Financial Risk

Management
Nguyen Thu Hang
nguyenthuhang.cs2@ftu.edu.vn
 Outline
• Chapter 1: Overview of Financial Risk
Management
• Chapter 2: Forward and Futures and Applications
Chapter 3: Forward and Futures Pricing
• Chapter 4: Swap contracts, pricing and
applications
• Chapter 5 : Options and applications
• Chapter 6: Option Pricing and Module Wrap-up
 Assessment
• Performance: 10%
• Mid-term test: 30%
• Final term test : 60%
 Course material
• Options, Futures and other derivatives, 10e by
John Hull (2018):
• Main contents: Chapter 1,2,3,4,5,7,10,11,12
and 13.
 CHAPTER 1

INTRODUCTION TO RISK
MANAGEMENT
Outline
I. Interest rate, return and risk
1. Interest rate
2. Return
3. Risk
4. Risk preference
II. Risk management
1. Impact of financial risk management
2. Derivatives
- Concepts
- Ways derivatives are used
Interest rate
• For a simple loan

• For a fixed payment loan

• For a coupon bond:

Interest rate and time value of money
• The future value of PV after n years:
- Interest is paid once per year
- Interest is paid m time per year

- R : discrete /periodic interest rate

- Interest is paid continuously:

 R : continuous interest rate  Denoted as Rc

or r in the following slides
Effective annual rate:
• Ex: A bank quotes an interest of 8% per
annum (called simple annual rate) with
quarterly compounding. What is the
effective annual rate (equivalent annual
interest rate)?
Continuous compounding rate
• Ex: A bank quotes an interest of 8% per
annum (called simple annual rate) with
quarterly compounding. What is the
equivalent rate with continuous
compounding?
Effective annual rate and continuous
compounding rate
 Problems
1. What rate of interest with continuous
compounding is equivalent to 15% per
annum with monthly compounding?
2. A deposit account pays 12% per annum with
continuous compounding, but actually paid
quarterly. How much interest will be paid
each quarter on a $10000 deposit?
3. Techcombank quotes an interest rate of 14%
per annum compounded quarterly. What are
equivalent annual and continuous rates?
4. What rate of interest with continuous
compounding is equivalent to 15% per annum
with monthly compounding?
5. A deposit account pays 12% per annum with
continuous compounding, but interest is
actually paid quarterly. How much interest
with be paid each quarter on a $10,000
deposit.
• Bond pricing
- With a periodic interest rate

- With a continuous interest rate

- With different continuous interest rates for

each payment:
Ex: A 2 year T-bond with a principal of $100
provides coupon at the rate of 6% per annum
semiannually. Calculate the theoretical price
of the bond?

Maturity Zero rate (%)

(continuously
compounded)
0.5 5.0
1.0 5.8
1.5 6.4
2.0 6.8
 Bond Yield
• The bond yield is the discount rate that makes
the present value of the cash flows on the
bond equal to the market price of the bond
• Suppose that the market price of the bond in
our example equals its theoretical price of
98.39
• The bond yield (continuously compounded) is
given by solving

to get y=0.0676 or 6.76%.

 Problem
6. Suppose that 6-month, 12-month, 18-month,
24-month and 30-month zero rates are
respectively, 4%, 4.2%, 4.4%, 4.6% and 4.8%
per annum, with continuous compounding.
Estimate the price of a bond with a face value
of 100 that will mature in 30 months and pays
a coupon of 4% per annum semiannually.
 Zero Rates
A zero rate (or spot rate), for maturity T is the rate
of interest earned on an investment that provides
a payoff only at time T
Calculate Treasury zero rates- Bootstrap method
Bond Time to Annual Bond Cash
Principal Maturity Coupon Price
(dollars) (years) (dollars) (dollars)

100 0.25 0 97.5

100 0.50 0 94.9

100 1.00 0 90.0

100 1.50 8 96.0

100 2.00 12 101.6

 7. Calculate Treasury zero rates
Bond Time to Annual Bond Cash
Principal Maturity Coupon Price
(dollars) (years) (dollars) (dollars)

100 0.50 0 98

100 1.00 0 95

100 1.50 6.2 101

100 2.00 8.0 104

Half the stated coupon is assumed to be paid every six months

Calculate zero rates for maturities of 6 months, 12 months, 18
months and 24 months.
 Forward Rates

The forward rate is the future zero rate

implied by today’s term structure of interest
rates
 Calculation of Forward Rates

n-year Forward Rate

zero rate for n th Year
Year (n ) (% per annum) (% per annum)

1 3.0
2 4.0 5.0
3 4.6 5.8
4 5.0 6.2
5 5.3 6.5
 Formula for Forward Rates
• Suppose that the zero rates for time periods T1 and
T2 are R1 and R2 with both rates continuously
compounded.
• The forward rate for the period between times T1
and T2 is
 Problem
8. The 6-month, 12-month, 18-month and 24-
month zero rates are 4%, 4,5%, 4,75% and
5%, with semiannual compounding.
a. What are the rates with continuous
compounding?
b. What is the forward rate for the 6-month
period beginning in 18 months?
 9. The following table gives the prices of bonds
Bond Time to Annual Bond Cash
Principal Maturity Coupon Price
(dollars) (years) (dollars) (dollars)

100 0.50 0 98

100 1.00 0 95

100 1.50 6.2 101

100 2.00 8.0 104

Half the stated coupon is assumed to be paid every six months

a. Calculate zero rates for maturities of 6
months, 12 months, 18 months and 24
months.
b. What are the forward rates for the following
periods: 6 months to 12 months, 12 months
to 18 months and 18 months to 24 months.
c. Estimate the price and yield of 2 –year bond
providing a semiannual coupon of 7% per
annum.
10. Which security has a higher effective annual
interest rate?
(a) A three-month T-bill selling at $97, 645 with
face value of $100, 000.
(b) A coupon bond selling at par and paying a
10% coupon semi-annually.
 Return and Risk
• Defining Financial Risk & Return
• Define return as the total gain or loss
experienced on an investment over a given
period of time.  How to measure return?

• Define risk as the variability of returns

associated with a given asset.  How to
measure risk?
 Measures of return
• Simple return: Return measured as the change in an asset's value plus
any cash distributions (dividends or interest payments). (Holding period
return)
• Continuously compounded return: see in advanced materials

• Where Pt+1 = price (value) of asset at time t+1;

Pt = price (value) of asset at time t;
Ct+1 = cash flow paid by time t+1

Calculate yearly, monthly, daily holding period returns (HPR) 

Yearly return, monthly return, weekly return and daily return.
• Simple return for a single and multi-
periods (using dividend-adjusted prices)
Simple return for a period

Simple return for multi-periods

- Compute the annualized return from a one
-month return:

- Compute the annualized return from a

one-week return
• Ex: Stock A was bought for VND 30,000

in January 2013 and sold for VND40,000

in April 2013. Compute the simple return

for the three-month period and its

annualized return?
 Continuously compounded return
• rt : monthly continuous return.
• Rt: monthly simple return
• Compute the annualized return from a one -
month return/ a one week return:
Realized Return Versus Expected Return
• Realized (ex post) return is easily computed:
– Calculate yearly, monthly, daily holding period returns (HPR)
• Real financial decisions, however, are based on expected (ex ante)
returns, not realized returns:
– Realized return (at best) useful in estimating expected return
• Can specify conditional or unconditional expected returns
– Conditional expected return: “If the economy improves next year, the
asset’s return is expected to be 12%.” Or could be conditional on
return on overall stock market.
– Unconditional expected return: “The asset’s return next year is
expected to be 12%.”
 EXAMPLE 1: Expected Return
What is the expected return on an Exxon-Mobil bond if the return
is 12% two-thirds of the time and 8% one-third of the time?
Solution
The expected return is 10.68%.
R e = p1 R 1 + p2 R 2
where
p1 = probability of occurrence of return 1 = 2/3 = .67
R1 = return in state 1 = 12%= 0.12
p2 = probability of occurrence return 2 = 1/3 = .33
R2 = return in state 2 = 8% = 0.08
Thus
Re = (.67)(0.12) + (.33)(0.08) = 0.1068 = 10.68%
• Expected return – General equation

E(R) =Expected return
n = Number of states
Ri= return in state i
pi= Probability of occurrence of state i
• It’s rarely feasible to specify the full
distribution of possible returns.
• Use the average of historical returns as a
measure of expected return:

• Generate expected return based on a specific

asset pricing model, such as CAPM
Measures of Risk
• Standard deviation: is a measure of the
dispersion of a set of returns around their
expected value.
• Beta: (systematic risk) measures the degree to
which the stock moves with the overall market.

• Volatility
EXAMPLE 2: Standard Deviation (a)
Consider the following two companies and
their forecasted returns for the upcoming year:
EXAMPLE 2: Standard Deviation (b)
• What is the standard deviation of the returns
on the Fly-by-Night Airlines stock and Feet-on-
the-Ground Bus Company, with the return
outcomes and probabilities described above?
Of these two stocks, which is riskier?
EXAMPLE 2: Standard Deviation (c)
• Solution
– Fly-by-Night Airlines has a standard deviation of returns of 5%.
EXAMPLE 2: Standard Deviation (d)
• Feet-on-the-Ground Bus Company has a standard
deviation of returns of 0%.
EXAMPLE 2: Standard Deviation (e)
• Fly-by-Night Airlines has a standard deviation of
returns of 5%; Feet-on-the-Ground Bus Company has
a standard deviation of returns of 0%
• Clearly, Fly-by-Night Airlines is a riskier stock because
its standard deviation of returns of 5% is higher than
the zero standard deviation of returns for Feet-on-the-
Ground Bus Company, which has a certain return
• Standard deviation- general equation

• It’s rarely feasible to specify the full

distribution of possible returns and expected
variance.
– Must know all possible outcomes & associated
probabilities
• Instead, analysts usually gather historical data
and use these to generate expected return
and variance
• Uncorrected sample standard deviation/
standard deviation of the sample

• Corrected sample standard deviation

Return and risk of a portfolio
 The Historical Trade-Off Between Risk & Return
(1926-2000)

Average Annual Average Risk

Rate of Return Premium (Extra
Return vs. Treasury
Portfolio Nominal Real
Bills)

Treasury Bills 3.9 0.8 0

Government Bonds 5.7 2.7 1.8
Corporate Bonds 6.0 3.0 2.1
Common Stocks (S&P 500) 13.0 9.7 9.1
Small Firm Common Stocks 17.3 13.8 13.4
Figures are in percent per year.
The Historical Trade-Off Between Risk & Return
(1926-2000)

Portfolio Standard
Deviation Variance
Treasury Bills 3.2 10.1
Government Bonds 9.4 88.7
Corporate Bonds 8.7 75.5

Common Stocks (S&P 500) 20.2 406.9

Small Firm Common Stocks 33.4 1118.4
Real future value of $ 1 invested in 1926

Real returns

660
267

6.6
Index

5.0

1 1.7

Source: Ibbotson Associates Year

 % Historical returns, U.S., 1926-2000

Year
Source: Ibbotson Associates
 Average risk by period

Period Market St.Dev.

(NYSE) (sm)
1926-1930 21.7
1931-1940 37.8
1941-1950 14.0
1951-1960 12.1
1961-1970 13.0
1971-1980 15.8
1981-1990 16.5
1991-2000 13.4
 Histogram of Return on Portfolio of Large Company Stocks
1926-2000

Number of
years

Return %
 Histogram of Return on Portfolio
of Large Company Stocks, 1926-2000
 Normal distribution

x – 3s x – 2s x–s x x+s x +2s x + 3s

Approximately 68% of the measurements

Approximately 95% of the measurements

Approximately 99.7% of the measurements

 Two Assets With Same Expected Return But
Different (Continuous) Probability Distributions
Probability Density

Stock 1

Stock 2

0 5 6 7 8 9 10 11 12 13 14 15
Return %
 The Volatility
• The volatility is the standard deviation of
the continuously compounded rate of
return in 1 year
• The standard deviation of the return in
time Dt is
• If a stock price is $50 and its volatility is
25% per year what is the standard
deviation of the price change in one day?
 Estimating Volatility from
Historical Data
1. Take observations S0, S1, . . . , Sn at
intervals of t years
2. Calculate the continuously
compounded return in each interval
as:

3. Calculate the standard deviation, s ,

of the ui´s
4. The historical volatility estimate is:
 Nature of Volatility
• Volatility is usually much greater when the
market is open (i.e. the asset is trading) than
when it is closed
• For this reason time is usually measured in
“trading days” not calendar days when options
are valued
 Risk Preferences: Comparing Two Assets With The
Same Expected Return
• Stocks 1 & 2 both have an expected return of 10%.
– Both offer 10% return in an average economy
– Stock 2 would have higher return if economy booms
– Stock 1 has lower return variability; does better in bad times
• Whether an investor would consider them equally attractive depends on
his/her degree of risk aversion (utility function)
– Risk averse investor prefers lower variability for given E(R)
– Risk seeking investor prefers higher variability for given E(R)
– Risk neutral investor is indifferent about variability
• Finance theory, common sense, and observed behavior all suggest investors
are risk averse
– If two assets offer equal E(R), will pick one with less variability
– Must be offered higher E(R) to accept higher variability
II. Risk management:
 Use derivatives to decrease the volatility of
future cash flows

Impact of Financial Risk Management

on Cash Flow Volatility
Like lihood

Cash Flow
 The Nature of Derivatives
A derivative is an instrument whose value
depends on the values of other more basic
underlying variables
• Futures Contracts
• Forward Contracts
• Swaps
• Options
 Forward Contracts
• A forward contract is an agreement to buy or sell an
asset at a certain time in the future for a certain price.
• A forward contracts are traded in the OTC market.
• Forward contracts are popular on currencies and
interest rates.
• There is no daily settlement (but collateral may have to
be posted). At the end of the life of the contract one
party buys the asset for the agreed price from the other
party.
• By contrast in a spot contract there is an agreement to
buy or sell the asset immediately (or within a very short
period of time).
 Futures Contracts
• A futures contract is an agreement to buy or sell
an asset at a certain time in the future for a
certain price
• Available on a wide range of underlying assets
• Traded in futures exchanges
• A range of delivery dates.
• Futures contracts are standardized by the
exchange
• Settled daily
 Delivery
• Delivery or final cash settlement rarely takes place with
futures contracts. They are normally closed out before
maturity.
• If a futures contract is not closed out before maturity, it is
usually settled by delivering the assets underlying the
contract. When there are alternatives about what is delivered,
where it is delivered, and when it is delivered, the party with
the short position chooses.
• A few contracts (for example, those on stock indices and
Eurodollars) are settled in cash
• When there is cash settlement contracts are traded until a
predetermined time. All are then declared to be closed out.
 Margins
• A margin is cash or marketable securities
deposited by an investor with his or her broker
• The balance in the margin account is adjusted
to reflect daily settlement
• Margins minimize the possibility of a loss
through a default on a contract
 Example of a Futures Trade
• An investor takes a long position in 2
December gold futures contracts
– contract size is 100 oz.
– futures price is US$1250
– margin requirement is US$6,000/contract (US$12,000
in total)
– maintenance margin is US$4,500/contract (US$9,000
in total)
Operation of margin account
Profit from a Long Forward or
Futures Position

Profit

Price of Underlying
at Maturity
Profit from a Short Forward or
Futures Position
Profit

Price of Underlying
at Maturity
 Forward Contracts vs Futures
Contracts
Forward Futures
Private contract between two parties Traded on an exchange
Not standardized Standardized
Usually one specified delivery date Range of delivery dates
Settled at end of contract Settled daily
Delivery or final settlement usual Usually closed out prior to maturity
Some credit risk Virtually no credit risk
 Foreign Exchange Quotes

• Futures exchange rates are quoted as the number of

USD per unit of the foreign currency
• Forward exchange rates are quoted in the same way
as spot exchange rates. This means that GBP, EUR,
AUD, and NZD are USD per unit of foreign currency.
Other currencies (e.g., CAD and JPY) are quoted as
units of the foreign currency per USD.
 Problems
10. A trader enters into a one-year short forward
contract to sell an asset for $60 when the
spot price is $58. The spot price in one year
proves to be $63. What is the trader's gain or
loss?
11. A company enters into a long futures
contract to buy 1,000 units of a commodity
for $20 per unit. The initial margin is $6,000
and the maintenance margin is $4,000. What
futures price will allow $2,000 to be
withdrawn from the margin account?
 Problems
12. A company enters into a short futures
contract to sell 50,000 pounds of cotton for
70 cents per pound. The initial margin is
$4,000 and the maintenance margin is
$3,000. What is the futures price above
which there will be a margin call?
 Options
• A call option is an option to buy a certain
asset (underlying asset) by a certain date
(expiration date or maturity) for a certain
price (the strike price or exercise price)
• A put option is an option to sell a certain
asset (underlying asset) by a certain date
(expiration date or maturity) for a certain
price (the strike price or exercise price)
 Options vs Futures/Forwards
• A futures/forward contract gives the holder
the obligation to buy or sell at a certain price
• An option gives the holder the right to buy or
sell at a certain price
 American vs European Options
• An American option can be exercised at any
time during its life
• A European option can be exercised only at
maturity
Option Positions

• Long call
• Long put
• Short call
• Short put
 European Call option-example (a)
• A European call option with a strike price of
$100 to purchase 100 shares of a certain
stock. The current stock price is $98, the
expiration date of the option is in 4 months,
and the price of an option to purchase one
share is $5.
 European Call option-example (b)
• On the expiration date,
- If ST(stock price = $115) is above $100 
The investor will choose to exercise Makes
a gain of $15 per share or $1500  A net
profit of $1000.
- If ST is less than $100  The investor will
choose not to exercise.  Losses $5 per
share of $500.
 Long Call

Profit from buying one European call option: option price =

$5, strike price = $100.

30 Profit ($)

20

10 Terminal
70 80 90 100 stock price ($)
0
-5 110 120 130
 Short Call
Profit from writing one European call option: option price = $5,
strike price = $100
Profit ($)
5 110 120 130
0
70 80 90 100 Terminal
-10 stock price ($)

-20

-30
 European put option-example (a)
• A European put option with a strike price of
$70 to sell 100 shares of a certain stock. The
current stock price is $65, the expiration date
of the option is in 3 months, and the price of
an option to sell one share is $7.
 European put option-example (b)
• On the expiration date,
- If ST(stock price) is below $70 (let’s say
$55) The investor will choose to exercise
Makes a gain of $15 per share or $1500 
A net profit of $800.
- If ST is above $70  The investor will choose
not to exercise.  Losses $7 per share of
$700.
 Long Put

Profit from buying a European put option: option price = $7,

strike price = $70
30 Profit ($)

20

10 Terminal
stock price ($)
0
40 50 60 70 80 90 100
-7
 Short Put

Profit from writing a European put option: option price = $7,

strike price = $70
Profit ($)
Terminal
7
40 50 60 stock price ($)
0
70 80 90 100
-10

-20

-30
• Payoff of the four positions on the date of maturity T
Google Option Prices (July 17, 2009;
Stock Price=430.25)

Calls Puts
Strike price Aug Sept Dec Aug Sept Dec
($) 2009 2009 2009 2009 2009 2009
380 51.55 54.60 65.00 1.52 4.40 15.00
400 34.10 38.30 51.25 4.05 8.30 21.15
420 19.60 24.80 39.05 9.55 14.70 28.70
440 9.25 14.45 28.75 19.20 24.25 38.35
460 3.55 7.45 20.40 33.50 37.20 49.90
480 1.12 3.40 13.75 51.10 53.10 63.40
 Exchanges Trading Options
• Chicago Board Options Exchange
• International Securities Exchange
• NYSE Euronext
• Eurex (Europe)
• and many more (see list at end of book)
 Problems
13. A trader buys 100 European call options with a
strike price of $20 and a time to maturity of one year.
The cost of each option is $2. The price of the
underlying asset proves to be $25 in one year. What
is the trader's gain or loss?
14. A trader sells 100 European put options with a
strike price of $50 and a time to maturity of six
months. The price received for each option is $4. The
price of the underlying asset is $41 in six months.
What is the trader's gain or loss?
 Problems
15. The price of a stock is $36 and the price of a three-month call
option on the stock with a strike price of $36 is $3.60. Suppose
a trader has $3,600 to invest and is trying to choose between
buying 1,000 options and 100 shares of stock. How high does
the stock price have to rise for an investment in options to be
as profitable as an investment in the stock?
16. A one-year call option on a stock with a strike price of $30
costs $3; a one-year put option on the stock with a strike price
of $30 costs $4. Suppose that a trader buys two call options
and one put option.
(i) What is the breakeven stock price, above which the trader
makes a profit? ……….
(ii) What is the breakeven stock price below which the trader
makes a profit? ……….
 SWAPS
• A swap is an agreement to exchange cash
flows at specified future times according to
certain specified rules.
• See in Chapter 3.
Ways Derivatives are Used
• To hedge risks
• To speculate (take a view on the future
direction of the market)
• To lock in an arbitrage profit
• To change the nature of a liability
• To change the nature of an investment
without incurring the costs of selling one
portfolio and buying another
 Hedging Examples
• A US company will pay £10 million for imports
from Britain in 3 months and decides to hedge
using a long position in a forward contract
• An investor owns 1,000 Microsoft shares
currently worth $28 per share. A two-month put
with a strike price of $27.50 costs $1. The investor
decides to hedge by longing put options.
Value of Microsoft Shares with and
without Hedging
 Speculation Example

• An investor with $2,000 to invest feels that

a stock price will increase over the next 2
months. The current stock price is $20 and
the price of a 2-month call option with a
strike of $22.50 is $1
• What are the alternative strategies? What
are their returns?
 Problems
17. You would like to speculate on a rise in the price of
a certain stock. The current stock price is $29 and a
3-month call with a strike price of $30 costs $2.90.
You have $5,800 to invest. Identify two alternative
investment strategies, one in the stock and the other
in an option on the stock. What are the potential
returns of the strategies?
18. Describe the profit from the following portfolio: a
long forward contract on an asset and a long
European put option on the asset with the same
maturity as the forward contract and a strike price
that is equal to the forward price of the asset at the
time the portfolio is set up.
 Arbitrage Example
• A stock price is quoted as £100 in London
and $162 in New York
• The current exchange rate is 1.6500
• What is the arbitrage opportunity?
The Law of One Price and arbitrage
• In a competitive market, if two assets are
equivalent, they will tend to have the same
market price.
• The Law of One Price is enforced by a process
called arbitrage.
• Ex: if the price of gold in Tokyo is $1200 per
ounce, what is its price in Seoul?