk and market risk, banks second issuance, and so on.
must calculate capital using a standardized model and can
A risk with this strategy comes when the commercial paper can
(if they receive approval from their national regulators) also cal
not be rolled over in the way we have described. If the market
culate capital using an internal model. However, these internal
(rightly or wrongly) loses confidence in the bank, it is likely that
models cannot reduce total capital requirements below a mini
the maturing commercial paper cannot be replaced (or must
mum level that is set equal to a certain percentage of the capital
be replaced at much higher interest rates). Unless the bank has
given by the standardized approach. By 2027, this percentage
other guaranteed lines of credit, it could default on its debt and
will be 72.5% . This means that:
go bankrupt. Note that if the five-year loans had been financed
Required Capital = m ax(IM C,0.725 X SMC) with five-year debt, this problem would have been avoided
because the loan repayments could have been used to repay
where IMC is the capital given by the internal models and SM C
the debt.
is the capital given by the standardized (Basel Com m ittee)
models. The failure of Northern Rock in the United Kingdom can be
traced to this type of liquidity problem. The British bank had
a mortgage portfolio that it was partly funding with commer
Trading Book versus Banking Book
cial paper. W hile this m ortgage portfolio was not unduly risky,
When calculating regulatory capital, it is im portant to distinguish problems in the U.S. m ortgage market made investors nervous,
between the trading book and the banking book. The trading and the commercial paper could not be rolled over. Lehman's
book (as its name implies) consists of assets and liabilities that demise in 2008 was also largely a result of liquidity problems of
are held to trade. The banking book consists of assets and this type.
liabilities that are expected to be held until maturity. Items in
A s a result of the liquidity problem s encountered during the
the trading book are subject to market risk capital calculations,
crisis, the Basel Com m ittee has (as part of Basel III) developed
whereas items in the banking book are subject to credit risk
two liquidity ratios to which banks are required to adhere.
capital calculations. These calculations are quite different. In
The Liquidity C o vera g e Ratio is a requirem ent designed to
the past, there had som etim es been am biguity as to whether
ensure that banks have sufficient sources of funding to survive
a transaction (e.g ., a credit derivative) should be in the bank
a 30-day period of acute stress (e .g ., where it is dow ngraded,
ing book or the trading book. Banks tended to take advantage
loses deposits, or has drawdow ns on its lines of credit). The
of this ambiguity by putting each transaction in the book that
N e t Stable Funding Ratio is a requirem ent that limits the size
would lead to the lowest capital requirem ent (usually this was
of m ism atches betw een the m aturity of assets and the maturity
the trading book).
of liabilities.
The Fundam ental Review o f the Trading B ook mentioned ear
lier attem pts to clarify the Basel Com m ittee's rules concerning
whether an instrument should be in the banking book or the 1.3 DEPOSIT INSURANCE
trading book. If a bank has a desk for trading a specific instru
To maintain confidence in the banking system , many countries
ment, that instrument will normally be considered to be part of
have introduced deposit insurance. This typically provides a
the trading book. O therw ise, it will be part of the banking book.
certain amount of protection to a depositor against losses aris
ing from a bank failure. In the U .S., the amount is currently USD
Liquidity Ratios 250,000. In some jurisdictions, all banks pay the same insurance
premium per year per dollar of deposit insured. In other jurisdic
Many of the problems experienced during the financial crisis
tions (such as the U .S.), the insurance premium is based on an
were a result of a lack of liquidity, rather than a shortage of
assessm ent of each bank's risk.
capital. Consider a bank that wants to fund five-year loans. One
possibility is to issue five-year bonds so that the maturities of If deposit insurance were provided to a bank without any other
its assets and liabilities are m atched. A tem pting alternative measures being taken, the insurance might encourage the bank
that could lead to lower funding costs (in many interest rate to take on more risks than they would otherwise. For exam ple,
banks could offer slightly above average interest rates to deposi can be on a b e st efforts or a firm com m itm ent basis. As its name
tors and then use the funds to make risky loans at relatively high implies, best efforts means that the bank will do its best to sell
interest rates to borrowers. W ithout deposit insurance, this would the securities for the agreed upon price. However, there are
not be possible because depositors would withdraw their money no guarantees. The bank is paid a fee that usually depends
when the risks being taken became apparent. With deposit (to some extent) on how successful it has been in selling the
insurance, the strategy might be feasible because depositors securities for the agreed upon price.
know that they are protected in the event of bank failure and will
In the case of a firm commitment, the bank does guarantee that
appreciate the above average interest rates they are receiving.
the securities will be sold for an agreed upon price. The bank
This argument is an exam ple of what is known as a moral hazard, buys the securities at the agreed upon price and then attempts
which can be defined as the risk that the behavior of an insured to sell them for a higher price. Its profit is the difference between
party will change because of the mere existence of the insur the two prices. If it misjudges the market and is unable to sell the
ance, and thus the insurance contract will become riskier. It is a securities for more than the agreed upon price, it will incur a loss.
serious consideration in deposit insurance, because governments A firm commitment is sometimes referred to as a b o u g h t deal.
certainly do not want to set up a program that encourages a
A firm commitment arrangem ent is riskier for an investment
bank to take larger risks.
bank (but less risky for the issuing company) than a best efforts
Risk-based deposit insurance premiums reduce the moral hazard arrangem ent. Suppose that a company wants to issue 10 million
to some extent. The moral hazard is also lessened by regulations new shares. It is currently publicly traded, and its share price
that ensure that a bank's required capital increases with the risks (which has risen recently) is USD 58. In negotiations with its
it takes (see Section 1.2). investment bank, there are two offers on the table:
1. A best efforts arrangem ent where shares will be sold at the

best possible price and the bank will be paid USD 1.50 per
1.4 INVESTM ENT BANKING share sold (to keep the exam ple sim ple, we assume that
the bank's fee does not depend on the price at which the
A major activity of a bank's investment banking arm is rais
shares are sold); and
ing capital for com panies in the form of debt, equity, or more
com plicated securities (e.g ., convertible debt). This process is 2 . A firm commitment arrangem ent where the bank guaran
referred to as underwriting. Typically, a company will approach tees that the shares can be sold for USD 50.
the investment bank to discuss its plans to issue securities. Once Table 1.1 summarizes these alternatives from the perspective
the plans have been agreed upon, the securities are originated of the investm ent bank and considers two outcom es. In the
along with docum entation itemizing the rights of investors who first one, the shares can be sold for USD 55; in the second
purchase the securities. A prospectus detailing the company's one, they can be sold for USD 48. The best efforts alternative
past perform ance and future prospects is also produced. This is certain to give the bank a gross profit of USD 15 million.
includes a discussion of risks, any outstanding lawsuits, and On the other hand, the firm com mitment alternative is much
other relevant information. There is usually a road show in which riskier. If the shares can be sold for USD 55, the bank will make
senior m anagem ent from the issuing company and executives USD 50 million. If the shares can only be sold for USD 48,
from the investment bank attem pt to persuade investors to buy however, the bank will lose USD 20 million.
the securities. Finally, a price for the securities is agreed upon
The decision taken by the bank will depend on the subjective
between the bank and the issuing com pany, and the bank then
probabilities it assigns to different outcom es in conjunction
proceeds to market the securities.
with its risk ap p etite. For the com pany, the risks are less with a
There are two types of offerings. firm commitment because it knows it will realize USD 500 million
1. Private placem en ts: where the securities are sold (or placed) (regardless of the final m arket price). Under the best efforts
with a small number of large institutional investors arrangem ent, the m aximum am ount realized (after co nsid er
(e.g ., pension plans and life insurance companies). ing the bank's fee) would be USD 535 million for the first sce
nario in Table 1.1 and a maximum of USD 465 million for the
2 . Public offerings: where securities are offered for sale to the
second scen ario .8
general public.
In the case of a private placem ent, the investment bank receives

8 Table 1.1 may understate the risks of a best efforts arrangem ent to the
an agreed upon fee. In the case of a public offering, the agree com pany. If there is a dram atic m arket downturn, the issue may be w ith
ment between the investment bank and the issuing company drawn so that the com pany raises no capital.

Table 1.1 Profit to Bank from Best Efforts and Firm Commitment Alternatives to Sell 10 Million Shares
(USD Million)
Best Efforts, Fee Equals USD 1.50 Firm Commitment, Bank Buys Shares
per Share Sold for USD 50
Price Realized = USD 55 + 15 +50
Price Realized = USD 48 + 15 -2 0
IPOs the bids presented in Table 1.2. To evaluate the bids, it is nec
essary to sort bidders from the highest to the lowest. This has
An IPO (initial public offering) is a first-time offering of a com been done in Table 1.3. We then search for the maximum price
pany's shares to the public. Prior to an IPO , shares are typically at which 500,000 shares or more can be sold. From Table 1.3,
held by the company's founders, venture capitalists, and others we see that 30,000 shares have been bid for at USD 70 or
who have provided early stage funding. The shares being sold more, 130,000 have been bid for at USD 65 or more, 170,000
can be a mixture of existing and new shares, which can provide have been bid for at USD 63 or more, and so on. Furtherm ore,
additional capital for the company. Som etim es the founders 480.000 shares have been bid for at USD 56 or more and
retain control by arranging for the shares they keep in the com 680.000 have been bid for at USD 55 or more. The maximum
pany to have better voting rights than other shares. price at which 500,000 shares can be sold is therefore USD 55.
Because the company's shares do not yet trade on an exchange, All successful bidders pay this price. The seven highest bidders
it is difficult for an investm ent bank to accurately assess what the in Table 1.3 get the full amount of the shares for which they bid.
share price will be after the IPO. Bidder D gets 20,000 shares (the difference between the
500.000 being sold and the 480,000 for which a higher price
For exam ple, suppose the company wishes to raise USD 100
than USD 55 has been bid).
million. The investment bank must try and estim ate
An advantage of Dutch auctions is that (if all potential investors
Value of Com pany A fter USD 100 Million Cash Injection
in a company bid) the price charged is the one that balances
Number of Shares Post IPO
supply and demand in the market. In theory, the post-IPO price
Typically, the investm ent bank sets the offering price below its should be similar to the pre-IPO price.
best estim ate to make it more likely that it can sell the issue at
One high profile IPO that used the Dutch auction approach was
the offering price.
that of G oogle in 2004. This auction was a little different from
There is often a substantial increase in the share price after an
IPO . This means that the company could have probably issued
shares at a higher price, thereby raising more money. It also Table 1.2 Bids for Ten Participants in a Dutch Auction
indicates that IPOs tend to be good investm ents. Unfortunately, when 500,000 Shares are Being Sold
it is often difficult for small investors to buy IPOs.
Bidder Number of Shares Requested Price Bid (USD)
A 100,000 65
Dutch Auctions B 50,000 60
The advantage of using investm ent banks to handle an IPO is C 30,000 70

that they have the necessary expertise as well as relationships D 200,000 55
with potential investors. However, some issuers feel that they
E 70,000 58
would prefer for the market to decide the right price for their
company. One way they can do this is through a Dutch auction. F 150,000 61
This is a procedure where all investors (not just clients of an G 40,000 63
investment bank) are invited to submit bids indicating how many
H 40,000 56
shares they would like to purchase and at what price.
I 80,000 54
As a simple exam ple of how a Dutch auction works, suppose
J 100,000 50
that a company wants to sell 500,000 shares and has received
Table 1.3 Bids in Table 1.2 Sorted from Highest to Lowest
Bidder Number of Shares Requested Cumulative Number of Shares Requested Price Bid (USD)
C 30,000 30,000 70
A 100,000 130,000 65
G 40,000 170,000 63
F 150,000 320,000 61
B 50,000 370,000 60
E 70,000 440,000 58
H 40,000 480,000 56
D 200,000 680,000 55
I 80,000 760,000 54
J 100,000 860,000 50
the "plain vanilla" Dutch auction we have described. Instead, bankers will also advise com panies that are the subject of a
G oogle reserved the right to change (at the last minute) the takeover attem pt by another company.
number of shares that would be offered and the percentage of
In advising Com pany A on a potential takeover of Com pany B,
the requested amount allocated to each bidder. W hen it saw
it is necessary for an investment bank to value Com pany B and
the bids, it decided that the number of shares being offered
to assess any potential synergies (i.e., cost savings, economies
would be 19,605,052 at a price of USD 85. The total value of
of scale, market share, or other benefits from merging the two
the offering was therefore USD 1.67 billion, and investors who
com panies). It must also consider the type of offer that should
had bid USD 85 or more got 74.2% of the shares for which they
be made. This could be a:
had bid. This was a surprising decision. G oogle could have
raised USD 2.25 billion instead of USD 1.67 billion with a more • Cash offer: where the existing shares of Com pany B are pur
usual Dutch auction (where investors bidding USD 85 or more chased for cash,
got 100% of the shares for which they had bid). Perhaps found • Share-for-share exch an ge: where newly issued shares of
ers Sergei Brin and Larry Page anticipated (correctly as it turned Com pany A are exchanged for those of Com pany B so that
out) that they would be able to issue more shares at a much Com pany B's shareholders becom e shareholders of C om
higher price later on. pany A , or
On the first day of the new issuance, Google's shares closed at • Com bination of a cash offer and a share-for-share exchange.
USD 100.34 (i.e., 18% above the issue price). This was followed In a cash offer, the acquisition's risk and uncertainties are borne
by a further 7% increase on the second day. In this exam ple, the by the acquiring com pany. In a share-for-share exchange, they
use of a Dutch auction did not eliminate the IPO underpricing are shared between the two com panies.
problem we mentioned earlier. G oogle did use two investment
The initial offer is not usually the final offer, and the investment
banks (Morgan Stanley and Credit Suisse First Boston) to assist
bank must use its experience to develop a reasonable plan
in the issuance. However, the fee paid was less than it would
for the price negotiations. The investm ent bank must assess
have been for a regular IPO .
the best way to approach the m anagem ent of the target
company. The takeover may be hostile (i.e., opposed by existing
Advisory Services management) or friendly (i.e., supported by m anagem ent). In
some instances, it may be necessary for investment bankers to
In addition to handling securities issuances, investm ent banks
consider antitrust concerns and whether regulatory approval for
also offer advice to corporations on decisions involving mergers
the m erger will be necessary.
and acquisitions, divestm ents, and restructurings. Specifically,
they assist com panies in finding acquisition partners and in The com panies targeted by takeover attem pts are also advised
finding buyers for divisions that are to be divested. Investment by investment bankers. Som etim es a company (often with the

advice of an investment bank) will take steps to avoid being Banks also offer brokerage services to retail clients. They will
taken over. These are known as poison pills. Exam ples of poison take orders from a client and arrange for them to be executed
pills are on an exchange. A full-service broker offers investment research
and advice. O thers offer low-cost services where little or no
• Granting key em ployees attractive stock options that can
advice is given to clients. Som etim es a broker manages discre
be exercised in the event of a takeover— this could dissuade
tionary accounts where clients have entrusted them to make
a potential acquirer from proceeding because the key
investment decisions on their behalf.
em ployees will almost certainly leave;
• Adding a provision to the company's charter making it
impossible for a new owner to fire the existing directors for
a period of tim e; 1.5 CO N FLICTS O F INTEREST
• Issuing preferred shares that automatically get converted to
Banking gives rise to several potential conflicts of interest. The
regular shares in the event of a takeover;
following are exam ples.
• Allowing existing shareholders to purchase shares at a
• Suppose that an investment banker is handling a new equity
discount in the event of a takeover;
issue for a client and is finding it difficult to persuade inves
• Changing the voting structure so that m anagem ent has
tors to buy the shares. The investment banker might ask the
sufficient votes to block a takeover; and
bank's brokers to advise clients to buy the securities and to
• Adding a provision that allows remaining shareholders to sell buy them for clients' discretionary accounts.
shares to the new owner at a 50% premium in the event of a
• Suppose an investment banker is advising a client about a
successful takeover.
possible acquisition. It knows that the target company is a
Poison pills are illegal in many countries, but they are permitted client of the commercial banking arm of the bank. Investment
in the U.S. However, they must be approved by the majority of bankers might ask the commercial bankers for their im pres
shareholders. If shareholders feel that the poison pills will benefit sions of the target and thereby gain confidential information
management at the expense of shareholders, they are likely to that can be passed on to the acquiring company.
vote against them. One argument in favor of poison pills is that • Suppose an investment bank is hoping for lucrative business
they benefit shareholders by improving the negotiating position of from a company. It might contact researchers that work for
management in the event of a takeover attempt (potentially result the brokerage end of the bank and ask them to come up with
ing in a price more favorable to existing shareholders). a "buy" recommendation for the company's stock to please
the company's managem ent.
• Suppose that a bank's commercial banking arm has a large
Trading
loan outstanding to a client where there is a high probability
A nother activity of investm ent banking is trading. As discussed of a loss. It might suggest to the client that it replace the loan
earlier, regulations im plem ented since the 2007-2008 financial with a bond issue handled by the bank's investment banking
crisis have limited the extent to which banks can do speculative arm. If the investment banking arm agrees to this, the debt
trading. In the U.S., the Volcker rule (part of the D odd-Frank is likely to be taken over by investors who are less informed
Act) does not allow U.S. banks to engage in proprietary trad than the commercial bankers.
ing. In some other countries, proprietary trading must be ring
C o nflicts of interest are handled in part by w hat are term ed
fe n ce d to ensure that depositors are not adversely affected by
C h in ese walls. Th ese are rules within a bank preventing the
any losses.
tran sfer of inform ation from one part of the bank to another.
Banks offer market-making services where they quote bids and A cynic m ight argue that a bank will not (in practice) enforce
offers on a wide range of different products to meet the needs C hinese walls if they reduce profits. However, it is in the
of corporate treasurers and institutional investors. These prod bank's interest to enforce C hinese w alls. Big fines can be (and
ucts typically depend on exchange rates, interest rates, precious have been) levied fo r violations of conflict of interest rules.
metal prices, and so on. Some of the products involve options, A d d itio n ally, a bank's reputation is its most valuable asset.
forward contracts, and more com plex derivatives that will be A bank that is seen to be ignoring conflict of interest rules
explained in later chapters. Typically, a bank will enter into a can lose business. Th ese fines and reputational costs are
contract with a corporate end user and then hedge all or part of generally much greater than any gains arising from conflict of
its risk by trading with another financial institution. interest violations.
The types of conflicts of interest mentioned earlier are the rea
son why U.S. regulators had separated investm ent banking from
commercial banking. Under the G lass-Steagall A ct of 1933,
commercial banks could assist with the issues of Treasury and
municipal bonds and handle private placem ents, but they were
not allowed to handle public offerings and other investment
banking activities. Similarly, investm ent banks were not allowed
to take deposits and make commercial loans.
In 2007, there were five large investm ent banks in the U .S.:
Goldm an Sachs, Morgan Stanley, Merrill Lynch, Bear Stearns,
and Lehman Brothers. The 2007-2008 financial crisis led to a big
upheaval. Lehman declared bankruptcy, Bear Stearns was taken
over by JPM organ Chase, and Merrill Lynch was taken over by
Bank of Am erica. Goldm an Sachs and Morgan Stanley became
banking holding com panies with both commercial and invest of default. In 1999, they started to accept subprim e m ortgages,
ment banking interests. which are much riskier.
Since the 1990s, banks have used the originate-to-distribute

model for a wide range of loans without the help of a govern
1.6 THE ORIGINATE TO DISTRIBUTE ment agency (and in most cases without payment guarantees).
M ODEL This means that the loans originated by banks are converted
into securities and the investors who buy the securities bear the
Traditionally, banks have originated loans and kept them on credit risk. This process is known as securitization. It enables
their balance sheet. An alternative to this is what has becom e banks to remove loans from their balance sheets and frees up
known as the originate-to-distribute m odel. Under this model, funds so that more loans can be m ade. The bank earns a fee for
banks use their expertise to originate loans and then sell them originating the loan and a further fee if it services the loan after
(directly or indirectly) to investors. it has been originated.
Originate-to-distribute arrangem ents have been used in the Tranches are often created from loan portfolios so that each
U.S. m ortgage market for many years. The U.S. governm ent has tranche contains different exposures to losses on the portfolio.
sponsored the creation of three entities: A simplified exam ple is shown in Figure 1.1. A loan portfolio
1. Governm ent National M ortgage Association (GNM A) or with a total principal of USD 100 million is sold by the bank to a
Ginnie Mae, special purpose vehicle (SPV), which arranges for the cash flows
from the loans to be passed to three tranches. The senior, mez
2. Federal National M ortgage Association (FNM A) or Fannie
zanine, and equity tranches fund 70%, 25%, and 5% of the loan
Mae, and
portfolio, respectively. The returns received by tranche holders if
3 . Federal Home Loan M ortgage Corporation (FHLM C) or they do not bear any losses are 5% for the senior tranche, 8%
Freddie Mac. for the mezzanine tranche, and 25% for the equity tran ch e.10
These agencies buy m ortgage portfolios from banks and other Repaym ents of principal flow first to the senior tranche. When
m ortgage originators, package the cash flows into securities, that has been repaid, they flow to the mezzanine tranche, and
and sell the securities to investors. These investors are not sub when the mezzanine tranche has been repaid, they flow to the
ject to credit risk because the respective agency guarantees the equity tranche. Interest payments flow to the senior tranche
m ortgage paym ents. However, the securities are subject to pre until it has received its promised return of 5% on the outstand
paym ent risk. This is the risk that the mortgage principal will be ing principal. They then flow to the mezzanine tranche until it
paid off by the borrower earlier than exp ected .9 Prior to 1999, has received its promised return of 8% on outstanding principal.
the agencies only handled m ortgages that had a low probability Finally, they flow to the equity tranche.
9 This tends to happen when interest rates have decreased, and the
m ortgage can be refinanced at a lower interest rate (but it may also 10 As will be explained, losses can reduce these returns. The reduction
happen sim ply because a house has been sold). The investor then must can be particularly severe for the equity tranche because it bears the
reinvest his or her funds at a lower interest rate. first USD 5 million of losses.

From this description, we see that the equity tranche bears the investors, etc. Since 1988, global rules for regulating banks have
first 5% of losses, the mezzanine tranche bears the next 25% of been determ ined by the Basel Com m ittee. Bank regulatory capi
losses, and the senior tranche bears all losses in excess of 30%. tal is currently the sum of amounts for market risk, credit risk,
and operational risk.
The originate-to-distribute model played a role in the 2007-2008
crisis. Banks relaxed their mortgage lending standards so that To maintain confidence in the banking system , many countries
the quality of the mortgages being originated declined. Despite have developed deposit insurance procedures. These protect
this decline in quality, however, banks still managed to securitize depositors against losses arising from a bank failure. There is
them . In fact, they re-securitized mortgages by creating tranches usually a maximum amount that can be claimed by any one
from tranches. As defaults on the mortgages grew higher, losses depositor from the insurance fund.
were experienced by tranche holders. Unsurprisingly, for a
There are potential conflicts of interest in a bank's activities and
period after the crisis, the originate-to-distribute model could
banks have developed internal rules to avoid them . Violations of
not be used because it was not trusted by investors.
the conflict of interest rules are likely to be costly both in terms
of fines as well as loss of reputation.
SUMMARY Banks som etim es sell loans they have originated to investors.
This is known as the originate-to-distribute model. This model
Banks can engage in many different types of activities. They was heavily used for subprim e m ortgages in the period im m e
can take deposits, make loans, trade securities, arrange security diately before the 2007-2008 financial crisis and was partly
issues, provide advice to corporations, provide services to retail responsible for the crisis.
The following questions are i to help candidates understand the material. They are not actual FRM exam questions.
QUESTIONS
Short Concept Questions

1.1 W hat is the difference between commercial banking and 1.6 Explain the difference between best efforts and firm com
investment banking? mitment when an equity issue is made.
1.2 W hat are the three main risks in banking mentioned in the 1.7 W hat is an IPO?
chapter?
1.8 W hat is a broker's discretionary account?
1.3 W hat is the difference between regulatory capital and
1.9 W hat is the originate-to-distribute model?
econom ic capital?
1.10 W hat are the two types of offers that can be made by the
1.4 W hat are the two liquidity ratios introduced as part of
acquiring company in a takeover?
Basel III?
1.5 Explain the difference between a private placem ent and a

public offering.
Practice Questions
1.11 W hat was the effect of the G lass-Steagall A ct in the 1.18 The bids and bidders in a Dutch auction to sell 20,000
U .S.? When was it repealed? W hat changes along the lines shares are as follows:
of G lass-Steagall have been made since the 2007-2008
crisis? Bidder Number of Shares Price (USD)
1.12 W hat is the difference between a bank's trading book and
A 3,000 80
its banking book? W hy is the distinction between the two
B 2,000 73
important from a regulatory perspective?
C 5,000 90
1.13 W hat is meant by moral hazard? Explain why deposit
insurance can give rise to moral hazard. D 6,000 85
1.14 W hat is a poison pill? Give three exam ples. E 9,000 70
1.15 G ive two exam ples of conflicts of interest that could arise F 4,000 84
from communication between investm ent bankers and G 8,000 86
commercial bankers. W hy do the senior m anagem ent of
banks want to avoid conflicts of interest? A t what price are the shares sold? How many shares does
1.16 W hat is the difference between standardized models and each bidder get?
internal models in regulation? W hat is the formula that will 1.19 An investment bank has been asked to underwrite an
be used from 2027 going forward to determ ine regulatory issue of 20 million shares. The share price is currently USD
capital using calculations from both models? 24. The bank is trying to decide between a firm com m it
1.17 Explain the risks in funding long-term needs with short ment at USD 20 versus a best efforts where it will charge
term instruments. W hat steps have been taken to reduce 40 cents for each share sold regardless of price. Discuss
these risks in Basel III? the pros and cons of the two alternatives.

The following questions are intended to help candidates understand the material. They are not actual FRM exam questions.
ANSW ERS
1.1 Com m ercial banking involves the traditional banking trading book face market risk capital. Items in the banking
activities of taking deposits and making loans. Investment book face credit risk capital.
banking involves underwriting new issues of securities,
1.13 Moral hazard is the risk that the existence of an insur
advising com panies on major financial decisions, and trad ance contract will change the behavior of the insured
ing activities. party thereby making the insurance contract riskier.
1.2 M arket risks, credit risks, and operational risks Deposit insurance could encourage a bank to take more
risks because it is less likely to lose depositors when they
1.3 Regulatory capital is capital determ ined by regulators.
becom e aware of the risks being taken.
Econom ic capital is a bank's own estim ate of the capital
it needs. 1.14 A poison pill is a change made by a company to make it
more difficult for another company to acquire it. Exam ples
1.4 Liquidity coverage ratio (to ensure that the bank can sur
vive a 30-day period of extrem e stress) and the net stable of poison pills are (a) granting key em ployees stock
options that can be exercised in the event of a takeover,
funding ratio (to determ ine whether the maturities of
(b) making it impossible to fire directors for a period of
assets and liabilities are reasonably well matched).
time after a takeover, (c) issuing preferred shares that
1.5 A private placem ent is the placem ent of an issue of
automatically becom e common shares in the event of
securities with institutional investors without offering the
a takeover, (d) allowing existing shareholders to buy
securities to the general public. A public offering is an
new shares at a bargain price in the event of a takeover,
offering of securities to the general public.
(e) changing the voting structure to give m anagem ent
1.6 In a best efforts underwriting, the investment bank does more control, and (f) allowing remaining shareholder to
its best to sell securities at the agreed price but there is sell shares to the new owners in the event of a takeover at
no guarantee. In a firm com m itm ent, the bank guarantees a premium price.
that the issue can be sold at a certain price.
1.15 One possibility: when advising about a possible acquisi
1.7 An IPO (initial public offering) is an issue of a company's tion, an investment banker might get confidential informa
equity to the market for the first tim e. tion about the target company from a commercial banking
1.8 A discretionary account is an account where the broker arm of the bank if the target happens to be a client of the
can trade without consulting the investor. bank. Another possibility: a commercial banker might use
the investment banking operation to replace a loan that
1.9 The originate-to-distribute model describes the situa
contains too much credit risk with a bond issue. Viola
tion where a bank originates loans and then sells them to
tions of conflicts of interest can give rise to big fines and
investors. O ften the cash flows from a portfolio of loans
reputational losses that far exceed the extra profits earned
are channelled to different tranches.
from the violations.
1.10 A cash offer and a share-for-share exchange
1.16 Standardized models are those determ ined by the Basel
1.11 The G lass-Steagall A ct established a separation between Com m ittee. Internal models are those developed by the
commercial banking and investment banking in the U.S. bank. By 2027 the required capital will be the maximum of
It prevented commercial banks from engaging in some (a) that given by approved internal m odels, and (b) 72.5%
investment banking activities, and vice versa. It was of that given by standardized models.
repealed in 1999. Since the 2007-2008 crisis, rules have
1.17 Short-term funding must be rolled over frequently. If the
been im plem ented in the U.S. to prevent commercial
market rightly or wrongly loses confidence in the bank
banks from engaging in proprietary trading.
this could becom e im possible— even if the assets being
1.12 A bank's trading book includes instruments that the bank financed are sound. The Net Stable Funding Ratio require
expects to trade. The banking book includes instruments ment in Basel III is designed to ensure that banks do not
the bank expects to keep until maturity. Items in the make excessive use of short-term funding.
1.18 19,000 shares can be sold for 85 or more. 23,000 shares 1.19 Assuming that the shares can be sold for the same price,
can be sold for 84 or more. Shares are therefore sold for the best efforts alternative will lead to a gross profit for
84. Bidders C , D, and G get the shares they have bid for. the bank of USD 8 million. The bank's profit from the firm
Bidder F gets 1,000 of the 4,000 shares it has bid for. com mitment could be much higher. For exam ple, if shares
Bids sorted from highest to lowest: can be sold at the current price of USD 24, the bank will
make USD 80 million. However, there is a chance that the
price will collapse. For exam ple, if the post-issue price is
Cumulative
only USD 17, the bank will lose USD 60 million.
Number Number
of Shares of Shares Price Bid
Bidder Requested Requested (USD)
C 5,000 5,000 90
G 8,000 13,000 86
D 6,000 19,000 85
F 4,000 23,000 84
A 3,000 26,000 80
B 2,000 28,000 73
E 9,000 37,000 70

Insurance
Companies
and Pension Plans
Learning Objectives
Describe the key features of the various categories Calculate and interpret loss ratio, expense ratio, combined
of insurance com panies and identify the risks facing ratio, and operating ratio for a property-casualty insurance
insurance com panies. company.
Describe the use of mortality tables and calculate the Describe moral hazard and adverse selection risks facing
premium payment for a policy holder. insurance com panies, provide exam ples of each, and
describe how to overcom e the problems.
Distinguish between mortality risk and longevity risk and
describe how to hedge these risks. Evaluate the capital requirements for life insurance and
property-casualty insurance com panies.
Describe a defined benefit plan and a defined
contribution plan for a pension fund and explain the Com pare the guaranty system and the regulatory
differences between them . requirements for insurance companies with those for banks.
15
Insurance provides protection against sp ecific adverse that he or she is alive at the beginning of the year). The table
events. The com pany or individual obtaining protection is shows that a man aged 70 has a 2.338% probability of dying
known as the p o licy h o ld er. Typically, the policyholder must within the next year, whereas for a woman this probability is a
make regular paym ents known as prem ium s. The most likely more favorable 1.5612% .
outcom e in a year is that the adverse events do not occur
The survival probability for Year n is the cumulative probability
and so there is no cost to the insurance com pany. If one of
that an individual will live to Year n. The survival probability for
the specified adverse events does occur, however, there is
Year zero is 1. The survival probability for Year n + 1 can be
usually a relatively large paym ent from the insurance com
calculated from the survival probability for Year n and the prob
pany to the policyholder that covers all or part of the
ability of death within one year for Year n.
losses exp erien ced .
For exam ple, the probability that a man will survive until age 71
Most insurance contracts can be categorized as either life
is the probability that he survives until age 70 and does not die
insurance or p ro p erty and casualty insurance.1 The sim plest type
within the next year. This is
of life insurance contract requires regular monthly or annual
premiums for the life of the policyholder. When the policyholder 0.73427 X (1 - 0.023380) = 0.71710
dies, the premiums stop, and his or her beneficiaries receive a which agrees with the survival probability for age 71 in the table.
lump sum paym ent. A property and casualty insurance policy The probability that a man aged 71 will die in the following year
typically lasts one year and provides the policyholder with is 0.025549. Thus, the probability that a man aged 70 will die
compensation for losses arising from accidents, fires, thefts, and between the first and second year is2
similarly insured adverse events. The policy is usually renewed
(1 - 0.023380) X 0.025549 = 0.024952
annually. W hile the premiums in a life insurance policy do not
usually change from year to year, those in a property and casu Similarly, the probability that he will die between the second
alty policy generally do. and third year is
There are sim ilarities betw een pension plans and the con (1 - 0.023380) X (1 - 0.025549) X 0.027885 = 0.026537
tracts offered by life insurance com panies. In an em ployer- and so on. Assuming that (on average) a person dies in the
sponsored pension plan, it is typ ically the case that both the
middle of a year, the life expectancy of a man aged 70 (shown in
em ployee and the em ployer m ake regular contributions to the
the third column) can be calculated as:
plan. The contributions are used to fund a lifetim e pension
for the em ployee follow ing the em ployee's retirem ent. We 0.023380 X 0.5 + 0.024952 X 1.5 + 0.026537 X 2.5 + • • •
d escrib e pension plans after presenting the activities of life

insurance com panies.
2.2 LIFE INSURANCE
There are many different types of life insurance policies. Here
2.1 MORTALITY TABLES
we describe some of the more common ones.
M ortality tables are used extensively by actuaries for setting life

insurance contract premiums and assessing pension plan obliga Whole Life Insurance
tions. We review the information in these tables before discuss
ing life insurance contracts and pension plans. As its name implies, whole life insurance provides insurance for
the whole life of the policyholder. The policyholder makes regu
Table 2.1 shows an extract from mortality tables produced by
lar monthly or annual payments until he or she dies. A t that
the U.S. Social Security Adm inistration. The table is based on
tim e, the face value of the policy is paid to the designated ben
the death rates observed for men and women of different ages
eficiary. This means that there is certain to be a payment by the
in 2014.
insurance company to the policyholder,3 and the only uncer
The Probability of Death within 1 Year column is the probability tainty for the insurance company is when the payment will occur.
that an individual will die during the following year (providing
2 This means that individual will survive the next year (while he is 70) and
die during the second year (when he is 71).
1 Property and casualty insurance is som etim es referred to as non-life
insurance. Health insurance is som etim es considered to be a third 3 Life insurance is som etim es referred to as life assurance when a payout
category. is certain.

Table 2.1 Extract from Mortality Tables Published by U.S. Social Security Administration Based on 2014 Data.
(See www.ssa.gov/OACT/STATS/table4c6.html)
Males Females
Probability of Probability of
Death within Survival Death within Survival
Age (Years) 1 Year Probability Life Expectancy 1 Year Probability Life Expectancy
30 0.001498 0.97520 47.86 0.000673 0.98641 52.06
31 0.001536 0.97373 46.93 0.000710 0.98575 51.10
32 0.001576 0.97224 46.00 0.000753 0.98505 50.13
33 0.001616 0.97071 45.07 0.000805 0.98431 49.17
50 0.004987 0.92913 29.64 0.003189 0.95794 33.24
51 0.005473 0.92449 28.79 0.003488 0.95488 32.34
52 0.005997 0.91943 27.94 0.003795 0.95155 31.45
53 0.006560 0.91392 27.11 0.004105 0.94794 30.57
70 0.023380 0.73427 14.32 0.015612 0.82818 16.53
71 0.025549 0.71710 13.66 0.017275 0.81525 15.78
72 0.027885 0.69878 13.00 0.019047 0.80117 15.05
73 0.030374 0.67930 12.36 0.020909 0.78591 14.34
90 0.164525 0.18107 4.08 0.129475 0.29650 4.85
91 0.181600 0.15128 3.79 0.144443 0.25811 4.50
92 0.199884 0.12381 3.52 0.160590 0.22083 4.18
93 0.219331 0.09906 3.27 0.177853 0.18536 3.88
Usually, both the payments and the face value of the policy so that the expected payout during the second year is USD
remain constant through tim e. 1,534. Similarly, the probability that the policyholder will die
during the third year is
Consider a USD 1 million whole life policy where the
policyholder is a 30-year old male. The insurance company can (1 - 0.001498) X (1 - 0.001536) X 0.001576 = 0.001571
calculate its expected payout each year. From Table 2.1, we see
so that the expected payout during the third year is USD 1,571.
that the probability of a male who is 30 years old dying within
one year is 0.001498. The insurance company's expected payout The expected payout increases year by year throughout the life
(in USD) during the first year is therefore: of the policy. From Table 2.2, we can calculate the probability
that a policyholder aged 30 will survive until 70 as:
1,000,000 X 0.001498 = 1,498
Probability of Survival from Birth Until Age 70 0 73427
The probability that the policyholder will die during the second ----------- ----------------------------------- ------= ---------- = 0 75294
Probability of Survival from Birth Until Age 30 0.97520
year is the probability that death does not occur during the
first year multiplied by the probability of dying during the The probability that the policyholder (currently 30 years old) will
second year. From Table 2.1 this is die between age 70 and 71 is therefore:
(1 - 0.001498) X 0.001536 = 0.001534 0.75294 X 0.023380 = 0.017604
Chapter 2 Insurance Companies and Pension Plans ■ 17

Table 2.2 Calculation of Present Value of Expected Payout per Dollar of Face Value for a Female Aged 50
in Three-Year Term Life Insurance
Time (Years) Expected Payout Discount Factor5 PV of Expected Payout
0.5 0.003189 0.9759 0.003112
1.5 0.003477 0.9294 0.003232
2.5 0.003770 0.8852 0.003337
Total 0.009681
The expected payout from the policy during its fortieth year
(i.e., when policyholder is 70) is therefore USD 17,604.
Suppose that the premium charged is USD 15,000 per year. It is

clear from these calculations that the insurance company has an
expected surplus during the early years and an expected deficit
in later years.
On any one policy, the results are uncertain. If the insur

ance company sells many similar policies to men aged 30,
however, the investm ent of the early premiums is going to be
an im portant part of how the insurer finances later payouts.
O f the USD 15,000 per USD 1 million face value received in the
Fiqure 2.1 Shown is a pattern of expected surpluses
first year, the expected USD amount available for investment is
and expected deficits on USD 1 million whole life
15.000 - 1,498 = 13,502 insurance policy. The age of the policyholder is
Similarly, the expected USD amount of the premium in the 30 and the premium is USD 15,000 per year.
second year that is available for investment is
15.000 - 1,534 = 13,466

Given that a whole life insurance policy involves funds being
By contrast, in the fortieth year the insurance company has a
invested for the policyholder, a natural developm ent is to
negative expected USD cash flow of:
allow the policyholder to specify how the funds are to be
17,604 - 15,000 = 2,604 invested. Variable life insurance is the name used to describe
This is the year when the policyholder (if still alive) will turn 70. policies where the policyholder can do this. Typically, there is
Figure 2.1 shows how a typical whole life insurance policy a minimum guaranteed payout on the policyholder's death.
generates expected surpluses and expected deficits during However, the payout can be greater than this if the investments
its life. From the insurance company's perspective, the invest do well.
ment income from the expected surpluses must be sufficient Norm ally, if the policyholder stops making premium pay
to finance the expected deficits. Figure 2.1 is illustrative of the m ents, the policy no longer provides coverage. The policy is
general pattern of expected surpluses and deficits in a whole life then referred to as lapsing. A nother variation on the standard
policy. It does not include investment income. whole life policy is where the policy holder can reduce the pre
In many jurisdictions, there are tax benefits associated with mium down to a specified minimum. This is called universal life
whole life insurance. When funds are invested by the insurance. W hile reducing the premium reduces the benefits,
policyholder, tax is paid on the investment income each year. it does not result in the policy lapsing. Variable-universal life
But when the funds are invested by the insurance com pany, no insurance incorporates the features of both variable and uni
tax is payable until there is a payout from the insurance policy. versal life insurance.
In some jurisdictions, taxes are not levied on the payout.4
4 Note that taxes are usually payable on the proceeds if the policy is 5 As will be explained in later chapters, the discount factor for tim e t is
sold to a third party before the policyholder's death. the value of one dollar received at tim e t. In this case it is 1/1 .05f.

Term Life Insurance The breakeven premium is calculated by equating the pres
ent value of expected premiums with the present value of the
Term life insurance lasts a specified number of years. If the expected payout:
policyholder dies during the life of the policy, there is a pay
out equal to the face value of the policy. O therwise there is no 2.850327X = 0.009681
payout. M ortgages are a common reason for purchasing term
This gives X = 0.003396. The breakeven cost of three-year term
life insurance. For exam ple, a 30-year old person with a 15-year
insurance for a 50-year old fem ale is therefore 0.003396 per dol
m ortgage might choose to buy a 15-year term life insurance
lar of face value. For a policy with a face value of USD 1 million,
policy with a declining face amount equal to the amount out
the breakeven premium is therefore USD 3,396 per year. (The
standing on the m ortgage. If the policyholder dies, the insur
insurance company will of course charge more than this to cover
ance will provide dependents with the funds necessary to pay
adm inistrative costs and earn a profit.)
off the mortgage.
Mortality tables can be used to determ ine breakeven premiums

on term life insurance. As a simple exam ple, consider a three- Endowment Life Insurance
year term life insurance contract offered to a woman aged 50. Endow m ent life insurance is a typ e of term life insurance
The probability that she will die in the first year is given by
w here there is alw ays a payout at a pre-specified contract
Table 2.1 as 0.003189. The probability that she will die during m aturity. If the policyholder dies during the life of the policy,
the second year is the payout occurs at the tim e of the policyholder's death.
(1 - 0.003189) X 0.003488 = 0.003477 O th e rw ise , it occurs at the end of the life of the policy.
Som etim es the payout is also m ade when the policyholder
And the probability that she will die during the third year is
has a critical illness. In the case of a with p ro fits e n d o w
(1 - 0.003189) X (1 - 0.003488) X 0.003795 = 0.003770 m en t life insurance policy, the insurance com pany declares
We assume that death always occurs halfway through a year and bonuses that depend on the perform ance of its investm ents.
that the discount rate is 5% (with annual com pounding). This Th ese bonuses increase the final payout (assum ing that the
leads to the present value of the expected payout per dollar of p o licyholder lives until the end of the life of the policy). In
face value being calculated as indicated in Table 2.2. a unit-linked e n d o w m e n t policy, the p o licyholder chooses
a fund, and the payout depend s on the perform ance of
Now consider the premium paid by the policyholder. Suppose
that fund.
this is X and that (as is the case with most insurance contracts) it
is paid in advance. The insurance company is certain to receive
the first premium at tim e zero. It will receive the second pre Group Life Insurance
mium after one year if the policyholder is still alive. The prob
ability of this is 0.996811 (= 1 — 0.003189). It will receive the Group life insurance is typically purchased by com panies for
third premium after two years if the policyholder is still alive at their em ployees. The premiums may be paid entirely by the
that tim e. The probability of this is company or shared between the company and its em ployees.
Note that while individuals are normally required to undergo
(1 - 0.003189) X (1 - 0.003488) = 0.993334
medical tests when applying for life insurance, these tests are
The present value of the expected premiums received by the usually waived in group life insurance. This is because the insur
insurance company can therefore be calculated as indicated in ance company knows it will be taking some better-than-average
Table 2.3. and worse-than-average risks.
Table 2.3 Calculation of the Present Value of Expected Premiums for a Female Aged 50 in Three-Year Term
Life Insurance
Time (Years) Probability of Receiving Premium Discount Factor PV of Expected Premiums
1 1.000000 1.000000 X
2 0.996811 0.952381 0.949344X
3 0.993334 0.907029 0.900983X
Total 2.850327X

Annuity Contracts Longevity and Mortality Risk
Most life insurance com panies offer annuity contracts in addi Longevity risk is the risk that people will live for longer than
tion to life insurance contracts. W hile life insurance converts mortality tables indicate. Recall that Table 2.1 is based on the
regular payments into a lump sum, annuity contracts do observed probability of people of different ages dying in the
the reverse (i.e., they convert a lump sum into regular pay year 2014. It estim ates that a fem ale aged 30 has a life exp ec
ments). Typically, the payments in an annuity contract last for tancy of 52.06 years (i.e., she will, on average, die at age 82.06).
the rest of the policyholder's life. In some cases, the annuity However, advances in medical science, improved nutrition, and
starts as soon as a lump sum is deposited with the insurance other factors may lead to this being an underestim ate. O ver the
company. In the case of d e fe rre d annuities, it starts several last century, the life expectancy of people born in the U.S. have
years later. steadily increased to the point where som eone born today can
expect to live about 20 years longer than a similar person born
As with life insurance, taxes can be a consideration when o
100 years ago.
an annuity is purchased. Because the insurance company is
investing funds on behalf of the policyholder, tax is normally On the other hand, mortality risk is the opposite of longevity
payable only when the annuity is received. If the policyholder risk. It is the risk that wars, epidem ics, and other factors will lead
were investing the funds them selves, however, tax would to people dying sooner than expected.
be payable on the investment income each year. There are
The life insurance business of an insurance company should w el
therefore two advantages to this arrangem ent: Taxes are
come the possibility that people will live longer than expected.
deferred so that investments grow tax-free, and many policy
This will lead to premiums on whole life policies being paid for
holders have relatively low marginal tax rates when the annuity
longer periods of tim e and payouts being (on average) later
is received.
than expected. Mortality risk is more of a concern, however,
The amount to which the policyholder's funds grow in an annuity because it could lead to earlier payouts without sufficient col
contract is referred to as the accum ulation value. Funds can usu lected premiums to cover them.
ally be withdrawn early, but there may be penalties. Some annu
The annuity business of an insurance company has the oppo
ity contracts have em bedded options designed to ensure that
site exposure com pared to that of the life insurance business.
the accumulation value never declines. For exam ple, the con
If people live longer than expected, they receive the annuity to
tract may be structured so that the accumulation value tracks
which they are entitled for longer and thus make the contract
the S&P 500 with a lower limit on the return of zero and an
more expensive for the insurance company. If they die sooner
upper limit of 7% .6
than expected, the contract will prove to be less expensive for
In the United Kingdom , deferred annuity contracts som etimes the insurance company.
guarantee a minimum level for the future annuity paym ents. For
The longevity/mortality exposures on an insurance company's
exam ple, an insurance company might offer an annuity that will
whole life business should (to some extent) offset the exposure
start in ten years and pay at least 8% of the accumulation value
on its annuity business. However, there is unlikely to be a per
at that tim e .7*If interest rates fall and life expectancies increase,
fect offset and the residual exposure must be m anaged. One
however, the guarantee can prove to be very expensive. Equita
approach is the use of longevity derivatives. These instruments
ble Life, a huge U.K. life insurance company founded in 1762
have been structured in a number of different ways so that
that had 1.5 million policyholders at its peak, was an exam ple of
the payoff depends on the difference between a pre-specified
an insurance company that failed because of the generous guar
(expected) mortality rate for individuals in a certain age
antees it offered.
group and the actual mortality rate. A simple payoff could
be defined as:
Payoff = (Pre-Specified Fixed M ortality Rate

— Realized M ortality Rate) X Principal (2.1)
6 We will discuss options in a later chapter. By creating an upper and This is similar to a forward contract on the mortality rate. (For
lower limit on the return each year, the insurance com pany has sold ward contracts will be discussed later in this chapter.) Other
a put option to the policyholder and bought a call option from the
policyholder.
longevity derivatives involve bonds where either the principal
7 The specification of a minimum level for the annuity was regarded as a

necessary part of marketing the annuity and its cost was often not calcu 8 See for exam ple http://w w w .dem og.berkeley.edu/~ andrew /1918/
lated by the insurance com pany. figure2.htm l for life expectancy estim ates by year since 1900.

or the interest rate depends on the difference between a pre the em ployee's spouse (and possibly other dependents) will
specified mortality rate and an actual mortality rate. receive a (usually smaller) pension if they are still alive when the
em ployee dies.
Longevity derivatives are potentially of interest to defined ben
efit pension plans (which will be discussed later in this chapter) In a d efin ed contribution plan, the funds are invested by the
as well as insurance com panies because these plans also have employer. When the em ployee retires, he or she can begin w ith
an exposure to longevity risk. The mortality rate used to define drawing the fund s.10 Som etim es the em ployee can choose how
an instrument with the payoff in Equation (2.1) could be the the funds are invested, and som etim es he or she can opt for a
mortality rate of all members of a pension plan that are currently lump sum payment at retirem ent.
receiving pensions.
In a d efin ed b en efit plan, funds are also usually contributed by
the em ployer and em ployee. In this case, however, the contribu
Investments tions are pooled, and a formula is used to determ ine the pen
sion received by the em ployee on retirem ent. For exam ple, the
Life insurance policies and annuity contracts generate funds for pension plan might state that the pension is the em ployee's
investment. The investment strategy of a life insurance company average annual income during the last three years of em ploy
is therefore very important. ment multiplied by the number of years of em ploym ent multi
Many of these investments consist of long-term corporate bonds.9 plied by 2 % .111
2
Life insurance companies can try to match the maturities of the A defined benefit plan is much riskier for an em ployer than a
bonds with the maturities of their obligations. However, corporate defined contribution plan. In a defined contribution plan, the
bonds give rise to both market risk and credit risk. The market risk company is merely acting as an agent investing the pension plan
relates to interest rates (as rates increase, bond prices decline). contributions on behalf of its em ployees. The pension obtained
The credit risk relates to the possibility that bond issuers will by an em ployee depends on the extent to which the em ployee's
default. Life insurance companies could avoid credit risk by invest funds have grown.
10
In a defined benefit plan, the company has
ing in government bonds. Over the long term, however, the extra guaranteed to its em ployees that pensions will be calculated in
return earned on corporate bonds (over risk-free government a certain way.
bonds) more than compensates for losses arising from defaults.
Each year, actuaries assess the present value of a defined ben
Life insurance com panies have risks on both sides of their bal efit pension plan's obligations and com pare it with the plan's
ance sheets: market and credit risks on the asset side along assets. The company is required to provide contributions to the
with longevity and mortality risks on the liability side. In addi plan to make up for any shortfall. And while it may be able to
tion, they are subject to operational risks that are like those of spread these contributions out over several years, the shortfall
banks, which were discussed in the previous chapter. Regulators will remain a liability that reduces shareholders' equity on its
take all these risks into account in determ ining minimum balance sheet. The risk for the company and its shareholders in
capital requirem ents. defined benefit plans is the reason why defined benefit plans
are not initiated today. Most com panies that have defined ben
efit plans initiated them many years ago, and many firms have
2.3 PENSION PLANS switched from defined benefit plans to defined contribution
plans (at least for new em ployees) to avoid these risks.
Pension plans are like annuity contracts in that they are
designed to produce income for an individual for the remainder An important issue in estimating defined benefit plan obliga
of his or her life following retirem ent. Typically, contributions tions is the discount rate used. Plan outflows often stretch for
to the plan are made by the individual and the individual's many years into the future. For exam ple, the present value of
em ployer while the person is em ployed. These contributions are USD 1,000 paid out in 40 years with a discount rate of 2% is
deductible for tax purposes. about USD 453. If the discount rate is 5%, however, the present
There may be some indexation of the pension. For exam ple, it

might be agreed that the annuity growth rate will reflect 75% 10 In the U.S. a 401 (k) is a type of defined contribution plan. Some other
of the inflation rate. The term s of the pension plan state that countries have similar plans.
11 G overnm ent pension plans such as Social Security in the U.S. are a
type of defined benefit plan.
9 Insurance com panies also make equity investm ents. As discussed, a
particular policy could require equity or other investm ents by the insur 12 The only real risk for the com pany is a small operational risk that it will
ance com pany. m anage the funds badly and be sued by em ployees.

value is less than one third of this. Accounting standards now Unlike life insurance, property and casualty insurance is typically
require pension plan obligations to be discounted at the yield renewed from year to year. If the insurer feels that the risks have
on AA-rated bonds. increased, the premium may be raised. For exam ple, car insur
ance premiums are likely to increase if a driver has been con
We mentioned earlier that life insurance companies manage risks
victed of speeding.
by investing in long-term corporate bonds. Specifically, they
attempt to match the maturities of the bonds with the maturity of The risks to a property and casualty insurance company can be
their life insurance and annuity obligations. It might be thought divided into:
that defined benefit pension plans would follow the same invest
• Risks where the average payout can be predicted reasonably
ment strategy because their obligations are also fairly predictable.
well from historical data because the yearly payout arises
In practice, however, the returns provided by bonds are not nec
from many independent (or almost independent) claims by
essarily sufficient for pension plans to meet their obligations. As a
policyholders, and
result, pension plans typically put much of their assets into
i o • Risks where a single event (such as a hurricane or an earth
equities. If equity markets do well (as they have done in many
quake) can lead to many claims at the same tim e.
parts of the world since 1960), these plans should be able to
meet their obligations. If there is a prolonged decline in equity Car insurance is in the first category. For exam ple, assume a
prices (particularly if it is combined with an increase in life expec company has insured 100,000 car owners in a certain risk cat
tancy), however, these plans are likely to chalk up huge deficits. egory. Furtherm ore, it knows from experience that 10% of driv
ers will make a claim in a year, with an average claim being USD
Primarily, shareholders bear the cost of deficits in defined bene
3,000. As a result, it can expect its total annual cost in meeting
fit plans. In cases of bankruptcy, the cost may be borne by
claims to be about USD 30 million (= 3,000 X 0.1 X 100,000).
governm ent-sponsored entities.1
141
3 5In either case, there is a trans
W hile the total annual cost might vary slightly from year to
fer of wealth to retirees from the next generation. It has been
year, large differences are extrem ely unlikely statistically. The
argued that the term s of defined benefit plans must be altered
key point is that claims by different drivers are independent (or
so that there is some risk sharing between generations. If equity
almost independent) of each other. Independence implies that if
markets do well, some of the benefits can be passed on to the
Driver A has an accident, this does not increase the probability
next generation. If they do badly, some of the costs should be
a n
of Driver B having an accident. O f course, the insurance com
borne by retirees.
pany also monitors trends in the number of accidents per year,
trends in automobile repair costs, and trends in the dam ages
2.4 PROPERTY AND CASUALTY awarded to accident victims.
INSURANCE Risks in the second category are referred to as catastrophe risks.

For exam ple, a company that has insured 100,000 homes
Property and casualty insurance contracts are quite different against hurricane dam age in South Florida will have claims that
from life insurance contracts. Property insurance provides pro are not independent. Either a hurricane happens (and most poli
tection against dam age to property from fire, theft, flooding, cyholders file a claim) or it doesn't (and no policyholders do so).
and other loss-generating events. Casualty insurance provides Catastrophe risks are therefore a much greater threat for insur
coverage for liabilities arising from injuries and dam ages sus ance com panies. As a result, insurance com panies used models
tained by others due to the insured party's actions. O ften, produced by specialists to predict the probability of different
both property and casualty insurance are provided in a single events that they are insuring against. However, this does not
policy. For exam ple, hom eowners' insurance typically provides alter the fact that they are (by nature) all-or-nothing risks.16
insurance for losses from fire and theft as well as for liabilities
if others are injured on the property. M eanwhile, car insurance
typically provides coverage for theft as well as claims by others CAT Bonds
for dam ages caused.
When a company does not want to keep catastrophe risks, it
can pay a reinsurance company to take them on. It can also use
derivatives known as C A T (catastrophe) bonds.
13 A common portfolio mix is 60% equity, 40% debt.
14 For exam ple, in the U .S., the Pension Benefit G uarantee Corporation
insures private defined benefit plans.
15 See for exam ple K. P. Am bachtsheer, Pension Revolution: A Solution 16 Analysts often set premiums so that their coverage is three tim es the
to the Pension Crisis. Hoboken, N J: John W iley & Sons, 2007. largest cost given by simulations.

A C A T bond is a bond issued by an insurance company that Table 2.4 Calculation of Operating Ratio
pays a higher than normal rate of interest. If payouts by the for a Property-Casualty Insurance Company
insurance company for a specified risk are in a certain range,
Loss Ratio 70%
the bond's interest (and in some cases the principal) are used to
provide the payouts. Expense Ratio 26%
For exam ple, suppose an insurance company has USD 100 million Com bined Ratio 96%
of exposure to Florida hurricanes and wants to reduce this expo Dividends 1%

sure to USD 40 million. It could issue three bonds. Com bined Ratio A fter Dividends 97%
1. Bond A has a principal balance of USD 20 million and cov Investment Income (2%)
ers claims in the range USD 40 to 60 million.
O perating Ratio 95%
2. Bond B has a principal balance of USD 20 million and covers
claims in the range USD 60 to 80 million.
3 . Bond C has a principal balance of USD 20 million and cov and casualty insurance com pany. The loss ratio is 70% . The
ers claims in the range USD 80 to 100 million. expense ratio is 26% . The co m b in ed ratio (which is the sum of
the loss ratio and the expense ratio) is 96% . Som etim es a
Bond A is riskier than Bond B, which is in turn riskier than Bond
small dividend is paid to policyholders. Table 2.4 assum es
C. As a result, the interest rate offered on Bond A will be greater
that this is 1%. The co m b in ed ratio a fter d ivid en d s is
than that on Bond B, which in turn will be greater than that on
therefore 97% .
Bond C.
The insurance company's operating ratio is a gross profitability
With these bonds, the principal is at risk in the event of a hur
measure. From the loss ratio, expense ratio, and dividends it
ricane. As an alternative, the insurance company could issue
appears that the operating ratio is 97%. However, this does
bonds with much higher principal balances so that the dollar
not consider the interest earned. Premiums are typically paid
amount of promised interest is much higher and can be used to
at the beginning of a year and payouts happen later. The
cover claims without putting the principal at risk.
policyholder's funds can therefore be invested in short- and
It is natural to ask why anyone would buy a C A T bond. The main medium-term bonds to earn interest. Table 2.4 assumes that
reason is that the risk is likely to be uncorrelated (or almost the interest earned is 2%. This changes the operating ratio
uncorrelated) with other risks in an investor's portfolio. C A T to 95%.
bonds therefore provide diversification benefits for investors.
Capital market theory would suggest that the expected return
on a security should be the risk-free rate when the security's 2.5 HEALTH INSURANCE
return is uncorrelated with the return provided by the market.
C A T bonds are attractive because they can offer a higher return A final category of insurance is health insurance. In many coun
than that suggested by capital market theory. tries, health care is provided almost entirely by the governm ent.
In other countries, public and private health care system s run
side by side. In the U .S., private health insurance is a necessary
Loss Ratios expenditure for many people.
A key statistic for a property-casualty insurance company is One difference between health insurance and other types of
its loss ratio. This is the ratio of payouts to premiums. A loss insurance concerns the circum stances when premiums increase.
ratio of 70% would mean that for every USD 100 of premiums In whole life insurance, premiums typically remain constant.
received, the insurance company pays out USD 70 in claims. Even if it becom es known that a policy holder has a short life
The remaining 30% is to cover expenses and (hopefully) make a expectancy because of a terminal disease, premiums cannot
profit. Two major expenses are be increased. In property-casualty insurance, risks are reas
1. Selling expenses, and sessed every year, and premiums may increase or decrease. In
health insurance, premiums may increase because the overall
2. Expenses related to determ ining the validity of a claim
costs of health care have increased. However, they cannot do
(referred to a loss adjustm ent expenses).
so because the policyholder develops unexpected health
The expense ratio is total expenses divided by premiums problems that were unknown at the tim e the policy was
received. Table 2.4 shows an incom e statem ent for a property initiated.

2.6 M ORAL HAZARD AND A D V ERSE fram ework known as Solvency II was im plem ented in 2016 and
applies to all insurance com panies.17
SELECTIO N
Solvency II specifies a minimum capital requirement (MCR) and a
In this section we discuss two im portant risks facing insurance solvency capital requirement (SCR). If capital falls below the SCR,
com panies: moral hazard and adverse selection. an insurance company is required to formulate a plan to bring it
back up above the SCR level. If it falls below the MCR level, the
insurance company may be prevented from taking new business,
Moral Hazard
and existing policies might be transferred to another insurance
Moral hazard is the risk that the behavior of the policyholder will company. The MCR is typically between 25% and 45% of the SCR.
change as a result of the insurance. We mentioned moral hazard
As in the case of the Basel Com m ittee rules, there are both stan
regarding deposit insurance in Chapter 1. We noted that banks
dardized approaches and internal model-based approaches to
might be tem pted to follow riskier strategies when depositors
determ ining capital requirem ents. There are capital charges for:
have governm ent insurance because the insurance makes it less
likely that depositors will transfer their funds elsewhere. O ther • Investment risk (this relates to the asset side of the balance
exam ples of moral hazard include the following. sheet),
• If your house insurance fully insures you against burglaries, • Underwriting risk (this relates to the liabilities side of the bal
you may be less likely to install alarm system s and cam eras. ance sheet), and
• As a result of buying health insurance, you may visit the doc • Operational risk.
tor more frequently. Investment risk is divided into credit risk and market risk. M ean
Moral hazard is not a big problem in life insurance. It is difficult while, the capital requirements for property-casualty underwrit
to imagine that an individual would take up a risky pursuit (such ing tend to be higher than those for life insurance underwriting.
as sky diving) because he or she has bought life insurance. This is because the catastrophe risks associated with the form er
are greater than the longevity/m ortality risks associated with the
Insurance com panies attem pt to reduce moral hazard in a num
latter.
ber of ways. The following are exam ples.
In the U .S., insurance is regulated at the state level rather than
• Most insurance policies have a deductible so that the policy
at the federal level. The National Association of Insurance C om
holder bears the first part of any loss.
missioners is a national forum for insurance regulators to
• Som etim es there is co-insurance where the insurance com exchange ideas. For exam ple, it provides statistics on loss ratios
pany pays only a certain percentage of any loss. for insurance com panies across the country.18 However, there
• There is nearly always a limit on the amount that can be claimed. are differences from state to state in the way insurance com pa
nies are regulated, and a large U.S. insurance company may
Adverse Selection have to deal with 50 different regulators.
The guaranty system for policyholders in the U.S. is different

Adverse selection is the risk that insurance will be purchased
from the deposit insurance system for bank depositors. As
only by high-risk policy holders. If an insurance company offers
mentioned in Chapter 1, premiums paid by banks create a fund
the same car insurance rates to everyone, for exam ple, the
administered by the Federal Deposit Insurance Corporation
rates will seem more attractive to those who have bad driving
(FD IC) that is used to com pensate depositors for losses. The
records. It is therefore im portant that insurance com panies know
federal governm ent has added to the fund when necessary.
as much as possible about the risks they are taking on before
quoting a premium. The risk assessm ents should be updated as In contrast, there is no perm anent fund to provide protection for
more information is obtained. policyholders. And insolvencies are handled on a state-by-state
basis. W hen one insurance company fails, others are required to
make contributions to a fund, and there are limits on what can
2.7 REGULATION be claimed as well as delays in settlem ent.
In Chapter 1, we explained that the Basel Com m ittee deter

17 More information can be found in https://ec.europa.eu/info/
mines global regulatory requirements for banks. In contrast, business-econom y-euro/banking-and-finance/insurance-and-pensions/
there are no similar global regulatory requirements for insurance risk-m anagem ent-and-supervision-insurance-com panies-solvency-2_en
com panies. In the European Union, however, a regulatory 18 For more information see http://w w w .naic.org/

SUMMARY insurance), the average payout per policy on a large number of
policies can be predicted with reasonable accuracy. For other
The two main categories of insurance are life insurance and risks (such as those associated with natural disasters), the insur
property-casualty insurance. Life insurance com panies offer a ance company is in the position where either its underwriting
number of products whose costs are a function of how long the business will prove to be very expensive or it will cost nothing
policyholder lives. W hole life insurance has mortality risk in that at all.
it becom es more expensive if the policyholder dies at a young Health insurance has some of the features of life insurance and
age. Annuities have longevity risk in that they becom e more some of the features of property-casualty insurance. Health
expensive the longer the policyholder lives. insurance premiums can increase because the cost of provid
Pension plans have much in common with life insurance com ing health care increases (just as property-casualty insurance
panies. They offer annuity-type contracts whose costs depend premiums can increase), but they cannot increase because a
on how long pension plan members live. There are two types policyholder's health deteriorates. In the latter respect, health
of pension plans: defined contribution and defined benefit. insurance is like whole life insurance.
Offering em ployees a defined contribution plan poses very little Insurance com panies must consider moral hazard and adverse
risk for com panies because they merely invest the pension plan selection when assessing risks and setting premiums. Moral haz
contributions made on behalf of their em ployees (who are then ard is the risk that the policyholder will take more risks because
able to withdraw the funds when they reach the retirem ent age). of the existence of insurance and thereby make payouts more
A defined benefit plan is much riskier because the funds are likely. A dverse selection is the risk that individuals who are most
pooled, and it is promised that each pension will be calculated likely to suffer losses will tend to buy the insurance.
(and paid) in a certain way. Actuaries assess defined benefit
Insurance com panies are regulated to ensure that they keep
plans annually and determ ine whether they are overfunded
enough capital for the risks they are taking. A property-casualty
or underfunded.
company must typically keep more capital for the risks it under
Property-casualty insurance com panies are concerned with pro writes than a life insurance company. In the European Union,
viding protection for loss or dam age to property and for liabili capital for all insurance com panies is determ ined by Solvency II
ties arising from injuries and dam ages resulting from the insured rules. In the U .S., regulation is a responsibility of states, not the
party's actions. For some risks (such as those associated with car federal governm ent.

QUESTIONS

2.1 Which of these is the biggest risk for whole life insurance? 2.7 W hat is universal life insurance?
A. Longevity risk A. W hole life insurance where the policyholder chooses
B. M ortality risk how funds will be invested
C. Policies being sold to a third party B. W hole life insurance where the policyholder can vary
D. Inflation premiums from year to year
C. Term life insurance that can be renewed
2.2 Which of these is the biggest risk for property-casualty
D. Life insurance where the policyholder can ask for an
insurance?
early payout in certain circumstances
A. Longevity risk
B. M ortality risk 2.8 Is the following statem ent true or false? "D efined contri
C. Natural disasters bution pension plans are more risky for a company than
D. Inflation defined benefit plans."
2.3 Is the following statem ent true or false? "Fo r a woman 2.9 How is the loss ratio of a property-casualty insurance com
aged 70, five-year term insurance will have lower prem i pany defined?
ums than whole life insurance." A. Ratio of payouts to premiums
2.4 W hat two activities of life insurance com panies have B. Ratio of payouts plus selling expenses to premiums
opposite exposures to longevity risk?

C. Ratio of all costs to premiums
A. Term life insurance and whole life insurance D. Ratio of all costs to premiums plus interest income
B. Term life insurance and endowm ent life insurance 2.10 Which of the following is true?
C. W hole life insurance and annuities A. There is a single regulator for all insurance companies
D. None of the above globally.
2.5 W hat are the most popular investments for life insurance B. The National Association of Insurance Com m issioners
regulates all insurance com panies in the U.S.
com panies?
C. Insurance com panies operating in the state of Maine
A. Equities
B. Treasury bonds are regulated by the state of Maine.
C. Com m ercial paper D. Germ an insurance com panies and Italian insurance
com panies are subject to different regulations.
D. Long-term corporate bonds
2.6 W hat is variable life insurance?

A. W hole life insurance where the policyholder chooses
how funds will be invested
B. W hole life insurance where the policyholder can vary
premiums from year to year
C. Term life insurance that can be renewed
D. Life insurance where the policyholder can ask for an
early payout in certain circumstances

Practice Questions
2.11 Using Table 2.1, what is the probability that a woman 2 .1 7 Suppose that in a certain defined benefit plan the following
aged 70 will live to age 90: simple situation exists. Employees work for 40 years with a
A. 0.29650 salary that increases exactly in line with inflation. The pension
B. 0.35801 is 60% of the final salary and increases exactly in line with
C. 0.31166 inflation. Employees always live for 25 years after retirement.
D. It is impossible to tell from the information in the The funds in the pension plan are invested in bonds that
table. earn the inflation rate. Which of the following is the best esti
mate of the percentage of the employee's salary that must
2 .1 2 W hat is the minimum USD annual premium that an insur
be contributed to the pension plan? (Hint: You should do all
ance company should charge for a two-year term life
calculations in real rather than nominal terms so that salaries
insurance policy with face value of USD 1 million when
and pensions are constant and the interest earned is zero.)
the policyholder is a woman aged 71? (Use Table 2.1 and
assume an interest rate of 3% com pounded annually.)
A. 37.5%
B. 43.75%
A. 18,153
C. 22.5%
B. 17,874
C. 17,996 D. 27.5%
D. 17,767 2 .1 8 A life insurance company issues whole life insurance poli

cies where the annual premiums are USD 1 million. A
2 .1 3 An insurance company offers a policy that pays out if a
property-casualty insurance company issues fire, theft,
worker becom es unem ployed. Which of the following risks
are applicable? and flood house insurance policies with annual premiums
of USD 1 million. Is the following statem ent true or false?
A. Moral hazard, but not adverse selection
"It is likely that the life insurance company will have to
B. Adverse selection, but not moral hazard
C. Moral hazard and adverse selection keep more capital for the risks it faces."
D. Neither moral hazard nor adverse selection 2 .1 9 Consider two bonds. One is a C A T bond where there is
no default risk. The other is a regular corporate bond. An
2 .1 4 A company's defined benefit plan invests primarily in equi
analysis has shown that the expected loss from default
ties. Which of the following creates risks for the company?
risks on the corporate bond is the same as the expected
(Assume that equity markets are unaffected by A , B, C,
loss from insurance claims on the C A T bond. The bonds
and D.)
have the same coupon and the same price. Which bond
A. High interest rates
would be most attractive to a fund manager with an exist
B. Low interest rates
ing portfolio of corporate bonds?
C. Em ployees working past retirem ent age
A. The bonds are likely to be equally attractive
D. An epidem ic that shortens life expectancy
B. The C A T bond is likely to be more attractive
2 .1 5 Is the following statem ent true or false? "D efined benefit
C. The corporate bond is likely to be more attractive
plans have similar exposures to life insurance com panies.
D. Any of A , B, and C could be true.
Like life insurance com panies, they invest primarily in
2 .2 0 In Solvency II, which of the following is true?
bonds matching the maturities of assets and liabilities."
A. If capital falls below the solvency capital requirement,
2 .1 6 In health care insurance, is it usually the case that:
an insurance company is not allowed to take on further
A. Premiums cannot increase from year to year
business.
B. The insurance company can change premiums as it
B. If capital falls below the solvency capital requirem ent,
reassesses risks
an insurance company's policies may be transferred to
C. Premiums increase with the age of the policyholder
another insurance company.
D. Premiums can increase as the overall cost of health
C. Insurance com panies are only required to form ulate a
care increases
solvency capital plan if capital falls below the minimum
capital requirement.
D. None of the above

ANSW ERS
2.1 B. The biggest risk for whole life insurance is that the 2.12 B. The probability of a payout in the first year (time 0.5
policyholder dies earlier than expected. This is referred to years) is 0.017275. The probability of a payout in the sec
as mortality risk. Longevity risk is the risk that the policy ond year (time 1.5 years) is
holder will live longer than expected. Sale of the policy to
(1 - 0.017275) X 0.019047 = 0.018718
a third party should make no difference and policies are
not inflation-linked. The PV of the expected cost of the policy is therefore:
2.2 C. Natural disasters such as hurricanes and earthquakes 1 7 ,2 7 5 /(1 .03°-5) + 1 8 ,7 1 8 /(1 .0 3 1-5) = 34,928
can lead to a high volume of claims in a year. A , B, and D
The first premium is at time zero. The second premium, at
are not relevant to property-casualty insurance.
time one year, has a probability of 1 — 0.017275 = 0.982725
2.3 True. Term insurance provides a payout if she dies in the of being made. If the premium is X, the expected present
next five years. W hole life insurance provides a payout value is
w henever she dies.
X + 0.982725X/1.03 = 1.954102X
2.4 C. Most forms of life insurance have mortality risk (i.e.,
The minimum premium is given by solving:
the risk that policyholders will die earlier than expected).
1.954102X = 34,928
Annuities have longevity risk (i.e., the risk that policyhold
ers will live longer than expected). It is 17,874.
2.5 D. Life insurance com panies tend to invest in long-term 2.13 C. There is both moral hazard and adverse selection. The
corporate bonds. This allows them to match assets and behavior of a policyholder with a job might change (he or
liabilities. Equities and commercial paper do not allow she will not be as concerned about being fired as he or
them to do this. They prefer long-term corporate bonds she would be if there were no policy) and an em ployee
to long-term Treasury bonds because on an actuarial basis without a job will not look as hard to find one. This is
they get well com pensated for taking credit risk. moral hazard. Also, the policy is likely to attract people
with less secure jobs. This is adverse selection.
2.6 A. In variable life insurance, the policyholder chooses
how surplus funds are invested, and the final payout may 2.14 B. Low interest rates will reduce the discount rate used to
increase if investments do well. assess liabilities and therefore increase the present value
2.7 B. In universal life insurance, premiums can be varied from of the liabilities. High interest rates, em ployees working
year to year. past retirem ent age, and a shortening of life expectancy
all lessen risks. (The question states that the impact of A,
2.8 False. A defined benefit plan is riskier because the com
B, C , and D on equity markets should not be considered.
pany is responsible for any shortfall assessed by actuaries.
In practice a reduction in interest rates tends to lead to an
2.9 A. Loss ratio is the ratio of payouts to premiums. increase in equity prices.)
2.10 C. Insurance com panies are different from banks in that 2.15 False. Defined benefit pension plans invest in equities
there are no global regulations. Regulations are enforced as well as bonds. Bonds do not provide a large enough
by the individual states in the U.S. The National A ssocia return to m eet their obligations and so they have no
tion of Insurance Com m issioners is a forum for exchanging choice but to hope that equity markets will continue to
ideas but does not impose the regulations. In the Euro perform well in the long term .
pean Union there is a single set of regulations, so Germ an
2.16 D. Health insurance premiums can increase from year to
and Italian insurance com panies should be subject to the
year but the increase should only reflect overall increases
same regulations.
in health care costs.
2.11 B. It is
2.17 A. The em ployee's salary is constant in real (inflation-
Probability of Survival from Birth Until A g e 90 0.29650 adjusted) term s. Suppose it is X. The pension is 0.6X.
Probability of Survival from Birth Until A g e 70 ~ 0.82818 The real return earned is zero. The pension plan con
= 0.35801 tributions therefore grow to 40XR where R is the

(employer + employee) contribution rate as a percentage 2.19 B. The C A T bond is likely to be more attractive because it
of the em ployee's salary. The present value of the benefits offers better diversification benefits. The corporate bond's
is 25 X 0.6X = 15X. We therefore require 40XR = 15X so return will be som ewhat correlated with the return on
that R = 15/40 = 0.375. Contributions equal to 37.5% market indices. The C A T bond has the advantage that its
of salary are therefore necessary. This simple exam ple return is almost entirely uncorrelated with the return on
illustrates the problems with defined benefit plans. Total market indices.
contributions of em ployer and em ployee are, in practice,
2.20 D. If capital falls below the solvency capital requirem ent,
15% or less, but much higher contributions are necessary a plan to bring it back up must be form ulated. If it falls
to pay promised pensions with certainty. below the minimum capital requirem ent, the insurance
2.18 False. The property-casualty insurance company's risks company may not be allowed to take on more business
are likely to be higher because the policies include some and its policies may be transferred.
catastrophe risks.
Chapter 2 Insurance Companies and Pension Plans 29

Fund Management
Learning Objectives
Differentiate among open-end mutual funds, closed-end Describe various hedge fund strategies, including long/
mutual funds, and exchange-traded funds (ETFs). short equity, dedicated short, distressed securities,
m erger arbitrage, convertible arbitrage, fixed income
Identify and describe potential undesirable trading behav arbitrage, emerging markets, global macro, and managed
iors at mutual funds. futures, and identify the risks faced by hedge funds.
Calculate the net asset value (NAV) of an open-end mutual Describe characteristics of mutual fund and hedge fund
fund. perform ance and explain the effect of m easurem ent
biases on perform ance m easurem ent.
Explain the key differences between hedge funds and
mutual funds.
Calculate the return on a hedge fund investment and

explain the incentive fee structure of a hedge fund includ
ing the term s hurdle rate, high-water mark, and clawback.
31
Fund managers invest money on behalf of individuals and com by asset managers that are not banks (e.g ., Fidelity), while oth
panies. Funds from different clients are pooled, and the fund ers are offered by the asset m anagem ent divisions within banks
managers choose investments in accordance with stated invest (e.g ., JP Morgan A sset M anagement).
ment goals and risk appetites. There are several advantages to
The assets of an open-end mutual fund are valued at 4 p.m.
this approach.
each day. The net asset value (NAV) is the value of the assets of
• Fund managers may have more investment expertise than the fund divided by the number of shares in the fund. All sales
their clients. and purchases of shares take place at the 4 p.m . NAV after
• Transaction costs (as a percentage of the amount traded) are the order is placed. An investor might issue instructions to a
usually lower for large trades than for small trades. broker to buy or sell shares at 10 a.m . on a business day, but the
instructions will not be carried out until 4 p.m . that day.
• It is difficult for a small investor to be well diversified, but a
large fund with billions of dollars should not have any diffi When discussing insurance contracts in Chapter 2, we noted
culty in achieving diversification. several tax advantages. For exam ple, tax is deferred in many
jurisdictions if funds are invested to provide an annuity or a
In this chapter, we consider mutual funds, exchange-traded
pension. However, there are no tax advantages associated with
funds (ETFs), and hedge funds. Mutual funds and ETFs cater
investments in mutual funds. The investor pays taxes as though
to individual investors, whereas hedge funds generally have
he or she owns the investments of the fund. For exam ple, if
high minimum investment thresholds that limit participa
shares are bought at USD 70 and sold at USD 90, the investor
tion to wealthy individuals and institutions. Hedge funds are
has a USD 20 capital gain that is subject to taxation.
also subject to less regulation than mutual funds or ETFs and
are free to follow a wide range of trading strategies. Further Open-end funds can be categorized as follows:
more, hedge funds are not required to disclose their holdings • Money market funds,
on a continuous basis (unlike mutual funds or ETFs). However,
• Bond funds, and
they are subject to some additional restrictions on how they can
solicit funds from investors. • Equity funds.
Funds that invest in more than one type of security are referred
to as hybrid funds (or m ulti-asset funds).
3.1 MUTUAL FUNDS
Money market funds invest in fixed-incom e instruments that
have a life of less than one year (e.g ., commercial paper). Bond
Mutual funds (which are called unit trusts in some countries)
mutual funds invest in fixed-incom e securities that last more
have been a popular investment vehicle for small investors for
many years. There has been considerable growth in the assets than one year. For many investors, money market funds are an
alternative to a savings account at a bank. In fact, som etimes
managed by mutual funds. In the U.S. for exam ple, the assets of
funds allow clients to write checks on their funds. And while
mutual funds have grown from USD 0.5 billion in 1940 to about
USD 19 trillion in 2 0 1 7 .1 there is no deposit insurance, the return on money market funds
is usually higher than on bank deposits.
There are two types of mutual funds: open-end and closed-end.
Most2 money market funds in the U.S. keep constant NAV of
Open-end funds are by far the most popular and account for
USD 1, with each day's gains being returned to the investor. A
over 98% of mutual fund assets in the U.S.
negative return is therefore referred to as breaking the buck
(because it causes the NAV to fall below USD 1). Breaking the
Open-End Funds buck is very unusual, but it did happen to the Reserve Primary
Fund (a large money market fund) in Septem ber 2008. This
The key feature of open-end funds is that the number of shares
happened because the fund held commercial paper issued by
(and the size of the fund) expand and contract as investors
Lehman Brothers. To avoid a panic, the governm ent stepped in
choose to buy and sell shares. If more investors decide to buy
with a guaranty.
shares rather than sell shares, the number of shares in the fund
increases; if the reverse happens, and more investors sell, the
number of shares decrease. Some open-end funds are offered
2 Recent regulations by the Securities and Exchange Com m ission have

1 The Investm ent C om pany Fa ctb o o k 2018, The Investm ent Com pany imposed a floating NAV requirem ent for some money m arket funds.
Institute, 2018 https://w w w .ici.org/pdf/2018_factbook.pdf See: https://personal.vanguard.com /pdf/VG M M R.pdf

Equity funds are the most popular type of mutual funds. They Between 1993 and 2017 the net assets of actively man
can be subdivided into: aged funds grew from about USD 1.5 trillion to about USD
12.5 trillion. During the same period, the net assets of index
• Actively managed funds, and
funds grew at a faster rate (though sm aller in absolute terms)
• Index funds.
from USD 28 billion to USD 3.3 trillion.6 The popularity of index
In the case of an actively managed fund, the fund manager funds can be attributed to research (to be discussed later in this
uses his or her skill to achieve the fund's objectives. One objec chapter) showing that actively managed funds generally do not
tive might be to invest in stocks that provide a high dividend outperform the market.
income. The fund would then trade in a manner consistent with
this objective. Stocks in the portfolio that reduce their dividend
would be sold, while those that increase their dividend will be
Closed-End Funds
purchased. Closed-end funds are funds where the number of shares remains
Index funds, on the other hand, attem pt to track a specific index constant through tim e. In essence, a closed-end fund is a regular
such as the S&P 500 or the FT S E 100. A simple way of doing company whose business is to invest in other com panies. Thus,
this would be to buy all the shares in the index in amounts that buying a share in a closed-end fund is like buying a share in any
reflect their weight in the index. Som etim es, a sm aller represen other com pany. Price changes balance supply (investors wanting
tative set of shares are bought instead. to sell shares) with demand (investors wanting to buy shares).
Tracking error measures how well a fund tracks its intended W hereas open-end funds are bought and sold at their NAV,
index. A popular tracking error is the square root of the aver share prices for closed-end funds are typically lower than their
age squared difference between the fund's return and the NAV. For exam ple, the price of a closed-end fund's shares might
index's return (referred to as the root-m ean-square error). For be USD 30 even though the NAV is USD 32 per share. This
exam ple, consider a mutual fund that is designed to track the means that an arbitrageur could (in theory) profit by buying all
S&P 500. Suppose that the returns in successive years on the the shares of the closed-end fund at the current market price
S&P 500 are 4.0% , 12.0% , 13.0% , —6.0% , and 2.0% and that the and then selling the assets of the fund. In practice, this arbitrage
returns on the fund are 3.3% , 11.1%, 13.2% , —7.0% , and 2.0% opportunity would drive up the fund's share price and eliminate
(respectively). The difference between the fund's return and the profit.
the S&P 500 return is —0.7% , —0.9% , + 0.2% , —1.0% , and 0.0% Researchers have investigated the reason for the discrepancy
(respectively). The average difference is —0.48% (i.e., on aver between the NAV and the share price for closed-end funds.
age the fund has underperform ed the S&P 500 by 0.48% ).3 The Stephen Ross has argued that m anagem ent fees explain the
standard deviation of the differences is 0.545% and the tracking discount.7
error is therefore 54.5 basis points per year.
We can illustrate the argument with a simple exam ple. Assum e
The expense ratio is the m anagem ent fee charged on a yearly that an investor plans to keep a closed-end fund for five years
basis to the value of the assets being m anaged. In addition, and then sell it. Assum e further that the fund pays no dividends
investors som etim es must pay a front-end load (a fee when they and that the expected return from the fund's assets is 10%.
buy) and/or a back-end load (a fee when they sell). The fees
An investment of USD 100 in the assets of the fund (not the
charged vary from country to country. They are relatively low in
fund itself) has an expected value of USD 161.05 (= 100 X 1,15)
the U .S. and Australia and relatively high in Canada and most
in five years. The correct risk-adjusted discount rate to apply to
European countries.4 Fees are also much higher for actively
this expected value is 10% because this gives a present value
managed funds than for index funds. Some index funds in the
of USD 100 and is consistent with the current USD 100 market
U.S. charge no fe e s,5 whereas actively managed funds tend to
value of the assets.
charge 1% to 2% per year.
3 The underperform ance may be a result of m anagem ent fees, to be

discussed shortly.
4 See A . Khorana, H. Servaes, and P. Tufano, "M utual fund fees around 6 See Investm ent Com pany Institute w w w .ici.org
the w o rld ," R eview o f Financial Stu dies, 22 (March 2009): 1279-1310.
7 See, for exam ple, S. Ross, "N eoclassical finance, alternative finance,
5 Fidelity Investm ents began offering no-fee funds in 2018 https:// and the closed-end puzzle," European Financial M anagem ent, 8 (2002):
fundresearch.fidelity.com /m utual-funds/sum m ary/31635T708 129-137.
Chapter 3 Fund Management ■ 33

Now consider what happens when the investm ent is in a closed- For example, consider an ETF where there are 10,000 shares
end fund and the fund invests in the USD 100 of assets. We worth USD 25 each. The value of the fund's assets is USD 260,000
suppose that the m anagem ent fee is 1% of the fund's value each and therefore the NAV is USD 26 (= 260,000/10,000).
year. This means that the value of the fund grows at 9% rather An institutional investor owning 1,000 shares could choose to
than 10%. The expected value of the fund after five years is thus exchange the shares for one tenth of the fund's portfolio. This
USD 153.8 (= 100 X 1.095). The correct discount rate is still 10% would reduce the number of shares of the E T F to 9,000 and
and therefore the value of the investment in the fund is provide the institutional investor with an immediate gain of
USD 1 per share.
USD 100 X 1.095
USD 95.54
1. 105 Now suppose that the shares are worth USD 26 each and the
We therefore expect the market value of the fund to be 4.46% value of the assets of the fund is USD 250,000. In this case,
an institutional investor can acquire (from elsewhere) a port
less than the value of the underlying assets (USD 95.54 instead
folio that is the same as one tenth of the fund's portfolio and
of USD 100.00).
exchange that portfolio for 1,000 new shares in the fund. The
As already m entioned, closed-end funds are much less number of shares in the fund would then increase to 11,000, and
popular than open-end funds. However, they do have certain the institutional investor should gain USD 1 for each new share.
advantages. Unlike open-end fund shares, shares of closed-end
The fact that institutional investors have these opportunities
funds can be bought and sold at any time of day; they can even
means that in practice the share price and the net asset value do
be shorted.8 Also, unlike open-end funds, closed-end funds
not diverge (at least not significantly or for long periods of time).
do not need to keep enough liquid assets to handle possible
W hile many ETFs are designed to track an index, they can be
redem ptions. This is because closed-end fund investors trade
actively managed as well. However, to ensure that the m echa
with each other, whereas open-end fund investors trade with the
nism for equalizing the NAV to the share price works, they must
fund itself.
disclose their assets twice a day. Mutual funds disclose their
assets much less frequently.
3.2 EXCH AN GE-TRADED FUNDS E T Fs are gaining in popularity. By 2017, their total net assets in
the U.S. was USD 3.4 trillio n .9 As m entioned earlier, they com
Exchange-traded funds (ETFs) combine features of open-end bine features of open- and closed-end funds. Like closed-end
mutual funds with features of closed-end mutual funds. In a 2008 funds they can be traded at any tim e, they can be shorted, and
survey of investment professionals, ETFs were voted the most they do not have to keep liquid assets to m eet redem ptions.
innovative investment product of the previous two decades. But unlike closed-end funds, E T Fs trade with no discount to
They have been trading in the U.S. since 1993 and in Europe the NAV.
since 1999. A well-known ETF is the SPDR S&P 500 ETF, which
tracks the S&P 500 and trades under the symbol SPY.
An E T F is created when an institutional investor deposits a block 3.3 UNDESIRABLE TRADING

of shares with the E T F and in exchange receives shares in the BEHAVIOR
ETF. The shares of the E T F are then traded on an exchange just
like the shares of any other company. In the U .S., mutual funds and ETFs are heavily regulated by the
Securities and Exchange Commission (S E C ).10 Com plete and
Institutional investors have the right to give up shares in the
accurate financial information must be provided to prospective
E T F and in exchange receive their share of the ETF's underlying
investors. There are also rules to prevent conflicts of interest and
assets. They also have the right to do the reverse (i.e., obtain
fraud. Despite these safeguards, there have been instances of
additional shares in the E T F by adding assets). The assets added
undesirable behavior.
must have the same composition as the ETF's current assets.
These features act as a control mechanism preventing the NAV a) Late Trading. As m entioned, all trades to buy and sell
and the share price from diverging. shares in an open-end mutual fund are at 4 p .m ., and thus
9 See Investm ent Com pany Institute w w w .ici.org

8 Shorting is a procedure where an investor borrows shares from another
investor and sells them in the m arket, hoping to buy them back more 10 For Security and Exchange Regulations, see https://w w w .sec
cheaply later. .gov/fast-answers/answersm utfundhtm .htm l

trade instructions should reach a broker before 4 p.m . 3.4 H ED G E FUNDS
For adm inistrative reasons, however, those trades may
not be passed to the mutual fund until well after 4 p.m . Hedge funds (a form of alternative investm ents) are subject to
If there are m arket developm ents soon after 4 p .m ., a less regulation than mutual funds and ETFs. W hile mutual funds
trader might call his or her broker to cancel a trade due and ETFs cater to the needs of small investors, hedge funds
to be carried out at the 4 p.m . price or to put through a usually accept only large investments from wealthy private
new trade at that price. Some brokers in the U.S. have individuals or institutions. There are several other differences.
been known to (dishonestly) accept orders after 4 p.m . The following are exam ples.
This activity is known as late trading and is not perm itted
• A mutual fund or E T F allows investors to redeem their shares
by the S E C . In fact, there have been several prosecutions
on any day. A hedge fund may have a lock-up p e rio d during
leading to large fines and the involved em ployees losing
which time funds cannot be withdrawn. Lock-up periods of
their jobs.
one year are common.
b) M arket Timing. Not all the assets of an open-end mutual
• The NAV of a mutual fund or E T F must be calculated and
fund trade actively. This may lead to the prices used to
reported at least once a day. In contrast, hedge funds have
calculate NAV being stale (i.e., not reflecting recent infor
no such requirem ents, and their NAVs are reported much less
mation). The prices of securities that trade on overseas
frequently.
markets may also be stale because of tim e zone differences.
• Mutual funds and ETFs must disclose their investment
For exam ple, suppose that it is now 3:45 p.m . and prices
in markets have been rising during the last few hours. The strategies. Hedge funds generally follow proprietary strate
existence of stale prices means that shares in the mutual gies that they see as fundamental to their com petitiveness
and/or value proposition. They give prospective clients some
fund are probably worth slightly more than the NAV, and
information to explain their value proposition, but do not
therefore buying at the 4 p.m . NAV is attractive. Similarly,
if prices in markets have been falling, selling at the 4 p.m. disclose everything. Furtherm ore, they are not obligated to
stick to one strategy.
NAV is attractive. M arket timing trades of this sort are
not illegal, but they must be quite large to be worthwhile. • Mutual funds and ETFs may be restricted in their use of
If the mutual fund allows the trades, the size of the fund leverage. A hedge fund is only restricted by the amount
will whipsaw up and down. This could lead to costs for all banks are willing to lend to it.
investors as the fund may have to keep additional cash to • Hedge funds charge an incentive fee as well as a
accom m odate redem ptions. Regulators are likely to be m anagem ent fee. A typical hedge fund fee is 2 plus 20%.
concerned if special trading privileges are offered to market This means that the investors are charged 2% of the value
timers. of their investm ent per year along with 20% of the profits
c) Front Running. If a trader working for a mutual fund (or any (if these net profits are positive).11
other type of fund) knows that the fund will execute a big The descriptor h ed g e fund arises from the long-short strategies
trade that is likely to move the market, it is tem pting for followed by many hedge fu n d s.12 This type of strategy involves
the trader to trade on his or her own account im m ediately taking long positions in stocks expected to provide good returns
before putting through the fund's trade. For exam ple, if and short positions in stocks expected to provide poor returns.
a fund is going to buy 1 million shares of a certain stock, However, there are also hedge fund strategies that involve little
the trader might buy 10,000 for his or her own account to no hedging whatsoever.
first. The trader could also inform favored custom ers or
Some hedge funds have done very well. Jim Simons is a
other fund em ployees about what is about to happen and
form er math professor who founded hedge fund Renaissance
allow them to trade ahead of a predicted price increase or
Technologies in 1982. Its flagship Medallion fund has had a truly
decrease. Front running (also known as tailgating) is illegal
amazing record— returning an average of 35% per year over
in fund m anagem ent.
d) D irected Brokerage. This involves an informal arrange

ment between a mutual fund and a brokerage house. The 11 Fund of funds have been developed to help investors choose a
portfolio of hedge funds. This creates an extra layer of fees for the
unwritten agreem ent is that the mutual fund will use the
investor. Fund of funds used to be able to charge as much as 1 plus
brokerage house for its trades if the brokerage house 10%, but their fees are now usually much less than this.
recommends the mutual fund to its clients. The practice is 12 It was coined by Carol Loomis in 1966 in an article about the first
frowned upon by regulators. hedge fund, A .W . Jo n es & Co.

a 20-year p erio d .13 It uses com plex mathematical models and The expected returns for strategies A 1, B1, and C1 are 3%,
huge volumes of data to determ ine its trading, much of which 2.5% , and 0% (respectively). Strategies B1 and C1 make no
is autom ated. As previously m entioned, a common hedge fund sense because the risk-return trade-off is negative (i.e., the
fee schedule is 2 plus 20%. It is reported that Renaissance has expected return decreases as risk increases). N evertheless, strat
charged as much as 5 plus 44% . In 2017, Jim Simons' wealth was egies B1 and C1 could be attractive to hedge fund managers (as
estim ated to be USD 18 billion.14 we will now show).
O ther well-known hedge funds are Bridgew ater (founded by Ray The fees to the hedge fund for each of the strategies are calcu
Dalio), Soros (founded by G eorge Soros), and Citadel (founded lated in Table 3.1. The m anagem ent fee for all three strategies is
by Ken Griffin). All three founders have becom e very rich from USD 2 million (i.e., 2% of USD 100 million). For the first strategy,
the incentive fees charged to investors. the profit after the m anagem ent fee is USD 1 million and the
incentive fee is thus USD 0.2 million (i.e., 20% of this profit).
In this chapter, we will assume that the management fee is cal
For the second strategy, there is a 50% chance that the incen
culated on the assets at the beginning of the year and that the
tive fee will be zero, and a 50% chance that it will be 20% of
incentive fee is calculated after subtracting management fees.
USD 3 million. For the third strategy, there is a 50% chance that
But it should be noted that some hedge funds do try to use a
the incentive fee will be zero, and a 50% chance that it will be
more aggressive fee structure where the management fee is cal
20% of USD 8 million.
culated based on the end-of-year asset value (making it greater if
the fund's value has increased) and the incentive fee is calculated The table shows that the expected incentive fee increases as the
before subtracting management fees (which is also more valuable strategy becom es riskier even though the expected gross return
to the hedge fund managers if the value of fund has increased). declines as this occurs.
The incentive fee can be thought of as a call option on the net If the hedge fund manager does better as more risks are taken,
profit produced by the hedge fund for an investor in a given the investor is likely to do worse. These total returns are summa
year. Consider the 2 plus 20% fee schedule. Using the assum p rized in Table 3.2.
tions mentioned above, we can calculate the incentive fee as:
This simple exam ple illustrates that a hedge fund has an incen
0.2 X max (R X A — 0.02 X A , 0) tive to take risks even if the risks are not justified by a higher
where A is the assets under m anagem ent at the beginning of expected return.
the year and R is the return on the assets during the year. This It is also the case that hedge funds can benefit from risks even
is the payoff from a call option on the dollar return with a strike
when there is an acceptable risk-return trade-off (i.e., the hedge
price equal to 2% of the assets under m anagem ent. This creates
fund earns a higher expected return from taking more risks).
a situation where a hedge fund has an upside for the managers,
but no downside.
Table 3.1 Hedge Fund Fees for Strategies A1, B1,
As an exam ple, suppose that a hedge fund has USD 100 million and C1 (USD Million)
of investors' funds and its fees are 2 plus 20%. Consider the fol
Management Expected Total
lowing three strategies.
Strategy Fee Incentive Fee Expected Fee
1. Stra teg y A 1 : Choose a safe investment that will produce a
profit of USD 3 million with certainty. The expected return is A1 2 0.2 2.2
3% (= 3/100). B1 2 0.3 2.3
2. Strategy B1: Choose a riskier strategy that has a 50% C1 2 0.8 2.8
chance of producing a profit of USD 5 million and a 50%
chance of producing a profit of zero. The expected return is
Table 3.2 Returns to Hedge Fund and Investors
2.5% (= 0.5 x 5% + 0.5 x 0%).
from Strategies A1, B1, and C1
3. Stra teg y C 1: Choose a highly risky strategy that has a 50%
Return to Return to
Strategy Hedge Fund Investor Total Return
chance of a loss of USD 10 million. The expected return is
0 % [= 0.5 X 10% + 0.5 X (-1 0 % )]. A 2.2% 0.8% 3.0%
B 2.3% 0.2% 2.5%

13 This includes a 98.2% return in 2008, a year when the S&P 500 lost 38.5%.
C 2.8% - 2 .8 % 0.0%
14 According to the Forbes Billionaires List.

For exam ple, suppose that we change the above exam ple so He was betting that prices in the winter would rise relative to
that the three strategies becom e the following. those in the summer. This type of strategy worked well the pre
vious year because hurricanes had adversely affected natural gas
1. Stra teg y A 2 : Choose a safe investm ent that will produce a
supplies. Hunter's bonus from his trading in that year is reported
profit of USD 3 million with certainty.
to have been about USD 100 million. In 2006, however, Hunter's
2. Strategy B2: Choose a riskier strategy that has a 50% strategy lost about two thirds of the USD 9 billion that had been
chance of producing a profit of USD 10 million and a 50% invested with Am aranth. But while these losses caused the fund
chance of producing a profit of zero. The expected return is to be wound down, Hunter was able to keep the bonus he had
5% ( = 0.5 X 10% + 0.5 X 0%). received the previous year.
3. Strategy C 2: Choose a highly risky strategy that has a 50%
Investors are naturally wary of the one-sided nature of incentive
fees and some hedge funds have adjusted the term s they offer
chance of a loss of USD 12 million. The expected return is
to reflect this. The following are exam ples.
9% (= 0.5 x 30% + 0.5 x (-1 2 % )).
• Som etim es incentive fees are payable only on returns above
In this case, there is a positive risk-return trade-off because the
a certain level. This level is referred to as a hurdle rate.
expected total returns from strategies A 2, B2, and C2 are 3%,
• Som etim es the agreem ent between hedge funds and inves
5%, and 9% (respectively). The fees for the hedge fund are now
tors states that incentive fees only apply when cumulative
calculated as shown in Table 3.3.
profits for an investor are positive. This is known as a high-
Strategy A2 is the same as strategy A1 so that the investors w ater mark clause. If USD 100 million is invested, and the
earn 0.8% . The investor's expected returns after fees from strat hedge fund loses USD 10 million, for exam ple, it must make
egy B2 is 2.2% (= 5% — 2.8% ), and the investor's expected USD 10 million before incentive fees kick in. There may also
return after fees from strategy C2 is 4.2% (= 9% — 4.8% ). The be a proportional adjustm ent clause stating that the high-
hedge fund's expected return is thus higher than the investor's w ater mark only applies to funds that are not withdrawn.
expected return. Furtherm ore, it is never negative. These results If the investor in our exam ple withdraws half of his or her
are summarized in Table 3.4. remaining funds after the loss, the hedge fund only needs
An exam ple of a hedge fund that tried to increase the value of to make USD 5 million for the investor before incentive fees
the call option em bedded in its incentive fees by taking huge apply.
risks is Am aranth. Brian Hunter, a star trader at the firm, took • There is som etim es a claw back clause where incentive fees
large highly leveraged positions in natural gas futures in 2006. paid by the investor can be used to offset future losses.
When hedge fund managers incur substantial losses, however,
Table 3.3 Hedge Fund Fees for Strategies A2, B2, they have an incentive to close a fund and start a new one to
and C2 (USD Million) avoid high-water marks and clawbacks.
Management Expected Total

Strategy Fee Incentive Fee Expected Fee Prime Brokers
A2 2 0.2 2.2 A hedge fund's prime broker is the bank that handles its trades
and lends it money. Many hedge funds take short positions, and
B2 2 0.8 2.8
the prime broker will handle these for them as well. The bank
C2 2 2.8 4.8
may provide risk m anagem ent and hedging services as well.
Furtherm ore, the prime broker can carry out stress tests on the
hedge fund's portfolio to decide how much it is prepared to
Table 3.4 Returns to Hedge Fund and Investors lend. The hedge fund can then post its securities with the bank
from Strategies A2, B2, and C2 as collateral.
Expected Return Expected Return Total
As m entioned, hedge funds are subject to very little regula
Strategy to Hedge Fund to Investor Return
tion. However, their activities may be constrained by their prime
A2 2.2% 0.8% 3.0% broker. There is always a danger that in some adverse econom ic
environments (e.g., those experienced during the 2007-2008
B2 2.8% 2.2% 5.0%
crisis), the prime broker will reduce the borrowing limit of the
C2 4.8% 4.2% 9.0%
hedge fund and force it to close out positions.

Some hedge fund strategies can be certain to make money in There are many variations on the traditional long-short strat
the long term while risking short term losses. If these losses egy. If the hedge fund manager thinks that the market is more
occur, the bank could require additional collateral. Hedge funds likely to go up than down, the value of the long position may
must therefore consider the extent to which their prime bro be greater than the value of the short position. If the reverse is
kers are prepared to fund short-term losses. Long Term Capital true, the m anager may choose to have a larger short position.
M anagem ent is an exam ple of a hedge fund that took positions Hedge funds often use fundamental analysis, as pioneered by
that could reasonably be expected to be profitable if held for Ben Graham , to choose the underpriced and overpriced shares.
several years. However, there were huge short-term losses in
1998 because of the impact of Russia's default on its debt. The
increased collateral requirem ents that followed led to the fund's
Dedicated Short
failure. A t any given tim e, it is reasonable to suppose that there are as
Large hedge funds may use more than one prime broker. This many overvalued shares as undervalued shares. A dedicated
gives hedge funds additional flexibility and ensures that no one short fund devotes its attention to picking overvalued stocks.
bank is able to see every trade. Prior to 2008, it was considered Hedge funds using dedicated short strategies look for com
that nearly all the risks in the prime broker-hedge fund relation panies that are experiencing difficulties not recognized by the
ship were borne by the prime brokers. When Lehman defaulted, market. However, they are not hedged against the overall per
however, many hedge funds that used Lehman Brothers as their form ance of the market. It is therefore not surprising that dedi
prime broker found that they could not access the securities that cated short strategies perform badly during bull markets.
they had posted as collateral. This made the market realize that
both sides were subject to risks.
Distressed Debt
Some hedge funds specialize in trading distressed debt. They
3.5 TYPES O F H ED GE FUNDS use their understanding of the bankruptcy process to find situa
tions where they can take a big position in the debt and benefit
Hedge funds follow many different trading strategies. Here we from reorganization proposals.
discuss a few of the more common o n es.15
Merger Arbitrage
Long-Short Equity
When a company announces that it is prepared to buy another
The original hedge funds were long-short funds that purchased company, there is usually some uncertainty about whether the
stocks that were considered underpriced and shorted those acquisition will proceed. The share price of the target company
considered to be overpriced. A long-short fund's perform ance usually increases, but not to the price being offered. A merger
should depend entirely on the fund manager's ability to pick arbitrage hedge fund might consider that there is an 80%
winners and losers (and not on what happens to the market as a chance that the acquisition will be successful and that the acqui
whole) so long as: sition price will be higher than the current price. Buying the
• The value of the shares shorted equals the value of those target company's shares could then be a good trade. If the offer
bought, and on the table is a share-for-share exchange and the hedge fund
expects an im provem ent in the term s before a deal is finally
• Both the long and short portfolios have the same sensitivity
announced, it could buy the shares in the target company and
to market movements.
sell shares in the acquiring company in a ratio that reflects the
For exam ple, consider a situation where General Motors and current offer.
Ford are considered to have the same sensitivity to the S&P 500.
It should be emphasized that m erger arbitrage is not about
However, Ford is considered to be undervalued, while General
trading on inside (non-public) information (which is illegal).16 It is
Motors is considered to be overvalued. A long-short strategy
about assessing the probability of a m erger being successful and
could therefore involve buying USD 100,000 of Ford stock and
the likely final price (or exchange ratio in the case of a share-for-
selling USD 100,000 of General Motors stock.
share exchange) at the time of the m erger announcement.
15 Renaissance Technologies does not fit into any of the standard hedge
fund categories. It could be categorized as a Q uant Fund. 16 Ivan Boesky was sentenced to three years in prison for insider trading.

Convertible Arbitrage Managed Futures
Convertible bonds are bonds issued by a company that can Managed futures strategies attem pt to predict future com m od
be converted into a predeterm ined number of the company's ity prices and take positions that will be profitable if the predic
shares at a future tim e. Convertible arbitrage hedge funds tions are correct. Several different models are used and trading
use sophisticated models to value convertible bonds. They rules are usually b a ck-tested by seeing how well they would
hedge the risks associated with the company's share price, have performed if they had been used in the past. However,
credit spreads, and interest rates. If the market price is cur there is a danger in this. Back-testing does not differentiate
rently different from the hedge fund's model price, the strat between strategies with a fundamental understanding of the
egy can be profitable if the market price converges to the markets and strategies that were just lucky (and thus not neces
model price. sarily bound to be successful in the future).
Fixed-Income Arbitrage 3.6 RESEARCH ON RETURNS

A t any given tim e, some traded bonds are likely to be relatively
The key question for an investor is: "Is it worth paying a pro
expensive com pared with other similar bonds, while others are
fessional to invest my money for m e?" It is not clear that the
relatively cheap. In a fixed-incom e arbitrage strategy, the hedge
answer to this question is yes. Some mutual funds and hedge
fund manager buys bonds that seem relatively cheap and shorts
funds have produced excellent returns for investors, but there is
the ones that are relatively expensive. Typically, they use a lot of
always the all-too-true small print in any solicitation: "Past per
leverage to make the strategy worthwhile.
form ance is no guarantee of future results."
Emerging Markets Mutual Fund Research

Hedge funds that specialize in emerging markets attem pt to
Questions a prospective mutual fund investor might reasonably
gather information about little-known equity securities in devel
ask are as follows.
oping countries. When they consider a security to be under
valued (overvalued), they buy (short) it in the local markets. • On average, do actively managed mutual funds outperform
An alternative is to use Am erican Depositary Receipts (ADRs), the market?

which are certificates backed by shares of a foreign company • Do actively managed funds that outperform the market in
and traded on an exchange in the U.S. Any discrepancies one year have a high probability of doing so in the next year?
between AD R prices and local prices can give rise to arbitrage
Michael Jensen investigated these questions in the 1960s, and
opportunities. 17
the answer to both appears to be no. O ver the past half cen
Hedge funds can also invest in emerging market sovereign debt. tury, his results have been confirmed by many other researchers
However, this is fraught with risks. Countries such as Russia, using more recent data.
Argentina, Brazil, and Venezuela have defaulted multiple times
Jensen found that actively managed mutual funds (on average)
(as will be discussed in Chapter 5 of Valuation and Risk M o d els).
do not beat the market after expenses. This is not surprising.
On average, the returns to all investors (before expenses) is
Global Macro the market's return. Because mutual funds and ETFs hold over
30% of all U.S. corporate equity, any outperform ance by these
Global macro hedge funds use m acroeconom ic analysis to securities would imply system atic underperform ance by other
determ ine their trades. Specifically, they look for situations investors.
where markets are not in equilibrium using models based on
To answer the second question, Jensen calculated the prob
factors such as exchange rates, interest rates, balance of pay
ability that a mutual fund that has beaten the market for one or
ments, inflation rates, etc. Som etim es the results are spectacu
lar: The Quantum Fund managed by G eorge Soros made a
profit of USD 1 billion in 1992 by betting that the British pound
was overvalued. However, not all global macro trades are that
17 See M. C . Jen sen , "Risk, the pricing of capital assets, and the
successful, and econom ies can remain in disequilibrium for long evaluation of investm ent portfolios," Journ al o f Business, 42 (April
periods of tim e. 1969): 167-247.

more years will do so again the following year. This is referred Table 3.5 Performance of Hedge Funds
to as testing for p ersisten ce. He found that only 50% of mutual and the S&P 500
funds that beat the market one year did so again the follow
BarclayHedge Index S&P 500 Return
ing year. Mutual funds that beat the market two years in a row
Year Net Return (%) Including Dividends (%)
also had a roughly 50% probability of beating the market in the
following year. Similar results were obtained for mutual funds 2008 - 2 1 .6 3 - 3 7 .0 0
that had beaten market for three, four, five, and six years. W hile 2009 23.74 26.46
there may be some mutual funds that can consistently beat the
2010 10.88 15.06
market, Jensen's research (which has been confirmed by other
researchers using more recent data) indicates that there cannot 2011 - 5 .4 8 2.11
be very many of them . 2012 8.25 16.00
This research has led many investors to invest in index funds 2013 11.12 32.39
rather than actively managed funds. Index funds are term ed 2014 2.88 13.38
passive investm ents (as opposed to active m anagem ent). Index
2015 0.04 1.38
funds charge lower fees than actively managed funds and (on
average) perform better. 2016 6.10 11.96
These comments may seem to contradict the impressive returns 2017 10.36 21.83
advertised by many mutual funds. However, the fund featured
in a mutual fund advertisem ent may be one out of many differ
ent funds offered by a fund manager. Because Jensen's research
perform ed quite well. Table 3.5 com pares returns of all hedge
would suggest that a fund has a 1/2 probability of beating the funds with returns on the S&P 500 from 2008 to 2017. In 2008,
market in one year, the probability that it will beat the market
which was a watershed year for equity m arkets, hedge funds
every year for four years is therefore 1 /16 (= (1 /2 )4). If a com
on average lost money but outperform ed the S&P 500. From
pany has 16 different funds, there is a good chance that one will 2009 to 2017, hedge funds on average underperform ed the
beat the market every year for the last four years. That is the
S&P 500.
fund that will be advertised.
A possible reason for these results is that hedge funds tend to
underperform in bull markets and outperform in bear markets.
Hedge Fund Research As is apparent from the descriptions of different strategies in
Section 3.5, most strategies are not designed to follow market
It is not as easy to assess hedge fund perform ance as it is to
trends. For exam ple, the long-short strategy is designed to
assess mutual fund perform ance. There are com panies that col
either attenuate or eliminate the effects of market moves.
lect data on returns and provide return statistics for different
types of hedge funds, but participation by hedge funds is (to In view of the statistics in Table 3.5, a surprising fact is that
some extent) voluntary and not all of them report their results. hedge funds have been quite successful in attracting investors.
In fact, one can speculate that funds that incur losses (and funds Hedge fund assets under m anagem ent exceeded USD 3 trillion
that are closed) will be less inclined to report returns. This ten for the first time at the end of 2016.
dency would in turn create an upward bias in average reported
returns.
SUMMARY
The strategies followed by hedge funds are more sophisticated
than those followed by mutual funds, and a few have created a
An advantage of mutual funds and ETFs for a small investor
lot of wealth for investors (we mentioned the outstanding suc
is that they provide an easy way for the investor to obtain a
cess of Jim Simons and Renaissance Technologies earlier). But
well-diversified portfolio. Open-end mutual funds are by far the
others have failed badly. Som etim es hedge funds report good
most popular type of mutual fund. In these funds, the number of
returns for a few years and then lose a large percentage of funds
shares increases as new investors are attracted to the fund and
under m anagem ent (e.g ., Long Term Capital M anagem ent and
decreases as investors redeem their shares. All trades happen
Am aranth).
at the net asset value calculated each trading day at 4 p.m.
BarclayHedge provides an index tracking the perform ance of all A closed-end fund has a fixed number of shares that trade
hedge funds. Prior to 2008, the index shows that hedge funds throughout the day.

Exchange-traded funds are proving to be popular alternatives Research shows that, on average, actively managed mutual
to mutual funds. They trade on an exchange throughout the funds do not outperform the m arket. Furtherm ore, there is very
day. They are different from closed-end funds in that there is a little persistence: fund managers who have performed well for
mechanism whereby the share price matches the net asset value several years have probably done so by chance, and it would be
of the fund's investments. a mistake to assume that they have a better than even chance
of doing so in the future. These observations have increased
Hedge funds are used mostly by wealthy individuals and institu
the attractiveness of funds whose objective is to mimic an index
tional investors. They follow innovative strategies and charge an
of equity returns such as the S&P 500. Some hedge funds have
incentive (performance) fee, which is a percentage of profits, as
performed extrem ely well and have shown persistence, but
well as a m anagem ent fee. However, they are criticized because
on average hedge funds have underperform ed the S&P 500 in
the incentive fee is (in essence) an option on the performance
every year between 2009 and 2017.
of the fund and encourages the fund to take risks that are not in
the best interests of investors.

QUESTIONS

3.1 W hat is the difference between the share structures of 3.6 Explain the meaning of late trading and front running.
open-end and closed-end mutual funds?
3.7 Explain the meaning of market timing and directed
3.2 How is NAV defined? brokerage.
3.3 Explain how trades by investors to buy or sell shares in the 3.8 W hat are (a) hurdle rates, (b) high-water marks, and
fund are handled by an open-end mutual fund. (c) clawbacks in the contract between hedge funds and
3.4 W hat is the difference between an E T F and a closed-end investors?
mutual fund? 3.9 W hat does a hedge fund's prime broker do?
3.5 Give three exam ples of restrictions placed on mutual 3.10 W hat is meant by persistence in mutual fund returns?
funds that do not apply to hedge funds.
Practice Questions
3.11 The fees of a hedge fund are 2% plus 20%. W hat is the dividends of USD 1 per share in the first year and USD 2
investor's return as an algebraic function of the hedge per share in the second year. These dividends are rein
fund's return? Consider all possible values of the hedge vested by the fund. The capital gains in the first year are
fund's return. Assum e that the incentive fee is applied USD 3, and the capital gains in the second year are USD 4.
after the m anagem ent fee has been subtracted and that The investor sells the shares for USD 43 in the third year.
the m anagem ent fee is applied to the beginning-of-year Explain how the investor is taxed.
assets under m anagem ent. 3.15 A mutual fund investor tells you, "I only invest in funds
3.12 Repeat Question 3.11 assuming that the hedge fund that have beaten the market over the last three years."
follows the more aggressive strategy mentioned in the How would you respond?
chapter where the incentive fee is applied before the 3.16 W hat are index funds? W hy has their popularity increased
m anagem ent fee has been subtracted, and the man in recent years?
agem ent fee is applied to the end-of-year assets under
3.17 W hy does an E T F not have to worry about the liquidity of
managem ent.
its fund, whereas an open-end mutual fund does?
3.13 Consider how Renaissance Technologies might justify a
3.18 W hat is the mechanism that leads to the share price of an
5 plus 44% fee schedule. Assum e that the incentive fee is
ETF being close to its NAV?
applied after the m anagem ent fee has been subtracted,
that the m anagem ent fee is applied to beginning-of-year 3.19 W hy does the incentive com ponent of the fee of a hedge
assets, and that Renaissance has averaged a 35% return fund involve an option? How can the hedge fund increase
on assets under m anagem ent in recent years. the value of the option?
3.14 An investor buys 100 shares in an open-end mutual 3.20 W hy do hedge funds tend to beat a bear market and per
fund on January 1 of a year for USD 30. The fund earns form less well than a bull market?
42 Financial Risk Manager Exam Part I: Financial Markets and Products

ANSW ERS
3.1 An open-end fund has a number of shares that increase and 0 .8 R h - 0.02(1 + RH) if RH > 0
decrease as investors buy new shares or redeem existing R h - 0.02(1 + Rh) if R h < 0
shares. A closed-end fund has a fixed number of shares.
3.13 Renaissance Technologies has performed spectacularly well,
3.2 NAV is the value of the assets under m anagem ent divided with returns averaging 35% per year. If it continues to earn
by the number of shares of the mutual fund.
35%, investors will earn 0.56 X 30% or 16.8% per annum.
3.3 All trades are carried out at a price equal to the NAV cal 3.14 The investor will pay tax on the investm ent income from
culated at 4 p.m. dividends of USD 100 in the first year and on USD 200 in
3.4 In an E T F there is a mechanism to ensure that the share the second year. The investor will also pay tax on capital
price equals the NAV. gains of USD 300 in the first year and USD 400 in the
second year. The investor's basis at the start of the third
3.5 Mutual funds must be redeem able on any day. They must
year will be USD 40 per share ( = 3 0 + 1 + 2 + 3 + 4) or
calculate net asset value at least once a day. They must
USD 4,000 in total. The shares are sold for USD 4,300. Tax
disclose their investm ent policies. They may be restricted
is therefore payable on USD 300 in the third year.
in the leverage they can take on.
3.15 Research shows that mutual funds that have beaten the
3.6 Late trading is the illegal practice of finding a way of
market for three years only have a probability of about
trading an open-end mutual fund at the 4 p.m . price
50% of beating the market in the fourth year (i.e., there is
after 4 p.m . Front running is the illegal practice of trad
very little persistence in returns).
ing to profit from the fact that a large trade by a fund is
expected to move the market.
3.16 Index funds are funds that attempt to mirror the perfor
mance of a stock index such as the S&P 500. Their popular
3.7 M arket timing involves an open-end mutual fund inves ity has increased because investors have realized that, on
tor embarking on large trades motivated by the fact that
average, actively managed funds do not consistently beat
some of the security prices used to calculate an NAV are
the market and there is very little persistence in their returns.
stale. Directed brokerage is an arrangem ent between a
3.17 When an investor redeems shares in an open-end mutual
fund and a broker where the broker recommends the fund
fund, he or she trades with the fund. If the fund does not
and the fund uses the broker for trading.
have liquid assets, it must sell part of its portfolio to pro
3.8 A hurdle rate is the return that must be exceeded for an vide the investor with cash for the shares. When an investor
incentive fee to apply. A high-water mark is a loss that redeems shares in an ETF, he or she trades with another
must be recouped before incentive fees apply. A clawback investor, so the fund does not have this problem.
clause is a clause allowing investors to apply previously
3.18 Institutional investors have the right to exchange shares
paid incentive fees to losses.
in the fund for their share of the assets of the fund. They
3.9 The hedge fund's prime broker is a bank that carries out can also obtain new shares in the fund by contributing to
trades for it and lends it money. Typically, the securities of the fund a portfolio that reflects the fund's current asset mix.
the hedge fund reside with the prime broker as collateral
3.19 The fund receives an incentive fee if its return is positive
for loans and trades carried out.
(or above a hurdle rate). The incentive fee is never nega
3.10 Persistence measures the extent to which a fund manager tive. The fund's fees are therefore a call option on the
who has performed well in the past will continue to do so. return of the fund. The value of call options increases as
volatility increases. This means that the incentive fee actu
3.11 The investor's return as a function of the hedge fund's
return, RH, is ally incentivizes the fund to take as much risk as possible.
3.20 Many hedge funds follow strategies that have little to do

0 .8 (R h - 0.02) if R h > 0.02
with the return from the market. If the market does very
R h - 0.02 if R h < 0.02
well, this may not be reflected in the hedge fund's returns.
3.12 The investor's return R as a function of the hedge fund's If the market does badly, this also may not be reflected in
return, RH, is the hedge fund's returns.

Introduction
to Derivatives
Learning Objectives
Define derivatives, describe features and uses of deriva Calculate and compare the payoffs from hedging strate
tives, and com pare linear and non-linear derivatives. gies involving forward contracts and options.
Describe the over-the-counter market, distinguish it from Calculate and compare the payoffs from speculative strat
trading on an exchange, and evaluate its advantages and egies involving futures and options.
disadvantages.
Calculate an arbitrage payoff and describe how arbitrage
Differentiate between options, forwards, and futures opportunities are tem porary.
contracts.
Describe some of the risks that can arise from the use of
Identify and calculate option and forward contract payoffs. derivatives.
Differentiate among the broad categories of traders:

hedgers, speculators, and arbitrageurs.
45
The next three chapters introduce derivatives and the ways they financial assets). It is now a common practice to consider real
are traded. Derivatives are contracts whose values depend on options when valuing of capital investment opportunities.
(or derive from) the values of one or more financial variables • Homeowners som etim es have derivatives em bedded in their
(e.g ., equity prices, exchange rates, and interest rates). These m ortgages. For exam ple, a hom eowner can have the right to
variables are referred to as underlyings. repay the mortgage early and refinance at a lower rate. (We
Derivatives can be categorized into linear and non-linear prod will discuss this option in some detail in Chapter 18.)
ucts. Linear derivatives provide a payoff that is linearly related The underlyings in derivatives are often financial variables (e.g.,
to the value of the underlying asset(s). Forward contracts are an interest rates, exchange rates, and stock prices). However,
exam ple of linear derivatives. Specifically, they are agreem ents almost every observable variable can be an underlying for a
between two parties to buy or sell an asset at a specified price derivative. Exam ples include
at a future tim e. As we will see in later chapters, the value of a
• The price of hogs,
forward contract prior to maturity (as well as its payoff at matu
rity) is linearly dependent on the value of the underlying asset. • The price of electricity in a particular region,
• The amount of snow falling at a certain ski resort,

O ptions, on the other hand, are non-linear derivatives (i.e., their
payoff is a non-linear function of the value of their underlying • The tem perature at a w eather station,
assets). They are contracts where the holder has the right (but • Earthquake dam age claims made by an insurance company's
not the obligation) to buy or sell an asset for a specified price at policyholders, and
a future tim e. The payoff for an option is a non-linear function of • The lifespan for a representative group of 1,000 people.
the value of its underlying(s).
Derivatives can be used for either hedging or speculation. If a
The value of the underlying(s) is central to the valuation of both trader has an exposure in an asset class, a derivatives trade can
linear and non-linear derivatives. That being said, other variables reduce that exposure. If the trader has no exposure, however,
(e.g., interest rates and volatilities) can also play an important role. that same trade is speculative. As we will see in this chapter,
speculation can be extrem ely risky.
Derivatives have existed for many years. Before the develop
ment of money, for exam ple, goods could be som etim es be Derivatives have been criticized for their role in the 2007-2008
exchanged for crops that would be harvested in the future. In credit crisis. Note that in the years leading up to the crisis, banks
the nineteenth century, the Chicago Board of Trade was set up in the U.S. relaxed their lending standards on m ortgages. This
to facilitate agreem ents to exchange com m odities in the future. created a huge number of subprim e m ortgages (i.e., m ortgages
Derivatives now trade on exchanges as well as in over-the-counter to less credit worthy borrowers). Furtherm ore, banks did not
markets. Both types of markets have experienced unprecedented simply keep these m ortgages on their balance sheets. Instead,
growth as derivatives have become widely used. they bundled them into portfolios and created com plex deriva
tives whose values depended on the losses from defaults on
Derivatives are used in several ways.
these m ortgages.
• Derivatives are used by com panies to manage interest rate
The increased availability of m ortgages led to a significant
risk, foreign exchange risk, and the risk arising from com m od
increase in dem and for housing that in turn led to a sharp
ity price changes, etc.
increase in house prices. M eanwhile, defaults and foreclosures
• Derivatives are som etim es added to corporate bond issu began to rise as many subprim e borrowers realized they could
ances. They can give bond issuers the right to repay bonds not afford their m ortgages. (This was exacerbated by a rise in
early, or give bond holders the right to demand early repay m ortgage interest rates.) This led to an increase in the supply
ment. Som etim es, bond holders have the right to convert a of houses for sale and a reduction in housing prices. This in
bond into equity of the issuing company. turn led to negative equity positions (i.e ., situations where the
• Em ployee compensation plans som etim es give em ployees am ount owed on a m ortgage was greater than the value of
options to buy shares from the company at future times for a the house).
predeterm ined price. Some homeowners with negative equity defaulted even though
• Capital investment opportunities often have em bedded they could afford to service their mortgages, depressing house
options. For exam ple, a company embarking on an prices even further. As a result, there were losses on many of the
investment may be able to abandon it if things go badly derivatives created from mortgages, and investors throughout the
or expand if things go well. These are referred to as real world (temporarily) lost their appetite for risky debt of any sort. As
options because they involve physical assets (rather than a result, the world was plunged into the worst recession in 75 years.

O f course, derivatives have many attractive features for society. of tim e, special-purpose calculators were developed and put
They allow risks to be transferred from one party to another in into use by options trad ers.2
ways that benefit both sides. The following are exam ples.
Traditionally, derivatives exchanges have used what is referred
• Corporate treasurers can manage exchange rate risk, interest to as the open-outcry system . This involves traders meeting on
rate risk, and com m odity price risk in ways that would other the floor of the exchange and indicating their proposed trades
wise not be possible. with hand signals and shouting. (Tall traders may have had an
• Fund managers can diversify their exposures using derivatives. advantage because it was easier for them to attract the atten
tion of other traders.) Most trading is now done electronically,
• Ski slope operators can avoid being forced out of business
however, with com puters being used to match buyers and sell
due to a single unseasonably warm winter.
ers. Som etim es electronic trading is initiated by com puter algo
The challenge for regulators is to find ways to benefit from rithms without any human intervention at all.3
derivatives while discouraging extrem e speculative behavior
(i.e., where huge risks are taken or unnecessarily com plex instru
ments are created). Over-the-Counter Markets
In this chapter, we take a first look at derivatives markets by An advantage of over-the-counter (O TC) markets is that the
examining futures, forwards, and options. We also describe the contracts traded do not have to be the standard contracts
types of trades used by hedgers, speculators, and arbitrageurs. defined by exchanges, and market participants can trade any
Later chapters will go into more detail on the markets and the contracts they like.
ways these financial instruments are traded. O T C market participants can be categorized as either end users
or dealers. End users are corporations, fund managers, and other
4.1 THE MARKETS financial institutions who use derivatives to manage their risks or
to acquire specific exposures. Dealers are large financial institu
Derivatives trade on exchanges as well as in over-the-counter tions that provide both bid and ask quotes for commonly traded
markets. An exchange is a market where investors trade standard derivatives. They are also prepared to make one-sided quotes for
ized contracts that have been defined by the exchange. Over- highly structured derivatives when requested. Dealers typically
the-counter markets are markets where participants contact each offset the risks from their trades with end users by trading with
other directly (or possibly by using a broker as an intermediary) to other dealers in what is referred to as the interdealer market.
trade. Chapters 5 and 6 describe the operation of the two mar While end users typically contact dealers directly, dealers often
kets in some detail. Here we provide a brief overview. use interdealer brokers when trading with other dealers. The
advantage of an interdealer broker is that a dealer does not have
Exchange-Traded Markets to indicate their desired trades to other dealers.4 When finding a
dealer to be a counterparty to its client, the broker does not
Derivatives exchanges have existed for many years. For
reveal its client's name until the trade has been finalized.
exam ple, the Chicago Board of Trade (C BO T) was established
in 1848 to allow farm ers to trade with merchants. Within a few Before the 2007-2008 credit crisis, the over-the-counter market
years, a forerunner of futures contracts known as "to-arrive" was largely unregulated. Since the crisis there have been several
contracts began to be traded. There are now futures exchanges new regulations. The following are exam ples.
in many parts of the world.
In 1973, the C B O T launched the Chicago Board Options 2 See F. Black and M. Scholes, "The pricing of options and corporate
Exchange (C B O E ).1 The C B O E established well-defined con liabilities," Journ al o f Political Econom y, 81 (M ay/June 1973): 637-659;
R. C . M erton, "Theo ry of Rational Option Pricing," Bell Journ al o f
tracts and a mechanism to minimize the probability of losses
Econ om ics and M an agem en t Scien ce, 4 (Spring 1973): 141-183.
from defaults. Today, the C B O E is one of many exchanges
3 This has not been without its problem s. For exam ple, the flash crash
around the world trading options on stocks and stock indices.
in 2010, where m arkets declined by about 7% in 15 minutes before
rebounding, was partly the result of the unintended consequences of a
The year 1973 also saw the publication of the famous Black-
trading algorithm.
Scholes-M erton options valuation model. W ithin a short period
4 This can be im portant. If it is known that a dealer must get rid of a
large exposure, other traders may trade ahead of the dealer because
1 A t first, the exchange only featured call options and it did not trade they know that the dealer's trade will potentially move the m arket price.
put options until 1977. Call and put options will be form ally defined later A good interdealer broker will try to feed the dealer's trades into the
in this chapter. m arket anonymously.
Chapter 4 Introduction to Derivatives ■ 47

• In the U .S ., standardized O T C
derivatives traded between deal
ers must (whenever possible) be
traded on platforms known as
swap execution facilities. These
are like exchanges and feature
market participants posting bid
and ask prices.
• A central counterparty (CCP) must
be used for standardized transac
tions between dealers. (CCPs are
discussed in the next two chapters.)
• All trades must be reported to

a central registry. (Previously,
trades in the O T C market were
considered private transactions,
and there were no reporting
requirements.) Fiaure 4.1 Size of Exchange-traded and OTC market measured in terms of
the value of underlying assets between 1998 to 2017.
Market Size S o u rce: https://stats.bis.org/statx/toc/D ER.htm l
The Bank for International Settlem ents began collecting data market, they measure the value of the underlying assets. If an
on derivatives markets in 1998. Note that statistics for the option gives the holder the right to purchase 100 shares worth
exchange-traded market show the value of the assets underly USD 40 per share for USD 45 per share, for exam ple, the size of
ing outstanding exchange-traded contracts, while statistics for the contract would be recorded as USD 4,000 (= 100 X 40). In
the O T C market show the total principal underlying outstanding the case of the O T C m arket, the statistics measure the principal
transactions. Table 4.1 shows how the two markets have grown underlying outstanding transactions. This means that a forward
between June 1998 and June 2017. contract to buy 1 million British pounds at an exchange rate of
USD 1.2500 in the future would be recorded at the current value
The statistics indicate that the O TC market grew by a factor of
of 1 million British pounds and not at the value of the forward
7.4 between 1998 and 2017, while exchange-traded market grew
contract (which might be only a few thousand dollars).
by a factor of six. To put these numbers in perspective, note that
the world's gross domestic product (GDP) in 2017 was about USD The decline in the size of the O T C market between 2014 and
75 trillion. This means that the value of the assets underlying out 2017 can be attributed to the widespread use of com pression.
standing exchange-traded derivatives in 2017 was roughly equal This is a procedure where two or more market participants
to the world's GDP, while the value of the assets underlying the restructure transactions with the result being that their under
O TC derivatives market was about seven times the world's GDP. lying principal (and therefore the amount of capital they are
required to keep) is reduced. Com pression will be explained in
Figure 4.1 illustrates how the sizes of the two markets have
the next two chapters.
changed between 1998 and 2017. In interpreting Figure 4.1, it
is important to emphasize that the size statistics do not provide
the value of the transactions. In the case of the exchange-traded 4.2 FORW ARD CONTRACTS
Table 4.1 Value of Underlying Assets in Derivatives A forward contract is an over-the-counter contract where one
Markets in June 1998 and June 2017 (Trillions of USD) party agrees to buy an asset for a predeterm ined price at a
Exchange-Traded Over-the-Counter future time and the other party agrees to sell the asset for the
Market (OTC) Market predeterm ined price at the future tim e. Forward contracts can
be contrasted with spot contracts, which are agreem ents to buy
June 1998 13.3 72.1
or sell an asset almost im m ediately.5
June 2017 81.0 531.9
S o u rce: https://stats.bis.org/statx/toc/D ER.htm l 5 Settlem ent may take a day or two in a spot contract.

The party that has agreed to buy has a long forw ard position
while the party that has agreed to sell has a short forw ard p o si
tion. The specified asset price in a forward contract is referred
to as the forw ard price. The determ ination of forward prices will
be discussed in later chapters.
Forward contracts on foreign currency are very popular.

If Com pany A knows it will receive a certain amount of a foreign
currency at a certain future tim e, it can use a forward contract to
lock in the exchange rate by selling the foreign currency at the
forward exchange rate. Similarly, if Com pany B knows that it will
pay a certain amount of a foreign currency at a certain future
Figure 4.2 Payoff from a long position in a forward
tim e, it can enter into a long forward contract to buy the foreign
contract. The delivery price is K and the value of the
currency at the forward exchange rate.
asset at maturity is ST.
Table 4 .2 shows quotes for spot and forw ard contracts on
the U SD -euro exchang e rate (as they m ight have been made
by a d erivatives dealer) on Ju n e 8, 2018. The bid price is
the price at which the d ealer is prepared to buy and the ask
price is the price at which the d ealer is prepared to sell. The
quotes show that the dealer is prepared to pay USD 1.1768
per euro and sell at a price of USD 1.1771 per euro in the
spot m arket. The second row shows that the d ealer is p re
pared to com m it to pay USD 1.1802 per euro in one month
or sell at a price of USD 1.1805 per euro in one m onth. The
other rows provide quotes for buying and selling in three and
six m onths. Note that forw ard prices are not equal to spot
prices. W e will explain how they are calculated in a Fiqure 4.3 Payoffs from a short position in a forward
later chapter. contract. The delivery price is K and the value of the
asset at maturity is ST.
Earlier, we mentioned the distinction between linear and non
linear derivatives. Forward contracts are linear derivatives
If this exchange rate is USD 1.2500, for exam ple, Com pany A
because their payoff is linearly related to the value of the under
buys 1 million euros for USD 1.1948 when one euro is worth
lying asset at maturity.
USD 1.2500. This leads to a payoff of:
Suppose that Com pany A enters into a long forward contract to 1,000,000 X (1.2500 - 1.1948) = 55,200
buy 1 million euros in six months. From Table 4.2, the exchange
M eanwhile, Com pany B sells 1 million euros for USD 1.1944
rate will be USD 1.1948. Suppose further that Com pany B enters
when they are worth USD 1.2500. This leads to a payoff of:
into a short forward contract to sell 1 million euros in six months.
1,000,000 X (1.1944 - 1.2500) = -5 5 ,6 0 0
Table 4.2 shows that the exchange rate applicable to this trade
will be USD 1.1944. The payoff from the contracts depends on In general, suppose that S T is the asset price at the maturity of
the actual exchange rate in six months. a forward contract and K is the delivery price (i.e., the forward
price when the contract was initiated).6 The payoff from a long
forward contract on one unit of the asset is
Table 4.2 Spot and Forward Quotes on the USD-
Euro Exchange Rate ST — K
This is shown in Figure 4.2. The payoff from a short forward con
Bid Ask
tract on one unit of the asset is:
Spot 1.1768 1.1771 K - ST
One-Month Forward 1.1802 1.1805 This is shown in Figure 4.3.
Three-Month Forward 1.1858 1.1862
6 K is the forward price at the tim e the contract is entered into, but is
Six-Month Forward 1.1944 1.1948
not necessarily the forward price for the contract at future tim es.

4.3 FUTURES CONTRACTS W hile it costs nothing to enter into a forward contract, an option
has a price (known as the premium) to be paid at the outset.
As m entioned, forward contracts are traded in the over-the- Table 4.3 shows price quotes provided by the C B O E for call
counter m arket. A futures contract provides a similar payoff to options on IBM on June 11, 2018. Table 4.4 does the same for
a forward contract, but it trades on an exchange. The exchange put options on IBM. The options on stocks traded by exchanges
defines the asset and specifies the maturity dates that can be are Am erican. O ptions trade with several different future expira
traded. As we will see in later chapters, the exchange organizes tion dates.7 The price of IBM stock at the time of the quotes was
trading so that there is very little credit risk (i.e., risk that the bid USD 147.03 and ask USD 147.04.
agreem ent will not be honored) even though the two parties to We see from the tables that call option prices decrease as the
a trade may not know each other. strike price increases, whereas put option prices increase as the
W hile forward contracts trade most actively on a small number strike price increases. The tables also show that an option price
of underlying assets (e .g ., exchange rates and interest rates), increases as the time to maturity increases. Bid-ask spreads (par
futures trade on a w ide range of other underlyings. ticularly if they are expressed as a proportion of the price)
This includes are much higher for options on stocks than they are for the
stocks them selves.
• The prices of agricultural products such as corn, wheat, and
live cattle; C onsider a European call option that can be exercised at
tim e T. Suppose that K is the strike price and S T is the option
• The prices of metals such as gold, silver, copper, and platinum;
price at tim e T. Consider first the position of the trad er that
• Equity indices such as the S&P 500 and the N A SD A Q 100;
has bought the option (i.e ., has a long position in the call
• The prices of energy products such as oil, natural gas, and option). If S j > K, the trad er exercises the option. This means
electricity; that he or she pays K fo r an asset that can be im m ediately sold
• Real estate indices; for S T. The payoff to the trader is therefore S T — K. If S T < K,
the option is not exercised and the payoff to the trader is zero.
• Tem peratures in particular cities; and
Putting the S T > K and S T K outcom es together, we see
• Cryptocurrencies like bitcoin.
that the option payoff is
Futures contracts are covered further in later chapters.
m ax(ST — K, 0)
This payoff is illustrated in Figure 4.4.

4.4 OPTIONS
To the trader with a short position (i.e., the trader who has sold
O ptions are derivatives that give the holder the right (but not the option), the payoff is
the obligation) to buy or sell an asset at a predeterm ined price
—m ax(ST — K, 0)
in the future. They trade on exchanges as well as in the over-the-
counter market. If Sy > K, this trader must sell an asset worth S T for K. If S T < K,
the trader is not required to make the sale. This payoff is shown
To see how options work, note that there are two sides to every
in Figure 4.5.
options contract. In the case of a call option, the party with a
long position has the right (but not the obligation) to buy an Next consider a European put option with strike price of K and
asset from the party with a short position for a certain price asset price at maturity of S T. In this case, the trader who has
(known as the strike price or exercise p rice) at one or more bought the option has the right to sell the asset for K. If S T < K,
future tim es. If the party exercises this right, the party with the the trader will exercise this right, and an asset worth S T is then
short position must sell the asset for the strike price. sold for K. This creates a payoff of K — S T. If S T > K, the option
is not exercised, and there is no payoff to the option owner. The
A put option is a contract where the party with a long position
payoff to the option owner is therefore
has the right (but not the obligation) to sell an asset to the party
with a short position for a certain price (the strike price) in m ax(K — Sy, 0)
the future.
The date specified in an options contract is known as the exp i

7 The expiration dates in Tables 4.3 and 4.4 are the third Friday of the
ration date (or maturity date). A European option can only be
month (by coincidence it is the twenty-first day for each of the months
exercised at expiration. An Am erican option can be exercised at shown). The exchange has introduced w eeklys and quarterlys that expire
any time up until expiration. at other tim es. These will be discussed in later chapters.

Table 4.3 Call Option Premiums on IBM on June 11, 2018, with Different Expiration Dates
September 21, 2018 December 21, 2018 June 21, 2019
Strike Price Bid Ask Bid Ask Bid Ask
140 9.70 9.80 11.45 11.65 14.20 14.45
145 6.35 6.45 8.40 8.55 11.45 11.65
150 3.80 3.90 5.90 6.05 9.05 9.25
155 2.14 2.18 4.00 4.10 7.00 7.25
Table 4.4 Put Option Premiums on IBM on June 11, 2018, with Different Expiration Dates
September 21, 2018 December 21, 2018 June 21, 2019
Strike Price Bid Ask Bid Ask Bid Ask
140 2.76 2.81 5.00 5.15 8.65 8.90
145 4.50 4.65 7.05 7.15 10.85 11.10
150 7.10 7.25 9.65 9.80 13.45 13.70
155 10.55 10.65 12.75 12.95 16.40 16.70
Fiqure 4.4 Payoff to trader who has a bought a Fiqure 4.6 Payoff to trader who has a bought a
European call option. K is the strike price and S T is the European put option. K is the strike price and S T is the
asset price on the expiration date. asset price on the expiration date.
This payoff is shown in Figure 4.6.
To the trader with a short position (i.e., the trader who has sold
the option), the payoff is
- m a x (K — S j, 0)
If S T < K, the trader (who is short the put option) must buy an
asset worth S T for K. If S T > K, and the trader is not required to
buy the asset. This payoff is shown in Figure 4.7.
O ptions are more com plex derivatives than forward or

Fiqure 4.5 Payoff to trader who has sold a European futures contracts. As Figures 4.4 to 4.7 show, their payoffs
call option. K is the strike price and S T is the asset price are non-linear functions of the underlying asset price. This has
on the expiration date. two consequences.

N ext, consider the treasurer who is due to receive euros in
six months. This treasurer could buy a European put option to
sell 1 million euros at an exchange rate of USD 1.1950. If the
exchange rate in six months' time is less than this, the treasurer
exercises the option and sells the euros that are received for
USD 1.195 million. If the exchange rate in six months is greater
than this, the option is not exercised. However, the treasurer can
benefit from a relatively favorable exchange rate when the
1 million euros are sold.
These exam ples show that while a forward contract locks in the
price applicable to a future transaction, an option provides pro
Fiaure 4.7 Payoff to trader who has sold a European
tection against adverse price movements.
put option. K is the strike price and ST is the asset
price on the expiration date. It is im portant to note, however, that it does not cost any
thing (excep t for the bid-ask spread) to lock in the forward
price. By contrast, an option requires that the buyer pay a
1. The value of an option is a non-linear function of the value
prem ium . In our exam ple, the call (put) option for 1 million
of the underlying.
euros with a strike price of USD 1.1950 might cost USD
2. The value of an option is dependent on the volatility of the 40,000. W hile the treasurer could reduce the cost by increasing
underlying. (reducing) the strike price, this would lower the protection
as w ell.
We will discuss the properties of options in more detail later in
this book. The Valuation and Risk M o d els book covers the way
options are valued and hedged.
Speculators
4.5 M ARKET PARTICIPANTS Derivatives also allow risks to be taken with a relatively small
upfront paym ent. In this sense, they have much the same effect
as leverage and can be attractive to speculators.
There are three main categories of traders in derivatives
markets. To illustrate this point, suppose a stock is currently worth
USD 40 and a trader is convinced that the price will increase in
1. Hedgers,
the next three months. Furtherm ore, a call option on the stock
2. Speculators, and
with a strike price of USD 42 has a price of USD 2. One strat
3 . Arbitrageurs. egy would be to buy 100 shares of the stock for USD 4,000.
An alternative strategy would be to use the USD 4,000 to buy
Hedgers 2,000 options.
Both strategies involve an initial investm ent of USD 4,000.

Hedgers use derivatives to reduce or eliminate risk exposure.
As indicated in Table 4.5, however, they provide very
We have already seen how traders can use forward contracts on
different outcom es.
a foreign currency to manage foreign exchange risk.
Consider the outcomes where the share price is USD 45 at the
O ptions can also be used to hedge currency risk. Unlike forward
end of the three months. The results from using the two specu
contracts, options allow traders get downside risk protection
lative strategies would be as follows.
while preserving some upside potential.
1. Share purchase stra teg y: There is a USD 5 profit on each
For exam ple, a U.S. corporate treasurer due to pay 1 million
share purchased. The total profit is USD 500
euros in six months could buy a European call option to buy
= ((45 - 40) X 100).
1 million euros in six months with a strike price of USD 1.1950. If
the exchange rate in six months is greater than 1.1950, the trea 2. O ption strategy: The options strategy allows 2,000 shares
surer will exercise the option and acquire 1 million euros at an worth USD 45 to be purchased for USD 42. This generates
exchange rate of USD 1.1950. If the exchange rate is less than a profit of USD 6000 (= 2000 X (45 - 42)). When the initial
USD 1.1950, the treasurer does not exercise the option and is cost of the options is accounted for, the profit is USD 2,000.
instead able to buy the euros at a favorable exchange rate. (We ignore the impact of discounting.)

Table 4.5 Profits from Two Alternative Strategies that when their mandate is to hedge risks or to search for arbitrage
are Speculating that the Price of an Asset will Increase opportunities. If controls are not in place, these traders
may start speculating without the knowledge of others in
Asset Price in Three Profit from Asset Profit from
their organization.
Months (USD) Purchase (USD) Options (USD)
A speculating trader who loses money may seek to offset losses
30 - 1 ,0 0 0 - 4 ,0 0 0
by taking increasingly large and risky positions. If the offsetting
35 -5 0 0 - 4 ,0 0 0 gains do not m aterialize, the trader may increase the risks taken
40 0 - 4 ,0 0 0 again and again until the losses reach catastrophic levels. Exam
ples of such traders include
45 500 + 2,000
50 1,000 + 12,000 • John Rusnak at Allied Irish Bank, who lost USD 700 million
trading in foreign currencies and managed to conceal his
losses by creating fictitious option trades;
Table 4.5 indicates that the option strategy is four tim es as prof
• Nick Leeson at Barings Bank, who lost about USD 1 billion by
itable as the share purchase strategy if the share price proves
making unauthorized large bets on the future direction of the
to be USD 45 and is tw elve tim es as profitable if the share price
Nikkei index and managed to hide his losses from his superi
ends up being USD 50. However, the option strategy leads to
ors for some tim e;
a loss of USD 4,000 if the share price proves to be less or equal
to than USD 42. In these scenarios, the option is not exercised, • Jerom e Kerviel at Societe G enerale, who lost about
and there is no payoff to offset the USD 4,000 option premium. USD 7 billion by speculating on equity indices while giving
In contrast, the share purchase strategy involves no option pre the appearance of being an arbitrageur;
mium and thus produces a profit (or loss) that is solely depen • Kweku Adoboli at UBS, who lost about USD 2.3 billion taking
dent on the difference between the share price at purchase and unauthorized speculative positions in stock market indices.
the share price in three months.
SUMMARY
Arbitrageurs
Derivatives trading takes place in both exchange-traded mar
Arbitrage involves taking advantage of inconsistent pricing across kets and over-the-counter m arkets. These markets have been
two or more markets. For exam ple, suppose that an asset that introduced in this chapter and are discussed in more detail in
provides no income is priced at USD 50, the borrowing rate is 3%, the next two chapters.
and the asset's one-year forward price is USD 52. An investor can
Forward contracts are over-the-counter agreements where two
• Borrow USD 50 million for one year at 3% to buy 1 million parties agree to trade a certain asset at a certain price on a given
units of the asset for USD 50 per unit, future date. These contracts can be used to lock in the price of an
• Enter into a forward contract to sell 1 million units of the asset that will be bought or sold on the future date. Futures con
asset for USD 52 per unit in one year, and tracts are similar to forward contracts but are traded on exchanges.
• Repay the loan in one year at a cost of USD 51,500,000 O ptions trade on both exchanges and in over-the-counter mar
(= 50,000,000 X 1.03) and sell the 1 million units of the asset kets. W hile forwards and futures commit a trader to buying or
for USD 52,000,000. selling an asset for a certain price in the future, options give
This is arbitrage leads to a profit of USD 500,000. the holder the right (but not the obligation) to buy or sell at a
certain price in the future. W hereas forwards and futures can be
used to lock in the price for a future transaction, options can be
4.6 DERIVATIVES RISKS used to buy protection against unfavorable price movements
while allowing the holder to benefit from favorable movements.
We have just shown that derivatives markets attract many kinds
Derivatives are versatile instruments that can be used for hedg
of traders. This is one of their strengths and a major reason for
ing, speculation, or arbitrage. This is one of their strengths, but
their success. However, the leverage that speculators can obtain
it also creates risks. For many traders, speculation is more fun
means that it is very easy for traders to take significant risks.
(and potentially more rewarding) than hedging. Unless there are
A key problem is that the rewards from successful speculation good controls in place, there is the danger of traders engaging
are very high and many traders are tem pted to speculate even in unauthorized speculation (with disastrous results).

Q UESTIO N S

4.1 W hat is meant by a linear and a non-linear derivative? 4.6 W hat is the payoff from a long position in a call option in
Give an exam ple of each. term s of the asset price, ST, and the strike price, K?
4.2 W hat is the difference between a long forward position 4.7 W hat is the payoff from a short position in a put option in
and a short forward position in an asset? term s of the asset price, ST, and the strike price, K?
4.3 W hat are the two main types of markets for trading 4.8 W hat are the three types of traders in derivatives markets?
derivatives?
4.9 Traders such as Jerom e Kerviel and Nick Leeson had a
4.4 Which of the following is closest to the ratio of the size of mandate to carry out hedging or arbitrage trades. How
the O T C market to size of the exchange traded market in did they lose billions of dollars?
2017: (a) 2, (b) 4, (c) 7, or (d) 16? 4.10 "O ptions provide leverage." Explain this statem ent.
4.5 Which of the following are traded on exchanges: (a) for
wards, (b) futures, (c) options?
Practice Questions
4.11 A trader enters into a short futures contract to sell 4.17 W hat is the payoff from a portfolio consisting of a short
100 units of an asset for USD 50. W hat is the trader's gain forward contract with maturity T and a long call option
or loss if the price of the asset at maturity is (a) USD 55 with maturity T? Assum e that the strike price for the
and (b) USD 48? option is the forward price.
4.12 For a corporate treasurer wanting to hedge exchange rate 4.18 W hat is the payoff from a portfolio consisting of a short
risk, what is an advantage of the O T C market over the forward contract with maturity T and a short put option
exchange-traded market? with maturity T? Assum e that the strike price for the
option is the forward price.
4.13 W hat is the difference between selling a call option and
buying a put option? 4.19 A trader thinks that the price of a stock currently priced
at USD 70 will increase. The trader is trying to choose
4.14 A trader buys a call option on a stock with a strike price
between buying 100 shares and buying European call
of USD 50 when the stock price is USD 49. The cost of
the option is USD 2. Under what circum stances does the options on 1,000 shares. The options cost USD 7 per
option and have a strike price of USD 70 with a maturity of
trader make a profit? (Ignore the impact of discounting.)
six months. Explain the difference between the two trad
4.15 A trader sells a put option on a stock with a strike price
ing strategies.
of USD 50 when the stock price is USD 51. The price of
the option is USD 3. Under what circum stances does the 4.20 In Question 4.19, what is the breakeven stock price for
trader make a profit? (Ignore the impact of discounting.) which the two strategies give the same result? (Ignore the
impact of discounting.)
4.16 W hy does compression reduce the size of the O TC
markets as measured by the Bank for International
Settlem ents?

ANSW ERS
4.1 A linear derivative is a derivative where the payoff at 4.14 The payoff is m ax(S j — 50,0). This is greater than the price
a future tim e is linearly dependent on the value of the paid when:
underlying at that tim e. A non-linear derivative is a deriva
S j — 50 > 2 or S j > 52
tive where the payoff is a non-linear function of the value
4.15 The payoff is max(50 — S T,0). This is greater than the price
of the underlying. A forward contract is a linear derivative.
paid when:
An option is a non-linear derivative.
50 — S j > 3 or S j < 47
4.2 A long forward position is a position where a trader has
agreed to buy an asset at a certain price at a certain future 4.16 Com pression reorganizes the trades between several mar
tim e. A short forward position is a position where a trader ket participants, with the result being that the underlying
has agreed to sell an asset at a certain price at a certain principal is reduced. The BIS measures the size of the mar
future tim e. ket as the total outstanding underlying principal.
4.3 Exchanges and the over-the-counter markets 4.17 With our usual notation, the payoff is
4.4 (c). The O T C market is about eight tim es as big as the K — S T + m ax(ST — K, 0) = max(0,/< — S T)
exchange-traded market. To make sure you understand why this is true, you should
4.5 Futures and options (Options are also traded in the O T C consider the S T > K and the S T < K cases separately. The
market.) payoff is the payoff from a long put option with a strike
price equal to the forward price.
4.6 m ax(ST — K, 0)
4.7 —m ax(K — S Tl 0) ST > K ST < K

4.8 Hedgers, speculators, and arbitrageurs (a) Long Call ST ~ K 0
4.9 They switched from being hedgers/arbitrageurs to (b) Short Forward K - St K - ST
being speculators without anyone realizing that this had
Net (a) + (b) 0 K - ST
happened.
4.18 With our usual notation, the payoff is
4.10 Buying a call option on an asset is like borrowing money
to buy the asset, in that it allows big risks to be taken K — S T — ma x(K — ST,0) = —m ax(0,ST — K)
with a small initial investm ent. The gains and losses are To make sure you understand why this is true, you should
accentuated. consider the S t < K and the S t > K cases separately. The
4.11 (a) Trader loses (in USD) 100 X (55 - 50) = 500. payoff is the payoff from a short call option with a strike
price equal to the forward price.
(b) Trader gains (in USD) 100 X (50 — 48) = 200.
4.12 The transaction does not need to have the standard fe a ST > K ST < K
tures determ ined by the exchange. The maturity date and
(c) Short Put 0 ~ (K ~ S T)
(in the case of options) the strike price can be negotiated
to meet the treasurer's precise needs. (d) Short Forward K - ST K - ST
4.13 When a trader sells a call option, the trader must sell Net (a) + (b) K - ST 0
the asset when the asset's price is greater than the strike
Note that in this case, S j > K implies that the short for
price. When a trader buys a put option, the trader has the
ward is losing money. Thus, the solution is —m ax(ST — K,0)
option to sell the asset when the asset price is less than
that is the same as min(/C — ST,0), because K — S T < 0.
the strike price.

4.19 The option trading strategy is considerably riskier as 4.20 We require

indicated by the following table: 100(ST - 70) = 1,000(ST - 70) - 7,000
Profit from Profit from or

Stock Price Buying Stock Buying Stock 900S t = 70,000
(USD) (USD) Options (USD)
The breakeven stock price is therefore 70,000/900 or
50 - 2 ,0 0 0 - 7 ,0 0 0 77.78.
55 - 1 ,5 0 0 - 7 ,0 0 0
60 - 1 ,0 0 0 - 7 ,0 0 0
65 -5 0 0 - 7 ,0 0 0
70 0 - 7 ,0 0 0
75 +500 - 2 ,0 0 0
80 + 1,000 + 3,000
85 + 1,500 + 8,000
90 +2,000 13,000

Exchanges
and OTC Markets
■ Learning Objectives
Describe how exchanges can be used to alleviate Identify the classes of derivative securities and explain the
counterparty risk. risk associated with them.
Explain the developm ents in clearing that reduce risk. Identify risks associated with O T C markets and explain
how these risks can be m itigated.
Describe netting and describe a netting process.
Describe the role of collateralization in the over-the-counter
Describe the implementation of a margining process and market and compare it to the margining system.
explain the determ inants of initial and variation margin
requirem ents. Explain the use of special purpose vehicles (SPVs) in the
O T C derivatives market.
Com pare exchange-traded and O T C markets and
describe their uses.
57
This chapter exam ines how derivatives trading is organized. As USD 40,000 in Sept.
-------------------------------- *
mentioned in the previous chapter, derivatives trading takes A B
place on exchanges and in the over-the-counter (O TC) markets.
10,000 bushels in Sept.
This chapter is devoted to explaining how both these markets
have operated in the past and how they operate today.
USD 38,000 in Sept.
This chapter also introduces central counterparties (CCPs) in the
____________________ ^ _________________________________________________________________________________________
A B
context of exchange-traded markets. In the next chapter, we -------------------------------- ►
explain how they operate in the O T C market and discuss the 10,000 bushels in Sept.
potential risks associated with them .
5.1 EXCH A N G ES
PV of USD 2,000 now
Exchanges have existed for many years. The contracts traded Fiaure 5.1 Trader A and Trader B net two offsetting
when exchanges were first set up were precursors of the futures contracts for the future delivery of corn.
contracts trading today.
An exchange is an organization with members who trade with Another developm ent to protect members from losses was
each other. Initially, the main role of an exchange was to sim netting. Netting is a procedure where short positions and long
ply provide a forum where members could meet and agree to positions in a particular contract offset each other. For exam ple,
trades. These early exchanges also defined standard contracts, if Trader A (from the prior exam ple) subsequently agrees to sell
resolved disputes between members, and expelled members 10,000 bushels of corn for Septem ber delivery to Trader B for
who reneged on agreed transactions. However, they provided 380 cents per bushel, the contracts could be netted by Trader
few other services. For exam ple, there were no mechanisms in A paying USD 2,000 to Trader B in Septem ber. Alternatively, the
place to protect members from losses associated with counter present value of USD 2,000 could be paid at the starting date of
party defaults. the second contract. The key point is that once the two traders
A natural developm ent was for members to protect themselves have entered into offsetting contracts, there is no need for
by requiring margin. Margin refers to the assets transferred from corn to be exchanged in Septem ber. This is illustrated in
one trader to another for protection against counterparty default. Figure 5.1.
Consider Traders A and B, who have entered a trade where O ther netting arrangem ents are more com plicated. For
Trader A agrees to buy 10,000 bushels of corn from Trader B in exam ple, suppose that:
Septem ber at a price of 400 cents per bushel in Septem ber. Sup • Trader A agrees to buy 10,000 bushels of corn from Trader B
pose further that the price of corn then declines by 5 cents soon for 400 cents per bushel, and
after the agreem ent is m ade.1 If there is a margin agreem ent in
• Trader B agrees to buy 10,000 bushels of corn from Trader C
place, Trader B can request USD 500 (= 10,000 X 5 cents) from
for 400 cents per bushel.
Trader A . If the price subsequently declined by a further 10 cents
per bushel, a further USD 1,000 would be requested, and so on. Suppose further that both contracts having the same deliv
If the price of corn moved in the other direction, however, then ery date. The contracts can then be collapsed into a single
Trader B would be required to provide margin to Trader A . For contract where Trader A agrees to buy 10,000 bushels of corn
exam ple, a 20-cent increase in the price of corn would lead to a from Trader C for 400 cents per bushel. This is illustrated in
cumulative transfer of USD 2,000 from Trader B to Trader A. Figure 5.2.
In both cases, margin transfers get rid of the incentive for a If the contract between Trader B and Trader C is for 380 cents
party to back out of the transaction to take advantage of more per bushel, rather than 400 cents per bushel, we can use a com
favorable market prices in the market. bination of Figure 5.1 and Figure 5.2 to collapse the two con
tracts into a single contract in one of two ways.
1 Ideally traders should base the calculation of margin on the price for 2 The standardization of contracts in term s of size, quality of the com
future delivery in Septem ber, rather than at the current (spot) price. modity, and delivery dates facilitated netting.

USD 40,000 in Sept. For exam ple, assume that M em ber A agrees to buy 5,000
------------------------------------------------------------------------ *
A B
bushels of corn (which is defined by the Chicago Mercantile
Exchange as one contract) from M em ber B for delivery in
Septem ber at 400 cents per bushel (USD 20,000 in total). The
exchange (through its C CP) then becom es the counterparty
USD 40,000 in Sept.
________________________________________________________________________________________________________________ fc._________________________
W
to both m em bers. W hat this means is that C C P agrees to buy
B C 5.000 bushels of corn from M em ber B at 400 cents per bushel
while M em ber A agrees to buy 5,000 bushels of corn from the
4 ------------------------------------------------------------------------

C C P at 400 cents per bushel.
Thus, when M em ber A and M em ber B agree on a certain

transaction, the exchange stands between them as illustrated in
USD 40,000 in Sept. Figure 5.3.
The key point is that M em ber A no longer needs to worry about

10,000 bushels in Sept. the creditworthiness of M em ber B (and vice versa). Indeed, the
Fiqure 5.2 Netting of contracts when more than two two members might agree on a trade (either on the floor of the
traders are involved. exchange or electronically) without even knowing each other.
The C C P becom es the counterparty to both and is a clearing
1. Trader A agrees to buy 10,000 bushels of corn from Trader C house for all transactions.
at 380 cents per bushel and agrees to make a payment Another advantage of C C P s is that it is much easier for
equal to the present value of USD 2,000 to Trader B. exchange members to close out positions. To see how this
2 . Trader A agrees to buy 10,000 bushels of corn from works, note that M em ber A has a long position in 5,000 bushels
Trader C at 400 cents per bushel, and Trader C agrees (one contract) of Septem ber corn in Figure 5.3. (S e p te m b e r corn
to make a payment of the present value of USD 2,000 to means corn that will be delivered in Septem ber.) If M em ber A
Trader B. decides to close out this position, he or she could agree to sell
5.000 bushels of Septem ber corn to any other m em ber of the
O ne issue arising from netting arrangem ents involving more
exchange. That trade will also be transferred to the C C P and
than two m arket participants (such as that in Figure 5.2) is
M em ber A would then have two trades with the C C P that off
that the parties involved may have different credit quali
set each other. W ithout a CCP, M em ber A would either have to
ties. For exam ple, Trader C may be w ary about changing
approach M em ber B to close out the position (and there is no
a contract with Trader B for one with Trader A if Trader A
guarantee that M em ber B will be interested in doing this) or
is more likely to default. This is where the margin arrange
short 5,000 bushels of corn with another member. In the latter
ments m entioned earlier might be useful. If Trader A agrees
case, M em ber A would have to worry about the creditw orthi
to post margin in the event that the price of corn declines,
ness of the two members of the exchange that he or she has
Trader C's credit exposure to Trader A would be reduced.
traded with.
Note that margin already posted by Trader A with Trader B
would have to be transferred to Trader C at the
tim e of the netting.
This can get quite com plicated, and a natural

market developm ent was for exchanges to handle
margin arrangem ents so that traders did not need
to worry about the credit quality of other traders.
Central Counterparties
Exchanges today are more heavily involved in
organizing trading than they were in the past.
Specifically, they operate what are known as cen
tral counterparties (CCPs) to clear all transactions
between members. two of its members and becomes the counterparty to each member.
Chapter 5 Exchanges and OTC Markets ■ 59

5.2 HOW CCPs HANDLE CREDIT RISK When a futures price for a contract increases from the close
of trading on one day to the close of trading on the next day,
O nce an exchange has decided to establish a CCP, it must find a funds flow through the C C P from members who have net short
way of managing the associated credit risk. It can do this with a positions to members who have net long positions. This is indi
combination of the following: cated in Figure 5.4.
• Netting, When a futures price for a contract decreases from the close of
trading one day to the close of trading the next day, funds flow
• Variation margin and daily settlem ent,
through the C C P from members who have net long positions
• Initial margin, and/or
to members who have net short positions. This is indicated in
• Default fund contributions. Figure 5.5.
The number of long positions always equals the number of short

Netting positions. This means that while funds are flowing between the
members of CCP, there is no net cash inflow or outflow to
A s m entioned earlier, netting means that long and short
the CCP.
positions are com bined to determ ine a C C P 's net exposure
to a member. For exam ple, suppose that M em ber X shorts Daily settlem ent has another important advantage: It makes
one Septem ber corn contract. If it enters into a trade to buy closing out futures contracts much simpler. A m em ber does not
four Septem ber corn contracts, this will also becom e a trade have to worry about when a contract was entered or what the
betw een M em ber X and the CCP. These two trades would be futures price was at that tim e.
collapsed to a net long position of three Septem ber corn For exam ple, suppose that a long contract is closed out at 11 a.m.
contracts (i.e ., contracts to buy 15,000 bushels of corn on a particular day by trading a short contract at 375 cents
in Septem ber). per bushel. Daily settlement means that the member's gain or loss
up to the close of trading on the previous day has already been
recognized. If the futures price was 372 cents per bushel at close
Variation Margin and Daily Settlement of trading the previous day, only a further gain of three cents per
A futures contract is not settled at maturity. Rather, it is settled bushel needs to be added to the member's account as a result of
day-by-day during the time to maturity. Consider Trader X the market movement.
from the previous exam ple (who is long three Septem ber
corn contracts) and suppose that the Septem ber futures price
is 400 cents per bushel at the close of trading on Day 1 and
395 cents per bushel at the close of Day 2. Trader X has lost
Exchange m em bers Exchange m em bers

15.000 X 5 cents
that are net long that are net short
or USD 750. This is because Septem ber corn is now worth five
cents less per bushel than it was worth at the close of trading on Fiaure 5.4 Flow of variation margin for a futures
Day 1. The trader is thus required to pay USD 750 to the CCP. contract when the futures price increases from the
If Septem ber corn is 405 cents per bushel at the close of Day 3, close of trading on one day to the close of trading the
Trader X has gained next day.
15.000 X 10 cents
or USD 1,500. In this case, the exchange pays the trader

USD 1,500.
Each day, members who have lost money pay an amount equal Exchange m em bers Exchange m em bers
to their loss to the exchange CCP, while members who have that are net long that are net short
gained receive an amount equal to their gain from the exchange

CCP. These payments are known as variation margin, and they
Fiaure 5.5 Flow of variation margin for a futures
typically occur once per day. However, variation margin may be
contract when the futures price decreases from the
exchanged more often than once a day when markets are
close of trading one day to the close of trading the
highly volatile.
next day.

Initial Margin of variation margin across contracts. Exchanges also typically
have rules that reduce initial margin requirements so that the
In addition to variation margin, the C C P requires initial margin. total initial margin for the long Septem ber and short D ecem ber
These are funds or m arketable securities that must be deposited contract is less than the sum of the initial margins for the two
with the C C P in addition to variation margin. contracts considered separately.
To understand the role of initial margin, suppose that a member
who has a net long position in 20 Septem ber corn contracts
Default Fund Contributions
is required to make a variation margin payment of USD 7,000
and fails to do so. The C C P then must close out the member's As a final safety net for the CCP, members are required to make
position by selling 20 Septem ber corn contracts. W ithout ini default fund contributions. If the initial margin is not enough to
tial margin, the C C P would be looking at an immediate loss of cover losses during a m em ber default, the default fund contri
USD 7,000 because the variation margin paid to members with butions of that m em ber are used to make up the difference. If
short positions would be USD 7,000 higher than that received those funds are still insufficient, the default fund contributions of
from members with long positions. Furthermore, the price of other members are used. In the unlikely event that the losses are
Septem ber corn could decline by one cent per bushel before the greater than the sum of the defaulting member's initial margin
C C P is able to close out the member's position. This would lead and the default fund contributions from all m em bers, the equity
to a further loss for the C C P of USD 1,000 (= 0.01 X 5,000 X 20) of the C C P becom es at risk.
and increase the total loss to USD 8,000.
Initial margin is designed to prevent these type of losses. In the

situation described, the C C P might set the initial margin equal
5.3 USE O F MARGIN ACCO UN TS
to USD 700 per contract. (The initial margin for a contract is the IN OTHER SITUATIONS
same regardless of whether the position is short or long). In this
situation, the C C P would have required an initial margin of USD The margin accounts we have talked about so far are those
between C C P s and their members. If a retail trader contacts
14,000 (= 20 X USD 700). This would be more than enough to
cover the USD 8,000 loss. a broker to do a futures trade, however, the trader will be
required to post margin with the broker. And if the broker is not
Note that the initial margin for a futures contract is set by the a m em ber of the CCP, the broker will have to pass the trade to
exchange and reflects the volatility of the futures price. The
a member, and there will be a margin account kept between the
exchange reserves the right to change the initial margin at any broker and that member.
time if market conditions change.
The margin accounts between retail traders and brokers are
C C P s do not pay interest on variation margin payments because som ewhat different from the those between C C P s and their
futures contracts are settled daily (and not at maturity). How m em bers. For instance, there is an initial margin as well as a
ever, C C P s do pay interest on initial margin because it belongs m aintenance margin.3
to that m em ber that contributed it.
To see how m aintenance margin works, note that contracts
If the interest rate paid by the C C P is considered unsatisfactory, are settled daily (as it is for members) with gains (losses) being
a m em ber may be able to post securities such Treasury bills added (subtracted) to margin accounts. Funds in a margin
instead of cash. In that case, the C C P would reduce the value of
account in excess of the initial margin requirem ent can be w ith
the securities by a certain percentage in determ ining their cash drawn. If the balance in the margin account falls below the main
margin equivalence. This reduction is referred to as a haircut. A tenance margin level, the trader is required to supply additional
haircut for a particular asset is usually increased if the price vola
margin to bring the account back up to the initial margin level.
tility of the asset increases. If the trader does not supply the additional margin, the broker
Multiple contracts on the same asset can affect variation and ini closes out the retail trader's position by entering an offsetting
tial margin requirem ents. For exam ple, suppose that a member trade on behalf of the trader. The m aintenance margin is typi
is long one Septem ber corn contract (to buy corn for delivery in cally 75% of the initial margin.3
Septem ber) and short one D ecem ber corn contract (to sell corn
for delivery in Decem ber). When the m em ber is required to pay
variation margin for the Septem ber contract, the m em ber will 3 The exchange specifies minimum levels for the initial margin and the
m aintenance margin. The initial margin for a trade by a retail trader is
probably also receive variation margin for the D ecem ber con higher than the initial margin would be for the sam e trade by a C C P
tract (and vice versa). Therefore, there is an automatic netting member.

Margin accounts are used to mitigate credit risk in many other Short Sales
situations. We will now review several exam ples. In the next sec
tion and the next chapter, we will explain how margin accounts Shorting a stock involves borrowing shares and selling them
are used by C C Ps in the O T C market. in the usual manner. A t some later date, the shares are repur
chased and returned to the account from which they were
borrowed. A retail trader who chooses to short a stock in the
Options on Stocks U.S. is typically required to post margin equal to 150% of the
stock price at the tim e the short position is initiated. The pro
A trader with a net long position in a particular exchange-traded
ceeds of the sale account for two-thirds (100%/150%) of the
stock option has no potential future liability. The options are
margin. The trader must therefore contribute a further 50% of
usually paid for upfront, and they may or may not be exercised
the stock price.
(as explained in Chapter 4). Therefore, there is no reason for the
exchange to require margin from a trader with a net long posi The account is adjusted for changes in the stock price. When the
tion in a call or put option. stock price declines, the balance in the margin account
increases; when the stock price increases, the balance in the
However, a trader with a net short position in an option contract
margin account decreases. A m aintenance margin is typically set
(i.e., a trader who has sold call or put options) does have poten
at 125% of the stock price. If the margin account balance falls
tial future liability. If the options are exercised, the trader must
below the m aintenance margin, additional margin is required to
sell or buy the underlying stock at an unfavorable price. Traders
bring it up to the maintenance margin level.4
with short positions are therefore required to post margin with
the CCP. For exam ple, suppose a trader shorts 100 shares when the stock
price is USD 30. The proceeds of the sale (USD 3,000) belong to
The Chicago Board O ptions Exchange calculates margin that
the trader. The margin that must be initially posted, however, is
must be maintained each day as follows.
USD 4,500 (i.e., 150% of USD 3,000). The trader must therefore
For a short call option, the margin requirem ent is the greater of: post margin equal to the proceeds of the sale plus an additional
• 100% of the value of the option plus 20% of the underlying USD 1,500.
stock price less the amount (if any) that the option is out-of- Suppose further that the share price rises to USD 35. This is bad
the-money, or news for the trader because a short position is designed to do
• 100% of the value of the option plus 10% of the underlying well (poorly) when the price decreases (increases). The value of
stock price. the shares that have been shorted is USD 3,500, and the main
tenance margin is now USD, 4,375 (= 1.25 X USD 3,500). The
For a short put option, the margin requirem ent is the greater of:
USD 4,500 initial margin covers this. If the share price rises again
• 100% of the value of the option plus 20% of the underlying to USD 40, however, the maintenance margin becom es USD
stock price less the amount (if any) that the option is out-of- 5,000 (= 1.25 X USD 4,000) and there is a USD 500 margin call.
the-money, or If this margin call is not met, the position is closed out. The mar
• 100% of the value of the option plus 10% of the strike price. gin account balance belongs to the trader, and interest should
be paid on the balance to the trader.
Consider the situation where 100 call options on a stock are
sold for USD 5 per option when the stock price is USD 47. If the
strike price is USD 50, the option is USD 3 out-of-the-money and Buying on Margin
the margin required per option (USD) is
Buying on margin refers to the practice of borrowing funds from
max(5 + 0.2 X 47 — 3,5 + 0.1 X 50) = 11.4
a broker to buy shares or other assets. As an exam ple, consider
The total margin requirem ent is therefore USD 1,140. A s the a situation where a retail trader buys 1,000 shares for USD 60
stock price and option price change, the margin requirem ent per share on margin. The trader's broker states that the initial
changes and a trader may be requested to contribute more margin is 50% and the maintenance margin is 25%. The initial
funds to the margin account. If the trader does not post margin is the minimum percentage of the trade cost that must
additional margin when required, the trader's position is
closed out. O ptions (unlike futures) are usually settled at
m aturity. The margin posted by a trad er therefore belongs 4 Note that the short seller is required to bring the balance up to the
m aintenance margin level. This can be contrasted with a retail trader in
to the trad er and interest is paid by the C C P on a cash futures who is required to bring the balance up to the initial margin level
margin balance. when it drops below the m aintenance margin level.

be provided by the trader at the tim e of the trade. In this case, positions in derivatives them selves. Dealers typically hedge their
the trader must therefore deposit at least USD 30,000 in cash or risks by trading with other d ealers.5
marginable securities with the broker.
The attractiveness of the O T C market is that derivatives are not
Suppose that the trader deposits USD 30,000 in cash. The standardized by an exchange and they can therefore be tailored
remaining USD 30,000 that is required to buy the shares is to meet the needs of end users.
borrowed from the broker, who keeps the shares as collateral.
O T C derivatives have traditionally been cleared bilaterally. This
The balance in the margin account is calculated as the value of
involves the two parties to a transaction agreeing how it will be
the shares minus the amount owed to the broker. Initially, the
cleared, what netting arrangem ents will apply, and what col
balance in the margin account is USD 30,000 (= USD 60,000 —
lateral (if any) will be posted. However, C C Ps have existed for
USD 30,000). Gains and losses on the shares (as well as inter
some time in the O T C m arkets, and they have been increasingly
est charged by the broker) are reflected in the margin account
used following the 2007-2008 global financial crisis. (This reason
balance (which can also be viewed as the trader's equity in the
for this is discussed in more detail in the next chapter.)
position). The maintenance margin (25% in this case) is the mini
mum margin balance as a percentage of the value of the shares Table 5.1 provides some statistics on the O T C market in
purchased. If the margin balance drops below this minimum, D ecem ber 2017. As mentioned in Chapter 4, the O T C market
there is a margin call requiring the trader to provide additional was about eight tim es the size of the exchange-traded market
margin in order to bring the balance up to the maintenance at that time (when measured in term s of the notional values of
margin level. underlying assets). Table 5.1 shows that the value of all transac
tions in the O T C market was only around 2% of the value of the
Now suppose that the price of the security declines by USD 5.
underlying assets.6
The value of the shares purchased falls to USD 55,000 and the
balance in the margin account falls to USD 25,000. The margin Interest rate derivatives are by far the most popular type of
as a percentage of the value of the shares purchased is around derivative in the O T C m arkets. They account for about 80% the
45% (= 25,000/55,000). This is more than 25%, so there is no value of the market (measured in term s of underlying assets)
margin call. (The calculations here ignore the interest that would and about 75% of the value of outstanding derivatives. Most
be charged by the broker.) interest rate derivative transactions are interest rate swaps.
These are agreem ents to exchange a fixed interest rate on
If the price of the security falls further to USD 39, the cumu
a certain notional principal for a floating interest rate on the
lative loss on the position is USD 21,000 (= USD 60,000 —
same notional principal. They will be discussed in more detail in
USD 39,000). The balance in the margin account therefore falls
Chapter 20. For now, note that the principal is notional because
to USD 9,000 (= USD 30,000 - USD 21,000), and the value of
it is not exchanged.
the shares is now USD 39,000.
The value of the underlying assets in Table 5.1 is the notional
The balance in the margin account has now fallen to 23.1%
principal of outstanding transactions. The transaction values are
(= 9,000/39,000) of the value of the shares. Because this
calculated from the difference between the two interest rates
is less than 25% , there is a margin call. This requires the
(that are exchanged) being applied to the notional principal.
trader to bring the margin balance up to 25% of the value
of the shares. To do this, the trader must add USD 750 A major disadvantage of the O T C markets has traditionally been
(= 0.25 X USD 39,000 — USD 9,000) to the margin balance. related to credit risk. As we have explained, the exchange-
If it is not provided, the broker sells the shares. If it is provided, traded markets use margin accounts and C C P s to mitigate
the position is m aintained, and the amount borrowed from the credit risk. In the early days of the O T C m arkets, however,
broker falls to USD 29,250. transactions were generally cleared bilaterally, and measures to
alleviate credit risks were relatively unusual. Instead, two market
participants would simply agree to certain contingent future
5.4 OVER-THE-COUNTER MARKETS cash flows. If one side experienced financial difficulties and
Over-the-counter derivatives markets have existed for many

years. Traditionally, they have involved two market participants
5 A s m entioned in C hap ter 4, interdealer brokers are often used for O T C
interacting directly and agreeing to a transaction. As m en
transactions between dealers.
tioned in Chapter 4, participants in the market can be catego
6 If a transaction has a positive value of X to one side and a value of —X
rized as end users and dealers. Dealers satisfy the needs of end to the other side, the transaction would contribute X to the statistics on
users by entering derivative transactions and som etim es take m arket value.

Table 5.1 Statistics Produced for the OTC Market by the Bank for International Settlements in December 2017
(See www.bis.org)
Value of Underlying Assets Value of Transactions Ratio of Value of Transactions

(USD Billions) (USD Billions) to Underlying Assets
Foreign Exchange 87,117 2,293 2.63%
Interest Rate 426,649 7,579 1.78%
Equity 6,570 575 8.75%
Com m odity 1,862 189 10.15%
Credit Default Swaps 9,577 312 3.26%
O ther 137 8 5.84%
Total 531,912 10,956 2.06%
was unable to meet its obligations, the other side was likely to For exam ple, suppose that A and B are two com panies trading
experience a loss. derivatives with each other in the O T C market and that at a
point in time there are the four outstanding transactions
It should be em phasized that the credit exposure on a deriva
between them as listed in Table 5.2.
tive (such as an interest rate swap with a certain notional prin
cipal) is much less than that on a bond or a loan with the same Suppose that Company B gets into financial difficulties and
principal. If no collateral is posted, the credit exposure to a declares bankruptcy. W ithout netting, Company B will default
trader on a derivative is max(\/, 0), where V is the value of the on Transactions 1 and 3, but keep Transactions 2 and 4. (The
derivative to the trader. If the derivative has a negative value liquidators of Com pany B might keep Transactions 2 and 4 in
there is no exposure. If the derivative has a positive value, existence or sell them to a third party.) The potential loss to Com
however, the potential loss equals that positive value. The pany A is then USD 60 million. With netting, all transactions will
Value of Transactions column in Figure 5.1 is therefore a be considered as a single transaction worth -U SD 20 million to
better indication of potential credit exposure than the Com pany B. Com pany B's default then leads to a potential loss
underlying principal. for Company A of only USD 20 million (instead of USD 60 million).
The total expected cost of defaults on a derivatives portfolio If C om pany A gets into financial d ifficu lties and d eclares
with a counterparty depends on the lives of the derivatives. This b ankrup tcy, there is a gain from netting to C om pany B.
is because of the interplay between the following factors. W ithout netting , C o m p any B has a potential loss of USD
40 million (on Transactions 2 and 4). W ith netting , there
• There is a greater probability of the counterparty experienc
is no potential loss (In fa ct, C om pany B will have to pay
ing financial difficulties during the life of a derivative as the
the liquidators of C o m p any A USD 20 m illion to settle
life of the derivative increases.
outstanding co n tra cts).7
• The market variables that determ ine the value of derivatives
are likely to move more during the life of a derivative as the The Bank for International Settlem ents estim ates that, when
life of the derivative increases. enforceable netting agreem ents are considered, the total
exposure of participants in derivatives markets was
USD 2,683 billion in D ecem ber 2017. This is about 25% of
Bilateral Netting in OTC Markets the market value of transactions, indicating that the beneficial
Earlier in this chapter, we discussed the use of netting in exchange- effect of netting agreem ents for credit exposures in derivatives
traded contracts. Netting was adopted fairly early in the develop markets is considerable.
ment of the bilaterally cleared O TC markets. Two market partici
pants would enter into a master agreement that would apply to all
the derivatives they traded. In the event of a default by one side,
7 The rules are a little more com plicated than this. If one side defaults,
all the outstanding derivatives transactions between the two par the other side must replace outstanding transactions and is allowed to
ticipants would be considered as a single transaction. value transactions at the bid or the ask, w hichever is more favorable.

Table 5.2 Outstanding Transactions between own X.) If Y goes bankrupt, X should be able to continue to fulfil
Company A and Company B its obligations (and vice versa). Com pany X will typically have a
A A A credit rating, but only after rating agencies have carefully
Value to Company A Value to Company B
exam ined the legal arrangem ents and mechanics of how the
Transaction (USD Million) (USD Million)
company operates.
1 +40 -4 0
SPVs and SPEs are frequently used to create derivatives from
2 -3 0 +30 portfolios of assets such as m ortgages or other types of loans.
3 +20 -2 0 The company setting up the SPV/SPE is not responsible for
the payoffs on the derivatives, and anyone who purchases the
4 -1 0 + 10
derivatives has payoffs that may be affected by defaults on the
underlying loan portfolio. Because of the SPV/SPE's high credit
Collateral rating, however, the payoffs will not be affected by the credit-
worthiness of the SPV/SPE.
The next stage in the m anagem ent of credit risk in the bilater
ally cleared O T C markets is the posting of collateral. The credit
support annex (CSA) of a master agreem ent between two par Derivative Product Companies
ties specifies how the required collateral is to be calculated and
what securities can be posted. Typically, outstanding derivatives One historic approach for handling credit risk in derivatives mar
are valued every day, and the net value is used to determ ine kets involved the use of derivative product com panies (DPCs).
the extra collateral that must be posted. The term inology of These were well-capitalized subsidiaries of dealers designed to
the exchange-traded markets is som etim es used (with collateral receive A A A credit ratings. When the dealer traded with Com
being referred to as margin). pany X, the D PC (rather than the dealer) becam e Com pany X's
counterparty. This means that while a dealer may have had a
For exam ple, suppose that Com panies A and B are trading
poor credit rating, a well-capitalized DPC would have a A A A
derivatives. On a given day, the net value of outstanding trans
credit rating, and therefore counterparties would be com fort
actions increases by USD 1 million to Com pany B (and therefore
able trading with it.
decreases by USD 1 million to Com pany A). Under the term s of
the C SA , Com pany A might be required to post collateral of D PCs were set up so that they took on virtually no market risk.
USD 1 million to Com pany B .8 When they traded with a counterparty, they normally entered an
offsetting trade with the parent so that the parent is responsible
As already m entioned, C C P s have becom e an important feature
for managing the risk.
of O T C markets and will be discussed in more detail in the next
chapter. A t this point, we discuss other ways in which O T C One point to bear in mind is that credit risk is not eliminated by
market participants have attem pted to handle credit risk. a D PC . The D PC itself may be virtually riskless, but the parent
company is exposed to risk, and a default by the parent com
pany would have had consequences for a DPC's counterparties.
Special Purpose Vehicles Typically, it would have led to the D PC being sold to another
entity or all transactions being closed out at mid-market prices.
Special Purpose Vehicles (SPV), also called Special Purpose
D PCs tried to alleviate the concerns of counterparties by docu
Entities (SPE), are com panies created by another company in
menting exactly what would happen if the parent experienced
such a way that the credit risks are kept legally separate. SPVs
financial difficulties.
and SPEs are som etim es created to manage a large project
without the organization setting it up being put at risk. D PCs have becom e virtually nonexistent since the credit crisis
of 2007-2008. However, they were made redundant well before
For exam ple, suppose that Com pany Y creates SPV/SPE Com
then by the increasing use of collateral in the O T C market.
pany X. Typically, Y transfers assets to X and may not control
those assets. (In some jurisdictions, Y is not even allowed to
Credit Default Swaps

One way of managing credit risk is to use credit default swaps.
8 The C S A may require collateral to be posted only when the value of These are insurance-like contracts between a protection buyer
outstanding transactions to one side exceed s a certain threshold level.
A lso, to avoid the adm inistrative costs of small transfers, a minimum and a protection seller. The buyer pays a regular premium to
transfer amount is usually specified. the seller, and if there is a default by a specified entity (not the

buyer or seller), the seller makes a payment to the buyer. The netting, central clearing, margin requirem ents, and default
credit default swap market grew rapidly between 2000 and fund contributions. Netting allows traders to offset long and
2007, but it has declined since then. short positions. Central clearing means that an exchange's C C P
is always a m em ber trader's counterparty. By requiring initial
Some com panies have specialized in selling credit protection
margin and variation margin, exchanges ensure that they are
using credit default swaps. Monolines are com panies with good
unlikely to lose money from a member's default. Default fund
credit ratings that do this as their main activity. Also, some insur
contributions are an added source of protection for exchanges.
ance com panies have sold protection as an extension of their
other insurance activities. The most well-known of these insur Over-the-counter markets have copied many of the tools first
ance com panies is A IG , which (through its subsidiary AIG Finan introduced in exchange-traded m arkets. For exam ple, netting
cial Products) sold a huge amount of protection on products has been a feature of bilaterally cleared O T C markets almost
created from m ortgage portfolios. from the beginning. Margin (sometimes called collateral in O TC
markets) has becom e a progressively more popular feature of
The 2007-2008 crisis led to many failures among monolines.
bilaterally cleared O T C m arkets. As we will discuss in the next
M eanwhile, AIG suffered severe losses arising from credit
chapter, C C P s have becom e more widely used in the O T C mar
default swaps and required a USD 180 billion bailout from the
kets following the 2007-2008 crisis.
U.S. governm ent (the funds have since been repaid). Banks
such as Citigroup and Merrill Lynch that bought protection from O T C markets have experim ented with several other ways of
monolines also lost several billion dollars. managing credit risks. Special purpose vehicles and special pur
pose entities are useful in some situations. Derivative product
com panies were once seen as a way for dealers to convince
SUMMARY their counterparties that there was no risk in trading with them.
Credit derivatives have also been used. However, none of these
Exchange-traded markets have developed procedures to signifi alternatives have stood the test of tim e. Increasingly, risks in the
cantly reduce credit risk and make it easy for market participants O T C market are being managed in much the same way as they
to close out positions before maturity. These features include are in the exchange-traded market.

Q UESTIO N S

5.1 W hat is the main role of an exchange C C P ? 5.6 W hat does buying on margin mean?
5.2 Explain why C C Ps make it easy for traders to close out 5.7 In the futures m arket, how is the margin account between
positions in exchange-traded markets. a retail trader and his/her broker managed differently from
5.3 W hat is margin in an exchange-traded futures contract? that between a C C P m em ber and the C C P ?
Explain the difference between variation margin and initial 5.8 W hat are the most actively traded O T C instruments?
margin.
5.9 How does netting reduce credit risk in the bilaterally
5.4 Explain how netting works in futures markets. cleared O T C market?
5.5 W hat is the role of default fund contributions in C C Ps? 5.10 W hat is the difference between collateral and margin?
Practice Questions
5.11 Suppose that A agreed to buy a certain amount of soy margin is USD 2,000. Under what circumstance is the
beans for future delivery from B before the existence of trader required to provide more margin? How much mar
C C P s. How could A have exited from this trade. W hat gin is required? Under what circum stances can USD 500
problems might A have encountered? be withdrawn from the margin account?
5.12 Before the existence of central clearing, Trader A agreed 5.16 A U.S. trader sells 500 put options. The option price is
to buy 10,000 bushels of wheat for delivery in March from USD 3, the strike price is USD 31, and the stock price is
Trader B for 500 cents per bushel. A few weeks later, USD 30. W hat is the margin requirement?
Trader A sold 5,000 bushels of wheat for March delivery 5.17 A trader shorts 100 shares when the price is USD 50.
to Trader C for 520 cents per bushel and 5,000 bushels
The initial margin and maintenance margin are 150%
to Trader D for March delivery for 530 cents per bushel. and 125%. W hat is the initial margin required? How high
W hat is Trader A's net profit or loss? can the share price go before further margin is required?
5.13 W hat factors are likely to influence a C C P's determination (Ignore interest payments.)
of initial margin for a futures contract?
5.18 The counterparty to a dealer in the O T C market has
5.14 W hat funds are available to a C C P to fund losses arising agreed to post margin equal to max(V, 0) where V is the
from a default by a m em ber? In what order are they used? value of outstanding transactions to the dealer. W hat
credit risk is the dealer taking?
5.15 A trader contacts a broker to enter into a futures contract
to sell 5,000 bushels of wheat for 600 cents per bushel. 5.19 Explain the operation of a derivative product company.
The initial margin is USD 3,000, and the maintenance

ANSW ERS
5.1 The C C P stands betw een the two sides when a d eriva transactions, and the resulting portfolio is considered as
tive is traded on an exchange. Suppose A trades a d eriv a single transaction. The counterparty cannot therefore
ative contract with B. The transaction is moved to the default on transactions with negative value to itself and
C C P so that A is buying from the C C P and B is selling to keep transactions with a positive value to itself.
the CCP. 5.10 Margin is the word used for collateral in the exchange-
5.2 A C C P becom es the counterparty to all traders. A long traded m arket. Increasingly, it is also used in the O TC
position entered into with one counterparty can therefore market.
be closed out by a short position in the same contract 5.11 A could approach B to agree on a payment that will
entered into with another counterparty. close out the transaction. If no agreement is reached,
5.3 Margin is the word used to describe cash or m arketable A can enter into an offsetting transaction with a third
securities that must be provided by a trader. Variation party, C. However, A is then taking on the credit risk of
margin is an amount paid or received to settle a futures both B and C.
position daily. Initial margin is cash or m arketable securi 5.12 The trader's net profit on the delivery date will be
ties provided by a trader to a C C P to reduce the possibil
5,000 X 20 cents + 5,000 X 30 cents = USD 2,500
ity of losses from a default.
Since the introduction of daily settlem ent in futures
5.4 Long positions to buy an asset at a future time are netted
m arkets, the net profits such as this have been realized
with short positions entered into by the same trader to
day-by-day.
sell the asset at the same tim e. For exam ple, if a trader
has five long contracts (to buy) and then enters into three 5.13 Initial margin will depend on the volatility of the futures
short contracts (to sell), the trader's position going for price and how long it will take the exchange to close out
ward is two long contracts. the m em ber if the m em ber defaults.
5.5 Default fund contributions are used to fund losses when 5.14 Funds are used in the following order:
the initial margin of the defaulting m em ber of the C C P 1. Initial margin provided by the member,
proves inadequate. First the default fund contributions of 2. Default fund contribution of the member,
the defaulting party are used. Then the default fund con 3. Default fund contributions of other m em bers, and
tributions of other members are used. 4. Equity capital of the exchange.
5.6 Buying on margin means borrowing part of the funds nec 5.15 The trader is required to provide more margin if
essary to buy an asset. The asset is held as collateral. If more than USD 1,000 is lost from the margin account.
its value declines sufficiently far, the borrower will have to This will happen if the price of w heat falls by more
either provide further funds or be closed out. than 20 cents because variation margin will then
5.7 In the case of a retail trader, there is an initial margin and have reduced the margin balance by more than
a maintenance margin. The margin account balance is 5,000 X 20 cents = USD 1,000. The trader is required to
adjusted each day to reflect gains and losses. If the bal bring the balance in the margin account up to the initial
ance in the margin account falls below the maintenance margin level of USD 3,000. USD 500 can be withdrawn
margin level, the trader is asked for funds to bring the bal from the margin account if the price rises by 10 cents
ance up to the initial margin level. For the m em ber of an or more (5,000 X 610 cents = USD 30,500, which is
exchange, the maintenance margin and the initial margin USD 500 more than the required initial margin of
are effectively the same. USD 30,000).
5.8 Interest rate derivatives, in particular, interest rate swaps. 5.16 The margin per option in USD is
5.9 In the event of a default by a counterparty, positive max(3 + 0.2 X 30,3 + 0.1 X 31) = 9
valued transactions are netted against negative-valued The total margin requirem ent is therefore USD 4,500.

5.17 The trader is initially required to contribute USD 2,500 particular, it has exposure to increases in value between
in addition to the USD 5,000 obtained from selling the the last time that the counterparty posts collateral and the
shares, creating a margin balance of USD 7,500. If the time when the dealer is able to close out its position with
USD share price rises to X, the maintenance margin the counterparty.
becom es 100 X 1.25X = 125X. This is greater than 5.19 A derivative product company is a wholly owned subsid
the 7,500 margin balance when X > 7,500/125, that is, iary of a dealer and the company through which it con
when X > 60. ducts its derivatives transactions. It is given enough capital
5.18 The risk to the dealer corresponds to a possible increase to be rated A A A and takes no market risk. When it enters
in the value of its transactions with the counterparty into a transaction with a counterparty, it also enters into
around the time that the counterparty defaults. In an offsetting transaction with the parent company.

Central Clearing
Learning Objectives
Provide exam ples of the mechanics of a central counter • Com pare and contrast bilateral markets to the use of
party (CC P). novation and netting.
Describe the role of C C Ps and distinguish between bilat Assess the impact of central clearing on the broader finan
eral and centralized clearing. cial markets.
Describe advantages and disadvantages of central clear Identify and explain the types of risks faced by C C P s.
ing of O T C derivatives.
Identify and distinguish between the risks to clearing
Explain regulatory initiatives for the O T C derivatives mar members as well as non-members.
ket and their impact on central clearing.
Com pare margin requirements in centrally cleared and

bilateral markets, and explain how margin can mitigate risk.
71
Central counterparties (CCPs) have been used for trading frequently to reflect changing market conditions. Furtherm ore,
derivatives in exchange-traded markets for many years. As initial and variation margin payments may be required from
explained in Chapter 5, a derivatives exchange is organized members during the day (as well as at the end of a day) when
so that its C C P is the counterparty to all its m em ber traders asset prices are especially volatile. This is im portant given that
(whether they are buyers or sellers). a m em ber typically has only one or two hours to meet a margin
call. If a margin call is not met, the member's position is closed
The rules developed by C C P s for members posting margin allow
out. The positions of members are monitored carefully by the
the exchanges to handle credit risks efficiently.1 As a result,
C C P and members may be required to reduce their exposures
failures of C C Ps handling exchange-traded products have
by the C C P in some circum stances.
been rare.
This chapter focuses on C C Ps in the over-the-counter (O TC)
One exam ple of such a failure is that of French clearing house
m arket.4 These C C P s operate similarly to the C C P s used by
Caisse de Liquidation in 1974. Following a steep decline in sugar
exchanges. As with exchange C C P s, members of O T C C C P s are
futures prices, the exchange was required to make large varia
required to post initial and variation margin as well as make con
tion margin payments to its members with short positions. How
tributions to the default fund. However, products being cleared
ever, some members with long positions failed to make their
in the O T C market differ from those being cleared by
variation margin paym ents. As a result, their initial margin bal
exchanges. For exam ple, most exchange-traded contracts last
ances were insufficient to make the variation margin payments
only a few months, and very few last more than three years. In
even when combined with other funds the C C P could access.
contrast, O T C contracts often last ten years or longer. The aver
A similar failure is that of the Kuala Lumpur Com m odity Clearing age futures trade on an exchange is also much sm aller than the
House. The Malaysian exchange was forced to close in 1983 average trade in the O T C market.
after a steep decline in palm oil futures prices left the C C P
Another key difference is that while exchange-traded futures
unable to pay margin owed to members with short positions. As
contracts trade continuously, O T C contracts trade only interm it
with Caisse, this inability to pay was the result of members with
tently. As a result, O T C contracts are less liquid than exchange-
long futures positions failing to post variation margin when
traded contracts. The differences in trading frequencies
required.2
between the two types of markets also affects variation margin
It should be emphasized that failures such as these are rare and calculations. W hile exchange-traded variation margin require
futures markets have almost always been able to survive periods ments can be determ ined directly from market prices, it is often
of market volatility. For exam ple, futures markets in the U.S. necessary to use a model when determ ining margin require
were tested on O ctober 19, 1987, when the S&P 500 index ments in the O T C markets.
declined by over 20% in one day. This created a stressful situa
Three large C C P s for clearing O T C transactions are
tion for exchanges trading S&P 500 futures, such as the Chicago
M ercantile Exchange (CM E). Despite the dram atic fall in prices, • Sw apCIear (part of LCH Clearnet in London),
however, the C M E was able to pay in full all members with short • ClearPort (part of the C M E Group in Chicago), and
futures positions.3
• ICE Clear Credit (part of the Intercontinental Exchange).
Exchanges have learned from the failures and near-failures of
These large C C Ps are critical for the smooth functioning of the
C C P s in the past and are now considered to be extrem ely safe.
global financial system and are regarded as "too big to fail" by
For exam ple, initial margin requirem ents are now adjusted more
financial regulators. This means that in the event of financial dif
ficulties, they would almost certainly receive some type of bail
1 As explained in C hapter 5, margin is a form of collateral collected by
out. It is therefore important to exam ine their risks.
the C C P to cover the potential losses associated with a counterparty
default. There are also several other smaller, more localized O T C C C Ps
2 For more details on C C P failures, see J . G regory, Central C ounterpar that might need to be bailed out in the event of financial dif
ties: M andatory Clearing and Bilateral Margin R equirem en ts fo r O T C ficulties. This is because some authorities regard it as important
D erivatives, W iley 2014, C hapter 14.
to have local O T C C C P s to service financial institutions in their
3 The Hong Kong futures exchange was not so lucky at the tim e of the
region and clear transactions denom inated in local currency.
1987 crash and had to be bailed out by the Hong Kong governm ent.
It is also worth noting that som e brokers within the U.S. w ent bankrupt W hile cooperation between C C P s is limited, there are signifi
because their clients with long S&P futures positions failed to provide
cant econom ies of scale involved with running a CCP. Thus, we
variation margin when requested. A s a result they were unable to m eet
their obligations on the long futures positions they had entered into on
behalf of their clients. 4 C hapter 5 discusses C C P s in the exchange-traded market.

should expect mergers between C C P s as well as takeovers of transactions betw een m em bers and the defaulting party
sm aller C C Ps by larger ones. Such consolidation has already at prices that leave the non-defaulting m em bers with
begun with exchanges. For exam ple, the Chicago Board of som e lo ss.5
Trade and the Chicago Mercantile Group have merged to
C C P s cover their costs by charging fees per trade. They may
form the C M E Group, which includes the New York Mercantile
also earn interest on initial margin in excess of that paid to
Exchange and the Kansas City Board of Trade. Another exam
m em bers. For C C P s owned by their m em bers (or a subset of
ple is the m erger between Euronext and the New York Stock
their m em bers), excess profits are distributed to m em ber-
Exchange that created N YSE Euronext (which itself was acquired
owners. O ther C C Ps are owned by outside investors and are
by the Intercontinental Exchange).
under pressure to generate profits. Com petition between C C Ps
can benefit users by providing choices and encouraging C C Ps
to improve their system s. However, there is a danger that C C Ps
6.1 THE OPERATION O F CCPS will try to com pete with each other by reducing initial margin
requirem ents and default fund contributions. That in turn would
C C P s clearing trades in the O T C m arkets operate in much the
increase the risks that C C P s pose within the financial system.
sam e way as C C P s clearing trades on exchang es. M em bers
are required to post initial margin and variation margin as O T C C C Ps are subject to a great deal of regulation. For exam
well as make contributions to the default fund. The variation ple, the Financial Services Authority in the United Kingdom
margin is paid or received daily (or even more frequently) by monitors risks taken by LCH Clearnet.
m em bers and reflects the change in the value of each m em It is im portant that O T C C C P s not be allowed to take risks unre
ber's portfolio of transactions with the CCP. In absence of a lated to their main activity of clearing O T C transactions. For
m em ber default, the variation margin received by the C C P exam ple, it would be inappropriate for them to engage in spec
should always equal the variation margin paid by the CCP. ulative trading activities. In this respect, an O T C C C P should
This is because the C C P has a m atched book and therefore behave like a public utility.
it takes no m arket risk. W hen there is a default, there is mar
ket risk as the C C P closes out the positions of the defaulting
m em ber. 6.2 REGULATION O F O TC
The initial margin required from each m em ber is calculated DERIVATIVES M ARKET
using historical data. The key question for a C C P is "H ow much
could be lost if the member defaults and market price m ove Regulations introduced since the 2007-2008 global financial
ments reduce the value of the member's position as it is being crisis have led to an increase in the use of C C P s in the O T C
closed out?" derivatives market. These regulations were prompted by the
belief that com plex O TC-traded derivatives, specifically those
Typically, the C C P sets the initial margin so that if it takes five
created from portfolios of subprim e (i.e., riskier than average)
days to close out the member's position, the C C P is 99% certain
m ortgages, played a role in causing the crisis.
that the initial margin will cover the losses. If the defaulting
member's initial margin proves insufficient to cover losses, how During the time these m ortgage derivatives were being traded,
ever, the default fund contributions from the defaulting member O T C markets were largely unregulated. M arket participants
are used. If this is still not enough, the contributions from other could execute and clear trades in any way they chose without
members are used. Only when this amount is insufficient does reporting their trades to a central authority.
the C C P's equity becom e at risk. So, when the G-206 leaders met in Pittsburgh in Septem ber
W hen a m em ber defaults, the exchange typically holds an auc 2009, they were anxious to rein in the O T C m arket. They were
tion inviting other m em bers to bid for transactions that offset particularly concerned about system ic risk. This is the risk that
the defaulting m em ber's transactions. M em bers have an incen a default by one derivatives dealer could lead to losses being
tive to cooperate; if a C C P can quickly close out a defaulting incurred by other derivatives dealers on their transactions with
m em bers positions, the remaining m em bers' default fund
contributions will be safe, and they can continue to clear
5 For a discussion of this, see ISD A, "C C P loss allocation at the end
transactions through the CCP. If the auction fails, however, the
of the w aterfall" https://w w w .isda.org/a/jTDDE/ccp-loss-allocation-
C C P may have the right to allocate losses to m em bers who w aterfall-0807.pdf
have made recent gains. A dditionally, the C C P may choose to 6 The G-20 (or Group of Twenty) is an international forum for the
tear up transactions. This procedure involves closing out governm ents and central bank governors from 20 countries.
Chapter 6 Central Clearing ■ 73

the defaulting dealer. This in turn could result in further defaults The first two of these regulations apply only to transactions
and further losses by other dealers. In the worst-case scenario, betw een two financial institutions (or betw een a financial
this interconnectedness of derivatives dealers would lead to a institution and a non-financial com pany deem ed systemi-
collapse of the global financial system. cally im portant due to the volum e of its O T C derivatives
trading). This means that derivatives dealers do not have to
The statem ent issued by the G-20 leaders after the Pittsburgh
use electronic platform s and C C P s when trading standard
summit included the following paragraph.
ized contracts with most of their non-financial end users.
A ll standardized O T C derivative contracts should be The requirem ent that C C P s be used for standard interdealer
tra d ed on exchanges or electronic trading platform s, transactions (e .g ., interest rate sw aps), however, has led to a
w here appropriate, and cleared through central counter huge growth in the volum e of O T C transactions being cleared
parties b y end-2012 at the latest. O T C derivative co n through C C P s.
tracts should b e re p o rte d to trade repositories.
N oncentrally cleared contracts should b e su b je ct to
higher capital requirem ents. We ask the F S B 7 and its rel 6.3 STANDARD VERSUS
evant m em bers to assess regularly im plem entation and NON-STANDARD TRANSACTIONS
w hether it is sufficient to im prove transparency in the
derivatives m arkets, m itigate system ic risk, and p ro te ct The meaning of the term standard transaction is obviously
against m arket a b u se.8*• im portant to the application of the rules we have just men
The G-20 Pittsburgh meeting resulted in three major regulations tioned. Standard transactions are transactions that C C P s are
prepared to clear. For a C C P to clear a product, several condi
affecting O T C derivatives.
tions must be satisfied.
1. A requirem ent that all standardized O T C derivatives be
• The legal and economic term s of the product must be stan
cleared through C C P s. Standardized derivatives include
plain vanilla interest rate swaps (which account for most dard within the market.
traded O T C derivatives) and credit default swaps on indi • There must be generally accepted models for valuing the
ces. The purpose of this requirem ent is to create an envi product (because the C C P needs to determ ine variation mar
ronment where dealers have less credit exposure to each gin at least once a day).
other, reducing interconnectedness and system ic risk. • The product must trade actively. If this is not the case, it may
2 . A requirem ent that standardized O T C derivatives be traded be difficult to unwind a member's position if the member
on electronic platforms to improve price transparency. If fails to produce margin when required. It may also be dif
there is an electronic platform for matching buyers and ficult to obtain up-to-date valuations for non-actively traded
sellers, the prices at which products trade should be read products. A related point is that C C P s will not consider it
ily available to all market participants. These platforms are worthwhile to develop the systems to support the clearing of
called sw ap execution facilities (SEFs) in the U.S. and orga a product if their members do not trade it frequently.
nized trading facilities (O TFs) in Europe. In practice, stan • Extensive historical data on the price of the product should
dardized products are passed autom atically to a C C P once be made available to enable initial margin requirements to be
they have been traded on these platforms. determ ined.
3 . A requirem ent that all trades in the O T C market be The main product categories currently classified as standard are
reported to a central trade repository. This requirement interest rate swaps and credit default swaps on indices. O ther
provides regulators with important information on the risks products, such as options on interest rate swaps and single
being taken by participants in the O T C market. name credit default swaps, may be added at some stage. How
ever, it seem s likely that most exotic derivative products will be
classified as non-standard for the foreseeable future.
7 The Financial Stability Board (FSB) is an international institution that
Transactions that are cleared bilaterally (rather than through a
works with the G-20 to prom ote global financial stability by coordinating
the developm ent of regulatory, supervisory, and other financial sector C C P) are referred to as uncleared.
policies. It monitors and makes recom m endations about the health of
Regulators recognized that derivatives dealers could avoid the
the global financial system .
intent of the regulations described above by adding features
8 The New Rules for O T C Derivatives | Derivatives Risk M anagem ent
Software & Pricing Analytics, (n.d.). Retrieved from https://fincad.com /
to standard transactions that would make them slightly non
blog/new-rules-otc-derivatives standard. A 2011 G-20 meeting resulted in uncleared derivatives

joining their CCP-cleared counterparts in being subject to initial
and variation margin requirements.
The new rules, which are being phased in between 2016 and
2020, require margin to be posted for uncleared derivatives
traded between two financial institutions (or between a financial
institution and a systemically important non-financial institution
engaging in a large volume of transactions). Under the new rules,
initial margin and variation margin must be posted. This is a sig
nificant change because initial margin between O T C market deal
ers in the pre-crisis period was rarely applied (unlike variation
margin, which was common in the bilaterally cleared O TC
markets).9 Fiaure 6.1 Six market participants trade derivatives
with each other bilaterally. Each line represents a
Variation margin on uncleared trades is usually transm itted from
master agreement between a pair of dealers.
one counterparty to the other directly. However, initial margin
cannot be handled in this way. To see why, suppose counter
party A transferred USD 1 million of initial margin to counter of credit default swaps were cleared through C C P s in June
party B and B transferred USD 1 million of initial margin to A . In 2 0 1 8 .11
practice, the transfers would cancel each other. The regulations To exam ine the impact of C C P s on O T C m arkets, Figure 6.1
therefore require initial margin on uncleared transactions to be shows a sim plified situation where there are six participants in
transm itted to a third party to be held in trust. an O T C derivatives market trading with each other. It assumes
The initial margin that A has to post should cover the greatest that all contracts are cleared bilaterally and (as explained in
Chapter 5) there are master agreem ents between every pair of
decrease in the value of the contracts to itself (and therefore the
greatest increase in their value to B) estim ated to occur over a participants covering all contracts between them . The agree
ten-day period with 99% confidence in stressed market condi- ments outline netting arrangem ents, collateral arrangem ents,
1n and what happens in the event of a default by one side. Stan
tions. This requirem ent recognizes that if A defaults, it could
dard documentation for master agreem ents has been provided
take B up to ten days to close out (or replace) its positions with
A . This means that the close out could be quite expensive if the by the International Swaps and Derivatives Association.
value of these positions to B increase during that tim e. Figure 6.2 shows the situation when all transactions between the
six market participants are centrally cleared through the same
C C P In this case, we assume that the six market participants are
6.4 THE M OVE TO CEN TRAL members of the CCP. The positions of market participants are
CLEARIN G transferred to the CCP, and each market participant agrees to
adhere to the term s set by the C C P .12*Specifically, each partici
As previously m entioned, the new regulations have led to a pant agrees to post initial margin and variation margin. The par
huge increase in the volume of O T C derivatives cleared though ticipants also agree to make contributions to the default fund as
C C P s. Statistics produced by the Bank for International Settle required.
ments shows that 76% of interest rate derivatives and 54%
In practice, the current environment for trading derivatives is
a mixture of what is shown in Figure 6.1 and what is shown in
Figure 6.2. Some transactions (i.e., non-standard transactions
9 W hen entering into a transaction with a much less creditw orthy coun
between financial institutions and a subset of transactions with
terparty, a derivatives dealer might insist on the counterparty posting
initial margin. Posting of initial margin by both sides was alm ost unheard non-financial end users) are cleared bilaterally as indicated in
of in the bilaterally cleared m arket pre-crisis. Figure 6.1. Standard transactions between financial institutions
10 M arket participants use a model referred to as SIMM (Standard Ini (and some standard transactions with non-financial end users)
tial Margin Model) to calculate initial margin for uncleared trades. This
model was developed by the International Swaps and Derivatives A sso
ciation in conjunction with m arket participants. We m entioned earlier
11 O T C derivatives statistics at end-June 2018. (2018, O cto b er 31).
that the initial margin for cleared transactions is based on a five-day
Retrieved from https://w w w .bis.org/publ/otc_hy1810.htm
close out period. O ne reason for the difference is that standard transac
tions are more liquid and can therefore be expected to be closed out 12 The word used to describe the transfer of a contract from one party
more quickly. to another is novation.

Financial institutions understand that the margin they post
could be put to a better use. However, the effect of the margin
requirements within the central clearing ecosystem is that these
firms are much less likely to lose money because of a default
by another financial institution. Furtherm ore, these benefits can
only be obtained if all financial institutions are part of the C C P
ecosystem .
Fiaure 6.2 Six members of the CCP trade derivatives C C P s may give rise to an increase in netting.14 To see how this
with each other and clear their transactions through a can be the case, consider the situation in Figure 6.3 where three
single CCP. Each line represents an agreement between market participants trade bilaterally. The arrows in this figure
a member and the CCP. indicate that Party A has transactions with Party C that are worth
+ USD 8 million to Party A and —USD 8 million to Party C . Party
A also has transactions with Party B that are worth +USD
are cleared though C C Ps as indicated in Figure 6.2. A further 5 million to Party B and —USD 5 million to Party A . Finally, Party B
complication is that there is more than one CCP. This means that has transactions with Party C that are worth +USD 2 million to
even if all trades by the six market participants are cleared cen
Party C and —USD 2 million to Party B. Positive values lead to
trally, they might be cleared though different C C P s. potential credit exposures, while negative values do not. In the
M embers of a C C P clear the trades of non-members bilaterally. absence of any credit mitigation procedures:
These non-members are small financial institutions and non- • Party A has a credit exposure of USD 8 million to Party C and
financial com panies. (Retail investors do not generally trade in none to Party B,
the O T C market.) The non-members must post initial and varia
• Party B has a credit exposure of USD 5 million to Party A and
tion margin with the m em ber who is clearing their trades. This
none to Party C, and
is similar to the manner in which the non-members' trades are
cleared by exchange C C P s. • Party C has a credit exposure of USD 2 million to Party B and
none to Party A .
O T C transactions are different from futures in that they are not
settled daily. Cash flows settling the contracts occur periodi The total credit exposure of all three parties is USD 15 million
cally and som etim es (e.g ., in the case of European options) all (= 8 million + 5 million + 2 million).
cash flows are settled at the end of the contract's life. However, N ext, suppose that all transactions are cleared through a single
C C P s value O T C contracts at least once a day and transfers the CCP. Figure 6.4 indicates how this would work. In this exam ple,
required variation margin reflecting the change in net value of a transaction between A and B would be converted into a trans
outstanding contracts. This means, for exam ple, that the varia action between A and the C C P as well a transaction between
tion margin transferred from Party A to Party B belongs to Party A the C C P and B.
until the contractual cash flows occur. A s a result, Party B must
The C C P would then net the transactions in the manner shown
pay interest on the variation margin it receives from A . If clear
in Figure 6.5.
ing is through a CCP, the interest is paid by Party B to the C C P
and by the C C P to Party A . Interest on initial margin balances is This dem onstrates that the C C P has increased netting.
paid by the C C P (as is the case with exchange-traded contracts). • Party A is able to net its transactions with Party B against its
Even though interest is paid on margin transfers as appropriate, transactions with Party C.
financial institutions still regard them as a cost. Typically, institu • Party B is able to net its transactions with Party A against its
tions compare the interest rate paid on their margin with their transactions with Party C.
average cost of borrowed funds and calculate a cost relating to
• Party C is able to net its transactions with Party A against its
the difference.13
transactions with Party B.
However, the overall the credit exposure in the system is lower.
13 In the case of variation margin, the cost (which can be negative) is Parties A and B each have a credit exposure to the C C P of
referred to as the funding value adjustm ent (FVA). In the case of ini 3 million USD and Party C has no credit exposure at all. If the
tial margin, the cost is referred to as margin value adjustm ent (MVA).
W hether the average cost of borrowed funds should be used in the
calculation is debatable. This is because margin is a relatively safe use of 14 See Sections 5.1 and 5.4 of C hap ter 5 for a discussion of netting in
the financial institution's funds (som ewhat like buying Treasury bills). exchange-traded and O T C m arkets.

be netted when cleared bilaterally, this is not the case if the
standard transactions go through a C C P (while the non-standard
transactions continue to be cleared bilaterally). This means that
the loss of netting could outweigh the extra netting provided by
diverting transactions through C C P s even when the same C C P is
used for all transactions.
6.5 ADVANTAGES
AND DISADVANTAGES O F CCPS
Fiqure 6.3 Three market participants clearing A key advantage of the central clearing model is that it is much
bilaterally. easier for market participants to exit a C C P transaction than a
bilaterally cleared transaction.
When transactions are cleared bilaterally, a market participant

can only exit a trade by approaching the original counterparty
and trying to negotiate a close-out (which usually would involve a
payment from one side to the other). If the original counterparty
does not offer reasonable close-out term s, the market participant
would need to enter into an offsetting transaction with another
counterparty. This arrangement creates credit risk because either
the original counterparty or the new counterparty may default.
However, the credit risk is eliminated if both the original trade
and the offsetting trade are cleared with the same CCP.
Fiqure 6.4 The transactions in Figure 6.3 are If the original transaction and the new transaction are cleared
presented to a CCP. with different C C P s, the market participant will have to post
initial margin with each one. In the future, it may be possible to
arrange for trades with different C C P s to be netted so that the
initial margin is avoided. This arrangem ent is known as interop
erability and is not (as of yet) common practice.
In bilateral clearing, the risk of a default by a market participant

is borne entirely by its counterparties. If trades are cleared
through a CCP, however, the risks are shared by all members
of the C C P (some of which may never have traded with the
defaulting counterparty). This sharing of credit risk is referred to
as loss mutualization and is attractive to regulators because it
has the effect of reducing system ic risk. It does this by dispers
ing the impact of a default by a market participant throughout
Fiqure 6.5 Situation after CCP has netted the
the market.
transactions shown above in Figure 6.4.
Another advantage of C C P s is that they manage the margining,
credit exposures were handled with variation margin and initial netting, settlem ent, and default resolution that would typically
margin, there would be a saving of initial margin. be handled by each market participant in the case of bilateral
clearing. In fact, with the vast resources devoted to these func
Clearing through C C P s usually increases netting, but this is not
tions and intense regulatory oversight, it is expected that a C C P
always the case. For exam ple, the transactions with one C C P
can handle these functions better than any single market partici
cannot usually be netted against transactions with another CCP.
pant in the bilaterally cleared m arket. However, any operational
Furtherm ore, some transactions (e.g ., non-standard transactions) problems experienced by the C C P could have widespread con
cannot be cleared through C C P s. And while standard transac sequences because it is responsible for a far larger number of
tions and non-standard transactions between two parties can transactions than any single market participant.

C C P s may also improve liquidity in the O T C market by making it typically see a much sm aller percentage of their trading part
much easier for market participants to net and exit from transac ners' portfolios.
tions. Another positive aspect of central clearing is that it
encourages the developm ent of standard documentation for
6.6 CCP RISKS
O T C derivative transactions.15
Chapter 2 explained two key risks faced by insurance com It can be argued that the new derivatives regulations do noth
panies: moral hazard and adverse selection. Recall that moral ing more than replace too-big-to-fail banks with too-big-to-fail
hazard is the risk that insurance encourages the insured party C C P s. It certainly would be a disaster for the financial system if a
to take on more risk than they would otherwise, while adverse major C C P were to fail.
selection is the tendency for an insurance company to attract
However, a critical feature of C C P s is that they are much sim
clients with more risks than the general population. A disadvan pler organizations than banks and are therefore much easier to
tage for C C P s is that they entail both moral hazard and adverse
regulate.
selection risks.
The key activities of C C P s are
• Moral hazard exists because market participants have less
• Adm itting members,
incentive to concern them selves with the riskiness of the
com panies they trade with when much of the risk will be • Valuing transactions,
passed on to the CCP. • Determining initial margin and default fund contributions,
• Adverse selection exists when a dealer has a choice between and
clearing a transaction through a C C P or clearing it bilaterally • Managing systems for netting, transferring variation margin,
(e.g ., when it is trading a standard derivative with a non- and so on.
financial end user). If the dealer considers the credit risk of
In contrast, large banks engage in many more com plex and less
the counterparty to be high, it might persuade the counter
transparent activities. For exam ple, the many different types of
party to clear through a CCP. If the counterparty is financially
loan agreem ents that banks enter into give rise to credit risk,
strong, however, the dealer may be com fortable clearing the
while bank trading activities lead to market risk. Derivatives
transaction bilaterally.
trading leads to both market risks and credit risks (which can be
Another disadvantage of C C Ps is that they tend to increase the quite difficult to evaluate).
severity of adverse econom ic events (i.e., they are pro-cyclical).
Additionally, a bank's funding strategy can give rise to liquidity
When markets are highly volatile or there is a financial crisis, for
risks. Banks also face many operational risks arising from cyber
exam ple, many financial institutions are likely to have liquidity
security, anti-money laundering legislation, internal fraud, exter
shortages. A t the same tim e, C C P s are likely to increase initial
nal fraud, and so on.
margin requirements and default fund contributions. These
actions would thereby exacerbate the liquidity shortages faced Thus, the rise of C C P s has seen risk transferred from companies
by financial institutions. that are very difficult to regulate to com panies whose activities
are more am enable to oversight.
It is also difficult for C C P members to evaluate the credit risk
they are taking. Note that a C C P member's default fund contri It should also be noted that O T C C C P s have (up to now) func
butions (and even some of its variation margin gains) may be at tioned well. When Lehman Brothers declared bankruptcy in Sep
risk if another member defaults. However, a C C P m em ber may tem ber 2008, for exam ple, it was the largest bankruptcy in U.S
have very little information about trades done by other mem history. However, C C Ps (both those clearing exchange-traded
bers. This can be contrasted with bilateral clearing, where exp o products and those clearing O T C transactions) managed to
sures are more concentrated but better understood. close out Lehman's positions within a matter of d ays.16 By con
trast, disputes concerning Lehman Brothers' bilaterally cleared
Conversely, a C C P has a great deal of information about the
O T C transactions has dragged on for many years at great cost
portfolios of its members and can act to address excessive risk
to everyone involved.
taking. If a m em ber has a particularly large exposure, for exam
ple, the C C P may limit trading or increase the required initial
margin. In the bilaterally cleared market, however, participants A /
See an account of this by the Global Association of Central Counter

parties in "C entral Counterparty Default M anagem ent and the Collapse
15 C C P s do not usually accept transactions with non-standard of Lehman Brothers" http://ccp12.org/w p-content/uploads/2017/05/
docum entation. C C PD efaultM anagem entandtheCollapseofLehm anBrothers.pdf

Despite these points, it would be a mistake to believe that O TC can lead to defaults, investments may perform poorly and litiga
C CPs pose no risks. C C Ps clearing O TC transactions are critically tion costs may increase.
important to financial markets, and some of them are very large.
For example, in 2018 LCH Clearness SwapCIear cleared interest
Model Risk
rate swaps with a total notional principal of over USD 300 trillion.17
If LCH Clearnet failed, it would have to be bailed out (likely by the As mentioned previously, O T C C C P transactions are different
British government).18 than exchange C C P transactions. O T C transactions last longer,
are less standard, have less price transparency, and trade less
One significant problem with C C P s is that there is a positive
frequently.
correlation among m em ber defaults. If one m em ber defaults
because of difficult econom ic conditions, others are likely to do As a result, O T C C C Ps are more reliant on valuation models in
so as well. Recognizing this, regulators ask C C P s to consider determ ining transaction values and clearing variation margin
scenarios where multiple members default at the same tim e. transfers. If these models function poorly, the operation of the
Regulators also require C C P s to conduct stress tests involving C C P may be com prom ised and m em ber disputes may follow.
imaginary adverse events to determ ine whether they would sur
O T C C C Ps also rely on models to determ ine initial margin
vive and conduct close-outs efficiently.
requirem ents. W hereas exchange C C P s tend to have standard
Note that C C P s treat all members in the same manner when cal rules for determ ining initial margin, an O T C C C P must run mod
culating initial margin and default fund contributions. A conse els to determ ine how much initial margin is appropriate for each
quence of this is that C C P s do not take the credit quality of its member's position. A lesson from the failures of exchange C C Ps
counterparties into account in the same way that a dealer does mentioned in the introduction is that initial margin requirements
in the bilaterally cleared market. O nce a dealer has been adm it should be updated regularly as volatilities change. Members
ted as a member, it is treated in the same way as all other should be required to post initial and variation margin almost
m em bers.19 im m ediately upon request.
A risk for C C P s is that the auction processes for closing out

defaulting members could fail in the turbulent m arkets. It may Liquidity Risk
then be com pelled to force other members to share in the
losses and thereby cause more defaults. It might also lead to Like all other businesses, C C Ps are subject to liquidity risk. A
resignations among members unwilling to stay in the central large C C P holding tens of billions of dollars of initial margin is
faced with a trade-off between the return it gets by investing
clearing m odel.20 This in turn could lead to a loss of reputation
for the C C P and further resignations. this cash and the liquidity constraints of its investments. Liquid
investments (e.g ., Treasury bills) tend to provide lower returns
O ther risks faced by a C C P are than less liquid investments (e.g ., corporate bonds). A t the same
• Fraud, tim e, C C Ps need some of their investments to be readily con
• Com puter systems failure/hacking, vertible into cash in the event one or more members default.
Furtherm ore, it is important that the liquidity of an investment
• Litigation costs, and
be assessed assuming stressed market conditions because mem
• Losses on investments of the initial margin and variation ber defaults are likely to be accom panied by turbulent market
margin. conditions (which typically reduce investment liquidity).
There may be correlations between losses arising from defaults
and these types of losses. In the stressed market conditions that
SUMMARY
As a result of regulations introduced since the 2007-2008 crisis,
17 LCH. Monthly Volumes— SwapCIear Global. Retrieved from https://www. the volume of O T C transactions cleared through C C P s has risen
lch.com/services/swapdear/volumes/monthly-volumes-swapdear-global
dram atically. As of mid-2018, 76% of interest rate derivatives
18 A s mentioned earlier, it is subject to a great deal of oversight from the and 54% of credit default swaps were cleared through C C P s.
Financial Conduct Authority in the U .K.
O T C C C Ps operate much like the exchange C C P s that were dis
19 Regulators may be com fortable with this approach as assessing the
quality of m em bers would involve som e subjectivity on the part of C C P s. cussed in Chapter 5. Arguably, O T C C C P s are more difficult to
20 Typically, a m em ber would have to elim inate its C C P exposures, and manage than exchange C C Ps because their instruments are less
give one month's notice, in order to resign. standard, last longer, and only trade interm ittently.

C C P s require members to post initial margin, variation margin, be disastrous consequences. W hile O T C C C P s have been tested
and to make contributions to a default fund. W hile this model by difficult markets (e.g ., disruption caused by the default of
increases the funds that must be provided to support trading, Lehman Brothers), the central clearing model still contains risks.
it reduces counterparty credit risk. C C Ps also make it easier for M arket turbulence could lead to defaults by several members
market participants to close out positions and tend to increase simultaneously and a loss of confidence in O T C central clear
the amount of netting that is available to market participants. ing. Like many other businesses, C C P s also face operational and
liquidity risks. They are also reliant on models for determining
Some O T C C C P s are extrem ely large and play a key role in the
variation margin transfers and initial margin requests.
global financial system. If one of them were to fail, there could

Q UESTIO N S

6.1 W hat types of transactions are required to be cleared 6.5 Has the use of bilateral clearing for O T C derivatives
through C C Ps? increased or decreased since the 2007-2008 crisis?
6.2 G ive two exam ples of transactions referred to as "stan 6.6 G ive two reasons why netting might not increase as a
dard" in the regulations governing O T C derivative result of the new rules for central clearing.
transactions. 6.7 W hat are tear ups?
6.3 W hat are "uncleared" transactions? 6.8 How do C C P s give rise to moral hazard?
6.4 W hy does initial margin for uncleared derivatives have to
6.9 How do C C P s give rise to adverse selection?
be posted with third parties?
6.10 W hat are pro-cyclical actions? W hy might the actions of a
C C P be pro-cyclical?
Practice Questions
6.11 G ive three differences between the properties of O T C 6.15 Explain why interest is paid on variation margin in O T C
derivatives and exchange-traded derivatives that make markets but not in exchange-traded futures markets.
central clearing more difficult. 6.16 W hat are the pros and cons regarding competition
6.12 Describe the O T C transactions that continue to be cleared between the C C P s that clear O T C products?
bilaterally. 6.17 W hat is the difference between the criteria for setting
6.13 G ive three reasons why a C C P requires the O T C deriva initial margin in the cleared market versus the uncleared
tives it clears to be traded actively. market?
6.14 X, Y, and Z have entered into many derivative transactions. 6.18 Explain what is meant by (a) loss mutualization and (b) sys
When transactions between X and Y are netted, the net tem ic risk.
value to X is 60. When transactions between Y and Z are 6.19 W hy are C C P s easier to regulate than banks?
netted, the net value to Y is 70. When transactions between
6.20 W hy is an O T C C C P more dependent on valuation models
Z and X are netted, the net value to Z is 80. Suppose that
than an exchange C C P ?
all transactions are cleared through a C C P rather than bilat
erally. W hat is the net position of X, Y, and Z?
Chapter 6 Central Clearing 81

ANSW ERS
6.1 All standard transactions between financial institutions 6.12 Non-standard transactions and some standard transac
must be cleared through C C P s. If a non-financial institu tions between financial and non-financial com panies con
tion is engaging in a large volume of transactions, its tinue being cleared bilaterally.
standard transactions must also be cleared through C C P s. 6.13 It needs to value transactions at least once a day. It
Standard transactions include interest rate swaps and needs to unwind a member's position expeditiously if the
credit default swaps on credit indices. m em ber defaults. It is not worth developing the systems
6.2 Interest rate swaps and credit default swaps on credit to clear a particular transaction type if members do not
indices. trade it very often. The C C P uses trading data to set initial
margin.
6.3 Uncleared transactions are transactions not cleared
though a C C P 6.14 When they are cleared centrally, they net to —20 for X, +10
to Y, and +10 to Z.
6.4 If equal amounts of initial margin were posted from
Party A to Party B and from Party B to Party A , the net 6.15 Futures contracts are settled daily, so when variation mar
initial margin posted by each side would be zero, and thus gin is paid from A to B, it belongs to B. O T C derivatives
the initial margin would not be available to cover default are not usually settled daily, so variation margin from A to
losses. B belongs to A until contractual payments on the deriva
tives are required.
6.5 It has decreased for interest rate derivatives and credit
default swaps because of regulatory requirem ents. Both 6.16 Com petition gives market participants choice and may
interest rate derivatives and credit default swaps are pop provide incentives for C C P s to becom e more efficient and
ular categories of O T C derivatives (see Table 5.1). improve their system s. However, it may also lead to initial
margin and default fund contributions being reduced to
6.6 Transactions might be cleared through different C C P s.
attract more clients and, thereby, increase the probability
Netting between standard and non-standard transactions
is no longer possible. of a C C P failure.
6.7 Tear ups involve closing out transactions of members that 6.17 In the cleared market, C C P s want initial margin to cover
were originally entered into with the defaulting party at five-day price changes with 99% certainty. In the
uncleared m arket, it is anticipated that close-outs will take
prices that leave the members with some loss.
longer and initial margin covers ten-day price changes
6.8 A dealer may not monitor the credit quality of counterpar
with 99% certainty.
ties carefully.
6.18 Loss mutualization involves spreading losses over many
6.9 When trading standard transactions with an end user,
different entities. System ic risk is the risk that the inter
a dealer is more likely to insist that the transactions are
connectedness of financial institutions will lead to a crisis
cleared through a C C P when the credit quality of the end
whereby a failure of one financial institution leads to losses
user is low.
by other financial institutions— leading to even more
6.10 Actions that reinforce the severity of econom ic cycles are failures.
referred to as procyclical. During stressed periods, a C C P
6.19 C C P s are engaged in activities that are sim pler than those
is likely to ask for more initial margin from its members.
of banks. Thus, they are easier to be supervised and
This could exacerbate liquidity problems already being
regulated.
experienced by members and therefore be procyclical.
6.20 An O TC C C P needs models to value the portfolios of its
6.11 O T C derivatives last longer and may be much larger than
members and to determine the initial margin that should be
exchange-traded derivatives. They are less standard. They
posted by its members. Exchange C C Ps can usually observe
trade less frequently, have lower volume and are less liq
prices from contracts being traded in the market and have
uid. They may have less historical data.
developed standard rules for determining initial margin.

Futures Markets
Learning Objectives
Define and describe the key features of a futures contract, • Explain the different market quotes.
including the underlying asset, the contract price and size,
trading volume, open interest, delivery, and limits. Describe the mechanics of the delivery process and
contrast it with cash settlem ent.
Explain the convergence of futures and spot prices.
Evaluate the impact of different trading order types.
Describe the rationale for margin requirements and
explain how they work. Describe the application of marking to market and hedge
accounting for futures.
Describe the role of an exchange in futures and over-the-
counter market transactions. Com pare and contrast forward and futures contracts.
Identify the differences between a normal and inverted

futures market.
83
Futures contracts are popular exchange-traded products. They Table 7.1 Futures and Options Contracts Traded
are agreem ents to buy or sell an asset at a future time for a cer by the Ten Largest Exchanges in 2017
tain price. We have already discussed how variation margin and
initial margin are used to handle credit risks in futures markets. Futures and Options
In this chapter, we further explain how contracts are specified Contracts Traded
and how futures markets operate. Exchange (Million)
C M E Group 4,089
National Stock Exchange of India 2,465

7.1 EXCH A N G ES
Intercontinental Exchange 2,125
Futures contracts are actively traded all over the world. The C B O E Holdings 1,810
largest futures exchange in the world is the C M E Group, which
B3 1,809
form ed the result of a m erger of the Chicago Board of Trade
(C BO T) and the Chicago Mercantile Exchange (CM E) along with N A SD A Q 1,677
subsequent acquisitions of the New York M ercantile Exchange Eurex 1,676
(N YM EX) and the Com m odity Exchange, Inc. (C O M EX).
M oscow Exchange 1,585
O ther large futures exchanges are the National Stock Exchange Shanghai Futures Exchange 1,364
of India, the Intercontinental Exchange, B3 (formed in 2017 by
Dalian Com m odity Exchange 1,101
the merger of Brazilian exchanges BM & FBO VESPA and C ETIP),
Eurex (based in Germ any), the Shanghai Futures Exchange, and S o u rce: w w w .fia.org
the Dalian Com m odity Exchange. Table 7.1 shows the number of
options and futures contracts traded in 2017 by the ten largest
exchanges (as reported by the Futures Industry Association).12 • Members post initial margin and contribute to a CCP's default
fund to protect the C C P and its members against losses.
• M embers clear the trades of non-members and maintain mar
7.2 OPERATION O F EXCH A N G ES gin accounts with them to provide protection for them selves
against non-member defaults.
The following list reviews what we have learned in earlier chap
ters about the operation of exchanges. Because it is so easy to close out futures contracts, most are
closed out before delivery. The number of contracts in existence
• Standard contracts are defined by an exchange.
at any time is referred to as the open interest. This is the num
• An exchange's C C P becom es the counterparty to all mem ber of net long contracts held by members and it is equivalent
bers of the exchange when they trade. to the number of net short contracts held by members (because
• Trades are settled daily by variation margin flowing from one the number of long positions must always equal the number
side to the other. of short positions). When trading in a certain contract starts, the
• Positions can be closed out by entering into offsetting posi- open interest is zero. As members trade with each other,
tions (e.g ., one long contract to buy an asset in Septem ber the open interest increases. Typically, the open interest peaks
can be closed out by entering into a short contract to sell the shortly before the delivery period specified in the contract.
asset in Septem ber). M embers then start to close out their positions, and the open
interest plummets.
A trade of a futures contract between two members has one of

the following effects on the open interest of that contract.
1 It is not always easy to distinguish the volume of trading in futures from
that in options from published statistics. O ptions on futures for exam ple • When both members are taking new positions, the open
are not always reported separately from futures. However, other statis
interest increases by one.
tics published by the Futures Industry Association show that more than
half the futures and options contracts traded w orldw ide are futures. It • When one m em ber is closing out a position while the other
is estim ated that in total 14.84 billion futures contracts and 10.36 billion m em ber is taking a new position, the open interest remains
option contracts w ere traded in 2017.
the same.
2 Recall that taking a long position is referred to as buying a futures
contract, while taking a short position is referred to as selling a futures • When both members are closing out their respective posi
contract. tions, the open interest decreases by one.

The number of contracts traded in a day is referred to as the exchange to adequately distinguish grades of the underlying
trading volum e. The trading volume can be greater than the asset could cause the contract to fail. This is because the party
open interest at the end of the day if many traders are closing with a short position will always choose to deliver the lowest
out their positions (which, as m entioned, tends to occur towards quality product possible, and this is likely to be considered unac
the end of the life of a contract). It can also happen if there is a ceptable by the party with a long position.
large amount of day trading, which is when trades are entered
into and closed out on the same day.
Contract Size
Each exchange is responsible for determining the size of its
7.3 SPECIFICATION O F CONTRACTS contracts. Exchanges want to attract both retail investors (who
usually want to do small trades) and large corporations (who
Exchanges must define what is being traded in detail. W henever
may have large positions to hedge). Typically, the value of what
there is a choice about what is to be delivered, where it is to be
is delivered for a contract on a financial asset tends to be much
delivered, and when it is to be delivered, it is nearly always the
greater com pared to that of an agricultural product. However,
case that the party with the short position (the party making the
exchanges have also developed smaller contracts that are meant
delivery) has the right to choose.3
to appeal to retail investors seeking to hedge sm aller exposures
or take smaller speculative positions.
The Underlying Asset For exam ple, the C M E Group offers both a regular N A SD A Q
In the case of financial assets (e.g., foreign currencies or stock contract (which is USD 100 multiplied by the N A SD A Q 100
indices), the definition of the underlying asset is usually straight index) and a mini N A SD A Q contract (which is USD 20 multiplied
forward. For exam ple, the asset underlying one contract on the by the index). Interestingly, the mini N A SD A Q contracts trade
euro traded by the C M E Group is 125,000 euros. The asset more actively than the regular N A SD A Q contracts.
underlying one contract traded by the C M E Group on the S&P
500 Index is USD 250 multiplied by the index.4
Delivery Location
When the underlying asset is a com m odity, the grade (in terms
of quality) of the com m odity that could be delivered must be In the case of com m odities, transportation costs make the
specified. For exam ple, the Intercontinental Exchange has speci specification of the delivery location very im portant. For exam
fied the asset in its orange juice futures contract as "frozen con ple, the C M E Group's crude oil futures contract specifies that
centrates that are U.S. G rade A with a Brix value of not less than "d elivery shall be made free-on-board ("F .O .B .") at any pipeline
62.5 degrees." or storage facility in Cushing, Oklahom a with pipeline access to
Enterprise, Cushing storage or En b rid g e." For som e contracts,
In some instances, different grades can be delivered with a cor
the delivery location may factor into the price of the underlying
responding price adjustm ent. In the C M E Group's corn futures,
asset.
for exam ple, the grade is specified as "N o. 2 Yellow." However,
"N o. 1 Yellow" is deliverable for a 1.5 cents higher price per
bushel than "N o. 2 Yellow," while "N o. 3 Yellow" is deliverable
Delivery Time
for 1.5 cents less per bushel than "N o. 2 Yellow ."5 Failure by the
Futures contracts are referred to by their delivery months. The
precise dates during the delivery month when a delivery can be
made varies from contract to contract. The party with the short
position can choose among the delivery dates specified by the
exchange. Some contracts (e.g ., the CM E's crude oil futures)
3 A rare exception is in the C M E Group live cattle futures contract where allow for delivery on any day during a given month. O ther con
the contract was changed in 1995 so that the buyer was given some tracts specify a more restricted set of delivery dates.
delivery options.
Exchang es determ ine the delivery months for which a contract
4 The C M E Group also trades an E-mini S&P futures contract on USD
trad es, the tim e when a contract starts trading , and the tim e
50 tim es the index to appeal to sm aller investors.
when it finishes trading . For exam p le, consider the corn
5 From 2019 the discount applicable to No. 3 Yellow will be "betw een
2 and 4 cents depending on broken corn and foreign material and dam futures contracts traded by the C M E G roup. Th ese contracts
age grade facto rs." have delivery months in M arch, May, Ju ly , Septem ber, and
Chapter 7 Futures Markets ■ 85

D ecem ber. Trading in the D ecem b er 2020 contract began on
O cto b er 09, 2017, and continues until D ecem b er 14, 2 0 2 0 .6
A day's settlem en t price is the futures price at the close of trad

ing and it is used for determining daily settlem ent (i.e., variation
margin transfers). If the settlem ent price increases from close of
one trading day to close of the next trading day, funds are taken
out of the accounts of traders with short positions and added
to the accounts of traders with long positions. If the settlem ent
price decreases, funds are taken out of the accounts of traders
with long positions and added to the accounts of traders with Fiqure 7.1 Futures price converges to spot price
short positions. from above.
The futures price converges to the spot price as the delivery
period approaches. Figure 7.1 shows the situation where the
futures price starts above the spot price, while Figure 7.2 shows
the reverse. If the futures price is above the spot price during
the delivery period, traders would have a clear arbitrage oppor
tunity that can be im plem ented by:
• Shorting futures,
• Buying the asset, and
• Making the delivery.
Arbitrage opportunities such as this will not last long as traders

take advantage of them . Also, if the futures price is below the
spot price during the delivery period, those wanting to acquire
Fiqure 7.2 Futures price converges to spot price
the underlying asset will find it profitable to take a long futures
from below.
position and wait for delivery to be made. As they do this, the
futures price will rise towards the spot price.
Price Quotes or down by the price limit (referred to as a limit m ove), trading
is normally halted for the day. If the price limit is reached with
Traders need to know how prices of contracts will be quoted a price increase, the contract is referred to as limit up. If it is
and how accurately they will know the value of their positions. reached with a price decrease, the contract is referred to as limit
An exchange specifies the quotation convention it will use as dow n. A typical limit move for the C M E Group's corn futures
well as the minimum price movements for each contract. In would be 25 cents (i.e., 200 tim es the minimum price m ove
the case of the C M E Group's corn futures, prices are quoted ment). However, the exchange does have the authority to step
as cents per bushel, and the minimum price m ovement is 0.15 in and increase the limit.
cents per bushel. Because one contract is on 5,000 bushels, the
The purpose of price limits is to prevent large price movements
minimum price m ovement per contract is USD 6.25.
resulting from speculation. However, these price limits can also
hinder the determination of true market prices if limit moves
Price Limit arise from new information reaching the market.
For most contracts, exchanges set limits on how much a futures

price can move in one day. However, these price limits may be Position Limits
changed from tim e to tim e. If the price during a day moves up
A position limit (as the name implies) is a limit on the size of a
position that a speculator can hold. Its purpose is to prevent
speculators from exercising an unreasonable influence on the
6 In the case of corn futures, trading finishes on the business day prior
to the fifteenth calendar day of the contract month. O ther rules apply to market. Position limits are often in the tens of thousands of con
other contracts. tracts and do not affect most traders.

7.4 D ELIVERY M ECHANICS and D ecem b er 15, 2020, (with delivery happening one busi
ness day later). The last trading day (as m entioned earlier) is
A s m entioned before, very few futures contracts lead to the D ecem b er 14, 2020.
delivery of an underlying asset because trad ers usually prefer
to close out contracts before the delivery period. If traders
wish to buy or sell the underlying asset, they can then do
Cash Settlement
so in the spot m arket. However, it is im portant to note that In theory, futures contracts could be designed so that they are
the futures price is tied to the spot price by the p o te n tia l for settled in cash. Traders would then avoid the (sometimes incon
final delivery of the specified asset. Th erefo re, the m echan venient) delivery process. However, regulators do not like cash
ics of the delivery process are an im portant aspect of futures settlem ent because it makes futures contracts seem like gam
m arkets. bling. As such, they prefer physical settlem ents (i.e., the delivery
The time when a delivery can be made varies from contract of the asset) to take place w henever possible.
to contract. The delivery process begins when a m em ber with However, the C M E Group's futures contracts on the S&P 500 are
a short position issues a notice of intention to deliver to the settled in cash. This is because physical delivery would involve the
exchange CCP. This notice indicates how many contracts will delivery of a portfolio containing the 500 stocks underlying the
be delivered. For com m odity contracts, the notice states where S&P 500; this would be an expensive and inconvenient undertak
delivery will be made and (if applicable) what grade will be ing. Instead, all contracts are settled on the third Friday of the
delivered. The exchange must then choose one or more parties delivery month by a final exchange of variation margin. The final
with long positions to accept deliveries. Standard procedure is settlem ent price is the opening value of the S&P 500 on that day.
for exchanges to allocate delivery notices to members who have
Exam ples of other cash settled contracts include those depen
had net long positions for the longest period of tim e. However,
dent on w eather and real estate prices. The C M E Group's popu
som etim es delivery notices are allocated at random. Members
lar Eurodollar futures contract (which will be discussed in a later
cannot refuse to accept delivery notices, but som etim es they are
chapter) is also settled in cash.
given a short period of time to transfer their contracts to other
members.
Delivery of com m odities can involve warehousing costs. In the

7.5 PATTERNS O F FUTURES PRICES
case of livestock, there may be costs associated with feeding
and looking after the animals. There are stories of futures trad
Table 7.2 shows settlem ent futures prices for gold on June 25,
ers who inadvertently held long livestock positions during the
2018. This is referred to as a normal m arket because the futures
delivery period and received notices of delivery from the
price increases as time to maturity increases. Specifically, the
exchange. They then ended up owning livestock in a warehouse
futures price increases from USD 1,265.6 (which is close to the
several thousand miles aw ay.7
spot price) for June 2018 delivery to USD 1,480.1 for delivery
The price to be paid for the asset is the most recent settle in D ecem ber 2023. In the case of gold futures, delivery can
ment price. In some cases, this is adjusted for grade, delivery take place on any day during the delivery month (with notice of
location, (etc). intention to deliver being made one day earlier).
Th e first n o tice day is the first day when a notice to deliver The C M E Group futures for crude oil had a very different pat
can be subm itted to the CCP. Th e last n o tice day is the last tern on June 25, 2018. As shown by Table 7.3, the future prices
day when this can happen. The last trading day is gener declined from USD 68.08 to USD 53.57 as maturity increased.
ally a few days before the last notice day. For the D ecem ber This is referred to as an inverted market.
2020 corn futures contract we considered earlier, delivery
Some assets have patterns that are partly normal and partly
notices can be issued any tim e betw een N ovem ber 30, 2020,
inverted. For exam ple, soybean futures prices on June 25, 2018,
increased for maturities out to Ju ly 2019 and then decreased.
Furtherm ore, futures prices do not always have the same pat
tern. Oil futures, for exam ple, som etim es exhibit a normal mar
ket rather than the inverted market as presented in Table 7.3.
The determination of futures prices and whether a normal
7 The delivery of financial assets can usually be handled electronically market or an inverted market is likely to be observed will be dis
and does not give rise to these types of problem s. cussed in a later chapter.

Table 7.2 Settlement Prices for Gold Futures Table 7.3 Settlement Prices for Selected Crude Oil
on June 25, 2018 Futures Contracts on June 25, 2018
Maturity Month Settlement Price (USD per Ounce) Maturity Month Settlement Price (USD per Barrel)
June 2018 1,256.6 August 2018 68.08
Ju ly 2018 1,267.2 O ctober 2018 66.18
August 2018 1,268.9 D ecem ber 2018 65.47
O ctober 2018 1,274.7 June 2019 63.36
D ecem ber 2018 1,280.9 D ecem ber 2019 61.64
February 2019 1,287.1 June 2020 60.16
April 2019 1,293.2 D ecem ber 2020 59.01
June 2019 1,299.6 June 2021 57.91
August 2019 1,306.0 D ecem ber 2021 56.97
O ctober 2019 1,312.6 June 2022 56.15
February 2020 1,325.8 June 2023 54.91
April 2020 1,332.2 D ecem ber 2023 54.52
June 2020 1,339.2 June 2024 54.08
June 2021 1,379.8 June 2025 53.60
June 2022 1,420.3 S o u rce: w w w .cm egroup.com
D ecem ber 2022 1,440.7
June 2023 1,460.4

they do tend to have a close relationship with a member who
D ecem ber 2023 1,480.1
clears their trades. Locals and the clients of futures commission
S o u rce: w w w .cm egroup.com brokers can be classified as scalpers, day traders, or position trad
ers. Scalpers generally hold futures positions for a very short
period (perhaps only a few minutes) before closing them out.8 Day
7.6 M ARKET PARTICIPANTS traders enter positions with the intention of closing them out dur
ing the same trading day and (like scalpers) hope to make a profit
Futures market participants can be classified as (a) those that from short-term intra-day price movements. However, they are pre
solicit or accept orders to trade from retail and other clients, pared to wait longer than scalpers before closing out. Finally, posi
and (b) those that trade for their own account. The form er are tion traders have views on how the market will move over much
term ed futures com m ission m erchants or introducing brokers. longer periods of time and take positions to reflect those views.
Under the U.S. Com m odity Exchange A ct, they are subject
to registration and minimum capital requirem ents. Futures
commission merchants manage custom er funds (including
7.7 PLACING ORDERS
margin requirements), whereas introducing brokers do not.
Traders of futures and other securities can place many different
Futures commission brokers may or may not be members of an
types of orders.
exchange. Those that are not exchange members must clear
their clients' trades through members.
8 Some exchanges use m arket makers. These add liquidity to the market
Market participants who trade for their own accounts are referred by always being prepared to quote a bid and an ask price. The trading
to as locals. While they are not typically members of an exchange, activity of m arket makers is similar to that of scalpers.

Market Orders a limit order that is executed at a price of USD 52.5 or lower.
Som etim es the stop price and the limit price are the sam e, in
The sim plest order is a m arket order, which is a request to buy which case the order is called a stop-and-lim it order.
or sell futures (i.e., take a long or short position) as quickly as
possible at the best available price. The disadvantage of a mar
ket order is that a trader may end up buying (selling) at a much Market-if-Touched Orders
higher (lower) price than expected.
A m arket-if-touched (MIT) order is an order that becom es a
market order if a trade occurs at the specified price or a more
Limit Orders favorable price. It is also known as a board o rd er and is a way in

which a trader can ensure that profits are taken if there are suffi
The main alternative to a market order is a limit order, where the ciently favorable price movements. Consider again a trader with
trader specifies a price limit. A limit order can only be executed a short futures position when the price is USD 50. If the price
at the specified price or at a price more favorable to the trader. specified in an MIT order is USD 45, the trader is indicating that
In the case of a buy order, the limit price is the maximum price he or she wants to take profits by exiting from the position as
at which the trader is prepared to buy; in the case of a sell order, soon as the price reaches USD 45.
the limit price is the minimum price at which the trader is pre
pared to sell. Limit orders usually remain in effect for one day,
but traders can specify a longer or shorter period. Discretionary Orders
To see how limit orders work, suppose that a trader sees that A discretionary order, also called a m arket-not-held order, is an
the current futures price is USD 32.5. He or she could then put order that the broker can delay filling in hopes of getting a bet
in a limit order to buy at USD 32.3, which would be executed ter price.
only if the price declines slightly. If the limit is USD 32.6,
however, the order is much more likely to be executed if the
exchange operates on a price-time priority basis (i.e., it executes
Duration of Orders
higher priced orders first). As m entioned, the default arrangem ent is that an order con
tinues to exist for one trading day. If it is not filled by the end
of the trading day, it is cancelled. Traders can specify other
Stop-Loss Order periods of time during which an order is active. A t one extrem e,
In a stop-loss o rd er (sometimes referred to as a sto p order), the a fill-or-kill o rd er is an order that is automatically cancelled if it
order becom es a market order once the asset reaches a speci is not fully executed im m ediately (i.e., within a few seconds).
fied or a less favorable price. Stop-loss orders (as the name A t the other extrem e, an open o rd er or a good-till-cancelled
implies) are orders that are designed to limit a trader's loss on a o rd er remains open for the remaining life of the futures contract
certain position. unless it is cancelled by the trader.
For exam ple, suppose that the price is currently USD 50 and
that the trader has a short futures position. The trader could
issue a stop-loss to buy at USD 52. This becom es a market order
7.8 REGULATION O F FUTURES
to buy as soon as the future price reaches USD 52. If the price M ARKETS
continues to rise, the order might be executed at USD 53. A lter
natively, if the price reaches USD 52 and then falls, the order Futures markets are regulated in different ways in different
could be executed at USD 51. countries. In the U .S., futures markets are regulated by the
Com m odity Futures Trading Commission (C FTC ). This govern
ment agency aims to ensure that futures markets are open,
Stop-Limit Orders transparent, com petitive, and financially sound. The C F T C is
responsible for licensing individuals who offer their services to
W hereas a stop-loss order becom es a market order when the
the public and handling com plaints that are raised by futures
stop price is reached, a stop-limit order becom es a limit order.
markets participants. It also oversees the setting of position
For a stop-limit order, two prices must therefore be specified:
limits.
the stop price and the limit price. In the exam ple just given,
suppose that the stop price is USD 52 and that the limit price Some of the responsibilities of the C F T C have been del
is USD 52.5. O nce the price equals USD 52, the order becom es egated to the National Futures Association (NFA). This is a

self-regulatory organization consisting of members who par If the gold producer we have considered qualifies for hedge
ticipate in futures markets. Its aim is to protect investors and to accounting, the entire gain of USD 2,200,000 (= (1,300 -
ensure that members meet their regulatory responsibilities. It 1,190) X 100 X 200) will be realized in the third year. This is
monitors trading, resolves disputes, and takes disciplinary action likely to be attractive to the company as it looks to reduce its
when appropriate. earnings volatility.
In earlier chapters, we have mentioned the regulatory changes The rules concerning the use of hedge accounting are strict. The
in the O T C market introduced since the 2007-2008 crisis. These hedge must be fully documented, for exam ple, with the hedged
have led to increased responsibilities for the C F T C , which now item and the hedging instrument being clearly identified. The
ensures that standard O T C derivatives (e.g ., swaps) are traded hedge must also be classified as effective, which means that an
and cleared in accordance with the new rules. economic relationship that is not dominated by the effect of credit
risk must exist between the hedging instrument and the hedged
item.*•9 The effectiveness of a hedge must be tested periodically.
7.9 ACCOUN TIN G Taxation raises similar issues for futures trading. In many jurisdic
tions, futures contracts are normally treated for tax purposes
Normal accounting rules call for gains and losses from futures
as though they are closed out at the end of the tax year. W hile
to be accounted for as they occur. For exam ple, consider a gold
hedging transactions in the U.S. are exem pt from this rule, the
mining company with a fiscal year ending in Decem ber. Suppose
definition of a hedging transaction for tax purposes is differ
that it sells 200 two-year futures contracts on gold in June when
ent from the definition used for accounting purposes. For tax
the futures price is USD 1,300 per ounce. Each contract is for
purposes, a transaction is a hedging transaction if it is entered
the sale of 100 ounces. Suppose further the following scenarios.
in the normal course of business primarily to reduce risk exp o
• In D ecem ber of the first calendar year, the futures price is sures. Given the different criteria, it is possible for a transaction
USD 1,240 per ounce. to qualify as a hedge for tax purposes but not for accounting
• In D ecem ber of the second calendar year, the futures price is purposes.
USD 1,160 per ounce.
• The contract is closed out at USD 1,190 per ounce in June of

the third calendar year.
7.10 FORW ARDS COM PARED
WITH FUTURES
The USD profit is then reported as follows.
First fiscal year: (1,300 - 1,240) X 100 X 200 = 1,200,000 Forward and futures contracts are similar in that both are agree
Second fiscal year: (1,240 — 1,160) X 100 X 200 = 1,600,000 ments to buy or sell an asset in the future. However, there are
Third fiscal year: (1,160 - 1,190) X 100 X 200 = -600,000 some key differences. For exam ple, most forward contracts
are on foreign exchange or interest rates. In contrast, futures
Futures are settled daily so that the cash corresponding to the
contracts are on a wide range of financial and non-financial
profits is realized in the years in which the profits are accounted
assets. We gave exam ples of how forward contracts on for
for. The valuation process is referred to as marking to market.
eign exchange can be used for hedging in Chapter 4. We will
Realizing and accounting for gains and losses year-by-year when describe forward contracts on interest rates later in the book.
hedging could lead to an increase in reported earnings volatility,
W hereas a futures contract is traded on an exchange, a for
rather than a reduction in volatility as would be expected when
ward contract is an over-the-counter product and is (in many
performing hedging activities.
situations) subject to more credit risk. There are several other
If the gold company is hedging gold that it expects to produce differences. A futures contract is settled daily, while a forward
in two years, however, the contracts it has entered may qualify contract is settled at the end of its life. Delivery of the underly
for hedge accounting. This provides an exception to the general ing asset is relatively rare for futures contracts because traders
rule we have just mentioned and allows the gain (or loss) from usually close out their positions before the delivery period speci
hedging transactions to be recognized at the same time as the fied in the contract. In the case of a forward contract, delivery is
loss (or gain) on the items being hedged. The Financial A ccount
ing Standard Board (FASB) has issued FAS 133 and A S C 815
to explain when hedge accounting can and cannot be used by
9 IFRS 39 has been replaced by IFRS 9, which eases som e the hedge
com panies in the U.S. The International Accounting Standards accounting requirem ents. Sim ilarly, A S C 815 eases some of the rules in
Board (IASB) has similarly issued IAS 39 and IFRS 9. FAS 133.

usually made. Closing out is not as easy as it is for a futures con In general, the futures price of an asset is not equal to its spot
tract because a company with a forward contract must approach price and instead converges to the spot price as the specified
its counterparty and negotiate a close out. delivery period approaches. With normal m arkets, the futures
price increases with the maturity of the contract. With inverted
Forward contracts usually specify a single delivery date. In con
m arkets, the futures price decreases with the maturity of the
trast, futures contracts specify a period (sometimes a whole
futures contract. Some futures prices show patterns that are nor
month) during which tim e delivery can be made. Because
mal for some periods and inverted for others.
futures contracts are traded on an exchange, they are standard
ized financial products. Forward contracts have the advantage in Many different types of orders can be placed in futures markets.
that the delivery date can be chosen to meet the precise needs The sim plest is a market order, which indicates that the trade is
of the client. to be executed as soon as possible at the best available price.
A limit order specifies how high the price can be in the case of
a buy order and how low it can be in the case of a sell order.
SUMMARY O ther orders indicate the circum stances in which a trader would
want to cut losses after adverse price movements or take profits
Futures allow parties to contract for the future delivery of a wide because of favorable movements.
range of com m odities and financial assets. They are designed
The over-the-counter alternative to a futures contract is a for
so that there is very little credit risk and it is very easy for market
ward contract. Unlike futures contracts, forward contracts have a
participants to close out their positions. Because of these quali
single delivery date and are not settled daily. Forward contracts
ties, futures are very successful financial products and trade on
are not standardized and, unlike futures contracts, usually lead
exchanges throughout the world.
to delivery of the underlying assets.
The specification of contracts is an important activity for a
We will discuss how futures are used for hedging in the next
futures exchange. It is necessary for an exchange to define what
chapter. Later chapters will discuss the nature and pricing of
can be delivered, where it can be delivered, and when it can be
com m odity, foreign exchange, and interest rate futures.
delivered. When there are choices, it is the party with the short
position that chooses. If a contract is inadequately specified, it is
likely to fail.

QUESTIONS

7.1 Name three of the largest ten futures and options has entered into a particular contract specify which grade
exchanges in the world. it wants?
7.2 W hat is meant by the open interest of a futures contract? 7.6 W hy do exchanges trade "m ini" contracts?
7.3 How does the open interest change following a trade 7.7 List three types of futures contracts that are cash settled.
between two members who are both closing out 7.8 W hat is meant by (a) a normal market and (b) an inverted
positions? market?
7.4 W hat is the difference between a market order and a limit 7.9 W hat is the difference between a local and a futures com
order?
mission merchant?
7.5 Several different grades of corn can be delivered in a 7.10 W hat is the difference between a scalper and a day
futures contract. Can the party with a long position who trader?
Practice Questions
7.11 A futures contract is still trading as the delivery period is one year to (b) a trader entering into a futures contract to
reached. W hat trades would you make if the settlem ent buy 1 million British pounds in one year.
futures price is above the spot price? 7.17 Suppose that on a particular day there are 3,000 trades
7.12 Futures contracts are referred to by their delivery month. in a futures contract. O f the buyers in those 3,000 trades,
When during the delivery month can the actual delivery be 1,800 were closing out positions and 1,200 were taking
made? new positions. O f the sellers in those 3,000 trades, 1,400
were closing out positions and 1,600 were taking new
7.13 W hat would happen to a contract if its underlying product
had an inadequately specified quality? positions. W hat is the change in open interest during the
day? Does it increase or decrease?
7.14 Explain the difference between a market-if-touched order
7.18 A company enters into a futures contract to buy 10,000
and a stop-loss order.
units of an asset for USD 50 per unit in two years. A t
7.15 If the options (concerning delivery times and what can
the end of the first year, the futures price is USD 45.
be delivered) available to the party with a short position
A t the end of the second year, the futures price is USD 52.
are increased, do you think the futures price increases or
The contract is closed out during the third year when the
decreases? [Hint: An option always has a positive value to
futures price is USD 54. W hat is the accounting for the
the holder of the option.]
profit or losses from the trade each year if the company
7.16 Suppose that both the one-year forward and one-year (a) uses hedge accounting and (b) does not use hedge
futures price of the British pound is USD 1.3000. During accounting?
the year the futures price decreases to USD 1.2000 and
7.19 Explain the relationship between the Com m odity Futures
then rises to USD 1.4000 at the end of the year. The for
Trading Commission and the National Futures Association.
ward price is also 1.4000 at the end of the year. How are
the timing of cash flows different for (a) a trader entering 7.20 List six differences between futures and forwards.
into a forward contract to buy 1 million British pounds in

ANSW ERS
7.1 Exchanges (as listed in Table 7.1) are the C M E group, price or a more favorable price. It is designed to enable a
the National Stock Exchange of India, C B O E Holdings, B3, trader to take profits in the event of sufficiently favorable
N A SD A Q , Eurex, M oscow Exchange, Shanghai Futures price movements.
Exchange, and Dalian Com m odity Exchange. 7.15 The more options that the party with a short position has
7.2 The open interest is the number of contracts outstanding. the more attractive the contract is to that party. They are
It equals the number of long positions or equivalently the therefore prepared to agree to a lower delivery price.
number of short positions. O ptions available to the party with a short position there
7.3 The open interest goes down by one. fore tend to lower the delivery price.
7.4 A market order is to be filled as quickly as possible at the 7.16 Both traders will make a profit of USD 100,000. The for
best available price. A limit order specifies a limit on how ward trader will make all the profit at the end of the year.
The futures trader will have negative cash flows followed
high the price can be when buying or how low it can be
when selling. by positive cash flows because of daily settlem ent. On a
present value basis, the forward trader will do better. Note
7.5 No. Traders with short positions choose what will be deliv
that if the price had increased to USD 1.5000 and then
ered. When issuing a notice of intention, a trader with a
reduced to USD 1.4000, the futures trader would do bet
short position specifies which grade will be delivered.
ter on a present value basis.
7.6 Mini contracts are designed to attract retail traders who
7.17 The open interest goes down by 200. This is because
want to take relatively small positions.
there are 1,200 new long positions and 1,400 long posi
7.7 The C M E Group contracts on indices such as the S&P 500 tions are closed out (with short trades). Alternatively, there
and the N A SD A Q 100 are cash settled, as is the Euro are 1,600 new short positions and 1,400 are closed out
dollar futures contract. Contracts on real estate and the (with long trades).
w eather are also cash settled.
7.18 If the company is able to use hedge accounting, all the
7.8 In a normal market, the futures price is an increasing total profit of USD 40,000 ((54 — 50) X 10,000) is realized
function of maturity, whereas in an inverted market it is a in the third year. If the company is not able to use hedge
decreasing function of maturity. accounting, there is a USD 50,000 loss in the first year
7.9 A local is a speculator trading for himself or herself. A ((45 — 50) X 10,000), a USD 70,000 profit in the second
futures commission merchant processes trades for clients. year ((52 — 45) X 10,000), and a USD 20,000 profit in the
third year ((54 — 52) X 10,000).
7.10 Both take positions and close them out during the same
day. A scalper is looking for very short-term trends in 7.19 TH E C FT C is tasked with regulating the futures market in
futures prices and often keeps a position open for only a the U.S., but some of its responsibilities have been del
few minutes. A day trader keeps the position open longer. egated to the National Futures Association, which is a self
regulating organization of futures market participants.
7.11 Short futures, buy the asset, and make delivery. Delivery
will be at the most recent futures settlem ent price. 7.20 Futures are available on a much wider range of assets than
forwards. Futures are traded on an exchange, whereas
7.12 This varies from contract to contract. Som etim es the
forwards are traded O T C . Futures are standardized by the
delivery period is a single day, som etim es it is the whole
exchange, while forwards can be non-standard. Forwards
month, and som etim es it is part of the month.
have a single delivery date, whereas futures usually have a
7.13 The contract would fail. When the delivery period is reached, range of delivery dates. Futures are settled daily, whereas
shorts would deliver the cheapest version (lowest quality) of forwards are settled at the end of their life. Forwards
the product they could, and longs would be dissatisfied. are usually held to maturity, whereas futures are usually
7.14 A stop-loss order becom es a market order when there is a closed out before maturity. In the case of futures, credit
bid or offer at a specified price or at a less favorable price. risk is very low because the exchange requires initial and
It is designed to limit losses. A market-if-touched order variation margin. Credit risk may be higher in the case of
becom es a market order if there is a trade at the specified forwards.

Using Futures
for Hedging
Learning Objectives
Define and differentiate between short and long hedges Com pute the optimal number of futures contracts needed
and identify their appropriate uses. to hedge an exposure, and explain and calculate the "tail
ing the hedge" adjustm ent.
Describe the arguments for and against hedging and the
potential impact of hedging on firm profitability. Explain how to use stock index futures contracts to
change a stock portfolio's beta.
Define the basis and explain the various sources of basis
risk, and explain how basis risks arise when hedging with Explain how to create a long-term hedge using a "stack
futures. and roll" strategy and describe some of the risks that arise
from this strategy.
Define cross hedging, and com pute and interpret the
minimum variance hedge ratio and hedge effectiveness.
Like other derivatives, futures can be used for either specula This strategy locks in the price received for the oil at the cur
tion or hedging. If a trader has exposures to exchange rates, rent three-month futures price. To illustrate this, suppose that
interest rates, equity indices, or com m odity prices, a position in the current spot price of oil is USD 59.50 per barrel and the
futures contracts can reduce that exposure. But if the trader has three-month futures price is USD 60.00 per barrel. Consider two
no exposure, that same position is speculative. In this chapter, situations.
we focus on the use of futures for hedging.
1. The spot price of oil at the time it is delivered equals USD 50.
Eliminating all risk exposures using futures is usually im pos
2. The spot price of oil at the time it is delivered equals USD 68.
sible. It is therefore important to develop a way of calculating
an optimal hedge (i.e., a hedge that reduces risk as much as In the first situation, the price received for the oil in the market
possible). Initially, we will treat futures as forwards and assume in USD is
that the hedge position is left untouched for the duration of the 2.000. 000 X 50 = 100,000,000
exposure. Later, we discuss the impact of daily settlem ent on W hile the gain (USD) on the 2,000 futures contracts is
the construction of a hedge and discuss how long-term hedges
2.000 X (60 - 50) X 1,000 = 20,000,000
can be created from positions in a series of short-term futures
contracts. Com bining the price received for the oil in the market with the
gain on the 2,000 futures contracts, the net price received is
USD 120 million (= 100,000,000 + 20,000,000). This corresponds
8.1 LONG AND SHORT H EDGES to USD 60 (= 120,000,000/2,000,000) per barrel.
In the second case, where the spot price in three months equals
We start by providing simple exam ples of situations where short
USD 68 per barrel, the price received in the market for the oil in
and long futures positions can be used as hedges.
USD is
2.000. 000 X 68 = 136,000,000
Short Hedge However, there is a USD loss on the futures position of:
A short futures position is appropriate in the following 2.000 X (68 - 60) X 1,000 = 16,000,000
situations.
When this loss is considered, the net price received for the oil is
• A company owns a certain quantity of an asset and knows again USD 120 million (= 136,000,000 - 16,000,000), or USD 60
that it will sell it at a certain time in the future. per barrel. Table 8.1 shows the result of the strategy we have
• A company knows that it will receive a certain quantity of an considered for a range of oil prices. Note that the hedger locks
asset in the future and plans to sell it. in the initial futures price (USD 60) rather than the initial spot
price (USD 5 9 .5 0 ).1
For exam ple, consider a company that will receive 2 million
barrels of crude oil in three months. Assum e that it plans to By assuming that the hedger locks in the futures price, we
sell the oil as soon as it is received and knows that it will lose assumed that the company can either:
(gain) USD 20,000 (= 2,000,000 X USD 0.01) from each one-cent
• Deliver the oil under the term s of the futures contract as soon
decrease (increase) in the price of oil. The company regards this
as it receives it, or
risk as unacceptable.
• Close out the futures contract at the spot price when the oil
Oil futures contracts traded by the C M E Group are for the is received with the oil then being sold in the usual manner.
purchase or sale of 1,000 barrels of oil. The company can
In practice, futures hedges rarely work as they do in this ideal
therefore hedge its position by taking a short futures position
ized scenario (as we will discuss in later sections).
in 2,000 (= 2,000,000/1,000) three-month oil futures contracts.
A hedge that involves taking a short futures position is known as
a short h ed g e. Long Hedges
We assume that the oil is received during the delivery period of A long h ed g e is the opposite of a short hedge and it can
the futures contracts. As explained in the previous chapter, the be used when a company knows it will have to buy a certain
futures price should equal (or be very close to) the spot price quantity of an asset in the future.1
during this period. The company can either deliver the oil under
the term s of the futures contract or close out the futures con 1 Because there is no upper bound to the price of oil in three months,
tracts and sell the oil in the usual manner. the potential loss on the short futures contract is unlim ited.

Table 8.1 The Impact of Hedging the Sale of 1. The spot price of copper in two months equals USD 3.30
2 Million Barrels of Oil: In all Cases the Price Received per pound.
for the Oil, when Adjusted for the Gain or Loss on the 2. The spot price of copper in two months equals USD 2.70
Futures Position, is USD 120 Million per pound.
Price of Oil in Price Received Gain on Short If the spot price of copper becom es USD 3.30, the 150,000
Three Months for Oil Futures Position pounds of copper can be purchased in the market for:
(USD per Barrel) (USD Millions) (USD Millions) 150.000 X USD 3.30 = USD 495,000
20 40 80 If we assume that the copper is purchased when the futures
30 60 60 price equals the spot price, the futures price can be closed out
at the time of purchase. There will then be a gain on the six cop
40 80 40
per futures contracts of:
50 100 20
6 X 25,000 X (USD 3.30 - USD 2.90) = USD 60,000
60 120 0
This reduces the net cost of the copper purchased to
70 140 -2 0 USD 435,000 (= 495,000 - 60,000), or USD 2.90 per pound.
80 160 -4 0 Consider next the situation where the spot price of copper in
two months is USD 2.70. The cost of the copper purchased in
90 180 -6 0
the market is
100 200 -8 0
150.000 X USD 2.70 = USD 405,000
In this case there is a loss on the futures contracts of:

Consider a company that will need to purchase 150,000 pounds
6 X 25,000 X (USD 2.90 - USD 2.70) = USD 30,000
of copper in two months and wants to hedge its price risk. The
spot price of copper is USD 3.00 per pound and the two-month This increases the cost of the copper purchase to USD 435,000
futures price is USD 2.90 per pound. O ne futures contract (= 405,000 + 30,000), or USD 2.90 per pound.
traded by the C M E Group is on 25,000 pounds of copper. The As in the case of the short hedge, we see that the long
company can therefore hedge its risk by taking a long position hedge locks in a price equal to the current futures price of
in 6 (= 150,000/25,000) two-month copper futures contracts. USD 2.90 per pound. Table 8.2 shows the results for other
Consider two situations. copper prices.
Table 8.2 The Impact of Hedging the Purchase of 150,000 Pounds of Copper: In all Cases, when it is Adjusted
for the Gain or Loss on Futures Contracts, the Price Paid is USD 435,000
Price of Copper in Two Months

(USD per Pound) Price Paid for Copper (USD) Gain on Long Futures Position (USD)
2.00 300,000 -135,000
2.20 330,000 -105,000
2.40 360,000 -7 5 ,0 0 0
2.60 390,000 -4 5 ,0 0 0
2.80 420,000 -1 5 ,0 0 0
3.00 450,000 15,000
3.20 480,000 45,000
3.40 510,000 75,000
3.60 540,000 105,000
3.80 570,000 135,000
Chapter 8 Using Futures for Hedging ■ 97

One alternative to using futures is to buy the copper in the spot can. As such, a company should diversify outside its traditional
market today. There are two reasons why the hedger would areas of expertise only when there is synergy (i.e., when it is
probably find this less attractive. combining two or more different business activities such that the
value of the whole is greater than that of the sum of the parts).
1. In this exam ple, copper is more expensive in the spot mar-
ket than in the futures market. Two arguments against leaving hedging to shareholders are the
following.
2. The hedger would bear the cost of financing the purchase
and storing the copper for two months. 1. The shareholders of a public company are less informed
than m anagem ent about the risks being taken by the com
If the futures price were higher than the spot price, buying the
copper today would be more attractive to the hedger. In that pany. Therefore, the company's managers are in the best
case, however, the financing and storage costs would be at position to assess risks and hedge them .
least as great as the difference between the cost of the asset in 2. Shareholders that do understand the risks faced by the com
the futures market and the cost of the asset in the spot market. pany may find it difficult to hedge because their positions
O therwise, any trader would execute a simple arbitrage strategy: may require only a small fraction of one futures contract.
• Buy the asset in the spot market and store it, and W hether a firm chooses to hedge or not, its hedging strategy
• Sell the asset in the futures market. should be set by its board and clearly com m unicated to its
shareholders so that they know and understand the risks they
are taking.
8.2 PROS AND CON S O F HEDGING For exam ple, consider gold mining com panies. Note that it can
take several years to develop a mine for extraction. Given that
As we have just shown, hedging can reduce the risk arising from the price of gold can move adversely during that period, some
changes in asset prices. Hedging can help firms reduce the gold mining com panies choose to hedge the price that will
volatility of their earnings and potentially make them selves more apply to their future production. On the other hand, some firms
attractive to investors. choose not to hedge. In either case, most gold mining com
In practice, however, many com panies do not hedge. Here we panies are careful to explain their hedging strategies to their
consider some of the reasons. shareholders. If investors want an exposure to the price of gold,
they will buy the shares of gold producers that do not hedge. If
investors do not want this exposure, they will buy the shares of
Shareholders May Prefer No Hedging com panies that choose to hedge.
It is som etim es argued that there is no reason for a company to

hedge its risks because shareholders can do their own hedging.
By not hedging, a company allows its shareholders to decide There May Be Little or No Exposure
whether they want to take on a particular risk. In determ ining the size of an exposure, it is im portant to con
An argument in favor of leaving things to the shareholders is sider a company's com plete risk profile.
that they typically invest across multiple com panies. Sharehold As an exam ple, consider a jew elry manufacturer that produces
ers can diversify risks by choosing a portfolio of com panies in 24 carat gold jew elry and buys 100 ounces of gold every two
different industries and operating in different geographical months. Because the firm is exposed to the price of gold, it
regions. This diversification can substantially reduce risks that chooses to lock in its purchases for the next two years with a
might otherwise be hedged. For exam ple, while one company in series of long futures contracts.
an investor's portfolio might suffer from a decrease in oil prices,
However, suppose that an analysis of available data shows
another company in the same investor's portfolio might gain
that econom ic pressures cause the wholesale price of jew elry
from such a price change.
to reflect the price of the gold it contains. When the price of
A related point is that firms can be tem pted to diversify by gold increases (decreases), there is a corresponding increase
entering new lines of business or buying other com panies. How (decrease) in the price of gold jew elry. This would mean that if
ever, a company should always consider whether its shareholders the price of gold declines during the two-year period, the manu
can do this type of diversification more easily than the company 2 facturer would lose money on the hedges and the expected
im provem ent to the manufacturer's gross margin would not
2 This is not always the case. m aterialize. Similarly, if the price of gold increases during the

two-year period, neither the manufacturer's gains on the hedges W hile the first assumption may be reasonable in some hedging
nor the expected worsening of the manufacturer's gross margin situations, the second assumption is rarely true.
would m aterialize. In this case, the hedging has the effect of
As mentioned in the previous chapter, most futures positions are
increasing rather (than reducing) risk.
closed out prior to the delivery period specified in the contract.
O f course, it can be argued that the demand for jew elry may be With this fact in mind, a sensible rule of thumb for hedgers is
affected by the price of gold and that this might justify a long that the maturity for a futures contract should be the earliest
gold hedge. However, the size of the hedge that is necessary possible month after the maturity of the desired hedge.
for that purpose will be quite different from the size necessary
For exam ple, consider a futures contract with maturity months
to hedge the jew elry manufacturer's gold purchases. If the firm
in March, May, Ju ly, Septem ber, and Decem ber. The March
overestim ates its exposure and takes a long position in ten gold
contract would be used for Decem ber, January, and February
futures contracts when its true exposure requires only one con
exposures; the May contract would be used for March and April
tract, the extra nine contracts are unintended speculation. If the
exposures; and so on.
price of gold declines, the hedger will incur a loss.
However, closing out a futures contract before maturity exposes
the hedger to what is called basis risk.
Hedging May Lose Money
The basis for a futures contract at a given time is defined by:3
It is often assumed that the purpose of hedging is to increase
profits. This is incorrect. The purpose of hedging is to reduce Basis = Spot Price — Futures Price
the variability of profits. The outcome with hedging will If the asset being hedged is not the same as the asset underly
som etim es be worse than the outcome without hedging, ing the futures contract, we can extend this definition to:
whereas other tim es it will be better. However, the outcome
Basis = Spot Price of Hedged A sset — Futures Price of A sset
should always be more certain as a result of hedging.
Underlying Futures Contract
The fact that hedging can sometimes lead to losses can make
some corporate treasurers reluctant to hedge. For exam ple, sup Basis risk is the risk associated with the basis at the tim e a
pose that a treasurer working for an oil producer sells oil in the hedge is closed.
futures market to lock in the price received for the company's
Suppose a hedger is due to sell an asset in the future and there
future production. If the spot price of oil decreases, the hedging fore implements a short hedge (similar to that of the oil pro
will result in increased profits. However, we can only expect this to ducer in Section 8.1). Defining term s:
happen some of the time. The rest of the time the price of oil will
increase and the hedging will result in decreased profits. The trea F0: Futures price at the tim e the hedge is initiated,
surer may then get criticized (or worse) because of these losses. Ft: Futures price at the tim e the hedge is closed,
The reluctance to make a decision that could adversely affect prof St: Spot price of asset being hedged at the time the
its has led some treasurers to prefer the use of options for hedging. hedge is closed, and
Instead of locking in a price, options provide insurance. If the trea
bt Basis at time t (= S t — Ft).
surer bought put options on the future price of oil, the company
would be protected in the event of fall in price and yet still be able A t tim e t, the price received for the asset is S t and the gain on
to benefit from a sharp increase in price. O f course, this result is not the short position is F0 — Ft. It follows that:
achieved without a cost. As explained in Section 4.4 of Chapter 4, Net price received when a short hedge is used = S t + (F q — Ft)
options require a premium to be paid upfront by the purchaser.
This can be written as:
Net price received when short hedge is used = F0 + (St - Ft)

8.3 BASIS RISK *•
= F0 + b t
In the exam ples we looked at in Section 8.1, we made several In the oil exam ple in Section 8.1, we assumed that the spot
simplifying assumptions. In particular were the following. price was the same as the futures price at the tim e the hedge
• We assumed that the asset underlying the futures contract is

the same as the asset price to which the hedger is exposed. Note that som etim es the alternative definition: Basis =
Futures Price — S p o t Price is used, particularly when the futures
• We assumed that the futures price and the spot price are contract is on a financial asset. However, we will use the
equal at the tim e the hedge is closed out. Basis — Spot Price — Futures Price definition throughout.

was closed out. This meant that the basis was zero at the time USD 1.30, it must buy the underlying asset at USD 1.31. With
of the close out and the net price received for the oil was always no basis, the hedger would pay the initial futures price of
the futures price at hedge initiation (i.e., F0). When the hedge is USD 1.25 (= 1.30 — 0.05). Instead, the basis increases this to
closed before the delivery period and/or the crude oil underly USD 1.26 (= 1.31 - 0.05).
ing the futures is different from the crude oil that will be deliv
As a second exam ple, suppose that a hedger plans to sell
ered, then b t is uncertain and the hedger is subject to basis risk.
50,000 bushels of corn in June and uses the Ju ly futures con
Consider next a long hedge used when an asset is going to be tract for hedging. Each contract is on 5,000 bushels of corn and
purchased in the future (e.g ., position of the manufacturer in therefore a total o fte n contracts are shorted. Suppose that the
Section 8.1 that knows it will need to purchase copper in the futures price at the time the hedge is initiated is 300 cents per
future). A t time t, the price paid for the asset is S t and the gain bushel and that the futures price is 320 cents per bushel when
on the long futures is Ft — F0. This means that: the hedge is closed out in Ju n e. The spot price for the corn
being sold in June (which may be a different type of corn from
Net cost of asset when long hedge is used
that underlying the futures contract) is 325 cents per bushel. The
= S t - (Ft - F0) USD price received for the corn when the loss from hedging is
This can be written as: considered is
Net cost of asset when long hedge is used 50,000 X 3.25 + 10 X 5,000 X (3.00 - 3.20) = 152,500
= F0 + (St - Ft) This shows the net price received is 305 (= 152,500/50,000) cents
per bushel. In this case consider the following.
= F0 + b t
• The price of corn increases, so the corn producer loses
This shows that the net price received when a short hedge is
20 (= 320 — 300) cents per bushel when it closes the futures
im plem ented for the future sale of an asset is the same as the
contracts.
net price paid when a long hedge is im plem ented for the future
purchase of an asset. Both are equal to F0 + b t. • However, the basis is 5 (= 325 — 320) cents per bushel. When
the hedger sells the corn, it does so at the spot price of
The futures price (F0) is known at the tim e the hedge is initiated.
325 cents per bushel. Com bined with the 20 cents per bushel
The uncertainty about the perform ance of the hedge is there
loss on the hedge, the net price received is 305 cents per
fore due entirely to the uncertainty about the future basis (bt).
bushel. This more than the initial futures price of 300 cents
This uncertainty tends to be greater for futures on com m odities
per bushel.
than for futures on financial instruments.
These exam ples em phasize that the purpose of hedging
As a first exam ple of basis risk, consider a U.S. company that is
is to make the outcom e more certain. However, basis risk
planning to hedge an impending purchase of 250,000 British
means that there are still some uncertainties associated with
pounds (GBP) in February by using the C M E Group's March
hedge.
futures contract. Given that each contract is on G B P 62,500, the
company needs a long position in four contracts.
Suppose that the futures price at the time the hedge is initiated is
1.25 USD per G BP and the futures price at the time of the close 8.4 OPTIMAL H ED GE RATIOS
out in February is 1.30 USD per GBP. However, the spot price in
February is 1.31 USD per GBP. After the gain from hedging has In the exam ples presented so far, we have assumed that futures
positions equal the exposures being hedged. Thus, when 50,000
been considered, the cost (in USD) of the British pounds is
bushels of corn were being sold in an earlier exam ple, we
250,000 X 1.31 - (1.30 - 1.25) X 4 X 62,500 = 315,000 assumed that the hedger would enter into futures contracts for
This indicates that the hedger's net exchange rate is USD 1.26 a total of 50,000 bushels. In this section, we explore situations
(= 315,000/250,000) per GBP. In this case consider the following. where the asset underlying the futures contract is different from
the asset being hedged.
• The exchange rate increases and so hedging improves
(reduces) the price paid (compared to the spot price) by The h ed g e ratio is the ratio of the position in the futures con
0.05 USD per GBP. tract to the position in the underlying asset.
• However, the basis is USD 0.01 (= 1.31 — 1.30). This means Hedging an exposure to the price of one asset with a futures
that while the hedger can cash out the futures contracts at position in another asset is referred to as cross hedging. Cross
hedging can be analyzed by considering the relationship shorter time periods are often used to increase the number of
between the following: observations that can be used in generating estim ates.
Defining term s:
AS: Change in spot price during a period equal to the
life of the hedge, and
Qa: Size of the position being hedged (units),
A F: Change in futures price during a period equal to
Qf: Number of units of the asset underlying one futures
the life of the hedge.
contract, and
Suppose that historical data, used in conjunction with linear N*: Optim al number of futures contracts for hedging.
regression, shows that the best fit linear relationship between
A S and A F is It follows that:
h*Q,
AS = a + b A F + e N* = ( 8 . 2)
~o7
where a and b are coefficients and e is an error term . Suppose A common exam ple of cross hedging is related to the use of
further that h is the hedge ratio. The change in the value of the heating oil futures by airlines to hedge jet fuel. Note that while
position per unit of the asset being hedged is then: jet fuel futures do exist, they do not trade as actively as heat
AS - h A F = a + (b - h)A F + s ing oil futures. As a result, airlines often hedge using heating oil
futures instead.
The variance of the right-hand side is minimized by setting
h = b (so that the second term is zero). Suppose that an airline estimates that the correlation between
monthly changes in heating oil futures prices and jet fuel prices is
Defining term s:
USD 0.9. The standard deviation of monthly changes in heating oil
futures price per gallon is USD 0.03, while the price of jet fuel per
p: Coefficient of correlation between A S and A F,
gallon is USD 0.025. The optimal hedge ratio from Equation (8.1) is
a s: Standard deviation of AS,
0.025
a F: Standard deviation of A F, and 0.9 X 0.75
0.03
h*: Optim al hedge ratio. This indicates that the optimal hedge ratio is 75%.
An expression for b is4 Suppose that the airline wishes to hedge the purchase of
1 million gallons of jet fuel in one month. Each heating oil
futures contract is on 42,000 gallons of heating oil. The number
It follows that this is also an expression for the optimal hedge of long contracts required for hedging can be obtained using
ratio h*: Equation (8.2):

= 0.75 X 1 000,000 =
h* = p — (8.1)
(Tf 42,000
The h ed g e effectiven ess is the proportion of the variance in AS or 18, when rounded to the nearest whole number.
that is eliminated by the hedging. This is usually referred to as
the R2 of the regression and equals p2 in the case of linear mod Tailing the Hedge
els with a single variable.
The analysis presented so far is correct when hedging is carried out
If we have perfect correlation (p = 1) and as = a F, then h* = 1. with forward contracts. A small adjustment to the analysis known as
If we have perfect correlation but as is 20% higher than a F then tailing the hedge is (in theory) necessary when futures contracts are
h* = 1.2. In this case, this is also as expected because spot price used.5 This recognizes that futures are settled daily so that the
changes are always 20% greater than futures price changes. hedger is effectively implementing a series of one-day hedges. The
The param eters, p, as, and a F, are estim ated from historical data hedge ratio for a particular day should be given by Equation (8.1),
on A S and A F. Ideally, the time period over which changes are where the correlation p is the correlation between one-day changes
measured should be equal to the life of the hedge. In practice, while as and a F are the standard deviations of one-day changes.
4 Recall that the slope coefficient in the linear model is given by

5 The term , "tailing the hed g e," com es from the fact that, when it is
cov(A S,A F) pal s o-AF ÂS recognized that a hedge using futures is a series of one-day hedges, the
b - b —p
CTAF
\ <XAF
a (Tap hedge should in theory be adjusted (tailed) with the passage of tim e.

A related point is that analysts often work with the standard It is necessary to allow for the tim e difference between
deviation of one-day returns (i.e., daily volatilities) rather than the settlem ent of each one-day hedge and the maturity of
the standard deviation of price changes. This is because the for the futures contract. This means a discount factor must be
mer can be expected to be less variable. Define: applied to the hedge ratio calculated for each day.
When applying Equation (8.4), VA/VF is usually assumed to be

<t s: Standard deviation of the one-day return in spot
constant at its initial value and discount factor adjustments are not
price (i.e., the percentage change in spot price);
usually made. This is what we will do in the following applications.
op: Standard deviation of the one-day return provided
by futures price (i.e., the percentage change in As an exam ple, suppose that the standard deviations of the
futures price); daily returns in the futures price and the spot price are 1% and
1.2% (respectively) while the correlation between the two is
p: Correlation between the one-day spot return and the
futures return; 0.88. The value of the assets being hedged is 1 million USD,
while the value of one futures contract is USD 20,000. In this
S: Spot price;
case, VA = 1,000,000 and VF = 20,000. From Equation (8.3):
F: Futures price;
0.012
Value of position being hedged, which equals SQ A h = 0.88 X — — = 1.056
0.01
where (as above) QA is the number of units of the
assets being hedged; and The optimal number of contracts is from Equation (8.4)
VF: Value of one futures contract, which is defined as „ 1, 000,000

1-056 X -b — = 52.8
FQp, where Q F is the number of units of the asset 20,000
underlying one futures contract.
or 53, when rounded to the nearest whole number.
The standard deviation of the daily spot price change is 6sS,
and the standard deviation of the daily futures price change is 8.5 HEDGING EQ U ITY POSITIONS
apF (both in absolute term s). This means that the optimal hedge
ratio given by Equation (8.1) can be shown as: Stock index futures are popular products that can be used to
change an investor's exposure to the stock market. For exam ple,
an investor who believes that the market will be particularly
volatile during the next three months might want to tem po
M eanwhile, applying Equation (8.2) for a one-day hedge gives
rarily decrease or eliminate their exposure to the market. On
the optimal number of contracts:
the other hand, an investor who is bullish about the market's
Oa A <xs V/ prospects over the next three months might want to use futures
N* = p
a FF Q F r o'p VF markets to tem porarily increase their exposure.
The hedge ratio is som etim es also defined as:
Stock index futures can be a way of increasing or reducing
exposure to the market without incurring significant transaction
h = P ^ (8.3)
(Tf costs. Take the exam ple of an investor who is good at picking
This leads to: stocks but has no views on the future direction of the market.
The investor might want to form a portfolio and then hedge the
c Va N *= return from theh-A
market so that he or she(8.4)
is left with the excess
VF
return of the chosen stocks over the return from the market.
Note the differences between Equations (8.1) and (8.3). In
Equation (8.1), crs and crF are the standard deviations of the To see how this could work, suppose an investor has a portfolio
change in S and F (respectively) over the life of the hedge, while p of USD 1 million that is well diversified and closely tracks the
is the correlation between these changes. In Equation (8.3) os and S&P 500. The hedger wants to have no exposure to the market
c}p are the standard deviation of daily returns in S and F (respec for the next two months and decides to use the mini C M E
tively), while p is the correlation between these daily returns. futures contract on the S&P 500. As you may recall, this contract
is on USD 50 multiplied by the index.6*
In theory the following are true.
• The values of VA and VF are appropriate for successive one-

6 The regular contract is on 250 tim es the index, but the investor might
day hedges. They change as the price of the asset and the choose the mini contract because it trades more actively and allows
futures price of the contract change. positions to be hedged more precisely.
In this case, the standard deviation of the daily return on the return provided by the futures price. The correlation between
futures price and the standard deviation of the daily return of the daily returns on the asset and the futures is approxim ately
the asset being hedged can be assumed to be the same. Th ere 1.0. Therefore, Equation (8.3) gives the optimal hedge ratio as:
fore, the correlation between the daily returns can be assumed
Yi
to be 1.0. Equation (8.4) therefore gives the optimal hedge N* = (3
v<
ratio as: Consider, again, a portfolio worth USD 1 million when the index
futures price for a six-month contract is USD 2,500. If the beta of
the portfolio is 1.25, the number of contracts required to hedge
the portfolio is
If the futures price is USD 2,500, the number of contracts that
1, 000,000
should be shorted is therefore: N* = 1.25 X = 10
50 X 2,500
1,000,000
We will explain in a later chapter that the futures price of an
50 X 2,500
index should equal its spot price com pounded forward at the
excess of the risk-free rate over the dividend yield for the life of
Managing Beta the futures contract. For now, suppose that the hedge lasts for
six months, the risk-free rate is 4% per year and the dividend
The beta [(3) of a portfolio is the sensitivity of its return to the
return of the market portfolio. If a portfolio has a beta of 1.0, yield on the index is 2% per year. In this case, the initial futures
it mirrors what the market does. If the portfolio has a beta of price is the initial spot price com pounded forward for six months
at 2% (= 4% — 2%) per year (i.e., it is about 1% (= 0.5 X 2%)
0.5, it is half as volatile as the market. W hen the beta is 2.0, it is
twice as volatile as the market. For an investor in U.S. stocks, we higher than the spot price). Because the futures price of the
index is USD 2,500, the spot price must be about USD 2,475
can define the market portfolio as the S&P 500 Index.
(= 2,500/1.01).
The capital asset pricing model (CAPM ) relates the expected
Consider the situation where the index level at the end of six
return on a portfolio to its beta. The model states th at:7
months is USD 2,300. The percentage return on the market con
E(R P) - R f = p (R M - Rf) (8.5) sists of a capital loss of —7.07% (= —175/2,475) and a dividend
where E (R P) is the expected return on a portfolio, RF is the risk gain of 1% (half of the annual dividend yield). The total return
free rate, R m is the return on the market portfolio, and (3 is the on the index is therefore —6.07%. From the capital asset pricing
beta of the portfolio. The param eter beta magnifies the excess model in Equation (8.4), the expected return of the portfolio
return over the risk-free rate. Therefore, a portfolio with a beta during the six-month period is
of 1.5 has a 50% higher excess return over the risk-free rate than
0.02 + 1.25 X (-0 .0 6 0 7 - 0.02) = -0 .0 8 0 9
the market.
Suppose that the risk-free rate is 3%. If the market return is 7%, or -8 .0 9 % .8 The value of the portfolio is therefore expected to
the excess return over the risk-free rate is 4%. For a portfolio with decrease from USD 1,000,000 to USD 919,100 (= 1,000,000 X
(1 — 0.0809)). The USD gain on the futures position should
a beta of 1.5, the expected excess return over the risk-free rate is
therefore 6% (= 1.5 X 4%), and the total expected return of the therefore be
portfolio is 9%. If the return on the market is —3%, however, then 10 X 50 X (2,500 - 2,300) = 100,000
the difference from the risk-free rate is —6% (= —3% — 3%), and
This would bring the value of the portfolio up to USD 1,019,100.
therefore the expected portfolio return relative to the risk-free
rate is - 9 % (= —6% X 1.5). This means that the expected return Table 8.3 summarizes this calculation and shows similar calcula
on the portfolio is —6% (= - 9 % + 3%). tions for other values of the S&P 500 in six months. It can be
seen that the hedge works very well. The total value of the
Suppose we wish to use S&P 500 futures to hedge a well-
hedger's position is close to 1,020,000 in six months in a variety
diversified portfolio with a beta of [3. In this case, the stan
of scenarios. Note that this is the value of the portfolio if it had
dard deviation of the daily return for the hedged asset can be
been invested in a risk-free asset.9
assumed to be [3 multiplied by the standard deviation of the
8 Note that the risk-free rate for six months is half the risk-free rate
per year.
7 For an explanation of the argum ents leading to CA PM see J . Hull,
"Risk M anagem ent and Financial Institutions," 5th edition, 2018. 9 The table ignores the im pact of daily settlem ent.

Table 8.3 Performance of Hedge of a Portfolio with a Beta of 1.25
Index Level in Six Months 2,100 2,300 2,500 2,700 2,900
Capital Gain -1 5 .1 5 % -7 .0 7 % 1.01% 9.09% 17.17%
Dividend 1.00% 1.00% 1.00% 1.00% 1.00%
Total Index Return -1 4 .1 5 % -6 .0 7 % 2.01% 10.09% 18.17%
Portfolio Return -1 8 .1 9 % -8 .0 9 % 2.01% 12.11% 22.21%
Value of Portfolio in Six Months 818,100 919,100 1,020,100 1,121,100 1,222,100
Gain on Futures 200,000 100,000 0 -100,000 -200,000
Final Value 1,018,100 1,019,100 1,020,100 1,021,100 1,022,100
Our exam ple assumes that the investor wants to totally hedge For exam ple, suppose that it is currently January of Year 1 and
the risks in a portfolio with a beta of 1.25 (i.e., the hedger wants a company knows that it will have to sell 5,000 ounces of gold
to reduce the portfolio beta to zero). Stock index futures can in 18 months. However, it finds that only the contracts with
also be used to modify beta. sufficient liquidity are those with maturities of seven months
or fewer. The C M E Group's gold futures contracts trade with
Suppose that the current beta is [3 and the desired beta is [3**.
maturities in the months of February, April, Ju n e, August,
When (3 > /3*, the number of futures contracts the investor
October, and Decem ber. If one contract is on 100 ounces of
should short is
gold, a possible hedging strategy is as follows.
VA
03 - j3*)
VF January, Year 1: Sell 50 futures contracts maturing in
If the hedger we have just considered wants to reduce beta from August of Year 1.
1.25 to 0.5, for exam ple, then 6 (= 1.25 — 0.5 X [1,000,000/ Ju ly, Year 1: Close out the futures maturing in August
(50 X 2,500)]) contracts should be shorted. of Year 1 and sell 50 futures contracts
maturing in February of Year 2.
When (3 < (3*, a long position in contracts is required:
January, Year 2: Close out the futures maturing in February
VA
03* - jB) of Year 2 and sell 50 futures contracts
VF maturing in August of Year 2.
For exam ple, a long position in six contracts would increase
Ju ly, Year 2: Close out contracts maturing in August of
beta to 2.0 (because 6 = (2 — 1.25) X [1,000,000/(50 X 2,500)]). Year 2.
Suppose the following.

8.6 CREATING LONG-TERM H ED GES • The August of Year 1 contracts are sold at USD 1,300 and
closed out at USD 1,250 in Ju ly of Year 1.
Som etim es hedgers are faced with a lack of liquid (i.e., actively
• The February of Year 2 contracts are sold for USD 1,270 and
traded) futures contracts for the required hedge maturities.
closed out at USD 1,210 in January of Year 2.
Because the most liquid futures contracts are those with rela
tively short maturities, a hedger can work around this issue by • The August of Year 2 contracts are sold for USD 1,230 and
following what is term ed a stack and roll strategy. This involves closed out in Ju ly of Year 2 at USD 1,240, when the spot
price is USD 1,235.
• Implementing a short-maturity futures hedge,
The gain per contract on the short futures positions in USD is
• Closing the hedge out just prior to the delivery period and
replacing it with another short-maturity futures hedge, and 100 X (1,300 - 1,250) + 100 X (1,270 - 1,210) + 100
X (1,230 - 1,240) = 10,000
• Closing the new hedge out just prior to the delivery period
and replacing it with yet another short-maturity futures so that the total gain from the short futures positions is
hedge, and so on. USD 500,000 (= 50 X 10,000). The futures hedge is worth
USD 100 (= 10,000/100) per ounce so that the price realized that will be sold in the future. If the asset underlying the futures
for the gold is USD 1,335 (= USD 1,235 + USD 100) per ounce. contract and the asset being hedged are the sam e, then the
This is a good result and probably close to the result that would number of futures contracts traded should be calculated by
have been achieved from shorting futures contracts maturing dividing the size of the exposure by the quantity of underlying
in August of Year 2 (if that had been possible). However, the assets per contract.
hedger is subject to multiple risks. This is because there is some
However, there are several reasons why many exposures are
uncertainty as to the difference between the futures prices every
left unhedged. Som etim es it may sim ply not be in sharehold
time an old contract is closed out and a new one is entered.
ers' best interests to hedge a particular risk. In other cases,
In practice, com panies usually have exposure to the price of an the m arket price of a product may reflect the cost of its inputs,
asset every month, rather than in just one future month. The and therefore the m anufacturer of the product has little e xp o
hedger must then enter into enough short-maturity contracts to sure to the cost of the inputs. Finally, hedging using futures
hedge each future maturity month and roll them forward in the may be unattractive to som e treasurers because of the pos
way we have described (thus the term stack and roll). sibility that the hedge results in a loss. H edging, it should be
rem em bered, is about making an outcom e more certain— not
about increasing profits.
8.7 CASH FLOW CONSIDERATIONS
Basis risk arises from the difference between the spot price of
Because futures are settled daily, there is a mismatch between the the hedged asset and the futures price for the contract used for
cash flows from a futures contract used for hedging and the cash hedging at the tim e the hedge is closed out. This difference can
flows from the exposure being hedged. This difference can be either improve or worsen the position of a hedger.
especially important in the case of long-term hedges (such as the When there are differences between the asset underlying the
ones just considered). A company should therefore ensure that the futures contract and the asset whose price is being hedged, an
losses on its futures contracts can be financed without difficulty until optimal hedge ratio with minimum variance can be calculated.
the corresponding gains are made on the position being hedged. This ratio depends on the correlation between the change in
In the early 1990s, Germ an company M etallgesellschaft sold a the futures price and the change in the price of the asset being
huge volume of 5- to 10-year heating oil and gasoline fixed-price hedged, as well as on the standard deviation of each. A djust
contracts to its customers. It hedged its exposure by rolling ments to the minimum variance hedge ratio can be made to
over long positions in short-maturity futures contracts in the way allow for daily settlem ent and to express results as percentages
described in the previous section. The price of heating oil and (i.e., returns) instead of in absolute term s.
gasoline then began to fall, so the company expected to eventu Hedging using stock index futures is popular. Stock index
ally benefit from its fixed-price contracts. However, the im medi futures are a way of reducing or increasing an investor's exp o
ate losses on the futures contracts led to huge cash outflows that sure to the market for a period of tim e. These futures are likely
could not be financed. As a result, the fixed-price contracts were to be attractive to an investor who has no views on the future
abandoned at a cost to the company of over USD 1 billion. direction of the market but believes that it is possible to pick
stocks that will outperform the market.
SUMMARY Stack and roll is a procedure where sequences of short-maturity

futures contracts are used to hedge relatively long maturity
A long futures position can be used to hedge the price of an exposures. Under this approach, each contract is periodically
asset that will be purchased in the future, whereas a short closed out and replaced with the next one for as long as the
futures position can be used to hedge the price of an asset hedge is needed.
Chapter 8 Using Futures for Hedging 105

QUESTIONS

8.1 When should (a) a short hedge and (b) a long hedge be used? optimal hedge ratio is calculated from actual, rather than
8.2 Is hedging always profitable? Explain. proportional, changes in the asset price and the futures
price.
8.3 W hat is basis risk?
8.7 Explain why hedging using futures is a series of one-day
8.4 Consider the situation where the cost of an asset to be hedges.
purchased in the future is being hedged. W hat is the
8.8 G ive two exam ples of situations where an investor might
impact of a positive basis when the hedge is closed out?
want to use stock index futures.
8.5 W hat should the hedge ratio be when the asset price and
8.9 Explain what the beta (/3) of a portfolio measures.
the futures price have of a correlation of 0?
8.10 How could a hedging strategy using futures lead to cash
8.6 How is the number of contracts that should be used cal
flow problem s?
culated from the optimal hedge ratio? Assum e that the
Practice Questions
8.11 Explain why hedging is som etim es in the best interests of V,
N* = h
shareholders and som etim es it is not. Vf
Explain the meaning of QA, Q F VA and VF. How are h* and
8.12 It is now February. A company knows that in May it will have
f defined?
to sell 10,000 barrels of crude oil. It uses the CM E Group June
futures contract for hedging. Each contract is on 1,000 barrels 8.18 A company has a portfolio of stocks worth 1 million dol
of light sweet crude. What position should it take? What are lars with a beta of 1.5. An index futures price is currently at
the price risks that it is exposed to after taking the position? 3,000, and each contract is for delivery of 50 times the index.
How many contracts are necessary to hedge the market risk
8.13 20 futures contracts are used to hedge an exposure to
of the portfolio? Should long or short contracts be used?
the price of soybeans. Each futures contract is on 5,000
bushels. A t the time the hedge is closed out, the basis is 8.19 In Question 8.18, how can beta be reduced to 0.9? How
20 cents per bushel. W hat is the effect of the basis on the can it be increased to 1.8?
hedger if (a) the purchase of soybeans is being hedged 8 .2 0 On January 15 of Year 1, a company decides to hedge the
and (b) the sale of soybeans is being hedged? purchase of 100,000 bushels of corn on February 15 of
8.14 The standard deviation of quarterly (three-month) changes Year 2. The following table gives futures prices (cents per
in the price of a commodity is 80 cents, and the standard bushel) of three selected contracts on four different dates.
deviation of quarterly changes in the futures price of a Explain how the company can use the contracts to create
related com m odity is 90 cents. The correlation between the required hedge. W hat is the net (after hedging) price
the two changes is 0.81. W hat is the optimal hedge ratio paid for the corn as a function of the spot price on Febru
for a three-month hedge? How should it be interpreted? ary 15 of Year 2? Each corn contract is on 5,000 bushels.
8.15 In Question 8.14 the amount of the commodity being

January 15, April 15, August 15, February 15,
hedged is 200,000 units, and one futures contract is on
Year 1 Year 1 Year 1 Year 2
5,000 units of the commodity. How many contracts should
be used in hedging? (Round to the nearest whole number.) May, Year 1 300 320
Futures
8.16 "In theory tailing the hedge involves adjusting the number Price
of contracts used with the passage of tim e." Explain this
Sep. Year 1 330 320
statement.
Futures
8.17 Two alternative formulas for the hedge ratio are Price
March, Year 2 325 300

Futures Price
ANSW ERS
8.1 A short hedge should be used when an asset will be sold 8.12 The company should short 10 (= 10,000/1,000) contracts.
in the future. A long hedge should be used when an asset It is exposed to basis risk. There are two com ponents to
will be purchased in the future. this: the excess of the spot price of light sw eet crude over
8.2 Hedging is designed to make outcomes less variable. the futures price when the hedge is closed out in May and
the difference between the spot price of light sweet crude
Som etim es hedging leads to a profit relative to the
no-hedging situation. Som etim es hedging leads to a loss and the crude oil that the company is selling.
relative to the no-hedging situation. 8.13 The basis increases the net price after hedging by
8.3 The basis is the spot price minus the futures price. Basis 20 X 5,000 X USD 0.20 or USD 20,000. In (a) this is an
extra cost to the hedger. In (b) it is an extra amount
risk is the risk associated with the uncertainty about what
received from the sale of soybeans.
the basis will be when a hedge is closed out.
8.4 The cost of the asset is the initial futures price plus the 8.14 The optimal hedge ratio is
basis. A positive basis therefore worsens the hedger's 0.80

0-81 X — — = 0.72
situation. 0.90
This the proportion of the exposure that should be
8.5 The hedge ratio should be zero. This is common sense
hedged.
and is what is given by the Equation (8.1) when we substi
tute p = 0. 8.15 The number of contracts is
8.6 The number of contracts that should be used is h*Q A/Q F, 200,000
0.72 X 28.8
where h* is the optimal hedge ratio, QA is the number of 5,000
units of the asset being hedged, and Q F is the number of or 29, when rounded to the nearest whole number.
units of the asset underlying a futures contract.
8.16 Because futures contracts are settled daily, a hedge
8.7 Because futures contracts are settled daily a hedge using using futures contracts is actually a series of one-day
futures contracts can be regarded as a series of one-day hedges. In theory, the hedge ratio changes because the
hedges. Note that if forward contracts are used for hedg asset price and the futures price change, and because a
ing, daily settlem ent is not an issue. discount factor is necessary to account for the tim e dif
8.8 Possible exam ples: an investor wants to be out of the ference betw een each one-day hedge and the end of
market for a period; an investor wants to change the beta the life of the hedge. In practice, it is often the case that
of his or her portfolio; an investor specializes in picking the hedge ratio is calculated throughout the life of the
stocks but does not want an exposure to the return on the hedge using the initial values of the asset price and the
market portfolio. futures price, and the effects of the discount factors are
ignored.
8.9 The beta of a portfolio measures sensitivity of the return
from a portfolio to the return from the market. 8.17 Q a is the quantity of the assets to be hedged, is the
value of the assets to be hedged, Q F is the quantity of
8.10 Futures are settled daily. Losses on a futures hedge should
the assets underlying one futures contract, and VF is the
eventually be matched by gains on the price of the asset
value of the assets underlying one futures contract. The
being hedged, but the timing difference can give rise to
hedge ratio, h*, is calculated from the standard devia
cash flow problems.
tion of actual changes in the asset price and the futures
/V
8.11 A company's m anagem ent knows more about its risks price during the life of the hedge and the hedge ratio, h,
than the shareholders and is therefore in the best position is calculated from the standard deviation of percentage
to hedge these risks. Also, the size of a futures contract changes (returns) in the asset price and the futures price
may be too large to be useful to a single shareholder over one day.
who wants to hedge a particular risk faced by a company.
8.18 The number of contracts that should be shorted is
However, diversification is one way by which risks can be
reduced, and it is easier for shareholders to diversify than 1,000,000
it is for com panies to do so. 50 x 3,000

8.19 To reduce beta to 0.9 the number of contracts that should 20 May contracts on April 15 of Year 1. It should short
be shorted is 20 Septem ber contracts on April 15 of Year 1 and close
1, 000,000 them out by buying 20 Septem ber contracts on August

(1.5 - 0.9) X 15 of Year 1. It should short 20 March contracts on August
50 x 3,000
15 of Year 1 and close them out on February 15 of Year 2.
To increase beta to 1.8 the number long contracts
The gain on the short positions in cents per bushel is
required is
(300 - 320) + (330 - 320) + (325 - 300) = 15
1, 000,000
(1.8 - 1.5) X
50 x 3,000 The price paid is therefore S — 15 cents per bushel, where
8.20 The company should short 20 May contracts on S is the spot price on February 15 of Year 2. In total, the
January 15 of Year 1 and close them out by buying cost in USD is 1,000(S-15).
Foreign Exchange
Markets
Learning Objectives
Explain and describe the mechanics of spot quotes, forward Calculate and explain the effect of an appreciation/depre-
quotes, and futures quotes in the foreign exchange mar ciation of a currency relative to a foreign currency.
kets, and distinguish between bid and ask exchange rates.
Explain the purchasing power parity theorem and use this
Calculate bid-ask spread and explain why the bid-ask theorem to calculate the appreciation or depreciation of a
spread for spot quotes may be different from the bid-ask foreign currency.
spread for forward quotes.
Describe the relationship between nominal and real inter
Com pare outright (forward) and swap transactions. est rates.
Define, com pare, and contrast transaction risk, translation Describe how a non-arbitrage assumption in the foreign
risk, and econom ic risk. exchange markets leads to the interest rate parity theo
rem, and use this theorem to calculate forward foreign
Describe exam ples of transaction, translation, and eco exchange rates.
nomic risks, and explain how to hedge these risks.
Distinguish between covered and uncovered interest rate
Describe the rationale for multi-currency hedging using parity conditions.
options.
Identify and explain the factors that determ ine exchange

rates.
109
6000 Exchange rates can have a large effect on reported profits for
Billions of Dollars
firms that operate in multiple countries. Specifically, these firms
5000-
are subject to foreign exchange gains and losses from both
4000 -
operating cash flows in foreign currencies as well as from the
3000 - translation of asset and liability values in those currencies.
2000 -
1000 -
9.1 Q U O T E S
Year
0 --- I I I I I
1995 2000 2005 2010 2015 2020 When an exchange rate is quoted, there is a base currency and
Fiqure 9.1 Growth of foreign exchange trading a quote currency. Currency pairs are typically indicated as
through time. X X X Y Y Y or X X X /Y Y Y (with X X X as the base currency and Y Y Y as
S o u rce: BIS Triennial Central Bank Survey, Septem ber 2016. the quote currency). The exchange rate shows how much of the
quote currency is needed to buy one unit of the base currency.
The foreign exchange market (also referred to as the Forex, FX, For exam ple, a EURUSD quote of 1.2345 indicates that
or currency market) is the market where participants exchange 1.2345 U.S. dollars are needed to buy one euro. A U SD SEK
one currency for another. As discussed in Chapter 4, we can quote of 8.7654 would indicate that 8.7654 Swedish kronor are
distinguish between spot trades (where there is an agreem ent needed to buy one U.S. dollar. The three-letter abbreviations for
for the im m ediate or almost immediate exchange of currencies) some traded currencies are shown in Table 9 .1 .3
and forward trades (where there is an agreem ent to exchange
The most common exchange rate quotes are between USD and
currencies at a future tim e).1 The Forex market attracts both
another currency. Other quotes (e.g., between G BP and EUR) are
hedgers and speculators.
known as cross-currency quotes. Currency traders have conven
In term s of notional trading volum e, the foreign exchange mar tions about which currency is the base currency when exchange
ket is by far the largest market in the world. The 2016 Triennial rates are quoted. In the case of the exchange rate between the
Central Bank Survey of the Foreign Exchange and O T C D eriva U.S. dollar and the British pound, for example, the U.S. dollar is the
tives Markets Activity by the Bank for International Settlem ents quote currency. This is also the case when the U.S. dollar is quoted
(BIS) shows that trading in foreign exchange markets averaged with the euro, the Australian dollar, and the New Zealand dollar. In
USD 5.09 trillion per day in April 2016.2 Figure 9.1 shows that most other cases, however, the U.S. dollar is the base currency and
the volume of trading has grown quite rapidly since the BIS the other currency is the quote currency. The quote for the
started producing statistics in 1998. In 2016, 88% of the trading Canadian dollar, for example, is USDCAD, and it indicates the
was between the U.S. dollar (USD) and another currency. The number of Canadian dollars that are equivalent to one U.S. dollar.4
seven most popular currency pairs (listed by their global Forex
Spot exchange rates are typically quoted with four decimal
market shares) were
places. The bid-ask spread faced by corporations when they
USD and the euro (23.0%), trade large amounts of a currency is quite small. On Ju ly 6,
2018, for exam ple, EURUSD was quoted as bid 1.1744 and ask
USD and the Jap anese yen (17.7%),
1.1746. (For the small currency exchanges necessary when trav
USD and the British pound (9.2%), eling, however, bid-ask spreads are much larger.)
USD and the Australian dollar (5.2%), Forward exchange rates are quoted with the same base
USD and the Canadian dollar (4.3%), currency as spot exchange rates. They are usually shown as
points that are multiplied by 1/10,000 and then added to the
USD and the Chinese yuan (3.8% ), and
spot quote. Table 9.2 shows the points quoted for EURUSD on
USD and the Swiss franc (3.5%). Ju ly 6, 2018.
1 The standard settlem ent time for foreign exchange transactions is two
days. This is referred to as T + 2. A spot trade is therefore actually a two- 3 The euro (EUR) is the official currency of the European Union. It is used
day forward trade. A few currency exchanges, such as exchanges between by 19 of the 28 member states: Austria, Belgium, Cyprus, Estonia, Finland,
the U.S. dollar and the Canadian dollar, settle in one day (i.e., are T + 1). France, Germ any, G reece, Ireland, Italy, Latvia, Lithuania, Luxembourg,
Malta, Netherlands, Portugal, Slovakia, Slovenia, and Spain.
2 Triennial Central Bank Survey of foreign exchange and O T C derivatives
markets in 2016. (2016, D ecem ber 11). Retrieved from https://www.bis 4 News reports frequently quote the exchange rate the other way
.org/publ/rpfx16.htm around as C A D U SD .
Table 9.1 Currency Abbreviations Table 9.2 EURUSD Forward Rates on July 6, 2018;
the Spot Rate is Bid 1.1744, Ask 1.1746
Country/Currency Abbreviation
Maturity Bid Ask
Argentina Peso ARS
Australian Dollar AUD 1 W eek 5.74 5.90
Brazil Real BRL 2 W eeks 11.67 11.75
Britain Pound GBP 3 W eeks 17.55 17.65
Canadian Dollar CAD 1 Month 26.10 27.20
China Yuan/Renm inbi5 CNY 2 Months 52.87 53.87
Colum bia Peso COP 3 Months 80.87 82.07
Czech Koruna C ZK 4 Months 112.58 113.98
Denmark Krone D KK 5 Months 138.06 139.46
Egypt Pound EG P 6 Months 172.70 174.70
Euro EUR 7 Months 203.89 206.39
Hong Kong Dollar HKD 8 Months 231.79 234.29
Iceland Krona ISK 9 Months 263.36 265.86
India Rupee INR 10 Months 294.45 296.95

Iran Rial IRR 11 Months 326.60 329.10
Iraq Dinar IQD 1 Year 359.50 362.00
Israel New Shekel ILS 15 Months 460.01 461.77
Japan Yen JP Y 21 Months 662.53 665.83
Malaysia Ringgit MYR 2 Years 754.50 759.45
M exico Peso MXN 3 Years 1,119.70 1,135.70
New Zealand Dollar NZD 4 Years 1,452.40 1,472.40
Norway Krona NOK 5 Years 1,754.80 1,778.80
Pakistan Rupee PKR 6 Years 2,031.00 2,063.00
Philippines Peso PHP 7 Years 2,284.00 2,324.00
Poland Zloty PLN 8 Years 2,516.00 2,556.00
Russia Ruble RUB 9 Years 2,729.00 2,768.00
Saudi Arabia Riyal SAR
10 Years 2,929.00 2,969.00
Singapore Dollar SG D
12 Years 2,731.00 2,806.00
South Africa Rand ZAR
15 Years 3,139.00 3,239.00
South Korea Won KRW
20 Years 3,834.00 3,984.00
Sweden Krona S EK
30 Years 5,259.40 5,277.40
Switzerland Franc CHF
Taiwan Dollar TW D
Thailand Baht TH B As an exam ple, consider the EURUSD three-month forward

quote with bid 80.87 and ask 82.07. Because the spot rate was
USA Dollar USD
bid 1.1744 and ask 1.1746, this means that the three-month
forward bid quote is
5 The renminbi is the name of the official currency of China. The yuan is
the unit of account. 1.1744 + 0.008087 = 1.182487
Chapter 9 Foreign Exchange Markets ■ 111

M eanwhile, the three-month forward ask quote is Table 9.3 Forward Quotes for USDCAD on July 6,
1.1746 + 0.008207 = 1.182807
2018. Spot Exchange Rate is Bid 1.3082, Ask 1.3083
Maturity Bid Ask
The bid-ask spread for the points is 1.20. This increases the
bid-ask spread for the three-month forwards by 0.00012 1 W eek -2.21 -1.81
relative to the bid-ask spread for spot trades and makes it
2 W eeks - 3 .9 8 -3 .4 8
so that the bid-ask spread for the forward quote is 0.00032
3 W eeks -5 .8 5 -5 .2 5
(= 0.0002 + 0.00012). The bid-ask spread increases as the
maturity of the forward contract increases. 1 Month - 8 .2 9 -7 .6 7
For exam ple, the 20-year forward rate is bid: 2 Months -1 5 .2 0 -1 4 .2 9
1.1744 + 0.3834 = 1.5578 3 Months -2 2 .1 0 -20.91
4 Months -28.91 -2 7 .4 2
and ask
5 Months -3 5 .9 0 -3 3 .9 5
1.1746 + 0.3984 = 1.5730
6 Months -4 3 .6 0 -4 1 .0 8
for a bid-ask spread of 0.0152.
7 Months -5 3 .8 4 -4 9 .8 4
The quotes indicate that EUR, when purchased with USD, is
more expensive in the forward market than in the spot market. 8 Months -5 7 .8 8 -5 5 .0 8
We will discuss the reason for this later in this chapter in the 9 Months -6 3 .0 0 -6 0 .0 0
context of covered interest parity. 10 Months -7 1 .2 7 -6 8 .0 7
To provide another exam ple, Table 9.3 shows U SD CA D forward 11 Months -7 6 .0 0 -7 2 .4 0
rates on Ju ly 6, 2018. The spot exchange rate is bid 1.3082, ask
1 Year -8 2 .1 5 -7 8 .1 5
1.3083. In this case, the points are negative so that the forward
exchange rate is less than the spot exchange rate. Note that the 15 Months -1 0 4 .0 0 -9 4 .0 0
magnitude of negative ask points is always less than the magni 21 Months -1 4 6 .0 6 -1 3 1 .3 8
tude of negative bid points. This is consistent with the bid-ask
2 Years -1 6 2 .7 0 -1 4 2 .7 0
spread for forward quotes being greater than the bid-ask spread
3 Years -2 1 4 .0 0 -1 8 3 .0 0
for spot quotes. The ten-year forward bid quote is
4 Years -2 8 5 .7 0 -2 0 5 .7 0
1.3082 - 0.0521 = 1.2561
5 Years -2 9 0 .1 0 -2 7 0 .1 0
M eanwhile, the ten-year forward ask quote is
6 Years -3 8 5 .0 0 -2 6 5 .0 0
1.3083 - 0.0241 = 1.2842
7 Years -4 4 1 .0 0 -2 4 1 .0 0
The bid-ask spread is therefore 0.0281. However, note that the
8 Years -4 9 8 .0 0 -2 1 8 .0 0
bid-ask spread for ten-year forwards on EURUSD from Table 9.2
is a much lower 0.0042. O ne reason for the difference is that 9 Years -5 2 1 .0 0 -2 4 1 .0 0
long-dated forward contracts on EURUSD are more actively 10 Years -5 2 1 .0 0 -2 4 1 .0 0
traded than those on U SD C A D .
For exam ple, a U .S. com pany can fund its European o p era
Outrights and Swaps
tions by borrowing in USD and buying 1 million EUR today
A forward foreign exchange transaction, where two parties while at the sam e tim e agreeing to sell 1 million EUR for USD
agree on an exchange at some future date, is term ed an o u t in one m onth. This has the effect of funding the European
right transaction or a forw ard outright transaction. It can be operation in the dom estic currency. W e see from Table 9.2
contrasted with an FX sw ap transaction, where currency is that the bid quote for one-month forw ard EUR is 26.1 points.
exchanged on two different dates. Typically, an FX swap involves This is the am ount by which EUR is more valuable in the for
a foreign currency being bought (sold) in the spot market and ward m arket. In this case, the points reduce the net funding
then sold (bought) in the forward market. An FX swap is a way cost in USD because more USD are going to be received for
of funding an asset denom inated in a foreign currency by paying EUR in one month com pared to the am ount that could have
interest in the dom estic currency. been received today.
Table 9.4 Daily Volume of Different Types of Forex This section exam ines three categories of risk:
Trading in 2016 1. Transaction risk,
Daily Volume (Billions 2 . Translation risk, and
Type of Transaction of USD)
3 . Econom ic risk.
Spot 1,654
O utright Forwards 700 Transaction Risk

FX Swaps 2,383 Transaction risk is the risk related to receivables and payables. For
Currency Swaps 96 example, consider a British company that imports goods from
O ther Products (Incl. Options) 254 South Africa and pays for the goods in South African rand. It is
therefore exposed to G BPZAR (i.e., the number of rand per British
Total 5,087
pound) risk. If ZAR strengthens relative to GBP, the company will
S o u rce: BIS Triennial Central Bank Survey, Septem ber 2016. find that its profits suffer when it must buy ZAR to pay its suppliers.
Suppose further that the British company sells goods in Portugal

We mentioned earlier that Forex trading was estim ated to
and prices its goods in euros. In this case, it is exposed to EURGBP
be worth USD 5.09 trillion per day in 2016. Table 9.4 shows a
risk. If EUR weakens relative to GBP, the company will find that its
breakdown of this trading between spot trades, forward trades,
profits suffer when it exchanges its EUR revenues to GBP.
FX swaps, currency swaps, and other products. W hile an FX
swap involves the exchange of currency on two different dates Transaction risk can be hedged with outright forward transac
(as has been described), a currency swap (also known as a cross tions. For the company in the previous exam ple, buying ZAR
currency swap) involves the exchange of principal and a stream forward would lock in the exchange rate paid to South African
of interest payments in one currency for principal and a stream suppliers, while selling EUR forward would lock in the exchange
of interest payments in another currency. Currency swaps will be rate applicable to EUR revenues.
discussed in detail with exam ples in Chapter 20.
An FX swap is useful when a company owns foreign currency
that will be used for purchases at a future time and yet wants to
Futures Quotes earn interest in its dom estic currency. The swap would enable
the company to sell the foreign currency in exchange for its
Forex futures trade actively on exchanges throughout the world.
dom estic currency in the spot market and buy it back at a future
The C M E Group in the U.S. trades many different futures con
time in the forward market.
tracts on exchange rates between the U.S. dollar and other
currencies. These are always quoted with USD as the base cur
rency. This is because (from the perspective of the exchange) Translation Risk
a foreign currency is treated like any other asset and is valued
Translation risk arises from assets and liabilities denom inated in
in U.S. dollars. For exam ple, a six-month forward quote for the
a foreign currency. These must be valued in a firm's dom estic
U SD CA D of 1.3000 corresponds to a six-month futures quote of
currency when financial statem ents are produced. This can lead
0.7692 (= 1/1.3000) USD per C A D .
to foreign exchange gains or losses.
Popular contracts traded by the C M E Group are on 100,000
For exam ple, suppose that a U.S. company has a manufacturing
A U D ; 62,500 G B P; 100,000 C A D ; 125,0000 EUR; 12.5 million JPY;
facility in the U.K. A t the end of Year 1, the facility is valued at
and 125,000 CHF. The maturity months available on a given date
G B P 10 million and the G BPU SD exchange rate is 1.3500. A t the
include the following three months along with March, Ju n e, Sep
end of Year 2, the value of the facility in G B P has not changed
tem ber, and D ecem ber for the next 20 months. The C M E Group
and it is still valued at G B P 10 million. However, the G BPU SD
also trades contracts on several cross rates.
exchange rate is now 1.25. The company will record a foreign
exchange loss (in USD) of:
9.2 ESTIM ATIN G F X RISK
(1.3500 - 1.2500) X 10,000,000 = 1,000,000
Firms need to quantify their exposures to exchange rates at Borrowings in a foreign currency can also lead to foreign exchange
different times in the future. Once this is done, they must then gains and losses. To see how this is the case, suppose that a U.S.
decide whether their exposures are acceptable or whether some company has a loan of 20 million euros that will be paid back in five
hedging is necessary. years. Interest is paid in euros, and thus the firm is exposed to

transaction risk. However, the loan principal to be paid back also Consider again the U.S. company with a G B P 10 million
gives rise to translation risk, which can be much greater.6 manufacturing facility in the U.K. If the translation risk is consid
ered unacceptable, the facility can be financed by G BP 10 million
Suppose that the EURUSD exchange rate at the end of Year 1 is
of borrowings. There will then be no net translation gain or loss.9
1.2000 and that at the end of Year 2 it is 1.1500. Assuming the
loan is valued at par, loan will be valued in USD at the end of
Year 1 as: Economic Risk
20.000. 000 X 1.2000 = 24,000,000
Econom ic risk is the risk that a company's future cash flows will
Assuming the loan is valued at par, value of the loan at the end be affected by exchange rate movements. For exam ple, a U.S.
of Year 2 is firm that sells software in Brazil and denom inates the price of
20.000. 000 X 1.1500 = 23,000,000 the software in USD has no transaction risk. However, the firm
does have econom ic risk. If the real (BRL) declines in value rela
The company has a foreign exchange gain of USD 1 million
tive to the USD, the company's custom ers in Brazil will find its
because the euro has w eakened. If the euro had strengthened
software more expensive. As a result, either the demand for the
during the year, the company would have incurred a foreign
software will decrease or the firm will find it necessary to reduce
exchange loss.
the USD price of the software when it is sold in Brazil.
Translation risk is fundam entally different from transaction risk.
Som etim es exchange rate movements can affect a firm's
W hereas transaction risk directly affects a company's cash flows,
com petitive position in its dom estic market. Consider a U .K. firm
translation risk does not. However, it can have a big effect on its
with no production or sales overseas. Exchange rate movements
reported earnings.
might make it more profitable for a foreign com petitor to
It is som etim es recom m ended that translation risk be netted increase its activities in the U .K. in a way that adversely affects
against transaction risk, but this is not appropriate. As we have the firm.
described, forward contracts are very useful for hedging transac
Econom ic risk is more difficult to quantify than transaction or
tion risk. Transaction risk exposures should be estim ated month
translation risk, but possible exchange rate movements should
by month, and each month's exposure can be hedged sepa
be considered when key strategic decisions are being m ade. For
rately. However, it only makes sense to hedge translation exp o
exam ple, foreign exchange considerations might play a role in a
sure on one future date. For exam ple, it would be over-hedging
decision to move production overseas.
to hedge the FX exposure to the value of the assets in one and
in two years because the price increase (or decrease) over the
first year is then considered tw ice.
9.3 M U LTI-CU RREN CY H ED G IN G
Hedging translation risk with forward contracts on a reporting
U SIN G O P TIO N S
date makes accounting profits less volatile on that reporting
d ate.7 However, it is questionable whether this is a good idea Multinational com panies have exposures to many different
unless there is a plan to sell foreign currency assets or retire currencies. These exposures can lower FX risk because
foreign currency liabilities at a particular tim e in the future. This exchange rate movements across different currencies are not
is because hedging replaces accounting risk with cash flow risk perfectly co rrelated .10 In general, the volatility for an investment
(because forward contracts do affect future cash flows). W hile portfolio with multiple stocks is less than that of one with a
translation risk is reduced, the transaction risk relating to the single investment. In the same way, volatility arising from
cash flows from the forward contracts is increased. exposures to many different currencies is usually less than that
A better way of avoiding translation risk is to finance the assets arising from an exposure to a single currency.
in a country with borrowings in that country.8 In that case, gains As mentioned in a previous chapter, treasurers often prefer
(losses) on assets are offset by losses (gains) on liabilities. options to forward contracts when hedging. This is because
6 The translation risk becom es a transaction risk when the loan has to be
9 The interest on the borrowings would give rise to transaction risk,
repaid.
which if desired can be hedged separately with forward contracts.
7 It may also hedge translation risk on earlier reporting dates because n
A number of factors influence exchange rate m ovem ents, but even
1
the forward contracts will be marked to m arket on those dates.

after these factors have been taken into account, there is still a huge
8 W hen a debt instrum ent in a foreign currency m atures, it can be amount of uncertainty about the relative values of different currencies in
replaced with a new debt instrum ent denom inated in the same currency. the future.
options provide downside protection against adverse exchange There are some equilibrating forces at work here. As Country A
rate movements while still allowing a firm to benefit from increases its exports to Country B, its currency strengthens. This
favorable movements. One FX hedging strategy is to buy options causes its exports to becom e more expensive for custom ers in
on individual currencies to cover each possible adverse exchange Country B. As a result, there is lower demand for those exported
rate movement. A less expensive alternative, however, is for a firm goods in Country B. Similarly, Country A imports increase as its
to identify the portfolio of currencies to which it is exposed and currency weakens. As a result, goods imported from Country B
buy an option on that portfolio in the over-the-counter market. becom e more expensive, and this in turn reduces the demand
for goods imported from Country B.
For exam ple, a corporation could buy an option on a portfolio
consisting of: An illustration of the im portance of trade flows is provided by
the U SD CA D exchange rate. Because Canada is an oil export
• A long position in 100,000 units of currency A,
ing nation, the value of the Canadian dollar is influenced by the
• A long position in 200,000 units of currency B, and
price of oil. For exam ple, the Canadian dollar was worth more
• A short position in 75,000 units of currency C. than the U.S. dollar from 2011-2014, when the price of crude oil
This type of option is known as a b a sket option. was high. When the price of oil declined, so did the Canadian
dollar.
Multinational firms usually have exposures to an exchange rate
in every month of the year. One way to limit this exposure is to
trade options with monthly maturities. A less expensive alterna
Inflation
tive is to trade options on the average exchange rate during the If U SD CA D is 1.2500, we would expect the C A D price of goods
year. These options are known as Asian o p tio n s. 11* in Canada to be 25% higher than the USD price of goods in
the U.S. If this is not the case, there is a theoretical arbitrage
opportunity.
9.4 D ETER M IN A TIO N O F E X C H A N G E
For exam ple, suppose that a product costs USD 100 and
RATES
C A D 130. An arbitrageur can buy the product in the U.S. and
Exchange rates are determ ined by many interrelated factors. sell it in Canada for a profit of C A D 5 per unit. Similarly, if the
product costs USD 100 and C A D 120, the arbitrageur can
W hile this section describes some of those factors, note that
future exchange rates cannot be predicted with any precision. buy the product in Canada and sell it in the U.S. for a similar
Exchange rates (like the prices of all financial assets) are ulti profit.
mately determ ined by supply and dem and, which are in turn These arbitrage opportunities may not exist in practice due to
influenced by many factors. several costs that the arbitrageur may incur (e.g ., transportation
costs and possibly tariffs). However, this relationship is the basis
W hat follows is a discussion of the most important econom ic
of what is known as purchasing p o w e r parity.
variables that influence exchange rates.
To see how purchasing pow er parity w orks, suppose that
Balance of Payments and Trade Flows inflation is 3% per year in the U .S. and 1% per year in
Sw itzerland. The cost of a representative basket of goods in
The balance of payments between two countries measures the U .S. m easured in USD increases at 3% per year, while the
the difference between the value of exports and the value of cost of the sam e basket of goods in Sw itzerland increases
imports. For exam ple, suppose exports from Country A to at 1% per year. Suppose further that the initial U SD C H F
Country B increase. When exporters exchange their foreign exchange rate is 1.05. If purchasing pow er parity holds, the
currency-denominated revenues for their dom estic currency, cost of a basket of goods worth 100 USD is now worth 105
it will increase the demand for Country A's currency and CHF. A fte r one year, the sam e basket of goods will be worth
strengthen it relative to Country B's currency. If imports from USD 103 (= 100 X 1.03) and C H F 106 (= 105 X 1.01). Purchas
Country A to Country B increase, however, Country A's currency ing pow er parity suggests that the exchange rate should
will weaken relative to that of Country B (because importers becom e
would have to buy Country B's currency to pay for the goods
106.05
they are importing). 1.0296
103
11 The two ideas mentioned here can be com bined so that the m ultina
This is an (approximate) 2% strengthening of C H F relative to
tional firm hedges risks with an Asian basket option. USD.

The general (approxim ate) purchasing power parity formula is In the previous exam ple, using this approxim ation yields a real
rate of interest of 1% instead of 0.97% .
Percent Strengthening of Dom estic Spot Rate = Foreign
Inflation Rate — D om estic Inflation Rate The real interest rate (and som etim es even the nominal interest
rate) can be negative. For exam ple, the nominal interest rates in
In our exam ple, the foreign (CHF) inflation rate minus the
the Swedish krone, Japanese yen, Danish krone, euro, and Swiss
dom estic (USD) inflation rate is —2%, and thus the dom estic
franc were negative in period following the 2007-2008 financial
spot rate weakens by about %. 2
crisis. Additionally, it was estim ated by Fitch Ratings that about

USD trillion of governm ent bonds worldwide offered yields
Monetary Policy
8
below zero in 2 0 1 8 .13 It is som etim es argued that interest rates
The value of a country's currency is also influenced by the cannot becom e negative because it is always possible to hold
monetary policy of its central bank. If Country A increases its cash (which has an interest rate of zero). However, storage costs
money supply by 25% while Country B keeps its money supply for large amounts of cash are not trivial, and thus negative rates
unchanged, the value of Country A's currency will tend to do not necessarily present arbitrage opportunities.
decline by 25% relative to Country B's currency (with all else
being equal). This is because 25% more of Country A's currency
9.6 C O V ER ED INTEREST PARITY
is being used to purchase the same amount of goods.
Purchasing power parity, which was discussed in Section 9.4,
9.5 REAL VERSUS NOMINAL provides results that are at best approxim ately true in the long
term . O ver short periods of tim e, there can be significant devia
INTEREST RATES tions from purchasing power parity. We now explain a result
(illustrated by Figure 9.2) relating to forward exchange rates,
Analysts distinguish between nominal interest rates and real
spot exchange rates, and interest rates that holds much more
interest rates. Nominal interest rates are usually quoted in the
precisely because it is based on arbitrage argum ents
market and indicate the return that will be earned on a currency.
. 1 4
An interest rate of 4% per year for a currency of a country indi Suppose a U.S. trader starts with 100 G B P and wants to end up
cates that 100 of that currency will grow to 104 in one year. with USD in T years. As indicated in Figure 9.2, there are two
ways to do this.
Real interest rates are adjusted for inflation. As an exam ple, con
sider a basket of goods that costs 1 0 0 at the beginning of the 1. The trader can invest the funds at the G B P risk-free rate
year. If inflation in the country is 3%, then the basket of goods (R q bp) s o that they grow to 100(1 + Rq bp)7 at tim e T. A t the
will cost 103 at the end of the year. An individual that starts with same tim e, the trader can enter into a forward contract to
1 0 0 can buy either buy one basket of the goods at the begin exchange 100(1 + R G B P ) 7 f ° r USD at tim e T. This leads to : 1 5
ning of the year or invest at 4% and buy 100(1 + Rgbp)t F

104 where F is the T-year forward G BPU SD exchange rate.
1.0097
103
2. The trader can exchange the funds im m ediately for USD
baskets at the end of the year. This shows that the investor's real
and then invest the USD funds at the USD risk-free rate. The
purchasing power has increased by only 0.97% . This is referred
to as the investor's real interest rate.
In general:
13 Carson, R„ & M ogi, C . (2018, D ecem ber 18). "The USD 7.9 Trillion
1 + R nom Pile of Negative-Yielding Debt Is Growing Fast." Retrieved from https://
w w w .bloom berg.com /new s/articles/2018-12-18/japan-yields-on-cusp-of-
1 + R: nfl
zero-as-world-can-t-quit-negative-rates
where Rrea| is the real interest rate,Rnom is the nominal interest rate, 14 It is interesting to note that there have been departures from covered
and Rinf| is the rate of inflation . 1 2 This is often approximated as: interest parity since the 2007-2008 crisis. See for exam ple C . Borio,
R. N. M cCauley, P. M cGuire, and V. Susko, "C o vered Interest Parity
= Rnom Lost: Understanding the Cross-Currency Basis," Bank for International
Settlem ents, 2016: https://w w w .bis.org/publ/qtrpdf/r_qt1609e.htm .
The authors attribute this to changes in the dem and for FX hedging and
banks not having the balance sheet capacity to engage in the type of
12 This assum es that interest rates and inflation rates are expressed with
arbitrage required.
annual com pounding. See C hapter 16 for a discussion of com pounding
frequency issues. 15 Interest rates are here expressed with annual com pounding.
If F is greater than the exchange rate given by Equation (9.1):
100 G B P at
tim e zero t
+ û s d )
F > S (9.3)
( 1
T
( 1 + Rg bp)
an arbitrageur can
100 (1+RGBP)T 100S USD
G B P at tim e T at tim e zero • Borrow 100S USD for T years at RGBP,
• Convert the funds to 100 GBP,
• Invest the USD at RGBP for T years to obtain 100(1 + Rq bp ) 7
USD at tim e T, and

100(1 +Rgbp)tF 100S(1+RUSD)T
USD at tim e T USD at tim e T • Enter into a forward contract to convert this to 100(1 + RGBP)TF
USD at time T.
Fiqure 9.2 Two ways of converting 100 GBP to USD
at time T. S and F are the GBPUSD spot and forward From Inequality (9.3):
exchange rates, respectively. The variables RUSD and 100(1 + RGBP)TF > 100S(1 + RU5D)T
Rgbp are the risk-free interest rates in USD and GBP.
Thus, the trader has more USD than required to repay the funds
borrowed in USD.
100 G B P first becom es 100S USD, where S is the current
G BPU SD spot rate. A t tim e T, this becomes This analysis assumes that the trader can borrow and lend at the
same interest rate (i.e., that the borrowing and lending risk-free
iooso + r u s d )t
interest rates in the U.S. are the same and equal to Ru sd )- For
where Rusd is the annual USD risk-free rate. a large bank, this is close to true. If we take the bank's spread
There is no uncertainty about the amount of USD that will be between borrowing and lending rates into account, however, we
obtained at time T for either scenario. In the absence of arbi would obtain a narrow range of possible values for F (instead of
trage opportunities, they must therefore give the same result: just one value).
100(1 + RGBP)TF = 100S(1 + RU5D)T In general, for an exchange rate XXXYYY, we have
T
so that (1 + Ryyy )
F= S T (9.4)
T + ffxxx)
(1 + R )
( 1
u sd
F= S (9.1)
d + RGbp)T If the risk-free rate for currency X X X is higher than that for
Equation (9.1) shows what is commonly known as the co ve red currency YYY, X X X is w eaker in the forward market than in the
interest parity. If F is less than the exchange rate given by spot market (i.e., it takes less units of Y Y Y to buy one unit of
Equation (9.1): X X X in the forward market than it does in the spot market). If
T the risk-free rate for currency X X X is lower than that for currency
+ Ru sd )
F < S (9.2)
( 1
YYY, X X X is stronger in the forward market than in the spot mar

(1 + Rg bp)T
an arbitrageur can ket (i.e., it takes more units of Y Y Y to buy one unit of X X X in the
forward market than it does in the spot market).
• Borrow 100 G B P for T y e a rs at RGBP,
Table 9.2 shows that the USD, when exchanged for EUR, was
• Convert the funds to 100S USD,
w eaker in the forward market than in the spot market on Ju ly 6 ,
• Invest the USD at Rusd for T y e a rs to obtain 100S(1 + Ru sd ) 7
2018. For exam ple, the mid-market spot exchange rate is
USD at tim e T, and USD 1.1745 (= (1.1744 + 1.1746)/2), while the mid-market
• Enter into a forward contract to convert this to point for a 12-month forward is 360.75 (= (359.5 + 362)/2)
100S(1 + Rusd)t/ F G B P at tim e T. and therefore the mid-market forward rate is USD
1.2106 (= 1.1745 + 360.75/10000). From Equation (9.4):
From Inequality (9.2):
100S(1 + R u sd )t T R usd
1.2106 = 1.1745------^
1
100(1 + RGBP)r 1 + R eu r
F
which reduces to
Thus, the trader would have more G B P than required to repay
the funds borrowed in GBP. + Rusd
1.0307
1
1 + R eu r

This indicates that the one-year USD interest rate was about 3% as a percentage of the spot rate) is approxim ately equal to the
higher than the euro interest rates on Ju ly 6 , 2018. Indeed, this interest rate differential applied to time T.
was the case: The one-year interbank borrowing rate was about
Consider the previous exam ple where the interest rates in
2.8% in USD and -0 .2 % in EUR.
currencies X X X and Y Y Y are 3% and 5% per annum (respectively),
A similar analysis based on Table 9.3 shows that: the spot rate is 1.25, and the forward rate is 1.2621.
In this case, the forward rate would be quoted as
+ ^c a d
1.0062
1
121 points (= (1.2621 — 1.25) X 10,000) and:

1 + Rusd
121/ 10,000
This indicates that the one-year interbank USD interest rate was 0.0097
1.25
about 0.6% higher than the one-year C A D borrowing rate on
Ju ly 6 , 2018. This corresponds to 0.97% , which is approxim ately equal to the
% per year interest rate differential applied to six months.
As an application of Equation (9.2), suppose that interest rates
2
in currencies X X X and Y Y Y are 3% and 5% per annum (respec

tively) and that the X X X Y Y Y spot rate is 1.2500. The six-month
9.7 U N CO VERED INTEREST PARITY
X X X Y Y Y forward rate will be
1,05°-5 Covered interest parity concerns forward exchange rates and

1.2500-----= 1.2621
1.03° 5
can be expected to hold well because it depends on arbitrage
arguments.
Note that it takes 1.25 units of Y Y Y to buy X X X in the spot mar
ket and 1.2621 units of Y Y Y to buy X X X in the forward market. Uncovered interest parity is an argument concerned with
X X X is therefore stronger than Y Y Y in the forward market than it exchange rates them selves and is just one of the many interact
is in the spot market. ing factors that determ ine how exchange rates move. It argues
that investors should earn the same interest rate in all currencies
when expected exchange rate movements are considered.
Interpretation of Points
a / Consider currencies X and Y with risk-free rates of 2% and 6 %
From Equation (9.4), it is approxim ately true that when T < 1:
(respectively). In equilibrium, the two currencies should be
F 1 + RyyyT equally attractive. According to uncovered interest rate parity,
5 1 + RxxxT this means that the investor should expect the value of currency Y
to weaken by about 4% relative to the value of currency X.
This can be written
Arguably, there are many potential violations of uncovered inter
F_ 1+ Ry y y T _
est parity. In 2018, the interest rate in USD was much higher
S 1 + RxxxT
than the interest rate in EUR. However, the U.S. econom y was
or generally considered to be much stronger than many European
F —S 1 + RYYY____________
yyyT — 1 RxxxT econom ies. Furtherm ore, many market participants did not con
sider the interest rate differential to be indicative of a stronger
1 + R XXXT
euro in the future.
so that
If both covered and uncovered interest rate parity held, the
F —S Ry y y T — RxxxT
‘XXX
forward exchange rate would equal the expected future spot
+ Rxxx
exchange rate. We will discuss the relationship between the for-
1
or approxim ately ward/futures price of an asset and its spot price in Chapter 11.
F - S
— (r y y y ~ Rxxx)T
SUMMARY
This provides an interpretation of the forward rate points. The
term F - S is the points divided by 10,000 and (when expressed 1
6 The foreign exchange market has developed standard ways
of quoting exchange rates. For exam ple, each currency is pre
sented as a three-letter abbreviation. When an exchange rate
16 This is exactly true if the interest rates are expressed with a com
is referred to as X X X Y Y Y or XXX/YYY, it is the number of units
pound period of T. of currency Y Y Y (the quote currency) that is equal to one unit
of currency X X X (the base currency). USD is the quote currency and liabilities when financial statements are produced. Economic
when traders quote exchange rates between USD and the Brit risk arises from longer-term changes to a company's competitive
ish pound, the euro, the Australian dollar, or the New Zealand environment arising from future exchange rate movements.
dollar. In most other cases, USD is the base currency. Forward
Large multinationals often have exposures to many different
exchange rates are usually quoted as points that are added to
exchange rates and have risk diversification advantages. Some
or subtracted from spot exchange rates.
use fairly com plex derivatives that focus on the residual risk from
Several different types of trades are carried out in the foreign multiple exchange rate exposures or average exposures over
exchange markets. A spot trade is an exchange of one currency several months.
for another that takes place immediately (or almost immediately).
Many factors are involved in the determination of exchange
An outright forward is an agreem ent to exchange currencies
rates, and there is no precise way estimate a future exchange
that will take place at a certain time in the future. An FX swap is
rate. In the long term , however, a country's balance of payments
a transaction that involves a currency exchange at one time and
with another country will affect the exchange rate between its
the opposite exchange at a later tim e. A currency swap involves
currency and that of the other country. Monetary policies and
interest and principal in one currency being exchanged for inter
inflation rates are also important, but exchange rates can be
est and principal in another currency. The FX market is large and
immune to theoretical macroeconomic forces for periods of time.
involves transactions worth several trillion dollars every day.
There is a no-arbitrage relationship between forward exchange
A company has three types of foreign exchange risk: transaction,
rates and spot exchange rates that involves interest rates. If
translation, and economic. Transaction risk relates to the risk
the interest rate for currency A is less than that for currency B,
associated with the exchange rate that will apply when
for exam ple, currency A will be stronger in the forward market
(a) a foreign currency is purchased with the domestic currency in
than in the spot market. The percentage amount by which the
the future to buy goods and services overseas, or (b) future rev
forward exchange is better than the spot exchange is approxi
enues in foreign currencies are converted into the domestic cur
mately equal to the interest rate differential.
rency. Translation risk arises from the need to value foreign assets
Chapter 9 Foreign Exchange Markets 119

QUESTIONS

9.1 Suppose that the quote between currency A and currency B 9.6 How can a multinational that uses options for hedging for
is 1.2000 and that currency B is the base currency. How eign exchange risk reduce the cost of its hedging?
many units of currency A should be exchanged for 100 units 9.7 When Country A increases its exports to Country B, what
of currency B?
tends to happen to the exchange rate between the cur
9.2 Name three European countries that do not use the euro. rencies of A and B?
9.3 A three-month forward foreign exchange rate is specified 9.8 When inflation increases faster in Country A than in Coun
as bid 38.5, ask 40.5. W hat does this mean? try B, what tends to happen to the exchange rate between
the currencies of A and B?
9.4 W hat is an FX swap?
9.9 W hat is the difference between real and nominal interest
9.5 W hat is the difference between transaction and translation
risk? rates?
9.10 Under what circum stances is a currency weaker in the for

ward market than in the spot market?
Practice Questions
9.11 The exchange rate between USD and some foreign cur 9.16 W hat is econom ic FX risk?
rencies is quoted by traders with USD as the quote cur 9.17 If the nominal interest rate is % and the rate of inflation
2
rency. Give three foreign currencies for which this is the is 3%, what is the real interest rate? How should it be
case. interpreted?
9.12 In futures markets in the U .S., is the exchange rate 9.18 Interest rates in a currency X X X increase and interest
between USD and currency X X X specified as U SD XXX or rates in currency Y Y Y stay the sam e. The exchange rate
X X X U SD ? Explain.
is expressed as XXXYYY. Do forward rates increase or
9.13 The spot rate for X X X Y Y Y is 1.4251 and the one-year decrease?
forward rate is 200 basis points. How would the spot rate
9.19 The interest rate in X X X is 1% and in Y Y Y 4%. The X X X Y Y Y
and the forward rate have been expressed if the currency spot rate is 1.3000. How would three-month forward rate
had been YY YX X X ? be quoted using points?
9.14 Suppose there is a cash need for one year in a foreign cur 9.20 Until recently, Country A and Country B had similar inter
rency. Explain how a company can borrow dom estically est rates. The central bank of Country A has just increased
and use an FX swap to fund the cash need. W hat is the interest rates. A speculator thinks this will lead to interna
difference between that and borrowing funds in the for tional investors moving funds from Country B's currency to
eign currency to fund the cash need? Country A's currency to earn the higher interest rate. This
9.15 A company calculates its translation risk for its three- will increase the demand for currency A , and as a result,
month and six-month statem ents and hedges each with a currency A will strengthen relative to currency B. W hat
forward contract. Is this a sensible strategy? Discuss. spot or forward trades should the speculator do?
ANSW ERS
9.1 A is the quote currency. The quote indicates that 120 units 9.12 It quotes as XXXU SD because the exchange rate is the
of A should be exchanged for 100 units of B. number of units of USD per unit of the foreign currency.
9.2 Am ong the possibilities are the U .K ., Sweden, Norway, 9.13 The spot rate and forward rate are 1.4251 and
Denm ark, Czechoslovakia, Iceland, Poland, and 1.4251 + 0.0200 = 1.4451. If the currency had been
Switzerland. quoted the other way around, the spot and forward
would be 1/1.4251 = 0.7017 and 1/1.4451 = 0.6920. The
9.3 0.00385 should be added to the spot bid quote to get the
forward bid exchange rate. 0.00405 should be added to forward quote would be —97 because the forward rate is
the spot ask quote to get the forward ask exchange rate. 97/10,000 less than the spot rate.
9.4 An FX swap is an agreem ent to buy (sell) a certain amount 9.14 When the company borrows dom estically, an FX swap
of a currency at one time and sell (buy) it at another later can convert the borrowing to the foreign currency at the
beginning of the loan and back to the dom estic currency
tim e.
at the end of the loan. The difference between this and
9.5 Transaction risk exposure arises from cash inflows and
borrowing in the foreign currency is that interest is paid in
outflows in a foreign currency. Translation risk exposure
the dom estic currency instead of the foreign currency.
arises from FX gains and losses when assets and liabilities
denom inated in a foreign currency are converted to the 9.15 W hether or not balance sheet risks should be hedged with
forward contracts is debatable. (Some argue that reducing
dom estic currency for the purposes of producing financial
statem ents. the volatility of earnings is desirable; others argue that
non-cash-flow risks should not be hedged.) If they are, it
9.6 The multinational has an exposure each month to a basket
only makes sense to hedge the risk at one future tim e.
of currencies. It can hedge by buying an option on the
O therwise there is over-hedging. This is a key difference
basket rather than on each currency. It can also buy an
between transaction and translation risk.
option on the average exposure that will apply across sev
eral months, rather than buying one option for its exp o 9.16 Econom ic FX risk is the risk that the company's com peti
sure in each month. tiveness in either dom estic or foreign markets may be
affected by FX movements.
9.7 Country A's currency tends to strengthen relative to Coun
9.17 The real interest rate is —1%. When an investor earns 2%,
try B's currency because im porters in Country B will find it
necessary to buy Country A's currency to pay for goods. the investor's purchasing power actually decreases by % 1
per year because of inflation.

9.8 Purchasing power parity suggests that Country A's cur
rency will weaken relative to Country B's currency in 9.18 Currency X X X becom es w eaker in the forward market.
order to equalize the cost of a basket of goods in the two Exchange rates are expressed as the number of units of
countries. Y Y Y that would be exchanged for one unit of X X X . As a

result of the increase in interest rates, it takes less units of
9.9 A nominal interest rate is the interest rate usually quoted.
Y Y Y to buy one unit of X X X in the forward market and so
It is the rate earned in units of the currency. The real inter
the forward rate decreases.
est rate is the nominal interest rate minus the inflation
rate. It allows for the fact that inflation is causing the value
9.19 The forward rate is
of the currency to decline. (1.04)0-25

1.3000 X -------- = 1.3095
9.10 Currency A relative to currency B is w eaker in the forward ( 1. 01)025
market than in the spot market if interest rates are higher The forward rate would be quoted as 95.
in currency A than in currency B.
9.20 The speculator thinks that currency A will strengthen.
9.11 Possible answers are GBP, A U D , NZD, and EUR. The However, interest rate parity indicates that it is weaker in
exchange rate for these currencies is quoted by trad the forward market. If the speculator is right, he or she will
ers as the number of units of USD per unit of the foreign make money by buying currency A with currency B in the
currency. forward market and then selling it on the delivery date.

Pricing Financial
Forwards
and Futures
Learning Objectives
Differentiate between investment and consumption assets. Distinguish between the forward price and the value of a
forward contract.
Define short-selling and calculate the net profit of a short
sale of a dividend-paying stock. Calculate the value of a forward contract on a financial
asset that does or does not provide income or yield.
Describe the differences between forward and futures
contracts and explain the relationship between forward Explain the relationship between forward and futures prices.
and spot prices.
Calculate a forward foreign exchange rate using the inter
Calculate the forward price given the underlying asset's est rate parity relationship.
spot price, and describe an arbitrage argument between
spot and forward prices. Calculate the value of a stock index futures contract and
explain the concept of index arbitrage.
123
Chapter 9 exam ined the relationship between the forward Table 10.1 Profit from a Short Sale (USD)
price and the spot price of a foreign currency using the covered
May: Shares shorted +5,000
interest parity no-arbitrage argument. This chapter uses similar
no-arbitrage arguments to estim ate forward and futures prices July: Dividend paid - 2 0 0
for financial assets (e.g ., stocks and bonds). Sept: Position covered -3 ,0 0 0
A financial asset is an asset whose value derives from a claim Total 1,800
of some sort. An investm ent asset is an asset held by market
participants for investm ent purposes. All financial assets
(and a small number of non-financial assets) are investment
Table 10.2 Profit from a Long Position (USD)
assets. (Non-investment assets are som etim es referred to as May: Shares bought -5 ,0 0 0
consum ption assets.)
July: Dividend paid + 2 0 0
This chapter considers three types of financial assets: Sept: Shares sold +3,000
1. Assets providing no income, Total -1 ,8 0 0
2 . Assets providing a known income that is a fixed amount, and
3 . Assets providing a known income that is a percentage of margin), the broker can borrow the shares without asking the
their value. investor's permission. Small investors may find it difficult to short
We will confirm that the results for the above assets are if their brokers do not have access to shares that have been
consistent with those for forward contracts on currencies bought on margin by other investors. Large investors can bor
and use exam ples to show how they can be applied to for row the financial assets they want to short from financial institu
ward contracts on bonds and stock indices. We argue that (in tions (e.g ., State Street Corporation) for a fee. The fee is usually
theory) futures prices and forward prices for contracts with quite low (less than 50 basis points per year), but it may rise if
the same maturity on the same asset should be approxim ately the asset is scarce.
equal. This means that results produced for forward prices are
As an exam ple, suppose 100 shares are shorted in May when
approxim ately true for futures prices.
the share price is USD 50. The position is then covered in Sep
tem ber when the share price is USD 30. Suppose further that a
10.1 SH O RT S E LLIN G dividend of USD 2 per share is paid on the shares in July. The
profit from the short position (ignoring brokerage fees and
Some of the no-arbitrage arguments used in this chapter involve assuming there is no borrowing fee) is USD 1,800 (calculated as
shorting (short selling) an asset. As discussed in earlier chapters, indicated in Table 10.1). Note that the cash flows for an investor
to short a futures or forward contract means to agree to sell the with a long position are the mirror image of the cash flows of
underlying asset for a certain price in the future. The short sell an investor with a short position. Thus, an investor with a long
ing addressed here is the sale of an asset that is not owned with position in 100 shares between May and Septem ber would lose
the intention of buying it back later. USD 1,800 as indicated in Table 10.2 (again, ignoring brokerage
costs).
Short selling is profitable if the asset price declines but incurs
losses if the asset price increases. For exam ple, suppose an Short selling attracts the attention of regulators from tim e to
investor contacts his or her broker to short 1 0 0 shares of tim e. During the 2007-2008 global financial crisis, several coun
Com pany X. The broker will borrow the shares from another tries banned investors from short selling financial stocks because
investor and proceed to sell them in the market in the usual way. regulators believed such trades would exacerbate the crisis.
A t a later tim e, the investor would contact the broker again to However, many financial analysts believe short selling is
cover (i.e., close-out) the short position. The broker would then an im portant part of the price discovery process.
buy 100 shares of Com pany X and replace them in the account
from which they were borrowed. If a dividend is paid while the
10.2 TH E N O IN C O M E C A S E
shares were shorted, the investor would be required to pay that
dividend, and the funds would be passed on to the account
This section considers the relationship between the spot price
from which the shares were borrowed.
and the forward prices of a financial asset that provides no
In the U .S., if another investor has bought shares of Com pany income. The asset could be a non-dividend-paying stock, a
X on margin (see Section 5.3 for a description of buying on Treasury bill, or a zero-coupon bond.
For exam ple, suppose a financial asset that will provide no Table 10.3 Trading Possibilities for an Arbitrageur
incom e is priced at USD 70 per unit and a large bank can when an Asset Provides no Income and is Priced at
borrow (or lend) at 5% per year. W e will consider two scenarios USD 70 with a One-Year Borrowing/Lending Rate of
that give rise to an arbitrage opportunity for the bank's 5% per Year
traders.
Scenario A: One-Year Scenario B: One-Year
Scenario A : The one-year forward price of the asset is Forward Price Is USD 80 Forward Price Is USD 65
USD 80.
Buy asset for USD 70 and Short asset for USD 70 and
Scenario B: The one-year forward price of the asset is enter into a short forward enter into a long forward
USD 65. contract to sell it for USD 80 contract to buy it in one year
in one year. for USD 65.
Consider Scenario A . A trader can buy the asset today for USD
Profit equals USD 6.50 Profit equals USD 8.50
70 and sell it in the forward market for USD 80. The cost of
funding the purchase of the asset for one year is
0.05 X USD 70 = USD 3.50

forward price is less than USD 73.50. If there are to be no arbi
The trader therefore makes a profit of: trage opportunities, the forward price must therefore be equal
USD 80.00 - USD 70.00 - USD 3.50 = USD 6.50 to USD 73.50.
Clearly, the forward price of USD 80 is too high in this exam ple. We have assum ed that banks can borrow and lend at the same
In fact, any forward price above USD 73.50 would lead to the rate. W hile this is not exactly true, it is close to being true.
trader making a zero-risk profit. As a result, we can expect the The relevant rate for a bank trader is som etim es referred to
actions of arbitrageurs to drive down the forward price to a as the bank's opportunity cost of capital. In practice, it is the
point where the arbitrage is no longer profitable . 1 If there are to interbank rate (i.e ., the rate at which banks borrow and lend
be no arbitrage opportunities, the forward price must be less betw een them selves). The spread betw een the borrowing rate
than or equal to USD 73.50. and lending rate in the interbank m arket is typically around
0.1% . If the borrowing and lending rates in our exam ple are
Now suppose the forward price is USD 65 (i.e., USD 5 less than
5.1% and 5.0% , respectively, then the no-arbitrage argum ent
the spot price). A trader can short the asset and enter into a
presented with Scenario A shows that the forward price must
forward contract to buy it back in one year. He or she can then
be less than:
sell the asset for USD 70 im m ediately and buy it back for USD
65 in one year under the term s of the forward contract. This 70 X 1.051 = 73.57
trade leads to an initial profit of USD 5 (= USD 70 — USD 65). The no-arbitrage argument presented in Scenario B is unchanged
Additionally, the short position generates USD 70 in cash, which and we can conclude that the forward price must lie between
can be invested at the risk-free rate for one year. This gives a USD 73.50 and USD 73.57.
profit of:
We will continue to ignore the distinction between borrowing
0.05 X USD 70 = USD 3.5 and lending rates and refer to the interest rate at which a bank
The trading strategy therefore generates a total profit of USD 8.50 can borrow or lend as the risk-free rate. There is some ambiguity
(= USD 5.00 + USD 3.50). here (as risk-free rates are often assumed to be governm ent bor
rowing rates), but for bank traders the risk-free rate corresponds
This trading strategy is profitable for any forward price below
to the interbank rate.
USD 73.50. As traders take advantage of this arbitrage opportu
nity, however, the forward price is driven up until the strategy is The no-arbitrage argum ents presented here also assum e that
no longer profitable. the asset can be shorted with no borrow ing-related fe e s. The
shorting (borrow ing) fee w ould m ake the Scenario B trading
Table 10.3 summarizes these scenarios. The first trading strategy
strateg y more e xp e n sive . H ow ever, to avoid the borrow ing
generates profits when the forward price is greater than USD
fee w e can co n sid er an investor who already owns the asset
73.50. The second trading strategy generates profits when the
(and there m ust be such investors for any financial asset).
The investor in Scenario B can then sell the asset fo r USD 70,
release the funds fo r investm ent at the risk-free rate, and
1 In practice, the existence of arbitrageurs means that arbitrage

buy it back in one year for USD 65 to produce USD 8.50
opportunities (if they exist at all) are very limited. in profit.
Chapter 10 Pricing Financial Forwards and Futures ■ 125

If F < S ( 1 -F R) , then we are in Scenario B of Table 10.3. An
BOX 10.1 JO SEP H JET T 'S LOSS arbitrageur can lock in a profit of S(1 + R)T — F by shorting the
asset (or selling the asset if it is owned) and taking a long posi
In the early 1990s, Kidder Peabody trader Joseph
Je tt engaged in rogue trading activities related to the tion in the forward contract.
arbitrage strategies mentioned above. Je tt would buy
Another way to understand this result is by noting that a trader
zero-coupon Treasury bonds (known as strips) and sell
can exchange a cash flow of —S today for a cash flow of + F at
them in the forward m arket. Due to a flaw in Kidder
Peabody's com puter system s, however, he was able to time T with 100% certainty by:
im m ediately record the difference between the forward
• Buying the asset today, and
price and the spot price as a profit.
• Entering into a long forward contract to sell it at tim e T.
Consider the exam ple in Table 10.3. If the forward
price was USD 73.50 (as it should be for no-arbitrage Because the cash flows are certain, S must be the present value
opportunities), he would buy the asset for USD 70, sell of F received at time T discounted at the risk-free rate, so that:
a one-year forward for USD 73.50, and record a profit of
USD 3.50. S = F(1 + R)- t
The USD 3.50 is (of course) the cost of financing the or

trade and should have not be recorded as a profit. In
F = S(1 + R)J
fact, the firm's systems would account for this cost and
reverse the profit once the forward contract was a settled.
Je tt therefore had to continually increase the size of his Example:
position to maintain the appearance of profit.
Consider a forward contract to sell a non-dividend-paying stock
Unaware of the fictitious nature of his profits, Kidder
in three months. The current stock price is USD 50 and the
Peabody rewarded Je tt with over USD 13 million in
bonuses between 1992 and 1994. Eventually, when the three-month risk-free rate (annually com pounded) is 4% per
nature of his trading was understood, the firm had to year. Equation (10.1) gives a forward price of:
revise its profits downward by over USD 300 million.
F = 50(1 + 0.04)025 = 50.49
10.3 TH E K N O W N IN C O M E C A S E
Generalization
We now move on to consider a forward contract on a financial asset
We will now generalize the relationship between the forward
paying a known cash income. The asset could be a bond with a
price and the spot price of a financial asset that provides no
known coupon or a stock providing a dividend known in advance . 23
income. Defining term s:

Reconsider the example in Section 10.2 where an asset has a
F: Forward price of asset, price of USD 70. Now suppose the asset will provide a cash flow
T: Tim e to maturity of the forward contract, of USD 5 in six months. We further assume that interest rates are
not the same for all maturities. In particular assume the following.
S: Spot price of asset, and
• The one-year interest rate is 5% per year annually
R: Risk-free interest rate per year for maturity T
(compounded annually). com pounded (as in our earlier exam ple).
• The six-month interest rate is 4% per year annually
As mentioned earlier, the risk-free rate R is the relevant oppor
com pounded.
tunity cost of funds. This rate is usually an interbank borrowing/
lending rate.
2 It would be more natural to express the three-month interest rate
The no-arbitrage forward price is given by: with quarterly com pounding rather than annual com pounding. We use
annual com pounding because R is defined with annual com pound
F=S(1 + R ) 7 (10.1) ing in Equation (10.1). If the rate w ere 4% per annum with quarterly
com pounding, the forward price would be
The forward price is the spot price com pounded forward at the
50 X 1.01 = 50.50
interest rate R for time T.
We will discuss com pounding frequency issues in Chapter 16.
If F > S ( 1 -f- R) , then we are in Scenario A of Table 10.3. An
3 The dividend could be known because it has been declared before the
arbitrageur can lock in a profit of F — S(1 + R)T by buying the
ex-dividend date has been reached. A lternatively, m anagem ent may
asset and shorting the forward contract. have indicated that it plans to maintain the current level of dividends.
First, consider Scenario A (where the forward price is USD 80): USD 68.352, while the second trading strategy generates profits
a trader can follow the strategy described in Section 10.2 and when the forward price is less than USD 68.352. For there to be
buy the asset for USD 70 while selling a one-year forward for no-arbitrage opportunities, the forward price must therefore be
USD 80. In this case, the trader will receive USD 5 in six months. exactly USD 68.352.
The present value of the income is
Generalization
(1.04)1/2
Define / as the present value of the incom e from the asset. The
The trader can therefore borrow USD 4.903 for six months using
rest of the notation is the same as in the previous section (i.e .,
the income from the asset (equal to USD 5) to pay off this loan
S is the spot price of the asset, F is the forward price of the
in the sixth month when the income is received. He or she can
asset, T is the tim e to m aturity of the forward contract, and R
then borrow the remaining USD 65.097 (= 70 — 4.903) at 5% for
is the interest rate for m aturity T.) The relationship betw een F
one year. A t the end of the year, the amount required to pay off
and S is
this loan is
F = (S - /)(1 + R)t (10.2)
USD 65.097 X 1.05 = USD 68.352
In the exam ple in Table 10.4:
The profit to the trader is then:
USD 80 - USD 68.352 = USD 11.648 4.903
This trading strategy is profitable for any forward price above

USD 68.352. For there to be no arbitrage, it follows that the for
ward price must be less than USD 68.352. F = (70 - 4.903) X 1.05 = 68.352
Now consider Scenario B (where the forward price is USD 65). If F > (S - Od -F R) , buying the asset and selling it in the for
As in Section 10.2, a trader can short the asset and enter into a ward market will lead to a profit.
forward contract to buy it back for USD 65. In this situation, the If F < (S — 0(1 + R)T, however, shorting the asset and buying it
trader must pay the income of USD 5 provided by the asset to in the forward market will lead to an arbitrage profit.
maintain the short position. The trader can do this by investing
For another way of deriving Equation (10.2), consider the follow
USD 4.903 from the proceeds of the short sale for six months
ing trading strategy.
at 4% per year and the remaining USD 65.097 for one year at
5% per year. A t the end of the sixth months, the first invest • Buy the asset for S at tim e zero.
ment generates USD 5 (= 4.903(1.04)1/2), which is paid as a • Sell it for F at tim e T.
dividend to the asset owner. The second investment generates
This is certain to lead to a cash outflow of S at tim e zero, cash
USD 68.352 (= 65.097 X 1.05) in one year. The trader therefore
inflows with a present value of / during the life of the forward
makes a profit of:
contract, and an inflow of F a t tim e T. Setting the present
USD 68.352 - USD 65 = USD 3.352 value of inflows equal to the present value of the outflows
This strategy is profitable for any forward price below USD 68.352. yields
Table 10.4 summarizes these scenarios. The first trading strategy F

generates profits when the forward price is greater than (1 + R)t
Table 10.4 Trading Possibilities for an Arbitrageur. The Asset Price is USD 70 and an Income of USD 5 is
Expected in Six Months. The Six-Month Interest Rate (Borrowing or Lending) is 4%, whereas the One-Year Interest
Rate (Borrowing or Lending) is 5%
Scenario A: One-Year Forward Price is USD 80 Scenario B: One-Year Forward Price is USD 65
Buy asset for USD 70 and enter into a short forward contract to Short asset for USD 70 and enter into a long forward contract to
sell it for USD 80 in one year. In six months, collect USD 5.00 of buy it in one year for USD 65. In six months, pay income on the
income. asset of USD 5.00.
Profit equals USD 11.648 Profit equals USD 3.352

This can be written as: current spot price of the asset is USD 30. The forward price
F = (S - /)(1 + R)t (USD) is
Example: 30.88
Consider a ten-month forward contract on a bond paying a

USD 4 coupon in three months and in nine months. We assume
the risk-free rate for all maturities is % per year and the cash price
6
10.5 V A LU IN G FO RW A RD C O N TR A C TS
of the bond is USD 105. The present value of the coupons in is
The value of a forw ard contract is quite different from the

+ USD 7.771
1.06 0.25 1.06 0.75 forw ard p rice. W hen a forw ard co ntract on a financial asset
is first en tered , its forw ard price is calculated in the m an
The forward price of the bond is therefore:
ner described in Sectio ns 10.2 to 10.4. However, the value
(105 - 7 .7 7 1 )0 .0 6 # = USD 102.067
of the forw ard co n tract itself is zero (or very close to zero).
If this w ere not so, one party would require a paym ent
10.4 TH E K N O W N Y IE L D C A S E from the other at the outset (akin to the prem ium paid for
options). Forw ard contracts are norm ally structured so that
Now consider the case where a financial asset provides a known there is no such paym ent. A s tim e passes, how ever, the
yield during the life of the forward contract. This means the asset price changes and the value of the forw ard contract
known income is expressed as a percentage of the price of the may becom e positive or neg ative. W hile the value of the
asset (rather than in cash). co ntract changes, the price at which the asset will be even tu
ally bought or sold continues to equal the original forw ard
We assume that the yield is Q per year with annual com pound
price.
ing .4 If the income is reinvested in the asset, the number of units
held grows at rate Q. One unit of the asset grows to (1 + Q) Suppose that we are valuing a long forward contract to buy
units of the asset in one year and to (1 + Q )r units by time T. an asset for price K. We assume this is not a new contract, but
Consider the following trading strategy. rather one entered some time ago. The value of the contract
depends on movements in the price of the underlying asset and
• Buy one unit of the asset for S at time zero.
can be positive or negative.
• Enter into a forward contract to sell (1 + Q ) 7 units of the
asset for price F per unit at time T. We can value the contract by comparing it with a similar con
tract that could be entered today. Define K as the forward price
This strategy exchanges S at time zero for F(1 + Q at time T. If
at the tim e the contract was originally entered, F a s the current
) 7
we continue denoting the risk-free interest rate for maturity T as

forward price for the contract, and T as the contract's current
R, this means that S is the present value of F(1 + Q so that:
time to maturity.
) 7
F + Q )t
The two different forward contracts we consider are
( 1
(1 + R)r
or 1. A forward contract to buy the asset for price K at tim e T
(the contract we are interested in valuing), and
+ r V
(10.3)
1
1 + Q 2. A forward contract to buy the asset for price F at tim e T (a

contract that could be entered today).
Example: The only difference between these two contracts lies in the price
Consider an asset expected to provide a 3% yield per year over paid at time T. The value of the second contract minus the value
the next three years. The risk-free rate is 4% per year and the of the first contract is the present value of F — K (which can be
positive or negative.) However, the second contract is worth
zero because it is entered at the current forward price. Th ere
4 As mentioned in Footnote 2, com pounding frequencies will
fore, the first contract is worth the present value of F — K.
be discussed in C hapter 16. Ju st as an interest rate can be
expressed as 4% with semi-annual com pounding or equivalently as Specifically:
4.04% (= 1.02 X 1.02 — 1) with annual com pounding, a yield can be
F - K
expressed with different com pounding frequencies. A s the com pound Value of Long Forward Contract (10.4)
ing frequency increases, the numerical value of the yield decreases. (1 + R)t
Similarly: contract. It can be shown that if interest rates are constant (or if
they change in a perfectly predictable way), the theoretical
Value of Short Forward Contract = ------------
no-arbitrage forward and futures prices are the sam e
7
(1 + R)t . 5
_ K - F In practice, however, interest rates do vary unpredictably and

(10.5)
~ (1 + R)t futures prices are therefore different from forward prices. This
These formulas are true for all forward contracts on all assets difference is due to the correlations between the returns from
(not just financial assets). the underlying assets and interest rates.
In the case of financial assets, we can use Equations (10.1), As an example, suppose that the price of an asset is positive cor
(10.2), or (10.3) to calculate F. For exam ple, in the case of an related with interest rates. If the asset price increases, there will be
asset providing no income: an immediate gain from a long futures contract. This gain can then
be invested at a relatively high interest rate because (on average)
S(1 + R)t - K
Value of Long Forward Contract both the interest rate and the asset price will have increased. If
(1 + R)t
the asset price decreases, there will be an immediate loss on the
K
= S - T
( 10. 6 ) long futures contract. However, the funds can be financed at a low
+ R) 0
interest rate because interest rates (on average) decline when the
When an income with a present value of / is to be paid during asset price declines. This makes the long futures contract slightly
the remaining life of the forward contract: more attractive than a long forward contract and it would there
(S-/)(1 + R)t - K fore have a slightly higher price. When the correlation between
Value of Long Forward Contract
(1 + R)t the return from the underlying asset and interest rate is negative,
K however, this argument is reversed and the theoretical futures
I - (10.7)
0 + R)
T price is slightly lower than the theoretical forward price.
When a yield at rate Q is provided The differences between futures and forward prices are small
and can usually be ignored. However, an exception is the case of
S(1 + R)r/(1 + Q )t —K
Value of Long Forward Contract = Eurodollar futures contracts (which we will discuss in Chapter 19).
(1 + R)t
S____________ K Another issue is that while futures contracts can have a range of
( 10. 8)
(1 + Q ) t (1 + R)t delivery dates, forward contracts do not. As noted in previous
chapters, it is the party with the short position that chooses the
delivery tim e.
Example:
Consider the case of financial assets. If the interest rate is
Let us return to the forward contract we first considered (where
greater than the income generated by a financial asset, it is
the asset price is USD 70, there is no income, and the one-year
optimal for the party with the short position to deliver as early
interest rate is 5%). The current no-arbitrage forward price F is
as possible to avoid financing costs. If the income is greater than
USD 73.50. Suppose some time ago a long forward contract was
the interest rate, the reverse is true and the party with the short
entered to buy the asset at USD 78. Equation (10.6) gives the
position will deliver as late as possible to earn the maximum
value of the contract as:
income on the asset. This observation can be used to determ ine
78
70 - — - = -4 .2 8 6 the equivalent maturity for a forward contract.
1.05
To summarize, Equations (10.1) to (10.3) can be used
Alternatively, we can use Equation (10.4) to get
to determ ine futures prices as well as forward prices.
73.5 - 78 Equations (10.4) to (10.8) are irrelevant for futures prices
-4 .2 8 6
1.05 because daily settlem ent ensures the value of a futures contract
at the end of each day during its life is zero.
10.6 FO R W A R D V ER S U S FU TU R ES The fact that the forward price equals the futures price (to a
good approximation) does not mean the profits or losses from
Recall that futures contracts are settled daily, while forward con the two contracts are the same. Consider a situation where the
tracts are settled at maturity.
5 See J . C . C o x, J . E. Ingersoll, and S. A . Ross, "Th e relation between
Now consider two contracts that are the same in every aspect forward prices and futures p rices," Journ al o f Financial Econ om ics, 9
except that one is a futures contract and the other is a forward (D ecem ber 1981): 321-346.

one-year forward and futures price for an asset is USD 2.00. the N A SD A Q -100, the w eight of a stock is proportional to
Suppose that during a single day, the forward and futures prices its m arket capitalization (i.e ., its share price m ultiplied by the
increase to USD 2.10. Trader A , who has a long futures contract num ber of shares outstanding). In the case of the Dow Jo n es
on 1 , 0 0 0 units of the asset, makes an im m ediate profit of Index, the w eight is proportional to the share price.
USD 100 (because of daily settlem ent). Trader B, who has a long
The two S&P 500 futures contracts traded on the C M E are on
forward contract, also gains USD 100. However, this is in one
USD 50 and USD 250 multiplied by the index. The two CM E futures
year's tim e (because there is no daily settlem ent for forward
contracts on the N A SD A Q are on USD 20 and USD 100 multiplied
contracts). The accounting system s will therefore show Trader
by the index. The two CM E futures contracts on the Dow Jones are
A's position has increased by USD 100, whereas Trader B's
on USD 5 and USD 10 multiplied by the index. All contracts are set
position has increased by the present value of USD 100. This
tled in cash (rather than by delivering the portfolio). For example,
difference between forwards and futures is sym m etrical. If the
final settlement of the S&P 500 contract is the opening price of the
forward/futures price declined by USD 0.10, Trader A would lose
S&P 500 on the third Friday of the delivery month.
USD 100 im m ediately, whereas Trader B would lose slightly less
on paper (i.e., the present value of USD 100). Similar futures contracts trade actively in other countries. For
exam ple, futures on the CSI 300 Index (a market-capitalization
weighted portfolio of 300 Chinese stocks) trades on the China
10.7 E X C H A N G E RATES R EV ISIT ED Financial Futures Exchange (C FFEX ).
A stock index can be regarded as a financial asset that pays

Chapter 9 showed that the no-arbitrage argument led to the
d ivid en d s .6 7 Because the asset is the portfolio of stocks under
relationship between forward rates and spot rates for the G BP/
lying the index, the dividends are the dividends received by an
USD exchange rate being:
investor holding the portfolio. It is therefore possible to go
(1 + Rusd )7 through the stocks in the portfolio and estim ate the dividend
( 10 . 9 )
(1 + RGBp)T on each to produce the total estim ated cash incom e on the
index. This would allow the use of Equation (10.2). In practice,
Recall that G BP/U SD represents the exchange rate as the
it is usually assum ed the dividends on an index provide a
number of USD (U.S. Dollar) per G B P (British Pound Sterling).
known yield so that Equation (10.3) ap p lies . 7 The value of Q is
From the perspective of a U.S. investor, G B P can be treated the average dividend yield during the life of the forw ard/
in the same way as any other asset valued in USD. In this case, futures contract.
the asset provides income equal to the G B P interest rate. How
ever, the income is received in G B P and not USD, and thus
the value of the income (to a U.S. investor) is the G B P interest
Example:
rate multiplied by the value of G B P in USD. In other words, the Consider an index that is 2,500 in a situation when the risk-free
income from one unit of the asset is R q bp multiplied by its price. rate is 5% per year and the dividend yield is 3% per year (both
This makes the foreign risk-free rate a yield (rather than known are assumed to be the same for all maturities). The futures price
income), and thus Equation (10.3) applies. It is therefore not for a contract where the final settlem ent will be in six months is
surprising that when we set R equal to and Q equal to Rq bp/
/1 .0 5 V 5
Equation (10.3) becom es Equation (10.9).

2'500 X l 03 = 2'524
10.8 S T O C K IN D IC ES Index Arbitrage

If an index futures price is greater than its theoretical value, an
A stock index tracks the value of a hypothetical stock portfolio
arbitrageur can buy the portfolio of stocks underlying the index
(i.e., if the value of the hypothetical portfolio increases by X%, the
index also increases by X%). The CM E Group offers exchange trad
6 Dividend paym ents are not usually considered when an index is calcu
ing platforms for futures on several different stock indices. Among
lated and so the index does not represent the total return that would be
these are the S&P 500 (a portfolio of 500 stocks), the NASDAQ-100 earned by an investor in the stocks underlying the index. An exception
(a portfolio of 100 stocks traded on the Nasdaq Stock Market), and is a total return index, which is created by reinvesting dividends from the
the Dow Jones Index (a portfolio of 30 large stocks). hypothetical portfolio in the hypothetical portfolio.
7 Several financial institutions provide estim ates of the dividend yields

The w eight of a stock in a portfolio is the percentage of the on the indices that are commonly referred to by investors and on which
portfolio invested in the stock. In the case of the S&P 500 and futures contracts trade.
and sell the futures. If the futures price is less than the theoreti to fluctuating exchange rates, however, the trader cannot trade
cal price, the arbitrageur can short the stocks underlying the a portfolio that is always worth Z dollars.
index and take a long futures position. Both of these trading
strategies can be categorized as index arbitrage.
SUMMARY
Typically, a com puter program is used to send all the required
trades (for the stocks underlying the index) to an exchange at This chapter has considered futures prices and forward prices
the same tim e as the futures contract is traded. This is known for financial assets. The results are summarized in Table 10.5.
as program trading. Som etim es index arbitrage is done by In most circum stances, the futures prices and forward prices
trading a representative subset of the stocks underlying the for contracts on the same asset with the same maturity are
index. (approximately) the same.
Index arbitrage typically ensures that the theoretical relationship Interest rate futures are an important category of financial
between the index and futures on the index holds. O ccasionally, futures, which have not been considered in any detail so far.
there are exceptions. On O ctober 19, 1987, (Black Monday) the Although similar no-arbitrage approaches can be used to deter
market declined by more than 2 0 % and the number of shares mine their prices, interest rate futures have special features and
traded on the New York Stock Exchange easily exceeded all pre therefore deserve special attention. They are covered in greater
vious records. This led to delays in processing orders and index detail in Chapter 19.
arbitrage strategies could not be carried out efficiently. As a
result, the futures price of the index was well below the theoreti
cal price given by Equation (10.3).
Table 10.5 Summary of Results for a Forward
Contract on a Financial Asset; S is Current Price of Asset,
T is Maturity of Contract, and R is the Interest Rate
Indices Not Representing Tradable Value of Long
Portfolios Forward or Forward with

Asset Futures Price Agreed Price K
For the no-arbitrage Equation (10.3) to be applicable, it must
Provides No
be possible to trade a portfolio whose price always equals the S(1 + R)r
Income S (1 + R)t
index. W hile this is usually the case, there are scenarios where
the index (although well defined) does not correspond to the Provides Known
value of a tradable portfolio. Income with Pres (S - 0(1 + R)r S - 1 -------
ent Value / (1 + R)t
For exam ple, the Chicago M ercantile Exchange offers a futures
contract on the Nikkei 225 Index that is settled in dollars rather S K
Provides a Known J 1 + R \T
than yen. Suppose the Nikkei Index is denoted by Z. A trader Yield Equal to Q + q / (1 + Q )r (1 + R)t
can (in theory) trade a portfolio that is always worth Z yen. Due
S \ 1

Q U ES T IO N S

10.1 Explain what happens when shares are shorted. 10.6 W hat is the formula for the futures price of a financial
asset that provides a constant yield at rate Q?
10.2 "If you buy a non-income-producing financial asset for
1 0 0 and enter into a forward contract to sell it for 1 1 0 in 10.7 W hat is the difference between a forward price and the
one year, you have made a profit of 10." Is this statem ent value of a forward contract?
true? W hy or why not? Consider the situation where the 10.8 If the return from an asset is positively correlated with
interest rate is 4% per year and 10% per year. interest rates, would you prefer to enter into a long for
10.3 W hat is the formula for the futures price of a financial ward contract or a similar long futures contract? Explain.
asset that provides no income? 10.9 Exp Iain why the interest rate earned in a foreign currency
10.4 W hat is the formula for the futures price of a financial can be regarded as a yield.
asset that provides income with a present value of /?
10.10 Explain how index arbitrage is accom plished when the
10.5 W hat is meant by an asset that provides a constant yield? futures price is higher than its theoretical value.
Practice Questions
10.11 Suppose that you enter into a two-year forward contract compounding). W hat is the (a) forward price and
on a non-dividend-paying stock when the stock price is (b) value of the forward contract?
USD 40 and the risk-free rate (annually compounded) is
10.17 Six months after the forward contract in Question 10.16
10% per year. W hat do you expect the forward price to was entered into, the spot price is USD 56 and the risk
be? free rate is still 5% per year. W hat is the (a) forward price
10.12 Is the futures price of a non-dividend paying stock likely and (b) value of the forward contract?
to be greater or less than the expected future stock 10.18 If the contract in Questions 10.16 and 10.17 had been a
price? Explain your argument.
futures contract, how would your answers change?
10.13 The cash price of a bond is USD 90. It is expected to 10.19 If the return from an asset is negatively correlated with
provide a coupon of USD 3 in six months and 12 months.
interest rates, would you expect the forward price to be
The risk-free rate for all maturities is 5% per year (with greater than or less than the futures price? Explain.
annual com pounding). W hat is the 15-month forward
10.20 If a stock index, interest rate, and dividend yield remain
price of the bond?
constant, derive a formula for the futures price at time t
10.14 A stock index is 3,000, the risk-free rate is % per year,
8
in term s of the futures price at time zero. Suppose that
and the dividend yield on the index is 3% per year (both the risk-free rate is 5% per year and the dividend yield on
expressed with annual compounding). W hat should the an index is 3% per year. If the stock index stays constant,
one-year futures price of the index be? at what rate does the futures price grow? (All rates are
10.15 Explain the argument used for calculating the value of a expressed with annual com pounding.)
forward contract.
[Hint: If the futures contract initially has a time to matu
10.16 A one-year forward contract to buy a non-dividend rity equal to T, it has a time to maturity equal to T — t at
paying stock is entered into when the stock price is time t. Use Equation (10.3) to calculate the futures price
USD 50 and the risk-free rate is 5% per year (with annual F0 at tim e zero and the future price Ft at tim e t ]
A N SW ER S
10.1 The stock is borrowed from another investor and sold 10.14 The one-year futures price is
in the market in the usual way. A t a later tim e, it is pur
chased and returned to the account from which it was 3,000 X = 3145.63
I •U vJ
borrowed. Any income on the security has to be paid by 10.15 The forward contract is compared to a forward contract
the borrower. Som etim es there is a small fee for borrow with the same maturity that would be entered into today.
ing the stock. The present value of the difference between their payoffs
10.2 The profit is less than 10 because the investment of 100 can be calculated. We know that the value of a forward
must be financed. If the interest is 4% per year, the profit contract that would be entered into today is zero and
is only 6 . If it is 10% per year there is no profit. This was can therefore deduce the value of the other forward
the issue that led to Joseph Jett's false recording of profit. contract.
10.3 F = S(1 + R)t 10.16 The forward price is 50 X 1.05 = 52.5. The value of the
forward contract is zero.
10.4 F = (S - /)(1 + R)t
10.5 The income when expressed as a proportion of the asset 10.17 The forward price is 56 X 1.05°5 = 57.383. The value of
the forward contract is
price is constant. If a yield at rate Q is reinvested in the
asset, the holding of the asset grows at rate Q. 57.383 - 52.5
= 4.77
+ r Y 1.05°-5
10.6 F = Si
1
10.18 The value of the contract in Question 10.17 would be

1 + Q
zero because of daily settlem ent. O ther answers are the
10.7 The forward price is the delivery price that would be
same.
negotiated today in a forward contract. The value of a
forward contract is initially zero and then moves up or 10.19 W hen there is a negative correlation between return on
assets and interest rates, when the asset price increases,
down as the asset price changes.
funds tend to be invested at a relatively lower rate. On
10.8 You would prefer to own a long futures contract because
the other hand, when the asset price decreases, funds
daily gains will tend to be invested at a relatively high
can be invested at a relatively higher rate. This makes a
rate and daily losses will tend to be financed at a rela
long futures contract less attractive than a long forward
tively low rate.
contract and the futures price will be lower than the for
10.9 If interest is earned in the foreign currency, its value to a ward price.
dom estic investor is proportional to the value of the for
10.20 The relationship between the futures price, Ft, at time t
eign currency.
and the spot price is with the notation in the chapter:
10.10 A trader buys the stocks underlying the index and sells
J 1 + / ? \ T_t / i + /?V/i + q V / i + q Y
futures.
t_sVi + oj - S vi + q J vi + F/ +
10.11 The forward price is USD 40 X 1.1 2 or USD 48.4.
where S, R, and Q are the index level, risk-free rate, and
10.12 The futures price is the spot price com pounded forward dividend yield, respectively and T is the initial time to
at the risk-free rate. Most stocks can be expected to pro maturity. This shows that the futures price grows at:
vide a return greater than the risk-free rate. Hence, the
1 + Q „
expected future stock price is greater than the forward --------------- 1
1 + R
price.
When R = 5% and Q = 3%, the growth rate of the
10.13 The present value of the income (USD) is
futures price per year is
5.785 1 + Q 1.03
- 1 = -0 .0 1 9
1 + R 1.05
The forward price of the bond is
(90 - 5.785) X 1.05125 = 89.51 or —1.9%.

Commodity
Forwards and
Futures
Learning Objectives
Explain the key differences between com m odities and Com pute the forward price of a com m odity with storage
financial assets. costs.
Define and apply commodity concepts such as storage Com pare the lease rate with the convenience yield.
costs, carry m arkets, lease rate, and convenience yield.
Explain how to create a synthetic com m odity position,
Identify factors that impact prices on agricultural and use it to explain the relationship between the forward
com m odities, metals, energy, and w eather derivatives. price and the expected future spot price.
Explain the basic equilibrium formula for pricing Explain the relationship between current futures prices
commodity forwards. and expected future spot prices, including the impact of
system atic and nonsystematic risk.
Describe an arbitrage transaction in commodity forwards,
and com pute the potential arbitrage profit. Define and interpret normal backwardation and contango.
Define the lease rate and explain how it determ ines the
no-arbitrage values for com m odity forwards and futures.
Describe the cost of carry model and illustrate the impact

of storage costs and convenience yields on commodity
forward prices and no-arbitrage bounds.
135
This chapter considers forward and futures contracts on com • A com m odity held for investment purposes (e.g ., gold or
m odities. Chapter 10 showed how the price of a futures contract silver) can be borrowed for shorting (to be discussed later).
and the price of a forward contract with the same underlying However, the borrower must pay what is called a lease rate.
asset and maturity are approxim ately equal. This is true whether This lease rate can exceed the fees charged when financial
the asset is a com m odity or a financial asset. This chapter will assets are borrowed for shorting (see Section 10.1).
therefore treat futures as forwards (i.e., we will ignore the daily • A financial asset provides investors with an expected financial
settlem ent ) . 1
return that reflects its risk. Most commodities do not have
Most com m odities are consumption assets. This means that this property. Indeed, it can be argued that the prices of most
they are rarely held for purely investment reasons (metals such commodities are mean reverting. This means that although
as gold and silver are exceptions). Com m odity owners usually the price of a commodity can be quite volatile, it tends to
intend to use the com m odity in some way, after which it ceases get pulled back toward some central value. When the price
to be available for sale. The no-arbitrage arguments presented of a commodity is relatively high, its production will become
in Chapter 10 did not account for this and therefore are not fully attractive and its supply will therefore tend to increase. A t
applicable to consumption assets. the same tim e, users of the commodity may search for less
expensive alternatives. Com bined, these actions will tend
Chapter 10 showed that the no-arbitrage forward/futures price
to reduce the price of the commodity. If the price is at a
of a financial asset can be com puted from risk-free interest rates
relatively low level, on the other hand, the production of the
and the income generated by the asset. For most com m odities,
commodity will become less attractive, while its use will tend
however, no-arbitrage arguments can only be used to obtain an
to increase. As a result, its price will tend to rise.
upper bound for the futures price. An additional unobserved
parameter, known as the convenience yield, is required to deter As a result of these distinctions, the futures prices of commodities
mine the com m odity futures price in the market. can behave quite differently from those of financial assets.
11.1 W HY COM M ODITIES ARE 11.2 TYPES O F COM M ODITIES

D IFFEREN T
This section reviews the properties of various commodity
categories.
There are several im portant differences between com m odi
ties and financial assets. Some differences in particular are as
follows. Agricultural Commodities
• The storage costs associated with financial assets (e.g ., stocks
Agricultural com m odities with futures contracts include products
and bonds) are negligible. The storage costs for com m odi
that are grown (e.g ., corn, wheat, soybeans, cocoa, and sugar)
ties, however, can be quite substantial. These costs include
as well as livestock (e.g ., cattle and hogs).
insurance, which can vary over tim e. Some com m odities
deteriorate with time and require special (i.e., costly) care in It is expensive to store agricultural com m odities; even under
storage. Assets such as corn and natural gas are frequently optimal conditions, some products can only be stored for a lim
stored for use at a particular time of the year. O ther assets, ited period of tim e. There is also an interdependence among
such as oil and copper, are consumed throughout the year. various agricultural com m odities. For exam ple, the cost of fe ed
ing livestock can depend on the prices of grown agricultural
• Com m odities can be costly to transport and thus their prices
com m odities (e.g ., corn).
can depend on their location. By contrast, financial assets are
usually transported electronically at virtually no cost. The prices of agricultural commodities can be seasonal. For
exam ple, consider the prices of corn and soybeans. A t harvesting
time (O ctober to November), their prices tend to be relatively
1 We discussed in Chapter 10 how the maturity date for a financial
low. Outside of this period, however, their prices may be higher
futures contract should be chosen when there is a range of possible
delivery dates. The rule that can be applied to com m odity futures: If the as farmers and other distributors incur storage costs. This season
futures price for contracts is a decreasing function of m aturity, the short ality is sometimes reflected in futures prices, causing them to
will choose to deliver at the latest tim e because (as we will discuss later)
display a mixture of normal and inverted pricing patterns .2 *
the benefits of holding the com m odity in inventory exceed the financ
ing and storage costs; if the futures price for short-maturity contracts is
an increasing function of maturity, the short will choose to deliver at the 2 See Section 7.5 for a discussion of normal and inverted patterns for
earliest tim e to avoid financing and storage costs. futures prices.
There are also other factors that may affect the market's view on The crude oil market is the largest commodity market in the
the future price of a com m odity. If there has been (or if there is world and global demand is estim ated to be nearly 1 0 0 million
expected to be) a good (or bad) harvest, market participants will barrels per day .4 There are many grades of crude oil reflecting
expect prices to be relatively low (or relatively high). Political variations in gravity (density) and sulfur content. Two important
considerations can also affect futures prices .3 benchmarks are Brent crude oil (which is sourced from the North
Sea) and W est Texas Interm ediate (WTI) crude oil.
W eather is also an important consideration for many agricultural
com m odities. Bad w eather in Florida, for exam ple, can increase Natural gas is used for heating and generating electricity. It can
the futures price of frozen orange juice. M eanwhile, frosts in be stored (either above ground or underground) for indefinite
Brazil are liable to drastically reduce Brazilian coffee production periods, but its storage costs are quite high. Natural gas is also
and increase coffee futures prices. expensive to transport, and hence the price of natural gas can
vary regionally.
Metals C M E Group offers a futures contract on 10,000 million British

thermal units (BTUs) of natural gas. The contract requires deliv
Com m odity metals include gold, silver, platinum, palladium, ery to be made at a roughly uniform rate to a hub in Louisiana.
copper, tin, lead, zinc, nickel, and aluminum. Their proper Intercontinental Exchange (ICE) also offers a futures contract on
ties are quite different from those of agricultural products. For natural gas.
exam ple, metal prices are not affected by the w eather and are
The demand for natural gas is high in the winter (for heating
not seasonal (because they are extracted from the ground).
purposes) and to a lesser extent the summer (to produce elec
Additionally, their storage costs are typically lower than those of
tricity for air conditioning). This creates seasonality in the futures
agricultural products.
prices as illustrated by Table 11.1 and Figure 11.1.
Some metals are held purely for investment purposes. This
Electricity is an unusual com m odity because it is almost im possi
means (as we will explain later) that their futures price can be
ble to store The maximum supply of electricity in a region is
more easily obtained from observable variables. For someone
. 5
determ ined by the capacity of all electricity generating plants in

looking to hold a metal for investm ent, owning a futures con
that region. In the U .S., there are 140 such regions and the
tract can be an acceptable alternative to owning the physical
wholesale price of electricity is determ ined by the price charged
asset itself.
in each of them .6 The non-storability of electricity can lead to
As in the case of agricultural com m odities, inventory levels are huge price swings. For exam ple, heat waves in the summer can
important in determ ining prices. Most metals are extracted in drive up the demand for air conditioning and have been known
one country and consumed in another. As a result, exchange to increase the cost of electricity by as much as 1,000% . O nce
rates may affect prices. O ther crucial factors include the scope the heat wave is over, however, the price quickly returns to nor
of a metal's industrial application and the rate at which new mal levels (an exam ple of the mean reversion tendencies we
sources are found. Changes in extraction methods, actions by mentioned earlier).
governm ents (and/or cartels), and environmental regulations can
Futures contracts on electricity do exist, but they are less
also impact metal prices. Som etim es metal prices can even be
actively traded than futures contracts on natural gas and crude
affected by recycling processes. For exam ple, a metal used in a
oil. Electricity futures contracts also trade in the over-the-
production process one year can be recycled and re-enter the
counter market. A typical contract allows one side to receive a
market many years later.
specified number of m egawatt hours for a specified price at a
specified location during a particular month. In a 5 X 8 contract,
Energy power is received during the off-peak period (11 p.m . to 7 a.m.)
from Monday to Friday. In a 5 X 16 contract, power is received
Energy products are another important category of com m odi
ties. There are futures contracts on crude oil and crude oil
extracts (e.g ., petroleum and heating oil). Futures also trade on 4 International Energy A gency, (n.d.). Oil M arket Report. Retrieved from
natural gas and electricity. http://w w w .iea.org/oilm arketreport/om rpublic
5 There some artificial ways of storing electricity. For exam ple, electricity
can be used to pump w ater to the top of a hydroelectric plant so that it
can be released in a way that produces electricity at a later tim e.
3 For exam ple, escalating trade tensions betw een the U.S. and China
in 2018 made it difficult for U.S. farm ers to export com m odities such as 6 W hether electricity can be sold across regions depends on the
corn and soybeans. existence of, and the capacity of, transm ission lines.
Chapter 11 Commodity Forwards and Futures ■ 137

Table 11.1 Seasonality of Natural Gas Futures Prices during the on-peak period (7 a.m . to 11 p.m .) from Monday to
on July 24, 2018 Friday for the specified month. In a 7 X 24 contract, power is
received 24 hours a day, seven days a week for the specified
Maturity Month Settlement Price
month. Prices are highly seasonal, reflecting increased demand
August 2018 2.732 in the summer.
Septem ber 2018 2.719
O ctober 2018 2.739 Weather

Novem ber 2018 2.791
Derivative contracts on w eather are available in both the
D ecem ber 2018 2.904 exchange-traded and over-the-counter m arkets. The most popu
January 2019 2.989 lar contracts are those with payoffs contingent on tem perature
(which are used by energy com panies as hedges).
February 2019 2.958
Two im portant w eather derivative variables are heating degree
March 2019 2 . 8 6 6
days (HDDs) and cooling degree days (CDDs). The HDD and
April 2019 2.590 CD D for a day are (respectively) defined as:
May 2019 2.562 HDD = max(0, 65 - A)
June 2019 2.594
and
Ju ly 2019 2.627
CD D = max(0, A — 65)
August 2019 2.634
where A is the average of the highest and lowest tem perature
Septem ber 2019 2.615 during a given day at a specified w eather station (as measured
O ctober 2019 2.632 in degrees Fahrenheit). For exam ple, if the minimum
Novem ber 2019 2 . 6 8 6

tem perature during a day (midnight to midnight) is 40 degrees
Fahrenheit and the maximum is 60 degrees Fahrenheit,
D ecem ber 2019 2.813
A = (60 + 40)/2 = 50. The daily C D D is therefore zero, and the
January 2020 2.908 daily HDD is 15. Contracts are usually defined in term s of the
February 2020 2.871 cumulative HDD and CD D for all the days in a given month.
March 2020 2.783
April 2020 2.497 11.3 C O M M O D IT IES H ELD

May 2020 2.463 FO R IN V ES T M EN T
June 2020 2.493
Some precious metals are held for investment purposes. Gold
S o u rce: w w w .cm egroup.com
and silver (and to a lesser extent platinum and palladium) are in
this category. W hile these metals have industrial applications,
some purchasers hold these com m odities purely for investment
purposes. For these people, owning a futures or forward con
tract can be a practical substitute for owning the commodity
itself.
The storage costs of the metals held for investm ent are gener
ally low com pared to metal prices and can therefore be ignored.
Additionally, there is a generally small lease rate associated with
metals held for investm ent. For exam ple, gold is like a financial
asset in that it can be lent from one entity to another to earn
interest.
In analyzing the futures prices for investment commodities, we

Fiqure 11.1 Plot of futures prices for natural gas first ignore the lease rate. The commodities are then akin to finan
from Table 11.1. cial assets providing no income and therefore Equation (10.1)
of Chapter 10 should apply. This means that the relationship The lease rate of gold varies with the supply of gold that can be
between the forward price and the spot price is borrowed along with the demand to borrow gold. Recall that
when gold producers hedge future production, the banks on the
F = S( 1 + R)t
other side of the transaction borrow gold. As gold producers
where T is the tim e to maturity of the forward (measured in hedge more (less), the demand for borrowed gold will increase
years) and R is the annually com pounded risk-free interest rate (decrease) and the lease rate will rise (fall). Similarly, as asset
for maturity T. The no-arbitrage arguments are as follows. owners becom e more (less) willing to lend gold, the lease rate
• A rb itra g e A : If the forward price is greater than S(1 + R y , will tend to fall (rise). O ccasionally the lease rate is negative and
a trader can buy the investm ent com m odity at price S (by may therefore allow arbitrageurs to buy the metal and sell it
financing the purchase at rate R) and at the same time enter forward for a profit.
into a forward contract to sell it at time T.
• A rb itra g e B: If the forward price is less than S( 1 + R)T, a
trader who owns the investm ent commodity will find it prof
11.4 C O N V E N IE N C E Y IE LD S
itable to sell it at price S and at the same tim e enter into a
Now consider com m odities that are held almost always for con
forward contract to buy it back at time T.
sumption purposes (e.g ., crude oil, copper, and corn). Assume
that there are no storage costs. If F S(1 T R) , traders can
Lease Rates undertake Arbitrage A in Section 11.3. Even though the com
modity is not an investment asset, traders can still buy the asset
The lease rate for an investment commodity is the interest rate
by financing the purchase at rate R and entering a forward con
charged to borrow the underlying asset.
tract to sell it at price F at tim e T.
For exam ple, a gold producer might enter into a contract with
Therefore, the upper bound to the forward price is the spot
an investment bank to sell its future production forward at a pre
price com pounded forward at the risk-free rate:
determ ined price. The bank will then hedge the risk it is taking
on (with respect to the price of gold) by borrowing gold for a F < S(1 + R)t
time period equal to the life of the forward contract and selling
When the storage costs with present value U are considered,
it in the spot market. This synthetically creates a short forward the arbitrageur has extra costs to finance, and there is an extra
contract that offsets the contract it has with the gold producer.
repaym ent of U(1 + R)Trequired at time T. This means that the
The main lenders of gold are central banks
upper bound is given by:
. 7
Let € be the lease rate. The relationship between the forward

F < (S + U)(1 + R)t (11.2)
price and the spot price is determ ined from Equation (10.3) as
A storage cost can be regarded as negative income. Chapter 10
1 + r \t showed that when there is income with present value /, the no
F = SI
1+e arbitrage condition is
This equation can be used to calculate an implied lease rate:
1 F < (S - /)(1 + R)T
T(1 + R) - 1 ( 11 . 1)
Equation (11.2) can be obtained by setting I = —U.
Now consider Arbitrage B in Section 11.3. This arbitrage
Example: involves entities that own the asset choosing to sell it and enter
into a forward contract to buy it back at tim e T. Because this is
Assum e that the spot price of gold is USD 1,240, the six-month
a consumption asset, individuals and com panies that own the
futures price is USD 1,250, and the six-month risk-free rate is 4%
asset must plan to use it in some way. If they sell the asset, they
(with annual com pounding). The implied lease rate is given by:
can no longer do that. Therefore, asset owners are usually not
prepared to take this arbitrage opportunity.
( 5 x 1.04 - 1 = 0.023
V 1.250,/ To summarize, all that can be deduced for a consumption asset
or 2.3% . is that Equation (11.2) holds, giving an upper bound to forward/
futures prices.
7 The Federal Reserve Board does not lend gold, but other central gov Despite this limitation, however, we can measure the extent
ernm ents do. to which the owners of a consumption asset value having it

readily available in their inventory by using what is called the As we will explain later, the futures price of an asset can be
convenience yield. related to the expected future spot price. In our exam ple, it
appears that the market was expecting the price of oil to decline
The convenience yield essentially addresses the following
over the six-month period. Under such circum stances, hold
question: If the underlying com m odity were an investment asset,
ing oil as an investment makes no sense. (Indeed, if oil were an
what yield Y would be necessary to explain the observed futures
investment asset, its price would reflect market sentim ent.) The
price? In other words, Y (measured with annual compounding)
only reason som eone would hold oil is to use it. An expectation
satisfies
that the price of an asset will decline sharply is therefore consis
1 + R \t tent with the physical asset having a high convenience yield.
F = (S + U)
1 + Y
so that
. 1
11.5 C O S T O F C A R R Y
y= i R1 + R) — 1 (11.3)
The cost of carry for an asset reflects the impact of:
The convenience yield measures the benefits to the asset holder
• Storage costs,
of having it in their inventory as a protection against future
shortages or delivery delays. • Financing costs, and
Suppose there is exp ected to be a plentiful supply of the • Income earned on the asset.
asset during the life of a futures contract. This means it can In the case of a financial asset, there are no storage costs. If
be ordered at any tim e for alm ost im m ediate delivery. In the financing cost is R and the yield is Q (both expressed with
this case, the convenience yield is likely to be close to zero annual com pounding), the cost of carry per year is
and F = (S + U)(1 + R)T might be a reasonable approxim a
1 + R
tion. If inventory levels of the asset are low and there are (11.4)
1 + Q
concerns about shortages, however, the convenience yield
will likely be quite high. In that case, F is likely to be less than which is often simply approxim ated as R — Q.
(S + L0(1 + R)T.
Example:
Example: Consider a foreign currency. If the dom estic interest rate is 4%

and the foreign interest rate is 3%, the cost of carry is approxi
Suppose we are in a market like that of June 25, 2018. (See
mately 1%. If the foreign interest rate were 6 %, the cost of carry
Table 7.3.) The spot price of oil is USD 69 per barrel, and the six-
would be around — % (this is referred to as a negative carry).
2
month futures price is USD 65 per barrel. The cost of storing oil
for six months has a present value of USD 1 per barrel, and the In the case of a com m odity, there is usually no incom e .9 The
risk-free rate is 2% per year. The convenience yield satisfies cost of carry therefore consists of the interest rate and the stor
age cost. In the exam ple at the end of Section 11.4, the cost of
1/2
1.02
65 = (69 + 1) carrying oil for six months is given by:
1+y
so that • A 2% per annum interest cost, and
• A storage cost of about 1/69 = 1.45% per six months (or
- 1 = 0.183 about 2.9% per year ) . 1 0
The total cost of carry is therefore around 4.9% (= 2% + 2.9%)

That is, the convenience yield is 18.3%.
per year.
One way of interpreting this number is to view it as the cost of
borrowing oil (if that were possible). Because physical oil
9 The lease rate on investm ent com m odities is an exception.
provides benefits to the holder at the rate of 18.3% per
year, this is arguably the rate that would be charged to 10 This is an approximation because the USD 1 of storage costs is a pres
ent value and not the amount paid in arrears for storage. Storage costs
borrow it
would be 2.9% per year with annual compounding if they equaled 2.9%
.8
of the value at the beginning of the year and were paid at the end of each
year. Storage costs would be 2.9% per year with semiannual com pound
8 In this context, note that Equation (11.3), when there are no storage ing if the storage costs were paid at the end of each six-month period
costs (U — 0), is the sam e as Equation (11.1). and equaled 1.45% of the value at the beginning of the six-month period.
A Note on Compounding Frequencies 11.6 EX P EC TED FUTURE SPOT PRICES
for Interest Rates
The expected future spot price of an asset is the market's aver
Chapter 16 discusses compounding frequencies in detail and intro age opinion about what the spot price will be in the future.
duces continuously compounded interest rates (which are widely Natural questions are as follows.
used in the valuation of options and other derivatives). A continu
• Does the futures price of an asset equal the expected future
ously compounded interest rate can be thought of as an interest
spot price?
rate where compounding takes place very frequently (to the point
where the balance grows by a small amount every instant). A • Is the futures price a good forecast of the future spot price?
continuously compounded rate is equivalent to an annually com The futures price converges to the spot price at maturity of the
pounded rate if both provide the same result at the end of a year. contract. If an investor thinks the spot price at maturity will be
When R and Q are expressed with continuous com pounding, greater than the current futures price, the investor can take a
Equation (11.4) becom es long futures position. Similarly, if an investor thinks the spot
price at the maturity will be less than the current futures price,
R - Q
the investor can take a short futures position. In either case, an
without any approxim ation. M eanwhile, the forward price for investor that is correct will be able to close out the futures con
mula in Equation (10.3) becom es tract near the maturity for a profit.
F = Se(R- Q)T These trading strategies do not involve storing the commodity
where e is the mathematical constant that is approxim ately or investing in a physical asset. They only involve trading futures
equal to 2.71828. When the storage cost (as a fraction of the contracts. In the exam ple where the six-month futures price of
asset price) is expressed with continuous com pounding, the cost oil is USD 65, investors who think this is too low will take long
of carry for a consumption com m odity is the interest rate plus futures positions, and investors who think it is too high will take
the storage cost. The relationship between the futures price and short futures positions.
the spot price is
Early Work
F = S e (C~Y)T (11.5)
Econom ist John Maynard Keynes was one of the first research
where C is the cost of carry and Y is the convenience yield (both ers to consider the relationship between futures prices and
expected future spot prices. He argued that speculators
1 0
expressed with continuous compounding).

require compensation for the risks they are bearing, whereas
W hen the annually com pounded rate is low, it more closely
hedgers derive benefits from futures and thus do not require
resem bles the equivalent continuously com pounded rate
any such com pensation. Indeed, the hedgers might be prepared
and thus it is often reasonable to ignore com pounding fre
to lose money (on average) because their overall market risks
quency issues. As the rate increases in m agnitude, there is
are reduced by hedging.
a greater approxim ation error in treating these two rates as
equivalent. If hedgers tend to hold short positions, and speculators tend to
hold long positions, Keynes' argument suggests the longs will
We can illustrate this with the oil exam ple. The interest rate
tend to make money on average and the shorts will tend to lose
was 2% per annum and the storage cost was about 2.9% per
money on average. The futures price should therefore be less
annum. As an approxim ation, we treat these as continuously
than the expected future spot price. If the reverse is true (i.e.,
com pounded and set C = 4.9% in Equation (11.5). However,
hedgers tend to hold long positions and the speculators hold
Chapter 16 notes that the annually com pounded convenience
short positions), the shorts will make money on average, and the
yield on oil of 18.3% (which we calculated earlier) is equivalent
futures price will be greater than the expected future spot price.
to a continuously com pounded rate of 16.8% .111
2To see that
Equation (11.5) is correct, substitute for the continuously com
pounded estim ates of C and Y to get Modern Theory
F = 69 ©(0-049 - 0 . 168) x 0.5 More recent work on this subject has involved the capital
asset pricing model (CAPM ). Using C A PM , it is argued that
which is close to the assumed futures price of USD 65.
11 This is because ln(1.183) — 0.168. 12 See J . M. Keynes, A Treatise on M oney, London Macmillan, 1930.

an investor should earn a return greater than the risk-free rate This shows that the relationship between E(ST) and F depends
when the system atic risk of his or her portfolio is positive. Sys on X and R. If X > R, then E [ST) > F. If X < R, however, then
tem atic risk is the risk related to the perform ance of the market E (St) < F.
as a whole and cannot be diversified away. Non-systematic risk,
The variable X is the return from the synthetically produced
by contrast, can be diversified away by taking on a sufficiently
position. The system atic risk of this investm ent depends on the
diverse portfolio.
correlation between the underlying asset return and the market
In this context, the "m arket" is generally considered to be a return. If this correlation is positive, the system atic risk of the
well-diversified stock portfolio (such as that underlying the S&P investment will be positive (which implies X > R and therefore
500 Index). CAPM implies that if the return from an investment E (St) > F). If the correlation is negative, the system atic risk
is positively correlated with that of the S&P 500, it should earn of the investm ent is negative (which implies X < R and thus
an expected return greater than the risk-free rate. If the return E (S t) > F). W hen the correlation is zero, the synthetic trade has
from the investment is uncorrelated with the S&P 500, the risk no system atic risk, and the futures price should be equal to the
is non-systematic, and the expected investm ent return should expected future spot price.
equal the risk-free rate. Moreover, if the return from the invest
These results apply to the FX forwards and futures considered
ment is negatively correlated with the S&P 500 (so that there is
in Chapter 9, the financial futures considered in Chapter 10, and
negative system atic risk), the expected investm ent return should
the com m odity futures considered in this chapter. They are sum
be less than the risk-free rate.
marized in Table 11.2.
Consider our prior notation: S is the spot price of an asset, F is
An exam ple of an asset with positive system atic risk is the S&P
the futures price for maturity T, and R is the annually com
500 itself. We therefore expect this asset to fall into the middle
pounded risk-free interest rate for maturity T Suppose that P
category in Table 11.2 with F < E(ST). From Equation 10.3, we
. 1 3
is the present value of the futures price when discounted from

know the forward/future price of a stock index is given by:
time T to tim e zero at the risk-free rate:
1 + RV
F = SI
1 + Q
(1 + R)t
where Q is the dividend yield on the index.
By investing P at the risk-free rate, the trader can be sure of
Suppose that the index equals the forward price at time T. If
obtaining F a t tim e T. The trader can therefore synthetically cre
dividends are reinvested in the index, the position in the index
ate a long position in the asset at time T by:
will grow at rate Q, and value of the investment at time T is
• Investing P at the risk-free rate, and
/ 1 + D \T
• Entering into a long futures contract to buy the asset for F at F = s( ) <1 + Q)T = S(1 + R)T
time T.
This shows that the return realized by the investor is the risk-free
Let S T be the spot price at tim e T. The cash flows from the rate (R). However, because the index (by definition) is positively
trader's strategy are
correlated to itself, the investor should expect to earn more
Tim e 0: —P, and than R. This means that the expected value of the index at time
T should be greater than F (which is consistent with Table 11.2).
Tim e T: + S j.
Many com m odities have positive system atic risk because they
Let E denote the expected value operator. Thus, the expected
tend to cost more when the economy is doing well. An excep
cash flow at tim e T is E( S j). Suppose the expected return from
tion may be gold, which is often referred to as having negative
the trades we have just considered is X (with annual com pound
system atic risk. When the econom y is doing poorly, investors
ing). It follows that:
increase their holding of gold, and its price increases. When the
E(St) = P( 1 + X)T
econom y starts to recover, gold is exchanged for stocks, and its
Substituting P gives price declines. Assuming this is true, theory suggests that the
(1 + X )T futures price of gold overstates the expected future spot price:
E(St) = F
(1 + R)t
F > E(St)
For those assets with little or no system atic risk, CAPM argues
13 As indicated earlier, the relevant risk-free interest rate is the opportu
nity cost of funds to the trader. This could be an interbank borrowing/
that the futures price should be equal to the expected future
lending rate. spot price. But has this been confirmed em pirically?
Table 11.2 Relationship Between Futures Price and Expected Future Spot Price Suggested by CAPM
Underlying asset return and the Synthetic trade has no system atic X = R Futures price equals expected
market return are uncorrelated. risk. future spot price.
Underlying asset return is positively Synthetic trade has positive X > R Futures price is less than
correlated with the market return. system atic risk. expected future spot price.
Underlying asset return is nega Synthetic trade has negative X < R Futures price is greater than
tively correlated with the market system atic risk. expected future spot price.
return.
The answer: The futures price probably provides the best avail There are three main categories of com m odities: agricultural,
able forecast of the spot price. However, different industry metals, and energy. These com m odities all come with futures
experts could reasonably come up with different forecasts, and contracts that are commonly traded on exchanges.
it is difficult to em pirically determ ine the best forecast.
Unlike financial assets, most com m odities are not held for purely
investment purposes. (Exceptions include some metals such as
Normal Backwardation and Contango gold and silver.) Thus, a no-arbitrage argument can only give an
upper bound for a com m odity futures price. The convenience
When the futures price is below the expected future spot
yield of a com m odity measures the benefits of owning the
price, the situation is known as normal backwardation. When
physical com m odity over the holding of a corresponding futures
the futures price is above the expected future spot price, the
contract. These benefits include the ability to keep a produc
situation is known as contango. However, som etim es the term
tion process running and to obtain supplies of the asset on short
normal backwardation is used to refer to the situation where the
notice.
futures price is below the current spot price. Contango is then
used to refer to the situation where the futures price is above The cost of carry for a commodity is the cost of storage plus the
the current spot price. cost of financing the asset. For a com m odity held for consum p
tion, the relationship between the futures price and the spot
price depends on the convenience yield as well as the cost of
SU M M A RY carry.
The theoretical relationship between the futures price and the

Com m odities differ from financial assets in several ways. For
expected future spot price depends on w hether the return
exam ple, they are subject to transportation costs and storage
on the underlying asset is positively or negatively correlated
costs. Com m odities are also usually held in inventory so that
with the return on the stock m arket. A positive correlation
they can be used in some way.
implies the futures price is less than the expected future
In order to understand a commodity's price, it is therefore nec spot price. A negative correlation implies the futures price is
essary to understand how the commodity is used as well as how greater than the expected future spot price. W hen the correla
it is influenced by factors such as the seasonality of supply and tion is zero, the futures price is equal to the expected future
dem and. spot price.

Q U ES T IO N S

11.1 W hat differences between com m odities and financial 11.7 Explain why some equations show the relationship
assets are important for explaining why their futures between the forward price F a n d the spot price S for a
prices are determ ined differently? non-dividend-paying stock to be F = S(1 + R)T, while
others show it to be F = S e RT.
11.2 W hat are the three main energy products on which
futures contracts trade? 11.8 Suppose that speculators tend to take short futures posi
11.3 Explain how C D D and HDD are defined for w eather tions on an asset, while and hedgers take long futures
futures. position. W hat would Keynes argue about the ability of

futures prices to predict expected future spot prices?
11.4 Define the lease rate of a com m odity, such as gold.
11.9 How can a long position in an asset at a future time be
11.5 W hat is meant by the convenience yield of a commodity?
created synthetically?
11.6 W hat is meant by the cost of carry of a commodity?
11.10 Under what circum stances does CAPM suggest that the
futures price equals the expected future spot price?
Practice Questions
11.11 W hat is meant by mean reversion? price will increase im m ediately. If news reaches the
market that the price will almost certainly decrease by
11.12 Expla in the main differences between agricultural and
metal com m odities. 2 0 % in the next three months, the price does not neces
sarily decrease im m ediately." Discuss this statem ent.
11.13 A gold producer enters into a forward contract with an
investm ent bank to sell 1 0 , 0 0 0 ounces of gold for 11.18 Suppose that the storage costs for crude oil are USD 2
per barrel per year payable at the beginning of the
USD 1,200 per ounce. Explain how the bank would
year. The current spot price is USD 75, and the two-year
hedge its risk without using futures contracts.
futures price is USD 72. The risk-free interest rate is %8
11.14 If the current spot price of gold is USD 1,250, the one-
per annum (compounded annually). Estim ate the conve
year futures price of gold is USD 1,300, and the risk-free
nience yield of crude oil per year.
rate is 5% per annum (annually com pounded), what is an
estim ate of the implied gold lease rate?
11.19 If the futures price equals the future spot price for a
financial asset, what is the return of that asset?
11.15 W hy is it not possible to calculate the futures price for a
com m odity such as copper in the same way as a futures
11.20 Suppose that F-, and F are the futures prices on the
2
price for a financial asset? same com m odity with maturities t and t with t > t|.
-1 2 2
Storage costs are negligible. The risk-free rate is R for all

11.16 "The convenience yield is the lease rate for an asset if it
m aturities. Use an arbitrage argument to show that:
were possible to borrow it." Discuss this statem ent.
F < F-, (1 + R)t2_tl
2
11.17 "If news reaches the market that the price of oil will
almost certainly increase by 2 0 % in three months, the
A N SW ER S
11.1 Com m odities have storage costs and transportation 11.13 The bank can borrow gold from a central bank (paying
costs. They are in most cases not held for investment the lease rate) and sell it in the market (investing the pro
purposes. M arket participants who own com m odities ceeds of the sale). When the forward contract matures,
want to use them in some way. the bank buys the gold from the gold producer and
11.2 O il, natural gas, and electricity returns it the central bank.
11.3 The C D D , cooling degree for a day, is defined as 11.14 Let € be the implied gold lease rate (annually com
max(A — 65, 0) where A is the average of the highest pounded). Then:
and lowest tem perature in degrees Fahrenheit during 1.05

1,300 = 1,250 X
the day. HDD, heating degree for a day, is defined as 1 + €
max(65 - A , 0). so that
11.4 The Iease rate is the rate at which the com m odity can be
X 1.05 - 1 = 0.0096
borrowed from its owner. 1,300
11.5 The convenience yield of an asset is the benefit of own that is, the implied lease rate is 0.96% .
ing the product so that it is readily available for use in a
11.15 Copper is not an investment asset. Individuals and com
production process.
panies that own copper plan to use it in some way. That
11 . 6 The cost of carry is the cost of owning the com m odity. It is, they need to have copper in inventory and are not
is sum of the cost of financing and the cost of storing it. prepared to sell copper and buy it back in the futures
11.7 T h e f i rst equation assumes that R is the annually com market when futures prices are relatively low com pared
pounded risk-free interest rate. The second equation to the spot prices.
assumes that R is the continuously com pounded risk-free 11.16 For an asset such as copper, one can argue that the lease
interest rate. (Continuous compounding will be discussed rate that would be charged com pensates the lender for
in Chapter 16.) benefits of having it on hand, that is, it com pensates the
11.8 Keynes would argue that the futures price overstates the lender for the convenience yield.
expected future spot price because speculators are tak 11.17 If oil prices are expected to increase sharply, speculators
ing risks and require an expected positive profit to com will buy oil and store it. This means that the price will rise
pensate for such risk. Hedgers are reducing risk and may until the financing cost plus the storage cost equals the
be satisfied with a negative expected return. expected profit. If oil prices are expected to fall sharply,
11.9 An investor can borrow the present value of the futures there is no similar trade involving spot oil because those
price from the bank and take a long position in the cor who hold oil in inventory need it to feed their refineries.
responding futures contract on one unit of the asset. 11.18 The present value of the storage costs per barrel over
11.10 The future price equals the expected future spot price two years is
when the return from the underlying asset is uncorrelated 2
+ 3.85
with the market.
2
1.08
11.11 Mean reversion is the tendency for a market variable to The convenience yield Y is given by solving:
be pulled back to some central value over time.
72 = (75 + 3.85) X
11.12 Agricultural com m odities have higher storage costs. m
Prices of agricultural products tend to be seasonal and

The solution is
are influenced by the weather. Metal price may depend
on the success of exploration to find new mines. Also, X 1.08 - 1 = 0.130
exchange rates, extraction methods, actions by foreign
governm ents, and cartels can all be possible driving fac that is, the convenience yield is about 13% per year.
tors of a metal price.

11.19 The return of the asset is the risk-free interest rate. This The loan is repaid at time t and the asset is sold for F2.
2
is evident from the equations for the futures price. For The cash flows are
exam ple, when there is no income generated by the
Tim e t-p —F- + F-| = 0, and
1
underlying asset, the futures price is the spot price com

Tim e t2: F — F-,( 1 + R)t2_\
pounding forward at the risk-free rate. When there is
2
income generated by the asset during the tenor of the This simple strategy is certain to lead to a profit at time
futures, the futures price is adjusted for the income and t if:
2
then com pounded forward at the risk-free rate. F > F-,(1 + R

2 ) t 2 _ t 1
11.20 A trader can enter into a long futures contract with Thus, the prices will adjust such that:
maturity t| and a short futures contract with maturity t2.
F < F-,(1 + R)t2~tl
A t time t| F-1is borrowed and the asset is bought for F-1.
2
Options Markets
Learning Objectives
Describe the types, position variations, payoffs and profits, Describe how trading, commissions, margin requirem ents,
and typical underlying assets of options. and exercise typically work for exchange-traded options.
Explain the specification of exchange-traded stock option Define and describe warrants, convertible bon ds, and
contracts, including that of nonstandard products. em ployee sto ck options.
Explain how dividends and stock splits can impact the

term s of a stock option.
147
BOX 12.1 A HISTORY O F O PTIO N S
O ptions have been traded for thousands of years. Greek understood the relationship between the prices of European
philosopher Thales of Miletus, who lived over 2,500 years put options and call options. (This is called the put-call parity
ago, may have been one of the first options traders. A cco rd and it will be explained in the next chapter.)
ing to legend, he predicted that a coming olive harvest would
The popularity of options gradually increased during the late
be larger than normal. Acting on his prediction, he paid olive
nineteenth and early twentieth centuries. During this tim e,
press owners a fee for the right to use the presses at harvest
options were traded over-the-counter by brokers who negoti
tim e. His forecast for the harvest proved correct and he was
ated prices and matched buyers and sellers. The Put and Call
able to sell his call options on the presses for a sizable profit.
Brokers and Dealers Association was form ed to help with this
The seventeenth century Dutch tulip bubble also involved matching process.
options trading. The so-called "tulip mania" led to exorbi
The Chicago Board of Trade, which was concerned about a
tantly high prices for tulip bulbs. As a result, tulip w hole
decline of trading interest in com m odity futures, decided to
salers would buy call options on tulips to protect against
create an options exchange in 1968. Five years later, the C h i
unexpected price increases. M eanwhile, tulip growers would
cago Board O ptions Exchange (C B O E) was form ed. Options
buy put options to protect them selves in case the price
(initially only calls) were standardized for trading and (more
decreased.
importantly) the O ptions Clearing Corporation (O CC ) was
In 1637, the bubble burst and tulip prices collapsed. Many established. The O C C required option sellers to post col
put option sellers subsequently defaulted, giving options a lateral (in the form of margin) as way to protect buyers from
bad name throughout Europe for some tim e. And while an default risk.
options market developed in England during the late seven
That same year, the famous Black-Scholes Merton options
teenth century, options were later banned in that country for
valuation model was published. The model, which is used to
over years.
value options under various volatility assumptions, proved to
1 0 0
In the nineteenth century, Am erican financier Russell Sage be a useful tool for traders. By 1974, 20,000 call options
sold put and call options for stocks listed on New York contracts were traded daily on the C B O E . Three year later,
Stock Exchange. He is believed to be the first person who the exchange began trading put options . 1
This chapter discusses how options markets are organized. 12.1 C A LLS A N D PUTS
Later chapters in this book cover the properties of options and
options trading strategies. A European call (or put) option gives the buyer the right to buy
There are important differences between options and forwards/ (or sell) an asset at a certain price on a specific date. An
futures. With forwards and futures, a trader is obligated to buy Am erican call (or put) option gives the buyer the right to buy (or
or sell the underlying asset at a certain price. As a result, it costs sell) an asset at a certain price at any time before and during the
Q
nothing to take a long or short forward/futures position .2 * By specified date. The date specified in the option is known as the
contrast, an option involves paying a certain amount (called the expiration date (also called the maturity date). Most (though not
premium) to obtain the right to buy (or sell) an asset at a certain all) exchange-traded options are Am erican. By contrast, many of
price in the future. However, this right does not have to be the options traded in the over-the-counter market are European.
exercised. The price at which an asset can be bought or sold using an
In the U .S., tens of millions of options are traded daily on option is referred to as the strike price (also called the exercise
exchanges such as the C B O E , N A SD A Q , the New York price). In the early days of options trading, the strike price was
Stock Exchange, and the International Securities Exchange. typically set as the asset's spot price at the time the option was
Exchanges trading options outside of the U.S. include sold. Today, it is possible to buy or sell options with a range of
the Eurex, the National Stock Exchange of India, and different strike prices.
BM & FBO VESPA . O ptions are also traded globally in the over- Am erican options are more difficult to analyze than European
the-counter market. options because they can be exercised at any time before
Note that the term s European and Am erican have nothing to do

1 See w w w .cboe.com for a history of the exchange.
with the location of the option traders or the exchange. Am usingly,
2 However, it is necessary for traders to post margin in the case of a Berm udan option is an option that is partly A m erican. It can be
futures trading as explained in earlier chapters. exercised on pre-specified dates during its life, but not at any tim e.
maturity. W hile European options can be valued using the
Black-Scholes Merton model (which will be discussed in
Valuation and Risk M odels), Am erican options can only be
valued by using numerical procedures such as binomial trees
(also discussed in Valuation and Risk M o d els). As discussed in
the next chapter, however, one can argue that in certain situa
tions an Am erican option should never be exercised early and
therefore must have the same price as its European counterpart.
Moneyness
O ptions can be in-the-money, at-the-m oney, or out-of-the-
m oney. This is referred to as their m oneyness. Under the
sim plest definition of these term s, it is imagined what would
happen if the option could be exercised today. (This is done
even if the option cannot be exercised today.) An option that
would give a positive payoff if exercised today it is referred to call option is shown. The cost of the call is USD 4,
as in-the-money. If it would give a negative payoff, the option is and the strike price is USD 60.
referred to as out-of-the m oney. An option that is at-the-m oney
would give a payoff of zero.
will buy the asset for USD 60 and im m ediately sell it for more
A call (put) option is in-the-money when the asset price is than USD 60). When considering the price paid to buy the
greater (less) than the strike price. This is because the option option, the trader's total profit (if the option is exercised) will be
(if exercised today) would allow an asset to be purchased (sold) USD S - 60 - 4 .5 If the option is not exercised, the trader's loss
for less (more) than its current price. An investor who exercises will be USD 4.
an in-the-money call option can sell the asset im m ediately for
The profit to the trader is shown in Figure 12.1. Note that
a positive payoff equal to the excess of the current asset price
som etim es the trader can exercise the option and still suffer a
over the strike price. An investor exercising an in-the-money put
net loss. This happens when S is between USD 60 and USD 64.
option could buy the asset and sell it for the strike price to make
Would it be better for the trader not to exercise the option
an immediate profit equal to the excess of the strike price over
when the asset price is in this range? The answer is no. The
the current asset price.
trader will always lose USD 4 if the option is not exercised.
Likewise, a call (put) option is out-of-the-money when the asset M eanwhile, the loss from exercising when S is greater than
price is less (greater) than the strike price. In these situations, USD 60 is always less than USD 4 (e.g ., when S = USD 61, the
immediate exercise (if allowed) would not be profitable because loss is USD 3).
the payoff would be negative. An option is referred to as at-the-
Now consider the situation of the call option's seller. The option
money when the strike price equals the asset price .4
seller will make a profit of USD 4 (the option premium) if S is less

than 60 because the option will not be exercised at that price. If
Profits from Call Options S is between USD 60 and USD 64, the option will be exercised
and the seller's profit will be less than USD 4. If S is above
As an exam ple, suppose a trader buys an out-of-the-money
USD 64, the seller will suffer a loss. Figure 12.2 shows the seller's
European call option with a strike price of USD 60 for an asset
profit as a function of S. It should be no surprise that Figure 12.2
currently worth USD 55. The option premium is USD 4 and the
is the mirror image of Figure 12.1. O ptions (like other deriva
expiration date is six months from the day of transaction.
tives) are zero-sum gam es in the sense that one side's gain
Let S be the asset price on the expiration date. If S is less than always equals the other side's loss.
USD 60, then the option will not be exercised. If S is greater
than USD 60, the call option will be exercised (i.e., the trader
5 Note that we simplify m atters by ignoring the im pact of discounting.

4 O ther definitions of m oneyness are som etim es used by traders as well. Because the cost of the option is paid at tim e zero and the payoff is six
(See J . Hull, O ptions, Futures and O th er D erivatives, 10th edition, months later, the present value of the profit is PV(S — 60) — 4 where PV
section 20.4.) denotes present value.
Chapter 12 Options Markets ■ 149

Fiaure 12.3 Profit as a function of asset price on the
expiration date for the buyer of a European put option
is shown. The cost of the put is USD 5, and the strike
Fiaure 12.2 Profit as a function of asset price on the price is USD 60.
expiration date for the seller of a European call option
is shown. The cost of the call is USD 4, and the strike
price is USD 60.
Profits from Put Options

Now suppose a trader buys an out-of-the-money European put
option with a strike price of USD 60 for an asset currently worth
USD 62. The option premium is USD 5, and the expiration date
is three months from the date of transaction.
If the asset price on the expiration date is less than USD 60,
the put option will be exercised (i.e., the trader will buy the
asset at price S when the option expires and im m ediately sell
Fiqure 12.4 Profit as a function of asset price on the
expiration date for the seller of a European put option
it at USD 60). O nce again, accounting for the price paid to buy
the option, the trader's profit is then USD 60 - 5 - S .6 If S is
is shown. The cost of the put is USD 5, and the strike
price is USD 60.
more than USD 60, the option will not be exercised and the
trader's loss will be USD 5.
The profit to the trader is shown in Figure 12.3. As in the case

of call options, som etim es the trader exercises the option and Payoffs
still suffers a loss. This happens when S is between USD 55
Instead of characterizing a European option by its profit, we can
and USD 60. Exercising the option within this price range will
plot the payoff as a function of the asset price on the expiration
not produce a profit, but it will reduce the trader's loss.
date. These plots show the value of the option at maturity and
As in the case of call options, the profit of the corresponding put do not account for how much was paid to purchase the option.
option seller is the mirror image of the profit of the option buyer. As illustrated in Figures 12.1 through 12.4, there are four pos
If S is greater than USD 60, then the option will not be exercised sible option positions:
and the seller will gain the USD 5 paid by the buyer (the option
1. Long call (Figure 12.1),
premium). If S is between USD 55 and USD 60, the trader will
make a profit of less than USD 5. If S is below USD 55,
2. Short call (Figure 12.2),
the trader suffers a loss. Figure 12.4 shows the trader's profit as 3. Long put (Figure 12.3), and
a function of the asset price. 4. Short put (Figure 12.4).
Selling (or shorting) an option is also referred to as writing the
6 As in the case of call options, the im pact of discounting is ignored. In
fact, the present value of the profit is PV(60 — S) — 5 where PV denotes option. Figure 12.5 shows the payoff from each of the option
present value. positions previously described when the strike price is 60.
Maturity
Stock options on the C B O E are assigned one of the following
maturity cycles:
• January, A pril, Ju ly, and O ctober (referred to as the January

cycle);
• February, May, August, and Novem ber (referred to as the
February cycle); and
• March, Ju n e, Septem ber, and D ecem ber (referred to as the
March cycle).
Prior to the third Friday of the current month, options trade

with m aturities in the current month, the following month, and
the next two months in the cycle. Following the third Friday
Fiaure 12.5 Payoffs from option positions when
of the current month, options trade with m aturities in the next
strike price is 60 are shown. The asset price on the
month, the month after that, and the next two months in the
expiration date is S.
expiration cycle.
For exam ple, IBM is on a January cycle. This means that at the
beginning of A pril, options trade with maturities in April, May,
Denoting the asset price on the expiration date by S and the Ju ly, and October. After the third Friday in April, they trade with
strike price by K, the payoffs from the option positions are as expirations in May, Ju n e, Ju ly, and October. The precise matu
follows: rity date is the third Friday of the month (e.g ., April 19, 2019, for
Long Call: max(S — K, 0), an April 2019 option). Trading takes place every business day
(8:30 p.m . to 3:00 p.m . Chicago time) until the maturity date.
Short Call: —m ax(S — K, 0) = min(/< — S, 0),
The C B O E also offers short-term options called W eeklys and
Long Put: m ax(« — S, 0), and
long-term options known as LE A P S (long-term equity anticipa
Short Put: - m a x (K — S, 0) = min(S — K, 0). tion securities). W eeklys mature on Fridays, other than the third
W hile the payoff from a European option is related to the asset Friday of the month, and are typically offered on several such
price at expiration, what is called the intrinsic value is based Fridays. LEA PS on individual stocks mature on the third Friday of
on the current asset price (which we denote by S0). Intrinsic January and provide maturities of up to three years.
value measures the value the option would have if it could

only be exercised im m ediately. The intrinsic value of a call is Strike Prices
max(S — K, 0) and the intrinsic value of a put is m ax(« — S0, 0).
0
The C B O E sets option strike prices as multiples of USD 2.50

when the current price of the underlying asset is between USD 5
12.2 EXCH AN GE-TRADED OPTIONS and USD 25. When the current price is between USD 25 and
USD 200, strike prices are usually multiples of USD 5. When the
ON STOCKS *
current price is greater than USD 200, they are usually multiples
of USD 10. Strike prices are also adjusted for dividends and
This section describes how stock options are traded on the
splits (as will be described in the next section).
C B O E . This is the largest options exchange in the world and
trades over a billion contracts each year. The options on individ Initially, the three strike prices closest to the current price are
ual stocks are Am erican-style (i.e., they can be exercised at any typically listed. As the stock price moves, more strike prices
time before their maturities). When a trader with a long position are introduced. For exam ple, suppose the stock price is USD 19
decides to exercise, the O ptions Clearing Corporation uses a when trading for a new options contract begins. Initially, strike
random procedure to choose a trader with a short position to be prices of USD 17.50, USD 20.00, and USD 22.50 would be
exercised against. (This trader is referred to as being a ssig n ed .) traded. If the stock price moved below USD 17.50, options with
O ptions that are in-the-money at maturity are usually exercised a strike price of USD 15 would start trading. If the price moved
autom atically. A single option contract on the C B O E is the right above USD 22.50, options with a strike price of USD 25 would
to buy or sell 1 0 0 shares. start trading.

These rules can result in many different options contacts being ETP Options
available (e.g ., if there are five strike prices for each o fte n
maturity dates, there could be 50 tradeable calls and 50 trade- There are also C B O E options on many exchange-traded
able puts). All options of the same type (i.e., calls or puts) are products (ETPs) such as exchange-traded funds (ETFs). An
referred to as a class. All options of a particular class with a E T F can be designed to track an equity index, bond index, a
specific maturity date and strike price are referred to as an com m odity, or a currency.
option series. For exam ple, call options on IBM maturing in April ETP options are like options on individual stocks in that they are
2019 form an option series. Am erican-style and involve physical settlem ent (i.e., settlem ent
by delivery of the underlying asset).
Dividends and Stock Splits

Non-Standard Products
Cash dividends usually do not affect the term s of a stock
option .7 However, exceptions are som etim es made when the To compete with the over-the-counter options market, the C B O E
cash dividend is unusually large. Specifically, if the cash dividend also offers some non-standard options. For example, FLE X options
is more than 10% of the stock price, the O ptions Clearing Cor are options with non-standard terms that sometimes include
poration forms a com m ittee to determ ine whether adjustments • Non-Standard strike prices,
should be made.
• Non-Standard maturity dates, and
In contrast, stock splits do lead to strike price adjustm ents.
• Variations in style (American or European).
For exam ple, if a company announces a 5-to-1 stock split (i.e.,
each share is replaced by five new shares), the strike price will O ther non-standard options on the C B O E include Asian and
be reduced to one fifth of the original price and the number of Cliquet options on indices. Asian options provide a payoff based
options is multiplied by five. on the average price of the underlying asset during the life of
the option. Cliquet options provide a payoff equal to the sum
Sim ilarly, stock dividends also lead to strike price adjustm ents.
of the monthly capped returns provided by the asset (if this sum
Note that a 10% stock dividend means shareholders receive
is positive). Further discussion of non-standard options can be
one new share for each ten shares owned (this is identical to
found in Chapter 15.
an 11 -to-10 stock split). Using the stock split rule, the strike
price will be reduced to ten-elevenths of its original level Some non-standard contracts introduced by the C B O E have not
and the number of options is m ultiplied by 11/10. All these been particularly successful. For exam ple, credit event binary
adjustm ents are designed to keep the positions of buyers options (C EBO s) and deep-out-of-the-money (DO O M ) put
and sellers unchanged by incidents of stock splits or stock options were attem pts to com pete with the credit derivatives
dividends. market. However, both options have been discontinued .8
Index Options 12.3 TR A D IN G

In addition to options on individual stocks, the C B O E also offers In recent years, options exchanges have replaced the
options on equity indices. Among these include the S&P 500 open-outcry trading system with electronic matching platforms.
Index, the S&P 100 Index, the Dow Jo nes Industrial Average, However, exchanges still rely on market makers to provide
the Russell 1000, the FT S E 100, and the FTSE China 50. Many liquidity to the options market. These market makers will quote
of these options are European-style rather than Am erican-style. option bid and ask prices when they are asked to do so, while
Final settlem ent is in cash and calculated with reference to exchanges set upper limits on the size of the bid-ask spread
the value of the index at a pre-specified tim e. Index W eeklys (i.e., the difference between the bid and ask quotes).
(maturing on W ednesdays and Fridays) and LEA PS (maturing in
To take an option position, a trader customarily places an order
D ecem ber and June) are also traded.
through a broker, who charges a commission. The types of orders
that can be placed are like those for futures described in Chapter 7
7 In the early days when options w ere traded over-the-counter, the strike 8 C E B O s give positive payoff when the underlying com pany defaults
price was reduced by the dividend per share when a cash dividend was before the option matures and D O O M options have strike prices that
declared. are extrem ely low com pared with the initial price.
(e.g., market order, limit order, stop order, and so on). Commissions This calculation is repeated each day, with the current option
vary from broker to broker and can be as low as USD 5 per trade. premium replacing the proceeds of the sale in determining
the new margin requirements. If the margin requirement has
Like futures, exchange-traded options can be closed out by tak
increased, funds must be added to the trader's margin account. If
ing an offsetting position. For exam ple, a trader owning a put
the margin requirement has decreased, funds can be withdrawn.
with a certain strike price and maturity can exit from the position
by selling a put with the same strike price and maturity. As with The C B O E Margin Manual has special rules governing traders
the futures m arket, the open interest in the options market m ea with portfolios consisting of long and short option positions
sures the number of outstanding contracts. (which are perhaps combined with positions in the underlying
asset). For exam ple, no margin is required on a covered call posi
The C B O E imposes position limits and exercise limits on traded
tion (i.e., a written call plus a long position in the asset underlying
options. These are designed to prevent the market from being
the call). This is because there is no reason to suppose that the
unduly influenced by one investor (or a group of investors acting
trader will default; as long as the position is maintained, the asset
in concert). A position limit is defined as the number of con
is available to be delivered in the event the option is exercised.
tracts an investor can hold on the same side of the market (i.e.,
long calls and short puts are considered to be on one side of the As with futures, options margins are handled by the members of
m arket, while short calls and long puts are considered to be on the O ptions Clearing Corporation (O C C ). All option trades must
the other side of the market). The exercise limit is the maximum be cleared through an O C C member. Non-member brokers
number of contracts that can be exercised in five business days. must arrange to clear their clients' trades with a member. Thus,
It is usually the same as the position limit. all brokers maintain margin accounts with O C C m em bers, while
all end users maintain margin accounts with their respective
brokers. The margin requirements for these accounts must be at
12.4 MARGIN REQUIREM ENTS least as great as those specified by the O C C for its members.
Margin requirements have already been discussed in Chapter 5.

Recall that, in the U.S., a stock can be purchased by borrowing 12.5 OVER-THE-COUNTER M ARKET
up to 50% of its price. Such a transaction is known as buying on
margin. Options with maturities less than nine months cannot be Like the exchange-traded options m arket, the over-the-counter
bought on margin, and the full price must be paid upfront. Options (O TC) options market is substantial. W hile options on individual
with maturities greater than nine months can be bought on margin, stocks trade primarily on exchanges, options on foreign curren
but no more than 25% of the purchase price can be borrowed. cies, interest rates, and many other financial variables are traded
actively in the O T C market.
If a trader pays cash for an option, there would be no margin
requirem ents because the trader has no future liabilities. How The main advantage of the O T C market is that financial insti
ever, the seller of an option does have potential future liabilities. tutions can tailor options to meet the specific needs of their
As explained in Chapter 5, the C B O E margin requirements for clients. Option characteristics in the O T C market (e.g ., strike
a short call position include 1 0 0 % of the sale's proceeds along prices, maturity dates, and the tim es when options can be exer
with the greater of the following two values: cised) can differ from those available on the exchanges. The size
of the typical options transaction in the O T C market is large and
1 . 2 0 % of the share price less the amount the option is out of
the options often last longer than those traded on exchanges.
the money, and
O T C options can also be exotic (i.e., have non-standard struc
2. 1 0 % of the share price. tures). Exotic options will be discussed in Chapter 15.
The C B O E margin requirements for a short put include 100% of
the sale's proceeds along with the greater of the following two
calculations: 12.6 WARRANTS AND CO N VERTIBLES
. % of the share price less the amount the option is out of
W arrants are options issued by a corporation. They are usually
1 2 0
the money, and call options on the corporation's own stock, but they can also
2. 1 0 % of the strike price. be options to buy or sell another asset (e.g ., gold). O nce issued,
they are often traded on an exchange.
For options on indices, the 20% in this calculation is replaced by
15%. This recognizes that the volatility of a stock index is usually To exercise a warrant, the holder needs to contact the issuer.
less than that of an individual stock. When a call warrant on the issuing company's stock is exercised,

the firm issues more of its stock. O nce that happens, the warrant
holders can then buy the stock at the strike price. BOX 12.2 EM P LO Y EE STO C K
W arrants can be used by firms to make debt issuances more OPTION A C CO U N TIN G TRICKS
attractive to investors. For exam ple, suppose a company's stock It used to be the case that em ployee stock options had
price is currently USD 40 and the firm is planning a debt issue. no effect on a firm's profits if they had a strike price set to
It might choose to add two warrants to each USD 1,000 bond; current stock price. This made them an extrem ely attrac
tive form of compensation to senior m anagem ent. How
each warrant would give the holder the right to buy one share
ever, accounting rules have since changed and em ployee
at USD 45 on the expiration date. The bondholders would then stock options must now be valued (and expensed) at the
have a stake in the fortunes of the company beyond the desire tim e of issue. A s a result, they are not as widely used as
to see it avoid a default. they once w ere . *
9
•
A convertible bond (also referred to as a convertible) is similar As an exam ple of the way things used to work, suppose
that a firm's stock price has increased from USD 42 to
to a warrant. Specifically, a convertible is a bond that can
USD 50 over the past w eek. It was once tem pting for com
be converted into equity using a pre-determ ined exchange
pany executives to backdate an issue of at-the-money
ratio. For exam ple, suppose a company's current share price is em ployee stock options by three weeks and set the strike
USD 40. It might choose to issue ten-year bonds, each with a price to USD 42. This would increase the value of the
USD 1,000 par value, that can be converted into 20 shares at stock options for the executives while at the same time
any tim e after four years. When an investor chooses to convert, reducing its cost to the company (because at-the-money
options had no effect on the firm's profit). This practice
the company simply issues more of its stock to be exchanged
was (and still is) illegal, but academ ic studies have
for the bonds. Like warrants, convertible bonds are often traded produced evidence showing that it was once
on exchanges. w idespread As a result, the U.S. Securities and
. 1 0
Exchange Commission now requires option grants be

reported within two business days.
12.7 EM PLO YEE STO CK OPTIONS

Em ployee stock options are call options granted by a company
to its em ployees. They differ from exchange-traded options in SUMMARY
several ways. The following are exam ples.
A call (or put) option gives the holder the right to buy (or sell)
• There is usually a vesting period during which options cannot
an asset for a certain price by a specific date. There are four
be exercised. These periods can last up to four years.
possible positions: long call, short call, long put, and short put.
• Em ployees may forfeit their options if they leave their jobs
Exchanges such as the Chicago Board O ptions Exchange
(voluntarily or involuntarily) during the vesting period.
(C B O E) have rules for determ ining option strike prices and
• When em ployees leave after the vesting period, they usually m aturities. On the C B O E , each stock option contract gives the
forfeit options that are out-of-the-money. They may also have holder the right to buy or sell 100 shares. These options are
to exercise their in-the-money options im m ediately. Am erican-style and the expiration date is the third Friday of the
• Em ployees are not perm itted to sell their stock options to a expiration month. The C B O E sets strike prices with intervals
third party. that are multiples of USD 2.50, USD 5, or USD 10. In addition to
stock options, the C B O E also trades options on market indices,
Because the em ployee options cannot be sold, they are usually
exchange-traded funds, and other products.
exercised earlier than their exchange-traded counterparts. This
point is discussed further in the next chapter.
It is debatable whether em ployee stock options align with the

interests of shareholders and managers. Note that if the stock
9 Many com panies now award shares of their stock instead of options to
price does well, both shareholders and managers will gain. If the
em ployees.
stock price declines, however, the shareholders typically suffer
10 See D. Yerm ack, "G o o d tim ing: C E O stock option awards and com
more than managers. Thus, granting managers shares (rather
pany news announcem ents," Journ al o f Finance, 52 (1987): 449-476;
than stock options) can create a better alignment of interests. E. Lie, "O n the timing of C E O stock option aw ards," M anagem ent S ci
en ce, 51, 5 (May 2005): 802-812; R. Heron and E. Lie, "D o es backdating
Em ployee stock options are commonly used by start-up com pa explain the stock option pattern around executive stock option grants,"
nies that cannot afford to pay a com petitive salary. Journal o f Financial Econom ics, 83, 2 (February 2007): 271-295.
Option term s are not adjusted for cash dividends, but they brokers who are not m em bers, and these brokers will in turn
are adjusted for stock dividends and stock splits in a way require margin from their clients.
that keeps the positions of both the option buyer and seller
W arrants are options issued by com panies, usually on their
unchanged.
own stock. Convertible bonds are bonds that give their holders
M arket makers are often used to improve the liquidity of the the right to convert the bond into a certain amount of equity
options market. Option sellers (i.e., writers) must post margin at pre-determ ined tim es. Em ployee stock options are options
to ensure that they will not renege on their obligations. The issued by a company on its own stock and are reserved for its
O ptions Clearing Corporation operates in a similar manner to a own em ployees. These securities all have the property that,
futures clearinghouse, in that it requires margin from its mem when the holder exercises his or her right to obtain shares in the
bers selling options. These members also require margin from com pany, the company issues more stock.

QUESTIONS

12.1 W hat is the strike price of an option? 12.6 W hat is a LEA P?
12.2 W hat is the difference between an Am erican and a 12.7 W hat is (a) a position limit and (b) an exercise limit in the
European option? options market? W hy are they used?
12.3 W hat is the difference between buying a call and selling 12.8 Does an option w riter have to keep the position until the
a put? end of the option's life? Explain your answer.
12.4 W hat is the difference between the profit and payoff 12.9 W hy is there no margin requirem ent for a covered call?
from an option? 12.10 W hat is the difference between a warrant and an option
12.5 W hat is the intrinsic value of an option? traded by the C B O E ?
Practice Questions
12.11 Explain the meaning of the term at-the-money. 12.17 An option has a strike price of USD 50. The company
declares a 3-for-1 stock split. W hat effect does this have
12.12 Suppose a European call option to buy a share for USD 50
in nine months costs USD 6 . Under what circumstances on the strike price?
will the holder of the option make a profit? Under what 12.18 An option has a strike price of USD 50. A cash dividend
circum stances will the option be exercised? of USD 0.50 is announced with an ex-dividend date
before the end of the life of the option. W hat effect does
12.13 If the option in Question 12.12 is a put, rather than a call,
how does your answer change? this have on the strike price?
12.14 A stock price is USD 41 when options for a certain maturity 12.19 W hy do you think em ployee stock options cannot be
traded?
are introduced. What strike prices would initially trade?
12.15 An option is on the March cycle. W hat options (in 12.20 How would you design an executive stock option that
addition to W eeklys and LEAPs) trade on August 1? pays off only if the company outperform s others in the
same industry?
12.16 An option has a strike price of USD 50. The company
declares a 20% stock dividend. If the O C C were to adjust
the strike price, what would be the new strike price?
ANSW ERS
12.1 The strike price of an option is the price at which the satisfy investors when they exercise the options. In an
holder of a call (or put) can buy (or sell) the underlying exchange-traded option, the number of options expands
asset at a future tim e. as there are more trades and contracts as traders close
12.2 A European option can be exercised only on its maturity out their positions. The O ptions Clearing Corporation
date. An Am erican option can be exercised any tim e up administers margins and ensures that investors are able
to the maturity date. to exercise the options they own.
12.3 Buying a call gives a trader the right to buy an asset at a 12.11 An at-the-money option is usually defined as an option
certain price in the future. Selling a put gives the other where the asset price equals the strike price.
side the right to sell the asset at a certain price at a 12.12 The option will be exercised if the share price on the
certain tim e in the future. Buying a call gives a payoff of expiration date is greater than USD 50. The holder of
m ax(S — K, 0) where S is the stock price when the option the option will make a profit if the share price at expiry
is exercised and K is the strike price. Selling a put gives a is greater than USD 56 and present value discounting is
payoff of min(S — K, 0). Margin has to be posted when a ignored.
put is sold but not when a call is bought.
12.13 The option will be exercised if the share price at expiry
12.4 A profit calculation considers the initial option price. A is less than USD 50. The holder of the option will make
payoff calculation does not involve the transacted option a profit if the share price on the expiry date is less than
price. USD 44.
12.5 The intrinsic value of a call option is max(So — K, 0) and 12.14 Strike prices of USD 35, USD 40, and USD 45 would
the intrinsic value of a put option is max(/C — S0, ) where
0 normally be introduced.
S0 is the current stock price and K is the strike price. 12.15 O ptions trade with maturities in August, Septem ber, and
12.6 LEA PS (Long-Term Equity Anticipation Securities) are D ecem ber of the current year and March of the following
long term options (with maturities up to three years) year. W eeklys and LEA PS may also trade. The maturity is
traded by the C B O E . the third Friday of the month.
12.7 The position limit is a limit on the size of the position 12.16 The strike price becom es USD 50 X (5/6) = USD 41.67.
that an investor (or group of investors acting together)
12.17 The strike price becom es USD 50 X (1/3) = USD 16.67.
can have in option contracts. For this purpose, the posi
12.18 There is normally no effect on the strike price when a
tions in long calls and short puts are summed and the
cash dividend is declared. An exception may be made
positions in long puts and short calls are summed. An
when the dividend is large— more than % of the stock
exercise limit is a limit on the number of options that can
1 0
price.
be exercised within a five-day period. The purpose of
the limits is to prevent an investor (or group of investors) 12.19 Em ployee stock options are intended to provide an
from unduly influencing the market. incentive for the em ployee to work hard and increase
value for shareholders. If the em ployees were allowed to
12.8 The w riter of an option contract can buy the same option
sell their options without constraints, this incentive would
contract to exit from the position.
no longer exist.
12.9 When a trader owns a stock and has written a call option
on the stock, the stock is available to be delivered w hen
12.20 One possibility is to make the strike price proportional to
an index of the prices of the stocks of other com panies in
ever the call option is exercised. Thus, the credit risk is
the same industry as the company. The company would
covered and no margin requirem ent is needed.
then have to outperform its com petitors for the options
12.10 In a warrant, a fixed number of options are issued by to move in-the-money.
a company and the company guarantees that it will

Properties of
Options
Learning Objectives
Identify the six factors that affect an option's price. and w ithout dividends, and express it in term s of for
ward prices.
Identify and com pute upper and lower bounds for option
prices on non-dividend and dividend paying stocks. Explain and assess potential rationales for using the early
exercise features of Am erican call and put options.
Explain put-call parity and apply it to the valuation of
European and Am erican stock options, with dividends
159
The price of stock option can depend on : 1 Despite this profit, however, the option should not be exercised
before maturity if interest rates are positive. To understand why
• The price of the underlying stock,
this is so, note that the option owner is in one of two situations.
• The strike price,
Situation 1: The investor wants to have a position in the stock.
• The risk-free rate,
Situation 2: The investor does not want to have a position in
• The volatility of the stock price,
the stock.
• The time to maturity, and
In Situation 1, the option holder will not sell the stock after
• The dividends to be paid during the life of the option.
exercising the option (because they want to have a position in
Chapters 14 and 15 of Valuation and Risk M o d els will discuss the stock). Thus, he or she is better off holding the option until
how these variables are combined to value European and expiry. To understand this, consider two outcomes for Situation 1.
Am erican options under the Black-Scholes Merton assumptions.
O utcom e 7: The stock price is greater than USD 30 on the
This chapter produces upper and lower bounds for options
expiry date.
prices that do not depend on the valuation model used. It
also considers factors that might lead a trader to exercise an O utcom e 2: The stock price is less than USD 30 on the expiry
Am erican option before maturity. date.
This chapter highlights an important result known as put-call Under O utcom e 1, there is no benefit to exercising the option
parity, which is the relationship between the price of a European early because the strike price (USD 30) will be paid regardless
call option and the price of a European put option with the of when the option is exercised. In contrast, holding the option
same strike price and time to maturity. until expiry would allow the investor to earn additional inter
For the most part, this chapter assumes that the underlying est by investing the strike price at the risk-free rate for two
months.
asset is a stock. However, Section 13.4 illustrates how to extend
the results to options on other assets by using forward prices. Now consider Outcom e 2. By exercising early, the investor
incurs a loss of USD 30 — S (where S is the stock price at expiry).
13.1 C A L L O P T IO N S If the investor waits until maturity, the option is not exercised
and thus the investor does not incur this loss.
This section considers the properties of call options and discusses
The key point here is that the optionality (i.e., the choice pro
the circumstances under which American call options should be
vided by the option) gives the option holder insurance against
exercised before maturity. Upper and lower bounds for both
the value of the stock falling below USD 30. As soon as the
American and European call option prices are also derived.
option is exercised, this optionality is lost.
Now suppose that the investor is in Situation 2 and does not

American versus European Options: want a position in the stock. Exercising the option immediately
No Dividends would yield a profit equal to intrinsic value of the option (which
Many exchanged-traded options (including options on individual in this case is USD 20). However, selling the option yields a profit
stocks) are American. An important question is whether an American of the intrinsic value plus what is called the time value. The time
option should ever be exercised before its maturity date. To investi value is the value of the optionality given by the option and is
gate this, first consider call options on stocks that pay no dividends. equal to the premium that would be paid if the option were
at-the-money.
Consider an American call option where the strike price is USD 30
and the stock price is USD 50. Suppose further that the underlying For a more formal proof that the option should not be exercised
asset pays no dividends and there are two months until maturity. until maturity, consider two portfolios that could be held prior
to the maturity of the option.
In this case, it is tempting for an investor to exercise the call and
sell the stock. The investor could buy the stock for USD 30, sell Portfolio A : A call option plus a cash amount equal to the
it for USD 50, and thus receive a profit of USD 20. Because each present value of the strike price. (The present value is calcu
options contract is for the right to buy 1 0 0 shares (as mentioned in lated by discounting the strike price from the maturity of the
the previous chapter), the potential gain is USD 2,000 per contract. option to today at the risk-free rate.)
Portfolio B: The stock.
1 The price of the option does not depend on the expected return from the In Portfolio A, the cash can be invested so that it becomes the strike
stock. Chapter 15 in Valuation and Risk M odels discusses the reasons for this. price K b y the maturity of the option. If the stock price at maturity
is greater than the strike price, the option in Portfolio A can be
exercised so that the portfolio becomes worth the stock price. If the
stock price is less than the strike price at option maturity, the option
is not exercised and Portfolio A is worth the strike price.
It follows that Portfolio A is worth at least as much as (and in

some circum stances more than) the stock price at the maturity
of the option. On the other hand, Portfolio B is always worth the
stock price at maturity of the option.
This means that at maturity of the option:

Fiqure 13.2 Bounds for European or American
Value of Portfolio A > Value of Portfolio B call option on non-dividend paying stock.
This must also be true today, as otherwise there would be an
arbitrage opportunity. (If Portfolio B were worth more than Port The general shape of a call option price as a function of maturity
folio A today, for exam ple, an arbitrageur could buy Portfolio A is shown in Figure 13.1 and the bounds we have just derived are
and short Portfolio B to create a position that would never lead illustrated in Figure 13.2.
to a loss and would som etim es lead to a profit.)
The value of Portfolio A today is Employee Stock Options

Call Price + PV(K) Recall that Chapter 12 introduced em ployee stock options.
Em ployee stock options on non-dividend-paying stocks are fre
where PV denotes present value, with discounting being done from
quently exercised before maturity. Is this suboptimal behavior
the option's maturity to today at the risk-free rate. Meanwhile, the
on the part of em ployees? The answer is not necessarily.
value of Portfolio B today is the stock price S. It follows that:
Suppose an em ployee has stock options that are vested and
Call Price + PV(K) > S
in-the-money. If the em ployee is in Situation 2 mentioned previ
or ously (i.e., the em ployee does not want to keep a position in the
Call Price > S - PV(K) (13.1) stock), the argument given earlier shows that selling the option
would be more profitable than exercising the option and selling
Because an option's price cannot be negative, this result can be
the stock. However, em ployee stock options cannot be traded
extended to:
and thus must be exercised for em ployees to gain a profit. This
Call Price > max(S — PV(K), 0) explains why em ployee stock options tend to be exercised ear
Assuming interest rates are positive: lier than similar exchange-traded stock options.
K > PV(K)
Equation (13.1) thus implies

Impact of Dividends
Now consider exchange-traded options when underlying assets
Call Price > S — K
pay dividends before the option matures. Because dividends
If the option were exercised today, the call price would equal S — K. reduce the price of a stock, it can be optimal to exercise an
It follows that the call option should never be exercised early. Am erican call option im m ediately before an ex-dividend date.
An upper bound to the call price is the stock price. This is because W hether this should be done depends on the size of the
a call option can never be worth more than the stock price. dividend and the extent to which the option is in-the-money.
It can be shown that it is never optimal to exercise on an
ex-dividend date, regardless of how high the stock price is,
if the dividend is less than :2
K - K*
where K * is defined as the present value of K, with discounting

being done at the risk-free rate from the next ex-dividend date
Fiqure 13.1 Variation of call option price with stock 2 Strictly speaking, it is the am ount the stock price will decline on the
price. ex-dividend date as a result of the dividend (see Footnote 3).
Chapter 13 Properties of Options ■ 161

(or option maturity if there are no further ex-dividend dates) In Portfolio A , PV(K) becom es K at option maturity. If the stock
to the current ex-dividend date. price is greater than the strike price, K is exchanged for the
stock. A t option maturity, Portfolio A is therefore worth:
If the dividend is greater than K — K*, exercise of the option is
optimal for a sufficiently high stock price. m a x (S j, K ) + FV(Divs)
As an exam ple, suppose a call option has a strike price of
where ST is the stock price at option maturity and FV(Divs) is
USD 40 and the current stock price is USD 45. Suppose further
the value to which the dividends grow until the option matures
there is one w eek to maturity. A USD 2 dividend has been
(assuming they are invested at the risk-free rate).
declared and the ex-dividend date is about to be reached .3 An
option holder may wish to exercise the option early because the On the other hand, Portfolio B is worth:
dividend is (most likely) greater than the small difference
S T + FV(Divs)
between K and K* over one w e ek .4
To gain a general understanding of this analysis, suppose an It follows that:
option is so deep-in-the-m oney that it is alm ost certain to be Portfolio A > Portfolio B
eventually exercised. Exercising the option im m ediately before
an ex-dividend date would be the optimal decision if the at option maturity. For no arbitrage, it also be true today so that:
reduction in the stock price arising from the dividend is greater
European Call Option Price + PV(/<) + PV(Divs) > S
than the gain from delaying the paym ent of the exercise price.
This shows Equation (13.2) provides a lower bound for the price
Lower Bound When There Are Dividends of a European call option. Because an option price can never be
negative, we can extend Equation (13.2) to:
For a stock paying no dividends, Equation (13.1) applies
European Call Price > max(S — PV(K) — PV(Divs), 0)
Call Price > S - PV(K)
This is true for an Am erican and European call options because As an exam ple, consider a one-year European call option where
Am erican call options on non-dividend-paying stocks should the current stock price is USD 64 and the strike price is USD 60.
never be exercised early. A lower bound for the price of a call A dividend of USD 1 is expected in three months, six months,
option on a non-dividend paying stock is therefore the stock and nine months, and the risk-free rate is 4% per annum (with
price minus the present value of the strike price. annual compounding) for all maturities. The present value of the
strike price is USD 57.69 (= 60/1.04). The present value of the
In the case where there are dividends, the lower bound for a
dividends is
European call option is
1 1 1
European Call Price > S — PV(K) — PV(Divs) (13.2) ----------- ------------ ----------- = 2 94
1 1
1.04025 1.0405 1.040'75

where PV(Divs) is the present value of the dividends where
The call option price must therefore be at least:
discounting is done from the ex-dividend dates to today at the
risk-free rate. 64 - 57.69 - 2.94 = 3.37
To see how this works, the definitions of Portfolio A and When there are dividends, the lower bound in Equation (13.2)
Portfolio B are changed to: applies only to European options (i.e., USD 3.37 in the
example above is not the lower bound for the corresponding
Portfolio A : A European call option plus an amount of cash
American call option). In fact, the American option could be
equal to PV(/<) + PV(Divs), and
exercised immediately and must therefore be worth at least
Portfolio B: The stock. USD 4 (= 64 - 60).
3 The ex-dividend date is defined so that those who own the stock
before the ex-dividend date receive the dividend. Those who becom e 13.2 PUT O P T IO N S
owners of the stock on or after the ex-dividend date do not receive the
dividend.
Now consider put options. Note that the properties of put
4 It might be expected that the stock will decline by USD 2 on the
options can differ from those of call options. For exam ple,
ex-dividend date. In practice it may decline by less than this because
of tax effects: capital gains/losses and dividend income are taxed Am erican put options are som etim es exercised before maturity
differently in some jurisdictions. even when there are no dividends.
American versus European Options: M eanwhile, Portfolio D is worth K at option maturity. It follows
No Dividends that Portfolio C is worth at least as much as Portfolio D at

maturity of the option. For there to be no arbitrage, the same
Consider a two-month put option on a non-dividend-paying must be true today. Hence:
stock when the current stock price is USD 1 and the strike price
European Put Price + S > PV(K)
is USD 20. The option gives the holder the right to sell the
underlying asset for USD 20. Should the option be exercised? so that
The answer is clear: The put option should almost certainly be European Put Price > PV(K) — S (13.3)
exercised . 5
A European put price cannot be negative and thus (13.3)

Now suppose an investor owns both the stock and the put becom es
option. Assum ing interest rates are positive, it is better to
European Put Price > max(PV(/C) — S, 0)
receive the USD 20 strike price now than in two months. Notice
the difference between the argum ents for call options and Because an Am erican put can be exercised at any tim e, the best
those for put options: call option holders pay the strike price lower bound we can determ ine is
(and can benefit from holding the option and paying later)
Am erican Put Price > m ax(K — S, 0)
whereas put option owners receive the strike price (and can
benefit from exercising the option and investing the proceeds The upper bound for the price of a European put on a non
at the risk-free rate). dividend-paying stock is PV(K) because we know it cannot
be worth more than K at maturity. The upper bound for an
W hat does an investor give up by exercising early? There is
Am erican-style put option is K.
a small chance the stock price in our exam ple will rise above
USD 20 in two months. By exercising early, an investor gives up The general shape of a put option price as a function of the
the opportunity to benefit from that. stock price is shown in Figure 13.3. The bounds for European
and Am erican put options on non-dividend paying stocks are
The decision to exercise an Am erican put option is therefore a
illustrated in Figures 13.4 and 13.5.
trade-off between:
• Receiving the strike price early so it can be invested to earn

interest, and
• Benefiting from the optionality in circum stances where the
stock price moves above the strike price.
Consider the following two portfolios:
Portfolio C : A European put option plus one share, and
Portfolio D: Am ount of cash equal to PV(/<),
where PV(K) is defined (as before) as the present value of the strike
Fiqure 13.3 General shape of put option price as
price discounted from option maturity to today. If the strike price is
a function of stock price.
greater than the stock price at the option's maturity, the option will
be exercised and the share will be exchanged for the strike price
so that Portfolio C is worth the strike price K. If the strike price is
less than the stock price, the option is not exercised and Portfo
lio C is worth the stock price. Portfolio C is therefore worth:
PV(fC)
m a x (S T, K )
at option maturity, where S T is the stock price at maturity (as

previously defined).
5 A s we will explain, the early exercise decision depends on the chance

of the stock recovering and being above USD 20 in two months. The
chance of this occurring is likely to be very small so that exercising is Fiqure 13.4 Bounds for price of a European put
alm ost certainly optim al. option on non-dividend paying stock.

In general, exercising becom es less attractive to the holder of a
put option when:
• Stock price increases,
• Interest rate decreases,

• Tim e to maturity increases, and
• Dividends expected during the life of the option increase.
13.3 P U T-C A LL PARITY

Figure 13.5 Bounds for American put option on
Put-call parity describes the relationship between the price of a
a non-dividend-paying stock.
European call option and that of a European put option with the
same strike price and time to maturity. Consider the following
two portfolios from previous sections:
The Impact of Dividends Portfolio A : European call option plus an amount of cash
equal to PV(/<) + PV(Divs), and
To incorporate dividends into the analysis of put options,
Portfolios C and D are redefined as: Portfolio C: European put option plus one share.
Portfolio C: A European put option plus one share, and Assum e the call and put options have the same strike price and
time to maturity.
Portfolio D: An amount of cash equal to PV(K) + PV(Divs),
As discussed in the previous sections, PV(K) in Portfolio A
where PV(Divs) is (again) the dividends' present value. Portfolio C
becom es K at the option maturity, while PV(Divs) becom es
becomes
FV(Divs). If the call option is exercised (because the stock price
m a x (S T, K ) + FV(Divs) is greater than the strike price), K will be exchanged for a share
and Portfolio A will becom e one share plus FV(Divs). If the call
at maturity, where FV(Divs) is the value to which the dividends
option is not exercised (because the stock price is less than the
grow at option maturity. In Portfolio D, PV(K) becom es K at
strike price), Portfolio A will simply be K plus FV(Divs). Com bin
maturity and PV(Divs) becom es FV(Divs). Portfolio D therefore
ing these two observations gives the value of Portfolio A at
becom es
maturity:
K + FV(Divs)
m a x (S j, K ) + FV(Divs)
Because Portfolio C is worth at least as much as Portfolio D at
N ext consider Portfolio C . The dividends on the stock grow to
maturity of the option, and there is no arbitrage, such inequality
FV(Divs) by investing at the risk-free interest rate. If the stock
must also hold for their present values. Hence:
price at option maturity is greater than the strike price, the put
European Put Price + S > PV(K) + PV(Divs) option will not be exercised and the value of Portfolio C will
becom e S T + F V (D iv s ). If the stock price at option maturity is
Equation (13.3) is modified to: less than the strike price, then the put option will be exercised
and the value of Portfolio C will becom e K + FV(Divs). Putting
European Put Price > max(PV(K) + PV(Divs) - S, 0)
these results together gives the value of Portfolio C:
W hereas dividends make it more likely a call option will be
m a x (S T, K) + F V (D iv s)
exercised early, they make it less likely that a put option will
be exercised early. If an investor knows there will be future Portfolios A and C are therefore worth the same at option
d ivid end s, he or she is likely to be less inclined to exercise maturity. For there to be no arbitrage, they must be worth the
because that would mean selling the stock and giving up same today and hence:
those d ivid ends.
European Call Price + PV(/<) + PV(Divs)
O ptions far from maturity are also less likely to be exercised
= European Put Price + S (13.4)
early because there will be more tim e for the stock to move
above the strike price. The above argument is summarized in Table 13.1.
Table 13.1 Values of Portfolio A and Portfolio C at Option Maturity Leading to Equation (13.4) are Shown.
STis the Stock Price at Option Maturity
ST > K ST < K
Portfolio A Call Option ST — K 0
PV(K) K K
PV(Divs) FV(Divs) FV(Divs)
Total S T + FV(Divs) K + FV(Divs)
Portfolio C Put Option 0 K - ST
Share of Stock S T + FV(Divs) S t + FV(Divs)
Total ST + FV(Divs) K + FV(Divs)
To illustrate the existence of arbitrage opportunities when p u t- Because the risk-free rate is 4% per year, the arbitrageur can
call parity does not hold, consider the following two exam ples. simply borrow USD 37 at 4% per annum to set the position. As
a result, he or she will have more than enough funds at matu
rity to repay the loan. The gain to the arbitrageur at maturity is
Example 1: therefore:
Suppose the current stock price is USD 38. The price of a 40 - 37 X 1.040 5 = 2.27
European call option with strike price USD 40 that will mature in
six months is USD 5. The price of a European put with the same
strike price and time to maturity is USD 4, while the six-month Example 2:
risk-free rate is 4% per year (annually com pounded). Assum e the
Suppose the current stock price is USD 38. The price of
stock on which the options are written provides no dividends.
a European call option, with a strike price of USD 40 and
Then the current value of Portfolio A is therefore 5 + PV(40). tim e to m aturity of six m onths is USD 5. The price of a
This can be written as: European put option with the sam e strike price and tim e to
m aturity is USD 9. Th e risk-free rate is 4% per year (annually
40
5 + 44.22 com pounded) for all m aturities and a dividend of USD 1 is
1.040-5
exp ected in three m onths.
M eanwhile, the value of Portfolio C is USD 42 (= 38 + 4). This
The present value (USD) of the dividends is
means that Portfolio C is worth less than Portfolio A , and it
implies that the put price is too low relative to the call price. An 1
0.99
arbitrageur should 1.04°-25
• Short the call,

The value of Portfolio A is
• Buy the put, and
40
• Buy the stock. 5 + ----- — + 0.99 = 45.21
1.040-5
The cost of this position (in USD) is
The value of Portfolio C is USD 47 (= 38 + 9). This means that
- 5 + 4 + 38 = 37
Portfolio C is worth more than Portfolio A and implies that the
If the stock price is greater than USD 40 at option maturity, the
put price is too high relative to the call price. An arbitrageur
call option will be exercised against the arbitrageur and the should
put option will be worthless. Then, the arbitrageur will deliver
the stock and will receive USD 40. The value of the portfolio • Buy the call,
is therefore exactly USD 40 at maturity. If the stock price is • Short the put, and
less than USD 40 at maturity, the put option will be exercised • Short the stock.
and the call option will be worthless. The arbitrageur sells the
This creates a cash inflow today equal to:
stock for USD 40, and again the value of the portfolio is exactly
USD 40 at maturity. - 5 + 9 + 38 = 42

This is invested at the risk-free rate to grow to: where F is the forward price for a contract maturing at the same
time as the options and PV(F) is the present value of F when dis
42 X 1.0405 = 42.83
counted from the options' maturity at the risk-free rate. The call
at option maturity. and put options have the same strike price K and the same time
If the stock price is greater than USD 40 at maturity, the call to maturity.
option will be exercised and the put option will be worthless. A ssum e PV(/<) and PV(F) are invested at the risk-free rate so
The arbitrageur will then buy the stock for USD 40 and will use that they becom e K and F (respectively) at option m aturity.
it to close out the short position in the stock. The cost of doing The value of Portfolio A at option m aturity is the European
this will be USD 40. If the stock price is less than USD 40 at call option plus an am ount of cash equal to K. If the asset
maturity, the put option will be exercised against the arbitrageur price is g reater than K at m aturity, then the option will be
and the call option will be worthless. The arbitrageur will again exercised and Portfolio A will be worth the asset price.
buy the stock for USD 40 and close out the short stock position. O th e rw ise , Portfolio A will be worth K.
Thus, the cash inflow from setting up the portfolio grows to A t the same tim e, Portfolio C becom es the European put
USD 42.83 at option maturity, and the cost of the position at option option plus the asset (purchased for F under the term s of the
maturity is always USD 40. Therefore, there is always a gain of: forward contract). If the asset price is less than K at maturity, the
42.83 - 40.00 = 2.83 option will then be exercised, and the portfolio will be worth K.
O therw ise, Portfolio C will be worth the asset price.
Note that put-call parity applies only to European options. There
is no exact relationship between the price of an Am erican call Because both portfolios will be worth the greater of the strike
option and that of the corresponding Am erican put option. price and the asset price at maturity (and if there are no arbi
trage opportunities), both portfolios must be worth the same
today. The current value of Portfolio A is
13.4 USE O F FORW ARD PRICES Call Price + PV(K)
The results we have produced can be generalized to other Because it costs nothing to enter into a forward contract, the
underlying assets besides stocks. The sim plest way of doing this current value of Portfolio C is
is to work with the corresponding forward contracts and avoid
Put Price + PV(F)
consideration of the income on the asset. We redefine Portfo
lios A and C as: The put-call parity relationship is therefore:
Portfolio A : A European call option plus cash equal to PV(K), European Call Price + PV(K) = European Put Price + PV(F)
and (13.5)
Portfolio C: A European put plus a forward contract to buy The arguments that lead to this equation are summarized in
asset for F at option maturity plus cash equal to PV(F), Table 13.2.
Table 13.2 Values of Portfolio A and Portfolio C at Option Maturity Leading to Equation (13.5) are Shown.
Sjis the Asset Price at Maturity
ST > K ST < K
Portfolio A Call Option ST — K 0
PV(K) K K
Total St K
Portfolio C Put Option 0 K - ST
Forward Contract St ~ F St ~ F
PV(F) F F
Total St K
Because the put price cannot be negative, a lower bound for a on a stock paying no dividends should never be exercised
European call price can be deduced from Equation (13.5) as: before maturity. When there are dividends, it might be optimal
to exercise a call option on a stock im m ediately before an
European Call Price > PV(F) - PV(K)
ex-dividend date, but it is never optimal to exercise at other
which is an extension of the result in Equation (13.1). tim es.
Similarly, because the call price cannot be negative, the lower In contrast, it can be optimal to exercise a put option before
bound of the European put price is maturity regardless of whether the underlying asset pays a
European Put Price > PV(K) - PV{F) dividend.
which again is an extension of the result in Equation (13.3). Similar arguments also derive the lower and upper bounds of
option prices. If option prices in the market are less than the
These results can be used in conjunction with the expressions
lower bound or greater than the upper bound, there will be
of forward prices in Chapters 9 and 10. Analysts often use them
straightforward arbitrage opportunities.
directly by interpolating the forward prices observed in the market
to obtain forward prices for the desired option maturity. An important result presented in this chapter is put-call parity,
which is the relationship between the price of a European call
option and that of a European put option sharing the same
strike price and time to maturity.
SUMMARY
The results in this chapter can be extended to other assets other
This chapter derives several im portant results on option prices than stocks. Analysts often work with forward prices because
without using any assumptions regarding the dynamics of the this avoids the need to make estim ates about future dividends
underlying asset. In particular, any Am erican-style call option or other income on an asset.

Q U ES T IO N S

13.1 G ive two reasons why it is not optimal to exercise an value of money and the desire to retain the optionality."
Am erican call option on a non-dividend-paying stock Explain this statem ent.
before maturity. O ne reason should involve the time 13.6 G ive an exam ple to show that the price of a European
value of money. The other should involve the loss of put option can decrease as its maturity increases.
optionality.
13.7 Does it becom e more likely that a put option will be
13.2 W hy is an em ployee stock option on a non-dividend exercised before maturity when (a) the interest rate
paying stock som etim es exercised before maturity?
increases and (b) the stock price increases?
13.3 G ive an exam ple to show that a European call option's 13.8 W hat is the lower bound for the price of a European put
price can decrease as its maturity is increased.
option on a non-dividend-paying stock?
13.4 W hat is the lower bound for the price of a European call 13.9 W hat is the put-call parity formula in term s of forward
option on a non-dividend-paying stock? prices?
13.5 "The early exercise of an Am erican put on a non- 13.10 W hat is the advantage of expressing put-call parity in
dividend-paying stock is a trade-off between the time term s of the forward prices?
Practice Questions
13.11 Is an in-the-money Am erican call option on a non- 13.16 The current price of a non-dividend-paying stock is
dividend-paying stock worth more than its intrinsic value? USD 29 and the price of a four-month call option on the
Explain. stock with a strike price of USD 30 is USD 2. The risk-free
rate is 4% per annum (annually com pounded). W hat is
13.12 A four-month European call option on a stock is cur
the price of a four-month put option on the stock with a
rently selling for USD 2.50. The current stock price is
USD 54, and the strike price is USD 50. A dividend strike price of USD 30? Assum e no arbitrage opportuni
ties exist.
of USD 1.50 is exp ected in one month. The risk-free
interest rate is 3% per annum (annually com pounded) 13.17 A European call and European put option on a stock
for all m aturities. W hat opportunities are there for an both cost USD 5 with a common strike price USD 30
arbitrageur? and a common time to maturity of one year. The current
13.13 A seven-month call option pays dividends of USD 0.5 in stock price is USD 30. W hat arbitrage opportunities does
this create? Assum e no dividend is paid and the interest
three months and six months. The strike price is USD 40.
rate is positive.
Assum e a constant risk-free rate of % per annum (annu
8
ally compounded) for all maturities. Is it ever optimal to 13.18 Use the results in Chapter 9 to determine put-call parity for
exercise the option before maturity? Explain. a currency options on the GBP/USD exchange rate. Express
13.14 W hat will be the lower bound for the price of a three- your answer in terms of the USD risk-free rate, RUSD, the
month European put option on a non-dividend-paying G BP risk-free rate, Rq bp/ and the time to maturity, T.
stock if the current stock price is USD 22, the strike price 13.19 Use the results in Chapter 10 to determ ine put-call parity
is USD 25, and the risk-free rate is % per year (annually
6 for an index option. Express your answer in term s of the
com pounded)? risk-free rate, R, the dividend yield, Q, and the time to
maturity, T.
13.15 As explained in Chapter 12, historically calls were traded
on exchanges before puts. Explain how you can syn 13.20 Under what circum stances does a European call on an
thetically create a European put from a European call. asset equal the price of a European put on the asset
Assum e no dividends are paid and no arbitrage opportu when both have the same strike price and tim e to
nities exist. maturity? Express your answer in term s of forward prices.
ANSW ERS
13.1 Delaying paying the strike price allows interest to be 13.11 Yes. An in-the-money Am erican call on a non-dividend
earned on the strike price for a longer time period. The paying stock is worth at least S — PV(K), which is greater
call option also provides insurance against the event than S — K. It is never worth exercising the call early and
that the stock price becom es less than the strike price so it must be worth more than its intrinsic value.
at maturity. O nce the option has been exercised, this 13.12 The lower bound for the option price is
optionality will be lost.
13.2 An em ployee stock option cannot be traded. Thus, if S - PV(K) - PV(Divs) = 5 4 ----- ^ n r ------^4— = 2.99
1.031/3 1.031/12
the em ployee does not want a position in the company's
The option is selling for less than its low er bound. An
stock, the only alternative will be exercising the option
arb itrag eu r can buy the option and short the stock for
and selling the stock.
an initial cash inflow of USD 51.50. The arb itrag eur
13.3 If a big dividend is due to be paid, the holder of the has to pay dividends of USD 1.50 after one m onth.
option would like to exercise im m ediately before the If the option is e xe rcise d , the cost of closing out the
ex-dividend date. A European option that matures just short position will be USD 50. If it is not e xe rcise d , the
before the ex-dividend date is then likely to be worth cost of closing out the short position will be less than
more than one that matures after the ex-dividend date. USD 50. Th e w orst-case scenario fo r the arb itrag eu r is
13.4 The lower bound is m ax(S — PV(K), 0) where S is the th erefo re:
stock price and K is the strike price. Today: + 51.50,
13.5 By exercising before maturity, the holder of a put option O ne month: —1.50, and
receives the strike price earlier. However, delaying the
Four months: —50.00.
exercise of the option will benefit the holder of the stock
if the stock price rises above the strike price later. Once When the discount rate is zero, the sum of these cash
the option is exercised, however, the benefit of this flows will have zero present value. Any positive discount
optionality is lost. rate gives a positive sum of present values.
13.6 An exam ple would be a deep-in-the-money European 13.13 It is only optimal to exercise im m ediately before a
put option that is (almost) certain to be exercised. The dividend paym ent. Im m ediately before the three-month
shorter the maturity of the option, the earlier the strike payment, the option holder should wait, because there
price is received, and the greater the value of the option. are three months until the next dividend payment and
13.7 (a) When interest rates increase, the put option will be more K — K* is greater than the dividend payment:
likely to be exercised before maturity because the value * 40

K - K = 40 --------—- = 0.76 > 0.5
of investing the profit at the risk-free rate after selling the 1.08025
stock at the strike price is increased, (b) When the stock
Exercise can be optimal im m ediately before the
price increases, the put option will be less likely to be exer
six-month dividend payment because there is only one
cised because the payoff from doing so will be smaller.
month to maturity and K — K* is less than the dividend
13.8 The lower bound is m ax(PV(K) — S, 0) where S is the payment:
stock price and K is the strike price.
40
13.9 Call price + PV(K) = Put price + PV(F) where K is the = 0.26 < 0.5
1.081/12
common strike price and F is the forward price for a
contract that matures at the same time as the options.
13.14 The lower bound (USD) is
13.10 It is not necessary to predict income on the asset as the 25

- 22 = 2.64
forward prices reflect all future income before maturity. 1.06°-25

13.15 From put-call parity: 13.19 From Equation (10.3):

Call + PV(K) = Put + S 1+ RV
F = Si
1 + Q
so that:
Put = Call + PV(K) - S Substituting this into Equation (13.5) and noting that:
The put can therefore be created by buying the call, K

PV(K) =
shorting the stock, and investing PV(/<) so that it grows to (1 + R ) T
K at maturity. (1 + R ) T 1
PV(F) = S
13.16 By put-call parity: (1 + Q ) r (1 + R ) T1 (1 + Q ) T
Put = Call + PV(K) - S, K

European Call Price +
the put price (USD) is thus given by: (1 + R ) T
30
2 + - 29 = 2.61 = European Put Price +
1.041/3 (1 + Q ) T
13.17 From put-call parity, the excess of the call price over 13.20 The put-call parity formula is
the put price is S — PV(K). In this case S = K = 30 and
European Call Price + PV(K)
so S — PV(/<) is positive. The call should be worth more
= European Put Price + PV(F)
than the put, but they are both worth the sam e. An
arbitrageur should buy the call, sell the put, and short PV(F) = PV(/<) when F = K. The equation therefore
the stock. shows that a European call has the same price as a
European put when F = K, that is, when the strike price
13.18 From Equation (9.1):
equals the forward price for a forward contract maturing
(1 + Ru sd ) t at the same time as the option.
F = S T
(1 + Rg bp)
Substituting this into Equation (13.5) and noting that:
K
PV(K) =
( 1 + Pu s d Y
_ 1 + R u s d ) 7 __________ 1________
PV(F) =
T
” (1 + K g b p ) T (1 + * u s d ) T (1 + ^GBp)
K
European Call Price +
(1 + Ru sd V
= European Put Price + T

(1 + Rg b p )
Learning Objectives
Explain the motivation to initiate a covered call or a Describe the use and calculate the payoffs of various
protective put strategy. spread strategies.
Describe principal protected notes (PPNs) and explain Describe the use and explain the payoff functions of
necessary conditions to create a PPN. combination strategies.
171
O ptions can be arranged to form a wide (a) (b)
spectrum of payoff patterns. In theory, an
investor with access to European options
with all strike prices for a given matu
rity could achieve any continuous payoff
function of the underlying asset price at
expiry.
The option trading strategies considered in

this chapter can be divided into four groups:
1. Strategies involving an option and the

underlying asset,
2 . Strategies involving two or more call

options,
Figure 14.1 An Illustration of Equation (14.1) (a) shows the final value
3 . Strategies involving two or more put
of a protective put (i.e., put plus the asset) and (b) shows the final value
options, and
of call plus PV(K).
4 . Strategies involving both call and put
options.
• Equation (14.1) shows that a put option combined with the
Strategies involving only call options or only put options are
asset gives a position equivalent to a call option combined
term ed spreads. Those involving both call and put options are
with an amount of cash equal to PV(K). Figure 14.1 illustrates
term ed com binations.
this by showing the values of the two portfolios at option
For ease of exposition, it is assumed that (except where maturity. Buying a put option when the underlying asset is
otherwise stated) the underlying asset provides no income and held is referred to as a p ro tective p u t strategy. A trader who
all options are European. sets up a protective put is usually bullish about the asset
price (hence the position in the asset), but also wishes to limit
losses in the event of an unexpected price decrease (hence
14.1 STRATEGIES INVOLVING the put option).
A SIN GLE OPTION • Equation (14.2) shows that an asset combined with a short
call is equivalent to a short combined with an amount of
The put-call parity describes the relationship between the price
cash equal to PV(K). Figure 14.2 illustrates this by showing
of a European put option and that of a European call option
the values of the two portfolios at option maturity. An asset
with the same strike price and time to maturity. As derived in
plus a short call is known as a co ve red call. The call is usually
Chapter 13, the put-call parity result (assuming that the
out-of-the-money, allowing the asset owner to obtain a cash
underlying asset provides no income) is
inflow equal to the call option premium in exchange for giv
p + S=c+PV(K) (14.1) ing up some of the potential upside from an increase in the
where p is the price of a European put option with strike price asset price.
K and c is the price of a European call option with strike price • Equation (14.3) is the reverse of a protected put. It shows
K (with both options having the same tim e to maturity). S is the that a short put combined with a short position in the asset is
current asset price, and PV denotes present value from option equivalent to a short call combined with a liability of PV(/<).
maturity to today at the risk-free rate. This result can also be • Equation (14.4) is the reverse of a covered call. It shows that
written in the following ways: a long call com bined with a short position in the asset is
- c = PV(K) - p (14.2) equivalent to a long put com bined with a liability of PV(/<).
- S = - c - PV(K) (14.3)
- S = p - PV(K) (14.4) Principal Protected Notes

Equations (14.1) to (14.4) capture the various ways in which an A principal protected note (PPN) is a security created from a
option position can be combined with a position in the asset. single option such that the investor benefits from any gain in the
Consider the following specific points. value of a specified portfolio without the risk of losses.
(a) (b) Full participation PPNs (i.e., the owner
receives 1 0 0 % of the upside) are only possible
for portfolios that provide an income. To see
this, note that Equation (14.1) gives
p + S = c + PV(K)
Because the call is currently at the money (so

that K = S), put-call parity gives
c = p + S - PV(S)
The above equation implies
c > S - PV(S)
This shows that the call always costs more

than the funds available to pay for it. (In our
exam ple, the call will always cost more than
USD 1,837.02). However, Portfolio B's income
reduces the value of the call. If the income is
sufficiently high, a PPN can be created.
Fiqure 14.2 An Illustration of Equation (14.2) (a) shows the final value
of a covered call (i.e., asset plus short call) and (b) shows the final value Returning to our exam ple, suppose that
of a short put plus PV(K). Portfolio B provides a yield of 2%. An
extension of the Black-Scholes Merton model (which will be
covered in Valuation and Risk M o d els) shows that a PPN can be
To show how a principal protected note is created, suppose created if its volatility is less than 18%. (This is because the call
the three-year interest rate is 7% (annually com pounded). This option can then be created for less than USD 1,837.02.)
means that the present value of USD 10,000 is
USD 10,000 14.2 SPREAD TRADING STRATEGIES

--------- ^----= USD 8,162.98
1.073
Now consider trading strategies involving positions in two or
Suppose that Portfolio A consists of: more call (or put) options.
• A three-year zero-coupon bond that will pay USD 10,000 in

Bull Spread
three years; and
• A three-year call option on Portfolio B, which is currently A bull spread (as its name implies) is a position appropriate for
an investor expecting an increase in the price of an asset. To
worth USD 10,000 with a strike price of USD 10,000.
create the spread, the trader buys a European call option with
The holder of Portfolio A will to benefit from any increase in the strike price /C, and sells a European call option with strike price
value of Portfolio B without incurring any loss of principal if its K2. In this situation, K2 > /C, and both options have the same
value declines. This feature is likely to be attractive to risk-averse time to maturity. The payoff from a bull spread is calculated in
investors. Table 14.1 and illustrated in Figure 14.3.
Note that the first item in Portfolio A costs USD 8,162.98.
If the option can be purchased for less than USD 1,837.02 Table 14.1 A Payoff from a Bull Spread is Shown.
(= USD 10,000 — USD 8,162.98), then the portfolio will cost less Sjis the Asset Price at Maturity
than USD 10,000 and can be profitably offered to investors. Payoff from Payoff from
PPNs are possible because of the following. Long Call Short Call with Total
Range for S T with Price 1C, Strike Price K2 Payoff
• The investor is giving up three years of interest on the USD
1 0 , 0 0 0 investment. S t — Ki 0 0 0
K-\ < S T K2 S t ~ Ki S t ~ Ki
• The investor does not receive any income that the owners of
0
Portfolio B would receive during the next three years. ST > k2 S t ~ K, -(S r - K2) K2 ~ Ki
Chapter 14 Trading Strategies ■ 173

Put options can also be used to set up a bull spread. Suppose
c and c are the prices of call options with strike prices /C| and
-1 2
/<2 (respectively) and that p-\ and p are the prices of the corre
2
sponding put options (also respectively). By put-call parity:
c-, + PV(/C,) = pi + S
c + PV(/C2) = P2 + S
2 (14.5)
Thus:
q - c =
2 P 1 - p + PV(K2) - PV(/C-|)
2
Note that the left-hand side of this equation is the cost of

setting up the bull spread using calls. The right-hand side shows
the cost is the same if the call options are replaced by put
options combined with an amount of cash equal to the pres
ent value of K2 — Kf. As shown in Table 14.2, the payoff when
Fiaure 14.3 Payoff from a bull spread created from using puts and the cash is the same as in Table 14.1 (where calls
long and short call options. are used).
The advantage of a bull spread (compared with simply buying a

Bear Spread
call option that has a strike price K^ is that it is less expensive to
set up. The trader chooses to give up gains from the asset price A bear spread is a position where the trader buys a Euro
rising above K2. In return, there is a cost savings equal to the pean put option with strike price K2 and sells a European put
price of an option with strike price K2. option with strike price /C|. In this situation, K2 > K: and both
options have the same tim e to m aturity. The payoff from a
The choice of strike prices affects the potential returns to the
bear spread is calculated in Table 14.3 and is illustrated in
bull spread creator. If both options are out-of-the-money,
Figure 14.4.
the bull spread costs very little to set up. There is also a small
probability that both options will becom e in-the-money and thus Like a bull spread, a bear spread has a small probability of
generate a high return. If both options are initially in-the-money, attaining a large return if both options begin out-of-the-money
the bull spread is more expensive to set up and there is a high and a high probability of attaining a m odest return if both
probability that the strategy gives only a m odest return. options begin in-the-money.
Table 14.2 A Payoff from a Bull Spread Created from Puts Rather than Calls is Shown. S T is the Asset Price
at Maturity
Payoff from Long Put Payoff from Short Put

Range for ST with Strike Price K, with Strike Price fC2 Cash Amount Total Payoff
S T ^ /C| K-i - ST ~(K 2 - ST) K2 — K-j 0

Kf < S T — K2 0 —(K2 - ST) K2 ~ Kf S T — Ki
sT > k2 0 0 k2 - k ., K2 ~ Ky
Table 14.3 A Payoff from a Bear Spread is Shown. S T is the Asset Price at Maturity
Range for S T Payoff from Long Put with Strike Price K2 Payoff from Short Put with Strike Price K-i Total Payoff
S T — /C| k2 - ST -(K i - S T) K2 ~ Ki
/C| < S j — k 2 k 2 - ST 0 k2 - ST
sT > k 2 0 0 0
Box Spread
A box spread is a portfolio created from a bull spread (using call
options) and a bear spread (using put options). The strike prices
and tim es to maturity used for the bull spread are the same as
those used for the bear spread. The box spread aggregates the
payoffs from the portfolios in Figures 14.3 and 14.4 (the solid
lines) and produces a payoff that is always K2 — Kv This is also
dem onstrated in Table 14.5.
The cost of setting up a box spread should be PV ( /< 2 — /C|). If it

costs less than this, an arbitrageur can profit by buying the box
spread and earning more than the risk-free rate. If it costs more
than this, an arbitrageur can short the box spread by borrowing
it at less than the risk-free rate.
Figure 14.4 A payoff from bear spread created from Note that only European options can be used to construct a box
long and short put options is shown. spread with a known payoff at a future tim e. Because Am erican
options can be exercised early, the final payoff generated when
Call options can also be used to create a bear spread. From Am erican options are used can differ from K2 — K-\.
Equation (14.5):
p — pi = c —
2 2 C 1 + PV(K2) - PV(/C,) Butterfly Spread
The left-hand side of this equation is the cost of setting up the bear
A butterfly spread involves positions in three options. Like the
spread by using put options. The right-hand side shows the cost is
other spreads discussed in this chapter, it can be created from
the same if the put options are replaced by call options along with
either call or put options. If calls are used, a trader will take the
an amount of cash equal to the present value of K2 — K- . As shown 1
following positions:
in Table 14.4, the payoff is the same as in Figure 14.4.
• O ne long European call with strike price Kh
Note that while a bull spread (regardless of whether calls or puts
are used) involves buying at the low strike price and selling at • O ne long European call with strike price K3 where
the high strike price, a bear spread always involves buying at the K3 > Kh and
high strike price and selling at the low strike price. • Two short European calls with strike price K2 = (K + K3)/2. -1
Table 14.4 A Payoff from a Bear Spread Created from Calls Rather than Puts is Shown. S T is the Asset Price at
Maturity
Payoff from Long Call Payoff from Short Call

Range for S T with Strike Price fC2 with Strike Price fC| Cash Amount Total Payoff
St —Ki 0 0 k2 - k . « 2 - *1
/C, < S T < k2 0 - ( S r - Ki) k2 - k 1 k2 ~ sT
St > K2 St ~ K2 ~ (ST ~ Ki) « 2 - Ki 0
Table 14.5 Payoff from a Box Spread
Asset Price Range Payoff from Bull Spread (Table 14.1) Payoff from Bear Spread (Table 14.3) Total Payoff
S t — K-i 0 K2 — K-\ K2 — Kj
K-, < S T < k2 S t ~ K| k2 - ST k 2- k 1
St — K 2 K2 ~ K i 0 K2 - K i

Table 14.6 A Calculation of the Payoff from a Butterfly Spread Created from Calls is Shown. S T is the Asset Price
at Maturity
Payoff from Long Call Payoff from Two Short Payoff from Long Call
Range for ST with Strike Price fC, Calls with Strike Price K 2 with Strike Price K 3 Total Payoff1
S j — Ki 0 0 0 0
K: < S T < K2 S T - Ki 0 0 S t ~ Ki
k2 < ST < k3 S T ~ Ki —2(ST - K2) 0 2 K2 ~ K, - S T = K3 - S T
sT > 3 S T ~ Ki ~ (S t ~ K2)
N)
7s
S t ~ K3
O
k
I
i
All options have the same tim e to maturity. The payoff from
a butterfly spread as a function of the asset price is shown in
Table 14.6 and Figure 14.6.
Because a butterfly spread always provides a payoff that is

positive or zero, it must be the case that the cost of setting up
a butterfly spread is positive. This means that:
q + c > 3 2 c 2
Such a condition can be proved to be true under the Black-

Scholes Merton model (which will be covered in Valuation and
Risk M o d els). Under any model, the price of a call option is
always a convex function of the strike price (as illustrated in
Figure 14.6).
Usually, K2 is close to the current stock price. As indicated in

Figure 14.5, a butterfly spread is an appropriate trading strat
egy when an investor anticipates a small m ovement in the asset
price so that it is close to K2 at maturity. A big move in either
Fiqure 14.5 Payoff from a butterfly spread created
direction will lead to a payoff of zero. If a trader anticipates a
from call options.
big m ovement in the asset price and is uncertain whether the
price will go up or down, he or she can short a butterfly spread.
This will lead to a small profit if the trader is right and a small
loss if the trader is wrong.
To see the how a butterfly spread can be made using put

options, apply put-call parity three tim es:
c, + PV(/C,) = Pi + S
c + PV(K2) = p 2 + S
2
c + PV(K3) = p 3 + S
3
Adding the first of these equations to the third and then sub
Fiqure 14.6 Call option price as a function of the
tracting twice the second equation gives
strike price.
q + c - 2 c + PV(/C| + K3 - 2K2) = P t + p - 2p
3 2 3 2
Because K2 = 0.5 + K3), the above equation reduces to:

This shows that the cost of setting up the position should be
( /< 1
q + c — 3 2 c2 = pi + p - 3 2 p 2 the same whether the trader uses put options or call options.
Furtherm ore, the payoffs (as shown in Table 14.7) are the same
1 This uses the assumption that K2 — 0.5(Ki + K3). regardless of whether the trader uses puts or calls.
Table 14.7 Butterfly Spread Created from Put Options
Payoff from Long Put Payoff from Two Short Payoff from Long Put
Range for S T with Strike Price 1C, Puts with Strike Price K 2 with Strike Price K 3 Total Payoff
S t — Ki /C| - S T - 2 (K2 - S T) K3 ~ sT 2K2 - K, - K3 = 0
/<! < S-r < K2 0 - 2 (K2 - S T) K3 - sT S T + K3 — 2K2 = S t - K,
K2 < S T ^ K3 0 0 K3 - sT K3 ~ sT
S t > K3 0 0 0 0
Figure 14.7 Payoff from a calendar spread as seen at

the time the short-maturity option reaches maturity.
Fiqure 14.8 Payoff from a calendar spread
created from put options as seen at the maturity
Calendar Spread short-maturity option.
All positions considered thus far have involved options maturing
at the same tim e. To create a calendar spread, a trader buys a
where PV denotes the present value when discounting from
call option maturing at time T* and sells a call maturing at time
time T to today and PV* denotes present value when discounted
T. With this position, T* > T and the two calls have the same
from tim e T* to today. It follows that:
strike price of K.
c* - c = p* - p + PV(K) - PV(K*)
To examine the payoff from a calendar spread, consider the situa
Usually PV(/<) — PV*(K) is small so that the cost of a calendar
tion at the time when the short-maturity option matures (i.e., at time
7). A t this time, the option sold has just matured, while the option spread created from put options is close to that of a spread cre
bought will mature at T* — T. As mentioned earlier, the latter is a ated from call options. Figure 14.8 shows the payoff pattern of a
calendar spread when it is created from puts.
convex function of the asset price. This is shown in Figure 14.7.
The payoff from a calendar spread (as a function of the asset price)
is like that of a butterfly spread. There will be a small gain if the 14.3 CO M BIN A TIO N S
asset price at time T-| is near the strike price K. Otherwise, there will
be a loss approximately equal to the cost of setting up the spread. This section considers positions created using both call and put
Let c and c* be the prices of call options with strike price K and options.
maturities T and T* (respectively). M eanwhile, let p and p* be

the prices of call options with strike price K and maturities T and Straddle
T* (also respectively). By put-call parity:
A straddle is a position created from a long call and a long put
c + PV(K) = p + S with the same strike price and time to maturity. The strike price
c* + P V *(K ) = p* + S is usually close to the current asset price.

Table 14.8 Payoff from a Straddle Table 14.9 Payoff from a Strangle
Payoff from Payoff from Payoff from Payoff from

Long Call with Long Put with Total Long Call with Long Put with Total
Range for S T Strike Price K Strike Price K Payoff2 Range for S T Strike Price K 2 Strike Price 1C, Payoff3
ST < K 0 K - ST K - ST S t — Ki 0 /< 1 - ST K ~ ST
1
ST > K ST — K 0 ST — K /C, < ST — k 2 0 0 0
S T > K2 sT - k2 0 sT - k 2
Fiqure 14.9 Payoff from straddle.

Fiqure 14.10 Payoff from a strangle.
A straddle trader believes there will be a big move in the asset Continuing with our earlier exam ple, suppose the lower strike
price but is unsure whether the move will be up or down. price (/C|) used for the put option is USD 45 and the upper strike
price (K2) used for the call option is USD 55. The Black-Scholes
As indicated in Table 14.8 and Figure 14.9, a straddle provides a
Merton model gives prices for the call and put options as USD
V-shaped payoff.
0.52 and USD 0.32 (respectively). The cost of setting up the
A straddle can be fairly expensive to set up. This means that the strangle is therefore around USD 0.84, which is less than a quarter
asset price m ovement has to be quite large for a profit to be the cost of the straddle where both strike prices are USD 50.
realized. However, the asset price has to move even further for the position
Consider a three-month straddle where the asset price and the to be profitable. To make a profit, the asset price has to be
strike price (for both the call and the put) is USD 50. If the vola greater than USD 55.84 or less than USD 44.16 on the expiry date.
tility of the asset price is 2 0 % per annum and the risk-free rate is To summarize, both a straddle and a strangle are positions
2% per annum, the Black-Scholes Merton model (to be covered designed to provide profits when there is a large move in the
in Valuation and Risk M o d e ls) shows that the prices of a three- asset price. The larger the move, the greater the payoff. A stran
month European call option and a European put option are USD gle costs less than a straddle, but the asset price has to move
2.12 and USD 1.87 (respectively). This means that the asset price further for it to be profitable.
has to move by USD 3.99 (= 2.12 + 1.87) for there to be a profit.
If the asset price is between USD 46 and USD 54, the straddle
will incur a loss. 14.4 MANUFACTURING PAYOFFS
If options with all strike prices could be traded, a trader could
Strangle (in theory) create any payoff that is a continuous function of the
The cost of a straddle can be reduced by making the strike price asset price at a future tim e. To see this, note that the payoff
of the call greater than the strike price of the put. The position from a butterfly spread is a spike p a y o ff (see Figure 14.11). The
is then called a strangle. The payoff function is shown Table 14.9 spike can be made arbitrarily small by choosing strike prices
and Figure 14.10. that are close together. By constructing a portfolio from a large
3
2 This uses the assumption that K2 — 0.5(Ki + K3). This uses the assumption that K2 — 0.5(/C| + K3).
Payoff put options. C om b inations are created by using both calls
and puts.
Many other strategies are used by option traders in addition to

those discussed in this chapter. The following are exam ples.
Asset Price
• A diagonal sp rea d is created from a long call (or put) and a
Figure 14.11 Spike payoff from a butterfly spread. short call (or put) where both the strike prices and the times
to maturity are different. It can be considered as a cross
number of small spikes, any payoff function can be replicated between a bull/bear spread and a calendar spread.
to arbitrary accuracy. Because a butterfly spread can be con • A strip is like a straddle except that two puts are purchased
structed from either calls or puts, any continuous payoff function for every call. It is appropriate when a trader anticipates a big
can be created by using either type of option. move in the asset price and a downward m ovement is consid
ered more likely than an upward movement.
SU M M A RY • A strap is like a straddle excep t that two calls are purchased

for every put. It is appropriate when a trad er anticipates
This chapter d escrib es som e of the ways in which options a big move in the asset price and an upward m ovem ent is
can be used to produce various payoffs. Spreads are cre considered more likely than a downward m ovem ent.
ated from tw o or more call options as well as tw o or more

Q UESTIO N S

14.1 How is a covered call created? How can an equivalent 14.8 A range forward contract involves buying a call and
position be created using a put? selling a put where the strike price of the call is greater
than that of the put and the maturities of the two are the
14.2 How is a protective put created? W hy would a trader
create such a position? same. Sketch the payoff from the contract.
14.3 How is a bull spread created from call options? 14.9 A trader feels that there will be a big jump in a stock
price but is uncertain of the direction. Identify three
14.4 Expla in two ways a bear spread can be created.
trading strategies that the investor can im plem ent to
14.5 When should a trader consider creating a butterfly reflect his or her views.
spread?
1 4 .1 0 A trader creates three six-month butterfly spreads. The
14.6 W hat options positions are necessary to build a short first has strike prices of USD 30, USD 31, and USD 32.
position in a box spread? The second has strike prices of USD 31, USD 32, and
14.7 W hat is the difference between a straddle and a USD 33. The third has strike prices of USD 32, USD 33,
strangle? and USD 34. W hat is the resulting payoff pattern?
Practice Questions
14.11 "The higher interest rates are, the easier it is for a 14.16 Describe the payoff from a long position in a call with
bank to create a principal protected note." Explain this strike price K-\ and a long position in a put with strike
statem ent. price K2 where K2 > K |.
14.12 Explain why two of the options in a box spread create 14.17 A long strangle is combined with a short straddle. The strike
a long forward position and another two create a short price in the straddle is halfway between the strike prices in
forward position. the strangle. Describe the payoff from this position.
14.13 Three put options have the same expiration date and 14.18 A call option with a strike price of USD 40 costs USD 2
their strike prices are USD 45, USD 50, and USD 55. and a put option with a strike price of USD 30 costs
The market prices of the options are USD 2, USD 4, and USD 3. Both have the same time to maturity. Explain how
USD 7 (respectively). Explain how a butterfly spread a strangle can be created using these options, and con
can be created from these options. Provide a table struct a table showing the profit as a function of the asset
showing the profit as a function of the asset price at price at option maturity.
maturity.
14.19 A call and a put with a common strike price of USD 70
14.14 A trader creates a bear spread using put options with cost USD 7 and USD 5 (respectively). How is a straddle
strike prices of USD 25 and USD 30 and the same time created from these options? State the range of asset
to maturity. The options cost USD 2 and USD 4.50 prices leading to some profit on the straddle.
(respectively). Under what circum stances will the trade be
14.20 Com pany A has just announced a takeover offer for
profitable? Com pany B. The outcome of the takeover attem pt is
14.15 Suppose that put options on an asset with strike prices uncertain. If successful, a big increase in Com pany B's
USD 30 and USD 35 and the same time to maturity stock price can be anticipated. If unsuccessful, a big
cost USD 4 and USD 7 (respectively). How can these decrease can be anticipated. An investor buys a straddle
options be used to create (a) a bear spread and on Com pany B's stock where the strike price is close to
(b) a bull spread? the current market price. Is this a good trade? Discuss.
ANSW ERS
14.1 A covered call is created from a long position in an asset 14.8 The payoff of a range forward contract is of the form:
and a short position in a call. Equation (14.2) shows that
it is equivalent to a short put position together with
a cash position that equals the present value of the
strike price.
14.2 A protective put is created from a long position in both

an asset and a put option on the asset. The motivation
is setting up protection against downward movements
of the asset price while anticipating profit generated by
upward asset price movement.
14.3 A bull spread is created by buying a call with a certain

strike price and selling another call with a higher strike
price. Both calls have the same time to maturity. Note that when the strike prices are identical, a range
forward contract will be equivalent to a regular forward
14.4 A bear spread can be created by buying a put with a
contract.
certain strike p rice, K2, and selling a put with a lower
strike price, Kv Both puts have the sam e tim e to m atu 14.9 The trader can try a short butterfly spread, a short calen
rity. It can also be created by buying a call with strike dar spread, a strangle, or a straddle.
price K2, selling a call with strike price /C|, and adding 14.10 The payoff pattern is
an am ount of cash equal to the present value of the
difference betw een the strike prices.
14.5 A butterfly spread is appropriate when the price of the

asset at a certain future tim e is expected to be close to
a particular level. The spread is created from options
on three equally spaced strike prices where the middle
strike price equals this expected level.
14.6 Suppose K2 < K A long box spread is created by buying

-1
a call with strike price K-\, selling a call with strike price K2l
buying a put with strike price K2, and selling a put with
strike price K-\. A short box spread is therefore created by 14.11 In a principal protected note, the price of a call option has
selling a call with strike price K-\, buying a call with strike to be greater than the difference between S and PV(S). As
price K2, selling a put with strike price K2, and buying a interest rates increase, the difference becomes greater.
put with strike price K All options have the same time Thus, it is easier to create a principal protected note.
to maturity. 14.12 A box spread is created by buying a call with strike price
14.7 A straddle is constructed from a call and a put with /< , selling a call with strike price K2, buying a put with
1
the same strike price and time to maturity. For a strike price K2, and selling a put with strike price /C|. All
strangle, the strike price of the put is lower than the have the same maturity and K: < K2. Buying a call with
strike price of the call, but the tim es to maturity are still strike price K-\ and selling a put with strike price K-\ is
the same. equivalent to a long forward contract with delivery price

K- . Selling a call with strike price K2 and buying a put with

1 14.17 This is a butterfly spread minus some cash:
strike price K2 is equivalent to a short forward contract
with delivery price K2.
14.13 A butterfly spread is created by buying put options with

strike prices of USD 45 and USD 55 and selling two put
options with a strike price of USD 50. The cost is USD 1
(= 2 + 7 — (2 X 4)). A table showing the profit is as
follows:
Profit
Payoff: Payoff: Two Payoff: Including
Asset Price Long 45 Short 50 Long 55 the Cost of
Range Put Puts Put Setting Up 14.18 A strangle costs USD 5 and provides a profit as follows:
ST < 45 45 - S T -2(50 - S T) 55 - ST -1 Long Call Long Put Profit

Asset Price with Strike with Strike (Including
45 < S T < 50 0 -2(50 - S T) 55 - ST ST - 46
Range Price of 40 Price of 30 Initial Cost)
50 < S T < 55 0 0 55 - ST 54 - ST
ST < 30 0 30 - ST 25 - ST
ST > 55 0 0 0 -1
30 < ST < 40 0 0 -5
14.14 It costs USD 2.50 to set up the bear spread. If the asset
ST > 40 ST - 40 0 ST - 45
price is between USD 27.50 and USD 30, the payoff from
the spread will be less than USD 2.50. If it is less than 14.19 A straddle is created by buying the call and buying
USD 27.50, the payoff will be between USD 2.50 and the put. The cost is USD 12. It leads to a profit if
USD 5. Thus, the trade will be profitable if the price of the asset price at maturity is above USD 82 or below
the asset at maturity is less than USD 27.50. USD 58.
14.15 (a) A bear spread is created by buying the put option 14.20 The outcome of the takeover is uncertain, and we can
with a strike price of USD 35 and selling the put option expect a big move in the price of Com pany B's stock by
with a strike price of USD 30. (b) A bull spread is created the tim e the result of the takeover attem pt is known. A
by buying the put option with a strike price of USD 30, straddle would therefore appear to be a good trade.
selling the put option with a strike price of USD 35, and However, because knowledge of the takeover is in the
adding the present value of USD 5 to the portfolio. public domain, the prices of calls and puts will be higher
14.16 This has the same payoff as a strangle except that there than usual to reflect the fact that a big move in the stock
is an extra amount of cash equal to K-\ — K2. It is more price is likely. This em phasizes the point that having the
expensive to set up than a regular strangle where the put same view as the rest of the market is not usually suf
has a lower strike price than the call. ficient to make money trading options. It is necessary to
take a view that is different from the market consensus
(and be right!).
Exotic Options
Learning Objectives
Define and contrast exotic derivatives and plain vanilla • Identify and describe the characteristics and pay-off
derivatives. structure of the following exotic options: gap, forward
start, com pound, chooser, barrier, binary, lookback, Asian,
Describe some of the factors that drive the developm ent exchange, and basket options.
of exotic derivative products.
Describe and contrast volatility and variance swaps.
Explain how any derivative can be converted into a
zero-cost product. Explain the basic premise of static option replication and
how it can be applied to hedging exotic options.
Describe how standard Am erican options can be
transform ed into nonstandard Am erican options.
183
Standard European and Am erican options (i.e., those usually For exam ple, consider a European call option. When converted
traded on exchanges) are term ed plain vanilla options. Options into a zero-cost product, it has a payoff:
with non-standard properties are term ed exo tic options (or sim
m ax{ST — K — A , — A)
ply exo tics). Exotic options are designed by derivatives dealers
to meet the specific needs of their clients and are usually traded where A = c(1 + R)T and c is the normal option premium.
in the over-the-counter m arkets. Exotic options can be very In essence, this is a forward contract where the holder agrees to
profitable for derivatives dealers because they have relatively buy the option payoff at maturity for A . One extension of this
large bid-offer spreads. idea is to make the zero-cost product a futures contract (rather
Exotic options arise for several reasons. In some situations, they than a forward contract) on the option payoff. This is known as a
can provide more efficient hedging than plain vanilla options. futures-style option.
Exotic options may also best reflect a firm's view on factors such For exam ple, a futures-style call is marked to market in the
as interest rates, exchange rates, and com m odity prices. O cca same way as a regular futures contract with the final settlem ent
sionally, exotic options are used for tax or regulatory purposes. of m ax(Sx — K, 0), where S j is the final price of the underlying
asset and fC is the strike price. Futures-style options are offered
by exchanges such as the C M E Group and Eurex.
15.1 E X O T IC S IN V O LV IN G A S IN G LE
ASSET Futures-style options can be contrasted with the more usual equity-
style options, which involve the buyer paying an upfront premium.
In this section, we describe some non-standard options that pro
vide payoffs based on the price of a single asset. Non-Standard American Options
Exchange-traded Am erican options can be exercised at any time
Packages at the pre-determ ined fixed strike price. Variations on this stan
A package is a portfolio consisting of plain vanilla options on dard product are traded in the over-the-counter m arkets. The
an asset. We presented several packages in Chapter 14: bull following are exam ples .2
spreads, bear spreads, butterfly spreads, calendar spreads, • The exercise may be restricted to certain dates. Such an
straddles, and strangles. Packages are som etim es regarded as option is known as a Berm udan option. Interest rate options
exotic options because they are positions built to reflect a sp e are som etim es Bermudan (e.g ., an Am erican-style bond
cific market view and risk tolerance. option might only be exercisable on interest payment dates).
For exam ple, an investor who takes a long position in a butterfly • There may be an initial lock-out period during which tim e the
spread is acting on his or her belief that the future asset price will option cannot be exercised. As we saw in an earlier chap
be near the middle strike price. A t the same time, the investor is ter, em ployee stock options usually have a lock-out period.
building this position without taking on a great deal of risk. In con O nce the lock-out period is over, em ployee stock options are
trast, an investor with a similar view of the market who chooses to referred to as vested.
sell a straddle or a strangle is taking on much more risk. • The strike price may change during the life of the option. For
exam ple, when a corporate bond has a call option feature
Zero-Cost Products allowing the issuer to retire the bond early (i.e., buy it back
from the holder), the strike price at which the issuer can exer
Any derivative product can be converted into a zero-cost
cise the option tends to decline as tim e passes.
product by arranging for it to be paid for in arrears.
Valuation and Risk M odels explains how binomial trees can be used
Consider a derivative that matures at time T and has a premium
to value American options. This methodology can be adapted to
equal to f. Rather than requiring that the premium be paid
accommodate all the non-standard features just mentioned
upfront (which is normally the case), the derivative can be struc
.3
tured so that the buyer instead pays f(1 + R)T at maturity (where
A
R is the interest rate for maturity T). 2 W arrants (which w ere introduced in C hapter 12) can som etim es have
all these features.
A
A s we will see in Valuation and Risk M o d e ls, in the standard binomial
The seller of the option is taking some credit risk and (assuming no tree set up there is a test for w hether early exercise is optimal at each
collateral is posted) the interest rate R should be the interest rate that node of the tree. To value a non-standard Am erican option, we test
would be paid by the buyer of the option on a zero-coupon bond with for the optim ality of early exercise only at nodes where exercise of the
maturity T. option is allowed using the applicable strike price.
Forward Start Options
A forward start option is an option that will begin at a future time.
It is usually stated that the option will be at-the-money at the time
it starts. Employee stock options can be forward start options if
an employer promises that they will be granted on future dates.
Gap Options
A gap option is a European call or put option where the price
triggering a payoff is different from the price used in calculating Fiqure 15.2 Payoff from a gap put option when
the payoff. (a) K : K-, and (b) fC, > K 2.
Suppose the trigger price is K2 and the price used in calculating
the payoff is K-|. Therefore, consider the following.
(i.e ., the insured person must bear the first USD 500 of any
• The payoff from a call option is S T — K-, if S T > K2.
loss in value below USD 50,000). Suppose further that the
• The payoff from a put option is K-\ — S j if S j ^ K2. insurance com pany incurs an adm inistrative cost of USD 1,000
in assessing a claim . W ithout considering the adm inistrative
The payoffs from gap options are shown in Table 15.1 as well as
in Figures 15.1 and 15.2. Note that the payoff from a gap option cost, the insurance com pany has in effect sold a put option on
can be negative. For exam ple, if /C| = 15 and K2 = 10, the the value of the asset with a strike price of USD 4 9 ,5 0 0 . W hen
adm inistrative costs are considered, the cost to the insurance
payoff from a gap call option is negative when 1 0 — S r < 15.
Similarly, if /C, = 15 and K2 = 20, the payoff from a gap put com pany is
option is negative when 15 < S T < 20. In some circum stances, 1,000 + 49,500 — S t when S T < 49,500, and
the probability of a negative payoff is sufficiently high such that 0 when S T > 49,500,
the cost of a gap option is negative.
where S T is the value of the assets. The insurance company's
G ap options can be used to describe som e insurance con position is equivalent to selling a gap put option where (using
tracts. For exam p le, suppose an asset has been insured for the previous notation) K2 = 49,500 and K-, = 50,500.
USD 5 0,000. However, there is a deductible of USD 500
Cliquet Options
Table 15.1 Payoffs from Gap Call and Gap Put Options A cliquet option is a series of forward start options with certain
rules for determ ining the strike prices. For exam ple, a cliquet
Payoff from Gap Payoff from Gap
option might consist of five call options: a one-year option, a
Range for S T Call Option Put Option
one-year option starting in one year, a one-year option starting
S t — K2 0 /< 1 - ST in two years, a one-year option starting in three years, and a
ST > k2 S T - K, 0
one-year option starting in four years. This is therefore a portfo
lio consisting of a regular one-year option plus four forward start
options. A simple rule for the strike prices could be that each
option is initially at-the-money.
cliquet options arise in some annuity contracts. For exam ple, an

investor could be offered a series of options (such as those
previously described) on the S&P 500. To use the term inology
introduced in Chapter 14, this would be a series of principal pro
tected notes .4
4 W hen the interest and dividends foregone by the investor are less than
the value of the at-the-money options offered to the investor, the return
Fiqure 15.1 Payoff from a gap call option when to the investor is som etim es capped so that the investor gives up some
(a) K : fC, and (b) K, > K 2. upside. The cliquet option then consists of a series of bull spreads.
Chapter 15 Exotic Options ■ 185

Chooser Options
With a chooser option, the holder has a period of time (after
purchasing the option) where he or she can choose whether it
is a put option or a call option. For exam ple, the holder of two-
year European option might be allowed to choose whether it is
a call option or a put option at the end of the first year.
One feature of chooser options is that they can be viewed as

(a) (b)
packages of call options and put options with different strike
prices and tim es to maturity. To see this, suppose the time Fiaure 15.3 The payoff from (a) a cash-or-nothing call
when the choice is made is T |. Assum e the value of the chooser and (b) a cash-or-nothing put is shown. K is the strike
option at this time (T- ) is
1
price.
max(c, p)
where c is the price of a call option and p is the price of a put

option. This can be rewritten as:
c + max( , p — c)
0
If both options are European options maturing at tim e T (where 2
T2 > T- ) with strike price K, we know from put-call parity that if

1
there is no income on the asset:

(a) (b)
p - c = PV(K) - S
Fiaure 15.4 The payoff from (a) asset or nothing call
where PV denotes present value from T to T- . This means that
2 1
and (b) an asset or nothing put is shown. K is the strike
the value of the chooser option is price.
c + max(0, PV(K) — S- ) 1
Traditional European options can be thought of as combinations

where S-| is the asset price at tim e T This shows that the option
of binary options.
is a package consisting of:
A long position in a European call option is a combination of:
• A call option with strike price K maturing at time T2, and
• A put option with strike price PV(K) maturing at tim e Tv • A long position in an asset-or-nothing call, and
• A short position in a cash-or-nothing call with a payoff equal

to the strike price.
Binary Options Similarly, a long position in a European put option is
There are four types of binary options. • A short position in an asset-or-nothing put, and
1. Cash-or-nothing call: This pays a fixed amount if the • A long position in a cash-or-nothing put with a payoff equal
asset price is above the strike price at maturity and zero to the strike price.
otherwise.
Cash-or-nothing options are som etim es referred to as digital
2 . Cash-or-nothing p u t: This pays a fixed amount if the options.
asset price is below the strike price at maturity and zero
Binary options, like gap options, have discontinuous payoffs. For
otherwise.
exam ple, suppose a cash-or-nothing call pays off USD 100,000
3 . Asset-or-nothing call: This pays an amount equal to the
if the price of a stock at a certain tim e is above USD 29. If the
asset price if it is above the strike price at maturity and zero
price is USD 28.9, there is zero payoff; if it is USD 29.1, the pay
otherwise.
off is USD 100,000. As a result, traders may have an incentive to
4 . Asset-or-nothing p u t: This pays an amount equal to the engage in illegal price manipulation (especially if the underlying
asset price if it is below the strike price at maturity and zero asset is thinly traded).
otherwise.
To see how this would work, suppose that there is only a little
The payoffs are illustrated in Figures 15.3 and 15.4. time until maturity and the price is just below USD 29. In this
case, a trader with a long position in the option might enter buy Lookback Options
orders to move the price above USD 29. M eanwhile, a trader
that has sold the option might enter sell orders to try to keep The payoff from a lookback option depends on the maximum or
the price below USD 29. Note that there is much greater incen minimum asset price reached during the life of the option. There
tive to manipulate the asset price com pared to when the payoff are four types of lookback options.
is a continuous function of the underlying asset (as with a plain 1. A floating lookback call gives a payoff equal to the amount
vanilla call or put). by which the final asset price exceeds the minimum asset
Cash-or-nothing options are notorious for their association with price.
fraudulent activities and are banned in several countries. This type 2 . A floating lookback p u t gives a payoff equal to the amount
of fraud can take several forms, but it often features firms that :5
by which the maximum asset price exceeds the final asset
1. Do not allow custom ers to withdraw funds from their price.
accounts, 3. A fixed lookback call gives a payoff equal to m ax(Smax — K, 0),

2 . Make unauthorized use of personal data (e.g ., credit cards, where Smax is the maximum asset price in a given time frame
passports, and drivers licenses), and and K is the strike price.
3 . O verstate the average expected return and then structure 4 . A fixe d lookback p u t gives a payoff equal to m ax(/C-Smin, 0),
the product so that the expected return is negative. where Smin is the minimum asset price in a given time frame
and K is the strike price.
Asian Options Lookback options are more expensive than regular options. To
illustrate this, consider options on a non-dividend paying stock
Asian options provide a payoff dependent on an arithm etic aver
when the stock price is USD 30, the strike price is USD 30, the
age of the underlying asset price during the life of the option.
risk-free rate is 4%, the volatility is 20% , and the time to maturity
The average price is usually calculated using periodic observa
is one year.
tions (e.g ., at the end of each day).
Using the Black-Scholes Merton assumptions (as discussed in
Suppose K is the strike price, S T is the final asset price, and Save
Chapter 15 of Valuation and Risk M o d els), the value of a Euro
is the average asset price. There are four types of Asian options.
pean call is USD 2.98 and that of a European put is USD 1.80.
1. A vera g e price calls: These provide a payoff at maturity With the same assumptions, a one-year fixed lookback call and
equal to max(Save - K, 0). put with the same strike price are worth USD 5.61 and USD 3.85
2 . A vera g e price p u ts: These provide a payoff at maturity (respectively). A one-year floating lookback call and put are
equal to m ax(K — Save, 0). worth USD 5.03 and USD 4.44 (also respectively).
3 . A vera g e strike calls: These provide a payoff at maturity The value of a lookback option depends on how often the price
equal to m ax(S j — Save, 0). of the underlying asset is observed. (These observations are
4 . A vera g e strike p u ts: These provide a payoff at maturity used to calculate the maximum or minimum price.) Specifically,
equal to max(Save — S j, 0 ). a lookback option increases in value as the observation fre

quency is increased . 6
An Asian option is less expensive than a regular option with the

same strike price and can be more appropriate for hedging. For Lookback options can make an investor appear prescient. For
exam ple, consider a company that repatriates its earnings in a exam ple, a floating lookback call provides a payoff that makes
foreign currency every w eek. Because its yearly profits depend it look as though an asset purchased at the end of a period was
on the average exchange rate during the year, an Asian put purchased at the minimum price during the period. M eanwhile,
option on the value of the foreign currency can ensure that a floating lookback put provides a payoff that makes it look
the average exchange rate will not be worse than a particular as though an asset sold at the end of a period was sold at the
exchange rate (i.e., the option's strike price). This strategy is maximum price during the period. A fixed lookback call or put is
much less expensive than entering into 52 plain vanilla options akin to an Am erican option where an investor can use hindsight
(i.e., one option for every w eek of the year). to choose the best possible exercise date.
5 Securities and Exchange Com m ission, (n.d.). Binary options. Retrieved

from https://www.investor.gov/additional-resources/general-resources/ 6 The results just given assume that the asset price is observed
glossary/binary-options continuously.

Barrier Options Parisian options are similar to standard barrier options, but they
have one important difference: The asset price must remain
Barrier options have payoffs that depend on whether the asset above or below the barrier for a specified number of days
price reaches a particular barrier. There are four types of barrier before the option is knocked in or out. For exam ple, suppose
options. the number of days specified is ten. In one type of Parisian
1. Down-and-out: This is a European (call or put) option that option, the ten days must be consecutive. In another type, they
ceases to exist if the asset price moves down from its initial can be any ten days during the life of the option.
level to the barrier level during the life of the option.
2 . Down-and-in: This is a European (call or put) option that Compound Options

comes into existence if the asset price moves down from its
A compound option is an option on another option. Thus, there
initial level to the barrier level during the life of the option.
are two strike prices and two maturity dates.
3 . Up-and-out: This is a European (call or put) option that
Suppose the maturity dates are T-1 and T (with T > T- ) and
ceases to exist if the asset price moves up from its initial
2 2 1
the strike prices corresponding to those maturity dates are /C|

and K2 (respectively). Four types of compound options are as
4 . Up-and-in: This is a European (call or put) option that comes follows.
into existence if the asset price moves up from its initial
1. Call option on call option: The holder has the right to pay
/Ci at time T in order to obtain a long position in a call
-1
O ptions that cease to exist when a barrier is reached are som e option on an asset. This call option allows the asset to be
tim es referred to as knock-out options, whereas options that bought for K2 at tim e T2.
come into existence when a barrier is reached are referred to as
2 . Put option on call option : The holder has the right to
knock-in options.
receive K-\ at tim e T-\ and obtain a short position in a call
Barrier options can be attractive to market participants because option on an asset. This call option allows the asset to be
they are less expensive than regular options. For exam ple, sup bought for K2 at tim e T2.
pose an asset price is currently USD 30 and a trader wants a
3 . Call option on put option: The holder has the right to pay
one-year call option with strike price USD 32. Furtherm ore, sup
/Ci at time T-\ in order to obtain a long position in a put
pose that the trader considers it unlikely that the price will fall to
option on an asset. This put option allows the asset to be
USD 27 during the year. In this case, he or she could be tem pted
sold for K2 at time T2.
to buy a down-and-out call with a strike price of USD 32 and a
barrier at USD 27. This call will provide the same payoff as a 4 . Put option on p u t option : The holder has the right to
(more expensive) regular call with a strike price of USD 32 as receive /C| at tim e T-, and obtain a short position in a put
long as the stock price does not fall below USD 2 7 .7 option on an asset. This put option allows the asset to be
sold for K2 at time T2.
Barrier options have some interesting properties. As with binary
options and gap options, their payoff is discontinuous. Consider Com pound options are attractive to traders who w ant more
an up-and-out call with a strike price of USD 50 and a barrier at leverage than that provided by plain vanilla options. For
USD 60. If the stock price stays below the barrier and reaches exam p le, suppose that the current price of a non-dividend
USD 59.90 at maturity, the payoff is USD 9.90. If the stock price paying stock is USD 50, its volatility is 20% per year, and
is 0 . 1 higher, the payoff is zero. the risk-free rate is 2% per year. The Black-Scholes Merton
assum ptions (covered in Valuation and Risk M o d e ls) show that
Normally, options prices increase as volatility increases.
a one-year call option on the stock with a strike price of USD
However, this is not necessarily the case with barrier options.
55 should cost USD 2.47. Thus, com pound option could be
For exam ple, when a knock-out option is close to the barrier, an
structured as follow s.
increase in the volatility may lower the price because increases
the probability that the barrier will be hit. 1. A t the six-month point, the investor has the right to pay
USD 3 for a call option.
2 . The call option, if purchased, gives the investor the right to

7 Using Black and Scholes' assum ptions, if volatility is 20% and the risk buy the stock for USD 55 at the one-year point.
free rate is 2 %, with no dividends, the value of the regular option is
USD 1.82. W hen there is a down-and-out barrier at USD 27 the price The Black-Scholes Merton assumptions give the value of this
reduces to USD 1.56. option as USD 0.99.
Under what circum stances will the call option be purchased highly correlated returns is more expensive than a similar call
at the six month point in this exam ple? Assuming no changes option where the asset returns are uncorrelated (due to portfolio
in the volatility or the risk-free rate, the Black-Scholes Merton diversification effects).
model shows that the value of the call option is greater than
USD 3 if the stock price at the six-month point is greater than
USD 54.30. The call option will therefore be purchased if the 15.3 EXO TICS D EPEN D EN T
stock price is greater than USD 54.30. ON VOLATILITY
Com pared to regular options prices, the prices of compound
options are more sensitive to changes in volatility. For exam ple, Some exotic options are dependent on the volatility of an asset
if the volatility increases from 20% per year to 30% per year, price, rather than the asset price itself. An asset's volatility per
day is generally measured as the standard deviation of its daily
the Black-Scholes Merton assumptions show that the price of
the compound option in the previous exam ple increases by returns. The annualized volatility is usually calculated by multi
160% (from USD 0.99 to USD 2.57). M eanwhile, using the same plying the daily volatility by the square root of 2 5 2 .8
assumptions, the price of the regular option increases by only

79% (from USD 2.47 to USD 4.43).
Volatility Swap
A volatility swap is appropriate for a trader who wants to take a
15.2 EXO TICS INVOLVING position dependent only on volatility. W hile plain vanilla options
MULTIPLE ASSETS provide an exposure to volatility, they also depend on the price
of the underlying asset. A potential advantage of volatility swaps
We now move on to consider options with payoffs that depend is that their payoffs depend solely on realized volatility.
on the prices of two or more assets. A volatility swap is a forward contract on the realized volatility
of an asset during a certain period. A trader agrees to exchange
Asset-Exchange Options a pre-specified volatility for the realized volatility at the end of
the period (with both being multiplied by a certain amount of
In an asset-exchange option, the holder has the right to exchange principal).
one asset for another. Asset-exchange options can arise in several
Suppose that L^0\ is the principal, a K is the pre-specified yearly
ways. From the perspective of a U.S. investor, an option to exchange
X euros for Y Australian dollars is an option to exchange one asset volatility, and cr is the realized yearly volatility (calculated as
for another. An offer by Company X to acquire Company Y through described above). A swap where the trader pays fixe d provides
a payoff at the end of the period equal to:
the exchange of a certain number of its own shares for shares of
Company Y is another type of asset-exchange option. ^ / o l(cr - (T k )
Asset-exchange options are closely related to options where a Similarly, a volatility swap where the trader receives fixed
trader will receive the more valuable of two assets (i.e., Asset A provides a payoff at the end of the period equal to:
and Asset B). This is because the trader's position can be regarded
l-y o l ( ^ K - c r)
as a position in Asset A combined with an option to exchange
Asset A for Asset B. Similarly, it can be regarded as a position in Suppose that the principal (Lvo|) for a pay-fixed volatility swap on
Asset B combined with an option to exchange Asset B for asset A. the S&P 500 during the next three months is USD 1 million and
ok is 15% per year. Suppose further that the realized volatility of
the S&P 500 during the three-month period is 1% per day and
Basket Options thus the yearly volatility is 15.87% (= 1% X V^252 = 1 5 .8 7 % ).
A basket option is an option on a portfolio of assets. These The payoff from the volatility swap would then be USD 8,700
portfolios can contain assets such as stocks, stock indices, ( = (0.1587 — 0 .1 5 ) X USD 1 ,0 0 0 ,0 0 0 ). The corresponding
and currencies. Basket options can be appropriate hedging payoff from a receive-fixed volatility swap is USD —8,700.
instruments for firms seeking to reduce costs by hedging their
aggregate exposure to several assets with a single trade.
8 252 is an estim ate of the num ber of trading days in a year. It is
Basket options are dependent on the correlation between the assum ed (as will be discussed in Valuation and Risk M o d e ls) that the
volatility of an asset price during a period is dependent on the number
returns from the assets in the basket. For exam ple, a call option of trading days during the period, rather than the num ber of calendar
on the future value of a portfolio consisting of ten assets with days during the period.

Variance Swap A sset
Price, S Boundary
The payoff from a variance swap is calculated analogously to the . W here Value = 0
payoff from a volatility swap. As a reminder, the variance rate for jL.___________
an asset is the square of its volatility. Thus, the payoff for a pay-
Boundary W here
fixed variance swap is Value = M ax(S - 30, 0)
l-va r ( o - 2 - V K)
where Lvar is the principal for the variance swap and VK is the Tim e (Yrs)
specified fixed variance. The payoff or a receive-fixed variance 0 1
swap is similarly: F ia u re 15.5 Possible boundary for an up-and-out call

option when initial asset price is 30 and barrier is 50.
U ,r (V K - a 2)
Som etim es Lvo| and <« are specified in a variance swap with the
7
The up-and-out option is worth zero if the horizontal boundary is
understanding that VK = and Lâr =
l-vol 9 reached and max(S — 30, 0) if the vertical boundary is reached.
( 2 crK) The task in static options replication is to choose a portfolio of
plain vanilla options that match the payoff as closely as possible
15.4 HEDGING EXO TICS on the boundary.
The natural first option to use is a one-year European call option

Com pared to regular options, some exotics are easier to with a strike price of 30. This matches the payoff on the vertical
hedge, while som e are more difficult. For exam ple, an average part of the barrier. To match payoffs on the horizontal part of
price option is easier to hedge than a regular option because the barrier, choose N equally spaced points on the horizontal
as its m aturity approaches, more of the asset prices used to barrier and an additional N European options with a maturity of
calculate Save are observed and thus the payoff becom es pro
A A A Q
less than one year and value each option. Then, solve a set
gressively more certain. The value of the option therefore has of sim ultaneous equations to ensure that the portfolio formed
very little sensitivity to asset price m ovem ents occurring late from the all the options has a value of zero at each of the N
in its life. points on the horizontal boundary.
By contrast, barrier options can be quite difficult to hedge To express this algebraically, suppose Cj is the value of a one-
because when the asset price is close to the barrier, uncertainty year European call option with a strike price of USD 30 at the jth
about the payoff increases. Traders have developed a hedging point on the horizontal boundary ( 1 < j < N) and that fj; is the
procedure known as static options replication for dealing with value of the fth additional option at the jth point on the horizon
n
such exotic options. The key principle underlying the proce
1
tal boundary (1 < /, j < N). Define a,- as the size of the position
dure is that two portfolios that are worth the same on some in the fth additional option. The a,- are determ ined by solving the
boundary (which is a function of asset price and time) must also N simultaneous equations:
be worth the same at all interior points (i.e., points that could be N
reached before the boundary is reached). Therefore, if we can
cj +2 ai fij = 0 o ^j ^ N )
find a portfolio of plain vanilla options worth approxim ately the ;= 1
same as an exotic option on a boundary, the exotic option can As N is increased, the horizontal boundary is matched more
be hedged by shorting the portfolio. precisely and the hedge created by shorting the portfolio
To illustrate the nature of static options replication, consider improves.

a one-year up-and-out call option. The strike price is USD 30, A static options replication hedge can be left unchanged until
the barrier is USD 50, and the current stock price is USD 25. the boundary is reached. The trader must then unwind the
The natural boundary to use is the one shown in Figure 15.5. hedge portfolio and create a new hedge.
11 The options must have a maturity of less than one year so that they
9 From calculus, the change in <x2 is approxim ately equal to 2cr tim es the do not affect the value of the portfolio on the vertical part of the bound
change in er. This explains why the value of Lvar that corresponds to Lvo| ary. We could choose call options with strike price 50 and m aturities
is approxim ately given by Lvar — Lvo/(2crK). 1/(N + 1), 2/(N + 1 ) , . . . , N/(N + 1) years.
10 See E. Derm an, D. Ergener, and I. Kani, "Static O ptions Replication," 12 If the option expires before a tim e corresponding to a barrier point,
Journ al o f D erivatives, 2, 4 (Summer 1995): 78-95. the value of the option at the barrier point is zero.
SU M M A RY The adage "you get what you pay for" applies to options. How
ever, exotic options generally are less com petitively priced than
Derivatives dealers have been very creative in designing exotic regular options, so in their case it might be more accurate to
options. And while this chapter has covered some of the most say, "you get less than you pay for." Traders should bear this in
common exotic options, it has not mentioned other derivatives mind and avoid being overly optim istic in assessing potential
with intriguing names such as shout options, Madonna options, returns from exotic options.
Himalaya options, and pyramid options. In addition to trading exotics (whose future payoff depends
Some options (such as Asian options and basket options) can on the price of an asset), investors can trade instruments that
be efficient hedging instruments. However, it is not entirely depend on an asset's future volatility. A volatility swap provides
clear why some of the other options we have mentioned in this a payoff proportional to the difference between a realized vola
chapter are used by market participants. Some traders with a tility and a pre-specified volatility. A variance swap provides a
particular view of the market may find a certain type of exotic payoff proportional to the difference between a realized vari
option attractive. For exam ple, if a short-term spike in an asset ance rate and a pre-specified variance rate.
price is expected for some reason, a long position in a lookback Some exotic options are easier to hedge than plain vanilla
put position could be considered. If a trader wants to speculate options, while the opposite is true for other exotics. A pro
on an increase in a stock price with a highly levered transaction, cedure known as static options replication can be used as an
a compound option might be a reasonable choice. If a trader alternative to the G reek letter hedging procedures discussed
wants to reduce the cost of a regular call option and considers in Valuation and Risk M o d els. This involves shorting a portfolio
it unlikely that the asset price will fall to a certain level, a down- of regular options designed to replicate the value of an exotic
and-out call option could be appropriate. option on a boundary.
C h ap ter 15 Exotic Options 191

Q U ES T IO N S

15.1 W hat is a Bermudan option? 15.7 List four types of binary options.
15.2 Explain how a futures-style option works. 15.8 Does a basket option becom e more or less valuable as
15.3 W hat is a gap option? the correlation between the asset returns in the basket
increases?
15.4 Explain how (a) a chooser option and (b) a forward-start
option work. 15.9 How is a pay-fixed volatility swap defined?
15.5 List four types of compound options. 15.10 W hy are average price Asian options easier to hedge
than plain vanilla options?
15.6 How is a floating lookback put defined? Does it become
more or less valuable as the number of asset price obser
vations (used to calculate the payoff) increases?
Practice Questions
15.11 W hat is equivalent to a portfolio consisting of an up-and- 15.16 Exp Iain why a gap call option is a regular call option plus
in put option and an up-and-out put option where the a binary option when K2 > K using the same notation
barrier, strike price, and tim e to maturity are the same for as in the chapter
the two options?
15.17 Provide an alternative decom position of the chooser
15.12 W hat is the put-call parity relation between the following option to that given in the chapter so that it is a call
compound options: call on a call and a put on a call? maturing at tim e T-, plus a put maturing at time T2.
15.13 Does a down-and-out put option becom e more or less 15.18 Is a one-year at-the-money call option on a basket of
valuable as we increase the frequency with which we three stocks more or less valuable than a portfolio con
observe the asset prices in determ ining whether the bar sisting of three one-year at-the-money options, one on
rier has been hit? each stock?
15.14 W hat is the payoff from a portfolio consisting of a long 15.19 W hat view of the market would a trader have if he or she
position in both a floating lookback call and a floating chose a receive-fixed variance swap?
lookback put?
15.20 W hat is the key theoretical result underlying static
15.15 Derive expressions for the payoffs from a: options replication?
a. Long position in an average price call and short posi
tion in an average price put,
b. A long position in an average strike call and short
position in an average strike put, and
c. A long position in a plain vanilla European call and
short position in a plain vanilla European put.
d. All options have the same strike price and time to
maturity. Use the results to derive a relationship
between the prices of the six options you have
considered.
A N SW ER S
15.1 A Bermudan option is an option that can be exercised 15.12 We use the notation in the chapter and assume that the
only at certain pre-specified discrete tim es. K i, K2i T i , and T are the same for both the call on the
2
1 5 .2 A futures-style option is a futures contract on the option's call and the put on the call. A long position in the call on
a call plus the present value of K (with discounting from
payoff. The final settlem ent of the futures contract equals
this payoff. T to today) equals a long position in a put on the call
-1
plus the value of a European call maturing at time T2. If

1 5 .3 A gap option is a European call or put option where
c-| is the value at time T-| of a European call maturing at
the trigger determ ining whether there will be a payoff
tim e T2f the result can be seen from:
is different from the strike price used to calculate the
payoff. m a x(ci — /Ci, 0 ) + /C| = m ax( /C| — c-i, 0 )
+ Ci = max(c-|, /C|)
1 5 .4 A chooser option is an option with a certain strike price
and time to maturity where at some stage during the 15.13 A down-and-out put becom es less valuable because
option's life the holder will choose whether it is a call or the chance of the barrier being hit, so that the option is
knocked out, is greater.
a put. A forward start option is an option that will come
into existence at some future tim e. 15.14 A floating lookback call provides a payoff of S T — Smin
and a floating lookback put provides a payoff of
1 5 .5 The four types of compound options are call on a call,
call on a put, put on a call, and put on a put. Smax — S j. The payoff from the portfolio is therefore the
excess of the maximum asset price over the minimum
1 5 .6 A floating lookback put is an option that pays off the
asset price ( S 7 Smin + Smax Sj Smax ■Smin)'
amount by which the maximum asset price during the
option's life exceeds the final asset price. It becomes
15.15 (a) A long average price call gives a payoff of
more valuable as the asset price is observed more fre m a x (S ave — K, 0 ) . A short average price put gives a pay
quently in the calculation of the payoff. off of —m ax( K — Save, 0 ) . The payoff in (a) is therefore
always Save — /C whether Save > K or Save ^ K. Similarly,
1 5 .7 The four types of binary options are cash-or-nothing call,
the payoff in (b) is always S j — Save and the payoff in (c) is
cash-or-nothing put, asset-or-nothing call, and asset-or-
always S T — K. From this, it follows that:
nothing put.
(ci - P i) + (c - p2) = (c - p)
2
1 5 .8 As the correlation between the returns increases, a bas

ket option becom es more valuable because large m ove where c-\ and p-| are the prices of the average price call
ments in the value of the portfolio becom e more likely. and put, c and p are the prices of the average strike call
2 2
and put, and c and p are the prices of the plain vanilla
1 5 .9 A pay-fixed volatility swap provides a payoff equal to
call and put.
/-^(cr — erK) where a is the realized volatility per annum
during a certain period, er« is a pre-specified volatility,
15.16 Consider a gap option where the trigger price is K2 and
and Lyoi is a pre-specified principal amount. the strike price for determ ining payoffs is /C|. The gap call
option is a plain vanilla option with strike price K2 plus a
1 5 .1 0 An average price Asian option depends on the aver
cash-or-nothing binary option that pays off K2 — K if the
age asset price during the life of the option. A s the end
asset price is above K2.
of the option's life approaches, the average becom es
increasingly more certain so that there is very little risk 15.17 With the notation in the text we can write
to be hedged. The same is not true of a plain vanilla max(c, p) = p + max(c — p, 0 )
option. = p + m a x (S — PV(K), 0 )
15.11 The portfolio is equivalent to a plain vanilla put option. If This shows that the chooser option is a portfolio consist
the barrier is hit, the up-and-in option provides the plain ing of:
vanilla option payoff and the up-and-out option provides • A put option maturing at time T2, and
no payoff. If the barrier is not hit, the up-and-out option
• A call option with strike price PV(/<0 maturing at
provides the plain vanilla option payoff and the up-and-in
tim e T-|.
option provides no payoff.

15.18 It is less valuable because the stocks are less than per of the volatility. The trader is therefore expecting a low
fectly correlated. volatility.
r\
15.19 Because the payoff is Lvar(V/< — a ) the trader would be 15.20 If two portfolios have the same value on some boundary
taking the view that the realized variance rate (a 2) will be in {S, t} space they must have the same value at interior
less than the pre-specified variance rate (VK) in the future points (i.e., all points that could be reached prior to the
period that is considered. The variance rate is the square boundary being reached).
Learning Objectives
Describe Treasury rates, LIBO R, and repo rates, and Calculate the duration, modified duration, and dollar
explain what is meant by the risk-free rate. duration of a bond.
Calculate the value of an investment using different Evaluate the limitations of duration, and explain how
compounding frequencies. convexity addresses some of them .
Convert interest rates based on different compounding Calculate the change in a bond's price given its duration,
frequencies. its convexity, and a change in interest rates.
Calculate the theoretical price of a bond using spot rates. Com pare and contrast the major theories of the term
structure of interest rates.
Derive forward interest rates from a set of spot rates.
• Derive the value of the cash flows from a forward rate

agreem ent (FRA).
195
Interest rates are central to finance. This chapter discusses Libor
different types of interest rates and how they are m easured,
analyzed, and used. The London Interbank O ffered Rate (Libor) has historically been
an im portant reference rate in financial m arkets. Libor interest
The interest rate term structure describes how interest rates
rates are compiled from the estim ated unsecured borrowing
vary depending on their maturity. In an upward-sloping term
costs of 18 highly rated global banks .2 These estim ates are
structure, long-term interest rates are higher than short-term
made daily (just prior to 11 a.m . in London) for five different cur
interest rates. In a downward-sloping term structure, the
rencies and seven borrowing periods ranging from one day to
opposite is true.
one year. The highest and lowest four quotes for each estim ate
Chapters 17 and 18 will discuss products that rely on interest are discarded and the rem ainder are averaged to determ ine the
rates. Chapter 19 will cover interest rate futures and day count daily Libor fixings.
conventions. Chapter 20 will explain interest rate swap markets.
Banks have been accused of trying to m anipulate Libor rates
Some of the topics introduced in this chapter will be covered
by providing overly high or low estim ates of their borrowing
further in Chapters 9 to 13 of Valuation and Risk M odels.
rates. A bank might have an incentive to do this when the
payoff from a derivative it has entered depends on a specific
Libor fixing for that day. A nother scenario would be a bank
16.1 C A T E G O R IE S O F RATES that reports excessively lower rates in order to appear more
creditw orthy.
An interest rate is the return earned by a lender when advanc
ing funds to a borrower. A major factor in determ ining an inter The underlying problem is that there is not enough Libor bor
est rate is the risk that the borrower will default and not repay rowing for estim ates to be based on actual transactions. As a
the lender in full; this risk is called credit risk. As the credit risk result, the person providing the estim ates has to use quite a bit
increases, the interest rate required by the lender from the of judgem ent.
borrower also increases. Another factor in determ ining inter
There are plans to begin phasing out Libor in 2021 and replace
est rates is liquidity; in this context, liquidity refers to the ease it with a rate based on actual transactions (rather than esti
with which an interest-bearing instrument can be sold from one mates). There are two main candidates for this replacem ent
investor to another at a com petitive price.
benchm ark rate: the repo overnight rate and the overnight
Interest rates (and the spreads between different interest rates) interbank borrowing rate.
are often expressed in basis points. O ne basis point is 0.01% The U.S. has proposed the use of the repo-based Secured O ver
and thus an interest rate of % is equal to
2 2 0 0 basis points. night Financing Rate (SO FR). M eanwhile, the United Kingdom
has proposed the use of the Sterling O vernight Index A verage
Government Borrowing Rates (SO N IA), which is based on interbank borrowing. Both are one-
day rates. Swaps (to be discussed shortly, as well as in more
One im portant interest rate is that paid by a governm ent on its detail in Chapter 20) are necessary to create a com plete interest
borrowings in its own currency. In the U .S., this is referred to as rate term structure from these rates.
the Treasury rate. Note that it is considered highly unlikely that
the governm ent of a developed country will default on debt
issued in its own currency. This is because a governm ent can Repo Rate
always create more currency to meet its obligations
In a repo agreem ent, securities are sold by Party A to Party B for
. 1
As a result, governm ent debt from developed countries is con a certain price with the intention of being repurchased at a later
sidered to be risk-free and the interest rates on these borrow time at higher price.
ings are generally below those on other borrowings in the same Suppose that the initial price paid by B to A for the securities
currency.
is X and the price at which they will be repurchased is X + e.
It is not unknown for a governm ent of a developing country to default

on debt issued in its own currency, as will be explained in Chapter 5 of
Valuation and Risk M o d els. The countries in the European Union that
use the euro are in the situation where they do not have control over the 2 C red it ratings are discussed further in Chapter 4 of Valuation and Risk
money supply of the currency they use and may therefore default. M o d els.
The effect of the repo is that Party B has lent X to Party A for an O ther swaps can be used to create an interest rate term structure
amount of interest equal to e. from observed one-day rates .3 A three-month overnight indexed
swap is an agreem ent to exchange the geom etric average of the
If Party A fails to repurchase the securities as agreed, Party B
daily effective federal funds rates over the following three
can simply keep the securities. This means that Party B takes
months for a pre-determined fixed rate (with both rates being
very little risk, provided that:
applied to the same principal). Meanwhile, a five-year swap is
• The value of the securities equals (or is very close to) X, and one in which it is agreed that the exchange just described will
• This value is fairly stable. take place every three months for the next five years.
Note that Party B is in a better position than it would be if the

securities had been merely pledged as collateral. This is because Risk-Free Rates
the repo leads to Party B owning the securities without having
to initiate legal action to gain possession. Repos lasting one day The risk-free rates used to value derivatives are determ ined from
(i.e., overnight repos) are the most common, but longer-term overnight interbank rates using overnight indexed sw aps .4 Trea
repos exist as well. sury rates are not used because they are considered to be artifi
cially low. Two reasons for this are as follows.
Overnight Interbank Borrowing 1. Banks are not required to keep capital to support an invest
ment in Treasury instruments, but they are required to keep
In many countries, banks are required to keep cash in reserve capital for other very low risk instruments.
with the central bank. This reserve requirem ent depends on
2. In some countries (e.g ., the U .S.), the income from Treasury
each bank's outstanding liabilities. A t the end of each day,
instruments is given favorable tax treatm ent
som e banks have excess reserves, w hile others have a
. 5
sh o rtag e. This leads to the p ractice of o vernight interbank

lending.
16.2 C O M P O U N D IN G F R E Q U E N C Y
In the U .S., the interest rate on this overnight lending is
referred to as the federal funds rate. The weighted average of In order to fully understand a quoted interest rate, it is necessary
the rates in these transactions is term ed the effective federal to know the compounding frequency with which it is measured.
funds rate. The Federal Reserve monitors this rate and periodi The compounding frequency defines the extent to which interest
cally intervenes with its own trades to increase or decrease the is earned on interest. Up to now, this book has usually assumed
rate. M eanwhile, the respective average overnight rates in the annual compounding (i.e., compounding that occurs once per year).
United Kingdom and the Eurozone are the Sterling O vernight
For exam ple, suppose USD 100 is invested at 5% for five years
Index A verage (SON IA) and the Euro O vernight Index A verage
and that the interest rate is expressed with annual com pound
(EO N IA ).
ing. This means that this USD 100 grows to:
100 X 1.05 = 105

Swaps
at the end of one year. Funds are then reinvested so that it
W hile swaps are discussed at length in Chapter 20, note here becom es
that swaps are a convenient way in which long-term rates are
105 X 1.05 = 110.25
created from short-term rates to create a com plete interest rate
term structure.
Historically, the most common swap has been one where Libor 3 If a one-day rate such as S O FR , SO N IA , or EO N IA is to be used as a
benchm ark for lending or defining paym ents in derivative transactions,
is exchanged for a fixed interest rate. In a five-year swap, for
it is im portant that the m arket be able to define longer term rates that
exam ple, the three-month Libor might be exchanged for a fixed correspond to the one-day rate. This is what is achieved with swaps,
interest rate of 3% per annum. This means that the observed as will be discussed in more detail in C hapter 20. For exam ple, three-
month borrowing in the U .K. could be defined as the three-month
Libor rate is exchanged for a rate of 3% per annum every three
SO N IA rate plus 80 basis points.
months during the five years (with both rates being applied
4 Prior to the 2007-2008 crisis, Libor was used as a proxy for the risk
to a pre-specified principal). As we will explain in Chapter 20,
free rate. A s Libor rates soared during the crisis, m arket participants
swaps of this sort allow a com plete Libor term structure to be becam e less com fortable with this practice. O vernight interbank rates
determ ined despite the fact that Libor rates are (by their nature) are considered to provide a better proxy.
short-term rates. 5 Interest from U.S. Treasuries is not taxed at the state level in the U.S.
Chapter 16 Properties of Interest Rates ■ 197

at the end of two years. Further reinvestm ent means that it Table 16.1 Value to Which USD 100 Grows at 5% as
becom es the Compounding Frequency is Varied
110.25 X 1.05 = 115.76 Value of USD 100 at the
Compounding Frequency End of Five Years (USD)
at the end of three years. A t the end of year five, the investor
will receive Annual 127.63
100 X 1.05s = 127.63 Semi-annual 128.01
Q uarterly 128.20
Now suppose that the 5% interest rate is expressed with sem i
annual com pounding. This means that half of 5% (i.e., 2.5%) is Monthly 128.34
earned every six months with the funds being reinvested. By the W eekly 128.39
end of six months, USD 100 grows to:
Daily 128.40
100 X 1.025 = 102.5
This is then reinvested for a further six months to becom e can be considered as alternative units of m easurem ent for
interest rates.
102.5 X 1.025 = 105.12
Suppose R is an interest rate when compounding is m times
-1 -1
This is then reinvested for a further six months to becom e

per year and R is the equivalent rate when compounding is
2
105.12 X 1.025 = 107.75 m tim es per year. From Equation (16.1), the future values are
2
the same if:

and so on.
R-, \ miT / R2 \ m*T
A t the end of five years, the investm ent becom es A I 1 + — = A 1 H-----
m-| m 2
100 X 1.02510 = 128.01

This means that:
From an investor's perspective, it is better that funds be invested
at 5% with semi-annual compounding than at 5% with annual
compounding (because it is better to receive USD 128.01 than
to receive USD 127.63). The opposite is true from a borrower's
This can be rewritten as:
perspective (because it is better to pay USD 127.63 than to pay
USD 128.01).
R = m
2 2 (16.2)
Q uarterly compounding means that reinvestm ent is assumed to
take place every three months. If the interest rate was 5% with
For exam ple, suppose an interest rate is 8 % per year with
quarterly com pounding, 1.25% (= 0.05/4) would be earned each
annual compounding and we want to know what it would be if
quarter and USD 100 would grow to:
it had been expressed with quarterly com pounding. In this case,
, 0.05 V ° R-i = 0.08, m = 1, and m = 4. The rate R (i.e., with quarterly
+ — = 128.20
-1 2 2
1 0 0 1
compounding) is
at the end of five years. Monthly, w eekly, and even daily 0.08 V
0.0777
/ 4
compounding frequencies are defined similarly. Table 16.1 — ) - 1J

shows how the compounding frequency affects the value to
or 7.77% per year.
which USD 100 grows to at the end of five years when it is
invested at 5% per annum. As a second exam ple, suppose a rate is 10% per year with
monthly compounding and we want to know how it would
The general formula for the future value of an amount A when it
be expressed with semi-annual com pounding. In this case,
is invested at rate R for T years when R is expressed with a com
R-i = 0.1, m = 12, and m = 2. The rate R2 (i.e., with sem i
pounding frequency of m tim es per year is
-1 2
annual compounding) is
/ D \m T
Future Value = A ( 1 -\---- j (16.1)
0.10V2/2
0.1021
+i t 1 ) ~
Ju st as pounds and grams are alternative units of m easurem ent

for weight, annual compounding and semi-annual compounding or 1 0 . 2 1 %.
Usual Conventions where e is the mathematical constant is approxim ately equal to
2.71828. The future value of USD 100 at the end of five years
The usual convention is that interest rates are expressed so that when the interest rate is 5% with continuous compounding is
the compounding frequency corresponds to the frequency with (in USD):
which payments are made. Instruments that last one year or less
(from date of issue) are term ed money market instruments; their 100e ° 0 5 x 5 = 128.40
principal and interest are both paid at maturity. An interest rate From Table 16.1, we see that this is (to two decimal places) the
on a three-month money market instrument is therefore typically same as the future value with daily com pounding.
quoted with quarterly com pounding, the interest rate on a one-
Continuous compounding is used in the valuation of options
year money market instrument is typically quoted with annual
and other, more com plex derivatives. It can also be convenient
com pounding, and so on .6
to work with continuously com pounded interest rates and yields

This convention can be misleading. If the one-year money when considering futures prices (see Section 11.5 of this book).
market interest rate is 1 0 % and the one-month money market The formulas involving interest rates expressed with continuous
interest rate is 9.8% , it may appear as though the one-year rate compounding are actually sim pler than those involving periodic
is greater than the one-month rate. However, the 9.8% is m ea compounding (e.g ., compare Equation (16.3) with (16.1)).
sured with monthly com pounding, while the 1 0 % is measured
Suppose Rm is an interest rate expressed with compounding
with annual com pounding. Equation (16.1) shows 9.8% with
m tim es per year and Rc is the same rate expressed with con
monthly compounding becom es 10.25% when measured with
tinuous com pounding. From Equations (16.1) and (16.3), we
annual com pounding. This means that the one-month rate is
must have
actually higher than the one-year rate.
In the U.S. and some other countries, bonds which last longer
than one year (from date of issue) normally pay interest every six
so that
months. The yield provided by a bond (which is defined later in
this chapter) is therefore normally expressed with semi-annual
com pounding.
Som etim es the compounding frequency does not reflect the

frequency of payments. In Canada, for exam ple, m ortgage inter
Rc = m Ini 1 + (16.4)
est rates are expressed with semi-annual compounding even
though payments are made every month (or every two weeks).
In the U .S., this mortgage interest rate would be expressed (16.5)
with monthly com pounding; in the United Kingdom , it would
be expressed with annual com pounding. These varied ways of
The continually com pounded interest rate Rc that corresponds
expressing interest rates are regulatory requirements designed
to a semi-annually com pounded rate of % per year is therefore:
6
to allow borrowers to compare interest rates with as little confu

sion as possible. 2 X In 1 + 0.0591
or 5.91% . The quarterly com pounded rate corresponding to a

16.3 C O N T IN U O U S C O M P O U N D IN G
continuously com pounded 9% per year is
Table 16.1 shows the impact of daily com pounding. We can 4 X (e 0 0 9 / 4 - 1) = 0.09 10
imagine increasing the compounding frequency further so that
or 9.10% per year.
we compound every hour, every minute, or every second. In
the limit, we obtain continuous com pounding. With continuous
com pounding, Equation (16.1) becomes
16.4 Z E R O RATES
Future Value = A e RT (16.3)
A zero-coupon interest rate (also called a zero rate or spot rate)

for maturity T is the applicable interest rate when an investor
6 There are other conventions associated with the way interest rates
are quoted on money m arket instrum ents. These will be discussed in receives the total return (interest and principal) at the end of
C hap ter 19. T years.

As mentioned previously, instruments with maturities of one Table 16.2 Zero-Coupon Interest Rates (Compounded
year or less usually provide all principal and interest at the ends Semi-Annually)
of their lives. They therefore provide zero rates directly. Most
Zero-Coupon Interest Rate
instruments that (when issued) last longer than one year provide
Maturity (Years) (Semi-Annually Compounded)
regular interest payments known as coupons .7 Zero rates must
then be determ ined using a procedure we will describe later in 0.5 3.0%
this chapter. 1 . 0 3.8%
1.5 4.4%
16.5 D ISC O U N TIN G 2 . 0 4.8%
2.5 5.1%
In Section 16.2 and 16.3, we saw how the future value of an
3.0 5.3%
amount A invested today can be calculated. As an alternative,
we can ask how much a certain sum of money received in the
future is worth today. This is known as discounting (or deter
mining the present value). The present value of an amount A Suppose further that the bond provides semi-annual coupons
received at time T when the interest rate R is expressed with a at the rate of % per year every six months. This means that
6
compounding frequency of m tim es per year is the bond provides interest of USD 3 every six months. These
/ ft \-mT cash flows are as shown in Table 16.3. To value the bond,
Present Value = A 1 -\---- (16.6) we discount the cash flows at the interest rates correspond
V m) ing to the tim es when they are received and then sum the
For exam ple, USD 500 received in three years when the interest results.
rate is 4% (compounded semi-annually) has a present value of:
The present value of the first cash flow (received at time
0.5 years) can be calculated using Equation (16.6) and is
500 1 + 443.99
3 X 2.9557
The discount factor is the amount by which the cash flow is mul
tiplied to get the present value. In this case it is
The present value of the second cash flow is
0.8880 -2x1
0.038
3 X 2.8892
When the interest rate R is com pounded continuously we obtain
These and other results are summarized in Table 16.4. Using
Present Value = Ae~RT
these present values, the value of the bond itself can be shown
to be USD 102.0695. This is greater than USD 100 because
16.6 BO N D VA LU A TIO N the interest rate earned on the bond is % per year (which is
6
higher than the rates at which the bond's cash flows are being
The valuation of a bond involves identifying its cash flows and discounted).
discounting them at the interest rates corresponding to their

maturities. As an exam ple, suppose that zero-coupon inter
est rates with semi-annual compounding are those shown in Table 16.3 Cash Flows from the Bond
Table 16.2. (How they can be calculated will be discussed later
Time (Years) Cash Flow (USD)
in this chapter.)
0.5 3
Consider a three-year bond with a principal of USD 100. The
principal is also known as the par value (also known as the face 1 . 0 3
value) and it is the amount that must be repaid at maturity. In 1.5 3

this exam ple, the maturity is three years. 2 . 0 3
2.5 3
7 An exception is provided by STRIPS, which are explained in C hapter 9
3.0 103
of Valuation and Risk M o d els.
Table 16.4 Calculation of Bond Value Defining terms:
Cash Flow Discount Present Value m: The frequency with which coupons are paid (m = 2 in
Time (Years) (USD) Factor (USD) our exam ple),
0.5 3 0.9852 2.9557 A : The value of an annuity that pays USD 1 on each
coupon paym ent date, and
1 . 0 3 0.9631 2.8892
d: The value of USD 1 received at bond maturity.
1.5 3 0.9368 2.8104
The value of coupons is A(c/m). The value of the final principal is
2 . 0 3 0.9095 2.7285
100d. If c is the par yield, we therefore require
2.5 3 0.8817 2.6451
3.0 103 0.8548 88.0407 ( m ) A + 100d = 1 0 0 (16.7)

Total 102.0695
This can be rewritten to have c equal to:
( 1 0 0 - 1 0 0 c/)m
Bond Yield A
The return earned by an investor on a bond is often described In our exam ple, m = 2, A = 5.5310, and d = 0.8548, so that
by what is termed the bond yield. This is the discount rate that the par yield is
equates the present value of all the cash flows to the market price. (100 - 100 X 0.8548) X 2
5.2517
5.5310
Suppose that the market price of the bond we have been con
sidering is the value calculated in Table 16.4 (i.e., USD 102.0695). which is the same as that given by solving the bond pricing
The bond yield y (semi-annually compounded) is the solution to: equation.
3 3 3 3 3
+ y + + y/ + + y/ + + y/ + + y/
16.7 DURATION
1 / 2 ( 1 2 ) 2 ( 1 2 ) 3 ( 1 2 ) 4 ( 1 2 ) 5
103
+ ----------- 7 = 102.0695
d + y/ 2 ) 6
Yield duration8 measures the sensitivity of a bond's price to a

The solution can be found using an iterative trial-and-error change in its yield. Defining term s:
procedure. (Solver in Excel is a useful resource.) The solution for
our exam ple is y = 5.2455% . B: The price of a bond,
We would therefore refer to the bond in Table 16.3 as providing D: The bond's duration,
a yield of 5.2455% (with semi-annual compounding). Ay: The change in the bond's yield, and
A B: The change in the bond's price resulting from the

Par Yield change in the yield.
The par yield of a bond is the coupon rate that would cause the The approxim ate duration relationship is 9
value of the bond to equal its par value. A B = -D B A y (16.8)

For exam ple, consider the rates in Table 16.2 with a par value This is equivalent to:
of USD 100. If the par rate (paid semi-annually) on a three-year
AB
bond is c, the par yield would require
B
c c c
The value of the duration (D) depends (slightly) on the
/ 2 / 2 / 2
1 + 0 .0 3 /2 + 0 .0 3 8 /2 )1
2 (1 + 0 .0 4 4 /2 )3
compounding frequency with which the yield (y) is measured.
( 1
c / 2 c / 2 1 0 0 + c / 2
1 0 0
(1 + 0 .0 4 8 /2 )4 (1 + 0 .0 5 1 /2 )5 (1 + 0 .0 5 3 /2 )6
8 We will cover duration in more detail in C hapter 12 of Valuation and
This can be solved to obtain c = 5.2517% . A bond with a Risk M o d els.
coupon of 5.2517% paid semi-annually would therefore be 9 The relationship is approxim ate because the bond price is a non-linear
worth par. function of its yield.

The Macaulay duration is the correct value of D if the yield is Table 16.5 Calculation of Duration for the Bond
measured with continuous com pounding. The m odified duration in Table 16.3
is the correct duration to use when D is measured with some
Present Value
other compounding frequency.
Time Cash Using Yield as
We will first consider the situation where y is measured with con (Years) Flow Discount Rate Weight Time X Weight
tinuous compounding (and therefore the Macaulay duration
0.5 3 2.9233 0.02864 0.01432
applies). The correct D to use in Equation (16.8) is then the aver
age time the bondholder has to wait before receiving the pres 1 . 0 3 2.8486 0.02791 0.02791
ent value . 1 0 1.5 3 2.7758 0.02720 0.04079
For exam ple, if a bond has a present value of USD 106 with a 2 . 0 3 2.7049 0.02650 0.05300
cash flow in one year providing a present value of USD and a
6
2.5 3 2.6357 0.02582 0.06456
cash flow in two years providing a present value of USD 100, the
3.0 103 88.1811 0.86393 2.59180
Macaulay duration would be
Total 102.0695 2.79238
6
1 . 0 0 0 0 0
X 1 + 1.9434
106
Equation (16.8) therefore indicates that a 5-basis point (0.05% or When there is a 10-basis point increase in the continuously com
0.0005) change in the continuously com pounded yield to give pounded yield (i.e., from 5.1779% to 5.2779% ), the bond's new
rise to a price change of: price can be calculated as:
- 1 .9 4 3 4 X 106 X 0.0005 = - 0 .1 0 3 2 e ~0.5x0.052779 + 3 e -1 .0 x0 .0 5 2 7 7 9 + q 0 - 1 .5x0.052779 + 3 e -2 .0 x0 .0 5 2 7 7 9
so that the price decreases from USD 106 to USD 105.897. + 3e ~ 2 -5 x 0 0 5 2 7 7 9 + i03e~ 30x0052779 = 101.7849
For another exam ple, consider the bond in Table 16.3. The price M eanwhile, Equation (16.4) predicts the bond price will be
of the bond is USD 102.0695 and the yield is 5.2455% with 102.0695 - 2.7924 X 102.0695 X 0.001 = 101.7845
semi-annual com pounding. With continuous com pounding, the
Thus, the duration relationship provides a good prediction of
yield is obtained from Equation (16.4) is
the change in the bond price resulting from a change in yield.
2 X ln(1 + 0.052455/2) = 0.051779
or 5.1779% .
Modified Duration
The Macaulay duration calculation is illustrated in Table 16.5.
The calculations so far have assumed that the yield (y) is
When discounted at the yield, the first cash flow has a present
measured with continuous com pounding. When rates are
value of USD 2.9233. This is 2.864% of the total present value . 1 1
com pounded m tim es per year, Equation (16.8) provides an

The amount of weight given to T = 0.5 in the calculation of
appropriate approxim ation if the duration (when calculated as
Macaulay duration is therefore 2.864% (= 2.9233/102.0695).
in Table 16.5) is divided by 1 + y/m. This is referred to as the
The second cash flow has a present value of USD 2.8486 and
m odified duration and hence:
weighting given to T = 1.0 is therefore 2.791 %. Doing this cal
culation for each of the cash flows, Table 16.5, shows that the Macaulay Duration
Modified Duration = ------ ------- ---------
total Macaulay duration is 2.27238. + y/m 1
To test the accuracy of Equation (16.8) when the yield is To illustrate the use of modified duration, consider again the
expressed with continuous com pounding, we set D = 2.79238. bond in Table 16.3. The yield (measured with semi-annual com
pounding) is 5.2455% and the duration is 2.79238. The modified
10 This explains the term inology duration. Roughly speaking, duration duration is therefore:
is an estim ate of how long in years an investor has to wait to receive
2.79238
returns. 2.7210
1 + 0.052455/2
11 Note that for the purposes of producing Table 16.5, it does not m at
ter w hether the yield is expressed with continuous com pounding, sem i To test the accuracy of Equation (16.8) when D is set equal to
annually com pounding, or any other com pounding frequency, provided
the modified duration, suppose the semi-annually com pounded
that present values are calculated in a way that reflects the yield's com
pounding frequency. However, the sensitivity of the bond price to the yield is increased by 10 basis points from 5.2455% to 5.3455% .
yield in Equation (16.4) does depend on how the yield is m easured. The bond price then becom es
3 3 3 16.8 C O N V E X IT Y
1 + 0.053455/2 + ( 1 + 0.053455/2)2 + (1 + 0.053455/2)3
__________ 3__________ __________ 3__________ 103 The impact of parallel shifts in the interest rate term structure
(1 + 0.053455/2)4 (1 + 0.053455/2)5 (1 + 0.053455/2)6 can be measured more accurately by considering convexity
in addition to duration. If C is convexity, the approxim ation in
= 101.7923
Equation (16.4) can be refined to:
M eanwhile, the duration relationship in Equation (16.8) predicts
1 9
102.0695 - 2.7210 X 102.0695 X 0.001 = 101.7918 A B = - D B A y + -C B (A y ) 2 (16.9)
Again, note that Equation (16.8) is reasonably accurate.

This equation allows relatively large parallel shifts to be considered.
Dollar duration is another measure that is som etim es used. This
The calculation of convexity is similar to the calculation of dura
is the product of the modified duration and the price of the
tion in Table 16.5, except that in the final column we calculate
bond. If dollar duration is D$, the relationship in Equation (16.4)
the weight multiplied by the square of time (rather than just
becom es
the time). Table 16.6 shows that the convexity of the bond in
A 8 = —D$Ay Table 16.3 is 8.13904.
W hereas D provides the sensitivity of proportional changes in a
The convexity calculated in Table 16.6 can be used in
bond's price to a change in its yield, D$ provides the sensitivity
Equation (16.9) with the Macaulay duration calculated in
of actual changes in a bond's price to changes in its yield.
Table 16.5 when rates are expressed with continuous com
pounding. When rates are expressed with compounding m
Limitations of Duration tim es per year, we can define
Duration provides a good approxim ation of the effect of a Convexity

Modified Convexity = ------------ ^
small parallel shift in the interest rate term structure. W hen all + y/m )2
( 1
interest rates change by a certain am ount, the yield on a bond

and use it together with modified duration in Equation (16.5).
changes by alm ost the sam e am ount. In the previous exam ple,
the change in the yield was 10 basis points (or 0.1% ). If all In our exam ple, the modified convexity is
interest rates increased by basis points, the yield would
8.1390
1 0
increase by alm ost exactly 10 basis points and Equation (16.4) 7.7283
(1 + 0.52455/2)2
would provide a good estim ate of the decrease in the bond
price. Now consider a 2% increase the semi-annually com pounded
yield for the bond in the previous exam ple. This reduces the
Equation 16.8 cannot be relied upon, however, if the change
bond price to USD 96.6951. The duration approximation in
in the bond yield arises from a non-parallel shift in the
Equation (16.8) is
interest rate term structure or the change being considered is
large. 102.0695 - 2.7210 X 102.0695 X 0.02 = 96.5149
Table 16.6 Calculation of Convexity for the Bond in Table 16.3
PV Using Yield
Time (Years) Square of Time Cash Flow as Discount Rate Weight Square of Time X Weight
0.5 0.25 3 2.9233 0.02864 0.00716
1 . 0 1 . 0 0 3 2.8486 0.02791 0.02791
1.5 2.25 3 2.7758 0.02720 0.06119
2 . 0 4.00 3 2.7049 0.02650 0.10600
2.5 6.25 3 2.6357 0.02582 0.16139
3.0 9.00 103 88.1811 0.86393 7.77539
Total 102.0695 1 . 0 0 0 0 0 8.13904

The duration plus convexity approxim ation in Equation (16.9) is When we are calculating the forward rate for the period
between one year and 1.5 years using the data in Table 16.2:
102.0695 - 2.7210 X 102.0695 X 0.02 + ^ X 7.7283
2
1.038361
X 102.0695 X 0.02 2 = 96.6726
This is a more accurate estim ate.

1.067463
Thus:
16.9 FO R W A R D RATES
V2 _ 1.067463
= 1.028027
V-, ~ 1.038361
Forward rates are the future interest rates implied by today's
zero-coupon interest rates. For exam ple, consider the rates in Again, we assume that the forward rate F is expressed with
Table 16.2. The six-month rate is 3% and the one-year rate is semi-annual compounding and obtain
3.8% (both semi-annually com pounded). The forward rate for
the period between six months and one year is the rate implied 1 + ^ = 1.028027
2
by these two rates. Specifically, it is the six-month rate that,
when com pounded with the current six-month rate of 3%, gives so that F = 0.05605. The forward rate here is 5.605% .
a one-year rate of 3.8% .
Forward rates for later six-month periods can be calculated
The general procedure for calculating the forward rate between similarly. The results are shown in Table 16.7.
time T| and time T (where T > T-|) is as follows.
When interest rates are expressed with continuous com pound
2 2
• Calculate the value to which one dollar grows by time ing, the equations for determ ining forward rates are less cum
Ty (= Vy). bersom e. If R-| and R2 are the zero-coupon interest rates for the
• Calculate the value to which one dollar grows by time period between tim es T and T2:-1
T (= V2).
2 Vy = eRlTl
• Calculate what one dollar at time T-\ is equivalent to at time Vo = e ^ 2
T2 (= V2/Vy).
Thus:
• C alcu late the interest rate betw een tim es T-| and T 2
that eq uates one dollar at tim e T to V2jV \ dollars at

-1
V.— — gR2T"2— R1 F
tim e T2. V1
If F is the continuously com pounded forward rate

Consider again the rates in Table 16.2. When we are calcula
ting the forward rate for the period between six months and e F(T2- T ,) = Y l = e R2T2-RiTi
one year: V1
Which can be rewritten as:
1.015
F (T 2 - Ty) = R2T2 - R, T,
0.038 V
1.038361
+— j
Table 16.7 Forward Rates Calculated
Thus:
from the Interest Rates in Table 16.2
1.038361
1.023016 Period Forward Rate
1.015
0.5 year to 1.0 year 4.603%
If the forward rate F is expressed with semi-annual
com pounding: 1.0 year to 1.5 years 5.605%
1.5 years to 2.0 years 6.005%

1 + - = 1.023016

so that F = 0.04603. Therefore, the forward rate is 4.603% .
or paid after 15 months (rather than after 12 months). By conven
tion, however, an FRA is usually settled at the beginning of the
_ R2T2 - ff-iTi
period covered by the FRA. The settlem ent amount is therefore
T2 - T \
the present value of the difference in interest amounts (with the
For exam ple, suppose the three-year zero-coupon interest rate
discount rate being the floating Libor rate). In our exam ple, this
is 5% and the four-year zero-coupon interest rate is % (both 6
means that the FRA will lead to a USD amount of:

expressed with continuous com pounding). The continuously
com pounded forward rate for the fourth year is 2,000
1,976.28
1 + 0 .0 4 8 /4
0.06 x 4 - 0.05 x 3
0.09
4 -3
being received at the one-year point. Expressed algebraically,
or 9%. Note that if rates were expressed with annual compounding the payoff is
(rather than continuous compounding), = 1.053, V2 = 1.064, and
(R ~ R k )t L
V /V = 1.0906. Thus, the forward rate with annual compounding (16.10)
2 1
1 + Rr
is 9.06%.
where R is the realized floating rate, R« is the fixed rate, L is the
principal to which the rates are applied, and r is the length of
Forward Rate Agreement the time period.
A forward rate agreem ent (FRA) can be thought of as an agree For the party on the other side of the transaction (who pays
ment to apply a certain interest rate to a certain principal for a floating and receives fixed), the payoff is
certain period in the future. Form ally, it is an agreem ent that a
(R k ~ R b L
pre-specified fixed interest rate will be exchanged for a floating 1 + Ki
interest rate (with both being applied to a certain principal for a
certain period). The floating interest rate is usually Libor. ln this exam ple, the payoff would be USD —1,976.28.
An interest rate swap (which will be discussed in Chapter 20) can An FRA can be valued by assuming that the forward Libor will
be regarded as a portfolio of FRAs. Each exchange in an inter occur. A t the initiation of the transaction we are considering,
est rate swap is an agreem ent to exchange a pre-specified fixed we can assume that the forward value of Libor is 4% (so that
interest rate for a floating interest rate (again, with both applied the transaction is worth zero to both sides). Suppose that six
to a certain principal). months later, the forward Libor rate for the period covered by
the FRA (which would then be the period between six months
Suppose an FRA involves paying a fixed interest rate of 4% and
and nine months in the future) is 4.4% . The USD payoff from
receiving the three-month Libor on a principal of USD 1 million for
the FRA at the beginning of the three-month period covered
the three-month period beginning in one year. If the three-month
by the FRA can be estim ated by substituting 4.4% for R in
Libor fixing in one year proves to be 4.8% , the FRA will involve
Equation (16.10):
interest being paid at a rate of 4% and received at a rate of 4.8%.
(0.044 - 0.04) X 0.25 X 1,000,000
1_____1_____ — n o n 1 n
Because these are money market rates, they are expressed with __________________ 1________________
1 + 0.044 X 0.25
a compounding frequency corresponding to their maturity. (In
this case, they are expressed with quarterly com pounding.) The The value of the FRA is the present value of USD 989.12 (with
USD fixed paym ent (referred to as the paym ent on the fixed leg) discounting at the risk-free rate). Expressed algebraically, the
is therefore: value of an FRA where floating is received and fixed is paid is
0.25 X 0.04 X 1,000,000 = 10,000 (R f ~ R k ) t L \

The USD floating payment is similarly: 1 + R F t J
where RF is the forward rate for the period underlying the FRA
0.25 X 0.048 X 1,000,000 = 12,000
and PV denotes the present value (using the risk-free rate)
Because the FRA involves a fixed rate being paid and a floating from the beginning of the period covered by the FRA to today.
rate being received, the holder of the FRA is due: Similarly, the value of an FRA where floating is paid and fixed is
USD 12,000 - USD 10,000 = USD 2,000 received is
Interest is normally paid at the en d of the period to which it (R k ~ R f ) t L \

1 + K ;
applies. The USD 2,000 in this exam ple is therefore due to be f t

16.10 D ETER M IN IN G Z E R O RATES 2.8 n
Zero Rate (%)
2.6 -
Analysts are tasked with determ ining zero-coupon interest rates

2.4 -
from the market prices of traded instruments. As explained
previously, money market instruments lasting less than one 2.2 -
year provide their entire return at the end of their lives. Thus,
2
they provide information about their zero-coupon interest rates
-
in a direct way. However, instruments lasting longer than one 1.8 -
year usually make regular payments prior to maturity. It is then Maturity (Years)
+ - ------ -------- ------- ------- ------- -------
necessary to calculate the zero-coupon rates implied by these
1 . 6 1 1 1 1 1 1 1
0 0.5 1 1.5 2 2.5 3 3.5

instruments.
Fiqure 16.1 Zero curve calculated from the data in
One way of doing this is by working forward and fitting the zero- Table 16.6 and a two-year bond price.
coupon rates to progressively longer maturity instruments. This
is called the b o otstra p method.
As a simple exam ple of how this can be done, suppose the zero- The calculations to produce a piecewise linear zero-rate curve are
coupon interest rates (semi-annually com pounded) for maturities not always as straightforward as in the case above. For exam ple,
of 0.5, 1.0, and 1.5 years in Table 16.8 have already been deter suppose the next price available was for a 2.4-year bond (instead
mined. Suppose further that a two-year bond with a par value of of a two-year bond). We would set the 2.4-year zero rate equal
USD 100 has a market value of USD 102.7 when it pays a (semi to R. Because a coupon is paid on the bond at the 1.9-year point,
annual) coupon at the rate of 4% per year. the zero rate for a maturity of 1.9 years has to be determined in
terms of R. By interpolating between the 1.5-year rate of 0.025
If the two-year zero-coupon interest rate is R, the equation that
and the 2.4-year rate of R, the 1.9-year rate becomes
equates the value of the bond to USD 102.7 is
0.025 X 0.5 + R X 0.4
2 2 2 102
09
1 + 0 .0 2 /2 ( 1 + 0 .0 2 3 /2 )2 (1 + 0 .0 2 5 /2 )3 (1 + R / 2 )4
The other coupons are at tim es 1.4, 0.9, and 0.4 and can be
= 102.7
determ ined by interpolation from the rates that have already
This can be solved to give R = 0.0261 and the two-year rate is been determ ined. An iterative search can then be used to deter
therefore 2.61% with semi-annual com pounding. mine the value of R that matches the bond price.
A zero curve defines the relationship between zero-coupon

interest rates and their m aturities. The zero-coupon interest
16.11 T H E O R IE S O F TH E TERM
rates in a zero curve are usually assumed to be constant until the
first maturity for which data is available. They are also usually STR U C TU R E
assumed to be constant beyond the longest maturity for which
There are several theories about what determ ines the shape
data is available.
of the zero-coupon interest rate term structure. O ne is known
The zero curve we have calculated would therefore be as in as the m arket segm entation theory. This argues that short-,
Figure 16.1. medium-, and long-maturity instruments attract different types
of traders. W hile short-maturity rates would be determ ined by
the activities of traders who are only interested in short-maturity
Table 16.8 Zero-Coupon Interest Rates for the First
investm ents, medium- and long-maturity instruments would only
1.5 Years
attract traders interested in those investments.
Zero-Coupon Interest Rate
It is now generally recognized that the market segm entation
Maturity (Years) Semi-Annually Compounded
theory is unrealistic. Market participants do not focus on just
0.5 2 . %
0 one segm ent of the interest rate term structure. For exam ple,
pension plans are primarily interested in trading long-maturity
1 . 0 2.3%
bonds, but will move to medium- or short-maturity instruments
1.5 2.5%
if their yields seem more attractive.
A nother theory is known as the e xp e cta tio n s theory. This long-term rates relative to the market's expectations about
argues that the interest rate term structure reflects where future short-term rates. As they do this:
the m arket is expecting interest rates to be in the future. If
• Long-term borrowing becom es less attractive because short
the m arket exp ects interest rates to rise, the term structure
term rates are more attractive to borrowers, and
of interest rates will be upward-sloping (with long-maturity
• Short-term lending becom es less attractive because long
rates being higher than short-m aturity interest rates). If
term rates are more attractive to lenders.
the m arket exp ects interest rates to decline, then the term
structure will be downward-sloping (with long-m aturity rates Liquidity preference theory therefore argues that the process
being lower than short-m aturity rates). Exp ectatio ns theory of matching borrowers and lenders leads to long rates being
argues that forward rates should be equal to exp ected future higher than the market's expectations would suggest. To be pre
spot rates. cise, it argues that forward rates consistently overstate expected
future spot rates.
In practice, interest rate term structures are upward-sloping
much more often than they are downward-sloping. This calls the To take a simple exam ple, suppose that only two rates are
expectations theory into question, because we can reasonably offered in the market: a three-month rate and a five-year rate.
hypothesize that the market expects interest rates to decrease Suppose further that both rates are 2.5% per year and that this
as often as it expects interest rates to increase . 1 2 If the exp ecta reflects the market's expectations (so that all expected future
tions theory were correct, we would expect to see downward- three-month rates are 2.5% per year). If the term structure of
sloping term structures occurring as often as upward-sloping interest rates is flat at 2.5% (consistent with expectations theory),
term structures. liquidity considerations will lead lenders to choose to commit
funds for only three months, while borrowers will choose the five-
A third theory, known as the liquidity p referen ce theory, is
year maturity. This will lead to a mismatch. As financial interm edi
capable of explaining this disparity. The theory argues that if
aries try to match borrowers and lenders, market forces will lead
the interest rate term structure reflects what the market expects
to the five-year rates being pushed above 2.5% . For exam ple,
interest rates to be in the future (i.e., the expectations theory
it might be found that making the five-year rate 3.5% (while
holds), most investors will choose a short-term investment over
keeping the three-month rate at 2.5%) will cause some borrowers
a long-term investm ent. This is because of liquidity consider
to switch from five-month borrowing to three-month borrowing
ations. A short-maturity investm ent means that the funds will be
and some lenders to switch in the other direction. The end result
available earlier to meet any needs (anticipated or unforeseen)
is that supply and demand are matched at both maturities.
that arise.
We can also look at the choice between long-maturity and

short-m aturity instrum ents from the perspective of the bor SU M M A RY
rower. Short maturity borrowing usually has to be rolled over
into new borrowing at the end of its life. This also entails liquid In any given currency, several different interest rates are
ity risks. If (rightly or wrongly) the market's view of the financial monitored. These include governm ent borrowing rates, repo
health of the borrower declines, it may not be possible for the rates, and interbank borrowing rates such as Libor and overnight
com pany to roll over short-term borrowing at a com petitive rates.
interest rate.
The way in which an interest rate is measured is defined by its
Liquidity considerations therefore lead to lenders wanting to compounding frequency. The difference between measuring an
lend for short periods of time and borrowers wanting to bor interest rate with annual compounding and one with monthly
row for long periods of tim e. In order to match borrowers and compounding is analogous to the difference between measur
lenders, financial interm ediaries (such as banks) must increase ing a tem perature in degrees centigrade and degrees Fahren
heit. Continuous compounding is commonly used when valuing
options and similar derivatives.
12 Short-term interest rates are mean reverting. This means that they
tend to get pulled back toward a long-run average level. O ver a long Zero-coupon interest rates (also referred to as zero rates or
period of tim e they are below the long-run average (and therefore spot rates) are rates where all the return is received at the end
exp ected to move up) roughly half the tim e and above the long-run
of the instrument's life. Zero-coupon interest rates must usually
average (and therefore expected to move down) roughly half the tim e.
For a further discussion of this see J . Hull, O ption s, Futures and O th er be implied from the values of instruments that pay interest on
D erivatives, 10th edition. a regular basis. A common approach is known as the bootstrap

m ethod; this is where an analyst works forward from short-matu Forward rates are the future spot rates implied by the current
rity instruments to longer-maturity instruments to determ ine the interest rate term structure. Forward rate agreem ents (FRAs)
rates that match prices. are agreem ents where a certain rate observed in the future will
be exchanged for a pre-determ ined fixed rate (with both rates
Zero-coupon interest rates can be used to value bonds. Each
being applied to the same principal).
paym ent (coupon or principal) on the bond is discounted at
the zero-coupon interest rate corresponding to the tim e it will One explanation for how interest rate term structures are deter
be received. The par yield for a bond with a certain m aturity mined by the market is the liquidity preference theory. This
is the coupon that leads to the value of the bond equaling theory argues that if rates merely reflect the market's exp ecta
its par value. The yield on a bond is the discount rate that tions about future interest rates, borrowers will borrow at a
(when applied to all paym ents received by the investor) leads fixed rate for long periods, whereas lenders will prefer to lend
to a present value equal to the m arket price of the bond. for short periods. To match borrowers and lenders, it is neces
Duration is a measure of the sensitivity of a bond's price to sary to increase forward rates relative to the expected future
its yield. short rates.
Q U E S T IO N S

16.1 How is the Libor fixing for a particular currency and 16.6 How is the par yield on a bond defined?
maturity determ ined on a day?
16.7 How is a bond yield defined?
16.2 W hat is a repo rate? 16.8 W hat is the difference between modified duration and
16.3 Which is better for an investor: 6 % per year com pounded dollar duration?
annually or % per year com pounded monthly?
6
16.9 W hat is a forward rate agreem ent? Explain how it is usu
16.4 W hat is the formula for calculating the value at a future ally settled.
tim e T of an investment of amount A when the interest
16.10 Explain what the liquidity preference theory and the
rate is R and the compounding frequency is m? expectations theory imply about forward interest rates
16.5 W hat is the formula for the present value of an amount A and expected future spot rates.
received at time T when the continuously com pounded
interest rate is R?
Practice Questions
16.11 How is the effective federal funds rate determ ined? 16.17 For the interest rates in Problem 16.16, what are
the (semi-annually com pounded) forward rates
16.12 The six-month and one-year zero rates are 3% and 4%
for a six-month periods beginning in six, , and 18
(both compounded semi-annually) and a 1.5-year bond
1 2
paying a coupon of 4% per annum semi-annually has a months?
yield of 5%. What is the 1.5-year zero-coupon interest rate? 16.18 A three-year bond with a face value of USD 100 pays
16.13 A rate is % with semi-annual com pounding. W hat would

8
coupons annually at the rate of 10% per year. Its yield is
7% with annual com pounding. W hat are (a) the Macaulay
the rate be if expressed with monthly com pounding?
duration, (b) the convexity, (c) the modified duration, and
16.14 A rate is 7% per year with continuous com pound
(d) the modified convexity?
ing. W hat is the rate when measured with quarterly
com pounding? 16.19 For the bond in 16.18, estim ate the price change if the
annually com pounded yield changes from 7% to 8.5% ,
16.15 A two-year bond that pays coupons semi-annually
using both the duration and the duration plus convexity
at the rate of 7% per year has a market price of 103.
approxim ations.
W hat is the bond's yield measured with semi-annual
com pounding?
16.20 In an FRA, an annualized rate of 3% will be received
and six-month LIBO R will be paid on a principal of
16.16 The six-month, 12-month, 18-month, and 24-month zero
USD 5,000,000 for a six-month period starting in
rates are 5%, 5.5% , %, and 6.5% (all measured with
18 months. If the annualized six-month forward rate in
6
semi-annual compounding) respectively. W hat is the

18 months proves to be 3.5% , what is the settlem ent on
two-year par yield for a bond paying coupons every six
the FRA? When is it made?
months?
Chapter 16 Properties of Interest Rates 209

A N S W ER S
16.1 Eighteen banks make estim ates of the rate at which they An FRA is normally settled at the beginning of the under
can borrow from other banks just prior to 11 a.m . U.K. lying period. The settlem ent amount is the difference
tim e. The top four and bottom four estim ates are dis between the two interest amounts discounted from the
carded and the remainder are averaged. end of the period to the beginning of the period at the
16.2 A repo rate is a short-term secured lending rate. The floating rate.
borrower agrees to sell securities to the lender and buy 16.10 Expectations theory assumes that forward rates equal
them back for a slightly higher price a little later. expected future spot rates. In liquidity preference theory,
forward rates are greater than expected future spot
16.3 6 % per year com pounded monthly is better. 6 % per year
com pounded monthly is rates.
(1 + 0 .0 6 /1 2 )12 - 1 = 0.0617 16.11 Banks borrow and lend between them selves to meet
their reserve requirements with the central bank. The
or 6.17% per year com pounded annually.
weighted average of the interest rate in these transac
16.4 The formula is tions is the effective federal funds rate.
/ ft \m T
16.12 The price of the 1.5-year bond with a face value of 100 is
Future Value = A f 1 -\---- j
102
+ + 98.5720
16.5 The formula is 1 + 0 .0 5 /2 ( 1 + 0 .0 5 /2 )2 d + 0.05/2)
Present Value = Ae~RT
If the 1.5-year zero rate is R we must have
16.6 A bond's par yield is the coupon rate on the bond that
causes the bond's price to equal its par value. + + = 98.5720
1 0 2
1 + 0 .0 3 /2 ( 1 + 0 .0 4 /2 )2 (1 + R/ 2)
16.7 A bond's yield is the discount rate that, when used for all
of the bond's cash flows, causes the total present value The solution to this equation is R = 0.05027. The
of the cash flows to equal the current market price of the 1.5-year zero rate is therefore 5.027% .
bond. 1 6 .1 3 From Equation (16.2), the rate with monthly compounding is
16.8 Modified d uration measures the relationship between

0.08 V
0.07870
/ 1 2
proportional decreases (increases) in a bond's price and

— ) - 1
increases (decreases) in its yield. Dollar duration m ea

sures the relationship between the actual decreases or 7.870% .
(increases) in a bond's price and increases (decreases) in 16.14 From Equation (16.5), the rate is
its yield. 4 ( e o.07/4 _ d = 0.07062
16.9 A forward rate agreem ent (FRA) is an agreem ent to
or 7.062% per year.
exchange
16.15 To determ ine the yield, y, we must solve
(a) An amount obtained when a pre-determ ined fixed
rate of interest is applied to a certain principal for a cer 3.5 3.5 3.5 103.5
tain future period, and 1 + y / 2 + ( 1 + y/ 2 ) 2 + ( 1 + y/ 2 ) 3 + ( 1 + y/ 2 ) 4
(b) An amount obtained when the actual (floating) rate By trial and error (or using Solver) the solution is
observed in the market is applied to the same principal y = 0.053975. The yield is 5.3975% .
for the same period.
16.16 The par yield is the coupon rate c satisfying The Macaulay duration is 2.7458, the convexity is 7.9021,
c / 2 c / 2 c / 2
and the modified duration is
1 + 0 .0 5 /2 + 0 .0 5 5 /2 )2 (1 + 0 .0 6 /2 )3 2.7458
2.5661
( 1
+ c
1 0 0 / 2
1.07
+ ------------ -— r = 1 0 0
(1 + 0 .0 6 5 /2 )4 The modified convexity is
It is 6.46% . Alternatively, we can use Equation (16.7). In 7.9021

6.9020
this case m = 2, d = 0.8799, and A = 3.7179. 1.072
16.17 The forward rates are 16.19 Using duration, the price change is
-2 .5 6 6 1 X 107.8729 X 0.015 = - 4 .1 5 2 2
2 X ( 1f n l z
V 1.025
“ A
J = 0.060012
Using duration and convexity, it is
(
1.033 \
2 X -------- - - 1 = 0.70037 -2 .5 6 6 1 X 107.8729 X 0.015 + 7- X 6.9020 X 107.8729
V 1.02752 ) 2
( 1.03254 \ X 0 .0 1 52 = -4 .0 6 8 5
2 X -------— - 1 = 0.80073
V 1.033 J The actual bond price decline is 4.0419, showing that
If all rates were continuously com pounded, the forward duration plus convexity gives a better estim ate than
rates would be 6%, 7%, and 8%. Because we are dealing convexity alone.
with a semi-annually com pounded rate, they are slightly 16.20 The USD settlem ent in 18 months is
different: 6.0012% , 7.0037% , and 8.0073% .
(0.03 - 0.035) X 0.5 X 5,000,000
16.18 The calculations are in the following table: -1 2 ,2 8 5 .
1 + 0 .0 3 5 /2
PV with It is settled in 18 months.

Time Cash 7% Disc
(Yrs.) T Flow Rate Weight W TXW T2 x W
1 1 0 9.3458 0.0866 0.0866 0.0866
2 1 0 8.7344 0.0810 0.1619 0.3239
3 1 1 0 89.7928 0.8324 2.4972 7.4915
Total 107.8729 1 . 0 0 0 0 2.7458 7.9021

Corporate Bonds
Learning Objectives
Describe features of bond trading, and explain the behav Describe the different classifications of bonds character
ior of bond yield. ized by issuer, maturity, interest rate, and collateral.
Describe a bond indenture and explain the role of the cor Describe the mechanisms by which corporate bonds can
porate trustee in a bond indenture. be retired before maturity.
• Define high-yield bonds, and describe types of high-yield Define recovery rate and default rate, differentiate
bond issuers and some of the payment features unique to between an issue default rate and a dollar default rate,
high yield bonds. and describe the relationship between recovery rates and
seniority.
Differentiate between credit default risk and credit spread
risk. Evaluate the expected return from a bond investm ent and
identify the com ponents of the bond's expected return.
Describe event risk and explain what may cause it in cor
porate bonds.
A bond is a debt instrument sold by the bond issuer (the bor • The issuance cost is lower.
rower) to bondholders (the lenders). The bond issuer agrees to • The issuance can be com pleted quickly.
make payments of interest and principal to bondholders. The
• The issuance can be relatively small.
principal of a bond (also called its face value or par value) is the
amount the issuer has promised to repay at maturity. Interest rates for private placem ent bonds are generally higher
than those for equivalent publicly issued bonds. The issuer must
Bonds p erceived to be riskier than others available require
therefore weigh the benefits of private placem ent against the
higher interest rates to attract investors. Th e interest rate on
paym ent of a higher interest rate.
a bond is term ed the coupon rate. In the U .S ., coupons are
usually paid every six m onths. In som e other countries, cou In a public issue, the investment bank acts as the underwriter.
pons are paid with other freq u en cies (e .g ., m onthly, quarterly, This means that it buys the bonds from the corporation and
or annually). A U .S. bond paying a coupon of 8% would pay then tries to sell them to investors. The investment bank's profit
interest to the holder equal to 4% of USD 1,000 (i.e ., USD 40) is earned from the difference between the price it pays to buy
every six m onths. For exam p le, the interest m ight be payable the bonds from the corporation and the price at which it sells
on Feb ruary 15 and A ug ust 15 of each year during the life of the bonds to investors. O nce issued, the bonds are traded and
the bond. given ratings by rating agencies (as will be discussed later in this
chapter).
The face value of a bond in the U.S. is usually USD 1,000 and
bond prices are typically quoted per USD 100 of principal. The The risks taken by the underwriter are defined in its contract
value of the global bond market is approxim ately USD 100 tril with the issuer. Among these risks include the possibility that
lion (as measured by the value of outstanding bonds), which is interest rates will increase, reducing the value of the bonds
larger than the global equity market. before the underwriter can to sell them to investors.
This chapter focuses on bonds issued by corporations. Bonds

issued by the U.S. governm ent are discussed in some detail in
17.2 BOND TRADING
Chapters 9-13 of Valuation and Risk M o d els, while the credit
risk discussion in this chapter is continued in Chapters 4 and 6 of
Bonds issued via private placem ents are often not traded.
that book. Instead, they are held by the original purchasers until maturity.
Bonds issued in a public offering, on the other hand, are typi
cally traded in the over-the-counter m arket.1 This is in contrast
17.1 BOND ISSUANCE to stocks, which are typically traded on exchanges.
Corporate bond issuances are typically arrangem ent by invest Th ere is a netw ork of dealers who buy and sell bonds, either
ment banks. These banks have connections with many potential for their own portfolios or for their clients. D ealers aim to
bond purchasers and will work with the issuing corporation to m ake a profit from the d ifference betw een the prices at which
determ ine the appropriate term s for the issuance. they buy and the prices at which they sell. Bond prices (like
those of any other financial security) are determ ined by su p
The issuing corporation can choose between a private p la ce
ply and dem and. If more investors w ant to buy a bond than
m ent and a public issue. In a private placem ent, bonds are
w ant to sell it, the price increases. If the reverse is true, the
placed with a small number of large institutions (e.g ., pension
price d ecreases.
funds and insurance com panies). Som etim es one institution will
buy the entire bond issuance; on other occasions, bonds are In Chapter 16, we explained how a bond's yield is defined. Bond
sold to several different institutions. In either case, the bonds yield can be thought of as the return earned on a bond over its
are not offered to the general public. life assuming that all interest and principal payments are paid
as promised. As indicated in Figure 17.1, the yield is composed
A private placem ent has several advantages to the issuer.
of a risk-free return (which is the return that would be earned
• There are few er registration requirem ents. In the U .S., for on a similar risk-free instrument) and a credit spread (which is
exam ple, private placem ent issuances do not need to regis the extra return to com pensate the investor for the possibility
ter with the Securities and Exchange Com m ission.
• Rating agencies are not involved because they don't usually 1 Convertible bonds (i.e ., bonds that can be converted to equity
rate non-public issuances. in certain pre-defined ways) are som etim es traded on exchanges.
Yield period at a reasonable price. Markets where the trading volume
is high tend to be highly liquid and have low bid-ask spreads.
Markets where the trading volume is low are usually less liquid
and have higher bid-ask spreads.
Liquidity in the bond market varies from bond to bond. Some

bonds trade only a few tim es a year, while others trade several
tim es a day.3 Part of the yield on a bond is compensation for its
liquidity risk (or lack thereof). As a bond's liquidity declines,
Bond M aturity
investors require a greater yield. The Volcker rule (part of the
Fiqure 17.1 Illustration of the relationship between D odd-Frank legislation in the U.S.) now restricts the extent to
the yield on a corporate bond and the yield on a which banks can trade bonds (or other securities) for their own
risk-free bond. account. Some banks in the U.S. have had concerns that the
Volcker rule is reducing bond liquidity because it restricts their
of a default). As the maturity of the bond increases, the credit ability to keep bonds in their inventories for market-making
spread for a bond with a good credit rating tends to increase. purposes.4
This is because the chance that the issuer will experience finan The price quoted for a bond is referred to as the clean price.
cial difficulties increases with the passage of tim e. The cash price for a bond is called the dirty price and is the
If the price of a bond increases, its yield declines (and vice clean price combined with accrued interest (i.e., the interest
versa). When it is considered likely that interest rates will earned by the seller since the last coupon paym ent date). For
increase, investors will demand higher yields. This will lead to corporate bonds in the U .S., this interest is calculated with
selling pressure in the bond market, and the prices of all bonds a 30/360-day convention (where it is assumed that there are
should decline. When it is considered likely that interest rates 30 days in a month and 360 days in a year).5
will decline, the yields required by investors will also decline Suppose that the coupon rate is 8% per year, today is June 4,
and the prices of all bonds will increase. News about the finan and the last coupon payment date was February 15. The
cial health of a bond issuer may cause the market to adjust the number of days between February 15 and June 4 using the
required credit spread. This can also increase or decrease the 30/360-day convention is 109 (= 15 + 30 + 30 + 30 + 4). The
price of a bond. accrued interest on June 4 is therefore:
Bond dealers are market makers who quote bid and ask prices 109
— - X 8% = 2.422%
for bonds on request as well as maintain inventory to facilitate 360
trading. If risk-free interest rates and/or credit spreads increase,
If a dealer quotes the price of the bond as USD 96.00, the price
bond prices can decrease and dealers may lose money on their
that would be paid for a bond with principal of USD 1,000 is
inventories.
960 + 24.22 = 984.22
Liquidity is an important issue in bond trading. It can be defined
as the ability to turn an asset into cash within a reasonable time Day count conventions can vary across countries.
2 D ebt issued by a governm ent in its own currency is often considered

to be risk-free because governm ents can always print more of their 3 This may depend on w hether the bond issuer is well known and
currency to avoid defaulting on the debt. However, developing countries w hether the credit rating of a bond is sufficiently high to attract
do som etim es default on debt issued in their own currency. (The reasons institutional investors. (Some institutional investors are restricted to
for this are discussed in C hapter 5 of Valuation and Risk M o d e ls.) In buying investm ent g ra d e bonds. We explain the meaning of this term
practice, bonds issued by governm ents of the O E C D m em ber countries later in the chapter.)
in their own currency are usually considered to be risk-free (see www
4 This seem s to be supported by research. See for exam ple J . Bao,
.oecdw atch.org for information about the O E C D and a list of m em ber
M. O 'H ara, and A . Zhou. "The Volcker Rule and Market-Making in
countries). An exception is bonds issued in euros by countries in the
Tim es of Stress," Journ al o f Financial Econom ics, 2018.
European Union, because these countries are not able to m eet their
obligations by printing euros. (The European Central Bank makes 5 Day count conventions and accruals are also discussed in C hap ter 19
decisions on the euro money supply.) of this book and in C hap ter 9 of Valuation and Risk M o d els.
Chapter 17 Corporate Bonds ■ 215

17.3 BOND INDENTURES 17.4 CREDIT RATINGS
A bond indenture is a legal contract between a bond issuer and Ratings agencies such as Moody's, S&P, and Fitch provide opinions
the bondholder(s) defining the important features of a bond issue. on the creditworthiness of bond issuers. The scales used by these
These features include the maturity date (i.e., when the principal rating agencies are shown in Table 17.1. The best rating assigned
must be repaid), the amount and timing of interest payments, call by Moody's is Aaa; bonds with this rating are considered to have
able and convertible features (if any), and the rights of bondholders virtually no chance of default. The ratings continue downwards as
in the event of contract violations. The bond indenture may also follows: Aa, A, Baa, Ba, B, Caa, Ca, and C. Meanwhile, S&P and
include several covenants. These can be categorized as: Fitch use a slightly different rating scale (e.g., A A A is used instead
of Aaa, A A instead of Aa, BBB instead of Baa, and so on).
• N egative covenants (also known as restrictive covenants),
which limit the issuer's ability to engage in further debt Moody's subdivides its ratings (except Aaa and Ca) to create a
financing, asset sales, dividend payouts, and share buybacks; finer scale. For example, the Aa rating is subdivided into Aa1, Aa2,
• Positive covenants, which require the issuer to produce regu and Aa3; the A rating is divided into A 1 , A2, and A3; and so on.
lar financial statem ents, maintain properties, carry insurance, S&P and Fitch have similar subdivisions (e.g., A A rating category is
and use the money raised by the bond issuance in the man divided into A A + , A A , and A A —; the A rating category is divided
ner stated in the offering docum ent; and into A + , A, and A - ; and so on).
• Financial covenants, which may require the issuer to maintain

certain financial ratios (such as interest coverage or leverage
ratios).6 Table 17.1 Rating Scales for Corporate Bonds
There may also be other covenants that outline what happens
that are not in Default
if there is a change of control (referred to as CoC) and a credit Moody's S&P and Fitch
rating downgrade.
Investment Grade Bonds
Bonds issued by highly creditworthy firms generally contain
Aaa AAA
few covenants, while those issued by riskier firms often have
an extensive list of covenants. Aa1 AA+
Aa2 AA
Corporate Bond Trustee
Aa3 AA-
In a public bond issue, it is unrealistic to expect individual bond A1 A+
holders to monitor the issuer and ensure it follows the bond
A2 A
indentures. The issuer therefore appoints (and pays for) a corpo
rate bond trustee. This is a financial institution (usually a bank or A3 A-
trust com pany)7 that looks after the interests of the bondholders B aal BBB +
and ensures that the issuer com plies with the bond indentures.
Baa2 BBB
The trustee reports periodically to the bondholders and will act
Baa3 BBB-
on behalf of bondholders if the need arises. It may also act as
Non-Investment (Speculative) Grade Bonds
paying agent and registrar (i.e ., it may handle record keeping
and the disbursem ent of interest and principal to the bond Ba1 BB +
holders). Its specific duties are item ized in the bond indentures Ba2 BB
and the trustee is under no obligation to exceed those duties.
Ba3 BB-
For exam ple, som etim es the indenture states that trustee can
rely on the issuer and the issuer's attorneys for inform ation on B1 B+
w hether some covenants are being adhered to. In such cases, B2 B
the trustee is not required to conduct its own investigations.
B3 B-
Caa1 CCC+
6 Interest coverage is the ratio of earnings before interest and taxes
to interest expense. Leverage ratios can be defined in many ways and
Caa2 CCC
reflect the percentage of a business financed with debt. Caa3 ccc-
7 In the U .S., a federal statute requires that trustees be appointed for
Ca C C /C
bond issues over USD 5 million involving interstate com m erce.
Public bond issuers normally pay rating agencies to rate their bonds • The issuer may have rights to call the bond from the
(although they are not required to do so). In theory, this would proceeds of an equity issue.
appear to be a conflict of interest; agencies have an incentive to
produce overly positive ratings to attract and retain paying clients
(bond issuers) instead of the objective ratings sought by the non
17.5 BOND RISK
paying users (bond investors).8 In practice, however, rating agencies
Credit ratings measure default risk. Rating agencies produce
know that their reputations would suffer if they allowed their rela
tables showing:
tionships with bond issuers to influence their ratings; this creates an
incentive for rating agencies to be as objective as possible. • The probability that a bond will default within n years after
being given a certain rating for various values of n, and
High-Yield Bonds • The probability that a bond will move from one rating category
to another during a certain period (e.g., one year or five years).
Bonds rated above a threshold (as seen in Table 17.1) are
referred to as investment grade. Bonds below this threshold are We discuss these tables further and provide exam ples in
given various names: high-yield, non-investment grade, specula Chapter 4 of Valuation and Risk M odels.
tive grade, or simply junk. Another risk faced by bondholders arises from changes in how
There are several circum stances that can give rise to high-yield the market prices credit risk (i.e., the credit spread). During
bonds. normal tim es, the credit spread for a seven-year A-rated
bond is approxim ately 100 basis points. This means that if the
• High-yield bonds may be sold by young and growing com pa
seven-year risk-free rate is 3%, the yield on a seven-year A-rated
nies. These firms may have good prospects, but
corporate bond is 4%. (See Figure 17.1). During stressed
they lack the track record and strong financial statem ents
periods (i.e., when investors are particularly averse to taking
of more established com panies.
risks), this credit spread could rise to 2% or even 3 % .12
• Investment-grade bonds may becom e high-yield bonds
Another measure som etimes used by analysts is spread duration.
as the financial situation of the issuing firm deteriorates.
This is (approximately) the percentage change in the bond price
Such com panies are som etim es referred to as fallen angels.
for a 100-basis point increase in the credit spread (assuming that
• A company with stable cash flows increases its debt burden the risk-free rate remains unchanged). For exam ple, a spread
to benefit shareholders.9
duration of four indicates that a 100-basis point increase in the
High yield bonds som etim es have unusual features, such as credit spread will reduce the bond price by 4%.
the following exam ples.
• A deferred-coupon b o n d is a bond that pays no interest for Event Risk

a specified tim e period, after which time a specified coupon
is paid in the usual manner. There are many events (e.g ., natural disasters or the death of
a C EO ) that could adversely affect bonds. An im portant type of
• A step-up bond is a bond where the coupon increases with time.
event risk is that of a large increase in leverage.
• A paym ent-in-kind bond is a bond where the issuer has the
Firms can increase their leverage through activities such as
option of providing the holder with additional bonds in lieu
leveraged buyouts,13 share buybacks,14 certain mergers and
of interest.10
acquisitions, and other types of restructurings that benefit
• An exten d a b le reset b o n d is a bond where the coupon
shareholders at the expense of bondholders.
is reset annually (or more frequently) to maintain the price
of the bond at some level (e.g ., USD 101).11 An exam ple of an event that hurt bondholders (and had a nega
tive effect on the whole bond market) is the 1988 leveraged
buyout of RJR Nabisco by Kohlberg, Kravis, Roberts & Co. The
8 O ne can make a similar argum ent about bond trustees, who are paid
by the issuer but act on behalf of the bondholders.
USD 25 billion buyout (and the resulting increase in leverage)
increased the credit spread on existing bonds from 100 basis
9 The RJR Nabisco leveraged buyout is an exam ple of a situation where
high-yield bonds were created from a buyout by a private equity firm. points to 350 basis points.
10 The issuer might do this if prohibited from paying interest by

covenants in other bond issues.
12 This phenomenon is referred to as a flight to quality.
11 This structure has a flaw. If the com pany experiences financial dif
13 This is where a private group of shareholders borrows funds to buy
ficulties, the coupon necessary to maintain a specified price may be
out the current shareholders.
extrem ely large. This will create cash flow problem s for the com pany
and the need for an even higher coupon. 14 This is where shares are repurchased from investors.

Som etim es bond indentures for lower-rated issues anticipate the Issuer
possibility of increased leverage and include a m aintenance o f
net worth clause. This clause requires the firm to keep its equity Issuers can be put into five broad categories:
value above a prescribed level. If it fails to do this, it must begin 1. Utilities: Exam ples include electric, gas, water, and com mu
to retire its debt at par until the equity moves above the pre nications com panies,
scribed level.
2 . Transportation com panies: Exam ples include airlines, rail
In some cases, the company merely has to offer to retire debt roads, and trucking com panies,
at par; bondholders would then choose to accept or decline the
3 . Industrials: Exam ples include manufacturing, retailing, min
offer. However, the bondholders will normally accept the offer
ing, and service com panies,
because it is unlikely that the market price of the bonds would
be above par if the maintenance of net worth covenant has 4 . Financial institutions: Exam ples include banks, insurance
been breached. com panies, brokerage firms, and asset management

firms, and
5 . Internationals: Exam ples include supranational organizations

Defaults such as the European Investment Bank, foreign
Default occurs when a bond issuer fails to make the agreed governm ents, and other non-domestic entities. The bonds
upon payments to the bondholders. Those who are owed that they issue are referred to as Yankee Bonds in the U.S.
money by the bond issuer have a claim against the issuer's
assets. The issuing company may then reorganize itself (in
Maturity
negotiation with its creditors) or sell its assets to meet creditor
(including bondholder) claims. Corporate bonds have an original maturity of at least one year.
(Instruments with an original maturity of less than one year are
The bankruptcy laws in the U .S. facilitate reorganizations.
referred to as commercial paper.) Bonds with maturities of up to
For exam p le, a C h ap te r 11 bankruptcy filing gives a com
five years are usually referred to as short-term notes, those with
pany tim e to negotiate a re-organization with bondholders
maturities between five and 12 years are referred to as medium-
and other cred ito rs. During this p erio d , the firm 's execu tives
term notes, and those with maturities of greater than 12 years
rem ain in control of the business. They are prevented from
are referred to as long-term bonds. As explained later, there are
taking certain actions, how ever, such as selling fixed assets,
instances when all (or part) of the bond principal is repaid prior
arranging new loans, and stopping (or expanding) business
to maturity.
o p eratio n s. Th e reorganization may involve the sale of all
or part of the business, a reduction in the am ount owed on
loans, and/or a reduction in the interest rate charged on
the loans. Reorganizations can also result in d eb tho ld ers
Interest Rate
becom ing equity holders. Many large U .S. com panies, includ Bonds can also be categorized by how they structure their inter
ing G eneral M otors, Km art, and United A irlin es have m ade est rates.
C h ap te r 11 filings and survived.
Fixed-rate bonds pay the same rate of interest throughout their
An important consideration when a default occurs is the ranking lives. O ccasionally, the interest is payable in a foreign currency.
of claimants (i.e., which claims get satisfied first from available For exam ple, entities outside the U.S. som etim es issue bonds in
funds). Bondholders always rank above equity holders. The U.S. dollars.
holders of some bond issues may rank ahead of others, while
Floating-rate bonds, also known as floating-rate notes (FRNs) or
some bonds may rank ahead of trade creditors (e.g ., suppliers of
variable rate bonds, are bonds where the coupon equals a float
goods to the company that are owed money). This is discussed
ing reference rate (e.g ., Libor) plus a spread.
in more detail in the next section.
To see how a floating-rate bond might work, consider a bond
that pays a coupon every six months with interest equal to the
17.6 C LA SSIFIC A TIO N O F BO N D S six-month Libor plus 20 basis points. Thus, the Libor rate at the
beginning of the six-month period would determ ine the size
In this section, we consider the ways in which corporate bonds of the coupon paid at the end of the six-month period, with
can be classified. the 20-point spread remaining fixed. Some floating-rate bonds
specify maximum or minimum (or both) levels for their coupons. A m ortgage bond provides specific assets (e.g ., homes and
As we will see in Chapter 20, a floating-rate bond can be cre commercial property) as collateral. In the event of a default, the
ated from a fixed-rate bond and an interest rate swap. bondholders have the right to sell the assets to satisfy unpaid
obligations (although it is usually necessary to get permission
Zero-coupon bonds (as their name implies) pay no coupons to the
of the courts first). The m ortgage bondholder usually wants to
holder. Instead, they sell at a discount to the principal amount.
ensure that its position will not be worsened by future bond
For exam ple, consider a five-year, zero-coupon bond that sells issues. It may therefore impose conditions concerning future
for USD 80. This means that an investor could pay USD 800 at bond issues and the extent to which assets acquired in the
the outset and get USD 1,000 in five years. The interest rate future can be used as collateral for future bond issues.
with annual compounding would be
Som etim es there is an after-acquired clause that requires prop
erty acquired after the bonds are issued to be used as collateral
for the bonds. This effectively prevents the collateral from being
or 4.56% . used for other m ortgage bond issues.
The holder of a coupon-bearing bond in the U.S. can usually A collateral trust bond is a bond where shares, bonds, or other
hold a claim on the principal in the event of a bankruptcy. In securities issued by another company are pledged as collateral.
contrast, the holder of a zero-coupon bond can hold a claim on Usually, the other company is a subsidiary of the issuer. A com
the original price paid plus accrued interest. pany that has pledged shares of a subsidiary as collateral would
One of the attractions of zero-coupon bonds is that they can like to be able to vote the shares when key decisions are made
turn one form of income into another under certain tax regimes. at shareholder m eetings. As a result, the issuer has the right to
If the USD 200 difference between the price at which the bond vote shares in the subsidiary if there has not been a default. But
in the previous exam ple is bought and the final repaym ent of if there has been a default, the corporate bond trustee votes the
its par value is treated as a capital gain, the investor will have shares.
essentially converted what would normally be interest income Note that the corporate bond trustee will act in the best inter
into capital gain incom e.15 This is advantageous if capital gains ests of the bondholders (which may not always be in the best
are taxed at a lower rate. interest of the company's shareholders). Som etim es, there are
Another advantage of zero-coupon bonds is that there is no provisions requiring additional collateral to be provided if the
reinvestm ent risk. With a coupon-bearing bond, however, the appraised value of the collateral falls below a certain level.
coupons received must be reinvested. If interest rates decline An equipm ent trust certificate (ETC) is a debt instrument used
(increase), investors will be forced to reinvest at a relatively low to finance the purchase of an asset. (They are commonly used
(high) interest rate and thus a coupon bearing bond will lead to to fund aircraft purchases.) The title to the property vests with
a worse (better) result than a zero-coupon bond. the trustee, who then leases it to the borrower for an amount
sufficient to provide the lenders with the return they have been
promised. When the debt is fully repaid, the borrower obtains
Collateral
the title to the asset.
Bonds can also be classified by the collateral provided, which
An advantage an ETC to the lender is that the asset is already
becom es im portant if a company defaults on its debt payments.
owned by the trustee. Thus, legal proceedings are not necessary
As previously m entioned, a default leads to either a reorganiza
to take possession of the asset in the event of a default. Instead,
tion or an asset liquidation. In either situation, a bondholder
the trustee (acting in the interest of the investors) can simply
with collateral should fare better than one without collateral. In
lease the asset to another company.
the case of a liquidation, bonds with collateral will be paid first
from the proceeds of the sale of the collateral; in the case of a Debentures are unsecured bonds (i.e., bonds where no collateral
reorganization, bonds with collateral will be in a stronger negoti has been posted by the issuer). They rank below mortgage
ating position than bonds without collateral. bonds and collateral trust bonds and are likely to pay a higher
interest rate. O ften, a debenture bond's indenture will include
provisions that limit the extent to which the issuing company
can issue more debentures in the future. These provisions are
15 The U.S. now taxes zero-coupon bonds on imputed interest that needed because a new debenture issue weakens the position of
accrues each year. existing debenture holders.

For exam ple, debenture holders might get 30 cents on the when bonds can be called and at what price. The price that the
dollar (i.e., an amount equal to 30% of their principal) in the issuer must pay for the bond is known as the call price. Usually,
event of a default or liquidation. If the company had been the call price is greater than par when the bond is issued and
allowed to issue twice as many debentures (and if there are no declines towards par as the bond approaches maturity. Typically,
other significant general creditors), the 30 cents on the dollar a bond is not callable for the first few years of its life; this gives
would becom e 15 cents on the dollar. Debentures som etim es bondholders some protection against the possibility of early
include a negative p le d g e clause preventing the issuer from interest rate declines.
pledging assets as security for new bond issues if doing so
Some bonds are convertible into equity on pre-agreed term s.
weakens the debenture holder's position.
The conversion option is usually combined with a call feature.
A subordinated debenture, as its name implies, ranks below other W ithout the call feature, it is likely to be optimal for bondhold
debentures and other general creditors in the event of a bank ers to delay conversion for as long as possible.16 To force con
ruptcy (which means that other debentures get paid first from version (so that bonds becom e equity and new debt can be
available funds). Subordinated debentures require a higher rate of raised), the issuer typically calls the bond as soon as (or shortly
interest than unsubordinated debentures to compensate the hold after) the price of its equity has risen to the point where it is bet
ers for their inferior standing in the event of a default by the issuer. ter for a bondholder to convert the bond rather than sell it back
to the issuer.
Som etim es a bond issued by one company is guaranteed by
another company (e.g ., the company's parent). The bondholder A m ake-whole call provision occurs when the call price is calcu
should then receive the promised interest and principal (unless lated instead of being set in advance. The call price is typically
both the issuer and the guarantor default). The value of the set equal to the present value of the remaining interest and
guarantee depends on the correlation between the financial principal cash flows owed to the bondholder. The discount rate
perform ance of the issuer and the guarantor. As the correlation is typically the risk-free rate plus a certain spread.
increases, the guarantee becom es less valuable.
For exam ple, the call price for a U.S. bond with five years
remaining could be the present value of the remaining cash
flows discounted at the five-year Treasury rate plus 10-basis
17.7 DEBT RETIREM ENT points. Make-whole bonds have been growing in popularity
in recent years. The advantage of a make-whole call provision
Bonds often last until the specified maturity date and are then retired
is that the call provision has no financial cost to bondholders
using the proceeds from a new bond issuance. If interest rates
and therefore they do not need to demand a higher return as
decline during the life of the bond, an issuer would prefer to replace
com pensation.
an older bond issue with a new one before maturity. This benefits
the bond issuer because its interest payments will be lowered. How A sinking fund is an arrangem ent where it is agreed that bonds
ever, the bondholder is worse off because the funds received from will be retired periodically before maturity. The issuer may pro
the early retirement must be reinvested at a lower rate. vide funds to the bond trustee so that the trustee can retire the
bonds (usually at par value). Alternatively, the issuer can buy
Bond covenants can also prompt an issuer to replace an old
bonds in the open market and present them to the trustee. The
bond issue with a new one. Changes in the nature of the issuer's
latter is likely to be attractive to the issuer when bonds are sell
business or im provem ents in its financial health can cause
ing below par. A ccelerated sinking fund provisions allow the
bond covenants to be unnecessarily burdensome (even if those
issuer to retire more bonds than the amount specified in the
covenants were reasonable at the tim e the bond was issued).
sinking fund.
Repaying bondholders early is a way to eliminate highly restric
tive covenants. Certain assets pledged as collateral (such as plant and equip
ment) can depreciate in value. If the principal amount of a bond
In this section, we exam ine ways in which bonds can be retired
issue remains the sam e, it will becom e less well covered by the
early by the issuer.
collateral as time passes. One advantage to sinking fund provi
sions is that they can be made so that the amount borrowed
declines in lockstep with the declining value of the collateral.
Call Provisions
Bond indenture can som etim es allow the issuer to call the bond
16 if the bondholder delays conversion, he or she keeps protection
(i.e., buy it back from the bondholder) at a certain price at a cer against the equity price declining. A s soon as the bond is converted, this
tain tim e. Specifically, a schedule in the bond indenture specifies protection is lost.
As an alternative to retiring bonds, sinking fund requirements can For exam ple, if a bond sells for USD 40 per USD 100 of face
sometimes be satisfied by adding to the property held as collat value im m ediately after a default, the recovery rate would
eral. Normally, the property added is greater than the collateral be calculated as 40% . The loss given default is one minus the
required; this improves the security backing the debt. Another recovery rate.
approach is the use of maintenance and replacem ent funds, which
Credit risk and recovery rates will be discussed in more detail in
requires the issuer to maintain the value of the collateral with
Valuation and Risk M o d els. A t this stage, note some empirical
property additions. If property additions are not made, cash can
properties of recovery rates.17
be used to retire debt. W hereas satisfying sinking fund require
ments by adding property will usually increase the collateral sup • The average recovery rate is 38%.
porting the corresponding debt, maintenance and replacement • The distribution of recovery rates is bimodal.
funds are usually designed to maintain the value of the collateral. • There is no relationship between recovery rates and issue size.
There are tim es that a company wants to sell assets that have • There is a negative relationship between recovery rates and
been pledged as collateral. The bond indenture will normally default rates.
allow a company to do this as long as the proceeds from the • Recovery rates are lower in an econom ic downturn or in a
sale are used to retire the bonds. Selling property can therefore
distressed industry.
be a way to retire debt early.
• Tangible asset-intensive industries have higher recovery rates.
A final way in which bonds can be redeem ed early is by the
The e x p e c te d loss rate on a bond in a given year can be
company making a tender offer to bondholders. A tender offer
defined as:
is simply an offer to purchase the bonds. The offer can be at a
fixed price or it can be calculated as the present value of future Probability of Default X (1 — Expected Recovery Rate)
cash flows. As in the case of a make-whole call price, the dis For exam ple, if the probability of default in a year is 0.5% and
count rate is the risk-free rate plus a pre-specified spread.
the recovery rate is 40% , the expected loss rate is 0.3% .
17.8 DEFAULT RATE AND R ECO V ER Y

17.9 EX P EC TED RETURN FROM BOND
RATE
INVESTM ENTS
Two important statistics published by rating agencies are the
The expected return from a bond is
default rate and the recovery rate. The default rate for a year
can be measured in two ways. Risk-Free Rate + C redit Spread — Expected Loss Rate
1. Issuer default rate: This is the number of bonds that have It might be thought that a bond with a credit spread of 100
defaulted in a given year divided by the number of issues basis points (i.e., 1%) would have an expected loss rate of 1%
outstanding. (and therefore the expected return on the bond is the risk-free
rate). This is not usually the case, however, and the expected
2 . Dollar default rate: This is the total par value of bonds that
loss rate is lower than the credit spread.
have defaulted in a given year divided by the total par value
of all bonds outstanding. Table 17.2 presents some results from Hull (2018)18*showing that
the excess of the spread over the loss rate tends to increase as
The issuer default rate does not consider the size of the issues
the credit quality of the issuer decreases. (An exception is that
that defaulted, whereas the dollar default rate does. For exam
Ba issuers tend to have an excess greater than that of B issuers.)
ple, suppose that there are 100 bonds with a total par value of
USD 1 billion and that two bonds with a combined par value of It can be argued that while the Treasury rate is used as a risk
USD 50 million default. The issuer default rate is 2% and the dol free benchmark by bond traders, it is not the right reference
lar default rate is 10%. point for corporate bond yields. Rather, research suggests that a
When a bond defaults, the bondholder typically does not lose

everything and some recovery is usually made. Because it is dif
17 See D. B. M adan, G . S. Bakshi, and F. X. Zhang, "U nderstanding the
ficult to track the value of what is eventually received by claim Role of Recovery in Default Risk M odels: Em pirical Com parisons and
ants in the event of a default, the recovery rate is calculated as Implied Recovery Rates," ssrn id:285940.
the value of the bond a few days after default as a percentage 18 See J . Hull, Risk M an ag em en t and Financial Institutions, 5th edition,
of its par value. W iley, 2018.

Table 17.2 Loss Rates and Credit Spreads by Rating non-system atic risk because the return from a bond includes
Category for Bonds with a Maturity of about Seven Years only a small probability of a d efau lt.20
Spread Over Loss Excess of Spread

Rating Treasuries (%) Rate (%) Over Loss Rate (%) SUMMARY
Aaa 0.78 0.02 0.76
Public bond issues are usually underwritten by investment
Aa 0.86 0.05 0.81
banks. The bonds are then traded in the over-the-counter mar
A 1.11 0.12 0.99 ket. The yield on a bond is com prised of the risk-free return and
Baa 1.69 0.25 1.44 a credit spread.
Ba 3.22 1.30 2.92 Bond indentures describe the promises of the issuer and the
B 5.23 3.40 1.83 rights of the bondholder. A corporate bond trustee looks after
the interests of bondholders by ensuring that the issuer is in
Caa 11.46 7.50 3.96
com pliance with the term s of the indenture.
Credit rating agencies are paid by the issuer to provide an opin

higher risk-free benchmark (e.g ., the inter-bank borrowing rate) ion on the credit quality of the bonds it issues. Investment grade
should be used instead. However, changing the risk-free bench bonds have a credit rating of BBB (Baa) or above. High-yield
mark does not change the nature of the results: bond traders bonds have a lower credit rating. Bondholders take several risks:
still have an expected return greater than the risk-free rate and
1. The risk that interest rates in the market will increase,
the credit spread they earn is always greater than their expected
loss from defaults. 2 . The risk that there will be a general increase in credit
spreads,
The poor liquidity of co rp o rate bonds is one explanation for
3 . The risk that the issuer's credit quality will decline, and
the results shown in Table 17.2. H owever, w hile bond trad ers
do require som e com pensation fo r the illiquidity of bonds, 4 . The risk that there will be decisions taken that improve the
research shows that the com pensation provided by the mar positions of equity holders at the expense of bondholders.
ket for bond illiquidity does not account fo r the results in
Bonds can be classified by the type of issuer, the bond maturity, the
Table 1 7 .2 .19
nature of the interest rate paid, and the collateral provided. Som e
The main explanation for these results is that bonds do not times the issuer can take advantage of call provisions to retire a
default indep end ently of one another. W hen the econom y is bond early. Other factors that can lead to early retirement of some
doing w ell, the default rate on bonds tend s to be low. H ow or all of a bond issue are sinking funds, maintenance and replace
ever, the default increases when there is a recession. This ment funds, the sale of assets, and tender offers by the issuer.
m eans that bonds have system atic risk (i.e ., risk related to the
Two measures of default rates include the issuer default rate and
overall perform ance of the m arket) that cannot be diversified
the dollar default rate. Rating agencies monitor recovery rates
aw ay. Th u s, bond trad ers require com pensation fo r taking
in addition to default rates. The expected loss rate is the default
system atic risk.
rate multiplied by one minus the expected recovery rate. The
It is also worth noting that the non-system atic (i.e ., idiosyn expected loss rate per year is less than the credit spread. This
cratic) risk of bonds can (in theory) be diversified away. But means that the expected return of an investor on a portfolio of
in practice, a very large portfolio is necessary to elim inate bonds is greater than the risk-free rate.
19 For exam ple, J . Dick-Nielsen, P. Feldhutter, and D. Lando, "C orporate

Bond Liquidity Before and A fter the O nset of the Subprim e C risis,"
Journ al o f Financial Econ om ics, 103, 3 (2012): 471-492, uses several 20 W hereas 30 well-chosen stocks are sufficient to elim inate all non-
different liquidity m easures and a large database of bond trades. system atic risk in equity m arkets, thousands of bonds are necessary to
It shows that the liquidity com ponent of credit spreads is relatively small. do the sam e thing in bond m arkets.
Q UESTIO N S

17.1 W hat are the advantages to a bond issuer of a 17.6 W hat is the difference between default risk, spread risk,
Chapter 11 bankruptcy filing com pared with a filing and event risk?
that involves the immediate liquidation of assets?
17.7 W hat is a debenture?
17.2 G ive two other words used to describe the principal of a 17.8 Under what circum stances is an issuer likely to exercise
bond.
its right to call a bond?
17.3 W hat happens to a bond's yield when its price increases? 17.9 W hat are the two different ways of measuring default
17.4 W hat is the relationship between a bond's clean price rates?
and its cash price?
17.10 W hy do bonds have system atic risk?
17.5 W hat is the role of the corporate bond trustee?
Practice Questions
17.11 Which of the following is likely to have a lower coupon: 17.16 A seven-year zero-coupon bond sells for USD 70. W hat is
(a) private placem ent or (b) a public bond issue? W hy do the bond's yield?
issuers not always choose the lower coupon alternative?
17.17 Explain what is meant by a collateral trust bond and an
17.12 W hy do banks argue that the Volcker rule will reduce equipm ent trust certificate.
bond liquidity? 17.18 Explain how the depreciation of property that has been
17.13 A corporate bond in the U.S. pays a coupon of 10% per pledged as security is handled in bond indentures.
year. Payment dates are January 15 and Ju ly 15. W hat is 17.19 How is the recovery rate on a bond usually calculated?
the accrued interest on a USD 1,000 bond on the start of
17.20 A bond has a credit spread over the risk-free rate of
trading on April 1 in a (a) leap year and (b) a year that is
1.5% , a default probability per year of 0.6% , and an
not a leap year?
expected recovery rate of 30%. W hat is the expected
17.14 Explain three types of covenants in bond indentures. return in excess of the risk-free rate?
17.15 How is an investment-grade bond defined? W hat are
four names given to bonds that are not investment
grade?
Chapter 17 Corporate Bonds

ANSW ERS
17.1 A Chapter 11 bankruptcy filing is a filing where a company 1 7 .1 4 A negative or restrictive covenant limits the issuer's
has a period of time to negotiate a reorganization with its ability to do such things as raise more debt, sell assets,
creditors. pay dividends, and buy back shares. A positive covenant
requires the issuer to do certain things (e.g ., produce
17.2 Face value and par value.
financial statem ents, maintain properties, and carry
17.3 The yield decreases. (Bondholders have to pay more to
insurance). A financial covenant requires that certain
receive the same cash flows.)
financial ratios be maintained.
17.4 The cash price is the clean price plus accrued interest.
17.15 An investm ent grade bond is a bond with a rating
17.5 The corporate bond trustee looks after the interests of from Moody's of Baa3 or above or a rating of B B B - or
bondholders. above from S&P or Fitch. The four names given to
17.6 Default risk is the risk that the bondholder will not bonds that do not meet the investm ent grade criterion
receive the promised paym ents. Spread risk is the risk are high-yield bonds, non-investment grade bonds,
that the credit spread required by the market for the speculative bonds, and junk bonds.
default risk will increase. Event risk is the risk that actions 17.16 The yield (with annual compounding) is
are taken that increase leverage and worsen the position
100 A1/7
of bondholders. - 1 = 0.0523
70
17.7 A debenture is a bond with no collateral.
or 5.23% .
17.8 The issuer will exercise its rights to call a bond if interest
17.17 A collateral trust bond is a bond backed by securities
rates have declined or if it wants to eliminate restrictive
that are owned by the bond issuer (e .g ., shares of a
covenants in the bond's indenture.
subsidiary). An equipm ent trust certificate is a leasing
17.9 The issuer default rate is the percentage of issuers arrangem ent for a large asset such as an airp lane. The
defaulting. The dollar default rate is the dollar value of tru stee owns the asset and leases it to the borrow er
bonds that defaulted as a percentage of the dollar value for a period of tim e for an am ount sufficient to provide
of outstanding bonds. the lenders with th eir required return. A t the end of
17.10 The default rate on bonds is higher when the economy the lease period, the borrow er obtains the title to the
is performing poorly and the stock market is producing asset.
poor returns. 17.18 It can be handled by arranging for bonds to be retired as
17.11 Public issues usually have a lower coupon but higher the value of the collateral declines. It can also be handled
issuance costs. by requiring more assets to be pledged as collateral as
the value of the original collateral declines. Sinking funds
17.12 The Volcker rule restricts a bank's ability to trade for its
and m aintenance and replacem ent funds are used to
own account. This makes it more difficult for a bank to
ensure that bondholders are not adversely affected by
keep an inventory of bonds and act as a market maker.
the depreciation of collateral.
17.13 W hether it is a leap year or not makes no difference. The
17.19 It is usually calculated as the value of a bond a few days
day count convention for bonds is 30/360 so that it is
after a default as a percentage of its face value.
always assumed that there are 30 days in a month. In this
case, there are 75 days in the accrual period, the interest 17.20 The expected loss rate is 0.6% X (1 — 0.3) = 0.42% . The
is USD 100 per year and the accrued interest (USD) is expected return in excess of the risk-free rate is therefore
1.5% - 0.42% or 1.08%.
75
X 100 = 20.83
360
Mortgages and
Mortgage-Backed
Securities
Learning Objectives
Describe the various types of residential mortgage Explain the mechanics of different types of agency MBS
products. products, including collateralized m ortgage obligations
(CM O s), interest-only securities (lOs), and principal-only
Calculate a fixed rate mortgage payment, and its principal securities (POs).
and interest com ponents.
Describe a dollar roll transaction and how to value a
Describe the m ortgage prepaym ent option and the fac dollar roll.
tors that influence prepaym ents.
Explain prepaym ent modeling and its four com ponents:
Summarize the securitization process of mortgage backed refinancing, turnover, defaults, and curtailm ents.
securities (M BS), particularly formation of m ortgage pools
including specific pools and to-be-announceds (TBAs). Describe the steps in valuing an MBS using Monte Carlo
simulation.
Calculate weighted average coupon, weighted average
maturity, single monthly mortality rate (SMM), and condi Define Option Adjusted Spread (O AS), and explain its
tional prepaym ent rate (CPR) for a m ortgage pool. challenges and its uses.
Describe the process of trading of pass-through

agency MBS.
225
This chapter covers the properties of m ortgages and mortgage- To calculate the monthly interest payments on a fixed-rate mort
backed securities. M ortgages are used to finance residential and gage, it is necessary to first convert the quoted rate to a rate with
commercial property. M ortgage-backed securities (MBSs) are monthly compounding. This can be done using Equation (16.2).
investments created from the cash flows provided by portfolios For exam ple, if a mortgage rate in Canada is quoted as 4% with
of m ortgages. This discussion will focus on the U.S. mortgage semi-annual compounding, the rate with monthly compounding is
market because it is a large market that is very important to
fixed-incom e investors. 0.03967
Residential m ortgages in the U.S. typically last 15 or 30 years.

or 3.967% .
Their interest rates can be fixed or variable. Variable-rate
m ortgages are term ed adjustable-rate m ortgages (ARM s). In an In the U .S., m ortgage rates are quoted with monthly com pound
ARM , the interest rate is typically fixed for several years and is ing and thus such a calculation is not necessary.
then tied to an interest rate index. Among the most common Consider a 30-year U.S. m ortgage where the fixed rate is 6%
indices are the one-year Treasury rate (which is referred to as the with monthly com pounding. The monthly payments (constant
constant maturity Treasury rate), a cost of funds index (which is each month for 30 X 12 = 360 months) must totally amortize
the average interest expense incurred by financial institutions the m ortgage. This means that the monthly payments must pro
in a region), and Libor (which was discussed in Chapter 16). vide the lender with a return of principal as well as interest at
ARM s are less risky than fixed-rate m ortgages for lenders and 0.5% (= 6%/12) per month on the outstanding principal.
riskier than fixed-rate m ortgages for borrowers. For this reason,
If the USD paym ent made at the end of each month is X, it
ARM s typically have lower initial interest rates than com parable
must be the case that the present value of the summed pay
fixed-rate m ortgages.
ments equals the amount borrowed. If the amount borrowed is
Most of this chapter is concerned with fixed-rate m ortgages. USD 250,000, we must have
An interesting aspect of fixed-rate m ortgages in the U.S.
360 y
is that borrowers have an Am erican-style option to pay off Y ------- -------- t = 250,000
i t 1 (1 + 0 .0 6 / 1 2 )
their outstanding m ortgage balances. This is referred to as
the borrower's p rep a ym en t option . Som etim es a m ortgage is Using the formula for the sum of a geom etric series, this
prepaid because the m ortgaged property is being sold. O ften, becom es1
it will be prepaid because interest rates have declined and thus
X 1
the property can be refinanced at a lower interest rate. There 1
1 + 0.005 1.005360
-
is usually no penalty for exercising the prepaym ent option, 250,000

1
and it can be quite valuable to the borrower. On the other 1
1 + 0.005
-
hand, the prepaym ent option can be quite costly to those who
or
invest in M BSs. This is because the prepaym ent am ounts must
be reinvested even though interest rates tend to be low when X
250,000
prepaym ents occur. Analysts must take the prepaym ent option 0.005 1.005360
into consideration when valuing m ortgage portfolios.
This equation can be solved to give X = 1,498.88. Payments
This chapter discusses how mortgage payments can be calcu of USD 1,498.88 per month therefore fully amortize (i.e., repay)
lated and then explains how pools of mortgages are form ed to borrowings of USD 250,000 over 30 years.
create tradeable investment vehicles. It also outlines how pre
In general, the relationship between the amount borrowed A ,
payments can be modeled to calculate the value of an invest
the interest rate R (compounded monthly), and the monthly pay
ment in a mortgage pool.
ment X is
X
(18.1)
R/12 (1 + R/12)127
18.1 CALCULATING MONTHLY where T years is the life of the m ortgage.
PAYMENTS
As mentioned in earlier chapters, the compounding frequency 1 The sum of a geom etric series: a + ab + ab2 + • • • abn 1 is
with which a m ortgage interest rate is expressed does not a (1 - bn)
always match the frequency of payments. 1- b
Table 18.1 An Amortization Table for a USD 250,000 30-Year Mortgage when the Interest Rate is 6%
with Monthly Compounding is Shown. The Monthly Payment is USD 1,498.88
Month End-of-Month Interest Payment End-of-Month Principal payment Balance at End-of-Month
0 250,000.00
1 1,250.00 248.88 249,751.12
2 1,248.76 250.12 249,501.00
3 1,247.51 251.37 249,249.63
4 1,246.25 252.63 248,997.00
356 36.92 1,461.96 5,921.30
357 29.61 1,469.27 4,452.03
358 22.26 1,476.62 2,975.42
359 14.88 1,484.00 1,491.42
360 7.46 1,491.42 0.00
Amortization Tables
An am ortization tab le shows the m onthly principal and
interest paym ents on a m ortgage (assum ing no prepaym ent
of principal). A t the beginning of the m ortgage, m ost of
the m onthly paym ent is interest. Toward the end of the
m ortgage, most of the m onthly paym ent is principal. In the
USD 250,000 m ortgage in the previous exam p le, the interest
rate is 6% per year (i.e ., 0 .5% per month) and the m onthly
paym ent is USD 1 ,4 9 8 .8 8 . In the first month the interest on
the m ortgage is
0.005 X USD 250,000 = USD 1,250.00

Fiqure 18.1 Decline of outstanding principal with
which leaves time in a 30-year mortgage.
USD 1,498.88 - USD 1,250.00 = USD 248.88
as repayment of principal. A t the beginning of the second Figure 18.1 shows the way in which the principal declines with
month, the principal is therefore: maturity in Table 18.1.
USD 250,000 - USD 248.88 = USD 249,751.12 As an alternative to using an amortization table, analysts can
calculate the outstanding principal by discounting the remaining
The interest during the second month is cash flows using Equation (18.1). For exam ple, when there
0.005 X USD 249,751 = USD 1,248.76 are ten years (i.e., 120 months) remaining in the life of the
m ortgage, the outstanding USD principal (assuming there have
The repaym ent of principal is therefore: been no prepayments) is3
USD 1,498.88 - USD 1,248.76 = USD 250.12 1,498.88
135,008.97
An extract from the full amortization table is shown in Table 18.1.2 0.005 1.005120
2 Note that the numbers in Table 18.2 are based on the exact solution to 3 This calculation uses the exact solution to Equation (18.1), that is,
Equation (18.1), that is, a m ortgage paym ent, X, of USD 1,498.876313. a monthly m ortgage paym ent of USD 1,498.876313.
Chapter 18 Mortgages and Mortgage-Backed Securities ■ 227

It should be noted that the calculations in this section are In addition to W A C and W AM , m ortgage pool statistics of
not for the valuation of m ortgages. Instead, they are focused interest are as follows.
on the way the principal is am ortized over the life of the
• The average loan balance is the total current outstanding
m ortgage when a rate is quoted. The value of a m ortgage's
principal of the m ortgages in the pool divided by the number
cash flow s to an investor at a sp ecific tim e depends on
of m ortgages.
the interest rate term structure at that tim e along with the
• The pool's factor is the total current outstanding pool
exp ected costs from the prepaym ent option being exercised .
principal as a percentage of the original pool principal.
This will be discussed later.
A pool's factor declines with tim e because part of each
monthly payment by the borrower is a repaym ent of
principal. It also declines as a result of prepaym ents.
18.2 M O R T G A G E P O O LS *
• The w eighted-average FIC O score (FIC O is short for
M ortgage portfolios (also known as m ortgage p o o ls) can be Fair Isaac Corporation) is a FIC O score that measures the
creditworthiness of the borrower. It can range from 300
created for investm ent purposes. The m ortgages in a pool
to 850, with a score above 650 being considered acceptable
are usually similar in term s of loan type, interest rate, and
origination date. The w eighted-average coupon (W AC) is the by many lenders. The weights are given by Equation (18.2),
but with Pj usually set equal to the original principal of the
weighted-average interest rate on the m ortgages in the pool,
with the weight assigned to each m ortgage being proportional /th m ortgage.
to its outstanding principal. • The w eig h ted -a vera g e loan-to-value ratio (LTV) for a
m ortgage is the principal am ount of the loan divided by
To take a simple exam ple, suppose a pool consists only of a
the assessed value of the m ortgaged property at the tim e
USD 200,000 m ortgage with an interest rate (referred to as a
of the loan. The w eights are given by Equation (18.2), but
coupon) of 4% and a USD 400,000 mortgage with an interest
with P; usually set equal to the original principal of the /th
rate of 5%. It would have a W A C of:
m ortgage.
200,000 400.000
W AC X 4% + X 5% = 4.667% • The geographical distribution o f the loans describes the loca
600,000 600.000
tion of the properties being financed by the m ortgages.
The w eighted-average maturity (WAM) is similarly calculated
Prepaym ents are monitored in mortgage pools. In any given
as the weighted-average of the number of months to maturity,
month, some m ortgages may totally prepay and some may
with the weight assigned to each m ortgage being proportional
curtail (i.e., partially prepay). The single m onthly m ortality rate
to its outstanding principal.
(SMM) is the percentage of the outstanding principal that was
Suppose that the two m ortgages we have just considered prepaid during a given month. This does not include scheduled
have m aturities of 340 and 280 months (respectively). A pool payments of principal, such as those shown in Table 18.1. The
consisting of the two m ortgages would have a W AM (in constant prepaym ent rate (also known as the conditional p re
months) of: paym ent rate) is the annualized SMM. The CPR is calculated as:
200,000 400.000 CPR = 1 - (1 - SM M )12

WAM X 340 + X 280 = 300
600,000 600.000
A fter one month, the proportion of the original principal remain
In general: ing (excluding the impact of scheduled repayments of principal)
is 1 - SMM.
W A C = 2 > ;c ,-
/= 1 A fter two months, it is
n
W AM = (1 - SM M )2
(=1
where n is the number of mortgages in the pool, c, is the coupon A fter 12 months, it is
of the rth m ortgage, Li is the remaining life of the /th m ortgage, (1 - SM M )12
and:
The amount of principal lost because of prepaym ents during
the 12 months (if an observed SMM continues for 12 months) is
(18.2)
therefore:
where P; is the remaining principal of the /th m ortgage. 1 - (1 - SM M )12
18.3 A G E N C Y M O R T G A G E -B A C K ED m ortgages. The pools are structured so that they offer returns
in 50-basis point increments (e.g ., 3%, 3.5% , and 4%). Although
S EC U R IT IES (MBSS)
the returns are referred to as coupons, they are different from
the coupons provided by Treasury bonds and corporate bonds.
In the U .S., there are three agencies that buy m ortgages from
MBS coupons are made at the end of each month (rather than
banks and create m ortgage polls. The agencies are
semi-annually) and the payments are a blend of interest and
1. Governm ent National M ortgage Association, referred to as principal on the underlying m ortgages.5
Ginnie Mae (GN M A);
2 . Federal National M ortgage Association, referred to as Trading of Pass-Throughs

Fannie Mae (FN M A); and
Pass-throughs are characterized by their issuers, their coupons,
3 . Federal Home Loan M ortgage Corporation, referred to as and their maturities. For exam ple, one pass-through security
Freddie Mac (FH LM C). could be an investm ent in a G N M A 30-year 4% pool. The
G N M A is a governm ent agency, whereas FN M A and FH LM C are "30-year" descriptor refers to the original lives of the mortgages
private com panies known as govern m en t-spon sored enterprises. in the pool (rather than their current lives).
FN M A and FH LM C loans are not explicitly guaranteed by the Pass-throughs are different from other risk-free investments in
U.S. governm ent, but most market participants consider that that they have prepaym ent risk. Prepaym ent behavior depends
there is an implicit guarantee.4 upon several factors, including interest rates and m ortgage bal
The three agencies enable banks to make long-term loans to ances. (This will be discussed later.)
home buyers without keeping the loans on their books. The Pass-through agency securities trade as sp e cifie d p o o ls and
funds banks receive from selling their mortgages can then to-be-announced (TBAs). In the specified pools market, buyers
be used to make new loans to home buyers. From the U.S. and sellers agree to trade a certain amount of a specified pool at
governm ent's perspective, the agencies perform a useful a specified price. In the T B A market, buyers and sellers agree on:
function because they ensure that banks always have funds
• Issuer (e.g ., FN M A);
available for new home buyers to obtain m ortgages.
• Maturity (original maturity) of the mortgages (e.g ., 30 years);
The mortgages bought by these agencies must satisfy certain cri
teria in areas such as size and credit quality. Once the mortgages • Coupon (e.g ., 4.5% );
are purchased, cash flows from the mortgage pools are used to • Price per USD 100 of par value (e.g ., USD 104.50);
create mortgage-backed securities that are then sold to investors. • Par value (e.g ., USD 100 million); and
The three agencies guarantee their m ortgages for a fee so • Settlem ent month (e.g., August).
that pool investors have protection against mortgage defaults.
The T B A market is a forward market and attracts more trading
However, there is no protection for prepaym ent risk. If a
than the specified pools market.
m ortgage is prepaid when interest rates are low (as is often the
case), the investor must reinvest the funds and earn a lower than For exam ple, consider a T B A m arket trade where the seller
expected rate of interest. must deliver m ortgages from a FM N A 30-year 4.5% pool with
a par value of USD 100 million for USD 104.50 million (plus
The securities created by the agencies are known as agency
accrued interest from the beginning of the month) in August.
m ortgage-backed securities (agency MBSs). They can be con
However, the particular m ortgage pool that will be delivered
trasted with non-agency M BSs, which are issued by private cor
is not sp ecified. Instead, the seller has w hat is term ed a
porations (typically financial institutions) and are not guaranteed
ch ea p est-to -d eliver option. This means the seller can choose
by governm ent-sponsored institutions.
to deliver those securities in A ugust from any FN M A 30-year
The sim plest agency M BSs are pass-throughs, with all investors pool where the coupon is 4.5% . The settlem ent dates
in a pool receiving the same return. Specifically, investors get within the settlem ent month for T B A s are provided by the
their share of the cash flows from the m ortgages in the pool Securities Industry and Financial M arkets Association (S IFM A ).6
minus the agency's fees for guaranteeing and servicing the
5 The way in which interest and principal are blended for an individual
m ortgage is illustrated in Table 18.1.
4 A fter the 2007-2008 financial crisis, these entities experienced 6 See https://www.sifm a.org/resources/general/m bs-notification-and-
financial difficulties and there was an injection of capital from the settlem ent-dates/ The settlem ent date is usually the twelfth or thirteenth
U.S. Treasury. of the month.

Two business days before the settlem ent day, the seller has to W hat is term ed the value o f the roll is thus calculated as:
announce from which pool (or pools) the M BSs will be
A - B + C - D
d e live re d .7 The seller then receives the agreed price plus
accrued interest from the beginning of the month. (Accrued For example, suppose that a USD 1 million par value of a 4.5%
interest is calculated assuming 30 days in a month.) pool is sold for USD 102.50 in March and repurchased for
USD 102.00 in April. We suppose that the payment date is the
The remaining maturity of the m ortgages in the delivered pool
twelfth of the month for both months. This means that the accrued
is generally less than 30 years and there are rules that define
interest is USD 1,500 (= (12/30) X (0.045/12) X 1,000,000) in
the range of maturities that are acceptable. For exam ple, the
total for both transactions. It follows that A = USD 1,026,500 and
m ortgages in a delivered 30-year pool must have remaining
B = USD 1,021,500.
maturities between 15 and 30 years.
Now assume that the proceeds of the sale in the first month can
be invested at 0.1% for the month so that C = USD 1,026.5.
Dollar Roll In calculating D, we assume that if the pool had been not been
sold, interest and principal payments on the pool during the
A trade known as a dollar roll involves selling a T B A for one
month of the roll would have amounted to 0.45% of the par
settlem ent month and buying a similar T B A for the following
value. This means that D = USD 4,500.0.
settlem ent month. For exam ple, a trader could sell a USD
100 million 30-year FN M A pool with a 4.5% coupon for August In this case, the value of the roll (USD) is
settlem ent and buy a USD 100 million 30-year FN M A pool with
1,026,500 - 1,021,500 + 1,026.5 - 4,500 = 1,526.50
a 4.5% coupon for Septem ber settlem ent.
Chapter 16 covered repo transactions (i.e., where one party Other Agency Products
sells securities to another party and agrees to buy them back at
a future tim e for a slightly higher price). A dollar roll is similar The agency securities mentioned thus far have been pass
(in some respects) to a repo. But there are two important throughs. Another type of product is called a collateralized
differences. m ortgage obligation (CM O ). In a C M O , classes of securities that
bear different amounts of prepaym ent risk are created. These
1. The securities purchased in the second month may not be
classes are referred to as tranches.
the same as the securities provided in first month. The party
on the other side of the transaction can sell back the same A s a simple exam ple, suppose that there are Tranches A , B, and
securities, but it may also deliver securities with worse pre C with the following properties:
payment properties. • Tranche A investors finance 30% of the MBS principal,
2 . No interest is added to the price at which the securities • Tranche B investors finance 50% of the MBS principal, and
are repurchased. The dollar roll transaction involves the
• Tranche C investors finance the remaining 20% of the MBS
initiating party losing one month of interest payments from principal.
a pool with the specified coupon, while the party on the
other side gains one month of interest. Cash flows to the tranches are defined so that each tranche gets
interest on its outstanding principal. However, there are special
Defining term s:
rules for principal payments (both scheduled and prepayments).
Payments are first channelled to Tranche A . When the principal
A : The price at which the pool is sold during the first
month (including accrued interest), of Tranche A has been fully repaid, principal payments are then
channelled to Tranche B. W hen the principal of Tranche B has
B: The price at which the pool is purchased during the
been fully repaid, all remaining principal payments are chan
second month (including accrued interest),
nelled to Tranche C.
C: The interest earned on the proceeds of the sale for
one month, and In this example, most of the prepayment risk is borne by Tranche A
and very little is borne by Tranche C. However, the distribution pre
D: The coupon and the principal repaym ent that would
have been received on the pool sold during the first payment risk can be adjusted by changing the percentage of the
month. pool financed by the different tranches.
Two other agency securities are interest-only securities (lOs) and
7 Typically, the securities delivered can be from at most three pools that principal-only securities (POs). These are also called stripped MBSs.
have the specified coupon, maturity, and issuer. As their names imply, all the interest payments from a mortgage
pool go to the IOs, while all the principal payments go to the POs. payments. However, there can be other reasons. For exam ple,
Both IOs and POs are risky instruments. the borrower's credit rating may have improved so that he or
she is able to obtain a lower rate even when interest rates have
As prepaym ents increase, a PO becom es more valuable because
not changed. Another reason for refinancing can be that the
cash flows are received earlier than expected. By contrast, IOs
value of the property has increased so that a higher loan can be
becom e less valuable because few er interest payments are
negotiated. (This is referred to as cash-out refinancing.)
made overall. As prepaym ents decrease, the reverse happens.
The extent to which refinancing is likely to occur is measured
by the incentive function. A simple incentive function / for a pool
Non-Agency MBSs could be
Agency M BSs should be distinguished from M BSs that are not / = W AC - R (18.3)
issued by G N M A , FN M A, or FH LM C. In a typical non-agency
where W A C is the weighted-average coupon and R is the
securitization, a m ortgage portfolio is sold by a bank to a special
current m ortgage rate available to borrowers. The incentive
purpose vehicle (SPV), which in turn passes the cash flows to the
function is then the amount by which refinancing allows
various securities it creates. In this case, there is usually no guar
borrowers to reduce their interest rates.
antee protecting investors against defaults. Indeed, default risk
(rather than prepaym ent risk) is the major risk being taken by A slightly more elaborate incentive function is
investors. Typically, the SPV creates several tranches that are
/ = (W AC - R) X ALS X A - K
subject to different amounts of default risk and promised
where A LS is the average loan size, K is the estim ated cost of
returns.8 This securitization model played a key role in the
2007-2008 crisis.9 refinancing a loan, and A is an annuity factor giving the present
value of one dollar of payments per year for a period equal to
the weighted-average maturity. This incentive function reflects
18.4 M O D ELIN G PR EPA YM EN T the amount by which refinancing allows borrowers to reduce the
total present value of their remaining payments. It reflects the
B EH A V IO R
empirical evidence that the prepaym ent rate increases as the
average loan size increases.
Calculating the cost of the prepaym ent option to investors is
more com plicated than valuing an interest rate option. This is For a given incentive function /, the annualized prepaym ent rate
because prepaym ent behavior depends on more than just inter is som etim es modeled as:
est rates. There are generally four reasons for prepaym ents:
1
1. Refinancing, a + be~cl
2 . Turnover, where a, b, and c a re param eters estim ated from empirical
3 . Defaults, and data. It is illustrated in Figure 18.2 using the incentive function
in Equation (18.2) with interest rates measured in basis points
4 . Curtailm ents.
and assuming a = 4, b = 0.02, and c = 25. Note that this is an
S-shaped function.
Refinancing The shape of the CPR in Figure 18.2 is broadly consistent with
prepaym ent experience. It shows that refinancing creates very
Refinancing arises when a borrower prepays a mortgage in order
little prepaym ent when m ortgage rates have increased (so
to refinance the underlying property. The most likely reason
that W A C - R is negative). As mortgage rates decrease, the
for this is a decline in interest rates. By refinancing in such
prepaym ent rate increases quite fast and then levels off.
an environment, the borrower can reduce his or her monthly
The param eters a, b, and c may depend on the econom ic
8 See Figure 1.2 of C hapter 1 for a sim ple exam ple. environm ent. For exam ple, we m entioned earlier that increases
in housing prices make prepaym ents more likely. Another
9 Banks relaxed their lending standards and the tranches created
provided lower returns than expected because the default rates on the phenom enon that may affect the param eters is term ed
underlying m ortgages w ere much higher than exp ected . It was also burnout. W hen a m ortgage pool has been in existence for
the case that in the run-up period prior to the crisis, Fannie Mae and
some tim e and interest rates have declined, the m ortgage
Freddie Mac guaranteed riskier m ortgages than previously and they
experienced severe financial difficulties as a result of the high default holders most likely to refinance (e.g ., those who are financially
rates during the crisis. sophisticated, have a good credit rating, or high loan balances)

Defaults
When a mortgage holder defaults and the mortgage is part of
an agency pool, the agency pays the outstanding balance on
the m ortgage. This is treated as a prepaym ent and therefore
defaults are relevant to the calculation of prepaym ents despite
agency guarantees. Defaults added considerably to the pre
paym ent experience of m ortgage pools during the 2007-2008
crisis. Models use average FIC O scores, LTVs, and the history of
housing price movements to predict the default com ponent of
prepaym ents.
Fiaure 18.2 Relationship between annualized Curtailments

conditional prepayment rate, CPR, from refinancing
Curtailm ents are partial prepaym ents. These tend to occur when
and the incentive function / = WAC - R.
loans are relatively old and balances are relatively low. Prepay
ments from curtailm ents can rise as high as 5% when the loans
will tend to have already done so. The remaining m ortgage in a m ortgage pool have only one or two years to maturity.
holders are (on average) less likely to refinance. The burnout

phenom enon shows that a prepaym ent function can be path
dependent; prepaym ents on a m ortgage pool that is a few
18.5 VA LU A TIO N O F AN MBS P O O L
years old can depend on where interest rates were in the past
The first step in valuing an MBS pool is to develop a prepay
as well as where they are today.
ment model. The previous section indicates (in a general way)
what the different com ponents of such a model might be. Two
Turnover variables that future prepaym ents may depend on are as follows.
Turnover prepaym ents arise when a borrower sells the house. 1. Level of interest rates: The burnout phenomenon shows
Turnover is higher in the summer months than the winter that the com plete pattern of interest rates because m ort
months. It is also lower early in the life of a m ortgage because gage origination is relevant to determ ining prepaym ents in
homeowners usually do not relocate im m ediately after taking a given month.
out a m ortgage. 2 . Housing prices: The history of housing prices since the
In the prepaym ent model developed by the Public Securities m ortgage origination may also be relevant. Sharp increases
Association (PSA) for analyzing Am erican M BSs, increasing in housing prices may lead m ortgage holders to refinance
prepaym ent rates are assumed for the first 30 months after (i.e., use cash-out refinancing). Sharp decreases in hous
m ortgage origination and constant prepaym ent rates are ing prices may lead to defaults. Both are liable to increase
assumed thereafter. In the standard model (called the 100% prepaym ents.
PSA), the annualized prepaym ent rate is 0.2% in month one. It It is necessary to develop models describing the uncertain future
then increases by 0.2% each month until reaching 6% in month behavior of these variables in order to calculate future prepay
30. A fter month 30, it remains constant at 6%. In the 150% PSA ment behavior.10
model, these numbers are 50% higher so that the annualized
In addition, there are several relevant param eters describing the
prepaym ent rate is 0.3% in the first month and rises to 9% in
m ortgage pool. The following are exam ples.
month 30.
• The prepaym ent rate tends to increase as the average loan
The turnover rate is liable to depend on the geographical
size increases.
location of the properties and on the average age of the
m ortgage holders. It does not depend on interest rates, except • The geographical distribution of the loans may affect the
to the extent that a hom eowner may be less inclined to move if model used to project housing prices and expected turnover.
he or she is paying below-market rates on his or her m ortgage.
(A reluctance to move when the mortgage rate is low is referred 10 The m odels should be in a risk-neutral world. This is explained in
to as the lock-in effect.) C hapter 16 of Valuation and Risk M o d els.
Table 18.2 Interest and Principal Cash Flows When there are No Prepayments
Month End-of-Month Interest Payment End-of-Month Principal Payment Balance at End-of-Month
0 100,000
1 333.33 406.35 99,593.65
2 331.98 407.71 99,185.94
3 330.62 409.07 98,776.87
4 329.26 410.43 98,366.44
177 9.78 729.91 2,204.35
178 7.35 732.34 1,472.01
179 4.91 734.78 737.23
180 2.46 737.23 0.0
• A verage FIC O scores and average LTVs affect default month's prepaym ents can depend on the history of interest
predictions. rates and the history of housing prices, as well as on their
• A verage loan age affects curtailm ent estim ates. current levels. O ther analytical tools, such as the use of trees,
cannot easily accom m odate this featu re.11
O nce a prepaym ent model has been specified, a technique
known as Monte Carlo simulation is used to value an M BS. This A s a sim ple exam ple to show the effect of prepaym ents
involves the following steps. on the valuation of a m ortgage pool, suppose that a pool
contains new 15-year m ortgages with paym ents being made
1. Random ly sam ple from p ro b ab ility distributions to d eter
at the end of each m onth. Suppose further that the coupon
mine a hypothetical month-by-month path for risk-free
(com pounded monthly) is 4% . The m ortgage paym ent for
interest rates and housing prices. The interest rate path
an M BS with USD 100,000 of the pool's principal is from
will be accom panied by estim ates of the spread of the
Equation (18.1):
m ortgage rate over the risk-free rate. Th e housing price
path may depend on interest rates and will reflect the USD 100,000 X 0.04/12 _ USD ^
geographical distribution of the m ortgage holders.
1 ______________________________ 1
_______________________
2. For each month, determ ine prepaym ent rates using the (1 + 0.04/12)1S0_
specified prepaym ent model, the path for interest rates,
and housing prices up to that month, as well as relevant W e suppose that the risk-free discount rate is alw ays 0.5%
param eters describing the mortgage pool. less than the m ortgage rate for all m aturities. If interest
rates do not change and there are no p repaym ents, we can
3 . Use the prepaym ent rates to calculate month-by-month
value the M BS by discounting USD 739.69 per month for 180
cash flows from the MBS.
m onths at 3.5% per year (or 3.5% /12 = 0.2917% per m onth).
4. Starting at the end of the life of the M BS, discount cash This gives USD 103,470. Th e interest cash flow s and principal
flows month-by-month back to today. The discount rate for cash flow s are shown in Table 18.2. The present value of the
a month is the risk-free interest rate sampled for that month. interest cash flow s with a 3.5% per year discount rate is
5. Repeat steps 1 to 4 many tim es. USD 2 7 ,7 5 9 , w hile the present value of the principal cash
flow s is USD 7 5 ,7 1 1 .
6 . Calculate the value of the MBS pool as the average of the
calculated present values.
The advantage of Monte Carlo simulation is that it can consider 11 See C hap ter 14 of Valuation and Risk M o d els for the use of trees to
what is referred to as path dependence. This means that a given value options.

Now suppose that there is uncertainty about the level of inter one where they are uncertain but have the same expected value
est rates over the next five years and that this uncertainty will be increases the expected present value of cash flows.
resolved soon. We suppose the following.
Table 18.3 shows the impact of the 2% (per month) prepaym ent
• There is a 50% chance that the mortgage rate will rise to 6% rate. A t the end of the first month, the scheduled principal pay
and there will be no prepaym ents. ment is USD 406.35 and the principal remaining after scheduled
• There is a 50% chance that m ortgage rates will drop to 2%. payments is USD 99,593.65 (as in Table 18.2). There is then an
Assum e that the prepaym ent at the end of a given month will additional principal repaym ent equal to 2% of USD 99,593.65
be 2% of the end-of-year m ortgage balance after the sched (i.e., USD 1,991.87) so that the total principal repayment
uled repaym ent has been considered. is USD 2,398.23 (= USD 1,991.87 + USD 406.35) and the
remaining principal is USD 97,601.77 (= USD 100,000 -
Assum e that there are no further changes to the interest rate
USD 2,398.23). Interest in the second month is
environment once this change has taken place and that the
USD 325.34 (= (0.04/12) X USD 97,501.77). The m ortgage pay
change takes place almost im m ediately after the pool of new
ment in the second month is 2% lower than in the first month
m ortgages is created.
(i.e., USD 724.89) because 2% of the m ortgage principal has
If the m ortgage interest rate is 6%, the cash flows will be as been lost from the pool. Because the scheduled principal pay
shown in Table 18.2 except that the discount rate will be 5.5% ment is 2% lower than in Table 18.2 (i.e., USD 399.55), the total
per year (i.e., 5.5% /12 = 0.4583% per month). The present principal at the end of month two (after scheduled repayments)
value of interest cash flows will be USD 25,259 and the present is USD 97,202.22 (= USD 97,601.77 - USD 399.55). An extra
value of the principal cash flows will be USD 65,269. The com 2% of this principal is then repaid, bringing the principal at the
bined present value will therefore be USD 90,528. end of month two down to USD 95,258.17. Calculations for the
remaining months continue in a similar manner.
If there were no prepaym ents when the interest rate dropped
to 2%, the total present value of the MBS cash flows (at 1.5% The present value of the interest paym ents in Table 18.3
discount rate) would be USD 119,162 and the value of the MBS discounted at 1.5% per year is USD 12,346, while that of
would be USD 104,845 (= 0.5 X (90,528 + 119,162)). the principal paym ents is USD 93,089 (for a total of USD
105,434). The results we have produced are sum m arized
This is greater than the USD 103,470 that was calculated assum
in Table 18.4. A s indicated in the final row, the value of the
ing no change in interest rates because of the non-linear relation
M BS is
ship between the present value of cash flows and the discount
rate. Moving from a world where interest rates are certain to 0.5 X (90,528 + 107,716) = USD 99,122
Table 18.3 The Cash Flows in the 2% Mortgage Interest Rate Environments are Shown. Prepayments are
2% per Month
End-of-Month End-of-Month Additional End-of-

Scheduled End-of-Month Scheduled Principal Month Principal Remaining Principal
Month Payment Interest Payment Payment Payment at Month End
0 100,000.00
1 739.69 333.33 406.35 1,991.87 97,601.77
2 724.89 325.34 399.55 1,944.04 95,258.17
3 710.40 317.53 392.87 1,897.31 92,968.00
4 696.19 309.89 386.30 1,851.63 90,730.07
177 21.13 0.28 20.85 1.26 61.70
178 20.70 0.21 20.50 0.82 40.38
179 20.29 0.13 20.16 0.40 19.82
180 19.88 0.07 19.82 0.00 0.00
Table 18.4 Summary of Results (USD)
Present Value of Present Value of

Interest Payments Principal Payments Total Present Value
No Interest Rate Change 27,759 75,711 103,470
M ortgage Rate Increase to 6% 25,259 65,269 90,528
M ortgage Rate Decrease to 2% 12,326 95,370 107,716
A verage of 6% and 2% M ortgage Environments 18,802 80,320 99,122
Interest rate uncertainty, together with the possible prepay The method of successive bisection can be used to create a
ments, reduce the price from USD 103.470 to USD 99.122 workable algorithm . First, initial high and low O A S estim ates are
(per USD 100 of principal). produced. This can be done by reducing O A S until the sim u
lated price is higher than the market price and then increasing
Table 18.4 also illustrates the general phenomenon that inter
O A S until the simulated price is lower than the market price.
est rate uncertainty (and the resulting prepaym ent uncer
The average of the high and low prices is then used in the sim u
tainty) reduces the value of lOs while increasing the value of
lation. If it proves to be too high, it becom es the new high O A S.
PO s. Note that the price of an IO constructed from the pool
If it proves to be too low, it becom es the new low O A S. The pro
decreases from USD 27.759 to USD 18.802, whereas that of a
cedure is then repeated. Note that the ranges between the high
PO constructed from the pool increases from USD 75.711 to
and low O A S are halved on each iteration.13
USD 80.320.
Assume that the pool in the previous example sells for USD 98.00.
The O AS is therefore the spread that must be added to the dis
18.6 O P TIO N A D JU S T E D SP R EA D count rates to give a present value of USD 98,000 per USD 100,000
of par value. This turns out to be 24.67 basis points. If we discount
The option-adjusted spread (OAS) is the excess of the expected at 5.7467% and 1.7467% (instead of 5.5% and 1.5%) in these
return provided by a fixed-incom e instrument over the risk-free two scenarios (respectively), the value of the pool changes from
return adjusted to account for em bedded options. The return on USD 99.12 to USD 98.00.
an MBS is adjusted for prepaym ent options as follows
O A S can be useful in determ ining the relative valuation of differ
O A S = Expected MBS Return - Return on Treasury Instruments ent MBS pools. For exam ple, an MBS with an O A S of 80 basis
We have outlined a procedure involving Monte Carlo simulation points (i.e., with a return of 0.8% over the Treasury rate) should
for determ ining the value of an M BS. This procedure can be be a better buy than one that provides an O A S of 40 basis
adjusted to determ ine the O A S provided by the M BS. The pro points (i.e., with a return of 40 basis points over the Treasury
cedure is as follows. rate). O f course, the O A S calculated depends on the extent to

which the underlying model correctly accounts for prepaym ents.
1. Make an initial estim ate of the O A S.
If the model is incorrect or has not been calibrated properly, the
2. Carry out a Monte Carlo simulation as described in the pre results cannot be relied upon. When an analyst finds a high-OAS
vious section but using discount rates equal to the Treasury pool, he or she should look for institutional or technical rea
rate plus the current estim ate of the O A S .12 sons why that pool might trade differently from the rest of the
3 . Com pare the price obtained with the market price. market. It may be that the analyst can find an assumption in the
model being used that leads to the high O A S. He or she should
4. If the market price is higher than the simulated price, the
then critically exam ine the validity of that assumption.
O A S estim ate is reduced. If the market price is lower than
the simulated price, the O A S estim ate is increased. The dependence of the prepaym ent model on interest rates
is critical. If an analyst feels that the model correctly describes
5. Continue changing the O A S estim ate until the simulated
prepaym ents, it should be possible to hedge the interest rate
price equals the market price.
exposure (at least approxim ately) to lock in an expected profit.
12 W hen the M BS is valued, an analyst might use a different proxy for

the risk-free rate other than the Treasury rate. The O A S , however, is 13 The sets of random num ber sam ples used to determ ine paths should
calculated relative to the Treasury rate. be the sam e on each iteration.

The natural instruments to hedge interest rate exposure are Three agencies create mortgage pools and provide investors
Treasuries. However, mortgage rates are not perfectly corrected with protection against defaults (but not against prepayments):
with Treasury rates and so even if an analyst has a perfect pre Governm ent National M ortgage Association (Ginnie Mae,
paym ent model that depends only on interest rates, there will G N M A ), Federal National M ortgage Association, (Fannie Mae,
always be some residual interest rate risk when Treasury instru FN M A), and Federal Home Loan M ortgage Corporation
ments are used as hedges. (Freddie Mac, FH LM C).
There is an active forward market in m ortgage pools created by

these agencies, known as the T B A market. A feature of this mar
SU M M A R Y ket is that the precise pool that will be delivered is not specified
at the tim e the forward contract is agreed. Param eters describ
The m ortgage market in the U.S. presents unique challenges for
ing the pool, such as the maturity of the m ortgages and coupon
analysts. The first point to note is that a mortgage is different
rate, are specified and the seller chooses which pool with the
from a bond in that the principal is not paid at the end. Instead,
specified param eters will be delivered. In practice, the seller is
the payments are made by the mortgage holder monthly and
likely to use a prepaym ent model to determ ine the least valu
are a blend of interest and principal so that by the end of the life
able pool that can be delivered.
of the m ortgage there is no principal outstanding.
A popular trade is the dollar roll, where a T B A for one month
M ortgages in the U.S. usually last 15 or 30 years. The holder
is sold and a similar T B A for the next month is purchased. This
has the option to prepay the m ortgage at any time (usually
can lead to funds being borrowed at an attractive interest rate.
without penalty). This creates uncertainty about when the lender
However, the pool received during the second month may be
will receive cash flows. In practice, analysts develop models
different from (and less highly valued than) the pool sold during
for estimating the rate at which prepaym ents will occur on in
the first month.
m ortgage pools. Two main reasons why homeowners prepay are
as follows.
1. Interest rates have declined so that the house can be refi

nanced at a lower cost.
2. The house has been sold.
Q U E S T IO N S

18.1 W hat is the difference between the pattern of cash flows 1 8 .5 W hat is a pass-through
provided by a fixed-rate mortgage and the pattern of 1 8 .6 Explain how a dollar roll works and why it is different
cash flows provided by a U.S. Treasury bond? Assum e no
from a repo as a way of using securities that are owned
prepaym ents and no defaults on the m ortgage. to borrow money.
1 8 .2 How is weighted-average maturity for a pool defined?
1 8 .7 List four reasons for prepaym ents.
Provide a formula.
1 8 .8 Explain what is meant by burnout.
1 8 .3 W hat is a pool's factor?
1 8 .9 W hy are mortgage defaults relevant to the prepaym ent
1 8 .4 Which of FN M A, G N M A , and FH LM C are private
rate calculated for an agency M BS?
com panies?
1 8 .1 0 W hat is a curtailment?
Practice Questions
18.11 W hat is the monthly payment on a 15-year mortgage 1 8 .1 5 Explain how CPR is defined.
where the rate is 5% with monthly compounding and the
1 8 .1 6 Explain how a T B A works.
amount borrowed is USD 100,000?
1 8 .1 7 W hat tends to happen to the values of lOs and POs as
18.12 How much is outstanding on the mortgage in Question 18.11
prepaym ents increase?
after seven years? Assume there is no prepayment.
1 8 .1 8 How is a prepaym ent function used in the valuation of an
1 8 .1 3 A pool consists of three m ortgages: MBS?
1. USD 100,000 mortgage with 330 months to maturity
1 8 .1 9 How is O A S defined for an M BS?
and a 4% rate
1 8 .2 0 In a dollar roll a USD 10,000,000 30-year 6% pool is
sold for 103.25 and bought for 102.80. The settlem ent
and a 5% rate
is on the fourteenth of the month for both months.
The proceeds of the sale can be invested at 4% per
and a 6% rate
year (com pounded m onthly), and interest and principal
Calculate the W A C and W AM . paym ents on the pool during the month of the roll
1 8 .1 4 W hat is the range of FIC O scores? W hat is mentioned in would am ount to 0.4% of the par value. W hat is the
the text as a typical minimum acceptable FIC O score? value of the roll?
Chapter 18 Mortgages and Mortgage-Backed Securities

A N S W ER S
18.1 A fixed-rate m ortgage provides monthly cash flows that of the period and leave the pool. The prepaym ent rate
are constant for the whole life of the m ortgage. A bond for the m ortgage holders who are left is then lower.
provides semi-annual coupons and a final payment of 18.9 When a mortgage holder defaults, the guarantor pays
principal.
the amount owing and the m ortgage is treated as a
18.2 The weighted-average maturity for the m ortgages in a prepaym ent.
pool is 18.10 A curtailm ent is a partial prepaym ent.
18.11 The monthly paym ent, X, satisfies

2
/=1
W 'L >
X
where n is the number of m ortgages, L, is the remaining 100,000
0.05/12 (1 + 0.05/12)12x15
life of the rth m ortgage, and w, is the proportion of the
total principal provided by the fth m ortgage. Solving this gives X equal to USD 790.79.
18.3 A pool's factor is the total current outstanding principal 18.12 The remaining principal (USD) is
divided by the total original principal for the m ortgages
790.79
in the pool. 62,464
0.05/12 (1 + 0.05/1 2)12x8
18.4 FN M A and FH LM C are private com panies.
18.13 W A C is
18.5 A pass-through is an MBS where the cash flows from
a pool of m ortgages are passed to investors after 100 200 300
X 4% + X 5% + — x 6% = 5.33%
subtracting the costs of guaranteeing and servicing the 600 600 600
m ortgages. All investors in the pool receive the same W A L (months) is

return.
100 200 300
X 330 + 310 X X 290 = 303.33
18.6 A dollar roll involves raising funds by selling a pool in 600 600 600
one month and buying a similar pool back the follow
18.14 FIC O scores range from 300 to 850. 650 is considered
ing month. Properties such as coupon, principal, and
the minimum acceptable score by many mortgage
maturity are specified but the party with a short posi
lenders.
tion chooses the actual pool that will be delivered. This
means that the pool that is received during the second
18.15 CPR is an annualized prepaym ent rate. If SMM is the pre
paym ent for a month, the CPR is defined as:
month may be different from the one sold during the
first month. This is one difference between a dollar roll CPR = 1 - (1 - SM M )12
and a repo. The other difference is that interest does not 18.16 A T B A is a forward contract to deliver an MBS pool with
have to be added to the price paid in the second month particular characteristics in a future month. The follow
because the trader who sells and then buys loses one
ing are specified: the issuer, the original maturity of the
month of MBS income, while the trader on the other side m ortgages, the coupon, the price per USD 100 of par
gains one month of MBS income. value, the par value, and the settlem ent month. The
18.7 Refinancing, turnover, defaults, and curtailments delivery day during the settlem ent month is specified by
SIFM A and is usually close to the twelfth day. Two days
18.8 Burnout occurs when interest rates decline over a
before the delivery day the seller chooses which pool (or
period of tim e. Tho se m ortgage holders who are
most likely to prepay (e.g . because their m ortgage pools) will be delivered.
balances are high, they have good cred it ratings, or 18.17 As prepaym ents increase, a PO tends to increase in
are financially so p histicated ) do so during the first part value, whereas an IO tends to decrease in value.
18.18 A Monte Carlo simulation is used. Paths are sam pled for 18.20 USD selling price is 10,325,000 + 10,000,000 X
interest rates and other relevant variables, such as house (14/30) X (0.06/12) = 10,348,333
prices. Prepayments are calculated for each month and
USD buying price is 10,303,000 + 10,000,000 X
the cash flows from the MBS are calculated. These are
(14/30) X (0.06/12) = 10,303,333
discounted at the interest rates along the path to obtain
a present value. This procedure is repeated many times Investment of proceeds yields in USD 10,348,333 X
and the value of the MBS is estim ated as the average of 0.04/12 = 34,494
the present values.
Payments on the pool would be USD 40,000.
18.19 The O A S is the spread between the discount rate and
The value of the roll (USD) is 10,348,333 — 10,303,333 +
the applicable Treasury rate that leads to the value of an
MBS equaling its market price. 34,571 - 40,000 = 39,494
Chapter 18 Mortgages and Mortgage-Backed Securities 239

Interest Rate
Futures
Learning Objectives
Identify the most commonly used day-count conventions, Calculate the theoretical futures price for a Treasury bond
describe the markets that each one is typically used in, futures contract.
and apply each to an interest calculation.
Calculate the final contract price on a Eurodollar futures
Calculate the conversion of a discount rate to a price for a contract.
U.S. Treasury bill.
Describe and com pute the Eurodollar futures contract
Differentiate between the clean and dirty price for a U.S. convexity adjustm ent.
Treasury bond; calculate the accrued interest and dirty
price on a U.S. Treasury bond. Explain how Eurodollar futures can be used to extend the
Libor zero curve.
Explain and calculate a U.S. Treasury bond futures contract
conversion factor. Calculate the duration-based hedge ratio and create
a duration-based hedging strategy using interest rate
Calculate the cost of delivering a bond into a Treasury futures.
bond futures contract.
Explain the limitations of using a duration-based hedging
Describe the impact of the level and shape of the yield strategy.
curve on the cheapest-to-deliver Treasury bond decision.
241
Earlier chapters have exam ined futures on assets such as Consider first the actual/actual convention, which is used for
com m odities, stock indices, and foreign currencies. This chapter Treasury instruments in the U.S. Under this convention, the
considers interest rate futures. These instruments are actively number of days in the period of interest is the actual number
traded and have several features that deserve special attention. of calendar days, while the number of days in the reference
period is the actual number of days between coupon payments.
Specifically, this chapter will focus on the Eurodollar and
The number of days between March 15 and Septem ber 15 is
Treasury bond/note futures contracts traded on the Chicago
184 (16, 30, 31, 30, 31, 31, and 15 in March, A pril, May, Ju n e,
M ercantile Exchange (CM E). Eurodollar futures contracts can
Ju ly, August, and Septem ber, respectively). The number of days
be used to hedge an exposure to (or speculate on) short-term
between March 15 and Ju ly 1 is 108 (16, 30, 31, 30, and 1 in
U.S. interest rates. M eanwhile, contracts on Treasury bonds and
March, April, May, Ju n e, and Ju ly, respectively). The accrued
notes can be used to do the same for longer-term U.S. inter
interest from March 15 to Ju ly 1 is therefore:
est rates. Note that similar contracts involving non-U.S. interest
rates are traded on exchanges in many other countries. 108
5 X — = 2.9348
184
Now consider the 30/360 convention, which (as mentioned in

19.1 D AY C O U N T C O N V E N T IO N S Chapter 17) is used for corporate bonds in the U.S. This
method assumes 30 days per month for the period of interest
Many bonds pay interest semi-annually in arrears. This means and 360 days in a year. This would mean that there are 180 days
that the first coupon payment is after six months and the last between March 15 and Septem ber 15 (15, 30, 30, 30, 30, 30,
coupon payment is at the end of the bond's life. and 15 in March, April, May, Ju n e, July, August, and Septem ber,
For exam ple, suppose a bond pays coupons on March 15 and respectively) and 106 days between March 15 and Ju ly 1 (15, 30,
Septem ber 15 at the rate of 10% per year. If a bond is pur 30, 30, and 1 in March, A pril, May, Ju n e, and Ju ly, respectively).
chased on Ju ly 1, it is important to know how much of the next The accrued interest between March 15 and Ju ly 1 is therefore
coupon (to be received in Septem ber 15) has already been calculated as:
earned. This is referred to as accrued interest, and it factors into 106
the overall price of the bond. 5 X —— = 2.9444
180
There are several ways to calculate accrued interest. These The 30/360 convention has some interesting features. For example,
methods, called day count conventions, can vary depending on the number of days calculated between February 28 and March 1
the underlying asset. is three, even though (in a non-leap year) only one calendar day has
passed. This means that a bond can accrue three days of interest in
just one calendar day. By the same token, a bond using the 30/360
Bonds convention would earn no interest on the thirty first of January,
Two popular day count conventions for bonds are March, May, July, August, October, and December. (Note that
the 30/360 convention was developed many years ago to simplify
1. Actual/actual, and
the arithmetic associated with day counts.)
2 . 30/360.
The 30/360 convention also applies to mortgage payments. This
Note that both methods have the form X/Y. The first part of the was implicitly assumed when calculating mortgage payments in
day count (X) describes how the number of days during a period Chapter 18. Recall that the accrued monthly interest (and the inter
of interest is determ ined. The second part of the day count (Y) est paid by the mortgage holder) for the examples in Chapter 18
describes how the number of days during some reference did not depend on the number of calendar days in a month.
period is determ ined.
The natural reference period for a bond is the time between

Money Market Instruments
coupon payment dates. In the exam ple given above, the refer
ence period is from March 15 to Septem ber 15. The interest In the case of money market instruments, interest is usually
earned during this period on a bond with a face value of USD 100 expressed per year. Thus, the "Y " in X/Y describes the number
is USD 5 (i.e., half of the annual coupon rate of 10% applied to of days in a year while the "X " describes how the number of1
*
USD 100). W hat we want to know is how much interest accrues
in the period between the last coupon date (March 15) and the 1 For corporate bonds denom inated in euros and British pounds the day
date when the bond is sold (July 1). count is usually actual/actual rather than 30/360.
days in the period over which accrued interest is calculated. For The quoted price is referred to as the clean price. The cash price
Treasury bills in the U .S., the day count convention is actual/360. paid by the purchaser (and received by the seller) is referred to
This means that the quoted interest rate applies to 360 days, as the dirty price. The dirty price is calculated by adding accrued
but the days in any holding period considered is the actual num interest to the clean price:
ber of days.
Dirty Price = Clean Price + Accrued Interest Since the
For exam ple, suppose the interest rate is 6% per year on an Last Coupon Date
actual/360 basis. This means that the interest earned in a full
For exam ple, suppose a trade is settled on May 8, 2019, for a
year of 365 days is
bond paying semi-annual coupons at the rate of 8% per year.*7
3
365 Suppose further that coupons are paid on April 1 and O ctober 1.
— X 6% = 6.0833%
360
As mentioned in the previous section, the actual/actual day count
Similarly, the interest earned in 90 days is 1.5%. convention applies to U.S. Treasury bonds. There are 183 days
between the two coupon dates (April 1 ,2 0 1 9 , and O ctober 1,
Day count conventions vary widely from country to country.
2019) and 37 days between the last coupon payment and the
For exam ple, money market instruments in Australia, Canada,
settlem ent date (April 1, 2019, and May 8, 2019). The payments
and New Zealand are quoted on an actual/365 basis, rather
on each coupon date (per USD 100 of principal) are USD 4 and
than actual/360 as in the U.S. Libor (which was discussed in
so the accrued interest is
Chapter 16) is quoted on an actual/360 basis in all currencies
except the British pound, where it is quoted on an actual/365
4 X ^ - = 0.8087
basis. 183
If the quoted price (clean price) is 105-08 (i.e., USD 105.25), the
dirty price is
19.2 P R IC E Q U O T E S FO R T R EA S U R Y
105.25 + 0.8087 = 106.0587
B O N D S A N D BILLS
The price paid for a bond with a face value of USD 100,000
There are several conventions for quoting the prices of would therefore be USD 106,058.70.
bonds and money m arket instrum ents. This section explains
the conventions used in the U .S. for Treasury bonds and Trea
sury bills. There is additional discussion of U .S. Treasury instru
Treasury Bills
ments in C hapters 9 to 13 of Valuation and Risk M o d els. Recall that interest rates on U.S. Treasury bills are calculated
using an actual/360 convention. However, the interest rate
is also expressed on a discount basis. This means that it is
Bonds
expressed as a percentage of the final value received by the
The bonds issued by the U.S. governm ent with original maturi investor (rather than the initial amount). The final value received
ties of ten years or less are referred to as Treasury notes, whereas by the holder of a Treasury bill is the face value of the bill.
bonds with longer maturities are referred to as Treasury bonds.
For exam ple, consider a 120-day Treasury bill with a face value
However, Treasury notes and bonds are collectively referred to as
of USD 100,000 that is quoted as bid 4.08, ask 4.05. The bid
bonds.
price indicates that interest will be paid at a rate of 4.08% per
U.S. Treasury bond prices are quoted per USD 100 of principal 360 days on USD 100,000. Because there are 120 days to matu
in dollars and 32nds of a dollar. Thus, a price quoted as 105-07 rity, the USD interest earned is
would be interpreted as USD 1 0 5 ^ and would mean that the

^ X 0.0408 X 100,000 = 1,360
360
cost of a bond with a face value of USD 100,000 is USD
105,218.75.2 The USD bid price of the Treasury bill is the face value minus the
interest:
100,000 - 1,360 = 98,640
Some very active issues are quoted to the nearest 1/64. If a plus
sign follows a quoted price it means that 1/64 must be added. Thus 3 The settlem ent date is two days after the trade date, and it is the
7 15 accrued interest to the settlem ent date rather than the trade date that
"1 0 5 — + " should be interpreted as 105— .
32 K 64 is used.
Chapter 19 Interest Rate Futures ■ 243

The Treasury bid price of 4.08 therefore means that the Treasury Table 19.1 Selection of the Treasury Bond and
bill could be sold for USD 98,640. Similarly, the ask quote of 4.05 Treasury Note Futures that Trade on the Chicago
means that the Treasury bill could be bought for USD 98,650. Mercantile Exchange
If Q is the quote and C is the corresponding cash price per
Futures Contract Deliverable
USD 100 of face value:
Two-Year Treasury Any Treasury note with an original
360
Q = ---- (100 - C) Note maturity not greater than five-years
n and three-months. The remaining
maturity must not be less than one-
or
year and nine-months on the first
day of the delivery month and not
C = 100 - -^ -Q greater than two-years on the last
360
day of the delivery month.
In this exam ple, we can use this formula to turn the bid quote
Five-Year Treasury Any Treasury note with an original
into a dollar price by setting n = 120 and Q = 4.08 to get Note maturity not greater than five years
and three-months and a current
120 maturity of not less than four-years
C = 100 - — X 4.08 = 98.64
360 and two-months on the first day of
the delivery month
Note that we have the seem ingly strange situation w here the
bid quote is higher than the ask quote. This is because the bid Ten-Year Treasury Any Treasury note with a remaining
quote is an interest rate and not a price. W hen interest rates Note maturity between 6.5 years and
ten-years
d ecrease, prices increase (and vice versa).
Treasury Bond Any bond with a remaining maturity
between 15 and 25 years
19.3 TREASURY BOND FUTURES Ultra Treasury Bond Any Treasury bond with a remaining
maturity greater than 25 years
Table 19.1 lists some of the Treasury contracts that trade on the
C M E. Delivery can take place at any time during the delivery
For nearly all Treasury bonds/notes,4 one futures contract is for
month. This party with the short position chooses which bond
the delivery of bonds with a face value of USD 100,000. Therefore,
will be delivered and when (during the delivery month) the deliv
the party with a short position in this example delivers bonds with
ery will occur.
a face value of USD 100,000 and receives USD 122,888 in return.
The price received for a delivered bond is determ ined (in part)
To calculate the conversion factor for bonds that are deliverable
by w hat is called the con version factor. Roughly speaking, the
in the last three contracts in Table 19.1:
conversion factor is hypothetical clean price for a bond with
a face value of one dollar given that all interest rates are 6% 1. Calculate the tim e to maturity from the first day of the
with semi-annual com pounding. Thus, the price received for a delivery month to the maturity of the bond,
bond is 2. Round the time to maturity calculated in Step 1 down to the

nearest three months, then
S f+ A
3. Calculate the clean price per one dollar of face value for a
where S is the most recent settlem ent price in the futures con
bond with the maturity calculated in Step 2 and a yield of
tract, f is the bond's conversion factor (as explained below), and
6% per annum (with semi-annual compounding).
A is the accrued interest calculated on an actual/actual basis (as
described earlier in this chapter). For exam ple, consider a Septem ber 2018 Treasury bond futures
contract on a 5% coupon bond maturing on May 15, 2037. The
As an example, suppose the most recent settlement price for a
time to maturity on the first day of the delivery month
Treasury bond futures contract is USD 105.50, the conversion fac
(Septem ber 1, 2018) is 18 years (i.e., Septem ber 1, 2018, to
tor is 1.1542, and the accrued interest on the bond at the time of
Septem ber 1, 2036) and 8.5 months (i.e., Septem ber 1, 2036, to
delivery is USD 1.12 per USD 100 of face value. The cash (USD)
May 15, 2037). This is rounded down to 18 years and six months.
received by the party with a short position (who is delivering the
bond) is
4 In the case of the two-year note futures, one contract is for the delivery
105.50 X 1.1542 + 1.12 = USD 122.888 of USD 200,000 face value.
The price of a bond with a maturity of 18 years and six months5 Table 19.2 Treasury Bond Futures Settlement Prices
with an annual yield of 6%, and a face value of USD 100 is on September 20, 2018
100 Contract Settlement Price
88.92
1.0337
Two-Year Treasury Note 105-105
This amount represents both the clean and the dirty price
Five-Year Treasury Note 112-117
(because there is no accrued interest), and the conversion factor
is therefore 0.8892. Ten-Year Treasury Note 118-180
Now consider the contract for D ecem ber 2018. A 5% coupon Treasury Bond 139-29
bond maturing on May 15, 2037, is also deliverable in this con Ultra Treasury Bond 153-08
tract. In this case, the time to maturity (measured from the first S o u rce: w w w .cm egroup.com
day of the delivery month) is 18 years and 5.5 months. This is
rounded down to 18 years and three months. The dirty price of Septem ber 20, 2018. The two-year settlem ent price (105-105)
the bond just b efo re the next coupon is received is should be interpreted as:
100
91.5839 1 0 5 ^ - = 105.328125
1.0336 32
when the yield is 6%. While this bond has 18 years to maturity, the M eanwhile, the five-year futures price (112-117) should be inter
dirty price when the bond is 18 years and three months from preted as:
maturity can be calculated by discounting this price for a further 11 75
three months. The discount factor is 1.014889 (i.e., V 1.03) so that 112— — = 112.367188
32
the dirty price is USD 90.24 (= 91.5839/1.014889). After subtract
Futures prices increase with the maturity of the contract. This
ing the accrued interest of USD 1.25, the clean price of the bond
can be understood by noting that the conversion factor system
is USD 88.99. The conversion factor is therefore 0.8899.6
means that the underlying asset is a bond with a coupon of 6%
The conversion factors for bonds delivered in the two- and five- per year (because that bond has a conversion factor of 1.0000).
year note futures contracts are calculated similarly except that the However, interest rates for all maturities were much less than
time to maturity is rounded down in Step 2 to the nearest month.7 6% on Septem ber 20, 2018. When this is the case, the value of
a bond paying a coupon of 6% per year increases with maturity
(because the higher 6% coupon is received for a longer period).
Quotes
Bond futures prices are quoted similarly to bond prices.
Cheapest-to-Deliver Bond Option
However, the shorter-maturity contracts are quoted to the
nearest 1/128 rather than to the nearest 1/32. Specifically, The cost of delivering a bond or note is
a 5 in the third decimal place indicates 0.5/32, a 7 in the M arket Price — Price Received
third decimal place indicates 0.75/32, and a 2 in the third
decimal place indicates 0.25/32. Table 19.2 shows the settle Not surprisingly, the party with a short position will choose the
bond for which this cost is least. This bond is referred to as the
ment prices for some bond futures contracts in Table 19.1 on
cheapest-to-deliver bond. As m entioned, the price received for
a bond is
S f+ A
5 This is assumed to be im m ediately after a coupon paym ent so that where S is the most recent settlem ent price in the futures con
there is no accrued interest.
tract, f is the conversion factor, and A is the accrued interest.
6 Both conversion factors we have calculated correspond exactly with The cash market price of the bond is
those published by the C M E G roup.
7 Suppose for exam ple that, when the tim e to maturity is rounded down Q + A
to the nearest month, there are three years and eight months to matu
where Q is the quoted price. The cheapest-to-deliver bond is
rity. We calculate the value of a 6% coupon bond with three years and
six months to maturity just before the paym ent of the coupon. We then therefore the one for which:
divide by 1/1.032/6 to add an extra two months to the life. Four months
Q - Sf
of accrued interest are then subtracted to move from the dirty price to
the clean price. is the least.

Suppose three bonds that can be delivered are those shown in asset providing known income. As explained in Chapter 10, the
Table 19.3 and that the most recent futures settlem ent price is futures price is
USD 115.75. As indicated by the calculations in the table, the
(s - /xi + r )t
first bond is the cheapest-to-deliver.
where / is the present value of the coupons that will be received
Several factors can determ ine which bond is the cheapest to
during the life of the futures contract, S is the current price of
deliver. For exam ple, if bond yields are greater than 6%, low-
the bond, T is the life of the futures contract, and R is the (annu
coupon long-maturity bonds will tend to be the cheapest to
ally com pounded) risk-free rate for maturity T. As explained in
deliver. If yields are less than 6%, however, high-coupon short-
Chapter 16, if R is expressed with continuous com pounding, the
maturity bonds will tend to be the cheapest to deliver. An
futures price becomes
upward-sloping yield curve tends to favor long-maturity bonds,
whereas a downward-sloping yield curve tends to favor short- (S - l)eRT (19.1)
maturity bonds. The current price (S) used these equations must be the dirty
The party with the short position chooses when to deliver. This cash price of the bond. (Thus, the calculated futures price is
gives rise to an option known as the wild card play. Note that also a dirty price). The steps in calculating the futures price are
the settlem ent price is the price at which the futures contract therefore as follows.
trades at 2 p.m . Chicago tim e. However, a notice of intention to 1. Calculate the current dirty price of the delivered bond from
deliver can be issued later in the day. The party with the short the quoted clean price by adding accrued interest.
position can therefore attem pt to wait for a day when bond
2. Calculate the present value of coupons that will be received
prices decline after 2 p.m . and therefore reduce the delivery
between the current time and the time when the bond will
cost. For exam ple, the bond can be bought at the 3:30 p.m.
be delivered.
price and sold at the 2 p.m . futures price (adjusted by the con
version factor). 3 . Obtain a dirty futures price by subtracting the amount in
Step 2 from the amount in Step 1 and compounding for
The wild card play and the ability to use the cheapest-to-deliver
ward at the risk-free rate to the time when the bond will be
bond make the futures contract more attractive to the party with
delivered.
the short position (because they provide that party with a high
degree of choice). Additionally, the party with the short position 4. Convert the bond's dirty futures price to a clean futures
can choose to deliver on any day during the delivery month. price by subtracting the accrued interest at the time of the
Together, these features tend to reduce the futures price; as delivery.
the contract becom es more attractive to the party with the 5. The clean futures price for the delivered bond equals
short position, that party becom es more prepared to accept a the quoted futures price m ultiplied by the conversion
lower price. factor. The quoted futures price is therefore calculated
as the value in Step 4 divided by the bond's conversion
factor.
Calculating the Futures Price
Suppose that it is known that a bond will be delivered in 250 days
The fact that the party with the short position has so many deliv under the terms of a futures contract. The timing of coupon pay
ery choices makes it difficult to determ ine futures prices (such as ments is shown in Figure 19.1. The last coupon on the bond was
those in Table 19.1). However, assuming that both the cheapest- paid 50 days ago, and the next coupon will be paid in 133 days.
to-deliver bond and the bond delivery time are known, the The coupon after the next one will be paid in 315 days (i.e.,
Treasury bond futures contract is simply a futures contract on an 65 days after delivery).
Table 19.3 Calculation of Cost of Delivery for Three Bonds When the Futures Price is 115.75
Bond Quoted (Clean) Bond Price (in USD) Conversion Factor Cost of Delivery (in USD)
1 98.75 0.8384 98.75 - 115.75 X 0.8384 = 1.7052
2 117.25 0.9874 117.25 - 115.75 X 0.9874 = 2.9585
3 145.75 1.2325 145.75 - 115.75 X 1.2325 = 3.0881
Coupon Today Coupon Delivery Date Coupon daily settlem ent takes place in the usual
m anner during the life of the contract
50 133 117
Days and is determ ined by the price at which
Days Days
the contract is trading at the end of the
Fiqure 19.1 Timeline for example
trading day. A final settlem ent price is
used to determ ine final transfers betw een
Assum e that the risk-free interest rate for all maturities is 5% those with long and short positions. All contracts then cease
with continuous compounding and that the delivered bond pays to exist.
a coupon of 7% semi-annually with a current quoted (clean)
The final settlem ent price for Eurodollar futures is USD 100 — R,
price of USD 108.00 and a conversion factor of 1.0400. The
where R is the Libor fixing for 90-day USD borrowings. For exam
cash (dirty) price is calculated by noting that the next coupon
ple, if the USD 90-day Libor fixing is 2.3% (using the actual/360
will be USD 3.5 (per USD 100 face value) and that that 50 of the
convention), the final settlem ent price of the corresponding
183 days between coupon payments have passed. It is therefore:
Eurodollar futures contract would be USD 97.70 (= 100 — 2.30).
108.00 + X 3.5 = 108.9563 The contract is designed so that a 1-basis point (0.01) move in
183
the futures price leads to a gain (or loss) of USD 25. Table 19.4
This is the variable S in Equation (19.1). The present value illustrates this by showing a situation where a long position
of the coupon that will be received after 133 days (133/365 in one contract is taken on Day 1 and final settlem ent is on
= 0.3644 years) is Day 5. During Day 1, the price increases by half a basis point
(0.005) for a gain of USD 12.50. Betw een Day 1 and Day 2, the
3.5e~005 x 0-3644 = 3.43 68
settlem ent price of the contract increases by 15 basis points
This is the variable / in Equation (19.1). (0.15) so that there is a gain of USD 375 (= 15 X USD 25).
The time to delivery for the futures contract is 250 days (i.e., The total gain over the five days is USD 500 (i.e ., the future
250/365 = 0.6849 years). From Equation (19.1), the cash (dirty) price increased by 20 basis points from USD 97.500 to
futures price is USD 97.700).
(108.9563 - 3.4368)e005 x 0-6849 = 109.1957 Note that increases in the futures price correspond to decreases
in interest rates (and vice versa). We can refer to 100 minus
To obtain the clean futures price, we must subtract the accrued
the quote as the futures interest rate. From Day 1 to Day 2,
interest. On the delivery date, 117 of the 182 days between pay
the settlem ent futures interest rate decreased from 2.495% to
ments will have passed and so the clean futures price is
2.345% and the futures price increased correspondingly from
USD 97.505 to USD 97.655.
109.5625 - ^ X 3.5 = 106.9457
182
Dividing by the conversion factor, we obtain an estim ated price

of the futures contract:
Table 19.4 The Results from Taking a Long Position
106.9457 in One Contract are Shown. Day 5 is the Final
102.83
1.0400
Settlement, Two Days Before the Third Wednesday of
the Contract Month
19.4 EURO DO LLAR FUTURES Settlement

Day Trade Price Price Change Gain (USD)
Eurodollar futures contracts have historically been popular con 1 97.500
tracts traded by the C M E Group. The exchange trades similarly
1 97.505 +0.005 + 12.50
designed one-month and three-month contracts. The discussion
in this section will center on the three-month contract, which 2 97.655 + 0.150 + 375.00
is often just referred to as the Eurodollar futures contract. This 3 97.715 + 0.060 + 150.00
contract is settled in cash two days before the third W ednesday
4 97.665 - 0 .0 5 0 -1 2 5 .0 0
of the contract month.
5 97.700 +0.035 + 87.50
Recall that C h ap ter 7 d escrib es the daily futures settlem ent
Total + 0.200 + 500.00
procedures. In the case of the Eurodollar futures contract,

A Eurodollar futures contract provides an approxim ate hedge Table 19.5 Settlement Prices for the Three-Month
for the interest payment on USD 1 million for three m onths.8 Eurodollar Futures Contracts on September 20, 2018
To see this, note that when the interest rate per year changes
by 1 basis point, the interest rate per three months changes by
Contract Month Settlement Price
one quarter of a basis point (i.e., 0.0025% ). On a principal of O ctober 2018 97.565
USD 1 million, the change in the interest paid is
Novem ber 2018 97.460
0.000025 X USD 1,000,000 = USD 25
D ecem ber 2018 97.350
January 2019 97.300

Quotes February 2019 97.225
Table 19.5 shows the settlem en t prices for all available three- March 2019 97.170
month Eurodollar futures contracts on Sep tem b er 20, 2018.
June 2019 97.005
A s ind icated , contracts trad e with every m aturity month for
the first few m onths. Th e m aturities after that are in M arch,
Ju n e , Septem ber, and D ecem ber. It can be seen that (with D ecem ber 2019 96.845
a few excep tio n s) settlem en t prices decline with m aturity. March 2020 96.825
This is consistent with a yield curve that is m ostly upward-
June 2020 96.825
sloping. The futures rate for 90 days beginning in the first
month (O cto b e r 2018) is 2.435% (= 100 — 97.565), w hile for Septem ber 2020 96.835
the D ecem b er 2024 co n tract it is 3.285% (= 100 — 96.715). D ecem ber 2020 96.830
C o ntracts are available with m aturities out to ten years,
March 2021 96.845
but the latest m aturity trad ed on Sep tem b er 20, 2018, was
D ecem b er 2024. June 2021 96.855
Comparison with FRAs D ecem ber 2021 96.850
March 2022 96.860

Forward rate agreem ents (FRAs) were introduced in Chapter 16.
There is an important difference between Eurodollar futures and June 2022 96.860
FRAs in the timing of the interest payments. Septem ber 2022 96.850
For example, suppose that the Libor interest rate on O ctober 15, D ecem ber 2022 96.835
2019 (two days before the third W ednesday of the month) is 2.3%. March 2023 96.830
Then, the interest for three months on USD 1,000,000 would be
June 2023 96.820
0.25 X 0.023 X USD 1,000,000 = USD 5,750
In an FRA, this interest is paid at the end of the three-month D ecem ber 2023 96.780
period (i.e., m id-January 2020).9 However, the Eurodollar futures
March 2024 96.770
contract provides a settlem ent at the beginning of the three-
month period (i.e., m id-October 2019). June 2024 96.755
Another difference between Eurodollar futures and FRAs is that Septem ber 2024 96.730
Eurodollar futures are (like all futures contracts) settled daily, D ecem ber 2024 96.715
whereas FRAs are settled only once at the end.
S o u rce: w w w .cm egroup.com
8 As we will discuss later, one reason why the hedge is approxim ate An FRA (by construction) provides a correct estim ate of the for
concerns the timing of the settlem ent in Eurodollar futures. ward rate (see Chapter 16). In order to estim ate the forward rate
9 As mentioned in C hapter 16, FRAs usually provide payoffs at the from Eurodollar futures quotes, analysts make what is referred
beginning of the three-month period. However, the payoff is calculated
to as a convexity adjustm ent. The adjustm ent is very small for
from the present value of the interest payable at the end of the period
(i.e., the FRA is econom ically equivalent to an instrum ent where interest contracts lasting only one or two years, but it becom es too large
is paid at the end of the period.) to ignore as for longer maturities.
This adjustm ent depends on the interest rate model that is where F is the forward rate for the period between T and T2l R
-1 1
assum ed. An adjustm ent based on the Ho and Lee interest rate is the zero-coupon interest rate for a maturity of T1; and R2 is the
model involves reducing the futures rate estim ate (when zero-coupon interest rate for a maturity of T2.
expressed in decimal form) by 0 .5 a 2T ( J + 0.25) to get a forward
This formula can be rearranged to give
rate o f:10
Future Rate - 0 .5 ct 2T(T + 0.25) (19.2) F(T2- T , ) + R1T1

(19.3)
T2
where a is the standard deviation of changes in the short-term
interest rate over one year, T is the maturity of the Eurodollar This m eans that if w e know the zero-coupon interest rate
futures as measured in years (so that the underlying rate lasts for a m aturity of T and the forw ard rate betw een T| and
-1
from T years to T + 0.25 years),*11 and the futures rate is T2, we can deduce the zero-coupon interest rate for a
100 — Q (where Q is the futures quote). When this formula is m aturity of T2.
applied, both the forward rate and futures rate should be
For exam ple, if the two-year zero rate is 4% and the forward
expressed using the actual/actual convention with continuous
rate for the period between two and 2.5 years is 4.4% , we can
com pounding.
deduce that the 2.5-year zero rate is
For exam ple, suppose the Eurodollar futures quote is USD
96.200, the standard deviation of the change in the one-month 0.044 X 0.5 + 0.04 X 2
0.0408
rate (a proxy for the short rate) in one year is estimated as 1.1%, 2.5
and the time to maturity is four years. Then Q = 96.2, er = 0.011,
or 4.08% .
and T = 4. The futures rate using actual/360 with quarterly
compounding is USD 3.8% (= 100 — 96.2). This is equal to Equation (19.2) can be used to extend the Libor curve using
3.853% ( = (365/360) X 3.8%) on an actual/actual basis with Eurodollar futures contracts.
quarterly compounding. Converting it to continuous com pound Note that a Eurodollar futures contract provides an estim ate
ing, the rate becomes of the futures rate for a 90-day period beginning at the end of
4 X ln(1 + 0.03853/4) = 0.03834 the contract's life. As described earlier, it can be converted to
an actual/actual continuously com pounded forward rate using a
The forward rate with continuous compounding is therefore: convexity adjustm ent.
0.03834 - 0.5 X 0 .0 1 12 X 4 X 4.25 = 0.0373 Contracts maturing in March, Ju n e, Septem ber, and D ecem ber
or 3.73% . are actively traded for several future years. The tim e between
contract maturities is usually 91 days (and occasionally 98 days).
The procedure is to assume that the forward rate estim ated for
The Libor Zero Curve a 90-day period (starting at the end of futures contract's life)
The Libor fixings (described in Chapter 16) are for maturities of is the forward rate for the 91 (or 98 days) between the maturi
one year or less. However, the Libor zero curve can be extended ties of Eurodollar futures. This allows the Libor zero curve to be
beyond one year using Eurodollar futures. (Another way to bootstrapped.
extend the Libor curve is to use swaps, which will be discussed Suppose, for exam ple, that it is April 15, 2019. Eurodollar
in the next chapter.) futures mature on:
As shown in Chapter 16, the relationship between forward rates June 17, 2019,
and spot rates when all interest rates are expressed with con
Septem ber 16, 2019,
tinuous compounding is
D ecem ber 16, 2019,
_ R2T2 ~ F-|T-|
t 2- t , March 16, 2020,
June 15, 2020,
Septem ber 14, 2020, and

10 This is based on the Ho and Lee model of interest rates. See T. S. Y.
Ho and S. Lee, "Term Structure M ovem ents and Pricing Interest Rate
D ecem ber 14, 2020.
Contingent C laim s," Journ al o f Finance, 41 (D ecem ber 1986): 1011-29.
11 The short rate is theoretically a rate for an infinitesim ally short period Note that the zero rate for March 16, 2020 (the 11-month rate)
of tim e. The one-month rate is a reasonable proxy. can be estim ated using interpolation from the Libor fixings on

April 15, 2 0 1 9 .12 First, the March 2020 Eurodollar futures can SOFR Futures
be used to estim ate the 90-day forward rate for a 90-day
period starting on March 16, 2020. This is assumed to apply to As discussed in Chapter 16, there are plans to replace LIBOR with
the 91-day period between March 16, 2020, and Ju n e 15, 2020, a benchmark rate that is based on actual trades rather than bank
and Equation (19.2) is used to estim ate the zero rate for June estimates. The U.S. plans to use the Secured Overnight Financing
15, 2020. The June 2020 forward contract is then used to pro Rate (SOFR), which is based on the rates in repo transactions (see
vide the forward rate for the 90-day period starting on Ju n e 15, Chapter 16).
2020. This is assumed to apply to the 91-day period between The C M E Group trades one-month and three-m onth contracts
Ju n e 15, 2020, and Septem ber 14, 2020, and Equation (19.2) is on S O FR . The three-m onth contract has a final settlem ent
used to estim ate the zero rate for Septem ber 14, 2020. This equal to 100 — R (where R is the com pounded interest rate
process is continued until the desired month is reached. that would have been earned over the previous three months
if funds w ere invested day-by-day at the overnight repo rate).
Hedging Assum ing that the overnight rate on the ith business day of the
three-m onth period is r(- and that the rate applies to d, days,
Suppose an investor wants to use Eurodollar futures to lock in then:
the interest rate on a three-month investment of USD 70 million.
The Eurodollar futures price is USD 95.300. This indicates R = (1 + c/1r1)( 1 + d2r2) • • • (1 + dnrn) - 1
that it should be possible to lock in a rate of approxim ately On most days, d, = 1. However, weekends and holidays
4.7% (= 100 — 95.3). Because one contract provides rate pro can lead to the overnight rates being applied to more than
tection for an investm ent of USD 1 million, a long position in one day. For exam ple, on Fridays d; is normally equal
70 contracts is required (recall that a long futures position gains to three.
when interest rates decline).
The three-month contract is structured like the Eurodollar
If the interest rate at maturity of the futures contract proves to futures contract so that a 1-basis point movement in the quote
be 3.5% , the final settlem ent will be USD 96.500 and the gain leads to a gain or loss equal to USD 25 and one contract can
will be USD 25 per basis point per contract. The basis point hedge a USD 1 million position for three months.
change is 120 (= 9,650 — 9,530) and thus the total gain is
M eanwhile, the one-month SO FR futures contract is structured
USD 25 X 120 X 70 = USD 210,000 like the three-month contract except that a 1-basis point
move leads to a gain or loss of USD 41.67 and the contract is
The interest earned at 3.5% is
designed to hedge a USD 5 million position for one month.
USD 70,000,000 X 0.25 X 0.035 = USD 612,500
The gain on the futures position brings this up to USD

822,500 (= USD 612,500 + USD 210,000), which is the inter 19.5 DURATION-BASED HEDGING
est at a rate of 4.7% on USD 70 million for three months (i.e.,
70,000,000 X 0.25 X 0.047 = 822,500). As discussed earlier, Recall the duration measure introduced in Chapter 16. If D is the
however, the futures settlem ent is at the end contract (not three modified duration of a bond portfolio worth B and A y is a small
months later) and so the hedge is im perfect. parallel shift in the yield curve, the change in the value of the
portfolio is approxim ately:
There is a small adjustm ent that can be made to consider
this timing difference. This involves discounting the exposure -B D A y
from the interest date to the end of the Eurodollar futures Suppose that a futures contract is used to hedge an exposure
contract's life (i.e., the tim e when the interest is determ ined). when the maturity of the hedge instrument does not match the
In the previous exam ple, the exposure is multiplied by maturity of the instrument underlying the futures contract. If it
0.9884 (= 1/(1 + 0.25 X 0.047)). This suggests that the number is assumed that only small parallel shifts in the yield curve will
of contracts should be 69.2 (= 0.9884 X 70). Rounding, we find occur, duration can be used to calculate an appropriate hedge
that 69 contracts should be bought. ratio. Defining term s:
Ep: The increase in value of one futures contract for a 1-basis

12 For exam ple, if the six-month LIBO R rate is 3% and the 12-month point downward parallel shift in the zero curve, and
LIBO R rate is 4% , the 11-month LIBO R rate would be
3 X (12 - 11) + 3.6 X (11 - 6) E v: The increase in value of a trader's position for a 1-basis
6 point downward parallel shift in the zero curve.
The number of long contracts that should be traded (with downward shift of A y in the yield curve, the futures price can be
a negative number indicating a short position) for a hedge expected to change by approxim ately:14
should be
FD A y
The impact of a 1-basis point downward parallel shift in the zero

curve on the futures price in our exam ple is therefore:
The gain (loss) on the futures position will then be offset by the
E F = 103,250 X 16 X 0.0001 = 165.2
loss (gain) on the trader's portfolio for a small parallel shift in the
interest rate term structure. Equation (19.3) gives the number of contracts required as:
For exam ple, suppose that a trader has a USD 2 million position 6,000
- 3 6 .3
in a nine-month money market instrument. The duration of 165.2
the instrument is 0.75 because it is a zero-coupon instrument. Rounding to the nearest whole number, 36 contracts should be
From the duration relationship, the trader's gain (in USD) from shorted.
a 1-basis point downward parallel shift in the zero curve is It should be em phasized that the duration-based hedges we
approxim ately: have considered in this section only provide protection against
small parallel shifts in the interest term structure. The hedges
E v = 2,000,000 X 0.75 X 0.0001 = 150
also assume that the interest rate to which the hedger is
The three-month Eurodollar futures contract is designed exposed is perfectly correlated to the interest rate underlying
so that a 1-basis point downward parallel shift in the yield the futures contract and (in the case of hedging using bond
curve gives rise to a gain of USD 25. This means that Ep = 25. futures) that the cheapest-to-deliver is known with certainty.
Equation (19.3) gives the number of contracts required as:
150 SUMMARY
25
A short position in six contracts will therefore provide an Eurodollar futures contracts provide payoffs dependent on
approxim ate h ed g e .13 movements in short-term interest rates. M eanwhile, Treasury
note and bond futures contracts provide payoffs dependent on
As a second exam ple, suppose that a fund manager has USD 5
movements in longer-term rates. Final settlem ent for Eurodollar
million position in bonds with a duration of 12. Assum e that the
futures contracts is in cash and happens on the Monday before
current Treasury bond futures price is USD 103.25 and that the
the third W ednesday of the delivery month. The payout equals
cheapest to deliver bond in a contract has an estim ated duration
USD 100 minus the Libor fixing on that day.
of 16 at maturity. The duration relationship estim ates the impact
of a 1-basis point downward parallel shift in the yield curve on In contrast, Treasury note/bond futures are settled by delivering
the fund manager's position as: a particular bond/note. The party with the short position can
choose which bond/note to deliver and decide when the deliv
E v = 5,000,000 X 12 X 0.0001 = 6,000
ery will take place. A conversion factor determ ines the price
Because one futures contract involves the delivery of bonds with received by the party with the short position from the party with
a face value of USD 100,000, value of one futures contract is a long position for the delivered instrument.
therefore USD 103,250. The futures rate underlying a Eurodollar futures contract for a
Assum e that the bond that will be delivered is known and that it specific future period can be compared with the forward rate
will have a duration of D at maturity. When there is a parallel underlying a forward rate agreement (FRA) for the same period.
However, the two instruments have important differences. One is
that the futures contract is settled daily, whereas the forward con
tract is settled once at the end. The other difference is that the
13 There are a num ber of approxim ations. 0.75 is not a modified dura futures contract settles at the beginning of the period to which
tion. It will therefore be evident from the discussion in C hapter 16 that the interest rate applies, whereas the forward rate agreement
the E v calculation is theoretically correct only when we are considering
recognizes that interest is paid at the end of the period. These
continuously com pounded yields. By contrast, the Ep assum es quarterly
com pounding. A lso , we do not consider the point discussed earlier that
futures are settled at the beginning of the underlying period whereas 14 Because there will be a proportional increase (D Ay) in the bond that
interest is paid at the end of a period. Finally, we do not consider the will be delivered, we can exp ect the sam e proportional increase in the
im pact of differences in day count conventions. futures price.

two differences mean that a convexity adjustment is necessary to When futures contracts are used for hedging, the duration of
estimate forward interest rates using Eurodollar futures rates. the hedge portfolio may not be the same as the duration of the
assets underlying the futures contract used for hedging. How
In the U .S., it is expected that the Libor benchm ark will be
ever, the number of futures contracts used for hedging can be
replaced by a one-day repo rate called the Secured O vernight
adjusted by multiplying the ratio of the duration for the portfo
Funding Rate (SO FR). The C M E Group trades one-month and
lio being hedged by the duration for the assets underlying the
three-month SO FR contracts that would be achieved by invest
futures contract.
ing day-by-day at this overnight rate.
Financial Risk Manager Exam Part I: Financial Markets and Products

Q U E S T IO N S

19.1 W hat is the day convention for (a) Treasury bonds, 19.6 How much does som eone with a short contract gain
(b) corporate bonds, and (c) Treasury bills in the U .S.? or lose when the three-month Eurodollar futures price
changes from 94.555 to 94.715?
19.2 W hat is the difference between the clean price and the
dirty price of a bond? 19.7 How is the three-month Eurodollar futures contract set
tled, and when is it settled?
19.3 How is a Treasury note defined in the U .S.?
19.8 G ive two reasons for differences between a Eurodollar
19.4 In an upward-sloping yield curve environment where
interest rates are above 6%, what types of bonds tend to futures rate and the corresponding forward rate in an
FRA.
be cheapest to deliver?
19.5 How should a quote of 103-132 for a Treasury bond 19.9 W hat is SO FR ? How is the three-month futures contract
futures be interpreted? on SO FR settled?
19.10 W hat protection is obtained when a Treasury bond

futures contract is used to hedge a bond portfolio using
duration analysis? W hat assumptions are necessary?
Practice Questions
19.11 A Treasury bond pays coupons at the rate of 7% per year 19.17 The Eurodollar futures price for a contract that matures
on June 1 and D ecem ber 1. W hat is the accrued interest in three years is 95.75. The standard deviation of the
between June 1 and Ju ly 31 per USD 100 of face value? change in the short rate in one year is 0.8% . Estim ate the
continuously com pounded forward rate between three
19.12 W hat is the answer to Question 19.11 if the bond is a
and 3.25 years.
corporate bond, rather than a Treasury bond?
19.13 W hat is the cash price of the Treasury bill that lasts for 19.18 If in Question 19.17 the continuously com pounded three-
year zero rate is 4.12% , what is the continuously com
200 days and has a quoted price of 4.12?
pounded 3.25-year rate?
19.14 A bond that can be delivered in the D ecem ber 2018
ten-year Treasury note futures contract is a bond with
19.19 Approxim ately how many three-month Eurodollar futures
maturity on April 15, 2026, that pays a coupon of 4% per contracts are necessary to hedge the six-month inter
est that will be paid on a USD 20 million bond? Assum e
annum. Calculate the conversion factor for the bond.
that the six-month period starts at the maturity of the
19.15 A futures price is 115.00. Three bonds that can be
futures contract that will be used. (Ignore the differences
delivered have quoted prices of 98-125, 103-127, and
between Eurodollar futures and FRAs mentioned in the
120-230. The conversion factors of the bonds are 0.8456,
chapter for this question.)
0.8844, and 1.0267. W hat is the cost of delivering each
19.20 It is March 10, 2019. The cheapest-to-deliver bond in
bond? Which is the cheapest to deliver?
the D ecem ber 2019 Treasury bond futures contract is
19.16 Assum e that the bond that will be cheapest to deliver
expected to be a bond with a duration of 12.5 at that
in a Treasury bond futures contract pays semi-annual
tim e. The current futures price is 104-127. A bond
coupons at the rate of 10% per annum on May 1 and
portfolio has a duration of 15 and is worth USD 30 million.
Novem ber 1 and will be delivered on Septem ber 1. The
How many bond futures contracts are necessary to
bond's quoted price on August 1 is 130.00 and its con
hedge the risk in the portfolio?
version factor is 1.2341. Estim ate the futures price on
August 1 assuming that all interest rates are 4% (continu
ously com pounded).

A N S W ER S
19.1 (a) actual/actual, (b) 30/360, (c) actual/360 and Decem ber, respectively). The number of days
between June 1 and Ju ly 31 is 59 (29, 30 in June and
19.2 The clean price is the quoted price. The dirty price is the
cash price that is paid. The dirty price equals the clean Ju ly, respectively). The accrued interest is therefore:
price plus accrued interest. 59

3.5 X — = 1.1472
180
19.3 A Treasury note is a bond with an original maturity less
than or equal to ten years. 19.13 The cash price per USD 100 of face value is
19.4 Low coupon long-maturity bonds will tend to be cheap 200
100 - X 4.12 = 97.7111
est to deliver. 360
_ „ 13.25 m 19.14 The bond's tim e to maturity on the first day of the deliv
19.5 103— — = 103.414063
32 ery months is seven years (Decem ber 2018 to Decem ber
19.6 The change is an increase of 16 basis points (94.715 — 2025) and 4.5 months (January 2026 to mid-April 2026).
94.555 = 0.16). A party with a short contract loses This is rounded to seven years and three months. The
16 X USD 25 = USD 400. dirty price of a seven year and three-month bond im m e
diately before the coupon payable in three months is
19.7 The contract is settled two days before the third
14
W ednesday of the delivery month. The final settlem ent is
9
100
Y — + = 90.7039
100 — R where R is the three-month USD Libor fixing for n o 1.03' 1.03 14
the day. when the yield is 6%. The dirty price of the bond three
19.8 The futures contract is settled daily, and final settlem ent months earlier is
is at the beginning of the period covered by the interest 90.7039
89.3732
rate, not at the end. V T03
19.9 SO FR is the overnight repo rate. The three-month futures Subtracting the accrued interest of 1, we get a clean
is settled by calculating the rate that would have been price of 88.3732 and the conversion factor is 0.8837.
earned over the previous three months by rolling an
19.15 The prices of the three bonds are
investm ent forward day-by-day at the SO FR rate.
12 5 12 75
9 8 ^ — = 98.3906, 1 0 3 - —— = 103.3984,
19.10 The hedge protects against small parallel shifts in the 32 32
zero curve. The following assumptions must be made: 23
and 120— = 120.7187
the cheapest-to-deliver bond is known and movements 32
in the rates to which the portfolio is exposed are very The costs of delivering the bonds are
similar to movements in the corresponding Treasury
98.3906 - 0.8456 X 115 = 1.1466,
rates.
103.3984 - 0.8844 X 115 = 1.6924, and
19.11 The number of days between June 1 and D ecem ber 1
is 183 (29, 31, 31, 30, 31, 30, 1 in Ju n e, Ju ly, August, 120.7187 - 1.0267 X 115 = 2.6482.
Septem ber, October, November, and Decem ber, respec The first bond is the cheapest to deliver.
tively). The number of days between June 1 and Ju ly 31
19.16 There are 92 days between May 1 and August 1 (30, 30,
is 60 (29, 31 in June and July, respectively). The accrued
31, and 1 in May, Ju n e, July, and August, respectively)
interest is therefore:
and 184 days between May 1 and Novem ber 1 (30, 30,
60 31, 31, 30, 31, 1 in May, Ju n e, Ju ly, August, Septem ber,
3.5 X — — = 1.1475
183 October, and November, respectively). The dirty price of
19.12 Using a 30/360 day count, the number of days between the bond is therefore:
June 1 and D ecem ber 1 is 180 (29, 30, 30, 30, 30, 30, 1
92
in Ju n e, Ju ly, August, Septem ber, October, November, 130 + 5 X 132.5
184
No coupons will be paid in the 31-day period between 19.18 Using Equation (19.17) we get
August 1 and Septem ber 1. The time to delivery is
0.04255 X 0.25 + 0.0412 X 3
31/365 = 0.0849 years. The dirty futures price is 0.0413
3.25
therefore:
or 4.13% .
132.5 e0 0849x0 04 = 132.95 09
19.19 The change in the value of the instrument for a 1-basis
The accrued interest on Septem ber 1 is 5 X 61/184
point parallel shift in the interest rate is
= 3.3423. The clean futures price is therefore:
USD 20,000,000 X 0.5 X 0.0001 = USD 1,000
132.9509 - 3.3423 = 129.6086
This is 40 tim es USD 25. It follows that 40 contracts
Dividing by the conversion factor we obtain the
should be shorted.
estim ated futures price as:
19.20 The futures price of the bond is
129.6086
105.0227 12 75
1.2341 104— — = 104.3984
32
19.17 The actual/360 futures rate is 100 — 95.75 = 4.25. This
and the value of one futures contract is USD 104,398.4. The
is 4.25 X 365/360 = 4.3090% on an actual/actual basis.
number of contracts necessary to hedge the portfolio is
This rate is com pounded quarterly. The rate with continu
ous compounding is 4 X ln(1 + 0.043090/4) = 0.042860 30,000,000 X 15
— '------------------ = 344.8
or 4.2860% . The convexity adjustm ent is 104,398.4 X 12.5
0.5 X 0.0082 X 3 X 3.25 = 0.000312 or 345 when rounded to the nearest whole number.
An estim ate of the continuously com pounded forward

rate is therefore:
0.042860 - 0.000312 = 0.042548
or 4.255% .
Chapter 19 Interest Rate Futures

Swaps
Learning Objectives
Explain the mechanics of a plain vanilla interest rate swap Calculate the value of a plain vanilla interest rate swap
and com pute its cash flows. from a sequence of forward rate agreem ents (FRAs).
Explain how a plain vanilla interest rate swap can be used Explain the mechanics of a currency swap and com pute its
to transform an asset or a liability and calculate the result cash flows.
ing cash flows.
Explain how a currency swap can be used to transform an
Explain the role of financial interm ediaries in the swaps asset or liability and calculate the resulting cash flows.
market.
Calculate the value of a currency swap based on two
Describe the role of the confirmation in a swap simultaneous bond positions.
transaction.
Calculate the value of a currency swap based on a
Describe the com parative advantage argument for the sequence of forward exchange rates.
existence of interest rate swaps and evaluate some of the
criticisms of this argument. Identify and describe other types of swaps, including com
modity, volatility, credit default, and exotic swaps.
Explain how the discount rates in a plain vanilla interest
rate swap are com puted. Describe the credit risk exposure in a swap position.
Calculate the value of a plain vanilla interest rate swap

based on two simultaneous bond positions.
Swaps are over-the-counter (O TC) derivatives contracts where rate of 2.24% is observed at time zero. This leads to a floating
the parties agree to exchange certain cash flows in the future. paym ent at tim e 0.25 years of:
These exchanges of cash flows generally depend in part on the
0.0224 X 0.25 X USD 10,000,000 = USD 56,000
future values of variables such as interest rates, exchange rates,
equity prices, and com m odity p rices.1 As a result, there is some The fixed payment received is at a rate of 3% per year (com
uncertainty associated with swaps. pounded quarterly):
Forward contracts can be treated as swaps where there will be 0.03 X 0.25 X USD 10,000,000 = USD 75,000
a cash-flow exchange on just one future date. However, swaps When these interest payments are netted, there is a net cash
often feature exchanges on many future dates. Furtherm ore, inflow at time 0.25 years of:
while the value of a swap is normally zero (or very close to
USD 75,000 - USD 56,000 = USD 19,000
zero) when it is initiated, the value of each exchange made in
the swap is typically not zero. It is usually the case that some A t time 0.25 years, the three-month Libor rate is 2.32% . This
exchanges have positive values while others have negative val determ ines the net cash flow one period later (i.e., at time
ues at the time the swap is initiated. 0.5 years). The floating payment to be made is
This chapter exam ines interest rate and currency swaps in some 0.0232 X 0.25 X USD 10,000,000 = USD 58,000
detail, and then briefly introduces some of the many other types
The fixed payment that will be received is USD 75,000. As
of swaps that are traded. Statistics produced by the Bank for
before, the net cash flow can be calculated as:
International Settlem ents show that interest rate swaps and
currency swaps accounted for around 60% and 5% of all O T C USD 75,000 - USD 58,000 = USD 17,000
derivatives contracts (respectively) at the end of 2017.23 Table 20.1 shows that for the scenario considered in the exam
ple, the net cash flows are positive for the first half of the swap's
life and negative for the second half.
20.1 M EC H A N IC S O F IN T ER EST
RATE SW APS Day Count Issues
o The cash flows in Table 20.1 are approxim ations because they
The most common interest rate swap involves Libor being
do not consider day counts.4
exchanged for a pre-determined fixed rate for several years.
For exam ple, Party A might agree to pay Party B a fixed rate of For exam ple, suppose that the swap agreement states that
interest of 3% per year (compounded quarterly) on USD 10 million exchanges take place on January 1, April 1, July 1, and O ctober 1,
for three years. In return, Party B agrees to pay Party A interest at the swap starts on April 1 of a given year, and the first cash-flow
the three-month Libor rate on the same principal over the same exchange is on July 1 of that year.
period. In this example, interest would be exchanged every three
As discussed in Chapter 19, USD Libor is quoted on an
months. The principal of USD 10 million is referred to as the notional
actual/360 basis. There are 91 (= 29 + 31 + 30 + 1) days
principal because it is never exchanged. It is merely used to calculate
between April 1 and Ju ly 1. The floating-rate interest accruing
the interest payments that are exchanged.
between April 1 and Ju ly 1 would therefore be
Table 20.1 shows one possible scenario from the perspective of 91
Party B. The three-month Libor rates are in the second column — - X 0.0224 X USD 10,000,000 = USD 56,622.22
360
and (except for the first) are not known at the tim e the swap
This is a little different from the approxim ate estim ate of
is agreed.
USD 56,000 in Table 20.1.
The exchange of funds related to a certain Libor takes place one
The fixed interest rate of 3% is also expressed with an accom
period (three months in the case of the previous exam ple) after
panying day count m ethod. In this case, the possible methods
the Libor rate is observed. In the scenario in Table 20.1, a Libor
are actual/365 and 30/360. If the day count is 30/360, then the
fixed-rate cash flows in Table 20.1 are correct. If the day count is
actual/365, the first fixed payment would be
A
The variables are often referred to as risk factors by market partidpants.

91
— - X 0.03 X USD 10,000,000 = USD 74,794.52
2 BIS. (2019, Ju n e 04). O T C derivatives outstanding. Retrieved from 365
https://w w w .bis.org/statistics/derstats.htm
3 Libor was introduced in C hapter 16. 4 We discussed day count conventions in C hapter 19.
Table 20.1 A Possible Scenario for a Company that has Agreed to Receive a Fixed Rate of Interest of 3% per
Year and Pay Libor, with Payments Being Exchanged Every Three Months on a Principal of USD 10 Million
3-Month Libor Floating Amount Paid Fixed Amount Received

Time (Years) (% per Year) (USD) (USD) Net Cash Flow (USD)
0.00 2.24
0.25 2.32 56,000 75,000 +19,000
0.50 2.44 58,000 75,000 + 17,000
0.75 2.60 61,000 75,000 +14,000
1.00 2.76 65,000 75,000 +10,000
1.25 2.92 69,000 75,000 +6,000
1.50 3.00 73,000 75,000 +2,000
1.75 3.12 75,000 75,000 0
2.00 3.20 78,000 75,000 -3 ,0 0 0
2.25 3.20 80,000 75,000 -5 ,0 0 0
2.50 3.32 80,000 75,000 -5 ,0 0 0
2.75 3.36 83,000 75,000 -8 ,0 0 0
3.00 84,000 75,000 -9 ,0 0 0
rather than the USD 75,000 in Table 20.1. Because there counterparty (C C P ).5 Each side then has an agreem ent with the
are 92 days between Ju ly 1 and O ctober 1, the second fixed C C P and will have to post the required initial margin and varia
payment would be tion margin. If one of the parties to a transaction is not a C C P
member, it must arrange to clear the transaction through a
92
— - X 0.03 X USD 10,000,000 = USD 75,616.44 member.
365
The precise payment dates are affected by weekends and holi If one party to an interest rate swap transaction is an end user,
days. One common workaround is the next business day con the transaction can be cleared bilaterally as described in earlier
vention, where payment takes place on the next business day chapters. The confirmation agreem ent will then typically state
after the specified day. Exam ples of other conventions include that a master agreem ent already entered into between the two
the p reced in g business day convention (where payment is on sides will apply to the swap. As discussed Chapters 5 and 6, a
the business day preceding the specified day) and the m odified master agreem ent applies to all bilaterally cleared transactions.6
follow ing business day convention (which becom es the preced
ing business day convention if the following business day is in a
Quotes
different month from the specified day).
Large financial institutions are market makers for interest rate
swaps. Table 20.2 shows a hypothetical list of bids and asks for an
Confirmations
exchange that takes place every three months. The bid quote is
The agreem ent in an O T C derivatives transaction is called a the rate the institution is willing to pay to receive Libor, whereas
confirm ation (also known as a confirm ). The confirm ation sp e c the ask quote is the rate the firm is willing to receive to pay Libor.
ifies the dates when paym ents will be exchanged, how the Note that the swap rate is the average of the bid and ask quotes.
paym ents will be calculated, w hat day count conventions will
be used, which country's holiday calendars will apply,
and so on.
5 Material on C C P s was presented in Chapters 5 and 6.
6 The International Sw aps and Derivatives Association (ISDA) is the
As discussed in earlier chapters, an interest rate swap between industry association for derivatives m arket participants and provides the
two financial institutions must be cleared by a central docum entation used in many m aster agreem ents.
Chapter 20 Swaps ■ 259

Table 20.2 Example of Bid and Ask Quotes that Might be Made by a Financial Institution
Maturity (Years) Bid (% per Year) Ask (% per Year) Swap Rate (% per Year)
2 2.95 2.97 2.960
3 3.06 3.09 3.075
5 3.08 3.12 3.100
7 3.11 3.15 3.130
10 3.14 3.18 3.160
30 3.20 3.26 3.230
Suppose that a non-financial corporation accepts the bid quote 20.2 TH E R ISK -FR EE RATE
for maturity of five years and a notional principal of USD 100 million.
This means that it will receive 3.08% and pay Libor for five years. The risk-free rate is an im portant input for valuing swaps and
If the financial institution is lucky, it will find another end user other derivatives. As discussed in Chapter 16, Treasury rates
who wants to enter into the opposite (i.e., pay fixed, receive might seem to be the obvious choice for the risk-free rate but
Libor) transaction. If not, it could choose to hedge its risk by for various reasons are too low.
trading a swap with another financial institution to receive fixed
Before the 2007-2008 crisis, derivatives traders usually used
and pay Libor.
Libor rates as proxies for risk-free rates. This was convenient
when valuing Libor-based interest rate swaps because the same
Swaps Based on Overnight Rates yield curve was used to determ ine both a swap's cash flows and
the discount rate used for those cash flows.
As explained in earlier chapters, Libor is expected to be
replaced as a benchmark by an overnight rate. It is then likely During the crisis, however, Libor rates soared. As a result,
that Libor-based swaps will becom e less popular compared to most traders began using the discount rates calculated from
swaps based on the new benchmark. In the United Kingdom , the overnight index swap rates (OIS rates).7 These rates are discussed
preferred overnight rate is an interbank rate known as SO N IA. in Chapter 16, and the swaps work in the way described at the
In the U .S., the preferred overnight rate is a repo rate known as end of Section 20.1. Swaps with maturities up to one year
SO FR (see Chapter 16 for a discussion of these rates). provide a direct estimate of zero-coupon risk-free interest rates
out to the maturity of the swap. Swaps with longer maturities
Swaps based on overnight rates are structured differently from
provide estimates of par yields; the bootstrap method described
swaps based on Libor. On each paym ent date, a fixed rate is
in Chapter 16 can be used to estimate zero-coupon rates from
exchanged for the rate that is obtained when the overnight rate
these par yields.
is com pounded forward day-by-day between the last payment
date and the current paym ent date.
Suppose that there are n business days between the last pay 20.3 R EA S O N S FO R TR A D IN G
ment date and the current payment date, the overnight rate on IN T ER EST RATE SW APS
the rth of these business days is r-, and the rate applies to d; cal
endar days. The floating rate that is exchanged for the fixed rate Interest rate swaps are popular products because they can be
on the current payment date is used to transform assets and liabilities. That is, a company with
R = (1 + d iq X I + d2r2) • • • (1 + dnrn) - 1 a floating-rate loan can use a swap to convert it to a fixed-rate

liability. Similarly, a company with a fixed-rate loan can use a
On most days d, = 1, but weekends and holidays can lead to
the overnight rate being applied to more than one day. For
exam ple, on a Friday d,- will normally be equal to 3.
7 It seem s to be generally agreed that O IS rates should be used for
collateralized transactions, and it is som etim es argued that this is appro
There is usually just one exchange at maturity for swaps lasting
priate because the rate of interest paid on cash collateral is based on
up to one year. For swaps lasting more than one year, there are the overnight rate. Some traders continued to use LIBO R rates for non-
usually exchanges every three months. collateralized transactions in the years following the financial crisis.
swap to convert it to a floating-rate liability. M eanwhile, a com 3.08%
Financial 3.5%
pany that has invested in a fixed-rate bond can use a swap to Borrower
Institution
exchange its fixed-rate income to floating-rate income. Similarly, LIBO R
a company with a floating-rate income can exchange it for a
Fiqure 20.1 Conversion of fixed-rate borrowing to
fixed-rate income with a swap.
floating-rate borrowing.
For example, suppose that a company has borrowed USD 10 million
for five years at a fixed interest rate of 3.5% (compounded quar
terly), with interest being paid quarterly. The company can use the 2.97%
Financial
4 LIBO R + 0.3%
five-year bid quote of 3.08% in Table 20.2 to convert a fixed-rate Borrower ----------- ►
Institution *
borrowing into a floating-rate borrowing. When it enters into the LIBO R
swap contract (in conjunction with its borrowings), it has three sets Fiqure 20.2 Conversion of floating-rate borrowing to
of cash flows. fixed-rate borrowing.
1. It pays a fixed rate of 3.5% on its borrowings of
USD 10 million. Now consider how swaps can be used to manage assets. As
2 . It pays Libor on USD 10 million under the term s of an exam ple, suppose that a company has a ten-year USD
the swap. 10,000,000 investm ent paying interest at 2.8% . If it accepts the
ten-year ask quote in Table 20.2, it will have the following inter
3 . It receives 3.08% on USD 10 million under the term s of
est cash flows.
the swap.
• It receives 2.8% on its investment.
These three sets of cash flows are shown in Figure 20.1 and net
out to an interest paym ent of • It receives Libor.
• It pays 3.18% .
3.5% + Libor — 3.08% = Libor + 0.42%
Thus, the swap enables a 3.5% fixed-rate liability to be
out to
exchanged for a Libor plus 42 basis points floating-rate liability.
Libor + 2.8% - 3.18% = Libor - 0.38%
Now suppose that a company has borrowed USD 10 million for
two years at a floating interest rate of three-month Libor plus The swap enables a ten-year investment paying 2.8% to be
30-basis points.8 The company can use the ask quote of 2.97% exchanged for one paying Libor minus 38-basis points.
in Table 20.2 to convert the floating-rate liability to a fixed-rate
Finally, suppose that a company has a three-year USD 10,000,000
liability. When it enters the swap (in conjunction with its borrow
investment paying Libor minus 0.1% . If it accepts the three-year
ings), it has three sets of cash flows.
bid quote in Table 20.2, it will have the following interest cash
1. It pays Libor plus 30 basis points on its borrowings of flows.
USD 10 million.
• It receives Libor minus 0.1% on its investment.
2 . It receives Libor on USD 10 million under the term s of
• It pays Libor.
the swap.
• It receives 3.06%
3 . It pays 2.97% on USD 10 million under the term s of
the swap.
out to
These three sets of cash flows are shown in Figure 20.2 and
Libor — 0.1% — Libor + 3.06% = 2.96%
net out to an interest payment of:
The swap enables a ten-year investment paying LIBO R minus
Libor + 0.30% - Libor + 2.97% = 3.27%
10-basis points to be exchanged for one paying 2.96% .
The swap thus enables a Libor plus 30 basis point floating-rate
liability to be exchanged for a 3.27% fixed-rate liability.
3.18%
Financial 2 .8%
Investor
Institution
LIBO R
8 This means that Libor is observed every three months and interest at
an annual rate that is 0.3% higher than Libor is applied to the USD 10 Figure 20.3 Conversion of fixed-rate investment to
million borrowings and paid three months later. floating-rate investment.

3.06% Table 20.3 Five-Year Rates Offered to Companies a
Financial L IB O R - 0 .1 %
Institution
Investor and B in Fixed and Floating Markets. Rates are Paid
LIBO R Quarterly
Fiaure 20.4 Conversion of floating-rate investment Fixed Floating
to fixed-rate investment.
Com pany A 3.5% Libor + 0.1%
Comparative Advantage Arguments Com pany B 4.9% Libor + 0.9%
It is som etim es argued that creditworthy com panies find it much

easier to borrow at fixed rates than less creditworthy com panies. If Com pany B borrows at the floating rate of 3.5% indicated in
For exam ple, Table 20.3 shows the five-year rates that might be Table 20.3, it has three sets of interest rate cash flows.
offered to a company with a high credit rating (Com pany A) and 1. It pays Libor + 0.9% .
a company with a lower credit rating (Com pany B). Assum e that
2 . It pays X% .
both rates involve quarterly payments.
3 . It receives Libor.
The interesting aspect of these rates is that the spread between
them in the two markets are not the same. Com pany A pays 1.4% These cash flows net out to:
less than Com pany B in fixed-rate markets, but only 0.8% less in X + 0.9%
floating-rate markets. Thus, it can be argued that Com pany A has
Com pany B can borrow at 4.9% directly. The interest rate
a comparative advantage in fixed-rate markets and that Company
im provem ent in the rate paid by Com pany B is therefore:
B has a comparative advantage in floating-rate markets.
4.9% - (X + 0.9%) = (4 - X)%
An interest rate swaps trader would argue that a company
should always raise money in the market where it has a com par The total interest rate im provem ent to both sides from the
ative advantage and then swap to the market it wants. swap is
For exam ple, suppose that Com pany A wants to borrow at a (X - 3.4)% + (4 - X)% = 0.6%
floating rate while Com pany B wants to borrow at a fixed rate.
W here does this im provem ent come from? It is the difference
First imagine that Com pany A and Com pany B get in touch with
between (a) the spread between the rates offered to A and B
each other directly to do a swap. Suppose that Com pany B pays
in fixed-rate markets (= 1.4%) and (b) the spread between the
a fixed rate of X% to Com pany A and that Com pany A pays
rates offered to them in floating-rate markets (= 0.8% ).
Libor. This swap is shown in Figure 20.5.
Both sides obtain the same im provem ent in the rate they pay if:
If Com pany A borrows at the fixed rate of 3.5% indicated in
Table 20.3, it has three sets of interest rate cash flows. (X - 3.4)% = (4 - X)%
1. It pays 3.5% . or X = 3.7% .
2 . It receives X% . Figure 20.6 shows the swap when X = 3.7% along with the
borrowing by the two com panies in the markets where they
3 . It pays Libor.
have a com parative advantage. The swap leads to Com pany A
These cash flows net out to borrowing at Libor — 0.2% (= Libor + 3.5% — 3.7% ), which is
Libor + 3.5% - X% 0.3% less than the Libor + 0.1% it would pay if it went straight
to floating-rate markets. M eanwhile, Com pany B borrows at
Because Com pany A can borrow at Libor + 0.1% directly, the
4.6% (= 3.7% + (Libor + 0.9%) — Libor), which is also 0.3%
interest rate im provem ent in the rate paid by Com pany A is
less than the 4.9% rate it would pay if it went directly to fixed-
(Libor + 0.1%) - (Libor + 3.5% - X%) = (X - 3.4)% rate m arkets. As mentioned above the total apparent gain to
3.70%
3.5% LIBO R + 0.9%
Com pany A Com pany B -------------- ►
LIBO R
Fiqure 20.5 Trial swap between Company A and Fiaure 20.6 Swap between Company A and
Company B. Company B together with borrowing.
the two sides is the difference between (a) the spread
between the fixed rates offered to the two companies
(1.4% in our exam ple) and (b) the spread between the
floating rates offered to the two com panies (0.8% in
our exam ple).
the two sides.
In practice, if Com pany A and B are non-financial orga
nizations, they are not likely to get together directly. In rate agreem ent (FRA) can be valued by comparing it to a new
this case, a swap trader working for a financial institution could FRA known to be worth zero. A swap can be valued in the
act as intermediary by standing between the companies in the same way.
way indicated in Figure 20.7. The figure shows that the swap
Consider a swap that was entered some time ago and now
trader would gain 0.04% and the interest rate paid by each com
has two years to maturity. Suppose that a fixed rate of 4% is
pany is reduced by 0.02% (i.e., Com pany A pays Libor minus
received and Libor is paid every three months on a principal of
18 basis points and Com pany B pays 4.62% ). The total gain to
USD 100 million. We also suppose that the quotes in Table 20.2
all three parties (Company A , Com pany B, and the intermediary)
are the ones made by derivatives dealers today. The swap rate
is 0.6% , as before.
(i.e., the average of the bid and offer quotes) for a new two-year
W hy does this opportunity exist in financial m arkets? The answer swap is 2.96% . This is the mid-market quote, and it is therefore
is that the result of this swap arrangem ent is not quite as simple reasonable to assume that a swap where 2.96% is paid and Libor
as it has been portrayed it so far. is received is worth zero today. For valuation purposes, we can
Note that if Com pany B accepts the fixed rate presented in imagine a trader taking two positions.
Table 20.3, it is certain that it will be able to borrow at 4.9% 1. A two-year swap where a fixed rate of 4% is received
for five years. If it accepts the floating rate, however, the 0.9% and three-month Libor is paid on a principal of
spread over Libor is only guaranteed for three months. If C om USD 100 million.
pany B's credit worthiness changes, the spread over Libor that it
2. A two-year swap where a fixed rate of 2.96% is paid
will pay is also likely to change.
and three-month Libor is received on a principal of
For exam ple, there is a possibility that Com pany B's credit rating USD 100 million.
declines so that by year tw o, it is paying Libor plus 2% instead of
These two swaps net out to a position where 1.04% is received.
Libor plus 0.9% . The swap in Figure 20.7 will then lead to it pay
Specifically, USD 260,000 (= 0.25 X 0.0104 X USD 100,000,000)
ing 5.72% at that tim e (which is more than the 4.9% it would pay
would be received every three months for the next two years.
if it went directly to the fixed-rate markets).
Now suppose that the two-year risk-free rate is 2.4%
Note that Com pany A does not have the same type of uncer
(compounded quarterly) for all maturities. This is equivalent
tainty as Com pany B. In the swap shown in Figure 20.7, it would
to 0.6% every three months. The (USD) value of the position is
have guaranteed borrowing at Libor — 0.18% for five years.
therefore:
Modifying the statement made in connection with Figure 20.2, we
260,000 260,000 260,000 260,000 260,000
can say that the swap succeeds in exchanging a floating-rate liability
1.006 1.0062 1.0063 1.0064 1.0065
for a fixed-rate liability if the company's creditworthiness does not
change so that it always borrows at the same spread above Libor. 260,000 260,000 260,000
, + 7 + 2,024,945
1.0066 1.0067 1.0068
As we have explained, the value of the second swap is zero. It

20.4 VA LU A TIO N O F IN T ER EST
follows that the value of the first swap (i.e., the one we are inter
RATE SW APS ested in) is USD 2,024,945.
A swap's value is zero (or close to zero) at the time it is entered.9 Swaps on overnight rates can be valued similarly. In general,
suppose that trader owns an N-year swap where a fixed rate of
A t later tim es, it may have a positive or negative value because
X% will be received (the floating rate paid can be Libor or the
of changes in interest rates. Chapter 16 showed how a forward
overnight rate). Suppose further that the mid-market fixed rate
for an N-year swap is Y% . The value of the swap is the value
9 Bid-ask spreads mean that it has a slightly positive value to one side
(the provider of the bid and ask quotes) and a slightly negative value to of an N-year annuity of (X — Y)% on the notional principal. If
the other side (the side that accepts either a bid or an ask quote). the fixed rate of X% is paid (rather than received), the value of

the swap is the value of an N-year annuity of (Y — X)% on the Valuation Using Forward Rates
notional principal.
The forward rates calculated from swaps in this way (possibly with
It is som etim es necessary to interpolate between quotes
some interpolation) can be used to value existing swaps where
to value a swap. For exam ple, if a swap has a remaining
Libor is exchanged for a fixed rate. The procedure is as follows.
life of 2.5 years and the quotes in Table 20.2 apply, one
could assume that a 2.5-year swap where the fixed rate is 1. Assum e that forward Libor rates will be realized.
3.0175% (= 0.5 X (2.96% + 3.075%)) has a value of zero. 2 . Calculate the net cash flows that will be exchanged.
3 . Discount the net cash flows at risk-free rates.

Libor Forward Rates Existing O IS swaps can similarly be valued. Note that forward
Swaps where Libor is exchanged for a fixed interest rate can be overnight rates are calculated from the OIS zero curve. In calcu
used to estim ate Libor forward rates. The procedure used is a lating the cash flows that will be exchanged, it is then assumed
bootstrap method where progressively longer maturity swaps that the forward overnight rates will be realized. The resulting
are considered. net cash flows are discounted at the risk-free rate.
Consider a swap where the 12-month Libor is exchanged for a

fixed rate. Suppose that the one-year Libor rate is 4% and that Libor Zero Rates
the Libor forward rate estim ate for the period between one year
Zero rates can be calculated from Libor spot rates and the Libor
and two years is 5% (with all rates expressed using annual com
pounding). Suppose further that the mid-market rate in a three- forward rates calculated from FRAs or swaps. For exam ple,
year Libor-for-fixed swap is 5.5% . suppose the following.
A three-year swap can be valued as a portfolio of FRAs. • The six-month Libor spot rate is 4%.
Chapter 16 showed that an FRA can be valued by assuming that • The forward Libor rate for the period between six months
the forward interest rates are realized. The same is therefore true and one year is 4.3% .
of a swap.
Assuming these rates are com pounded semi-annually, we can
Suppose that the one-, two-, and three-year risk-free rates calculate the (annually com pounded) one-year rate R by solving:
(annually com pounded) are 3.7% , 4.2% , and 5.3% (respectively).
1 + R = (1 + 0.04/2) X (1 + 0.043/2)
Consider a three-year swap on a notional principal of USD 100
where 5.5% is received and Libor is paid. We know that this This gives R = 4.193% .
swap is worth zero (because 5.5% is the three-year swap rate). W hat will the one-year Libor spot rate be? Recall that a Libor
Because the one-year Libor rate is 4%, the value of the first rate is designed to be the rate at which an AA-rated bank can
exchange in the swap is worth: borrow. Before the 2007-2008 crisis, analysts assumed that that
0.055 - 0.040 Libor rates were risk-free rates. This assumption meant that a sin
100 X 1.4465
1.037 gle Libor zero curve could be estimated and the one-year Libor
spot rate in the previous exam ple would be set equal to 4.193% .
Because the one to two year forward rate is 5%, the value of the
second exchange in the three-year swap is However, the crisis led banks to realize that that Libor lending is
0.055 - 0.050 not risk-free. It is now recognized that a bank would rather make
100 X 0.4605 two six-month Libor loans to AA-rated banks than a single one-
1.0422
year Libor loan to a single AA-rated bank.10 A consequence of
If we define the forward rate for the period between two years
this is that the one-year Libor spot rate in our exam ple would be
and three years as F, the value of the third exchange in the
greater than 4.193% .
three-year swap is
0.055 - F Banks now calculate different zero curves corresponding to one-
100 X month lending, three-month lending, six-month lending, and
1.053s
12-month lending. In particular they calculate
Because the swap is worth zero:
0.055 - F
1.4465 + 0.4605 + 100 X
1.0533
10 Note that a borrowing bank is AA-rated at the start of a Libor loan.
This can be solved to give F = 0.07727. The forward rate for the The longer the loan lasts the more likely that the bank will get into finan
two to three-year period is therefore 7.727% . cial difficulties and default.
• A Libor zero curve from the one-month Libor spot rate and Table 20.4 Cash Flows in a Fixed-for-Fixed Currency
forward one-month Libor rates, Swap Where Interest at 5% in USD on a Principal of
• A Libor zero curve from the three-month Libor spot rate and USD 10 Million is Exchanged for Interest at 4% on a
forward three-month Libor rates, Principal of 8 Million Euros
• A Libor zero curve from the six month Libor spot rate and for USD Cash Flow Euro Cash Flow
ward six-month Libor rates, and Time (Years) (Millions) (Millions)
• A Libor zero curve from the 12-month Libor spot rate and for
0 -1 0 +8
ward 12-month Libor rates.
1 +0.5 - 0 .3 2
If they were to be represented graphically, the first zero curve
2 +0.5 - 0 .3 2
would be below the second, which would be below the third,
which would be below the fourth. 3 +0.5 - 0 .3 2
4 + 10.5 - 8 .3 2
20.5 C U R R E N C Y SW APS
The euro outflow is converted to a USD outflow at the begin
This section discusses currency swaps. First, consider a fixed-for-
ning and subsequent euro inflows are converted to USD inflows.
fixed currency swap. This is a swap where a fixed interest rate
in one currency is exchanged for a fixed interest rate in another Recall that the com parative advantage benefits related to inter
currency. Two principal amounts (i.e., one for each currency) est rate swaps may be largely illusory because the spreads
must be specified. charged over a floating reference rate (e.g., Libor) are liable to
change with the creditworthiness of the borrower. In the case of
One difference between a currency swap and an interest rate
currency swaps, tax issues can create non-illusory com parative
swap is that principal amounts are exchanged in a currency
advantages.
swap, whereas they are not exchanged in an interest rate swap.
Specifically, principal amounts are exchanged in the opposite For exam ple, a company might have a low marginal tax rate in
direction to the interest rate payments at the beginning of the France and a high marginal tax rate in the U.S. If it wants to bor
swap's life and in the same direction as the interest rate pay row in euros, it could make sense for the company to borrow in
ments at the end of it. USD (and thus be able to deduct the USD interest for tax pur
poses) and then swap the borrowing to euro borrowings using
For exam ple, consider a swap where interest at 5% in USD
the swap presented in Table 20.4.
on a principal of USD 10 million is received and interest at
4% in euros on a principal of 8 million euros is paid. Both
interest rates are annually com pounded, and payments are Valuation
exchanged every year for four years. The cash flows are shown
The two sets of cash flows in a swap are referred to as legs.
in Table 20.4. The interest cash flows are USD 0.5 million
Using this term inology, the swap in Table 20.4 has a USD leg
(= 0.05 X USD 10 million). The interest cash flows are
and a euro leg. Currency swaps are usually designed so that
0.32 million euros (= 0.04 X 8 million euros).
they have a value close to zero when first negotiated. Later, the
A currency swap can be used to transform liabilities and assets remaining cash flows can be valued by considering each leg
in an analogous way to interest rates swaps. For exam ple, the separately.
swap in Table 20.4 can be used to transform borrowings at 5%
Suppose that a fixed-for-fixed swap involves exchanging
on USD 10 million to borrowings at 4% on 8 million euros. Note
principal and interest in currency X for principal and interest
that the initial exchange converts the USD principal received
in currency Y, and that the valuation is required in currency X.
at the beginning to a euro principal. Subsequent exchanges
The valuation procedure is as follows.
provide the necessary USD interest and principal payments
with euro payments. In each year, USD 0.5 million of interest is • Value the remaining currency X cash flows in currency
required on the USD loan. USD 0.5 million is received to make X term s.
this payment under the term s of the swap, while the party with • Value the remaining currency Y cash flows in currency Y
the liability pays the euro interest rate. term s.
The swap can also be used to convert an 8 million euro invest • Convert the value of the currency Y cash flows to currency X
ment earning 4% to a USD 10 million investm ent earning 5%. at the current exchange rate.

In the case of the swap presented in Table 20.4, suppose one The value of the contract is the present value of the three
year after initiation (just after the first exchange of payments): exchanges. It is
• The risk-free interest rate in USD is 4.5% for all maturities, 0.1244 0.1249 0.6520
USD 0.809 million
• The risk-free interest rate in euros is 3.5% for all maturities, 1.045 1.0452 1.045s
and This is the same value as the one obtained earlier. This illustrates
that valuing each leg separately as bonds gives the same result
• The exchange rate (USD per euro) is 1.15.
as valuing each exchange separately using forward exchange
The remaining cash flows (with time being measured from the rates.
valuation date rather than from the start of the swap) are shown
in Table 20.5. The value in millions of USD of the USD cash
flows is Other Currency Swaps
0.5 0.5 10.5 Two other types of currency swaps are those where the follow
10.137
1.045 1.0452 1.0453 ing is true.
The value in millions of euros of the euro cash flows is
1. A floating rate in one currency is exchanged for a fixed rate
0.32 0.32 8.32 in another currency.
d--------- d--------r — 8.112
1.035 1.0352 1.035s
2. A floating rate in one currency is exchanged for a floating
The value of the swap in millions of USD is therefore:
rate in another currency.
10.137 - 8.112 X 1.15 = 0.809
The first type of swap can be used to convert a floating-liability/
The swap can also be valued in term s of forward exchange asset in one currency to a fixed-liability/asset another currency
rates. For exam ple, suppose that the forward exchange rates (or vice versa). The second type of swap can be used to convert
(USD per euro) for year one, tw o, and three are a floating-liability/asset in one currency to a floating-liability/
1.045 asset in another currency.
Year 1: 1.15 X — — = 1.1611
1.035 These swaps can be valued by valuing each leg in its own cur
1.0452 rency. Fixed legs can be valued in the same way as the fixed legs
Year 2: 1.15 X ------ - = 1.1723
1.0352 of a fixed-for-fixed swap, while floating legs can be valued by
1.045s assuming forward interest rates in the currency will be realized.
Year 3: 1.15 X ------ - = 1.1837
1.035s
The USD equivalent of the first exchange (i.e., when USD 0.5 million
is received and 0.32 million euros is paid) is
20.6 O T H ER SW APS
0.5 - 0.32 X 1.1611 = USD 0.1284 million Swaps exchanges can be defined in many ways. A variation on
The USD equivalent of the second exchange (i.e., when USD 0.5 the interest rate swaps discussed thus far is when the notional
million is received and 0.32 million euros is paid) is principal changes in a pre-determ ined way through tim e. This
could be useful for a company embarking on a multi-year proj
0.5 - 0.32 X 1.1723 = USD 0.1249 million
ect that knows its borrowings will increase with tim e. With this
The USD equivalent of the final exchange (i.e., when USD 10.5 mil kind of swap, the company can transform its borrowings from
lion is received and 8.32 million euros is paid) is fixed to floating or from floating to fixed.
10.5 - 8.32 X 1.1837 = USD 0.6520 million An equity sw ap is a swap where a fixed return is exchanged for
the return generated when the notional principal is invested in
Table 20.5 Cash Flows in Swap in Table 20.4 as Seen pre-specified equity. For exam ple, it might be agreed that the
One Year After the Swap is Initiated return from USD 10 million invested in the S&P 50011 during
each six-month period for the next five years will be exchanged
USD Cash Flow Euro Cash Flow for USD 10 million invested at a fixed interest rate of 4%. In
Time (Years) (Millions) (Millions) some cases, the equity return is swapped for a floating (e.g .,
Libor-based) return instead of a fixed return.
1 +0.5 - 0 .3 2
2 +0.5 - 0 .3 2 11 The S&P Index used to define the return in this type of swap would
typically be the total return index, which is the index form ed when divi
3 + 10.5 - 8 .3 2
dends on the underlying stocks are reinvested in the index.
A com m odity swap is a swap where a certain amount of a com counterparty (CC P). The C C P requires both initial margin and
modity at a pre-determined fixed price is periodically exchanged variation margin from each side. Chapter 6 exam ined the risks
for the same amount of the commodity at a floating price (i.e., associated with a possible default by the CCP.
the market price at the time of the exchange). This is equivalent
O ther swaps between financial institutions are cleared bilaterally
to a portfolio of forward contracts on the price of the commodity.
and regulations require both initial margin and variation mar
In a volatility sw ap, the historical volatility observed during a gin to be posted. This largely elim inates the credit risk in these
certain period is exchanged for a pre-determ ined fixed volatility. transactions. The variation margin is transferred directly from
Both are multiplied by the notional principal. one side to the other. Initial margin must be posted by each side
with a third-party trustee.
A credit default swap (CDS) provides insurance against a default
by a company. The buyer of this protection makes periodic fixed There is no requirem ent that collateral (i.e., margin) be posted
payments to the seller of protection for an agreed period of for transactions between a financial institution and non-financial
tim e. If the company that is the subject of the protection (known com panies. When no collateral is posted, or when transactions
as the reference entity) defaults, there is a payment from the are only partly collateralized, the credit exposure must be moni
seller of protection to the buyer of protection. O therw ise, the tored carefully. The initial pricing of transactions should consider
seller of protection does not have to make any payments. In an expected credit losses by each sid e .14
index credit default sw ap, the single reference entity is replaced
by portfolio of com panies. In return for regular paym ents, the SU M M A RY
buyer of protection is com pensated for defaults by any of the
com panies in the portfolio. In an interest rate swap, one party agrees to pay interest at a
There are many other types of swaps and the exchanges in a floating rate on a notional principal and receive interest at a
swap can be defined in almost any way. Some can be overly fixed rate on the same notional principal. Interest payments are
exchanged periodically, and the swap has a pre-agreed life. In
com plicated.
a currency swap, interest payments on a pre-agreed principal
To see how complicated swaps can be, consider an exotic deal
in one currency are exchanged for interest payments on a pre
known as the 5/30 swap that was defined by Bankers Trust12 and agreed principal in another currency. Principal amounts in the
sold to several clients in 1993. For this type of swap, payments
two currencies are exchanged in the opposite direction to inter
depended in a com plex way on the 30-day commercial paper est payments at the beginning of the swap's life and in the same
rate, a 30-year Treasury bond price, and the yield on a five-year direction as interest payments at the end.
Treasury bond. Another exam ple of com plexity was Spanish bank
The popularity of interest rate and currency swaps arises from
Santander defining swaps where the payments depended in a
com plex way on whether a three-month reference rate stayed their ability to transform exposures connected to a firm's assets
within the 2% to 6% range. In both cases, it was alleged that the or liabilities. An interest rate swap can transform a fixed-rate
liability to a floating-rate liability (or vice versa). It can also trans
risks in the swaps were not fully understood by the banks' clients.
form a fixed-rate asset to a floating-rate asset (or vice versa).
Currency swaps can similarly transform assets/liabilities in one
20.7 C R E D IT RISK currency to assets/liabilities in another currency.
There are several approaches to valuing swaps. Interest rate

Swaps have the potential to give rise to credit risk. If Com pany X swaps can be valued by comparing them with similar swaps
gets into financial difficulties and defaults when its outstanding known to have a value of zero. Swap are som etim es valued as
swaps with Com pany Y have a positive value to Com pany Y, the difference between the value of the stream of payments that
Com pany Y may experience a loss.13 will be made by one side and value of the stream of payments
Key aspects of credit risk were discussed in earlier chapters. that will be made by the other side. Another approach is to
Recall that interest rate swaps and index credit default swaps value each exchange of payments and sum the results.
between financial institutions must be cleared through a central
14 Credit value adjustm ent (CVA) is a downward adjustm ent in the pricing
of derivatives made by a bank to reflect expected losses from a default
12 Bankers Trust was sold for USD 10.1 billion to Deutsche Bank in 1998
by the counterparty. Debit value adjustm ent (DVA) is an upward adjust
after it had suffered losses from its positions in Russian bonds.
ment to the price by the bank reflecting the possibility of a default by
13 A s discussed in C hapter 5, credit risk is not considered on a the bank (along with the accompanying credit losses). The calculations of
transaction-by-transaction basis. Transactions w hether cleared through a these quantities involve estimating the probability of defaults occurring
C C P or bilaterally are netted. during future periods and expected credit losses conditional on a default.

Q U ES T IO N S

20.1 An interest rate swap starts on June 1 ,2 0 1 9 , and 20.6 "Borrowing at a Libor floating rate for five years and then
involves a six-month Libor being paid and a fixed rate swapping floating-rate payments for fixed-rate payments
of 5% being received every six months on a principal of does not guarantee a fixed rate of interest on the bor
USD 5 million. When does the first exchange take place? rowings." Explain this statem ent.
How is it calculated? 20.7 Are principal payments exchanged in (a) interest rate
20.2 Explain what the five-year USD swap rate is? swaps and (b) currency swaps?
20.3 W hat day count is used for USD Libor? 20.8 Expl ain the two ways a fixed-for-fixed currency swap can
20.4 An interest rate swap is based on an overnight rate and be valued.
exchanges take place every three months for three years. 20.9 Expl ain how an equity swap works.
How are the payments calculated? 20.10 Expl ain how a volatility swap works.
20.5 W hat is the difference between the discount rate used to
value swaps (a) prior to the 2007-2008 crisis and (b) after
the 2007-2008 crisis?
Practice Questions
20.11 The quotes for a five-year interest rate swap are bid 3.20, involves paying a fixed rate of 5% at the end of each
ask 3.24. W hat swap would be entered by a company quarter and receiving the rate implied by the overnight
that can borrow for five years at 4.2% per year but wants rate when it is com pounded day-by-day during the quar
to borrow at a floating rate? W hat rate of interest does ter. The notional principal is USD 20 million. The current
the company end up borrowing at? quote for a three-year overnight index swap is bid 3.80,
20.12 Suppose that in Question 20.11 a company can borrow ask 3.88. The risk-free rate is 3.6% for all m aturities. All
rates are com pounded quarterly.
at Libor plus 50 basis points but wants to borrow for five
years at a fixed rate. W hat swap should the company 20.15 Suppose that the six-m onth Libor rate is 5%, the for
enter into? W hat rate of interest does the company end ward Libor rate for the period between 0.5 and 1.0 year
up borrowing at? (Assume that the spread above Libor at is 5.6% and the forw ard Libor rate for the period
which the company borrows does not change.) between 1.0 and 1.5 years is 6.0. The two-year Libor
swap rate is 5.7% . All risk-free rates are 4.5% . W hat is
20.13 Company A can borrow at a fixed rate of 4.3% for five
the forward Libor rate for the period between 1.5 and
years and at a floating rate of Libor plus 30 basis points.
Company B can borrow for five years at a fixed rate of 2.0 years? All rates are expressed with semi-annual
com pounding.
5.9% and at a floating rate of Libor plus 100 basis points.
As a swaps trader you are in touch with both companies 20.16 In Question 20.15, what is the 1.5-year Libor zero rate
and know that Company A wants to borrow at a float expressed with semi-annual com pounding?
ing rate and that Company B wants to borrow at a fixed 20.17 Consider a currency swap where interest on British
rate. Both companies want to borrow the same amount of pounds at the rate of 3% is paid and interest on euros at
money. Design a swap where you will earn 10-basis points,
2% is received. The British pound principal is 1.0 million
and which will appear equally attractive to both sides. pounds and the euro principal is 1.1 million euros. The
20.14 You are required to estim ate the value of an overnight most recent exchange has just occurred and the interest
indexed swap that has three years left in its life and is exchanged every six months. There are two years are
remaining in the life of the swap. The current exchange 20.19 A bank trades a swap where a fixed rate of 5% in cur
rate is 1.15 euro/pound. The risk-free rates in pounds rency A is paid and Libor in currency B is received. Show
and euros are 2.5% and 1.5%. Value the swap by consid that the swap can be considered as a fixed-for-fixed cur
ering it as the difference between two bonds. All rates rency swap plus an interest rate swap.
are com pounded semi-annually. 20.20 A bank is taking floating-rate deposits and using the
20.18 Value the swap in Question 20.17 by considering it as a funds to make five-year fixed-rate loans. W hat risks is it
portfolio of forward contracts. taking? How can they be hedged?
Chapter 20 Swaps 269

A N S W ER S
20.1 T h e fi rst exchange will take place six months after June 1, portfolio is swapped for the return obtained by investing
2019, on D ecem ber 1, 2019. It will be calculated as: the principal at a fixed rate (or possibly a floating rate).
0.5 X (0.05 - R) X USD 5,000,000 20.10 A t the end of each period, a pre-specified volatility
where R is the Libor rate on June 1 ,2 0 1 9 . multiplied by a pre-specified principal is exchanged for
the observed historical volatility during the same period
20.2 The five-year USD swap rate is the average of the bid
multiplied by the same principal.
and ask quotes for five-year swaps.
20.11 The company should arrange to receive fixed and pay float
20.3 Actual/360
ing to convert the fixed-rate loan to a floating-rate loan. It
20.4 The payments on a given date are the exchange of a pre will accept the bid quote of 3.20. Its cash flows will be
determ ined fixed rate for the three-month rate implied
• Receive 3.2% ,
by the overnight rates observed during the previous
• Pay 4.2% , and
three months. The implied three-month rate is
• Pay Libor.
(1 + d ^ X 1 + d2r2) (1 + c/nrn) - 1
These net to Libor plus 1%.
where n is the num ber of business days during the
previous three months, the overnight rate on the /th of 20.12 The company should arrange to pay fixed and receive
these business days is r„ and the rate applies to d , cal floating to convert the floating-rate loan to a fixed-rate
endar days. loan. It will accept the ask quote of 3.24. Its cash flows
will be
20.5 Libor was used as a proxy for the risk-free rate pre-crisis.
Interbank overnight rates have been used to determ ine • Pay 3.24% ,
discount rates since then. • Receive Libor, and
20.6 The spread added to the Libor floating rate is liable to • Pay Libor + 0.5% .
change if the creditworthiness of the borrower changes. These net to 3.74% .
This means that the fixed rate calculated using the cur
20.13 The spread between the fixed rates offered to Com
rent spread may not be what applies for all periods.
panies A and B is 5.9% — 4.3% or 1.6%. The spread
20.7 Principal payments are not exchanged in an interest rate between the floating rates is 70 basis points or 0.7% . The
swap (exchanging principals would not, of course, make difference between these two spreads is 1.6% — 0.7%
any difference when the principals used to determ ine or 0.9% . It should be possible to design a swap where
interest payments on both sides of the swap are the the parties are in aggregate 0.9% better off. The bank
same). Principal payments are exchanged in a currency (intermediary) wants 0.1% . This leaves 0.4% for each
swap (there are two different principals, one for each side. We should therefore be able to design a swap
currency). where Com pany A borrows at Libor + 0.3% — 0.4%
20.8 A fixed-for-fixed currency swap can be valued as the or Libor — 0.1% and Com pany B appears to borrow
difference between two bonds or as a portfolio of for at 5.9% — 0.4% = 5.5% . If the bank pays X% to A we
ward contracts, one corresponding to each exchange. require 4.3% + Libor — X% = Libor — 0.1% so that
X = 4.4. Similarly, if B pays Y% to the bank, we require
20.9 A principal is specified, and the return obtained per
Y% + 1% = 5.5% so that Y = 4.5. The swap arrange
period by investing the principal in a specified equity
ment is
20.14 The swap rate is the average of 3.80 and 3.88, or 3.84% . 100 X (1 + R /2)3 = 108.5311
The swap involves paying 5% when the market rate is
This can be solved to give R = 0.055329. The zero rate is
3.84% . The swaps value is the present value of:
5.5329% .
0.25 X (0.0384 - 0.05) X USD 20,000,000
20.17 The swap involves exchanging
= - U S D 58,000
0.5 X 0.03 X 1,000,000 = 15,000 pounds with
on every payment date for the next three years. Because 0.5 X 0.02 X 1,100,000 = 11,000 euros with a final
the risk-free rate is 3 .6 % /4 = 0.9% per quarter, the value is exchange of principal. The value of the British pound
l 2 58,000 bond in British pounds is
- V — — r = -6 5 6 ,9 3 8
& 1.009' 15,000 15,000
-------------------- + --------------------- r
20.15 A swap where 5.7% is paid and Libor is received is worth 1 + 0.5 X 0.025 (1 + 0.5 X 0.025)2
zero. Per 100 of principal, first FRA is worth: 15,000 1,015,000
---------------------------------------------------------
0.5 x (0.05 - 0.057) x 100 (1 + 0.5 X 0.025)3 (1 + 0.5 X 0.025)4

- 0 .3 4 2 = 1,009,695
1 + 0 .0 4 5 /2
The second FRA is worth: The value of the euro bond in euros is
0.5 x (0.056 - 0.057) x 100 11,000 11,000
- 0 .0 4 8
(1 + 0 .0 4 5 /2 )2 1 + 0.5 x 0.015 ( 1 + 0.5 x 0 .0 1 5)2
The third FRA is worth: 11,000 1, 111,000
H------------ --------------- 1--------- ----- -----------
0.5 x (0.060 - 0.057) x 100 (1 + 0.5 x 0 .0 1 5)3 (1 + 0.5 x 0 .0 1 5)4
= 0.140
(1 + 0 .0 4 5 /2 )3 = 1,110,797
If the required forward rate is R th e n :
The value of the swap in British pounds is therefore
0.5 X (R - 0.057) X 100 1,110,797/1.15 - 1,009,695 = -4 3 ,7 8 5 .
- 0.342 - 0.048
(1 + 0 .0 4 5 /2 )4
20.18 The forward rates corresponding to the exchanges at
+ 0.140 = 0
tim es 0.5, 1.0, 1.5, and 2.0 years are
This can be solved to give R = 0.0625. The forward rate 1.0075
1.15 X 1.1443
for the period between 1.5 and 2.0 years is 6.25% (semi 1.0125
annually com pounded). 1.00752
1.15 X 1.1387
20.16 Using six-month forwards, 100 would grow to: 1.01252
1.00753
100 x (1 + 0.05/2) x (1 + 0.056/2) x (1 + 0.06/2) 1.15 X 1.1330
1.01253
= 108.5311 1.00754
1.15 x 1.1275
If R is the 1.5 year zero rate with semi-annual 1.01254
com pounding: The exchanges are
Time Pounds Paid Euros Received Value in Pounds
0.5 15,000 11,000 11,000/1.1443 - 15,000 = - 5 ,3 8 7
1.0 15,000 11,000 11,000/1.1387 - 15,000 = - 5 ,3 4 0
1.5 15,000 11,000 11,000/1.1330 - 15,000 = - 5 ,2 9 2
2.0 1,015,000 1,111,000 1,111,000/1.1275 - 1,015,00 = - 2 9 ,5 9 2

The G B P value of the swap is 20.20 The bank is taking the risk that interest rates will increase
5,387___________________5,340 because, if that happens, it will have to pay a higher
1 + 0.5 X 0.025 (1 + 0.5 X 0.025)2 interest rate on the floating-rate deposits but will not
___________5,292___________________ 29,592 receive a higher rate of interest on the fixed-rate assets.

It can hedge its risk by entering into a swap where it is
(1 + 0.5 X 0.025)3 (1 + 0.5 X 0.025)4
= -4 3 ,7 8 5 receiving floating and paying fixed.
This agrees with the answer in Question 20.17.
20.19 The swaps are as follows.

• The fixed rate in currency A is paid, and a fixed rate
in currency B is received.
• The same fixed rate in currency B is paid, and Libor
in currency B is received.
IN D EX
A B
accum ulation value, 20 Bakshi, G . S „ 221
adjustable-rate m ortgages (ARM s), 226 Bankers Trust, 267
A doboli, Kweku, 53 Bank for International Settlem ents (BIS), 48, 75, 110, 258
after-acquired clause bond, 219 bankruptcy, 22
agency m ortgage-backed securities (agency MBSs) banks
dollar roll, 230 conflicts of interest, 9-10
non-agency, 231 deposit insurance, 5-6
other agency products, 230-231 investm ent banking
pass-throughs trading, 229-230 advisory services, 8-9
agency security bond, 230-231 Dutch auctions, 7-8
agricultural com m odities, 136-137 IPO s, 7
A IG , 66 offerings, 6
A llied Irish Bank, 53 trading, 9
alternative investm ents, 35 underwriting, 6
Am erican call/put options, 148 originate to distribute m odel, 10 -11
Am erican D epositary Receipts (ADRs), 39 regulation
Am erican option, 50 Basel Com m ittee, 4
Am erican vs. European options capital, 3-4
call options, 160-161 liquidity ratios, 5
put options, 163-164 standardized models vs. internal m odels, 4-5
amortization table, 227-228 trading book vs. banking book, 5
annuity contracts, 20 risks
arbitrageurs, 53, 125 credit risks, 2-3
Asian options, 115, 187 m arket risks, 2
asset-exchange option, 189 operational risks, 3
asset-or-nothing call, 186 Bao, J ., 215
asset-or-nothing put, 186 BarclayH edge, 40
at-the-money, 149 Barings Bank, 53
average loan balance, 228 barrier options, 188
average price calls, 187 Basel Com m ittee, 4
average price puts, 187 basket option, 115, 189
average strike calls, 187 bear spread, 174-175
average strike puts, 187 beta (jS), 103-104
bid-ask spreads, 110, 111, 263 modern theory, 141-142
bilateral clearing, 77 capital charges, risks, 24
binary options, 186-187 capital investm ent opportunities, 46
Black-Scholes Merton m odel, 47, 148, 149, 173, 189 car insurance, 22
board order, 89 Carson, R., 116
Boesky, Ivan, 38 cash dividends, 152
bond covenants, 220 cash incom e, 126
bond dealers, 215 cash offer, 8
bond duration cash-or-nothing call, 186
limitations of, 203 cash-or-nothing put, 186
m acaulay duration, 202 cash-out refinancing, 231
m odified duration, 202-203 cash settlem ent, 87
yield duration, 201-202 casualty, 22
bond indentures, 216 casualty insurance, 16
bond issuance, 214 catastrophe risks, 22
bond mutual funds, 32 C A T (catastrophe) bonds, 22-23
bond risk, 217-218 C C P s. See central counterparties (CCPs)
bonds central clearing
classification of, 218-220 CCP, 73
collateral, 219-220 advantages and disadvantages, 77-78
day count conventions, 242 six m arket participants trade derivatives, 75
default, 217-218 regulation of, O T C derivatives m arket, 73-74
event risk, 217-218 six m arket participants trade derivatives, 75
exp ected return from , 221-222 standard vs. non-standard transactions, 74-75
interest rate, 218-219 three m arket participants, 77
issuer, 218 central counterparties (C C Ps), 48, 58, 59, 72,
maturity, 218 259, 267
bond trading, 214-215 central clearing
bond valuation, 200-201 advantages and disadvantages, 77-78
bond yield, 201 liquidity risk, 79
bootstrap m ethod, 206 model risk, 79
Borio, C ., 116 six m arket participants trade derivatives, 75
box spread, 175 handling credit risk
Bridgewater, 36 daily settlem ent, 60
British thermal units (BTUs), 137 default fund contributions, 61
bull spread, 173-174 initial margin, 61
butterfly spread, 175-177 netting, 60
buying on margin, 153 variation margin, 60
margin accounts, in other situations
buying on margin, 62-63
C options on stocks, 62
calendar spread, 177 short sales, 62
call options, 50, 51 central registry, 48
Am erican vs. European options, 160-161 change of control (C o C ), 216
dividends im pact, 161-162 cheapest-to-deliver option, 229, 245-246
em ployee stock options, 161 Chicago Board of Trade (C B O T ), 46, 47, 84, 148
lower bound, 162 Chicago Board O ptions Exchange (C B O E ), 47, 62, 148
m oneyness, 149 index options, 152
payoffs, 150-151 margin manual, 153
profits from, 149-150 maturity, 151
call provision bonds, 220-221 stock options, 151
Canadian dollar, 110 strike prices, 151-152
capital asset pricing model (CA PM ), 103 Chicago M ercantile Exchange (C M E), 72, 84, 131,
futures price vs. exp ected future spot price, 143 242, 244
274 ■ Index
China Financial Futures Exchange (C F F E X ), 130 default rate and recovery rate, 221
chooser options, 186 expected return from bond investm ents, 221-222
Citadel, 36 indentures, 216
claw back clause, 37 issuance, 214
clean price, 215, 243 rating scales for, 216
cleared transactions, 74 risk, 217-218
cliquet options, 185 trading, 214-215
closed-end funds, 33-34 trustee, 216
C M E G roup, 85 vs. yield bond, 215
collateralized m ortgage obligation (CM O ), 230 corporate bond trustee, 216
collateral trust bond, 219-220 cost of carry, 140-141
com m ercial banking, 2 covered call, 172
com m odity covered interest parity, 116-118
contracts, 87 credit default swap (CD S), 74, 267
convenience yields, 139-140 credit event binary options (C E B O s), 152
cost of carry, 140-141 credit ratings, corporate bonds, 216-217
delivery of, 87 credit risks, 2 -3 , 21, 196
expected future spot price daily settlem ent, 60
contango, 143 default fund contributions, 61
early work, 141 initial margin, 61
modern theory, 141-143 netting, 60
normal backwardation, 143 sw aps, 267
vs. financial assets, 136 variation margin, 60
investm ent, 138-139 credit support annex (CSA ), 65
swap, 267 credit value adjustm ent (CVA), 3, 267
types of crude oil m arket, 137
agricultural com m odities, 136-137 C S A . S e e credit support annex (CSA)
energy, 137-138 cum ulative probability, 16
m etals, 137 currency sw aps, 265-266
weather, 138 curtailm ents prepaym ent, 232
Com m odity Exchange, Inc. (C O M EX ), 84
Com m odity Futures Trading Com m ission
(C FT C ), 89 D
com parative advantage argum ents, 262-263 Dalian Com m odity Exchange, 84
com pounding frequency, 197-199 Danish krone, 116
com pound interest rates, 141 day count conventions
com pound options, 188-189 interest rate futures, 242-243
conditional prepaym ent rate, 228 swap interest rates, 258-259
confirm ations, swaps, 259 day traders, 88
constant prepaym ent rate, 228 day trading, 85
consumption assets, 124 debenture bonds, 219
contango, 143 debit value adjustm ent (DVA), 267
continuous com pounding, 199 debt capital, 4
convenience yields, com m odity, 139-140 debt retirem ent bond, 220-221
conversion factor, 244 dedicated short, 38
convertible arbitrage, 39 deep-out-of-the-money (D O O M ), 152
convertible bonds, 154, 214 default prepaym ent, 232
convexity adjustm ent, 248 default rate bond, 221
convexity, bond, 203-204 deferred annuities, 20
cooling degree days (CD D s), 138 deferred-coupon bond, 217
corporate bonds, 21 defined benefit plan, 21
classification of, 218-220 defined contribution plan, 21
credit ratings, 216-217 deposit insurance, 5-6, 24
debt retirem ent, 220-221 derivative product com panies (D PCs), 65
Index
derivatives em erging m arkets, 39
forward contracts, 48-49 em ployee com pensation plans, 46
futures contracts, 50 em ployee stock options
linear, 46 accounting tricks, 154
m arket participants call options, 154, 161
arbitrageurs, 53 endow m ent life insurance, 19
hedgers, 52 energy com m odity, 137-138
speculators, 52-53 equipm ent trust certificate (ETC ), 219
markets equity capital, 3
exchange-traded, 47 equity funds, 32, 33
O T C , 47 equity investm ents, 21
size, 48 equity swap, 266
non-linear, 46 Ergener, D ., 190
options, 50-52 Eurex, 84
risks, 53 euro, 116
Derm an, E ., 190 Eurodollar futures
diagonal spread, 179 contract, 87
Dick-Nielsen, J ., 222 vs. forward rate agreem ents, 248-249
digital options, 186 hedging, 250
dirty price, 215, 243 libor zero curve, 249-250
discount basis, interest rate, 243 quotes, 248
discount factor, 200 SO FR futures, 250
discounting, 200 Euro O vernight Index A verag e (EO N IA ), 197
discretionary orders, 89 European call/put options, 148, 149
distressed debt, 38 European option, 50
dividends, 152 European vs. Am erican options
call options call options, 160-161
Am erican vs. European options, 160-161 put options, 163-164
im pact of, 161-162 exchanges
lower bound, 162 CCP, 59
paym ents, 130 margin, 58
put options netting of contracts, 59
Am erican vs. European options, 163-164 exchange-traded Am erican options, 184
im pact of, 164 exchange-traded funds (ETFs), 32, 34, 152
Dodd-Frank A ct, 9 exchange-traded m arkets, 47
D od d -Frank legislation, 215 exchange-traded options, 148
dollar duration, 203 vs. em ployee stock options, 154
dollar roll, 230 on stocks, 151-152
dom estic currency, 113 exchange-traded products (ETPs), 152
Dow Jo n es Index, 130 ex-dividend date, 162, 172
duration, bond exercise price, 50, 148
limitations of, 203 exotic options
m acaulay duration, 202 defined, 184
m odified duration, 202-203 hedging exotics, 190
yield duration, 201-202 m ultiple assets, 189
duration of orders, 89 single assets
Dutch auctions, 7-8 asian options, 187
barrier options, 188
binary options, 186-187
E chooser options, 186
econom ic capital, 4 cliquet options, 185
electricity, 137 compound options, 188-189
electronic trading, 47 forward start options, 185
276 ■ Index
gap options, 185 multi-currency hedging, using options, 114-115
lookback options, 187 quotes
non-standard am erican options, 184 Canadian dollar, 110
packages, 184 currency abbreviations, 111
zero-cost products, 184 EU RU SD forward rates, 111
volatility, 189-190 futures, 113
expectations theory, 207 outrights and swaps, 112-113
expense ratio, 33 U S D C A D , forward quotes, 112
expiration date, 50, 148 real vs. nominal interest rates, 116
extendable reset bond, 217 uncovered interest parity, 118
external fraud, 3 forex trading, 113
forward contracts, 46, 48-49
F forward outright transaction, 112
face value, 200 forward prices, 49, 166-167
Fair Isaac Corporation (FIC O ), 228 forward rate agreem ents (FRAs), 205, 248-249, 263
fallen angels bonds, 217 forward rates, 204-205, 264
Fannie M ae, 10, 229, 231 forward start options, 185
Federal D eposit Insurance Corporation (FD IC ), 24 FRA . See forward rate agreem ent (FRA)
federal funds rate, 197 Freddie Mac, 10, 229, 231
Federal Home Loan M ortgage Corporation (FH LM C), 10, 229 fund m anagem ent
Federal National M ortgage Association (FN M A), 10, 229 ETF, 34
Federal Reserve Board, 139 hedge funds
Feldhutter, P., 222 fees for strategies, 36, 37
F IC O . See Fair Isaac Corporation (FICO ) prime brokers, 37
fill-or-kill order, 89 types of, 38-39
Financial Accounting Standard Board (FASB), 90 mutual funds, 32-34
financial assets, 124, 136 undesirable trading behavior, 34-35
financial covenants, 216 fund m anagers, 32
Financial Stability Board (FSB), 74 future basis (bt), 100
firm com m itm ent, 6 future markets
first notice day, 87 accounting, 90
Fitch Ratings, 116, 216 contracts, specification of
fixed-incom e arbitrage, 39 contract size, 85
fixed-rate bonds, 218 delivery location, 85
fixed-rate m ortgages, 226 delivery tim e, 85-86
flight to quality, 217 position limits, 86
floating rate bonds, 218 price limit, 86
floating-rate notes (FRN s), 218 price quotes, 86
Ford, 38 underlying asset, 85
foreign currency, 113 delivery m echanics, 87
foreign exchange markets exchanges, 84
covered interest parity, 116-118 forward and futures contracts, 90-91
interpretation of points, 118 futures prices, patterns of, 87-88
estim ating FX risk m arket participants, 88
econom ic risk, 114 operation, of exchanges, 84-85
transaction risk, 113 placing orders, 88-89
translation risk, 113-114 regulation of, 89-90
exchange rates futures commission m erchants, 88
balance of paym ents, 115 futures price (F q), 100
inflation, 115-116 vs. expected future spot price, 143
m onetary policy, 116 patterns of, 87-88
trade flow s, 115 futures-style option, 184
growth, 110 FX swap transaction, 112
Index ■ 277
G Ho, T. S. Y., 249
Hull, J ., 103, 149, 207, 221
gap options, 185
Hunter, Brian, 37
G eneral Motors, 38, 218
hurdle rate, 37
Ginnie M ae, 10, 229
hybrid funds, 32
global financial crisis, 124
global macro, 38
good-till-cancelled order, 89 I
governm ent borrowing rates, 196
index arbitrage, 131
G overnm ent National M ortgage Association
index funds, 33
(G N M A ), 10, 229
index options, 152
governm ent-sponsored enterprises, 229
initial public offering (IPO s), 7
G-20 Pittsburgh summit, regulation, 74
insolvencies, 24
Graham , Ben, 38
institutional investors, 34
gross dom estic product (G D P), 48
insurance com panies
group life insurance, 19
life insurance
annuity contracts, 20
H endow m ent life insurance, 19
health insurance, 23 group life insurance, 19
heating degree days (HDDs), 138 investm ents, 21
hedge effectiveness, 101 longevity risks, 20-21
hedge funds term life insurance, 19
fees for strategies, 36, 37 whole life insurance, 16-18
perform ance of, 40 m ortality tables, 16, 17
prim e brokers, 37 regulation, 24
types of Intercontinental Exchange (IC E), 137
convertible arbitrage, 39 interdealer brokers, 47
dedicated short, 38 interdealer m arket, 47
distressed debt, 38 interest coverage, 216
em erging m arkets, 39 interest-only securities (lOs), 230
fixed-incom e arbitrage, 39 interest rate futures
global macro, 38 day count conventions, 242-243
long-short equity, 38 duration-based hedging, 250-251
managed futures, 38 Eurodollar futures
m erger arbitrage, 38 vs. forward rate agreem ents, 248-249
hedge ratio, 100 hedging, 250
hedgers, 52 libor zero curve, 249-250
hedging quotes, 248
basis risk, 99-100 SO FR futures, 250
cash flow considerations, 105 treasury bills, price quotes, 243-244
duration-based, 250-251 treasury bonds
equity positions, 102-104 cheapest-to-deliver bond option, 245-246
Eurodollar futures, 250 futures price calculations, 246-247
exotics, 190 price quotes, 243
exposure, 98-99 quotes, 245
im pact, 97 interest rates
long hedge, 96 bond, 218-219
long-term hedges, 104-105 bond valuation, 200-201
optimal hedge ratios, 100-102 com m odity, 141
profit, 99 com pounding frequency, 197-199
shareholders, prefering, 98 continuous com pounding, 199
short hedge, 96 convexity, 203-204
highwater mark clause, 37 defined, 196
high-yield bonds, 217 determ ining zero rates, 206
278 ■ Index
discounting, 200 known income case, 126-128
duration generalization, 127-128
limitations of, 203 known yield case, 128
m odified duration, 202-203 Kohlberg, Kravis, Roberts &. C o, 216
yield duration, 201-202 Kuala Lumpur Com m odity Clearing House, 72
forward rates, 204-205
level of, 232
rates category, 196-197 L
swaps, 74 Lando, D ., 222
based overnight rates, 260 last trading day, 87
confirm ations, 259 lease rate, investm ent com m odity, 139
day count issues, 258-259 Lee, S., 249
quotes, 259-260 Leeson, Nick, 53
trading, 260-263 Lehman Brothers, 32
valuation of, 263-265 leverage ratios, 216
term structure theories, 206-207 Libor, 258. See also London Interbank O ffered Rate
usual convention, 199 (Libor)
zero rates, 199-200 Libor-based sw aps, 260
internal fraud, 3 Libor forward rates, 264
International Accounting Standards Board (IASB), 90 Libor zero curve, 249-250
International Energy A gency, 137 Libor zero rates, 264-265
International Sw aps and Derivatives Association (ISDA), 75, 259 Lie, E „ 154
interoperability, 77 life insurance
in-the-money, 149 annuity contracts, 20
introducing brokers, 88 endow m ent life insurance, 19
inverted m arket, 87 group life insurance, 19
investm ents, 21 investm ents, 21
asset, 124 longevity risks, 20-21
banking term life insurance, 19
advisory services, 8-9 whole life insurance, 16-18
Dutch auctions, 7-8 limit down, 86
IPO s, 7 limit orders, 89
offerings, 6 limit up, 86
trading, 9 linear regression, 101
underwriting, 6 liquidity, 215
com m odity, lease rate for, 139 liquidity coverage ratio, 5
expected return from bond, 221-222 liquidity preference theory, 207
grade bonds, 215, 217 liquidity ratios, 5
incom e, 18, 19 loans, geographical distribution, 228
risk, 24 loans, securitization of, 10
loan-to-value ratio (LTV), 228
J locals, 88
lock-in effect, m ortgage rate, 232
Jap an ese yen, 116
London Interbank O ffered Rate (Libor), 196, 197
Jen sen , M ichael, 39
longevity risks, 20-21
Je tt, Jo sep h , 126
long hedge, 96
long-short equity, 38
K long-term corporate bonds, 21
Kani, I., 190 long-term equity anticipation securities (LEA P S), 151
Kerviel, Jero m e, 53 long-term hedges, 104-105
Keynes, J . M ., 141 lookback options, 187
Kidder Peabody, 126 loss mutualization, 77
Km art, 218 loss ratios, 23
knock-in options, 188 lower bound, dividends, 162
Index ■ 279
M multi-asset funds, 32
mutual fund research, 39-40
macaulay duration, bond, 202
mutual funds, 32
M adan, D. B., 221
closed-end funds, 33-34
m aintenance and replacem ent funds, 221
open-end funds, 32-33
make-whole call provision bond, 220
Malaysian exchange, 72
managed future strategies, 38 N
margin options, exchange-traded options, 153 Nasdaq contract, 85
market-if-touched (MIT) order, 89 Nasdaq Stock M arket, 130
market-not-held order, 89 National Association of Insurance Com m issioners, 24
m arket orders, 89 National Futures Association (NFA), 89
m arket participants, 88 natural gas, 137
M arket Risk Am endm ent, 4 futures prices, 138
m arket risks, 2, 21 negative covenants, 216
m arket segm entation theory, 206 negative pledge clause, 220
maturity, bond, 218 negative return, 32
maturity date, 50, 148 net asset value (NAV), 32
maturity options, 151 net stable funding ratio, 5
asset price, 166 netting, 60
stock price, 165 New York M ercantile Exchange (N YM EX), 84
M BSs. See m ortgage-backed securities (MBSs) New York Stock Exchange, 131, 148
M cCauley, R. N ., 116 Nikkei 225 Index, 131
M cGuire, P., 116 no-arbitrage argum ents, 124, 125, 139
m erger arbitrage, 38 no-arbitrage equation, 131
metal com m odity, 137, 138 no income case, 124-126
M etallgesellschaft, 105 nominal interest rates, 116
minimum capital requirem ent (M CR), 24 non-dividend-paying stock, 124
modern theory, 141-143 non-investment assets, 124
m odified duration, bond, 202-203 non-standard american options, 184
Mogi, C ., 116 non-standard products, 152
money m arket funds, 32 normal accounting, 90
money m arket instrum ents, 242-243 normal backw ardation, 143
m oneyness, 149 normal m arket, 87
Monte Carlo sim ulation, 233 notional principal, 258
M oody's, 216
moral hazard, 6, 24
m ortality tables, 16, 17 O
m ortgage-backed securities (MBSs) offerings, 6
agency m ortgage-backed securities private placem ents, 6
dollar roll, 230 public offerings, 6
non-agency, 231 O 'H ara, M „ 215
other agency products, 230-231 oil futures, 87
pass-throughs trading, 229-230 open-end funds, 32-33
monthly payments calculation, 226-228 open order, 89
m ortgage pools, 228 open-outcry system , 47
non-agency, 231 operational risks, 3, 24
option adjusted spread, 235-236 optimal hedge ratios, 100-102
pool valuation, 232-235 option adjusted spread (O A S), 235-236
prepaym ent m odel, 231-232 options
m ortgage pools, 228 call options, properties
m ortgage portfolios. See m ortgage pools Am erican vs. European options, 160-161
m ortgages, 19 dividends im pact, 161-162
280 ■ Index
em ployee stock options, 161 calendar spread, 177
lower bound, 162 cash-or-nothing call, 186
exchange-traded products (ETPs), 152 cash-or-nothing put, 186
history of, 148 gap options, 185
margins, 153 straddle, 178
properties strangle, 178
forward prices, 166-167 pension plans, 21-22
put-call parity, 164-166 defined benefit plan, 21
put options, 162-164 defined contribution plan, 21
O ptions Clearing Corporation (O C C ), 148, 151, 153 per yield, 201
options markets plain vanilla options, 184
calls and puts policyholder, 16
m oneyness, 149 pool's factor, 228
payoffs, 150-151 position traders, 88
profits from call options, 149-150 positive covenants, 216
profits from put options, 150 prem ium s, 16
convertible bond, 154 prepaym ent
em ployee stock options, 154 curtailm ents, 232
exchange-traded options on stocks, 151-152 defaults, 232
margin requirem ents, 153 loan, 228
over-the-counter m arket, 153 option, 226
trading, 152-153 refinancing, 231-232
warrants, 153-154 turnover, 232
organized trading facilities (O TFs), 74 present value, 200
O T C . See over-the-counter (O TC ) markets price limits, 86
out-of-the money, 149 pricing financial forw ards and futures
outright transaction, 112 exchange rates, 130
overnight interbank borrowing, 197 forward vs. futures, 129-130
overnight rate, interest rates, 260 known income case, 126-128
over-the-counter (O TC ) m arkets, 46-48, 58, 148. known yield case, 128
See also swaps no income case, 124-126
advantage of, 153 short selling, 124
bilateral netting, 64-65 S&P 500 futures, 130
collateral, 65 stock indices
confirm ation, 259 index arbitrage, 130-131
credit default sw aps, 65-66 indices not representing tradable portfolios, 131
D P C , 65 valuing forward contracts, 128-129
SPV, 65 prime brokers, 37
principal-only securities (PO s), 230
P principal protected note (PPN), 172-173
private health insurance, 23
packages, 184
private placem ent bonds, 214
Parisian options, 188
private placem ents, 6
par value, 200
profits
pass-through agency securities trade, 229-230
endow m ent, 19
payment-in-kind bond, 217
from long position, 124
payoffs, 150-151
from short sale, 124
asset-or-nothing call, 186
program trading, 131
asset-or-nothing put, 186
property and casualty insurance, 22-23
bear spread, 174-175
proportional adjustm ent clause, 37
box spread, 175
public bond issue, 214
bull spread, 173-174
public offerings, 6
butterfly spread, 176
Index ■ 281
Public Securities Association (PSA), 232 share-for-share exchange, 8
purchasing pow er parity, 115, 116 short forward position, 49
Put and Call Brokers and D ealers Association, 148 short hedge, 96
put-call parity, 148, 164-166 shorting an option, 150
put options, 50, 51 short selling, 124
Am erican vs. European options, 163-164 short-term interest rates, 207
m oneyness, 149 Simons, Jim , 35, 36, 40
payoffs, 150-151 single assets, exotic options
profits from, 150 asian options, 187
barrier options, 188
Q binary options, 186-187
chooser options, 186
Quantum Fund, 39
cliquet options, 185
R com pound options, 188-189

forward start options, 185
real vs. nominal interest rates, 116
gap options, 185
recovery rate bond, 221
lookback options, 187
refinancing prepaym ent, 231-232
non-standard american options, 184
regulatory capital, 4
packages, 184
reinsurance com pany, 22
zero-cost products, 184
Renaissance Technologies, 35, 38
single monthly m ortality rate (SM M ), 228
repo rate, 196-197
single option, trading strategies, 172-173
research on returns
sinking fund, 220
hedge fund research, 40
Societe G enerale, 53
mutual fund research, 39-40
SO FR . See Secured O vernight Financing Rate
Reserve Primary Fund, 32
(SO FR)
restrictive covenants, 216
solvency capital requirem ent (SCR), 24
risk-based deposit insurance prem ium s, 6
Solvency II, 24
risk-free interest rates, 142, 197, 260
Soros, G eorge, 36, 39
risk-return trade-off, 36
sovereign debt, 39
risks, in banking
special purpose entities (SPE), 65
credit risks, 2-3
special purpose vehicle (SPV), 10, 65, 231
m arket risks, 2
specified pools, M BSs, 229
operational risks, 3
speculators, 52-53
risks, insurance com panies
S&P 500 Index, 40, 142, 216, 266
adverse selection, 24
spot rate, 199-200
moral hazard, 24
spread trading strategies
RJR Nabisco, 217
Rusnak, Jo h n , 53
box spread, 175
S butterfly spread, 175-177
Sage, Russell, 148 calendar spread, 177
scalpers, 88 stack and roll strategy, 104
Secured O vernight Financing Rate (SO FR), 196, 250 standard contracts, 58
Securities and Exchange Com m ission (SEC ), 34, 187 Standardized models vs. internal m odels, 4-5
Securities Industry and Financial M arkets Association (SIFM A), 229 standard vs. non-standard transactions, 74-75
securitization, 10 step-up bond, 217
selling. See shorting an option Sterling O vernight Index A verage (SO N IA ), 196, 197, 260
settlem ent prices, 86 stock index futures, 102
crude oil futures, 88 stock price, option maturity, 165
for gold futures, 88 stock splits, 152
Shanghai Futures Exchange, 84 stop-limit orders, 89
282 ■ Index
stop-loss order, 89 cheapest-to-deliver bond option, 245-246
straddle, 177-178 futures price calculations, 246-247
strangle, 178 futures settlem ent prices, 245
strike prices, 50, 148, 151-152 price quotes, 243
strips, 126 quotes, 245
survival probability, 16 treasury contracts, 244
Susko, V., 116 treasury notes, 243
swap execution facilities (SEFs), 74 treasury rate, 196
swaps, 197 "tulip m ania,"148
credit risk, 267 turnover prepaym ents, 232
currency sw aps, 265-266
interest rate
U
based overnight rates, 260
UBS, 53
confirm ations, 259
day count issues, 258-259 uncleared transactions, 74
quotes, 259-260 uncovered interest parity, 118
trading, 260-263 underlying(s), 46

underlying assets, 48, 85
valuation of, 263-265
other, 266-267 underwriting, 6
risk-free rate, 260 risk, 24

undesirable trading behavior
Swedish krone, 116
directed brokerage, 35
Swiss franc, 116
front running, 35
system ic risk, 73
late trading, 34-35
m arket tim ing, 35
T United Airlines, 218
United States' Com m odity Exchange A ct, 88
tailing the hedge, 101
United States dollar (USD), 2
term life insurance, 19
United States Social Security Adm inistration, 16
term structure, interest rate, 196, 206-207
universal life insurance, 18
Thales of M iletus, 148
unsecured bonds, 219
tim e value, option, 160
U .S. dollar (USD), 110
to-be-announced (TBAs), 229, 230
tracking error, 33 U .S. m ortgage m arket, 5
U .S. Securities and Exchange Com m ission, 154
trading. See also dollar roll
U.S. Treasury, 229
bond, 214-215
usual convention, interest rates, 199
options m arkets, 152-153
strategy, 125
volum e, 85 V
trading book vs. banking book, 5
variability of profits, 99
trading strategies
variable life insurance, 18
manufacturing payoffs, 178-179
variable rate bonds, 218
single option, 172-173
variable-rate m ortgages, 226
spread trading strategies
variance swap, 190
volatility swap, 189, 267
box spread, 175
Volcker rule, 215
butterfly spread, 175-177
calendar spread, 177 W
straddle, 177-178 w arrants, 153-154
strangle, 178 w eather com m odity, 138
Treasury bills, 124, 243-244 w eeklys and long-term options, 151
treasury bonds w eighted-average coupon (W AC), 228
Index 283
w eighted-average FIC O score, 228
Z
w eighted-average maturity (W AM ), 228
zero-cost products, 184
W est Texas Interm ediate (WTI) crude oil, 137
zero-coupon bonds, 124, 184, 219
whole life insurance, 16-18
zero-coupon interest rate, 199-200, 206
wild card play, 246
zero-coupon treasury bonds, 126
writing option, 150
zero curve, 206
zero rates, 199-200, 206
Y Zhang, F. X ., 221
Yerm ack, D ., 154 Zhou, A ., 215
284 Index

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What are some of the main risks banks face?

What are some of the main risks banks face?

What are some types of life insurance discussed?

What are some types of life insurance discussed?

Financial Risk Manager

Printed in the United States of Am erica

Pearson ISBN 10: 0135968194

Chapter 10 Pricing Financial 11.5 Cost of Carry 140

Chapter 16 Properties 17.1 Bond Issuance 214

16.5 Discounting 200 17.7 Debt Retirement 220

Michelle McCarthy Beck, SMD Dr. Victor Ng, CFA, MD

William May, SVP

Describe investment banking financing arrangem ents

2 ■ Financial Risk Manager Exam Part I: Financial Markets and Products

Operational Risks cussed further in Chapter 7 of Valuation and Risk M odels.

Operational risk is defined by bank regulators as:4

3 This is explained in more detail in C hapter 6 of Valuation and Risk Capital

5 See C hapter 6 of Valuation and Risk M o d els for further discussion.

4 ■ Financial Risk Manager Exam Part I: Financial Markets and Products

1. A best efforts arrangem ent where shares will be sold at the

In the case of a private placem ent, the investment bank receives

6 ■ Financial Risk Manager Exam Part I: Financial Markets and Products

Price Realized = USD 55 + 15 +50

Price Realized = USD 48 + 15 -2 0

The advantage of using investm ent banks to handle an IPO is C 30,000 70

8 ■ Financial Risk Manager Exam Part I: Financial Markets and Products

Since the 1990s, banks have used the originate-to-distribute

10 ■ Financial Risk Manager Exam Part I: Financial Markets and Products

Short Concept Questions

1.5 Explain the difference between a private placem ent and a

1.14 W hat is a poison pill? Give three exam ples. E 9,000 70

12 ■ Financial Risk Manager Exam Part I: Financial Markets and Products

14 ■ Financial Risk Manager Exam Part I: Financial Markets and Products

d escrib e pension plans after presenting the activities of life

M ortality tables are used extensively by actuaries for setting life

16 ■ Financial Risk Manager Exam Part I: Financial Markets and Products

30 0.001498 0.97520 47.86 0.000673 0.98641 52.06

31 0.001536 0.97373 46.93 0.000710 0.98575 51.10

32 0.001576 0.97224 46.00 0.000753 0.98505 50.13

33 0.001616 0.97071 45.07 0.000805 0.98431 49.17

50 0.004987 0.92913 29.64 0.003189 0.95794 33.24

51 0.005473 0.92449 28.79 0.003488 0.95488 32.34

52 0.005997 0.91943 27.94 0.003795 0.95155 31.45

53 0.006560 0.91392 27.11 0.004105 0.94794 30.57

70 0.023380 0.73427 14.32 0.015612 0.82818 16.53

71 0.025549 0.71710 13.66 0.017275 0.81525 15.78

72 0.027885 0.69878 13.00 0.019047 0.80117 15.05

73 0.030374 0.67930 12.36 0.020909 0.78591 14.34

90 0.164525 0.18107 4.08 0.129475 0.29650 4.85

91 0.181600 0.15128 3.79 0.144443 0.25811 4.50

92 0.199884 0.12381 3.52 0.160590 0.22083 4.18

93 0.219331 0.09906 3.27 0.177853 0.18536 3.88

(1 - 0.001498) X 0.001536 = 0.001534 0.75294 X 0.023380 = 0.017604

Chapter 2 Insurance Companies and Pension Plans ■ 17

Time (Years) Expected Payout Discount Factor5 PV of Expected Payout

0.5 0.003189 0.9759 0.003112

1.5 0.003477 0.9294 0.003232

2.5 0.003770 0.8852 0.003337

Suppose that the premium charged is USD 15,000 per year. It is

On any one policy, the results are uncertain. If the insur

of exposure to Florida hurricanes and wants to reduce this expo Dividends 1%

In Chapter 1, we explained that the Basel Com m ittee deter

D. 17,767 2 .1 8 A life insurance company issues whole life insurance poli

d) D irected Brokerage. This involves an informal arrange

The date specified in an options contract is known as the exp i