BBFH202
BBFH202
BBFH202
Accounting (Honours)
Bachelor of Commerce in
Banking & Finance (Honours)
Bachelor of Commerce in
Internal Auditing (Honours)
Business Finance I
Module BBFH202
Authors: I. Kwesu
Master of Science in Economics (UZ)
Bachelor of Science in Economics (UZ)
S. Zhanje
Master of Science in Economics (UZ)
Bachelor of Science in Economics (UZ)
Editors: S. Murevanhema
Bachelor of Science in Mathematics and Physics (UZ)
Grad CE (UZ)
Itumeleng Magadi
Master of Science in Banking and Financial Services (NUST)
Bachelor of Commerce in Banking (Honours) (NUST)
IOBZ Diploma
Mount Pleasant
Harare, ZIMBABWE
Layout: C. Nhari
ISBN: 978-1-77938-728-8
the errors), they still help you learn the correct thing as the tutor may dwell on matters irrelevant to the
as much as the correct ideas. You also need to be ZOU course.
open-minded, frank, inquisitive and should leave no
stone unturned as you analyze ideas and seek
clarification on any issues. It has been found that Distance education, by its nature, keeps the tutor
those who take part in tutorials actively, do better in and student separate. By introducing the six hour
assignments and examinations because their ideas are tutorial, ZOU hopes to help you come in touch with
streamlined. Taking part properly means that you the physical being, who marks your assignments,
prepare for the tutorial beforehand by putting together assesses them, guides you on preparing for writing
relevant questions and their possible answers and examinations and assignments and who runs your
those areas that cause you confusion. general academic affairs. This helps you to settle
down in your course having been advised on how
Only in cases where the information being discussed to go about your learning. Personal human contact
is not found in the learning package can the tutor is, therefore, upheld by the ZOU.
provide extra learning materials, but this should not
be the dominant feature of the six hour tutorial. As
stated, it should be rare because the information
needed for the course is found in the learning package
together with the sources to which you are referred.
Fully-fledged lectures can, therefore, be misleading
Note that in all the three sessions, you identify the areas
that your tutor should give help. You also take a very
important part in finding answers to the problems posed.
You are the most important part of the solutions to your
learning challenges.
Module Overview............................................................................................................................9
Unit 1 ..............................................................................................................................................10
Nature and Scope of Business Finance.......................................................................................10
1.0 Introduction ..................................................................................................................... 10
1.1 Objectives ....................................................................................................................... 10
1.2 Definition of Business Finance ....................................................................................... 10
1.3 Goals of Financial Management ..................................................................................... 11
1.4 Wealth Maximisation versus Profit Maximisation .......................................................... 11
Activity 1.1 ........................................................................................................................... 11
1.5 Shareholder Wealth Maximisation and Social Responsibility ........................................ 11
1.6 The Roles of the Financial Manager ............................................................................... 12
1.7 The Agency Problem ...................................................................................................... 13
1.8 Dealing with Agency Problems: Motivating Managers .................................................. 14
Activity 1.2 ........................................................................................................................... 14
1.9 Forms of Business Enterprise.......................................................................................... 15
1.9.1 Sole proprietorship ................................................................................................... 15
1.9.2 Partnership ................................................................................................................ 16
1.9.3 The corporate form of business ................................................................................. 17
1.9.4 Hybrids ..................................................................................................................... 18
Activity 1.3 ........................................................................................................................... 19
1.10 Summary ....................................................................................................................... 19
References............................................................................................................................. 20
Unit 2 ..............................................................................................................................................21
Analysis of Financial Statements ................................................................................................21
2.0 Introduction ..................................................................................................................... 21
2.1 Objectives ....................................................................................................................... 21
2.2 The Statement of Comprehensive Income (The income statement)................................ 21
2.3 The Statement of Financial Position (The balance sheet) ............................................... 23
2.4 The Cash Flow Statements .............................................................................................. 25
2.4.1 The purpose of cash flow statements ....................................................................... 25
2.4.2 Uses of cash flow statements..................................................................................... 25
2.4.3 The form of cash flow statements as stated by FRS1 ................................................ 25
Activity 2.1 ........................................................................................................................... 27
2.5 Financial Ratio Analysis ................................................................................................. 27
2.6 Classification of Ratios ................................................................................................... 28
2.6.1 Short- term solvency or liquidity ratios ..................................................................... 29
2.7 The Role of the Operating Cycle ..................................................................................... 29
2.8 Measures of Liquidity ..................................................................................................... 31
2.9 Profitability and Activity Ratios ..................................................................................... 32
2.10 Financial Leverage Ratios ............................................................................................. 33
2.10.2 Coverage financial leverage ratios ......................................................................... 35
2.11 Return Ratios and the Du Pont System .......................................................................... 35
2.11.1 Return on investment ratios ..................................................................................... 35
2.11.2 The Du Pont System ............................................................................................... 36
2.12 Shareholder Ratios ........................................................................................................ 37
2.13 Common Size Analysis ................................................................................................. 38
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2.14 Limitations of Ratios ..................................................................................................... 40
2.15 Summary ....................................................................................................................... 42
References............................................................................................................................. 43
Unit 3..............................................................................................................................................44
The Financial Environment: Markets, Institutions, Interest Rates and Taxes ..................44
3.0 Introduction ..................................................................................................................... 44
3.1 Objectives ....................................................................................................................... 44
3.2 Financial Markets............................................................................................................ 44
3.3 Market Classification ...................................................................................................... 44
3.4 Participants in the Domestic Money Market ................................................................... 45
3.5 Money Market Instruments ............................................................................................. 45
3.6 Capital Market Instruments ............................................................................................. 45
Activity 3.1 ........................................................................................................................... 46
3.7 Term Structure of Interest Rates ..................................................................................... 46
3.8 The Determinants of Market Interest Rates .................................................................... 46
Activity 3.2 ........................................................................................................................... 47
3.9 The Yield Curve.............................................................................................................. 47
3.9.1 The market segmentation theory .................................................................................. 47
3.9.2 The expectations theory .............................................................................................. 48
3.9.3 Liquidity preference theory ........................................................................................ 48
3.10 Significance of the Shape of the Yield Curve ............................................................... 48
Activity 3.3 ........................................................................................................................... 48
3.11 Taxation and Financial Decisions ................................................................................. 49
3.11.1 Components of taxable income............................................................................... 49
3.11.2 The deductions that allowed for tax purposes include: .......................................... 49
3.11.3 Capital allowances are a special feature in finance and in Zimbabwe they are in
two forms namely: ............................................................................................................. 50
3.11.4 The allowances are computed on cost using straight-line technique or diminishing
method. The original cost includes the following items: .................................................. 50
3.11.5 Wear and tear allowance ........................................................................................ 50
3.11.6 Special initial allowance......................................................................................... 50
3.11.7 Scrapping allowance .............................................................................................. 51
3.11.8 Recoupment ............................................................................................................ 51
3.12 Summary ....................................................................................................................... 51
References............................................................................................................................. 52
Unit 4 ..............................................................................................................................................53
Basic Financial Calculations and Time Value of Money .........................................................53
4.0 Introduction ..................................................................................................................... 53
4.1 Objectives ....................................................................................................................... 53
4.2 Future Value.................................................................................................................... 54
4.3 Determining the Unknown Interest Rate ......................................................................... 56
4.4 Application: Determining Growth Rates......................................................................... 57
4.5 Determining the Number of Compounding Periods........................................................ 58
4.6 Frequency of Compounding............................................................................................ 61
4.7 Continuous Compounding .............................................................................................. 62
4.8 The Annual Percentage Rate ........................................................................................... 62
4.9 Periodic Rate ................................................................................................................... 62
4.10 Effective Rates of Interest ............................................................................................. 63
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4.11 Continuous Compounding ............................................................................................ 64
4.12 Present Value of a Lump Sum (single cash flow) ......................................................... 65
4.13 Discounting Simple Cash Flow..................................................................................... 65
4.14 Non-Annual Discounting .............................................................................................. 66
Activity 4.3 ........................................................................................................................... 67
4.15 Valuing Series of Cash Flows ....................................................................................... 67
4.16 Present Value of Uneven Series of Cash Flows ............................................................ 68
4.17 Future Value of Uneven Series of Cash Flows ............................................................. 69
4.18 Annuities ....................................................................................................................... 71
4.19 Ordinary Annuities........................................................................................................ 72
4.19.1 Annuities Given Present Values ............................................................................. 74
4.19.2 Future value of ordinary (end-of-the-period) annuities .............................................. 74
4.19.3 Effects of Annuities Due......................................................................................... 76
4.19.4 Future value of an annuity due ............................................................................... 77
4.19.5 Growing annuities .................................................................................................. 77
4.20 Perpetuities.................................................................................................................... 78
4.21 Growing Perpetuities..................................................................................................... 79
4.22 Deferred Annuities ........................................................................................................ 79
Activity 4.6 ........................................................................................................................... 81
Activity 4.7 ........................................................................................................................... 82
4.24 Summary ....................................................................................................................... 82
References............................................................................................................................. 84
Unit 5 ..............................................................................................................................................85
Bond Valuation .............................................................................................................................85
5.0 Introduction ..................................................................................................................... 85
5.1 Objectives ....................................................................................................................... 85
5.2 What is a bond? ............................................................................................................... 85
5.3 Types of bonds ................................................................................................................ 85
5.3.1 Treasury bonds ......................................................................................................... 85
5.3.2 Corporate bonds ....................................................................................................... 86
5.3.3 Municipal bonds ....................................................................................................... 86
5.3.4 Foreign bonds ........................................................................................................... 86
Activity 5.1 ........................................................................................................................... 88
5.4 Key Characteristics of Bonds .......................................................................................... 88
5.4.1 Par value ................................................................................................................... 89
5.4.2 Coupon interest rate ................................................................................................. 89
5.4.3 Maturity date ............................................................................................................ 89
5.4.4 Provisions to call or redeem bonds .......................................................................... 89
5.4.5 Sinking funds............................................................................................................ 90
5.4.6 Bond indentures........................................................................................................ 90
5.4.7 Trustee ...................................................................................................................... 90
5.4.8 Costs of bonds to the issuer ...................................................................................... 90
5.4.9 Impact of bond maturity ........................................................................................... 90
5.4.10 Impact of offering size ........................................................................................... 90
5.4.11 Impact of issuer’s risk ............................................................................................ 91
5.4.12 Impact of the cost of money ................................................................................... 91
Activity 5.2 ........................................................................................................................... 91
5.5 Bond Valuation ............................................................................................................... 91
5.5.1 Valuing a straight-coupon bond ............................................................................... 92
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Activity 5.3 ........................................................................................................................... 93
5.5.2 Different value, different coupon rate, but the same return ..................................... 93
5.5.3 Valuing a zero-coupon bond .................................................................................... 95
Activity 5.4 ........................................................................................................................... 96
5.5.4 Returns on bonds ...................................................................................................... 96
5.5.5 Yield-to-maturity ...................................................................................................... 97
Activity 5.5 ........................................................................................................................... 99
5.5.6 Value of bonds as they approach maturity ............................................................... 99
5.5.7 Value of bonds as yields change .............................................................................. 99
5.5.8 Yield-to-call ............................................................................................................ 100
Activity 5.6 ......................................................................................................................... 101
5.6 Summary ....................................................................................................................... 101
References........................................................................................................................... 102
Unit 6 ............................................................................................................................................103
Valuation of Securities ...............................................................................................................103
6.0 Introduction ................................................................................................................... 103
6.1 Objectives ..................................................................................................................... 103
6.2 Differences Between Debt and Equity .......................................................................... 103
6.2.1 Voice in management ............................................................................................. 104
6.2.2 Claims on income and assets .................................................................................. 104
6.2.3 Maturity .................................................................................................................. 104
6.2.4 Tax treatment.......................................................................................................... 104
Activity 6.1 ......................................................................................................................... 104
6.3 Common and Preferred Stock ....................................................................................... 105
6.3.1 Features of common stock...................................................................................... 105
6.3.2 Preferred stock........................................................................................................ 106
6.3.3 Basic rights of preferred stockholders .................................................................... 106
6.3.4 Features of preferred stock ..................................................................................... 107
Activity 6.2 ......................................................................................................................... 107
6.4 Valuation of Common Stock......................................................................................... 108
6.4.1 Zero-growth model ................................................................................................. 108
Activity 6.3 ......................................................................................................................... 110
6.4.3 Variable-Growth Model ......................................................................................... 110
6.4.4 Free cash flow valuation model ............................................................................. 113
6.5 Other Approaches to Common Stock Valuation ........................................................... 115
6.5.1 Book value approach .............................................................................................. 115
6.5.2 Liquidation value.................................................................................................... 116
6.5.3 Price/Earnings (P/E) Multiples............................................................................... 116
Activity 6.3 ......................................................................................................................... 117
6.5.4 The Dividend Valuation Model and the Price-Earnings (P/E) ratio .............. 117
6.6 Holding Period Returns (HPR) ..................................................................................... 117
6.6.1 Return with No Dividends ...................................................................................... 118
6.6.2 Return with dividends at the end of the period ...................................................... 118
Activity 6.4 ......................................................................................................................... 119
6.7 Valuation of Preference Stock ...................................................................................... 119
Activity6. 5 ......................................................................................................................... 120
6.8 Summary ....................................................................................................................... 120
References........................................................................................................................... 121
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Unit 7 ............................................................................................................................................122
Risk and Return Fundamentals ................................................................................................122
7.0 Introduction ................................................................................................................... 122
7.1 Objectives ..................................................................................................................... 122
7.2 Definition of Risk ......................................................................................................... 122
7.3 Return of an Asset ......................................................................................................... 123
7.4 Risk Preferences............................................................................................................ 123
Activity 7.1 ......................................................................................................................... 124
7.5 Types of Risk ................................................................................................................ 124
7.5.1 Cash flow risk......................................................................................................... 124
7.5.2 Business risk ........................................................................................................... 124
7.5.3 Financial risk .......................................................................................................... 126
7.5.4 Operating and financial risk ................................................................................... 126
7.5.5 Default risk ............................................................................................................. 127
7.5.6 Reinvestment rate risk ............................................................................................ 128
7.5.7 Interest rate risk ...................................................................................................... 129
Activity 7.2 ......................................................................................................................... 130
7.5.8 Purchasing power risk ............................................................................................ 130
7.5.9 Currency risk .......................................................................................................... 131
7.6 Return and Risk............................................................................................................. 131
7.6.1 Expected return of an asset..................................................................................... 132
7.6.2 Risk of an asset ....................................................................................................... 133
7.6.3 Return and the tolerance for bearing risk ............................................................... 136
Activity 7.3 ......................................................................................................................... 136
7.7 Selecting Among Different Investments ....................................................................... 137
7.7.1 The mean variance rule .......................................................................................... 137
7.7.2 Coefficient of variation (CV). ................................................................................ 138
7.8 Summary ....................................................................................................................... 139
References........................................................................................................................... 140
Unit 8 ............................................................................................................................................141
Portfolio Theory and Capital Asset Pricing Model ................................................................141
8.0 Introduction ................................................................................................................... 141
8.1 Objectives ..................................................................................................................... 141
8.2 Diversification and Risk................................................................................................ 141
8.3 Common Sense Diversification..................................................................................... 145
8.4 Correlation Coefficient ................................................................................................. 146
8.5 Portfolio Size and Risk ................................................................................................. 148
8.6 Modern Portfolio Theory and Asset Pricing ................................................................. 151
8.7 The Capital Asset Pricing Model (CAPM) ................................................................... 151
8.8 Security Market Line (SML) ......................................................................................... 156
8.9 Characteristics of Betas ................................................................................................. 157
8.10 Portfolio Beta............................................................................................................... 159
8.11 Criticism and Reality of CAPM .................................................................................. 161
8.12 The Arbitrage Pricing Model ...................................................................................... 161
8.13 Financial Decision-Making and Asset Pricing ............................................................ 162
Activity 8.2 ......................................................................................................................... 163
8.14 Summary ..................................................................................................................... 163
References........................................................................................................................... 164
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Unit 9 ............................................................................................................................................165
The Cost of Capital .....................................................................................................................165
9.0 Introduction ................................................................................................................... 165
9.1 Objectives ..................................................................................................................... 165
9.2 The Calculation of Weighted Average Cost of Capital (WACC) ................................. 165
9.3 Determination of the Cost of Different Sources of Financing....................................... 166
9.3.1 The cost of debt ...................................................................................................... 166
Activity 9.1 ......................................................................................................................... 168
9.3.2 The cost of preference stock................................................................................... 168
Activity 9.2 ......................................................................................................................... 168
9.3.3 The cost of common stock ..................................................................................... 168
Activity 9.3 ......................................................................................................................... 173
9.4 Determining the Proportion of each Capital Component .............................................. 173
9.5 Computation of the Weighted Average Cost of Capital ................................................ 174
9.6 The Marginal Weighted Average Cost of Capital (MWACC) ...................................... 176
9.7 Weighting Schemes ...................................................................................................... 177
9.7.1 Book Value versus Market Value .......................................................................... 177
9.7.2 Historical versus target ........................................................................................... 178
9.8 Factors that Affect WACC ............................................................................................ 178
Activity 9.4 ......................................................................................................................... 179
9.9 Summary ....................................................................................................................... 181
References........................................................................................................................... 182
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Module Overview
Business Finance I is an introductory course in the study of money and its management. The
primary purpose of this basic course in finance is to introduce you to the world of
finance. It is intended to be a true survey of the field with material selected from the three
principal areas of finance: financial institutions and markets, investments, and business
finance. The study of financial institutions and markets is concerned with the institutional
aspects and encompasses the creation of financial assets, the markets for trading securities,
and the regulation of financial markets. Investments focus on the analysis of individual
assets and the construction of optimal portfolios. Business finance relates to the financial
manager's role in raising and administering capital in an efficient and profitable manner.
The synopsis of the broad areas covered by this course is given below:
• In Unit One we describe the nature and scope of business finance, and the different
forms of business are discussed.
• In Unit Two we discuss the three major forms of financial statements namely income
statement, balance sheet and cash flow statement with respect to ratio analysis.
• In Unit Three we discuss the financial environment in which the roles of financial
institutions and the attendant various financial instruments are examined.
• In Unit Four we cover the basic financial calculations and the time value of money
concepts, culminating in the valuation of various forms of annuities.
• In Unit Five we introduce you to bond fundamentals and the valuation of bonds.
• In Unit Six we walk you through the valuation of securities which include debt and
equity securities.
• In Unit Seven we discuss the risk and return fundamentals. The concept of risk is
covered in detail and this extends to calculation of the mean, variance and standard
deviation and coefficient variation of securities.
• In Unit Eight we look at the Portfolio Theory and the Capital Asset Pricing Model.
Portfolio expected return, variance and standard deviation of a two-asset portfolio are
determined.
• Lastly, in Unit Nine we describe the techniques employed in the determination of a
firm’s cost of capital and the optimal capital structure.
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BLANK PAGE
Unit 1
The financial decisions of a business enterprise must be made with some objective in mind.
This applies whether the decision is an investment decision (for example, buying equipment),
or a financing decision (for example, issuing bonds). In this unit, we look at the different
forms of business enterprise; in order to understand the context in which decisions are made.
We also put focus on the objective of financial management: the maximisation of owners'
wealth and then take a look at the agency relationship - when managers represent owners'
interests - and some of the problems this agency relationship may create.
1.1 Objectives
By the end of this unit you, should be able to:
• define business finance
• discuss the objective(s) of business and financial management
• explain the functions of a business finance manager
• link the three interrelated areas of finance
• distinguish the alternative forms of business organisation, their characteristics,
and the advantages and disadvantages of each
• explain the concept and issues of “agency
Investment decisions are concerned with how the firm allocates resources to assets that
generate income now and in the future. Investment decisions include capital budgeting
processes and techniques, capital rationing, financial markets and security price analysis.
Financing decisions are concerned with the raising of funds to finance corporate assets.
Finance decisions in broad sense, are concerned with the process, institutions, markets and
instruments involved with the transfer of money between individuals, businesses and
governments. Areas of financing decisions include sources and forms of finance, capital
structure, cost of capital and dividend decisions.
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1.3 Goals of Financial Management
These goals or objectives should be considered carefully and within their proper context to
clarify financial management's approach for each goal.
Activity 1.1
1. Clearly distinguish between investment decisions and financing decisions.
2. List and explain the objectives of financial management.
3. Distinguish between wealth maximisation and profit maximisation.
Beyond the shareholders, managers have a responsibility to the firm's stakeholders (its
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employees, community, customers). It is possible to maximise owners' wealth and be socially
responsible.
When financial managers assess a potential investment in a new product, they examine the
risks, the potential benefits, and costs. If the risk-adjusted benefits do not outweigh the costs,
they will not invest. Similarly, managers assess current investments for the same purpose. If
benefits do not continue to outweigh costs, they will not continue to invest in that product, but
will shift their investment elsewhere. This is consistent with shareholder-wealth maximisation
and with the allocative efficiency of the market economy.
If a firm invests in the production of goods and services that meet the demand of consumers in
such a way that benefits exceed costs, the firm will be allocating the resources of the
community efficiently, employing assets in their most productive use. If later the firm must
disinvest (perhaps close a plant) then it will have a responsibility to assist employees and other
stakeholders who are affected. Failure to do so could tarnish its reputation, erode its ability to
attract new stakeholder groups to new investments, and ultimately act to the detriment of
shareholders.
The effects of a firm's actions on others are referred to as externalities and pollution is a very
current example that keeps increasing in importance. Suppose the manufacture of a product
creates air pollution. If the polluting firm takes action to reduce this pollution, it incurs a cost
that either increases the price of its product or decreases profit and the market value of its
stock. If competitors do not likewise incur costs to reduce their pollution, the firm is at a
disadvantage and may be driven out of business through competitive pressure. The firm may
try to use its efforts at pollution control to enhance its reputation, in the hope that this will lead
to a sales increase which is large enough to make up for the cost of reducing pollution. This is
what is called a market solution. A market solution is when the market places a value on the
pollution control and rewards the firm (or an industry) for it. If society really believes that
pollution is bad and that pollution control is good, the interests of owners and society can be
aligned.
It is more likely, however, that pollution control costs will be viewed as reducing owners'
wealth. Then firms must be forced to reduce pollution through government laws or
regulations. But such laws and regulations also come with a cost, that is, the cost of
enforcement. Again, if the benefits of mandatory pollution control outweigh the cost of
government action, society is better off. And if the government requires all firms to reduce
pollution, then pollution control costs simply become one of the conditions under which owner
wealth-maximising decisions are to be made.
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• financial analysis, planning and control;
• managing the firm's asset structure (effective employment of funds, the outcome of
which is reflected on the asset side of the balance sheet);
• managing the firm's financial structure (the effective acquisition of funds, the outcome
of which is reflected on the liabilities side of the balance sheet). See Figure 1.1 below.
Financial
Money
assets
market
Figure 1.1 Role of a Financial Manager (Source: Levy and Sarnat, 1991)
A finance manager addresses such varied issues as the decisions on the plant location, the
raising of capital, or simply how to get the highest return on the company's resources.
Actual corporate events provide a host of examples of some of the problems with the agency
relationship. For example, managers of a corporation may fight a takeover that would be in
the best interests of shareholders. As another example, managers adopt golden parachutes
which are lucrative compensation packages that take effect if a manager loses his or her job in a
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takeover.
These agency problems result in direct and indirect costs such as monitoring costs, bonding
costs, restructuring costs, opportunity costs and residual losses.
• Monitoring costs are costs incurred by the principal to monitor the actions of the agents
(for example, annual report to shareholders).
• Bonding costs are costs incurred by the agent to ensure they will act in the best interests
of the principals (for example, binding employment contract).
• Restructuring costs are expenditures to restructure the organisation so that the
possibility to undesirable managerial behaviour is limited.
• Opportunity costs are associated with lost profit opportunities because the organisational
structure does not permit managers to take actions on a timely basis as would be
possible if managers were also the owners.
• Residual loss is the implicit cost when management and shareholders' interests cannot
be aligned, even when bonding and monitoring costs are incurred.
Activity 1.2
1. Define the term business finance.
2. What role does the finance manager play in a firm?
3. Explain the scope of the agency theory. How best can shareholders deal with
the problems of agency relationships?
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1.9 Forms of Business Enterprise
There are three basic forms of business:
• Sole proprietorship
• Partnership
• Corporation
In Zimbabwe, the most common form of is the sole proprietorship. However, in terms of the
business revenues, the corporate form is dominant in Zimbabwe. We will look at each of these
three forms of business and then take a look at hybrid forms that borrow features from one or
more of these basic forms of business.
If more funds are needed to operate or expand the business than are generated by business
operations, the owner either contributes his or her personal assets to the business or borrows.
For most sole proprietorships, local banks represent the primary source of borrowed funds.
However, there is a limit to how much banks will lend to sole proprietorships, most of which
are relatively small businesses.
A proprietor is liable for all the debts of the business, in fact, it is the proprietor who incurs
the debts of the business. If there are insufficient business assets to pay a business debt, the
proprietor must pay the debt out of personal assets.
For tax purposes, the sole proprietor reports income from the business on his or her personal
income tax return. Business income is treated as the proprietor’s personal income. Therefore,
business income is taxed as the proprietor’s individual taxable income. Because the
proprietorship is dependent on the single owner, the life of a sole proprietorship ends with the
life of the proprietor, though the assets of the business pass to the proprietor’s heirs. The
assets may also be sold to some other firm, at which time the sole proprietorship ceases to
exist.
But the sole proprietorship form of business is not without its disadvantages.
Disadvantages include:
• Unlimited liability: the owner is liable for all debts of the business;
• Limited life of the proprietorship: the life is limited by the life of the owner; and
• The business has limited access to additional funds. Most of the funds are from the
proprietor's own assets or from lending arrangement with local banks.
1.9.2 Partnership
A partnership is a business enterprise owned by two or more persons who share the income
and liability of the business. There are different types of partnerships, including:
• General partnership
• Limited partnership
• Master limited partnership
These different types of partnerships differ in terms of the liability and control of the partners.
A general partnership is a partnership in which each partner is liable for the debts of the
business. Each partner is referred to as a general partner. The owners (that is, the partners)
share in the management of the business and share in the profits and losses of the business
according to the terms of the partnership agreement. In a general partnership, each partner is
liable for the debts of the partnership under the legal concept of joint and several liability.
Joint and several liability means that a creditor can sue one or more of the partners separately
or all of them together.
The life of the partnership may be limited by agreement or by life of partners, requiring a
reformulation of the partnership as partners exit and enter the partnership. Further, ownership
of a partnership interest cannot be freely transferred; this causes some problems in cases of a
partner wanting to leave or in the event of the death of a partner (because the partnership share
cannot be inherited).
The partnership can raise funds from either partner contributions or from borrowing from
local banks, though these sources may be quite limiting for a growing partnership. Because
the partners own 100% of the ownership, additional ownership interests cannot be sold. The
importance of this limitation is that a large, growing partnership may be constrained by its
limited sources of financing.
The income of partnership is taxed as income of partner (in portions agreed upon in
partnership agreement), flowing directly to the taxable income of the partner. Like the sole
proprietorship, the business income of a partnership is taxed only once as the individual
owners' income.
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There are a number of advantages of the partnership form of business. The advantages
include:
• The general partners are decision-makers.
• The owners (the partners) divide income according to partnership agreement.
• Income is taxed once.
A limited partnership is a partnership in which there are both general partners, who typically
are also the managers of the partnership; and limited partners, who have limited liability but
also limited say in the decision-making of the partnership (which is why limited partners are
often referred to as "silent partners").
A master limited partnership is a limited partnership in which limited liability ownership units
are publicly traded. The ownership interests, which represent a specified ownership percentage,
are traded in much the same way as the shares of a corporation. One difference, however, is
that a corporation can raise new capital by issuing new ownership interests, increasing the
number of shares of stock (but diluting existing shares), whereas a master limited partnership
cannot. It is not possible to sell more than a 100% interest in the partnership, yet it is possible
to sell additional shares of stock in a corporation.
If the board of directors decides to distribute cash to the owners, that money is paid out of
income left over after the corporate income tax has been paid. The amount of the cash
payment, or dividend, must then also be included in the taxable income of the owners.
Therefore, a portion of the corporation's income (that is, the portion paid out to owners) is
subject to double taxation: once as corporate income and once as individual owner's income.
The dividend declared by the directors of a corporation is distributed to owners in proportion
17
to the numbers of shares of ownership they hold. If owner A has twice as many shares as
owner B, he or she will receive twice as much money. The ownership interest of a corporation
(equity; stock), can be owned by a very few investors (that is, a closely-held corporation) or
publicly traded (that is, a public corporation or publicly-held corporation). A corporation can
raise funds by borrowing or by issuing additional ownership interests.
There are a number of advantages and disadvantages of the corporate form of business.
Advantages include:
• Limited liability - the most an owner can lose is the original investment in the
company;
• The business enterprise has a life in perpetuity, that is, out-living its owners;
• The corporation has access to additional funds through the sale of new share of stock;
• Income is distributed according to proportionate ownership.
Disadvantages include:
• The separation of ownership and decision-making.
• Double taxation on income. Income is taxed at both the corporate level and the
individual owner level when dividends are paid out.
1.9.4 Hybrids
There are a number of other forms of business enterprise that are hybrids of the three basic
forms. The hybrid forms that we will discuss are the professional corporation, the joint
venture, and the limited liability company.
A joint venture is a partnership or a corporation formed for a specific business operation. For
tax and other legal purposes, a joint venture partnership is treated as a partnership, and a joint
venture corporation is treated as a corporation.
Zimbabwean corporations are entering into joint ventures with foreign corporations,
enhancing participation and competition in the global marketplace. Joint ventures are an easy
way of entering a foreign market and of gaining an advantage in a domestic market. Joint
ventures are becoming increasingly popular as a way of doing business. Participants - whether
individuals, partnerships, or corporations, get together to exploit a specific business
opportunity. Afterwards, the venture can be dissolved. Recent alliances among communication
and finance firms have sparked thought about what the future form of doing business will be.
Some believe that what lies ahead is a virtual enterprise, that is, a temporary alliance without
all the bureaucracy of the typical corporation, that can move quickly and decisively to take
advantage of profitable business opportunities.
18
Activity 1.3
1. Explain the forms of business enterprises in Zimbabwe.
2. Which form do you think is ideal for low-income earners?
1.10 Summary
In this unit we discussed that Business Finance deals with financial analysis, planning and
control, acquisition and effective employment of funds in order to maximise the value of the
firm. Finance involves financing and investment decisions for the long-term (fixed assets
management) and short-term (current asset management) objectives. Investment decisions are
concerned with how the firm allocates resources to assets that generate income now and in the
future. Investment decisions include capital budgeting processes and techniques, capital
rationing, financial markets and security price analysis. Areas of financing decisions include
sources and forms of finance, capital structure, cost of capital and dividend decisions. The
main objective of financial management is to maximise shareholders’ wealth or value of the
firm which translates into maximising the price of the common stock. The financial manager
faces two fundamental problems: firstly, which assets to invest in and how to invest in them,
and secondly how to acquire the necessary cash to do so. An agency relationship is the
relationship between the principal and the agent, in which the agency acts for the principal. In a
corporation, the principals are the shareholders and the agents are the managers. Beyond the
shareholders, managers have a responsibility to the firm's stakeholders (its employees,
community, customers).
19
References
Block, S.B. and Hirt, G.A. (1994). Foundations of Financial Management. (Sixth Edition). Boston:
Irwin.
Ehrhardt, M.C. and Brigham, E.F. (2011). Financial Management, Theory and Practice.
(Thirteenth Edition). Mason: South-Western Cengage Learning.
Gitman, L. and Zutter, C.J. (2012). Principles of Managerial Finance. (Thirteenth Edition). Boston:
Pearson Education, Inc.
Kwesu, I., Nyatanga, E. and Zhanje, S. (2002). Business Statistics. Harare: ZOU.
Levy, H. and Sarnat, M. (1991). Capital Investments and Financial Decisions. (Fourth Edition). New
York: Prentice Hall.
McLaney, E..J. (2000). Business Finance: Theory and Practice. New York: Prentice Hall.
20
BLANK PAGE
Unit 2
Analysis of Financial Statements
2.0 Introduction
This topic, Analysis of Financial Statements, discussed in this unit is of paramount
importance in business finance and financial management because financial managers
need accurate forecasts of cash flows to make accurate investment and financing
decisions. Investment bankers and deal makers need to know how much to bid for a
company in an acquisition and/or merger. Managers need to know how much cash flows
are generated by assets to get feedback on their strategic decisions. Investors and creditors
need to know how much cash is generated from assets and operations to determine the
financial solvency of a company.
2.1 Objectives
By the end of this unit, you should be able to:
• explain the income statement, balance sheet and cash flow statement
• calculate and interpret financial ratios
• identify the sources of changes in returns by using the Du Pont System
• prepare a set of common size financial statements
• interpret the set of financial ratios
• assess the financial health of a firm
21
Sales or revenues: Represent the amount of goods or services
sold, in terms of price paid by customers
Though the structure of the income statement varies by company, the basic idea is to
present the operating results first, followed by non-operating results. In the case of the 'X'
Company Limited, whose income statement is presented below, the income from
operations is $190 million, whereas the net income was $100 million.
22
X' Company Limited’s Income Statement for the Period
Ending December 31, 2012
Description Amount in Millions $
Sales 1 000
Cost of goods sold (600)
Gross profit 400
Depreciation (50)
Selling, general, and administrative expenses (160)
Operating profit 190
Non-current assets comprise both physical and non-physical assets. Plant assets are
physical assets, such as buildings and equipment and are reflected in the balance sheet as
gross plant and equipment, and net plant and equipment. Gross plant and equipment is the
total cost of investment in physical assets. Net plant and equipment is the difference
between gross plant and equipment and accumulated depreciation, and represents the book
value of the plant and equipment assets. Accumulated depreciation is the sum of
depreciation taken for physical assets in the firm's possession.
Intangible assets are assets having no physical existence, such as patents and trademarks.
These assets may be amortised over some period, which is akin to depreciation. Liabilities
are obligations of the business enterprise. These are generally broken into two major groups:
current liabilities and long-term liabilities. Current liabilities are obligations due within one
year or one operating cycle (whichever is longer), while long-term liabilities are obligations
that are due beyond one year. Retained earnings are the accumulation of prior period's
earnings, less any prior period's dividends. Equity is the ownership interest of the business
enterprise. Book value of equity is the total value of the ownership of a firm, according to
23
accounting principles. Book value comprises:
Par value, which is a nominal amount per share of stock (sometimes prescribed by law), or
the stated value, which is a nominal amount per share of stock assigned for accounting
purposes if the stock has no par value.
Additional paid-in-capital is the amount paid for shares of stock by investors in excess of
par or stated value.
Treasury stock is the accounting value of shares of the firm's own stock bought by the
firm.
Retained earnings are the accumulation of prior periods' earnings, less any prior periods'
dividends.
2012 2011
Cash 60 50
650 600
Common stock 50 50
24
Knowledge of the preparation of financial statements is a pre-requisite to the preparation of
cash-flow statements and even the interpretation of financial statements. For now the cash
flow analysis is given preference.
The adjustment for changes in working capital accounts is necessary to adjust net income
that is determined using the accrual method to a cash flow amount. Increases in current
assets and decreases in current liabilities are positive adjustments to arrive at the cash
flow; decreases in current assets and increases in current liabilities are negative
adjustments to arrive at the cash flow.
Cash-flows for/from investing are the cash flows related to the acquisition (purchase) of plant,
equipment, and other assets, as well as the proceeds from the sale of assets. Cash flow
for/from financing activities is the cash flow from activities related to the sources of capital
funds (for example, buyback common stock, pay dividends, issue bonds).
The simplified cash flow statement for 'X' Company Limited is shown below:
The 'X' Company Limited cash flow statement for the period
ending December 31, 2012
$ million $ million
Operating Activities
Net income 100
Add depreciation 50
Less increase in accounts receivable (10)
Add decrease in inventory 20
Add decrease in accounts payable 50
Total cash from operations 210
Financing Activities
Retire debt (100)
Total cash flow from financing (100)
Investment Activities
Purchase of equipment (100)
Total cash flow for investment (100)
Net Cash Flow 10
The net cash flow is equal to the change in the cash account as reported on the balance
sheet. For the company, the net change in cash flow is a positive $10 million; this is equal to
the change in the cash account from $50 to $60 million. By studying the cash flows of a
company over time, we can gauge a company's financial health. For example, if a company
relies on external financing to support its operations (that is, cash flows from financing and
not from operations) for an extended period of time, this is a warning sign of financial
trouble up ahead.
26
Activity 2.1
From the following information shown below prepare a cash flow statement for XYZ
Limited for the year 2012.
2012 2011 2012 2011
$ $ $ $
Ordinary 600 000 550 000 Fixed assets 1 279 300 1 058 700
shares at cost
Besides information that companies are required to disclose through financial statements,
other information is readily available for financial analysis. For example, information such
as the market prices of securities of publicly traded corporations can be found in the
financial press and the electronic media daily. Similarly, information on stock price indices
27
for industries and for the market as a whole, are available in the financial press.
Another source of information is economic data, such as the Gross Domestic Product and
Consumer Price Index, which may be useful in assessing the recent performance or future
prospects of a firm or industry.
Besides financial statement data, market data, and economic data, in financial analysis you
also need to examine events that may help explain the firm's present condition and may
have a bearing on its future prospects. Current events can provide information that may be
incorporated in financial analysis.
The financial analyst must select the pertinent information, analyse it, and interpret the
analysis, enabling judgments on the current and future financial condition and operating
performance of the firm. In this unit, we introduce you to financial ratios-- the tool of
financial analysis. In financial ratio analysis, we select the relevant information (primarily
the financial statement data) and evaluate it. We show you how to incorporate market data
and economic data in the analysis and interpretation of financial ratios. And we show you
how to interpret financial ratio analysis and in the process warn you of the pitfalls that
occur when it is not used properly.
When we assess a firm's operating performance, we want to know if it is applying its assets
in an efficient and profitable manner. When we assess a firm's financial condition, we want
to know if it is able to meet its financial obligations.
There are six aspects of operating performance and financial condition we can evaluate
from financial ratios:
• A liquidity ratio provides information on a firm's ability to meet its short-term
obligations.
• A profitability ratio provides information on the amount of income from each
dollar of sales.
• An activity ratio relates information on a firm's ability to manage its resources (that
28
is, its assets) efficiently.
• A return on investment ratio provides information on the amount of profit, relative
to the assets employed to produce that profit.
• A financial leverage ratio provides information on the degree of a firm's fixed
financing obligations and its ability to satisfy these financing obligations.
• A shareholder ratio describes the firm's financial condition in terms of amounts per
share of stock.
We will cover each type of ratio, providing examples of ratios that fall into each of these
classifications.
The current investment in inventory, that is, the money "tied up" in inventory, is the
ending balance of inventory on the balance sheet. The average day's cost of goods sold is
the cost of goods sold on an average day in the year, which can be estimated by dividing
the cost of goods sold found on the income statement by the number of days in the year.
We compute the number of days of inventory by calculating the ratio of the amount of
inventory on hand (in dollars) to the average day's Cost of Goods Sold (in dollars per day):
29
If the ending inventory is representative of the inventory throughout the year, then the
number of days of inventory tells us the time it takes to convert the investment in
inventory into sold goods.
We can extend the same logic for calculating the number of days between a sale when an
account receivable is created to the time it is collected in cash. If the ending balance of
receivables at the end of the year is representative of the receivables on any day
throughout the year, then it takes, on average, approximately the "number of days credit"
to collect the accounts receivable, or the number of days receivables:
Accounts receivables
Number of days of receivables =
Average day's credit
What does the operating cycle have to do with liquidity? The longer the operating cycle,
the more current assets needed (relative to current liabilities) since it takes longer to
convert inventories and receivables into cash. In other words, the longer the operating
cycle, the more net working capital required.
We also need to look at the liabilities on the balance sheet to see how long it takes a firm
to pay its short-term obligations. We can apply the same logic to accounts payable as we
did to accounts receivable and inventories. How long does it take a firm, on average, to go
from creating a payable (buying on credit) to paying for it in cash?
�������� �������
Number of days of payables =������� ���� � ��������
The operating cycle tells us how long it takes to convert an investment in cash back into
cash (by way of inventory and accounts receivable). The number of days of purchases tells
us how long it takes us to pay on purchases made to create the inventory. If we put these
two pieces of information together, we can see how long, on net, we tie up cash. The
difference between the operating cycle and the number of days of payables is the net
operating cycle:
30
The net operating cycle, therefore, tells us how long it takes for the firm to get cash back
from its investment in inventory and accounts receivable, considering that purchases may
be made on credit. By not paying for purchases immediately (that is, using trade credit), the
firm reduces its liquidity needs. Therefore, the longer the net operating cycle the greater the
required liquidity.
The following list of items will be used to compute important ratios in this unit.
The ABC Company Limited’s List of Items from Financial Statement 2012
$ million
Current assets 15 889
Current liabilities 5 730
Closing stock 0
Sales 14 484
Cost of sales 1 197
Operating profit 6 414
Net profit 4 490
Accounts receivables 1 460
Total assets 22 357
Fixed assets 1 505
Shareholder’s equity 15 647
Application of' ratio analysis when the operating cycle is demonstrated below:
• The current ratio is the ratio of current assets to current liabilities. It indicates
a firm's ability to satisfy its Current liabilities with its current assets.
������� ������
Current ratio =������� �����������
• The quick ratio is the ratio of quick assets (generally current assets less inventory)
to current liabilities. It indicates a firm’s ability to satisfy current liabilities with its
most liquid assets.
������� ����������������
Quick ratio = ������� �����������
31
• The net working capital to sales ratio is the ratio is the ratio of networking capital
(current assets minus current liabilities) to sales. It indicates a firm’s liquid assets
(after meeting short term obligations) relative to its need for liquidity (represented
by sales).
Generally, the larger these ratios, the better the ability of the firm to satisfy its immediate
obligations.
Note: The quick ratio is the same as the current ratio because there is no inventory. Net
working capital-to-sales = ($15 889 – 5 730) / $14 484=0.701
The gross profit margin numerator is the ratio of gross profit to sales. It indicates how much
of every dollar of sales is left after costs of goods sold:
���������� �� �����
Gross profit margin = �����
The operating profit margin is the ratio of operating profit (EBIT, operating income,
income before interest and taxes) to sales. It indicates how much of each dollar of sales
is left over after operating expenses.
The net profit margin is the ratio of net income (a.k.a. net profit) to sales. It indicates
how much of each dollar of sales is left over after all expenses.
��� ������
Net profit margin = �����
32
Net profit margin = $4 490 / $14 484 = 31%
Activity ratios are measures of how well assets are used. Activity ratios (that are, for the
most part, turnover ratios) can be used to evaluate the benefits produced by specific
assets, such as inventory or accounts receivable. Or they can be used to evaluate the
benefits produced by a firm's total assets collectively.
• Inventory turnover is the ratio of cost of goods sold to inventory. This ratio
indicates how many times inventory is created and sold during the period.
���� �� �����
Inventory turnover ratio = ������� ���������
• Accounts receivable turnover is the ratio of net credit sales to accounts receivable.
This ratio indicates how many times in the period credit sales have been created
and collected on.
• Total asset turnover is the ratio of sales to total assets. This ratio indicates the
extent that the investment in total assets results in sales.
�����
Total assets turnover = ����� ������
• Fixed asset turnover is the ratio of sales to fixed assets. This ratio indicates the
ability of the firm’s management to put the fixed assets to work to generate sales:
�����
Fixed assets turnover ratio =����� ������
Financial leverage ratios are used to assess how much financial risk the firm has taken on.
There are two types of financial leverage ratios: component percentages and coverage
ratios. Component percentages compare a firm's debt with either its total capital (debt
plus equity) or its equity capital. Coverage ratios reflect a firm's ability to satisfy fixed
obligations, such as interest, principal repayment, or lease payments.
����� ����
Total debt – to-asset ratio = ����� �
• The long-term debt to assets ratio indicates the proportion of the firm's assets that
are financed with long term debt.
• The long-term debt to equity ratio indicates the relative uses of debt and equity as
sources of capital to finance the firm's assets, evaluated using book values of the
capital sources:
����
Debt-to-equity ratio=���� ����� �� ����������� � � ������
One problem with looking at risk through a financial ratio that uses the book value of
equity (the stock) is that most often there is little relation between the book value and its
market value. The book value of equity consists of:
• the proceeds to the firm of all the stock issued since it was first incorporated, less
any treasury stock (stock repurchased by the firm); and
• the accumulation of all the earnings of the firm, less any dividends, since it was first
incorporated.
The book value generally does not give a true picture of the investment of shareholders in
the firm because:
• earnings are recorded according to accounting principles, which may not reflect the
true economics of transactions; and
• due to inflation, the dollars from earnings and proceeds from stock issued in the
past do not reflect today's values.
Therefore, we may use the market value of equity in the denominator, replacing the book
value of equity. To do this, we need to know the current number of shares outstanding and
the current market price per share of stock and multiply to get the market value of equity.
34
2.10.2 Coverage financial leverage ratios
The interest coverage ratio indicates the firm's ability to satisfy interest obligations on its
debt.
����
Interest coverage ratio = �������� �������
This ratio tells us how well the firm can cover or meet the interest payments associated
with debt. The interest coverage ratio, also referred to as the times-interest-covered ratio,
compares the funds available to pay interest (that is, earnings before interest and taxes)
with the interest expense.
The fixed-charge coverage ratio is the ratio of earnings before interest and taxes + fixed
charges (before tax), divided by the fixed charges. For example, considering leases to be a
fixed charge:
���������� �������
Fixed charge coverage ratio =�������������� �������
This ratio indicates a firm's ability to satisfy fixed financing expenses, such as interest and
leases.
The cash flow interest coverage ratio is the ratio of cash flow from operations before
interest and taxes to interest:
This ratio indicates the firm's ability to satisfy its fixed financing obligations using the
cash flow that it generates.
The basic earning power ratio is the ratio of operating earnings to assets:
����
Basic earning power =����� ������
It is a measure of the operating income resulting from the firm's investment in total assets.
The return on assets is the ratio of net income to assets and indicates the firm's net profit
generated per dollar invested in total assets:
35
��� ������
Return on assets = ����� ������
The return on equity is the ratio of net income to shareholders' equity and represents the
profit generated per dollar of shareholders’ investment (that is, shareholders' equity).
��� ������
Return on equity = ���� ����� �� ������������� ������
The return on common equity is the ratio of net income available to common shareholders'
equity. This return is the profit generated per dollar of common shareholders' investment
(that is, common shareholders' equity).
The DuPont formula is the basic model used to describe the relationship of the respective
variables to return on equity and is given as:
As indicated in the expression, the output of the DuPont equation is the return on equity.
Example 2.1
A company has $5m pre-tax profit, a $50m sales volume, total assets of $30m, total
liabilities of $20m, a beginning of year owner's equity of $20m and a 40% income tax
rate. Calculate the company's return on equity.
Earnings per share (EPS) is the amount of income earned during a period per share of
common stock.
Book value equity per share is the amount of the book value of common equity per share
of common stock. As we have discussed earlier, the book value of equity may differ from
the market value of equity. The market value per share, if available, is a much better
indicator of the investment of shareholders in the firm.
The price-earnings ratio (P/E or PE ratio) is the ratio of the price per share of common stock to
the earnings per share of common stock. The P/E ratio is sometimes used as a proxy for
investors' assessment of the firm's ability to generate cash flows in the future. Dividends-
per- share (DPS) is the dollar amount of cash dividends paid during a period per share of
common stock.
The dividend payout ratio is the ratio of cash dividends paid to earnings for a period.
EPS
Dividend cover =
DPS
DPS
Dividend payout =
EPS
Example 2.2
Assume we are given the following information relating to Progress Company Limited in
Table 2.1 below.
37
Table 2.1
2012 2011
$ $
Earnings after interest and tax 124 000 77 000
Market price 4.50 2.50
Number of ordinary shares 250 000 250 000
Ordinary dividend paid 50 000 25 000
Table 2.2
2012 2011
EPS 49.6c 30.8c
P/E ratio 9.1 8.1
Dividend yield 4.44% 4%
Dividend cover 2.28 3.08
Dividend payout ratio 43.86% 32.47%
For the income statement, the benchmark is sales. For a given period, each item in the
income statement is restated as a percentage of sales.
For the balance sheet, the benchmark is total assets. For a given point in time, each item in
the balance sheet is restated as a percentage of total assets.
To see how this works, consider GM's financial statements in Table 2.3. The reported
financial data in the right-most columns has been converted into percentages of sales
revenues (that is, common size statements), shown in the middle column.
38
Table 2.3
Sales 100% 100% 100% 178 174 164 013 160 254
Cost of sales 73% 75.1% 75.7% 130 028 123 195 121 300
Gross profit 27% 24.9% 24.4% 48 146 40 818 38 954
Overhead expenses 9.1% 8.9% 7.8% 0 0 0
Net income 3.83% 3.0% 4.3% 6 698 4 963 6 881
Premium on exchange of
preference stocks 0.0% 0.0% 0.1% $26 $0 $153
Dividends on preference
stocks 0.0% 0.0% 0.1% $72 $81 $211
Earnings on common stock 3.7% 3.0% 4.1% 6 600 4 882 6 517
We can restate GM’s reported balance sheet items in terms of a percentage of total assets
as in Table 2.4:
Table 2.4
Table 2.5
Common size analysis is useful in analysing trends in profitability (using the common size
income statements) and trends in investments and financing (using the common size
balance sheet).
• Ratios only show the results of carrying on business, but they do not indicate the
causes of the ratios.
• Ratios can only be used to compare ‘like with like’.
• Ratios tend to ignore the time factor in seasonal businesses, for example
fluctuating stock level and debtor levels.
• Ratios can be misleading if accounts are not adjusted for inflation.
• The accuracy of ratios depends upon the quality of the information from which
they are calculated, the required information is not always disclosed in accounting
statements and accounting headings may be misleading.
40
Activity 2.1
Comparative Balance Sheets
Long-term liabilities:
Bonds payable (2019) 90 000 40 000
Total liabilities 300 000 220 000
Shareholders’ equity
Preferred stock, $100 par value 50 000 50 000
Common stock, $1 par value 100 000 100 000
Capital paid in excess of par 250 000 250 000
Retained earnings 300 000 250 000
Total shareholders’ equity 700 000 650 000
Total liabilities and shareholders’ equity
Using information above, compute and interpret the relevant financial ratios.
41
2.15 Summary
In this unit we discussed that financial statements are prepared on a historical basis, and
seldom, if ever, is inflation incorporated into the analysis. As a result the true market
value of owners’ equity, long-term, fixed assets and investments do not synchronise with
the figures disclosed in the balance sheet. Hence, analysis of financial statements should
be done with caution. Common size analysis is useful in analysing trends in profitability
(using the common size income statements) and trends in investments and financing
(using the common size balance sheet).
42
References
Block, S.B. and Hirt, G.A. (1994). Foundations of Financial Management. (Sixth Edition). Boston:
Irwin.
Ehrhardt, M.C. and Brigham, E.F. (2011). Financial Management, Theory and Practice.
(Thirteenth Edition). Mason: South-Western Cengage Learning.
Gitman, L. and Zutter, C.J. (2012). Principles of Managerial Finance. (Thirteenth Edition).
Boston: Pearson Education, Inc.
Kwesu, I., Nyatanga, E. and Zhanje, S. (2002). Business Statistics. Harare: ZOU.
Levy, H. and Sarnat, M. (1991). Capital Investments and Financial Decisions. (Fourth Edition).
New York: Prentice Hall.
McLaney, E..J. (2000). Business Finance: Theory and Practice. New York: Prentice Hall.
43
BLANK PAGE
Unit 3
The Financial Environment: Markets, Institutions,
Interest Rates and Taxes
3.0 Introduction
It is of paramount importance for finance managers to be able to understand the environment
and markets within which their organisations operate. Since finance managers make
investment and financing decisions, it is critical that whenever capital is required it is raised
using a least cost method and funds invested always should yield the greatest return in order
to be in line with the wealth maximisation principle. Financial markets bring together people
and organisations wanting to borrow money with those having surplus funds. There are
numerous financial instruments available to investors and financial managers. Each year new
instruments arrive on the market and some leave and in this unit we are going to explore all
these issues in greater detail.
3.1 Objectives
By the end of this unit, you should be able to:
44
• Primary and secondary markets
• Debt and equity markets
• Money and capital markets
The primary market is the financial market where securities are first issued and the
secondary market is a resale market where investors buy and sell previously issued
securities. Whenever a company, an individual, or the government issues or sells a debt
security like a bond, the transaction takes place in the debt market but issuing or selling of
common stock takes place in the equity market. Securities with maturity of one year or less
are traded in the money market whilst securities with greater maturity than one year are
traded in the capital market. It should be pointed out that money and capital markets may be
primary or secondary, organised (trading taking place in a central location) or over the
counter (having no central location).
Debentures are certificates issued to investors who are lending money to the company with the
intention of raising capital. The debenture pays a fixed interest to the investor which is
sometimes referred to as a coupon. The company redeems the debenture at a future date for a
fixed amount (par value). Preferred stock is a security with a defined, fixed periodic claim on
the income of a firm. Dividends on preferred stock must be paid before dividends on common
stock. Common Stock is the residual claim to the earnings of the corporation. Payment of
dividends is discretionary.
Activity 3.1
1. Distinguish between physical asset market and the financial asset markets.
2. Explain the different classifications of financial markets.
3. Why are financial markets essential for a healthy economy?
46
• The real risk free rate of interest, which would prevail on a risk free security if inflation
is not anticipated, and the Treasury bill rate is used as a proxy.
• The inflation premium, which is defined as the average inflation rate expected over the
duration of the instrument, is meant to compensate investors for the loss in
purchasing power.
• The default risk premium, which is meant to reward investors for the risk that the
borrower will not redeem the loan or pay interest on the principal.
• The liquidity premium, refers to the ability to sell the marketable security at fair price so
that the investor is able to recover the initial outlay.
• Maturity risk premium, rewards investors for interest rate risk.
All the five components have a positive impact on the prevailing interest rate at any moment
in time.
Activity 3.2
1. Formulate an equation for the nominal interest rate on any debt instrument.
2. Differentiate and explain the interest rate components.
3. Discuss the determinants of market interest rates.
• Normal, a normal yield curve implies that long-term rates will be higher than short
term interest rates.
• Inverse, an inverse yield curve indicates that short-term interest rates are higher than
long-term interest rates.
• Flat, a flat yield curve shows that short-term and long-term interest rates are the
same.
Three theories have been proposed to explain the shape of the yield curve and these are:
47
of the debt market relative to the short-term segment (Gitman and Zutter, 2012).
Activity 3.3
1. What is the term structure of interest rates? How is it related to the yield curve?
2. Discuss the three theories of the general shape of the yield curve.
3. Distinguish between the shapes of the yield curves using diagrammatic
illustrations
48
3.11 Taxation and Financial Decisions
Taxes play an important role in the amount of a firm's cash flows and the uncertainty of a
firm's cash flow. Taxable income is computed by deducting capital allowances and
permitted expenditures, any revenues of a capital nature, and any income that is not
recognised for tax purpose, from gross income. At this point there is a need to differentiate
taxable income from the firms' cash flows and profit, this is shown in the following tables:
It should be noted that capital allowances could only be claimed if the company has adequate
taxable income and allowances can be carried forward to future periods.
Scrapping allowance (+ve) = Cost of the asset - capital allowance - sale proceeds
If the amount is positive then the firm has wrapping allowance, which can be deducted from
taxable income.
3.11.8 Recoupment
Recoupment arises when an expenditure (loss) is claimed for tax purposes but is later
recovered; recoupment has a positive impact on taxable income since it is included as gross
income and also recoupment's calculation is similar to that of profit on disposal.
Recoupment is computed as follows:
Scrapping allowance (-ve) = Cost of the asset - capital allowance - sale proceeds.
If the amount is negative, there is recoupment and tax has to be paid on recoupment.
Activity 3.4
1. Compare and contrast taxable income with profits, taxable income with cash
flows, cash flows with profits.
2. How is taxable income calculated in Zimbabwe?
3. Explain the relevance of wear and tear allowance and special initial allowance
from a Zimbabwean perspective.
4. Differentiate scrapping allowance from recoupment.
5. Relate special initial allowance and wear and tear to scrapping allowance and
recoupment.
3.12 Summary
In this unit we highlighted that knowledge of the financial environment is critical since it
enables finance managers to pursue courses of action, which leads to the maximisation of the
value of the firm. If interest rates are low in the money market and unattractive, scarce
resources of the firm can be channeled to the capital market. Knowledge of financial
instruments traded in financial markets enable finance managers to diversify their firm's
portfolio and the yield (the term structure of interest rates) broadens the investor's horizon.
Finance managers are encouraged to know the Zimbabwean tax system in order for them to be
able to determine tax implications associated with investment and financing decisions they
make.
51
References
Block, S.B. and Hirt, G.A. (1994). Foundations of Financial Management. (Sixth Edition). Boston:
Irwin.
Ehrhardt, M.C. and Brigham, E.F. (2011). Financial Management, Theory and Practice.
(Thirteenth Edition). Mason: South-Western Cengage Learning.
Gitman, L. and Zutter, C.J. (2012). Principles of Managerial Finance. (Thirteenth Edition). Boston:
Pearson Education, Inc.
Kwesu, I., Nyatanga, E. and Zhanje, S. (2002). Business Statistics. Harare: ZOU.
Levy, H. and Sarnat, M. (1991). Capital Investments and Financial Decisions. (Fourth Edition). New
York: Prentice Hall.
McLaney, E..J. (2000). Business Finance: Theory and Practice. New York: Prentice Hall.
52
BLANK PAGE
Unit 4
Basic Financial Calculations and Time Value of Money
Well, then you should have deposited my money in the bank and I would have received it all back with
interest when I returned.
Matthew 25v 27
4.0 Introduction
In this unit we discuss that the basic idea of time value of money is that a dollar today is
worth more than a dollar tomorrow. Investors have time preference for money. This means
they will prefer to have their money today than in the future. This is so because money
received today can be reinvested at real interest to earn more money in the future. More so
money expected in the future is likely to be eroded its purchasing power by inflation. In
addition to real interest and inflation premium, there is no substantial accuracy that expected
incomes might materialise in the future. This is called default risk. It may be worse that one
may fail to convert his investment or claims into liquid cash at a fair and reasonable price.
This is called liquidity risk. Most finance decisions involve cash flows that materialise in
different periods. It is important for a project or investment analyst to consider the
implications of receiving these cash flows in different periods. In doing so they must
incorporate the time value of money. To make decisions today based on future income it is
important for one to convert future cash flows to their today equivalent value. The process of
doing so is called discounting. Alternatively the process of converting today's cash flows into
their future value equivalent is called compounding. Compounding or discounting depends
on the conditions of time value of money. In other words, they depend on real interest,
inflation premium, default, and liquidity risk.
4.1 Objectives
By the end of this unit, you should be able to:
• explain the mechanics of compounding: how money grows over time when it is
invested
• determine the future or present value of a sum when there are at least two
compounding periods per year
• discuss the relationship between compounding (future value) and bringing
money back to the present (present value)
• define an ordinary annuity and calculate its compound or future value
• differentiate between an ordinary annuity and an annuity due
• determine the future and present value of an annuity due
• calculate the annual percentage yield or effective annual rate of interest and
then explain how it differs from the nominal or stated interest rate
53
4.2 Future Value
Future value refers to the amount of money an investment will grow to over some length of
time at some given interest rate. The basic idea is to find how much some money will be
worth in the future. The process of leaving your money and any accumulated interest in an
investment for more than one period, thereby reinvesting the interest, is called compounding.
Compounding the interest means earning interest on interest, so we call the result
compound interest. Compound interest is earned on both the initial principal (original
amount invested) and the interest reinvested from prior periods. This is in contrast to cases
where only the principal is rewarded simple interest (interest earned only on the original
principal or amount invested).
Most financial transactions involve compound interest, though there are a few consumer
transactions that use simple interest.
The future value is the sum of the present value and interest:
Future value = Present value + interest
The basic valuation equation is:
FV = P (1 + r)n
where:
r is the rate of interest;
n is the number of compounding periods;
(1+r)n is the compound factor or the future value interest factor for given compound
period (n years) at a given interest rate (r%) which can be shortened FVIF (r%, n
years);
FV is the future value; and
P is the principal.
Solution
The basic assumptions we make are:
1) Investment of the principal is done at the beginning of the period. At this period the
principal is zero years old.
2) Interest payments accrue at the end of each compounding period. For example,
interest is paid 5 times, at the end of year 1, end of year 2, end of year 3 up to end of
year 5. Thus interest that accrues at the end of year 1 is the one paid on principal
(value at the beginning of year zero). Interest that accrues at the end of year 2 is the
one that is paid for the value at the beginning of year 2 (which is the same as at the
end of year 1). This process of annual compounding continues up to the terminal
period of the investment (year 5).
Now let us use figures in our example to explain this process. In our example, the principal
amount (P) is $1 000, annual interest rate (r%) is 6% (that is, r = 0.06) and number of periods
54
(n) is 5 years.
The value at the end of year 5 (maturity or terminal period) is given by:
FV5 = FV4 (1+0.06) = $1 000(1+0.06) 4 (1+0.06) = $1 000(1+0.06)5 = $1 338.20
Note that as we progress from year one to year 5, values at the end of each year will grow
by the factor (l + r).
For n years the interest factor (l + r)n can be determined using the Future Value Interest
Factor (FVIF) tables in the appendix. In this case the FVIF(r%, n years) is FVIF (6%,5
years) is 1.3382 (6% against 5 years in your future value tables).
In cases where future value is determined using tables the formula must be reinstated as
FV5=Principal x FV1F(r%, n years)
Example 4.2
Suppose you invest $1 000 in an account that pays 6% interest, compounded annually. How
much will you have in the account at the end of 10 years if you make no withdrawals?
Solution
Using a calculator:
FV10 = $1 000 (1 + 0.06)10 = $1 000 (1.7908) = $1 790.80
55
Using tables:
FVn = 1000 x FVIF (6%, 10 years) =1 000 x 1.7908 = $1 790.80
What would be the future values in examples I and 2 if interest was simple interest? We would
have a lower balance in the account:
FVn = $1 000 + [$1 000(0.06) (5)] = $1 300
FV10 = $1 000 + [$1 000 (0.06) (10)] = $1 600
Note: With simple interest only the principal amount is paid interest. Thus, future value is FVn = P(1+ rn)
Activity 4.1
What is the future value of $26 895 invested today for 20years if the interest rate is
10% per annum? Assume interest is
FV = P (1 + r)n,
Rearranging, we see that the ratio of the future value to the present value is equal to the
compound factor for five periods at some unknown rate:
$2 000 / $1 000 = (1 + r)5 or 2.0000 = (1 + r)5 where 2.0000 is the compound factor .
We therefore have one equation with one unknown, r. We can determine the unknown interest
rate either mathematically or by using the table of compound factors. Using the table of
factors, we see that for five compounding periods, the interest rate that produces a compound
factor closest to 2.0000 is 15% per year.
We can determine the interest rate more precisely, however, by solving for r
mathematically:
2 = (1 + r)5
Taking the fifth root of both sides, and representing this operation in several equivalent
ways,
1+ r = 21/5= 20.2
56
You'll need a calculator to figure out the fifth root:
(1 + r) = 1.1487
Therefore, if you invested $1 000 in an investment that pays 14.87% compounded interest
per year, for five years, you would have $2 000 at the end of the fifth year. We can
formalise an equation for finding the interest rate when we know PV, FY, and n from the
valuation equation and notation:
FV = P (1 + r)n,
Dividing both sides by PV and taking the nth root of both sides of the equation:
FV
r= n −1
P
As an example, suppose that the value of an investment today is $100 and the expected
value of the investment in five years is expected to be $150. What is the annual rate of
appreciation in value of this investment over the five year period?
150
r= 5 − 1 = 0.0845 = 8.45% per year
100
Solution
P = $1 000
FV = $1 750
n=3
r = (1 750/1 000)1/3-1
r = 20.51%
57
For example, consider an investment that has a value of $100 in year 0, a value of $150 at
the end of the first year and a value of $200 at the end of two periods. What is this
investment's annual growth rate?
P = $100
FV = $200
n =2
FV
r = n −1
P
r = (200/100)1/2-1
= 20.5-1
= 1.4142-1
= 41.42%
Let us start with the basic valuation equation and insert the known values of P, FV, and r:
FV = P(1 + r)n
$5 000 = 1 000(1 +0.10)n
Rearranging,
(1 + 0.10)n = 5.0000
58
Therefore, the compound factor is 5.0000.
Like the determination of the unknown interest rate, we can determine the number of
periods either mathematically or by using the table of compound factors. If we use the table
of compound factors, we look down the column for the 10% interest rate to find the factor
closest to 5.000 then we look across the row containing this factor to find n. From the table
of factors, we see that the n that corresponds to a factor of 5.000 for a 10% interest rate is
between 16 and 17, though closer to 17. Therefore, we approximate the number of periods
as 17.
Solving for the equation mathematically is a bit more complex. We know that:
5 = (1 + 0.10)n
We must somehow rearrange this equation so that the unknown value, n, is on one side of
the equation and all the known values are on the other. To do this, we must use logarithms
and a bit of algebra. Taking the natural log of both sides:
In 5 = n In(1 + 0.10)
or
In 5 = n In 1.10
where “In” indicates the natural log. Substitute the values of the natural logs of 5 and 1.10,
1.6094 = n(0.0953)
Since the last interest payment is at the end of the last year, the number of period years is
17. It would take 17 years for your investment to grow to $5 000 if interest is compounded
at 10% per year.
As you can see, given the present and future values, calculating the number of periods when
we know the interest rate is a bit more complex than calculating the interest rate when we
know the number of periods. Nevertheless, we can develop an equation for determining the
number of periods, beginning with the valuation formula:
FV = P(1 + r)n
and dividing both sides by P,
FV
= (1 + r)n
P
59
In FV – In PV = n In(1 +r)
Dividing both sides by In(1 + r) and exchanging sides,
InFV − InP
n=
In(1 + r )
Suppose that the present value of an investment is $100 and you wish to determine how long
it will take for the investment to double in value if the investment earns 6% per year,
compounded annually:
In200 − In100 5.2983 − 4.6052
n= = = 11.8885 = 1 2
In(1 + 0.06) 0.0583
You will notice that we round off to the next whole period. To see why, consider this last
example. After 11.8885 years, we have doubled our money if interest were paid 88.85% the
way through the twelfth year. But, we stated earlier that interest is paid at the end of each
period not part of the way through. At the end of the eleventh year, our investment is worth
$189.93, and at the end of the twelfth year, our investment is worth $201.22. So, our
investment's value doubles by the twelfth period with a little extra, $1.22.
The tables of factors can be used to approximate the number of periods. The approach is
similar to the way we approximated the interest rate. The compounding factor in this
example is 2.0000 and the discounting factor is 0.5000 (that is, FV/PV= 2.0000 and PV/FV
0.5000). Using the table of compound factors, following down the column corresponding to
the interest rate of 6%, the compound factor closest to 2.0000 is for 12 periods. Likewise,
using the table of discount factors, following down the column corresponding to the interest
rate of 6%, the discount factor closest to 0.5000 is for 12 periods.
Activity 4.2
• How long does it take to triple your money if the interest rate is 5% per year,
compounded annually?
• How long does it take to double your money if the interest rate is 12% per year,
compounded quarterly?
60
4.6 Frequency of Compounding
If interest is compounded more frequently than once per year, you need to consider
this in any valuation problem involving compounded interest. Consider the following
scenario:
$1 000 is deposited in an account at the beginning of the period and 12% interest is paid on
the account, with interest compounded quarterly. This means that at the end of the first
quarter, the account has a balance of:
FV1st quarter = $1 000 (1 +0.12/4) = $1 000 (1 + 0.03) = $1 030
The following quarters' balances are calculated in a like manner, with interest paid on the
balance in the account:
FV2nd quarter = $1 030.00(1+0.03) = $1 060.90
FV3rd quarter = $1 060.90(1+0.03) = $1 092.73
FV4th quarter = $1 092.73(1+0.03) = $1 125.51
Therefore, at the end of one year, there is a balance of
$1 000 (1 + 0.03)4 = $1 125.51.
When you face a situation in which interest is compounded more frequently than on an
annual balance, you need to adjust the number of period and the interest rate accordingly.
Thus, the future value of an investment in circumstances of non- annual compounding is
given by:
r
FVnx = P (1 + ) nx
x
r
where x stands for the number of times interest is compounded in a year thus is a
x
periodic interest rate.
0.12 5×12
20000(1 + ) =FV60 = 20,000(1+0.01)60 = $20,000(1.8167) = $36,333.93
12
61
4.7 Continuous Compounding
If interest is compounded continuously (that is, instantaneously), the compound factor uses
the exponential function, e, the inverse of the natural logarithm. The compound factor is:
Using a calculator, you need to use the maths function ex. For example, suppose you want to
calculate the future value of $1 000 invested five years at 4%, with interest compounded
continuously. The future value is:
FV5 = 1 000e(0.04)(5)
= 1 000e0.2
= 1 000(1.2214)
= $1 221.40
One obvious way to represent rates stated in various time intervals on a common basis is to
express them in the same unit of time; so we annualise them. The annualised rate is the
product of the stated rate of interest per compounding period multiplied by the number of
compounding periods in a year. Let r be the rate of interest per period and let n be the
number of compounding periods in a year. The annualised rate, also referred to as the
nominal interest rate or the annual percentage rate (APR), or the quoted rate is given by:
62
r
Periodic rate =
x
The effective rate of interest (effective annual rate or EAR) is therefore an annual rate that
takes into consideration any compounding that occurs during the year. The effective
interest rate is given by:
r
EAR= (1 + ) x − 1
x
Let's look at how the EAR is affected by the compounding. Suppose that the Safe Savings
and Loan promises to pay 6% interest on accounts, compounded annually. Because interest
is paid once, at the end of the year, the effective annual return, EAR, is 6%. If the 6%
interest is paid on a semi-annual basis -- 3% every six months -- the effective annual return
is larger than 6% because interest is earned on the 3% interest earned at the end of the first
six months. In this case, to calculate the EAR, the interest rate per compounding period is
0.03 (that is, 0.06 / 2) and the number of compounding periods in an annual period is 2:
EAR = (1 + r)x - 1
EAR = (1 + 0.03)2 - 1 = 1.0609 - 1 = 0.0609 or 6.09%.
Extending this example to the case of quarterly compounding with a nominal interest rate
of 6%, we first calculate the interest rate per period, r, and the number of compounding
periods in a year, n:
r = 0.06 / 4 = 0.015 per quarter, and
n = 12 months /3 months = 4 quarters in a year.
The EAR is:
EAR = (1 + 0.015)4 -1
= 1.0614 – 1
= 0.0614 or 6.14%
Suppose there are two banks: Bank A, paying 12% interest compounded semiannually, and
Bank B: paying 11.9% interest compounded monthly. Which bank offers you the best
return on your money? Comparing APR's, Bank A provides the higher return. But what
about compound interest? The EAR's for each account are calculated as:
Bank A:
EAR = (1 + 0.12/2)2 -1
= (1 +0.06)2
= 1.1236 - 1
= 0.1236 or 12.36%
63
Bank B:
EAR = (1 + 0.119/12)12- I
= (I + 0.0099)12 - 1
= 1.1257 - 1
= 0.1257 or 12.57%
Bank B offers the better return on your money, even though it advertises a lower APR. If
you deposit $1 000 in Bank A for one year, you will have $1 123.60 at the end of the year.
If you deposit $1 000 in Bank B for one year, you will have $1 125.70 at the end of the
year, providing the better return on your savings.
And the number of compounding periods in a year, n, is infinite. As the rate of interest, i,
gets smaller and the number of compounding periods approaches infinity, the EAR is:
APR ∞
EAR= 1 + ( )
∞
What does all this mean? It means that as the interest rate per period approaches 0 and the
number of compounding periods approaches infinity at the same time! For a given nominal
interest rate under continuous compounding, it can be shown that:
EAR = e x − 1
For the stated 6% annual interest rate compounded continuously, the EAR is:
EAR = e0.06 - 1
= 1.0618 - 1
= 0.0618 or 6.18%.
The relation between the frequency of compounding, for a given stated rate, and the
effective annual rate of interest for this example indicates that the greater the frequency of
compounding, the greater the EAR.
64
Effective Annual Rates of Interest Equivalent to an Annual
Percentage Rate of 6%
Calculation EAR
Frequency of compounding per year
Annual (1 + 0.060)1 - 1 6.00%
(1 + 0.015)4- 1 6.14
Quarterly
Continuous e0.06- 1 6.18
FV=P(1+r)n.
Making P the subject of the formula will make us obtain the present value.
CFn 1
n
= CFn ×
Present Value of Simple Cash Flow = (1 + r ) (1 + r ) n
Where:
CFn = Cash Flow at the end of time period n
r = discount rate
1
(1 + r) n = Present value interest or discount factor i.e. PV1F (r%, n years) and
this can be obtained in the present value tables in the appendix.
65
Using financial tables the present value is given by:
PV= CFn × PVIF(r%, n years)
Other things remaining equal, the present value of a cash flow will decrease as the discount
rate increases.
The present value of this cash flow can then be estimated as follows:
500,000
Present Value of Payment = ( = $192 750
(1.10)10
Using tables
Example 4.6
Calculate the present value of $10 000 to be received at the end of five years if the annual interest
rate is 6%, compounded semi-annually. If the annual rate is 6%, the semi-annual rate is 6%/2 =
3%.
66
Soluti o n
The number of semi-annual periods is 5 x 2 = 10
Therefore, the present value of this $10 000 is calculated as:
1
PV = 10 000×
0.06 5×2
(1 + )
2
= 10 000(PVIF3%, 10)
= 10 000(0.7408)
= 7 440.94
If interest was compounded continuously, the present value would be:
1
0.06×5
PV = 10 000× e
= 10 000(0.7408)
= 7 408.18
For a given annual percentage rate, the greater the frequency of compounding, the greater is
the effective interest rate and the smaller is the present value.
Activity 4.3
1. How much would you have to deposit today in an account that pays 4%
annual interest, compounded continuously, if you wish to have a balance of
$100 000 at the end of ten years?
2. Which of the following requires the least amount of a deposit today?
a. A balance of $10 000 four years from today that has grown from a
sum deposited in an account that pays 8% interest, compounded
quarterly.
b. A balance of $10 000 five years from today that has grown from a
sum deposited in an account that pays 7% interest, compounded
annually.
c. A balance of $10 000 ten years from today that has grown from a
sum deposited in an account that pays 4% interest, compounded
continuously.
d. A balance of $10 000 eight years from today that has grown from a
sum deposited in an account that pays 4% interest, compounded
semi-annually.
67
Translating a series of cash flows into a present value is similar to translating a single
amount to the present; we discount each cash flow to the present using the appropriate
discount rate and number of discount periods. Translating a series of cash flows into a
future value is also similar to translating a single sum: simply add up the future values of
each cash flow.
Because we are now dealing with cash flows or amounts that occur at different points in
time, it is useful to sort out these cash flows or values in a time line:
0 1 2 3 4 5
where the end of the period is marked with a period indicator. For example, if there is an
expected cash flow of $100 at the end of two years and a cash flow of $300 at the end of
five years, we would represent this as:
Today
0 1 2 3 4 5 period
where “0” represents today, the end of period 0. Though we could pose problems with cash
flows that occur either at the end of periods or the beginning of periods (as long as we have
consistently defined a period as a fixed unit of time), for consistency we generally consider
cash flows as end of period flows.
100
PV = = 100(0.9070) = 90.70
(1 + 0.05) 2
300
PV = = 300(0.7835) = 235.06
(1 + 0.05) 5
68
Exhibit 4.3: Present value of a series of uneven cash flows
0 1 2 3 4 5 period
$90.70
$235.06
$325.76
We can represent this problem in the time line shown in Exhibit 4.3.
The future value of the second cash flow is $300 (there is no compounding because the
$300 cash flow occurs at the end of the fifth period. Therefore, the future value at the end of
the fifth year is:
FV5 = $115.76 + 300 = $415.76.
69
Exhibit 4.4: Future value of a series of uneven cash flows
0 1 2 3 4 5 period
As shown in Exhibit 4.4, you will notice that we did not compound the last cash flow (the
$300 that occurs at the end of the fifth period); that is, because the last cash flow is
received at the end of the fifth period so there is no earned interest on that cash flow. We
have added the subscript to the FV notation to designate the future period (in this example,
that is, the end of period 5 so we use FV5).
0 0
1 0
2 $100 0.9070 $90.70
3 0
4 0
5 $300 0.78 $235.06
PV0=$325.76
FV5=$415.76
70
We can represent these calculations using formula for the present and future values.
The present value of a series of cash flows is:
n
CFt
PV0 = ∑ (1 + r )
t =1
t
where
PV0 is the present value at the end of period 0 (today),
t represents the period of receiving cash flow,
CFt is the cash flow at the end of period t,
r is the interest rate per period, and
n is the number of periods or maturity of the investment.
The future value (at the end of period n) of a series of cash flows is:
n
FV n = ∑ CF (1 + r )
t
t
n −t
You will notice that the cash flows are compounded n-t periods. Therefore, a cash flow
occurring at the end of the nth period is not compounded (that is, n-t = 0).
Suppose you deposit $100 today, $200 one year from today, and $300 two years from today
in account that pays 4% interest, compounded annually. What will be the balance in the
account at the end of (a) two years? (b) three years?
Solution
At the end of two years,
FV2 = 100(1+0.04)2 + 200(1+0.04)1 + 300 = $616.16
4.18 Annuities
If the cash flows are the same amount and occur at regular intervals of time (for example,
annually, semi-annually, monthly), we can use a short-cut to perform the calculations. An
annuity is a series or sequence of equal cash flows that meet the following criteria:
• The amount of the cash flow is the same each period.
• The cash flows occur (paid or received) at regular intervals of time.
If these are satisfied, the series is an annuity. The three distinct annuity patterns that are
71
found commonly in finance applications are ordinary annuities, annuity due and deferred
annuity.
Ordinary annuity is the annuity in which the first cash flow in the series is one period from
today. In other words, payments are made at the same time that interest is credited, that is
at end of the payment interval.
Annuity due is the annuity in which the first cash flow in the series occurs today. The
payment falls due at the beginning of the period.
Deferred annuity is the annuity in which the first cash flow in the series occurs beyond one
period from today.
If the payment of the annuity begins and ends on a fixed date, the annuity is known as an
annuity certain. On the other hand, if the payments continue forever the annuity is known as
perpetuity.
1
1 − (1 + r ) n
PV of an annuity = PV(A, r, n) = A
r
where
A = Annuity
r = Discount Rate
n = Number of years
Accordingly, the notation we will use in the rest of this module for the present value of an
annuity will be PV(A, r, n).
The present value of the instalment payments exceeds the cash-down price; therefore, you
would want to pay the $10 000 in cash now.
Alternatively, discounting each of the cash flows back to the present and aggregating the
present values as illustrated in Exhibit 4.5 could have estimated the present value.
0 1 2 3 4 5
$2 679
$2 392
$2 135
$1 906
$1 702
$10 814
Activity 4.4
1. Calculate the present value of a four- payment $1 000 ordinary annuity if
the interest rate is 5%.
2. Calculate the future value of a four- payment $1 000 ordinary annuity
if the interest rate is 5%.
Suppose you are the pension fund consultant to The Home Depot, and that you are trying to
estimate the present value of its expected pension obligations, which amount in nominal
terms to the following:
Years Annual Cash Flow
1 -5 $200 million
6 -10 $300 million
11-20 $400 million
If the discount rate is 10%, the present value of these three annuities can be estimated as
follows:
Present value of first annuity = $200 million × PV (A, 10%, 5) = $758 million
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Present value of second annuity = $300 million × PV (A, 10%, 5) / 1.105 = $706 million
Present value of third annuity = $400 million × PV (A, 10%, 10)/1.1010 = $948 million.
The present values of the second and third annuities can be estimated in two steps:
First, the standard present value of the annuity is computed over the period that the
annuity is received.
Second, that present value is brought back to the present.
Thus, for the second annuity, the present value of $300 million each year for 5 years is
computed to be $1 137 million; this present value is really as of the end of the fifth year. It is
discounted back 5 more years to arrive at today’s present value which is $706 million.
Cumulated present value = $758 million + $706 million + $948 million = $2.412 billion.
This monthly payment is an increasing function of interest rates. When interest rates drop,
homeowners usually have a choice of refinancing, though there is an upfront cost of doing
so.
74
(1 + r ) n − 1
FV of an Annuity = FV (A, r, n) = A
r
Thus, the notation we will use throughout this book for the future value of an annuity will
be FV(A, r, n).
The tax exemption adds substantially to the value because it allows the investor to keep the
pre-tax return of 8% made on the IRA investment. If the income had been taxed at say 40%,
the after-tax return would have dropped to 4.8%, resulting in a much lower
expected value:
(1.08) 40 − 1
Expected value of IRA set - aside at 65 if taxed = $2 000 = $230,127
0.048
As you can see, the available funds at retirement drops by more than 55% as a consequence
of the loss of the tax exemption.
r
Annuity given future value = A(FV, r, n) = FV n
(1 + r ) − 1
75
with bonds with a face value of $100 million coming due in 10 years would need to set aside
the following amount each year (assuming an interest rate of 8%):
0.08
Sinking fund provision each year = $100 000 000 10 = $6,902,950
(1.08) − 1
The company would need to set aside $6.9 million at the end of each year to ensure that
there are enough funds ($10 million) to retire the bonds at maturity.
Contrast this with an annuity of $100 at the beginning of each year for the next four years,
with the same discount rate.
Since the first of these annuities occurs right now, and the remaining cash flows take the
form of an end-of-the-period annuity over 3 years, the present value of this annuity can be
written as follows:
1
1 − (1 + r ) n−1
PV of beginning of period annuities over n years = A + A
r
1
1 − (1 + 0.10) 4−1
PV of $100 at beginning of each of next 4 years = $100 +$100
0.10
This present value will be higher than the present value of an equivalent annuity at the end
of each period.
76
4.19.4 Future value of an annuity due
The future value of a beginning-of-the-period annuity typically can be estimated by
allowing for one additional period of compounding for each cash flow:
(1 + r ) n − 1
FV of a beginning-of-period annuity = A (1 + r)
r
This future value will be higher than the future value of an equivalent annuity at the end of
each period.
Illustration IRA - Saving at the beginning of each period instead of the end
Consider the example of an individual who sets aside $2 000 at the end of each year for the
next 40 years in an IRA account at 8%. The future value of these deposits amounted to $
518 113 at the end of year 40. If the deposits had been made at the beginning of each year
instead of the end, the future value would have been higher as shown below:
(1.08) 40 − 1
Expected value of IRA (beginning of year) = $2 000(1.08) = $559,562
0.08
As you can see, the gains from making payments at the beginning of each period can be
substantial.
Activity 4.5
1. Calculate the future value of a four-payment $1 000 annuity due if the interest
rate is 5%.
2. Calculate the present value of a four-payment $1 000 annuity due if the interest
rate is 5%.
0 1 2 3 ……… n
Note that, to qualify as a growing annuity, the growth rate in each period has to be the same as the
growth rate in the prior period.
77
The process of discounting
In most cases, the present value of a growing annuity can be estimated by using the
following formula:
(1 + g ) n
1 −
(1 + r ) n
PV of a growing annuity = A(1+g)
r−g
The present value of a growing annuity can be estimated in all cases, but one - where the
growth rate is equal to the discount rate. In that case, the present value is equal to the
nominal sums of the annuities over the period, without the growth effect.
Note also that this formulation works even when the growth rate is greater than the discount
rate.
4.20 Perpetuities
A perpetuity is a constant cash flow at regular intervals forever. The present value of a
perpetuity can be written as:
A
PV of Perpetuity =
r
where A is the perpetuity. The future value of a perpetuity is infinite.
78
Illustration: Valuing a console bond
A console bond is a bond that has no maturity and pays a fixed coupon. Assume that you have
a 6% coupon console bond. The value of this bond, if the interest rate is 9%, is as follows:
A growing perpetuity is a cash flow that is expected to grow at a constant rate forever. The
present value of a growing perpetuity can be written as:
CF1
PV of Growing Perpetuity =
(r − g )
where CFI is the expected cash flow next year, g is the constant growth rate and r is the
discount rate.
While a growing perpetuity and a growing annuity share several features, the fact that a
growing perpetuity lasts forever puts constraints on the growth rate. It has to be less than the
discount rate for this formula to work.
The first problem is really a combination of a lump-sum (deposit today) and an annuity
problem (annual withdrawals once you retire), whereas the latter problem is a combination of
two annuities (an annuity of the deposits and an annuity of the withdrawals). Deferred
annuity problems can become quite complex, so it is recommended that you sort out the cash
flows by drawing a time line.
Starting with a relatively straightforward problem, consider a five-cash flow annuity of $100,
where the first cash flow occurs at the end of the third period. What is the present value of
this annuity? We could discount each cash flow individually, as shown in Exhibit 4.6.
0 1 2 3 4 5 6 7
$86.3838
$82.2702
$78.3526
$74.6215
$71.0681
392.6962
Solution
Step 1: Solve for the present value of the withdrawals. Given: CF = $10 000; r = 5%; n = 20
PV65 = $10 000 (present value annuity factor, r = 5%, n = 20)
PV 65 = $10 000 (12.4622) = $124 622
This is the balance required at the time of your 65th birthday (that is, at the end of 30
periods). Why 65th and 66th? Because we used an ordinary approach to solve this, which
means that the present value of the series occurs one period prior to the first cash flow. In
other words, by using the ordinary annuity short-cut the PV is on your 65th birth day, not on
your 66th birth day.
80
Step 2: Solve for the present value of your 65th birthday balance:
Given: FV65 = $124 622; n = 30; r = 5%
PV0 = $124 622/(1 + 0.05)30
PV0 = $124 622 (0.23138) = $28 834.72
Check it out:
PV = $28 834.72 (1 + 0.05)30 = $28 834.72 (4.32194) = $124 622
Activity 4.6
Suppose you wish to retire forty years from today. You determine that you need
$50 000 per year once you retire, with the first retirement funds withdrawn one
year from the day you retire. You estimate that you will earn 6% per year on your
retirement funds and that you will need funds up to and including your 25th
birthday after retirement.
1. How much must you deposit in an account today so that you have enough
funds for retirement?
2. How much must you deposit each year in an account, starting one year from
today, so that you have enough funds for retirement?
Therefore, the monthly payments are $4 387.14 each. In other words, if payments of
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$4 387.14 are made each month for twenty-four months, the $100 000 loan will be repaid
and the lender earns a return that is equivalent to a 5% annual rate on this loan. The
principal amount of the loan declines as payments are made, with the proportion of each
loan payment devoted to principal repayment increasing throughout the loan period from
$3 970.47 for the first payment to $4 368.94 for the last payment.
Activity 4.7
Consider a loan of $1 million that is paid off quarterly over a period of nine years.
Calculate the dollar amount of interest and loan principal repaid corresponding to
each payment if the interest rate is 6% per year, compounded quarterly.
4.24 Summary
In this unit we highlighted that future value refers to the amount of money an investment
will grow to over some length of time at some given interest rate. The process of leaving
your money and any accumulated interest in an investment for more than one period,
thereby reinvesting the interest, is called compounding.
The effective rate of interest (effective annual rate or EAR) is therefore an annual rate that
takes into consideration any compounding that occurs during the year. The effective
interest rate is given by:
r
EAR = (1 + ) x − 1
x
Present value is the current value of future cash flows discounted at the appropriate
discount rate.
CFn 1
Present value of Simple Cash Flow = n
= CFn ×
(1 + r ) (1 + r ) n
82
1
The present value interest factor for non-annual discounting is given by for discrete
r
(1 + )
x
1
discounting and for continuous discounting.
e rn
An annuity is a series or sequence of equal cash flows that meet the following criteria: The
amount of the cash flow is the same each period and the cash flows occur (paid or received)
at regular intervals of time. Ordinary annuity is the annuity in which the first cash flow in
the series is one period from today. In other words, payments are made at the same time
that interest is credited, that is at end of the payment interval.
Annuity due is the annuity in which the first cash flow in the series occurs today. The
payment falls due at the beginning of the period. Deferred annuity is the annuity in which
the first cash flow in the series occurs beyond one period from today. Perpetuity is a
constant cash flow at regular intervals forever. The present value of perpetuity can be
written as:
A
PV of Perpetuity =
r
Loan amortisation is the process of calculating the loan payments that amortise the loaned
amount. In other words, a loan is amortised when all liabilities (both interest and principal)
are paid by a sequence of equal payments made at equal intervals of time.
83
References
Block, S.B. and Hirt, G.A. (1994). Foundations of Financial Management. (Sixth Edition). Boston:
Irwin.
Ehrhardt, M.C. and Brigham, E.F. (2011). Financial Management, Theory and Practice.
(Thirteenth Edition). Mason: South-Western Cengage Learning.
Gitman, L. and Zutter, C.J. (2012). Principles of Managerial Finance. (Thirteenth Edition). Boston:
Pearson Education, Inc.
Kwesu, I., Nyatanga, E. and Zhanje, S. (2002). Business Statistics. Harare: ZOU.
Levy, H. and Sarnat, M. (1991). Capital Investments and Financial Decisions. (Fourth Edition). New
York: Prentice Hall.
McLaney, E..J. (2000). Business Finance: Theory and Practice. New York: Prentice Hall.
84
Unit 5
Bond Valuation
5.0 Introduction
Bonds are capital market instruments which are an important source of finance for
governments, municipalities and corporations. In this unit, we look at the different types
of bonds and at the key features associated with these instruments. We conclude the unit
by looking at the valuation process of bonds.
5.1 Objectives
By the end of this unit, you should be able to:
• define what a bond is
• describe the four major classifications of bonds
• distinguish traditional types bonds from conventional types of bonds
• describe the key characteristics of bonds
• use the basic bond valuation model
• calculate the price of a bond
• calculate the price of a callable bond
• calculate the yield to maturity (or call) of a bond
• determine interest rate risk, price risk and reinvestment rate risk
• calculate the impact on the price of a bond resulting from a change in the
market interest rate
We can further break the general classification into traditional bonds and conventional
bonds as shown in Tables 5.1 and 5.2 below:
86
Table 5.1: Characteristics and Priority of Lender’s Claim of Traditional Types of
Bonds
Bond Type Characteristics Priority of Lender’s Claim
Unsecured bonds
Debentures Unsecured bonds that only Claims are the same as those of
creditworthy firms can issue. any general creditor. May have
Convertible bonds are normally other unsecured bonds
debentures. subordinated to them.
Subordinated debentures Claims are not satisfied until Claim is that of a general creditor
those of the creditors holding but not as good as a senior debt
certain (senior) debts have been claim.
fully satisfied.
Income bonds Payment of interest is required Claim is that of a general
only when earnings are creditor. Are not in default when
available. Commonly issued in interest payments are missed,
reorganisation of a failing firm. because they are contingent only
on earnings being available.
Secured bonds
Mortgage bonds Secured by real estate or Claim is on proceeds from sale of
buildings. mortgaged assets; if not fully
satisfied, the lender becomes a
general creditor. The first-
mortgage claim must be fully
satisfied before distribution of
proceeds to second mortgage-
holders, and so on. A number of
mortgages can be issued against
the same collateral.
Collateral trust bonds Secured by stock and (or) bonds Claim is on proceeds from stock
that are owned by the issuer. and (or) bond collateral; if not
Collateral value is generally fully satisfied, the lender
25% to 35% greater than bond becomes a general creditor.
value.
Equipment trust Used to finance “rolling Claim is on proceeds from the
certificates stock”—airplanes, trucks, boats, sale of the asset; if proceeds do
railroad cars. A trustee buys the not satisfy outstanding debt, trust
asset with funds raised through certificate lenders become
the sale of trust certificates and general creditors.
then leases it to the firm; after
making the final scheduled lease
payment, the firm receives title
to the asset. A type of leasing.
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Table 5.2: Characteristics of Contemporary Types of Bonds
Bond Types Characteristics
Zero-(or low-) coupon A significant portion (or all) of the investor’s return comes from
bonds gain in value (that is, par value minus purchase price). Generally
callable at par value. Because the issuer can annually deduct the
current year’s interest accrual without having to pay the interest
until the bond matures (or is called), its cash flow each year is
increased by the amount of the tax shield provided by the interest
deduction.
Junk bonds Debt rated Ba or lower by Moody’s or BB or lower by Standard and
Poor’s. Commonly used by rapidly growing firms to obtain growth
capital, most often as a way to finance mergers and takeovers. High-
risk bonds with high yields—often yielding 2% to 3% more than the
best-quality corporate debt.
Floating-rate bonds Stated interest rate is adjusted periodically within stated limits in
response to changes in specified money market or capital market
rates. Popular when future inflation and interest rates are uncertain.
Tend to sell at close to par because of the automatic adjustment to
changing market conditions. Some issues provide for annual
redemption at par at the option of the bondholder.
Extendible notes Short maturities, typically 1 to 5 years, which can be renewed for a
similar period at the option of holders. Similar to a floating-rate
bond. An issue might be a series of 3-year renewable notes over a
period of 15 years; every 3 years, the notes could be extended for
another 3 years, at a new rate competitive with market interest rates
at the time of renewal.
Putable bonds Bonds that can be redeemed at par (typically, $1 000) at the option
of their holder either at specific dates after the date of issue and
every 1 to 5 years thereafter or when and if the firm takes specified
actions, such as being acquired, acquiring another company, or
issuing a large amount of additional debt. In return for its conferring
the right to “put the bond” at specified times or when the firm takes
certain actions, the bond’s yield is lower than that of a non-putable
bond.
Activity 5.1
1. Define what a bond is.
2. Describe the general classification of bonds.
3. Discuss the characteristics of the traditional and conventional bonds.
4. Explain why treasury bonds are considered riskless?
5. To what types of risk are investors of foreign bonds exposed?
88
5.4.1 Par value
The par value is the stated face value of the bond; for illustrative purposes, we generally
assume a par value of $1 000. In practice, some bonds have par values that are multiples
of $1 000 (for example, $5 000) and some have par values of less than $1 000 (Treasury
bonds can be purchased in multiples of $100). The par value generally represents the
amount of money the firm borrows and promises to repay on the maturity date.
A call provision is valuable to the firm but potentially detrimental to investors. If interest
rates go up, the company will not call the bond, and the investor will be stuck with the
original coupon rate on the bond, even though interest rates in the economy have risen
sharply. However, if interest rates fall, the company will call the bond and pay off
investors, who then must reinvest the proceeds at the current market interest rate, which
is lower than the rate they were getting on the original bond. In other words, the investor
loses when interest rates go up but does not reap the gains when rates fall. To induce an
investor to take this type of risk, a new issue of callable bonds must provide a higher
coupon rate than an otherwise similar issue of non-callable bonds.
Bonds that are redeemable at par at the holder’s option protect investors against a rise in
interest rates. If rates rise, the price of a fixed-rate bond declines. However, if holders
have the option of turning their bonds in and having them redeemed at par, then they are
protected against rising rates. If interest rates have risen, holders will turn in the bonds
and reinvest the proceeds at a higher rate.
89
5.4.5 Sinking funds
A sinking fund provision facilitates the orderly retirement of the bond issue. The issuing
firm may be required to deposit money with a trustee, which in turn invests the funds and
then uses the accumulated sum to retire the bonds when they mature. Usually, though, the
sinking fund is used to buy back a certain percentage of the issue each year. A failure to
meet the sinking fund requirement causes the bond to be thrown into default, which may
force the company into bankruptcy.
Sinking funds are designed to protect bondholders by ensuring that an issue is retired in
an orderly fashion. However, these can work to the detriment of bondholders. For
example, suppose that the bond carries a 10% interest rate but that yields on similar
bonds have fallen to 7.5%. A sinking fund call at par would require an investor to give up
a bond that pays $100 of interest and then to reinvest in a bond that pays only $75 per
year. This obviously harms those bondholders whose bonds are called. On balance,
however, bonds that have a sinking fund are regarded as being safer than those without
such a provision, so at the time they are issued sinking fund bonds have lower coupon
rates than otherwise similar bonds without sinking funds.
5.4.7 Trustee
A trustee is a third party to a bond indenture. The trustee can be an individual, a
corporation, or (most often) a commercial bank trust department. The trustee is paid to
act as a “watchdog” on behalf of the bondholders and can take specified actions on behalf
of the bondholders if the terms of the indenture are violated.
Activity 5.2
1. Why is a call provision advantageous to a bond issuer?
2. Discuss the four main factors that affect the cost of bonds to the issuer.
3. In what ways do sinking funds provide both benefits and disadvantages to
bondholders?
4. Explain why call provisions can be harmful to investors?
The value of a debt security today is the present value of the promised future cash flows (the
interest and the maturity value). Therefore, the present value of a debt is the sum of the present
value of the interest payments and the present value of the maturity value.
To figure out the value of a debt security, we have to discount the future cash flows (the
interest and maturity value) at some rate that reflects both the time value of money and the
uncertainty of receiving these future cash flows. We refer to this discount rate as the yield. The
more uncertain the future cash flows, the greater the yield. It follows that the greater the
yield, the lower the present value of the future cash flows (hence, the lower the value of
the debt security).
The present value of the maturity value is the present value of a future amount. In the case
of a straight coupon security, the present value of the interest payments is the present
value of an annuity. In the case of a zero-coupon security, the present value of the interest
91
payments is zero, so the present value of the debt is the present value of the maturity
value.
We can rewrite the formula for the present value of a debt security using some new
notation and some familiar notation. Since there are two different cash flows -- interest
and maturity value -- let C represents the coupon payment promised each period and M
represents the maturity value. Also, let n indicates the number of periods until maturity, t
indicates a specific period, and rd indicates the yield. The present value of a debt security,
V, is:
Value of a bond = PV of the interest + PV of the maturity value
C M
V = ∑t =1
N
+
(1 + rd ) (1 + rd )t
t
To see how the valuation of future cash flows from debt securities works, let us look at
the valuation of a straight coupon bond and a zero coupon bond:
12
$100 $1,000
Value of the bond = ∑ (1 + 0.05)
t =1
t
+
(1 + 0.05)12
= $886.33 + 556.84
= $1 443.17
This bond has a present value greater than its maturity value, so we say that the bond is
selling at a premium from its maturity value. Does this make sense? Yes: The bond pays
interest of 10% of its face value every year. But what investors require on their
investment, that is, the capitalisation rate considering the time value of money and the
uncertainty of the future cash flows, is 5%. So what happens? The bond paying 10% is
attractive, that is, so attractive that its price is bid upward to a price that gives investors
the going rate, the 5%. In other words, an investor who buys the bond for $1 443.17 will
get a 5% return on it if it is held until maturity. We say that at $1 443.17, the bond is
priced to yield 5% per year.
Suppose, instead, the interest on the bond is $50 every six months -- still considered a 10%
coupon rate -- instead of $100 once every year. Then,
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Interest, C = $50 every six months
Number of periods, n = 12 six-month periods
Maturity value, M = $1 000
12
50 1,000
Value of the bond = ∑ (1 + 0.05)
t =1
t
+
(1 + 0.05)12
The bond's present value is equal to its face value and we say that the bond is selling "at
par". Investors will pay face value for a bond that pays the going rate for bonds of similar
risk. In other words, if you buy the 5% bond for $1 000, you will earn a 5% annual return
on your investment if you hold it until maturity.
Activity 5.3
1. Why is it important for financial managers to understand the valuation
process?
2. What are the three key inputs to the valuation process?
3. Suppose a bond has a $1 000 face value, a 10% coupon (paid semi-annually),
five years remaining to maturity, and is priced to yield 8%. What is its value?
5.5.2 Different value, different coupon rate, but the same return
How can one bond costing $1 443.17 and another costing $1 000 both give an investor a
return of 5% per year if held to maturity? If the $1 443.17 bond has a higher coupon rate
than the $1 000 bond (10% versus 5%), it is possible for the bonds to provide the same
return. With the $1 443.17 bond you pay more now, but also get more each year ($100
versus $50). The extra $100 a year for 12 years makes up for the $443.17 extra you pay
now to buy the bond.
Suppose, instead, the interest on the bond is $20 every year, that is, a 2% coupon rate.
Then, Interest, C = $20 every year; Number of periods, n = 12 years; Maturity value, M=
$1 000 Yield, rd= 5% per year
12
20 1,000
Value of the bond = ∑ (1 + 0.05)
t =1
t
+
(1 + 0.05)12
= $177.27 + 556.84
= $734.11
The bond sells at a discount from its face value. Why? Because investors are not going
to pay face value for a bond that pays less than the going rate for bonds of similar risk. If
an investor can buy other bonds that yield 5%, why pay the face value, that is, $1 000 in
93
this case, for a bond that pays only 2%? They would not. Instead, the price of this bond
would fall to a price that provides an investor earn a yield-to-maturity of 5%. In other
words, if you buy the 2% bond for $734.11, you will earn a 5% annual return on your
investment if you hold it until maturity.
So when we look at the value of a bond, we see that its present value is dependent on the
relation between the coupon rate and the yield. We can see this relation in our example. If
the yield exceeds the bond's coupon rate, the bond sells at a discount from its maturity
value and if the yield is less than the bond's coupon rate, the bond sells at a premium.
Let's look at another example, this time keeping the coupon rate the same, but varying the
yield. Suppose we have a $1 000 face value bond with a 10% coupon rate that pays
interest at the end of each year and matures in five years. If the yield is 5%, the value of
the bond is:
V = $432.95 + $783.53 = $1,216.48
If the yield is 10%, the same as the coupon rate, 10%, the bond sells at face value:
V = $379.08 + $620.92 = $1,000
If the yield is 15%, the bond's value is less than its face value:
When we hold the coupon rate constant and vary the yield, we see that there is a negative
relation between a bond's yield and its value. This can be illustrated in the table below:
We see a relation developing between the coupon rate, the yield, and the value of a debt
security:
• If the coupon rate is more than the yield, the security is worth more than its face
value then it sells at a premium;
• If the coupon rate is less than the yield, the security is less that its face value then
it sells at a discount.
• If the coupon rate is equal to the yield, the security is valued at its face value.
Let us extend the valuation of debt to securities that pay interest every six months. But
before we do this, we must grapple with a bit of semantics. In Wall Street parlance, the
term yield-to-maturity is used to describe an annualised yield on a security if the security
is held to maturity. For example, if a bond has a return of 5% over a six-month period, the
annualised yield-to-maturity for a year is 2 times 5% or 10%. But is this the effective
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yield-to-maturity? Not quite. This annualised yield does not take into consideration the
compounding within the year if the bond pays interest more than once per year. The yield-
to-maturity, as commonly used on Wall Street, is the annualised yield-to-maturity.
Annualised yield-to-maturity = rd × 2
Suppose we are interested in valuing a $1 000 face value bond that matures in five years
and promises a coupon of 4% per year, with interest paid semi-annually.
This 4% coupon rate tells us that 2%, or $20, is paid every six months. What is the bond's
value today if the annualised yield-to-maturity is 6%? From the bond's description we
know that:
= $170.60 + 744.09
= $904.71
If the annualised yield-to-maturity is 8%, then:
Interest, C = $20 every six months
Number of periods, n = 5 times 2= 10 six-month periods
Maturity value, M = $1 000
Yield, rd= 4% for six-month period
10
20 1,000
The value of the bond = ∑
t =1 (1 + 0.04)
t
+
(1 + 0.04)10
= $162.22 + 675.56
= $837.78
The greater the yield, the lower the present value of the bond. This makes sense since an
increasing yield means that we are discounting the future cash flows at higher rates.
1,000
Value of the zero- coupon bond =
(1 + 0.05) 5
= $783.53
The price of the zero-coupon bond is sensitive to the yield. If the yield changes from 10%
to 5%, the value of the bond increases from $620.92 to $783.53.
Activity 5.4
1. The XYZ bond has a maturity value of $1 000 and a 10% coupon, with
interest paid semi-annually.
(a) If there are five years remaining to maturity and the bonds are priced
to yield 8%, what is the bond's value today?
(b) If there are five years remaining to maturity and the bonds are priced
to yield 10%, what is the bond's value today?
(c) If there are five years remaining to maturity and the bonds are priced
to yield 12%, what is the bond's value today?
2. What is the value of a zero-coupon bond that has five years remaining to
maturity and has a yield-to-maturity of 6%?
96
44.375 44.375 975 44.375 1,019.375
$965 = t
+ + = +
(1 + rd ) (1 + rd ) 2
(1 + rd ) 2
(1 + rd ) t (1 + rd ) 2
The rd is 5.1037%. The annualised return, that is, the convention used for reporting
yields on bonds is 5.104% × 2 = 10.2074%.
With the exception of the convention for annualising the return, the yield on the bond is
calculated in a manner similar to that of stocks. But there is another dimension to
consider with bonds that we need not consider with common stocks: bonds have a finite
life because they either mature or are called. Therefore, we are not just interested in the
annual yield, but also a yield if the bond is held to maturity or a yield to the point where a
bond is likely to be called.
5.5.5 Yield-to-maturity
The yield-to-maturity is the annual yield on an investment assuming you own it until
maturity. It considers all an investment's expected cash flows --in the case of a bond, the
interest and principal. When we look at yield-to-maturity, we once again see a relation
between a bond's yield and its value today.
The yield-to-maturity on a coupon bond is the discount rate, put on an annual basis, that
equates the present value of the interest and principal payments to the present value of the
bond. In the case of a bond that pays interest semi-annually, we first solve for the six-
month yield, and then translate it to its equivalent annual yield-to-maturity.
Let us look at yield-to-maturity on a coupon bond. Consider the Acme Corporation bonds
7
with 8 % coupon bonds maturing in 2020 and with interest paid-semiannually, what is
8
the yield-to-maturity on these bonds if you bought them on January 1, 2000 for 96.5
(which means 96.5% of par value)? Or, put another way, what annual yield equates the
investment of $965 with the present value of the twenty-two interest cash flows and
maturity value?
In this example, we know the following:
V = $1 000 x 96.5% = $965
7
C = 8 % / 2 × $1,000 = $44.375
8
M = $1 000
N = 11 years × 2 = 22 six-month periods
and t identifies the six-month period we're evaluating. Therefore,
N
C M
V =∑ +
t =1 (1 + rd ) (1 + rd ) N
t
22
44.375 1,000
V =∑ t
+
t =1 (1 + rd ) (1 + rd ) 22
97
If V = $965, we have one unknown: rd. Where do we start looking for a solution to rd? If
the bonds yielded 87/8%, they would be selling close to par (that is, $1 000). This would
be equivalent to a six month value of rd = 8.875% / 2 = 4.4375% for six months. But
these bonds are priced below par. That is, investors are not willing to pay full price for
these bonds since they can get a better return on similar bonds elsewhere. As a result, the
price of the bonds is driven downward until these bonds provide a return or yield-to-
maturity equal to that of bonds with similar risk.
Given this reasoning, the yield on these bonds must be greater than the coupon rate, so the
six-month yield must be greater than 4.4375%. Using the trial and error approach, we
would start with 5% and look at the relation between the present value of the cash inflows
(interest and principal) discounted at 5% and the price of the bonds (the $965):
22
$44.375 $1,000
Is $965 equal to ∑ (1 + 0.05)
t =1
t
+
(1 + 0.05) 22
?
$965 ≠ $925.96
In fact, using 5%, we have discounted too much, since the present value of the bond's cash
flows using 5% is less than the current value of the bonds. Therefore, we know that rd
should be less than 5%. We now have an idea of where the six- month yield lies: between
4.4375% and 5%. Using a financial calculator, we find the value of rd = 4.70%, a six-
month yield. Translating the six month yield into an annual yield, we find that these bonds
are valued such that the yield-to-maturity is 9.4%.
Alternatively the yield to maturity can be approximated using the following formulas
Vp
YTM ( rd ) = p + × (d − p )
V p + Vd
Where p is the rate which results into a bond trade at a premium, that is, V p and d is the
rate which will result into a discount value, Vd .
OR
Fd − Vd
I +( )
YTM (rd ) = n
Fd + 2Vd
3
Where I is the annual interest payment, Fd is the face or par value of the debt, Vd is market
value of the debt and n is the maturity period of the debt.
Note that the above formulae will not give exact yield but approximates it.
98
Activity 5.5
Suppose a bond with a 5% coupon (paid semi-annually), five years remaining to
maturity, and a face value of $1 000 has a price of $800. Calculate the yield to
maturity on this bond?
V = $531.10 + 437.48
V = $968.28
The value of the bond (that is, the sum of the present value of the interest and the present
value of the maturity value) increases ever so slightly until it approaches the maturity
value. The interest payments contribute less to the bond’s present value as time goes on
since there are fewer interest payments through time, yet the maturity value contributes
more as it nears in time, and hence more valuable as we get closer to maturity. The change
in the value of the bond as it approaches maturity is referred to as the time path of the
bond.
If yields did not change during the year and the bonds were valued to yield 9.4% per year
to maturity, the value of the bonds would have crept up to $966.4383 by the end of 2000
(simply because of the passage of time). But instead, the value of the bonds increased from
$965 to $975. Since the cash flows have not changed, the only thing that could cause the
value of the bonds to deviate from $966.4383 is the discount rate, that is, the yield. As we
saw from the calculations, the yield-to-maturity decreased from 9.4% to 9.264% per year.
As the yield decreased, the value of the bond increased.
22
$44.375 $1,000
V =∑ t
+
t =1 (1 + rd ) (1 + rd ) 22
If we hold C, T and M constant, we see that an increase in rd, the six-month yield,
decreases the present value of the bond. Likewise, a decrease in rd increases the present
value of the bond. The value of bonds is therefore sensitive to the yield.
5.5.8 Yield-to-call
Some bonds have a feature referred to as a call feature that allows the bond issuer to buy
back the bonds from the investor at a specified price called the call price during a specified
period prior the bond's maturity. If a bond is callable, investors are concerned with not just
its yield-to-maturity, but also its return if the bond is called away. Yield-to-call is a concept
similar to the yield-to-maturity. It is the yield to the point of when the bond is expected to
be called, instead of a yield to a bond's maturity. The yield-to-call is calculated like the
yield to maturity, except:
• instead of the number of periods to the maturity date, T is the number of periods to
some assumed point when the bonds are expected to be called, and
• the call price of the bond is used as the maturity value, M.
The call price is specified in the bond contract. When the bond may be called is also
specified in the contract, but it is usually a range of dates, so the precise date the firm will
actually call the bond is not specified. Therefore, some assumption has to be made
regarding when the bond will be called away.
1
Let us look at a callable bond to see how this works. Illinois Bell Telephone 8
4
debentures due in 2026 are callable in any year before maturity until the year 2021. But the
call price depends on the year called. If Illinois Bell calls in the bonds in 2001, the
company must pay 104.95, or $1 049.50 per bond; if called in 2021, it pays 100.24 or $1
002.40 per bond.
On January 1, 2001, the price was 88, or $880 per bond. A bond pays $82.50 interest each
100
year, or $41.25 every six months. As of January 1, 2001, there were 52 interest payments
remaining to maturity. We know that since this is a discount bond (the value is less than its
1
par value), the yield-to-maturity is greater than the 8 % coupon rate. Using a financial
4
calculator, the six-month rate is 0.0475 or 4.75%. Therefore, the yield-to-maturity as of
January 1, 2001 is 4.75% x 2, or 9.5%.
The yield-to-call is calculated is a similar manner. For example, if the bonds are called at
the end of 2001, Illinois Bell must pay $1 049.50 per bond at the call and prior to the call
has paid two interest payments since January 1, 2001: June and December of 2001. Using
this information (M= $1 049.50 and n = 2), the yield to-call is calculated by determining
the six-month yield and translating it into an annual yield. The six-month rate is 13.70%
and the yield-to-call is 29.28% per year.
We can calculate the yield-to-call for any possible call date. For bonds called at the end of
the year 2010, the yield-to-call is 10.64% per year.
Activity 5.6
1. The ABC Company bonds have a face value of $1 000, ten years
remaining to maturity, a 5% coupon (paid semiannually), and a current
value of $688.44. What is the yield-to-maturity of the ABC bonds?
2. Consider a bond that has a $1 000 face value, a 6% coupon (paid semi-
annually), matures in 6 years, and has a current price of 95. What is the
yield-to-maturity on this bond?
3. Consider a bond that has a 6% coupon (paid semi-annually) and a face
value of $1 000. What is the bond's yield-to-maturity if it has a current
price of 110 and:
(a) five years remaining to maturity?
(b) ten years remaining to maturity?
(c) twenty years remaining to maturity?
5.6 Summary
In this unit we discussed that bonds are capital market instruments that play a pivotal role
in the financing of government and corporate companies’ expenditures. These instruments
are generally classified into four major categories namely; treasury bonds, municipal
bonds, corporate bonds and foreign bonds. Bonds are distinctively debt securities that are
different from equity instruments. The key features associated with bonds include par
value, bond indentures, coupon interest rates, maturity date, call provisions, sinking fund
provisions. Bond valuation is an important process of computing bond values. There are
three key inputs to the valuation process, namely, cash flows, timing, and a measure of
risk. The value of a debt security today is the present value of the promised future cash
flows (the interest and the maturity value). If the coupon rate is more than the yield, the
security is worth more than its face value, that is, it sells at a premium. If the coupon rate
is less than the yield, and the value of the security is less than its face value, it sells at a
discount. If the coupon rate is equal to the yield, then the security is valued at its face
101
value.
References
Block, S.B. and Hirt, G.A. (1994). Foundations of Financial Management. (Sixth Edition). Boston:
Irwin.
Ehrhardt, M.C. and Brigham, E.F. (2011). Financial Management, Theory and Practice.
(Thirteenth Edition). Mason: South-Western Cengage Learning.
Gitman, L. and Zutter, C.J. (2012). Principles of Managerial Finance. (Thirteenth Edition).
Boston:
Pearson Education, Inc.
Kwesu, I., Nyatanga, E. and Zhanje, S. (2002). Business Statistics. Harare: ZOU.
Levy, H. and Sarnat, M. (1991). Capital Investments and Financial Decisions. (Fourth Edition).
New York: Prentice Hall.
McLaney, E..J. (2000). Business Finance: Theory and Practice. New York: Prentice Hall.
102
Unit 6
Valuation of Securities
6.0 Introduction
In this unit we discuss that when we value an investment, we need to know its expected
future cash flows and the uncertainty of receiving them. To value securities you must
understand the nature of the cash flows, their timing, and the uncertainty associated with
these future cash flows. We will look at the two types of equity securities: common stock
and preferred stock. These securities have different types of cash flows and the
uncertainty of each is different also.
6.1 Objectives
By the end of this unit you should be able to:
• differentiate between debt and equity
• discuss the features of both common and preferred stock
• explain the concept of common stock valuation
• use the zero-growth, constant-growth, and variable-growth versions of the dividend
discount model in computing the price or yield of a common stock
103
Table 6.1: Key Differences between Debt and Equity
Type of Capital
Characteristics Debt Equity
Voice in management a No Yes
Claims on income and assets Senior to equity Subordinate to debt
Maturity Stated None
Tax treatment Interest deduction No deduction
a
Debt-holders do not have voting rights, but instead they rely on the firm’s contractual
obligations to them to be their voice.
6.2.3 Maturity
Unlike debt, equity is a permanent form of financing for the firm. It does not “mature,” so
repayment is not required. Because equity is liquidated only during bankruptcy
proceedings, stockholders must recognise that, although a ready market may exist for their
shares, the price that can be realised may fluctuate. This fluctuation of the market price of
equity makes the overall returns to a firm’s stockholders even more risky (Gitman and
Zutter (2012).
Activity 6.1
Discuss the key differences between debt and equity.
104
6.3 Common and Preferred Stock
A firm can obtain equity capital by selling either common or preferred stock. All
corporations initially issue common stock to raise equity capital. Some of these firms later
issue either additional common stock or preferred stock to raise more equity capital.
Although both common and preferred stock are forms of equity capital, preferred stock
has some similarities to debt that significantly differentiate it from common stock. Here,
we first consider the features of both common and preferred stock and then describe the
process of issuing common stock, including the use of venture capital (Gitman and Zutter
(2012).
• Ownership
The common stock of a firm can be privately owned by private investors or
publicly owned by public investors. Private companies are often closely owned by
an individual investor or a small group of private investors (such as a family).
Public companies are widely owned by many unrelated individual or institutional
investors. The shares of privately owned firms are generally not traded; if the
shares are traded, the transactions are among private investors and often require
the firm’s consent. Large corporations are usually publicly owned, and their shares
are generally actively traded on stock exchanges (Gitman and Zutter (2012).
• Par value
The market value of common stock is completely unrelated to its par value. The
par value of common stock is an arbitrary value established for legal purposes in
the firm’s corporate charter and is generally set quite low, often an amount of $1
or less. Recall that when a firm sells new shares of common stock, the par value of
the shares sold is recorded in the capital section of the balance sheet as part of
common stock. One benefit of this recording is that at any time the total number of
shares of common stock outstanding can be found by dividing the book value of
common stock by the par value. Setting a low par value is advantageous in
countries where certain corporate taxes are based on the par value of stock. A low
par value is also beneficial in coutnries that have laws against selling stock at a
discount to par. For example, a company whose common stock has a par value of
$20 per share would be unable to issue stock if investors are unwilling to pay more
than $16 per share (Gitman and Zutter (2012).
• Preemptive rights
Gitman and Zutter (2012) postulate that the preemptive right allows common
stockholders to maintain their proportionate ownership in the corporation when
105
new shares are issued, thus protecting them from dilution of their ownership. A
dilution of ownership is a reduction in each previous shareholder’s fractional
ownership resulting from the issuance of additional shares of common stock.
Preemptive rights allow preexisting shareholders to maintain their preissuance
voting control and protects them against the dilution of earnings. Preexisting
shareholders experience a dilution of earnings when their claim on the firm’s
earnings is diminished as a result of new shares being issued.
The two authors further maintain that in a rights offering, the firm grants rights to
its shareholders. These financial instruments allow stockholders to purchase
additional shares at a price below the market price, in direct proportion to their
number of owned shares. In these situations, rights are an important financing tool
without which shareholders would run the risk of losing their proportionate control
of the corporation. From the firm’s viewpoint, the use of rights offerings to raise
new equity capital may be less costly than a public offering of stock.
• Dividends
The payment of dividends to the firm’s shareholders is at the discretion of the
company’s board of directors. Most corporations that pay dividends pay them
quarterly. Dividends may be paid in cash, stock, or merchandise. Common
stockholders are not promised a dividend, but they come to expect certain
payments on the basis of the historical dividend pattern of the firm. Before firms
pay dividends to common stockholders, they must pay any past due dividends
owed to preferred stockholders. The firm’s ability to pay dividends can be affected
by restrictive debt covenants designed to ensure that the firm can repay its
creditors.
106
sometimes allowed to elect one member of the board of directors (Gitman and Zutter
(2012).
• Restrictive covenants
The restrictive covenants in a preferred stock issue focus on ensuring the firm’s
continued existence and regular payment of the dividend. These covenants include
provisions about passing dividends, the sale of senior securities, mergers, sales of
assets, minimum liquidity requirements, and repurchases of common stock. The
violation of preferred stock covenants usually permits preferred stockholders
either to obtain representation on the firm’s board of directors or to force the
retirement of their stock at or above its par or stated value.
• Cumulation
Most preferred stock is cumulative with respect to any dividends passed. That is,
all dividends in arrears, along with the current dividend, must be paid before
dividends can be paid to common stockholders. If preferred stock is
noncumulative, passed (unpaid) dividends do not accumulate. In this case, only
the current dividend must be paid before dividends can be paid to common
stockholders. Because the common stockholders can receive dividends only after
the dividend claims of preferred stockholders have been satisfied, it is in the firm’s
best interest to pay preferred dividends when they are due.
• Callable or convertible
Preferred stock with a callable feature allows the issuer to retire outstanding shares
within a certain period of time at a specified price. The call price is normally set
above the initial issuance price, but it may decrease as time passes. Making
preferred stock callable provides the issuer with a way to bring the fixed-payment
commitment of the preferred issue to an end if conditions make it desirable to do
so. Preferred stock with a conversion feature allows holders to change each share
into a stated number of shares of common stock, usually anytime after a
predetermined date. The conversion ratio can be fixed, or the number of shares of
common stock that the preferred stock can be exchanged for changes through time
according to a predetermined formula.
Activity 6.2
1. What risks do common stockholders take that other suppliers of capital do
not?
2. How does a rights offering protect a firm’s stockholders against the
dilution of ownership?
3. What claims do preferred stockholders have with respect to distribution of
earnings (dividends) and assets?
4. Explain the cumulative feature of preferred stock. What is the purpose of a
call feature in a preferred stock issue?
107
6.4 Valuation of Common Stock
The value of a share or stock should be equal to the present value of all the future cash
flows (dividends) you expect to receive from that share. The basic valuation model for
common stock is given as follows:
D1 D2 D∞
P0 = + + ...
(1 + rs ) 1 (1 + rs ) 2 (1 + rs ) ∞
Where:
Since common stock never matures, today's value is the present value of an infinite
stream of cash flows. And also, common stock dividends are not fixed, as in the case of
preferred stock. Not knowing the amount of the dividends or even if there will be future
dividends, makes it difficult to determine the value of common stock.
In this unit, we will consider three models, namely, zero-growth, constant growth and
super-normal growth.
D1 = D2 = ... = D∞
When we let D1 represent the amount of the annual dividend, the above equation reduces
to:
∞
1 1 D
P0 = D1 × ∑ = D1 × = 1
t =1 (1 + rs ) t
rs rs
If dividends are constant forever, the value of a share of stock is the present value of the
dividends per share per period, in perpetuity.
Example 6.1
If the current dividend is $2 per share and that this dividend remains the same every year
forever. If the required rate of return is 10%, calculate the value of a share of stock.
108
Solution
$2
The required value is: P0 = = $20
0 .1
Therefore, the fairest price you can buy or sell this share today is $20. If you pay $20 per
share and dividends remain constant at $2 per share, you will earn a 10% return per year
on your investment every year.
Let D 0 indicate current period's dividend. If dividends grow at a constant rate, g, forever,
the present value of the common stock is the present value of all future dividends:
D0 (1 + g )
Ve =
ke − g
Example 6.2
If dividends just paid are $2 per share, and these will grow at a rate of 6% per year
forever, and the required return on similar dividends is 10% per annum. Calculate the
value of the share of stock.
Solution
The required value is:
$2(1 + 0.06)
Ve = = $53
0.1 − 0.06
If dividends are expected to grow in the future, the stock is worth more than if the
dividends are expected to remain the same.
109
Activity 6.3
1. The Pear Company has a current dividend of $3.00 per share. The dividends
are expected to grow at a rate of 3% per year for the foreseeable future. If the
current required rate of return on Pear Company stock is 10%, what is the
price of a share of Pear common stock?
2. ABC Company, from 2007 through 2012 period, paid the following per-share
dividends:
Assuming a required return of 15%, calculate the current value of its share.
Step 1
Find the value of the cash dividends at the end of each year, Dt , during the initial growth
period, years 1 through N. This step may require adjusting the most recent dividend, D 0 ,
using the initial growth rate, g 1 , to calculate the dividend amount for each year.
Therefore, for the first N years,
D t = D 0 × (1 + g 1 ) t
Step 2
Find the present value of the dividends expected during the initial growth period. Using
the notation presented earlier, we can give this value as
N
D0 × (1 + g 1 ) t N
Dt
∑
t =1 (1 + rs ) t
= ∑
t =1 (1 + rs )
t
110
Step 3
Find the value of the stock at the end of the initial growth period, PN = ( D N +1 ) /(rs − g 2 ) ,
which is the present value of all dividends expected from year to infinity, assuming a
constant dividend growth rate, g 2 . This value is found by applying the constant-growth
D1
model P0 = to the dividends expected from year N + 1 to infinity. The present
rs − g
value of PN would represent the value today of all dividends that are expected to be
1 D
received from year N + 1 to infinity. This value can be represented by × N +1
(1 + rs ) N
rs − g 2
Step 4
Add the present value components found in Steps 2 and 3 to find the value of the stock, P0
, given in the equation below:
N
D0 × (1 + g 1 ) t 1 D
P0 = ∑ + × N +1
t =1 (1 + rs ) t
(1 + rs )
N
rs − g 2
Example 6.3
An investor is considering purchasing the common stock of Warren Industries, a rapidly
growing boat manufacturer. The investor finds that the firm’s most recent (2012) annual
dividend payment was $1.50 per share. She estimates that these dividends will increase at
a 10% annual rate, g 1 , over the next 3 years (2013, 2014, and 2015) because of the
introduction of a hot new boat. At the end of the 3 years (the end of 2015), she expects
the firm’s mature product line to result in a slowing of the dividend growth rate to 5% per
year, g 2 , for the foreseeable future. The investor’s required return, rs , is 15%. To
estimate the current (end-of-2012) value of Warren’s common stock, she applies the four-
step procedure to these data.
Step 1
The value of the cash dividends in each of the next 3 years is calculated in columns 1, 2,
and 3 of Table 6.2 below. The 2013, 2014, and 2015 dividends are $1.65, $1.82, and
$2.00, respectively.
111
Table 6.2: Calculation of Present Value of Warren Industries Dividends (2013-2015)
Dt Present value
of dividends
t End of year D0 = D2012 (1 + g 1 ) t
(1)×(2) (1 + rs ) t
(3)÷(4)
(1) (2) (3) (4)
(5)
1 2013 $1.50 1.100 $1.65 1.150 $1.43
2 2014 1.50 1.210 1.82 1.323 1.37
Step 2
The present value of the three dividends expected during the 2013–2015 initial growth
period is calculated in columns 3, 4, and 5 of Table 6.2. The sum of the present values of
the three dividends is $4.12.
Step 3
The value of the stock at the end of the initial growth period (N=2015) can be found by
first calculating D N +1 = D2016
D2016 = D2015 × (1 + 0.05) = $2.00 × (1.05) = $2.10
By using , a 15% required return, and a 5% dividend growth rate, the value of the stock at
the end of 2015 is calculated as follows:
D $2.10 2.10
P2015 = 2016 = = = $21.00
rs − g 2 0.15 − 0.05 0.10
Finally, in Step 3, the share value of $21 at the end of 2015 must be converted into a
present (end-of-2012) value. Using the 15% required return, we get:
P2015 $21
3
= = $13.81
(1 + rs ) (1.15) 3
Step 4
Adding the present value of the initial dividend stream (found in Step 2) to the present
value of the stock at the end of the initial growth period (found in Step 3), the current
(end-of-2012) value of Warren Industries stock is:
P2012 = $4.12 + $13.81 = $17.93 per share.
The investor’s calculations indicate that the stock is currently worth $17.93 per share.
112
Activity 6.2
Suppose a stock has a current dividend of $5 per share and the required rate of return is
10%. If dividends are expected to decline 3% each year, what is the value of a share of
stock today?
Where
Because the value of the entire company, Vc , is the market value of the entire enterprise
(that is, of all assets), to find common stock value, Vs , we must subtract the market value
of all of the firm’s debt, VD , and the market value of preferred stock, V p , from Vc :
113
VS = VC - VD – VP
Because it is difficult to forecast a firm’s free cash flow, specific annual cash flows are
typically forecast for only about 5 years, beyond which a constant growth rate is assumed.
Here we assume that the first 5 years of free cash flows are explicitly forecast and that a
constant rate of free cash flow growth occurs beyond the end of year 5 to infinity. This
model is methodologically similar to the variable-growth model presented earlier.
Example 6.4
Dewhurst, Inc. wishes to determine the value of its stock by using the free cash flow
valuation model. To apply the model, the firm’s CFO developed the data given in Table
6.3. Application of the model can be performed in four steps.
Table 6.3 Dewhurst, Inc.’s, Data for the Free Cash Flow Valuation Model
Free Cash Flow
Year (t) (FCFt) Other Data
2013 400 000 Growth rate of FCF, beyond 2017 to infinity, g FCF = 3%
2014 450 000 Weighted average cost of capital, ra = 9%
2015 520 000 Market value of all debt, VD = $3,100,000
2016 560 000 Market value of preferred stock, VP = $800,000
2017 600 000 Number of shares of common stock outstanding = 300 000
Application of the model can be performed in four steps:
Step 1
Calculate the present value of the free cash flow occurring from the end of 2018 to
infinity, measured at the beginning of 2018 (that is, at the end of 2017). Because a
constant rate of growth in FCF is forecast beyond 2017, we can use the constant-growth
dividend valuation model to calculate the value of the free cash flows from the end of
2018 to infinity:
FCF2018
Value of FCF2018 − ∞ =
ra − g FCF
600,000 × (1 + 0.03)
=
0.09 − 0.03
618,000
=
0.06
= $10,300,000
Note that to calculate the FCF in 2018, we had to increase the 2017 FCF value of $600
000 by the 3% FCF growth rate, g FCF .
Step 2
Add the present value of the FCF from 2018 to infinity, which is measured at the end of
2017, to the 2017 FCF value to get the total FCF in 2017.
Table 6.4: Calculation of the Value of the Entire company for Dewhurst, Inc.
Year(t) Present Value of FCFt
Step 4
Calculate the value of the common stock, V S :
VS = VC − VD − VP
The value of Dewhurst’s common stock is, therefore, estimated to be $4 726 426. By
dividing this total by the 300 000 shares of common stock that the firm has outstanding,
we get a common stock value of $15.76 per share ($4 726 426/300 000).
It should now be clear that the free cash flow valuation model is consistent with the
dividend valuation models presented earlier. The appeal of this approach is its focus on
the free cash flow estimates rather than on forecasted dividends, which are far more
difficult to estimate given that they are paid at the discretion of the firm’s board. The
more general nature of the free cash flow model is responsible for its growing popularity,
particularly with CFOs and other financial managers.
Example 6.5
At year-end 2012, Lamar Company’s balance sheet shows total assets of $6 million, total
liabilities (including preferred stock) of $4.5 million, and 100 000 shares of common
stock outstanding. Its book value per share therefore would be:
$6,000,000 − $4,500,000
= $15 ��� �ℎ���
100,000
Because this value assumes that assets could be sold for their book value, it may not
represent the minimum price at which shares are valued in the marketplace. As a matter of
fact, although most stocks sell above book value, it is not unusual to find stocks selling
below book value when investors believe either that assets are overvalued or that the
firm’s liabilities are understated.
Example 6.6
Lamar Company found on investigation that it could obtain only $5.25 million if it sold
its assets today. The firm’s liquidation value per share therefore would be
5,250,000 − 4,500,000
= $7.50 per.share
100,000
Ignoring liquidation expenses, this amount would be the firm’s minimum value.
The use of P/E multiples is especially helpful in valuing firms that are not publicly traded,
but analysts use this approach for public companies too. In any case, the price/earnings
multiple approach is considered superior to the use of book or liquidation values because
it considers expected earnings.
Activity 6.3
1. Compare and contrast the following common stock dividend valuation models:
(a) zero-growth, (b) constant-growth, and (c) variable-growth.
2. Describe the free cash flow valuation model and explain how it differs from
the dividend valuation models. What is the appeal of this model?
3. Explain each of the three other approaches to common stock valuation: (a)
book value, (b) liquidation value, and (c) price/earnings (P/E) multiples.
Which of these is considered the best?
6.5.4 The Dividend Valuation Model and the Price-Earnings (P/E) Ratio
We can represent the Dividend Valuation Model in terms of a share's P/E ratio. Let us
start with the Dividend Valuation Model with constant growth in dividends:
D (1 + g )
P0 = Ve = 0
ke − g
If we divide both sides of this equation by earnings per share, we can represent the
dividend valuation model in terms of the Price-Earnings (P/E) ratio:
D0 (1 + g ) / EPS
P0 / EPS = Ve / EPS =
ke − g
The left hand side is the price earning ratio (P/E) while the right hand side is the dividend
payout ratio divided by ( k e − g )
This tells us the P/E ratio is influenced by the dividend payout ratio, the required rate of
return on equity and the expected growth rate of dividends.
Where Pn = new or current price of the share and P0 is the original or previous
price of the share.
If the period of time in which this spans is more than one year, we can determine the
annual return using time value of money maths, where:
FV = Ending price
PV= Beginning price
n= number of years in the period
Solving for r gives us the average annual return, which is the geometric average return.
n
FV
r= - 1
PV
Let us see how this works. At the end of 1990, Cyclops stock was $20 per share, and at the end
of 1999 Cyclops stock was $60 a share. The average annual return on Cyclops was:
$60
Return on Cyclops stock, 1990-9 = 9 -1 = 12.98%
$20
Cyclops stock has an average (that is, geometric average) return of 12.98% per year.
Example 6.6
To simplify our analysis, let us ignore our stock broker's commission. Suppose we bought
100 shares of Apple Computer common stock at the end of 1989 at $35.25 each, (a total of
$3 525 in Apple stock. During 1990, Apple Computer paid $0.45 per share in
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dividends, so we earned $45.00 in dividends. If we sold the Apple shares at the end of 1990
for $43 ($43.00 per share, or $4 300.00 for all 100 shares), what was the return on our
investment? It depends on when the dividends were received. If we assume that the
dividends were all received at the end of 1990, our return is calculated comparing the
stock's appreciation and dividends to the amount of the investment:
4,300 − 3,525 43
Return on Apple Computer stock = + 3,525
3,525
= 21.99% + 1.22% = 23.21%
Most of the return on Apple stock was from the capital yield (21.99%) -- the appreciation
in the stock's price.
But like most dividend-paying companies, Apple does not pay dividends in a lump-sum at
the end of the year, but rather pays dividends at the end of each quarter. If we want to be
painfully accurate, we could calculate the return on a quarterly basis and then annualise.
You will notice in many cases that the dividend yield is calculated by simply taking the
ratio of the annual dividend to the beginning period price. In the Wall Street Journal, the
dividend yield is the ratio of next year's expected dividend to today's share price. While
these short-cuts are convenient, remember that the true return should take into account the
time value of money. This is especially important when you are considering large
dividends relative to the stock price or when reinvestment rates are high.
Activity 6.4
Intermark, Inc. paid dividends of $0.055 per share in 1986. In 1990, Intermark paid
dividends of $0.09 per share. What was the average annual growth rate of Intermark’s
dividends from 1986 to 1990?
D
V ps =
k ps
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Example 6.7
If a preferred share has a $25 par value, a dividend rate of 10.25%, and a required rate of
return of 8%, what is its value?
$2.5625
V ps = = $32.03 =$32.03
0.08
The fairest price you can trade this share is $32.03.
Activity6. 5
At what price can you dispose your preference shares which are currently
receiving fixed dividends of $40 per share? These shares are issued in perpetuity
and have a yield rate of 15%.
6.8 Summary
In this unit we discuss the price of each share of a firm’s common stock is the value of
each ownership interest. Although common stockholders typically have voting rights,
which indirectly give them a say in management, their most significant right is their claim
on the residual cash flows of the firm. This claim is subordinate to those of vendors,
employees, customers, lenders, the government (for taxes), and preferred stockholders.
The value of the common stockholders’ claim is embodied in the future cash flows they
are entitled to receive. The present value of those expected cash flows is the firm’s share
value. The value of a share or stock should be equal to the present value of all the future
cash flows (dividends) you expect to receive from that share. There are three basic
dividend valuation models, namely, the zero-growth model, the constant-growth model
and the variable-growth model. The basic assumption of zero-growth valuation model is
that dividends paid to ordinary shareholders remain the same from the first period of
payment to infinity. The constant-growth model is based on the assumption that dividends
grow at a constant rate (g) forever and that the rate of growth is smaller than the required
return. The variable-growth model allows for a change in the dividend growth rate.
120
References
Block, S.B. and Hirt, G.A. (1994). Foundations of Financial Management. (Sixth Edition). Boston:
Irwin.
Ehrhardt, M.C. and Brigham, E.F. (2011). Financial Management, Theory and Practice.
(Thirteenth Edition). Mason: South-Western Cengage Learning.
Gitman, L. and Zutter, C.J. (2012). Principles of Managerial Finance. (Thirteenth Edition).
Boston: Pearson Education, Inc.
Kwesu, I., Nyatanga, E. and Zhanje, S. (2002). Business Statistics. Harare: ZOU.
Levy, H. and Sarnat, M. (1991). Capital Investments and Financial Decisions. (Fourth Edition).
New York: Prentice Hall.
McLaney, E..J. (2000). Business Finance: Theory and Practice. New York: Prentice Hall.
121
BLANK PAGE
Unit 7
"Risk is a most slippery and elusive concept. It is hard for investors -- let alone
economists -- to agree on a precise definition,"
Burton G. Malkiel, A Random Walk Down Wall Street, 1985, p. 187.
7.0 Introduction
In most important business decisions there are two key financial considerations: risk and
return. Each financial decision presents certain risk and return characteristics, and the
combination of these characteristics can increase or decrease a firm’s share price. Analysts
use different methods to quantify risk, depending on whether they are looking at a single
asset or a portfolio—a collection, or group, of assets. We will look at both, beginning with
the risk of a single asset.
7.1 Objectives
By the end of this unit, you should be able to:
• define risk
• explain different types of risk
• apply the concept of a probability distribution and the calculation of its expected
value
• determine the concepts and calculations of mean, variance, standard deviation, and
coefficient of variation for an asset held in isolation
• interpret standard deviation
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7.3 Return of an Asset
To assess risk on the basis of variability of return, there is need to know what return is and
how to measure it. The total rate of return is the total gain or loss experienced on an
investment over a given period. Mathematically, an investment’s total return is the sum of
any cash distributions (for example, dividends or interest payments) plus the change in the
investment’s value, divided by the beginning-of-period value. The expression for
calculating the total rate of return earned on any asset over period, t, rt , is commonly
defined as:
C + Pt − Pt −1
rt = t
Pt −1
Where:
rt = actual, expected, or required rate of return during period t
C t = cash (flow) received from the asset investment in the time period t - 1 to t
Pt = price (value) of asset at time t
Pt-1 = price (value) of asset at time t - 1
Example 7.1
Robin wishes to determine the return on two stocks that she owned during 2009, Apple
Inc. and Wal-Mart. At the beginning of the year, Apple stock traded for $90.75 per share,
and Wal-Mart was valued at $55.33. During the year, Apple paid no dividends, but Wal-
Mart shareholders received dividends of $1.09 per share. At the end of the year, Apple
stock was worth $210.73 and Wal-Mart sold for $52.84. Calculate the annual rate of
return, r, for each stock.
Solution
C t + Pt − Pt −1
rt =
Pt −1
$0 + 210.73 − 90.75
Apple: = 132.2%
90.75
Robin made money on Apple and lost money on Wal-Mart in 2009, but notice that her
losses on Wal-Mart would have been greater had it not been for the dividends that she
received on her Wal-Mart shares. When calculating the total rate of return, it is important
to take into account the effects of both cash disbursements and changes in the price of the
investment during the year (Gitman and Zutter, 2012).
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of most people most of the time, is called risk aversion. A person who is a risk-averse
investor prefers less risky over more risky investments, holding the rate of return fixed. A
risk-averse investor who believes that two different investments have the same expected
return will choose the investment whose returns are more certain. Stated another way,
when choosing between two investments, a risk-averse investor will not make the riskier
investment unless it offers a higher expected return to compensate the investor for bearing
the additional risk.
A second attitude toward risk is called risk neutrality. An investor who is risk neutral
chooses investments based solely on their expected returns, disregarding the risks. When
choosing between two investments, a risk-neutral investor will always choose the
investment with the higher expected return regardless of its risk.
Finally, a risk-seeking investor is one who prefers investments with higher risk and may
even sacrifice some expected return when choosing a riskier investment. By design, the
average person who buys a lottery ticket or gambles in a casino loses money. After all,
state governments and casinos make money off of these endeavours, so individuals lose on
average. This implies that the expected return on these activities is negative. Yet people do
buy lottery tickets and visit casinos, and in doing so they exhibit risk-seeking behaviour
(Gitman and Zutter, 2012).
Activity 7.1
1. What is risk in the context of financial decision making?
2. Define return, and describe how to find the rate of return on an investment.
3. Compare the following risk preferences:
(a) risk averse,
(b) risk neutral,
(c) risk seeking. Which is most common among financial managers?
124
Regarding expenditures: operating costs are comprised of fixed costs and variable costs.
The greater the fixed component of operating costs, the less easily a company can adjust its
operating costs to changes in sales.
We refer to the risk that comes about from the mixture of fixed and variable costs as
operating risk. The greater the fixed operating costs, relative to variable operating costs,
the greater the operating risk.
Let us take a look at how operating risk affects cash flow risk. Remember back in
economics when you learned about elasticity? That is a measure of the sensitivity of
changes in one item to changes in another. We can look at how sensitive a firm's operating
cash flows are to changes in demand, as measured by unit sales. We will calculate the
operating cash flow elasticity, which we call the degree of operating leverage (DOL).
The degree of operating leverage is the ratio of the percentage change in operating cash
flows to the percentage change in units sold. If a firm sells all its output in one period then,
Percentage.change.in.operating.cash. flows
DOL =
Percentage.change..in.units.sold
Suppose the price per unit is $30, the variable cost per unit is $20, and the total fixed costs
are $5 000. If we go from selling 1 000 units to selling 1 500 units, an increase of 50% of
the units sold, operating cash flows change from:
Operating cash flows doubled when units sold increased by 50%. What if the number of
units decreases by 25%, from 1 000 to 750? Operating cash flows decline by 50%. What
is happening is that for a 1% change in units sold, the operating cash flow changes by two
times that percentage, in the same direction. So if units sold increased by 20%, operating
cash flows would increase by 20%; if units sold decreased by 10%, operating cash flows
would decrease by 20%. Both sales risk and operating risk influence a firm's operating
cash flow risk. And both sales risk and operating risk are determined in large part by the
type of business the firm is in.
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7.5.3 Financial risk
Financial risk is the risk associated with how a company finances its operations. If a
company finances its operations with debt, it is legally obligated to pay the amounts
comprising its debts when due. By taking on fixed obligations, such as debt and long-term
leases, the firm increases its financial risk. A company that finances its business with
equity does not incur fixed obligations. The more fixed-cost obligations (debt) incurred by
the firm, the greater its financial risk.
The sensitivity of the cash flows available to owners when operating cash flows change is
referred to as the degree of financial leverage (DFL).
The cash flows to owners are equal to operating cash flows, less interest and taxes. If
operating cash flows change, how do cash flows change? Suppose operating cash flows
change from $5 000 to $6 000 and suppose the interest payments are $1 000 and, for
simplicity and wishful thinking, the tax rate is 0%:
A change in operating cash flow from $5 000 to $6 000; a 20% increase; increased cash
flows to owners by $1 000; a 25% increase.
The greater the use of financing sources that require fixed obligations, such as interest, the
greater the sensitivity of cash flows to changes in operating cash flows.
Combining a firm's degree of operating leverage with its degree of financial leverage
results in the degree of total leverage (DTL), a measure of the sensitivity of the cash flows
to owners to changes in unit sales:
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���������� ������ �� ���� ����� �� ������
DTL= ���������� ������ �� ����� ����
Suppose:
Then,
which we could also have gotten from multiplying the DOL, 2, by the DFL, 1.25. This
means that a 1% increase in units sold will result in a 2.5% increase in cash flows to
owners; a 50% increase in units sold results in a 125% increase in cash flows to owners; a
5% decline in units sold results in a 12.5% decline in cash flows to owners; and so on.
In the case of operating leverage, the fixed operating costs act as a fulcrum: the greater the
proportion of operating costs that are fixed, the more sensitive are operating cash flows to
changes in sales. In the case of financial leverage, the fixed financial costs, such as
interest, act as a fulcrum: the greater the proportion of financing with fixed cost sources,
such as debt, the more sensitive cash flows available to owners are to changes in operating
cash flows. Combining the effects of both types of leverage, we see that fixed operating
and financial costs together act as a fulcrum that increases the sensitivity of cash flows
available to owners.
Technically, default risk on a debt security depends on the specific obligations comprising
the debt. Default may result from:
• failure to make an interest payment when promised (or within a specified period);
• failure to make the principal payment as promised;
• failure to make sinking fund payments (that is, amounts set aside to pay off the
obligation), if these payments are required;
• failure to meet any other condition of the loan; or
127
• bankruptcy.
Financial managers need to worry about default risk because they invest their firm's funds
in the debt securities of other firms and they want to know what default risk lurks in those
investments. And, because they are concerned about how investors perceive the risk of the
debt securities their own firm issues. The greater the risk of a firm's securities, the greater
the firm's cost of financing.
Default risk is affected by both business risk and financial risk. We need to consider the
effects operating and financing decisions have on the default risk of the securities a firm
issues, since the risk accepted through the financing decisions affects the firm's cost of
financing.
Two types of risk closely related to reinvestment risk of debt securities are prepayment
risk and call risk. In the case of mortgage-backed securities, that is, securities that
represent a collection of home mortgages --a homeowner may pay off her or his mortgage
early. If paid off early, investors in mortgages get paid off early -- so they will have to
scramble to reinvest earlier than expected. Therefore, investors in securities that can be
paid off earlier than maturity face prepayment risk -- the risk that the borrower may
choose to prepay the loan-- which causes the investor to have to reinvest the funds. Call
risk is the risk that a callable security will be called by the issuer. If you invest in a
callable security, there is a possibility that the issuer may call it in (buy it back). While you
may receive a call premium (a specified amount above the par value), you have to reinvest
the funds you receive.
There is reinvestment risk for assets other than stocks and bonds, as well. If you are
investing in a new product -- investing in assets to manufacture and distribute it -- you
expect to generate cash flows in future periods. You face a reinvestment problem with
these cash flows: What can you earn by investing these cash flows? What are your future
investment opportunities?
If we assume that investors do not like risk then they will want to be compensated if they
take on more reinvestment rate risk. The greater the reinvestment rate risk, the greater the
expected return demanded by investors.
Reinvestment rate risk is relevant in our investment decisions no matter the asset and we
must consider this risk in assessing the attractiveness of investments. The greater the cash
flows during the life of an investment, the greater the reinvestment rate risk of the
investment. And if an investment has a greater reinvestment rate risk, this must be factored
into our decision.
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7.5.7 Interest rate risk
Interest rate risk is the sensitivity of the change in an asset's value to changes in market
interest rates. Market interest rates determine the rate we must use to discount a future
value to a present value. The value of any investment depends on the rate used to discount
its cash flows to the present. If the discount rate changes, the investment's value changes.
Interest rate risk is present in debt securities. If you buy a bond and intend to hold it until
its maturity, you don't need to worry about its value as interest rates change: your return is
the bond's yield-to-maturity. But if you do not intend to hold the bond to maturity, you
need to worry about how changes in interest rates affect the value of your investment. As
interest rates go up, the value of your bond goes down. As interest rates go down, the
value of your bond goes up.
Let us compare the change in the value of the Company Y bond to the change in the value
of the Company Z bond as the market interest rate changes. Suppose that it is now
January 1, 2001. If yields remain at 10%, the values of the bonds are:
= $90.91+82.64+75.13+751.31
= $1 000.00
and
$1,610.51
Value of Company Z bond = = $1,100.00
(1 + 0.10) 4
If market interest rates change causing the bonds to yield 12%, the value of the Company
Y and Company Z bonds are less:
And
$1,610.51
Value of Company Z bond = = $1,023.50
(1 + 0.12) 4
Looking at changes in the value of the bonds for different yield changes, we see that the
Company Z bond's value is more sensitive to changes in yields than is Company Y's:
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We can make some generalisations about the sensitivity of a bond's value to changes in
yields.
For a given coupon rate, the longer the maturity of the bond, the more sensitive the bond's
value to changes in market interest rates. Why? Because more or the bond's value is
farther out into the future (the principal payments) and the more the present value is
affected by a change in the discount rate.
For a given maturity, the greater the coupon rate, the less sensitive the bond's value to a
change in the yield. Why? The greater the coupon rate, the more of the bond's present
value is derived from cash flows that are affected less by discounting.
Activity 7.2
Compare the change in the value of two bonds that have the same coupon rate,
10% and the same face value, $1 000, with interest paid annually. If Bond SM has
five years remaining to maturity and Bond LM has ten years remaining to
maturity, what is the effect of a change in the yield on the bonds from 10% to
12%?
The extent to which inflation is anticipated is reflected in interest rates. Let us refer to the
return after considering inflation as the real return and refer to the return before removing
inflation as the nominal return. Therefore,
This relation between the nominal return, the inflation rate, and the real return is referred
to as the Fisher effect.
As you can see, the nominal return is comprised of three parts: the inflation rate, the real
return, and the cross product of the inflation rate and the real return. Since this cross-
product term is usually quite small we often leave it out and consider the nominal return
to be the sum of the inflation rate and the real return.
130
The difference between the nominal return and the real return is often referred to as the
inflation premium, since it is the additional return necessary to compensate for inflation.
Purchasing power risk is the risk that future cash flows may be worth less or more in the
future because of inflation or deflation, respectively, and that the return on the investment
will not compensate for the unanticipated inflation. If there is risk that the purchasing
power of a currency will change, investors -who do not like risk -- will demand a higher
return.
Financial managers need to assess purchasing power risk in terms of both their investment
decisions -- making sure to figure in the risk from a change in purchasing power of cash
flows -- and their financing decisions -- understanding how purchasing power risk affects
the costs of financing.
Consider a Zimbabwean firm making an investment that produces cash flows in British
pounds, £. Suppose we invest £10 000 today and expect to get £12 000 one year from
today. Further suppose that £l = $1.48 today, so you are investing $1.48 times 10 000 =
$14 800. If the British pound does not change in value, relative to the Zimbabwean dollar,
you would have a return of 20%:
But what if one year from now the British pound is worth $1.30 instead? Your return
would be less than 20% because the value of the pound has dropped against the
Zimbabwean dollar. You are making an investment of £10 000, or $14 800, and getting
not $17 760, but rather $1.30 times 12 000 = $15 600 in one year. If the pound loses value
from $1.48 to $1.30, your return on your investment is
Currency risk is the risk that the relative values of the domestic and foreign currencies
will change in the future, changing the value of the future cash flows. As financial
managers, we need to consider currency risk in our investment decisions that involve
other currencies and make sure that the returns on these investments are sufficient
compensation for the risk of changing values of currencies.
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7.6.1 Expected return of an asset
We refer to both future benefits and future costs as expected returns. The expected return
is a measure of the tendency of returns on an investment. This does not mean that these
are the only returns possible but just our best measure of what we expect.
Suppose we are evaluating investment in a new product. We do not know and cannot
know precisely what the future cash flows will be. But, from past experience, we can at
least get an idea of possible flows and the likelihood (the probability) that they will occur.
After consulting with colleagues in marketing and production management, we figure out
that there are two possible cash flow outcomes, success or failure, and the probability of
each outcome. Next, consulting with colleagues in production and marketing for sales
prices, sales volume, and production costs, we develop the following possible cash flows
in the first year:
But what is the expected cash flow in the first year? The expected cash flow is the average
of the possible cash flows, weighted by their probabilities of occurring:
Expected cash flow = 0.40 ($4 000 000) + 0.60(-$2 000 000) = $400 000
The expected value is a guess about the future outcome. It is not necessarily the most
likely outcome. The most likely outcome is the one with the highest probability. In the
case of our example, the most likely outcome is $2 000 000.
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Applying the general formula to our example,
n = 2 (there are two possible outcomes)
p1 = 0.40
p 2 = 0.60
x1 = $4,000,000
x 2 = −$2,000,000
E(cash flow) = 0.40 ($4 000 000) + 0.60 (-$2 000 000) = $400 000.
Considering the possible outcomes and their likelihood, we expect a $400 000 cash flow.
Since we are concerned about the degree of uncertainty (risk), as well as the expected
return, we need some way of quantifying the risk associated with decisions. Suppose we
are considering two products, Product A and Product B, with estimated returns under
different scenarios and their associated probabilities:
Product A
Scenario Probability of Outcome Outcome
Failure 25 -4
Product B
Scenario Probability of Outcome Outcome
Moderate success 30 30
Failure -5
60
We refer to a product's set of the possible outcomes and their respective probabilities as the
probability distribution for those outcomes.
We can calculate the expected cash flow for each product as follows:
133
Product A
Pi Xi Pi xi
Success 0.25 0.24 0.0600
Moderate success 0.50 0.10 0.0500
Failure 0.25 -0.04 -0.0100
Expected return 0.1000 or 10%
Product B
Pi Xi Pi xi
Success 0.10 0.40 0.0400
Moderate success 0.30 0.30 0.0900
Failure 0.60 -0.05 -0.0300
Expected return 0.1000 or 10%
Both Product A and Product B have the same expected return. However, the possible
returns for Product A range from -4% to 24%, where the possible returns for Product B
range from -5% to 40%. The range is the span of possible outcomes. For Product A the
span is 28%; for Product B it is 45%. A wider span indicates more risk, so Product B has
more risk than Product A.
But the range by itself doesn't tell us much about the possible cash flows at these extremes
or within the range. A measure of risk that tells us something about how much to expect
and the probability that it will happen is the standard deviation. The standard deviation is a
measure of dispersion that considers the values and probabilities for each possible outcome.
The higher the standard deviation, the greater would be the dispersion of possible
outcomes from the expected value. The standard deviation considers the deviation, or
distance, of each possible outcome from the expected value and the probability associated
with it.
σ ( x) = ∑ p (x
i i − E ( x )) 2
The deviation tells us how far each possible outcome is from the expected value. The
standard deviation is calculated in six steps as follows:
Step 2: Calculate the deviation of each possible outcome from the expected value.
Step 4: Weight each squared deviation by multiplying it with the probability of the outcome.
134
This is the variance of the possible outcomes, σ 2 ( x )
Step 6: Take the square root of the sum of the squared deviations.
Example 7.2
Calculate the standard deviation of the expected cash flows for Product A:
Step1: Calculate the expected value
Step 2: Calculate the deviation of each possible outcome from the expected outcome.
Step 4: Weight each of the squared deviations by multiplying the probability of the outcome by the
squared deviations.
Step 6: Take the square root of the sum of the squared deviations
135
The standard deviation for Products A and B are:
10% 18.57%
Product B
While the expected value of both products is the same, there is a different distribution of
possible outcomes for the two products. When we calculate the standard deviation around
the expected value, we see that Product B has a larger standard deviation. The larger
standard deviation for Product B tells us that Product B has more risk than Product A since
its possible outcomes are more distant from its expected value.
When we consider financing and investment decisions, we assume that most people are
risk averse. Managers, as agents for the owners, make decisions that consider risk "bad"
and that if risk must be borne, they make sure there is sufficient compensation for bearing
it. As agents for the owners, managers cannot have the "fun" of taking on risk for the
pleasure of doing so.
Risk aversion is the link between return and risk. To evaluate a return you must consider
its risk: Is there sufficient compensation (in the form of an expected return) for the
investment's risk?
Activity 7.3
There is a 50% probability that the Plum Company's sales will be $10 million next year, a
20% probability that they will be $5 million, and a 30% probability that they will be $3
million.
(a) What are the expected sales of Plum Company next Year?
(b) What is the standard deviation of Plum's next year's sales?
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7.7 Selecting Among Different Investments
The two basic approaches that can be used to choose between investments in
individual assets are the mean variance criterion and the coefficient of variation (C.V)
approach.
To us these assumptions are necessary but insufficient to come up with mean variance
efficient securities. To make the rule sufficient and for the purpose of this study, we will
add other assumptions as follows:
• Investment sets are complete, that is, investors are able to compare between or
among assets.
• Investors' choices are transitive, that is, if asset X is preferred to asset Y and asset
Y is preferred to asset Z, it, therefore, implies that asset X is better than asset Z.
If there are two assets X and Y then asset X is preferred to asset Y if:
(a) X has expected returns higher than or equal to that of Y and the risk of X is
lower than that of Y, that is,
E ( x ) ≥ E ( y ) and σ 2 ( x ) < σ 2 ( y )
(b) X has expected returns higher than that of Y and the risk of X is lower than or
equal to that of Y, that is,
E ( x ) > E ( y ) and σ 2 ( x ) ≤ σ 2 ( y )
Example 7.3
A firm is faced with the following investment alternatives that are mutually exclusive.
Which product is mean- variance efficient?
Product A 10% 9%
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Solution
Of the two investments, Product A is mean-variance efficient because it has the lower risk
though yield the same expected return as that of B.
Based on C.V we prefer an investment with lower C.V, that is, lower risk per each unit of
expected value.
Example 7.4
Of the two products in example 7.2, which one is preferred in terms of C.V?
Activity 7.4
Consider the following investments:
Investment Expected Return Standard Deviation
A 5% 10%
B 7% 11%
C 6% 12%
D 6% 10%
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7.8 Summary
In this unit, we demonstrated how risk and expected returns can be measured for assets
held in isolation. The expected return on an investment is the expected value of the
possible returns. Risk has been defined as the probability that some unfavourable event
might occur. Investment risk is related to the probability of earning a return, which is less
than what is being expected. The greater the probability of this happening, the riskier is
the investment. Rational investors will always hold a portfolio of assets and as a result
will not be too concerned about the risk of individual stocks, but about the risk of the
entire portfolio. Total risk, which is measured by the dispersion of returns about the mean,
is relevant only for assets held in isolation. Variance of returns, standard deviation of
returns or coefficient of variation of returns can measure an asset's total risk. The
coefficient of variation is the parameter normally preferred.
139
References
Block, S.B. and Hirt, G.A. (1994). Foundations of Financial Management. (Sixth Edition). Boston:
Irwin.
Ehrhardt, M.C. and Brigham, E.F. (2011). Financial Management, Theory and Practice.
(Thirteenth Edition). Mason: South-Western Cengage Learning.
Gitman, L. and Zutter, C.J. (2012). Principles of Managerial Finance. (Thirteenth Edition).
Boston: Pearson Education, Inc.
Kwesu, I., Nyatanga, E. and Zhanje, S. (2002). Business Statistics. Harare: ZOU.
Levy, H. and Sarnat, M. (1991). Capital Investments and Financial Decisions. (Fourth Edition).
New York: Prentice Hall.
McLaney, E..J. (2000). Business Finance: Theory and Practice. New York: Prentice Hall.
140
BLANK PAGE
Unit 8
Portfolio Theory and Capital Asset Pricing Model
"My ventures are not in one bottom trusted
Nor to one place; nor is my whole estate
Upon the fortune of this present year.
Therefore my merchandise makes me not sad"
-Merchant of Venice.
8.0 Introduction
In this unit we provide an overview of individual asset allocation. It is shown that an individual
can reduce the risk of his portfolio without sacrificing any expected return simply by spreading
his wealth over a number of assets in an appropriate way. This technique of diversification is
explained in some detail in terms of a simple two-asset example in order to build intuition.
8.1 Objectives
By the end of this unit you should be able to:
• compute the expected return of a portfolio
• compute the variance and standard deviation of the return of a portfolio
• find the composition of the minimum-variance two-asset portfolio
• explain the concept of diversification
• explain how to construct a diversified portfolio in practice
• discuss the concept of an efficient frontier
141
Consider Investment C and Investment D and their probability distributions:
We see that when Investment C does well, in the boom scenario, Investment D does poorly.
Also, when Investment C does poorly, as in the recession scenario, Investment D does well. In
other words, these investments are out of synch with one another.
Now let us look at how their "out-of-synchness" affects the risk of the portfolio of C and D. If
we invest an equal amount in C and D, the portfolio's return under each scenario is the
weighted average of C & D's returns, where the weights are 50%:
The calculation of the expected return and standard deviation for Investment C, Investment D,
and the portfolio consisting of C and D results in the following the statistics:
Normal 50% 0 0 0
Expected 2% 6% 4%
return
Standard 14.00% 19.97% 4.77%
deviation
___
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The expected return on Investment C is 2% and the expected return on Investment D is 6%.
The return on a portfolio comprised of equal investments of C and D is expected to be 4%. The
standard deviation of Investment C’s return is 14% and of Investment D's return is 19.97%,
but the portfolio's standard deviation, calculated using the weighted average of the returns on
Investment C and D in each scenario, is 4.77%. This is less than the standard deviations of each
of the individual investments because the returns of the two investments do not move in the
same direction at the same time, but rather tend to move in opposite directions.
The portfolio comprised of Investments C and D has less risk than the individual investments
because each moves in opposite directions with respect to the other. A statistical measure of
how two variables, in this case, the returns on two different investments, move together is the
covariance. Covariance is a statistical measure of how one variable changes in relation to
changes in another variable.
Cov XV = ∑ Pi ( xi − E ( x ))( yi − E ( y ))
Step 1: For each scenario and investment, subtract the investment's expected value from its
possible outcome;
Step 2: For each scenario, multiply the deviations for the two investments;
As you can see in these calculations, in a boom economic environment, when Investment C is
above its expected return (deviation is positive), Investment D is below its expected return
(deviation is negative). In a recession, Investment C’s return is below its expected value and
Investment D's return is above its expected value. The tendency is for the returns on these
portfolios to co-vary in opposite directions, producing a negative covariance of -0.0252.
Let us see the effect of this negative covariance on the risk of the portfolio. The portfolio's
variance depends on:
143
• the weight of each asset in the portfolio;
• the standard deviation of each asset in the portfolio; and
• the covariance of the assets' returns.
Let Cov 1, 2 represent the covariance of two assets’ returns. We can write the portfolio variance
as:
2 2 2 2
Portfolio variance = w1 σ 1 + w2 σ 2 + 2 cov 1, 2 w1 w2
Remember
Var(aX + bY)= a2Var(X) + b2Var(Y) + 2abCov(X,Y)
portfolio
Portfolio standard deviation =
var iance
We can apply this general formula to our example, with Investments C and D denoted by
subscripts 1 and 2 in the formulae respectively.
w2 = 0.50 or 50%
σ 1 = 0.1400 or14%
σ 2 = 0.1997 or 19.97%
Cov1, 2 = −0.0252
= 0.002275
and the portfolio standard deviation is 0.0477 or 4.77%, which is what we got when we
calculated the standard deviation directly from the portfolio returns under the three scenarios.
The standard deviation of the portfolio is lower than the standard deviations of each of the
investments because the returns on Investments C and D are negatively related: when one is
doing well the other may be doing poorly, and vice-versa. That is, the covariance is negative.
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Example 8.1: The Portfolio Variance and Standard Deviation
Consider a portfolio comprised of two securities, F and G:
invested
The covariance between the two securities' returns is 0.002. What is the portfolio's standard deviation?
Solution
Variance = 0.16(0.0025) + 0.36(0.0064) + [(2)(0.002)(0.40)(0.60)]
Variance = 0.0004 + 0.0023 + 0.00096 = 0.00366
Portfolio standard deviation = 0.06050
The investment in assets whose returns are out of step with one another is the whole idea
behind diversification. Diversification is the combination of assets whose returns do not vary
with one another in the same direction at the same time.
Diversify across industries: Investing in a number of different stocks within the same
industry does not generate a diversified portfolio since the returns of firms within an industry
tend to be highly correlated. Diversification benefits can be increased by selecting stocks from
different industries.
Diversify across industry groups: Some industries themselves can be highly correlated with
other industries and hence diversification benefits can be maximised by selecting stocks from
those industries that tend to move in opposite directions or have very little correlation with
each other.
Diversify across geographical regions: Companies whose operations are in the same
geographical region are subject to the same risks in terms of natural disasters and state or local-
tax changes. Investing in companies can diversify these risks whose operations are not in the
same geographical region.
Diversify across economies: Stocks in the same country tend to be more highly correlated
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than stocks across different countries. This is because many taxation and regulatory issues
apply to all stocks in a particular country. International diversification provides a means for
diversifying these risks.
Diversify across asset classes: Investing across asset classes such as stocks, bonds, and real
property also produces diversification benefits. The returns of two stocks tend to be more
highly correlated, on average, than the returns of a stock and a bond or a stock and an
investment in real estate.
• positively correlated if one tends to vary in the same direction at the same time as the
other
• negatively correlated if one tends to vary in the opposite direction with respect to the
other
• uncorrelated if there is no relation between the changes in one with changes in the other
By construction, the correlation coefficient is bounded between -1 and +1. We can interpret
the correlation coefficient as follows:
A correlation coefficient of +1 indicates a perfect, positive correlation between the two assets'
returns. A correlation coefficient of -1 indicates a perfect, negative correlation between the two
assets returns. A correlation coefficient of 0 indicates no correlation between the two assets
returns. A correlation coefficient falling between 0 and +1 indicates positive, but not perfect
positive correlation between the two assets returns, illustrated in Exhibit 8.1. A correlation
coefficient falling between -1 and 0 indicates negative, but not perfect negative correlation
between the two assets returns, illustrated in Exhibit 8.2.
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Exhibit 8.1: Positive Correlation between the Returns of Security Y and Security X
Y
Exhibit 8.2: Negative Correlation between the Returns of Security Y and Security X
Y
− 0.0252
Correlation of returns on Investments C & D = = −0.9014 .
(0.1400)(0.1997)
Therefore, the returns on Investment C and Investment D are negatively correlated with one
another. By investing in assets with less than perfectly correlated cash flows, you are getting
rid of some risk. The less correlated the cash flows, the more risk you can diversify away to
a point. Let us see how the correlation and portfolio standard deviation interact. Consider
two investments, E and F, whose standard deviations are 5% and 3%, respectively. Suppose
our portfolio consists of an equal investment in each; that is, w1 = w2= 50%.
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If the correlation between ... this means that the and the portfolio's
the assets' returns is ... covariance is ... standard deviation is ...
The less perfectly positively correlated are two assets' returns, the lower the risk of the
portfolio comprised of these assets.
Activity 8.1
The idea of diversification is based on beliefs about what will happen in the future: expected
returns, standard deviation of all possible returns, and expected covariance between returns.
Let us look at the effects of diversification with common stocks. As we add common stocks
to a portfolio, the standard deviation of returns on the portfolio declines to a point. After
around twenty different stocks, the portfolio's standard deviation is about as low as it is going
to get. Why does the risk seem to reach some point and not decline any farther? Because
common stocks' returns, in general, are positively correlated with one another. There just are
not enough negatively correlated stocks' returns to reduce portfolio risk beyond a certain
point.
We refer the risk that goes away as we add assets as diversifiable risk. The risk that cannot be
148
reduced by adding more assets is called non-diversifiable risk. The idea that we can reduce
the risk of a portfolio by introducing assets whose returns are not highly correlated with one
another is the basis of Modern Portfolio Theory (MPT). MPT tells us that, by combining
assets whose returns are not correlated with one another, we can determine combinations of
assets that provide the least risk for each possible expected portfolio return.
Though the mathematics involved in determining the optimal combinations of assets are
beyond this module, the basic idea is provided in Exhibit 8.3. Each squared point in the graph
represents a possible portfolio that can be put together comprising different assets and
different weights. The points in this graph represent every possible portfolio. As you can see
in this diagram:
• some portfolios have a higher expected return than other portfolios with the same level
of risk
• some portfolios have a lower standard deviation than other portfolios with the same
expected return
Since investors like more return to less and prefer less risk to more, some portfolios are better
than others. The best portfolios are those that are mean-variance efficient -- those that can't be
beaten in terms of either the level of return for the amount of risk or the amount of risk for the
level of return. The locus of mean-variance efficient investment will make up what is called
the efficient frontier. If investors are rational, they will go for the portfolios that fall on this
efficient frontier. All the possible portfolios and the efficient frontier are diagrammed in
Exhibit 8.3.
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Expected return
•
Key:
• optimal portfolio
portfolio
indifference curve
efficient frontier
150
So what is the relevance of MPT to financial managers? MPT tells us that we can manage
risk by judicious combinations of assets in our portfolios; and there are some combinations
of assets that are preferred over others.
The relevant risk to an investor is the portfolio's risk, not the risk of an individual asset. If an
investor holds assets in a portfolio and is considering buying an additional asset or selling an
asset from the portfolio, what must be considered is how this change will affect the risk of the
portfolio. This concept applies whether we are talking about an investor holding 30 different
stocks or a business that has invested in 30 different projects. The important thing in valuing
an asset is its contribution to the portfolio's return and risk.
The CAPM is a ceteris paribus model. It is only valid within a special set of assumptions.
These are:
• Investors are risk averse individuals who maximise the expected utility of their end-
of-period wealth. The implication is that the model is a one period model.
• Investors have homogenous expectations (beliefs) about asset returns. Implication is
that all investors perceive identical opportunity sets. That is, everyone has the same
information at the same time.
• Asset returns are distributed by the normal distribution.
• There exists a risk-free asset and investors may borrow or lend unlimited amounts of
this asset at a constant rate: the risk free rate ( rf ).
• There is a definite number of assets and their quantities are fixed within the one
period world.
• All assets are perfectly divisible and priced in a perfectly competitive market.
Implication: for example, human capital is non-existing (it is not divisible and it
cannot be owned as an asset).
151
• Asset markets are frictionless and information is available free and simultaneously to
all investors. The implication is that the borrowing rate equals the lending rate.
• There are no market imperfections such as taxes, regulations, or restrictions on short
selling.
We just saw that there is a set of portfolios that make up the efficient frontier, that is, the best
combinations of expected return and standard deviation. All the assets in each portfolio, even
on the frontier, have some risk. Now let us see what happens when we add an asset with no
risk, referred to as the risk-free asset. Suppose we have a portfolio along the efficient frontier
that has a return of 4% and a standard deviation of 3%. Suppose we introduce into this
portfolio the risk-free asset, which has an expected return of 2% and, by definition, a standard
deviation of zero. If the risk-free asset's expected return is certain, there is no covariance
between the risky portfolio's returns and the returns of the risk-free asset.
A portfolio comprised of 50% of the risky portfolio and 50% of the risk-free asset has an
expected return of (0.50) 4% + (0.50) 2% = 3% and a portfolio standard deviation calculated
as follows:
If we look at all possible combinations of portfolios along the efficient frontier and the risk-
free asset, we see that the best portfolios are no longer along the entire length of the efficient
frontier, but rather are the combinations of the risk free assets and one -- and only one--
portfolio of risky assets on the frontier. The combinations of the risk-free asset and this one
portfolio are shown in Exhibit 8.4. These combinations differ from one another by the
proportion invested in the risk free asset; as less is invested in the risk-free asset, both the
portfolio's expected return and standard deviation increase.
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Exhibit 8.4
Expected return
borrowing
lending
•
Key:
• Optimal portfolio
Efficient frontier
Capital market line
portfolio
William Sharpe demonstrates that this one and only one portfolio of risky assets is the market
portfolio; a portfolio that consists of all assets, with the weights of these assets being the ratio
of their market value to the total market value of all assets, if investors are all risk averse they
only take on risk if there is adequate compensation. And if they are free to invest in risky
assets as well as risk-free asset, the best deal lie along the line that is tangent to the efficient
frontier. This line is referred to as the capital market line (CML), shown in Exhibit 8.4.
If the portfolios along the capital market line are the best deals and are available to all
investors, it follows that the returns of these risky assets will be priced to compensate
investors for the risk they bear relative to that of the market portfolio. Since the portfolios
along the capital market line are the best deals, they are as diversified as they can get-- no
other combination of risky assets or risk-free asset provides a better expected return for the
level of risk or provides a lower risk for the level of expected return.
153
The equation to this line which represents the possible sets of portfolios of the riskless asset
and portfolio is
E [rm ] − rf
E [rf ] = rf + σ p
σm
E [rm ] − rf
where rf is the intercept and is the slope of the line and represent price of risk.
σm
We could also write
σp
E [rp ] = r f + [E [rm ] − rf ]
σm
The capital market line tells us about the returns an investor can expect for a given level of
risk. The CAPM uses this relationship between expected return and risk to describe how
assets are priced. The CAPM specifies that the return on any asset is a function of the return
on a risk-free asset plus a risk premium. The return on the risk-free asset is compensation for
the time value of money. The risk premium is the compensation for bearing risk. Putting
these components of return together, the CAPM says:
Expected return on an asset = expected return on a risk free asset + risk premium or, return
of any asset is given by a formula of the form:
We have established above that the appropriate measure of risk is cov( Ri , Rm ) and hence, the
equation can be rewritten as:
We already know the details surrounding two points on this line. The riskless asset has
expected return of rf and covariance with the market portfolio of zero (since rf is constant).
2
The market portfolio has expected return E[rm] and covariance with the market of σ p (the
covariance of a variable with itself is its variance). Since these two points must lie on the
line, the equation to the line must be:
154
E [rm − r f ]
E [ri ] = r f + σ i ,m
σ m2
E [rm − r f ]
E [ri ] = r f + 0
σ m2
E [rm − r f ]
E [ri ] = r f + σ m
σ m2
σ i ,m
It is common to standardise the units of this equation by defining β i = and
σ 2m
rewriting the equation as
E [ri ] = r f + β i . E [rm − rf ]
It is this equation that is known as the Sharpe-Lintner CAPM. The beta of the market
portfolio is one:
σ i ,m σ 2 m
βi = = =1
σ 2m σ 2m
This provides a reference point against which the risk of other assets can be measured. The
average risk (or beta) of all assets is the beta of the market, and its value is one. Assets or
portfolios that have a beta greater than one have above average risk. They tend to move
more rapidly than the market. For example, if the riskless rate of interest (T-bill rate) is 5%
p.a. and the market rises by 10%, assets with a beta of 2 will tend to increase by 15%. If,
however, the market falls by 10%, assets with a beta of 2 will tend to fall by 25% on
average. Conversely, assets with betas less than one are of below average risk and tend to
move less than the market portfolio. Assets that have betas less than zero tend to move in
the opposite direction to the market. These assets are known as hedge assets.
155
Since the market portfolio is made up of all assets, each asset possesses some degree of
market risk. Since market risk is systematic across assets, it is often referred to as systematic
risk and diversifiable risk is referred to as unsystematic risk. Further, the risk that is not
associated with the market as a whole is often referred to as company-specific risk when
referring to stocks, since it is risk that is specific to the company's own situation such as the
risk of lawsuits and labour strikes-- and is not part of the risk that pervades all securities.
The risk that reflects an asset’s The risk that reflects an asset’s
return’s movement with asset returns not moving along with asset
returns in general is referred to as returns in general is referred to as
… …
non-diversifiable diversifiable risk
market risk company-specific risk
systematic risk unsystematic risk
The measure of an asset's return sensitivity to the market's return, its market risk, is the
asset's beta. The expected return on an individual asset is the sum of the expected return on
the risk-free asset and the premium for bearing market risk. Let ri represent the expected
return on asset i, rf represented the expected return on the risk-free asset, and β i represent
the degree of market risk for asset i. Then:
E [ri ] = r f + β i E [rm − r f ]
The term, (rm − r f ), is the market risk premium, if you owned all the assets in the market
portfolio, you expect to be compensated (rm − r f ) for bearing the risk of these assets. β is
measure of market risk, which serves to fine tune the risk premium for the individual asset.
For example, if the market risk premium were 2% and the β for an individual asset were
1.5, you would expect to receive a risk premium of 3% since you are taking on 50% more
risk than the market.
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Exhibit 8.5
E(Ri)
Rm
Rf
1.0 β
• Assets that plot on the security market line are correctly priced (equilibrium). Along
the SML expected returns will be equal to required returns. In this case, it pays the
individual to hold on the security.
• Assets that plot off the SML are mispriced (disequilibria). Above the SML securities
will be under-priced because expected returns are greater than required returns. In this
case, it pays the investor to buy the security. On the other hand, assets that plot below
the SML are overpriced because expected returns will be lower than the required
return and hence the investor will prefer to sell the security.
As you can see in this graph the greater the β , the greater the expected return. If there were
no market risk (beta = 0.0) on an asset, its expected return would be the expected return on
the risk-free asset.
If the asset's risk is similar to the risk of the market as a whole (beta = 1.0), that asset's
expected return is the return on the market portfolio. For an individual asset, beta is a
measure of sensitivity of its returns to changes in return on the market portfolio. If beta is
one, we expect that for a given change of 1% in the market portfolio return, the asset's return
is expected to change by 1%. If beta is less than one, then for a 1% change in the expected
market return, the asset's return is expected to change by less than 1%. If the beta is greater
than one, then for a 1% change in the expected market return, the asset’s return is expected
to change by more than 1%.
157
Rit − R ft = α i + β i ( Rmt − R ft ) + ε it
In this regression, the beta is the ratio of the covariance to the variance of the market return.
The alpha is the intercept in the regression. This is not the CAPM equation. This is a
regression that allows us to estimate the stock's beta coefficient. The CAPM equation
suggests that the higher the beta, the higher the expected return. Note that this is the only type
of risk that is rewarded in the CAPM. The beta risk is referred to in some text books as
systematic or non-diversifiable or market risk. This risk is rewarded with expected return.
There is another type of risk which is called non-systematic or diversifiable, non-market or
idiosyncratic risk. This type of risk is the residual term in the above time-series regression.
R it − R ft = α i + β i ( R mt − R ft ) + ε it
158
The asset's characteristic line is the line of the best fit for the scatter plot that represents
simultaneous excess returns on the asset and on the market.
Exhibit 8.6
Realised
Excess
Return on Slope =
Asset i • • Beta
• • • •
•
• • •
• • •
• • • • •
• • •
• • •
• •
Intercept = Alpha
This is just the fitted value from a regression line. As mentioned above, the beta will be the
regression slope and the alpha will be the intercept. The error in the regression, epsilon, is
the distance from the line (predicted) to each point on the graph (actual). The CAPM
implies that the alpha is zero. So we can interpret, in the context of the CAPM, the alpha as
the difference between the expected excess return on the security and the actual return.
We typically estimate the beta for a common stock by looking at the historical relation
between its return and the return on the market as a whole. The betas of some firms' stocks
are close to one, indicating that the returns on these stocks tend to move along with the
market.
Cov ( Ri , Rm )
βi =
Var ( Rm )
159
n
Cov ∑ wi R p , Rm
Cov( R p , Rm ) i =1
βp = =
Var ( Rm ) Var ( Rm )
n
∑ w Cov( R , R
i i m ) n
= i =1
= ∑ wi β i
Var ( Rm ) i =1
β p = w1 β 1 + w2 β 2 + w3 β 3 + ... + wS β S
The beta of the portfolio is the weighted average of the individual asset betas where the
weights are the portfolio weights. So we can think of constructing a portfolio with
whatever beta we want. All the information that we need is the betas of the underlying
asset. For example, if I wanted to construct a portfolio with zero market (or systematic)
risk, then I should choose an appropriate combination of securities and weights that
delivers a portfolio beta of zero. Suppose we have three securities in our portfolio, with
the amount invested in each and their security beta as follows:
1 1.00 50%
2 2.00 25%
3 0.50 25%
The CAPM, with its description of the relation between expected return and risk and the
importance of market risk in asset pricing, has some drawbacks.
Though it lacks realism and is difficult to apply, the CAPM makes some sense regarding the
role of diversification and the type of risk we need to consider in investment decisions.
Anticipated factors are already reflected in an asset's price. It is the unanticipated factors that
cause an asset’s price to change. For example, consider a bond with a fixed coupon interest.
The bond's current price is the present value of expected interest and principal payments,
discounted at some rate that reflects the time value of money, the uncertainty of these future
cash flows, and the expected rate of inflation. If there is an unanticipated increase in inflation,
what will happen to the price of the bond? It will go down since the discount rate increases as
inflation increases. If the price of the bond goes down, the return on the bond will decrease.
Therefore, the sensitivity of a bond's price to changes in unanticipated inflation is negative.
Activity 8.2
1. The covariance of the returns on the two securities, A and B, is -0.0005. The
standard deviation of A's returns is 4% and the standard deviation of B's returns
is 6%. What is the correlation between the returns of A and B?
2. Consider a portfolio comprised of four securities in the following proportions
and with the indicated security beta.
8.14 Summary
In this unit we highlighted that when assets are combined into portfolios, the relevant risk is
an asset's market risk, which is the contribution of the asset to the risk of the portfolio. The
expected rate of return on a portfolio is the weighted average return on the component
assets, but the standard deviation of a portfolio is not the weighted average of the
component assets' standard deviation. Since most assets are not perfectly positively
correlated, combining assets into portfolios generally reduces risk. An asset's total risk
consists of the company specific risk which, however, can be eliminated by diversification,
and the market risk which cannot be eliminated by diversification. The feasible set of
portfolios represents all portfolios that can be constructed from a given set of assets. An
efficient portfolio is one that offers the most return for a given amount of risk or the least
risk for a given amount of return. The optimum portfolio for an investor is defined by the
tangency point between the efficient set of portfolios and the investor's highest indifference
curve.
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References
Block, S.B. and Hirt, G.A. (1994). Foundations of Financial Management. (Sixth Edition). Boston:
Irwin.
Ehrhardt, M.C. and Brigham, E.F. (2011). Financial Management, Theory and Practice.
(Thirteenth Edition). Mason: South-Western Cengage Learning.
Gitman, L. and Zutter, C.J. (2012). Principles of Managerial Finance. (Thirteenth Edition). Boston:
Pearson Education, Inc.
Kwesu, I., Nyatanga, E. and Zhanje, S. (2002). Business Statistics. Harare: ZOU.
Levy, H. and Sarnat, M. (1991). Capital Investments and Financial Decisions. (Fourth Edition). New
York: Prentice Hall.
McLaney, E..J. (2000). Business Finance: Theory and Practice. New York: Prentice Hall.
164
Unit 9
The Cost of Capital
9.0 Introduction
The concept in this unit, the cost of capital, is critical to the long-term success of the firm. In
order to maintain the market value for its stock, this is the rate of return a given project must
earn. Because a firm tries to maintain a target capital structure, which is the desired optimal mix
of debt and equity financing, it should use the weighted average cost to decide on investments.
In this unit, we specifically focus on the long-term fixed-asset investments, because they are
more permanent. There are four basic sources of long-term funds that businesses use. These are
long-term debt, preferred stock, common stock, and retained earnings. The specific cost of each
source is the after-tax cost of getting financing today. The cost of each source reflects the risk of
the assets the firm invests in. A firm that invests in assets having little risk in producing income
will be able to bear lower costs of capital than a firm that invests in assets having a higher risk
of producing income. Moreover, the cost of each source of funds reflects the hierarchy of the
risk associated with its seniority over the other sources. For a given firm, the cost of funds
raised through debt is less than the cost of funds from preferred stock which, in turn, is less than
the cost of funds from common stock. Why? This is because creditors have seniority over
preferred shareholders who have seniority over common shareholders. If there are difficulties in
meeting obligations, the creditors receive their promised interest and principal before the
preferred shareholders who, in turn, receive their promised dividends before the common
shareholders. If the firm is liquidated, the funds from the sales of its assets are distributed first to
debt holders, then to preferred shareholders, and then, to common shareholders (if anything is
left). For a given firm, debt is less risky than preferred stock, which is less risky than common
stock. Therefore, preferred shareholders require a greater return than the creditors and common
shareholders require a greater return than preferred shareholders.
9.1 Objectives
By the end of this unit, you should be able to:
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9.2 The Calculation of Weighted Average Cost of Capital (WACC)
The calculation of the cost of capital requires us to first determine the cost of each source of
capital that we expect the firm to use, along with the relative amounts of each source of capital
we expect the firm to raise. We can do this in three steps:
Step 1: Determine the cost of each source;
Step 2: Determine the proportions of each source to be raised as capital;
Step 3: Calculate the weighted average cost of capital given by the following formula:
WACC= ∑ wi ki
WACC = wd k d + w ps k ps + we k e
We will consider each step in this unit, but we first calculate the cost of each source, and then
determine the proportion of each source of capital to be used in our calculations. We then put
together the cost and proportions of each source to calculate the firm's cost of capital and we also
demonstrate the calculations of the marginal cost of capital.
kd = rd (1 - t)
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Example 9.1
A company has debentures with a par value of $100, coupon rate of 25%, with no maturity value.
Currently the debentures are trading at $75 and the corporate tax rate is 35%. What is the cost of
debt?
Solution:
I 25 1
rd = = = 33 %
PV 75 3
Where:
I = interest payment = face value × coupon rate
= $100 × 0.25 =$25
PV = Present value or the current market price of the debenture.
Therefore:
1 2
k d = 33 %(1 − 0.35) = 21 %
3 3
In most cases debentures are redeemable, that is, debentures have a maturity date. When the
yield to maturity is not given it is usually approximated using the following formula:
I+
(Fd − Vd )
Approximate yield to maturity = n
Fd + 2Vd
3
Where:
I= Interest payment
Fd = Face value
Vd = Market value
Example 9.2
A debenture has face value of $100 and is currently trading at $90. The debenture has 10 years to
maturity. What is the current yield of the debenture if the coupon rate is 20%? Hence calculate the
cost of debt given a tax rate of 40%.
Solution
20 +
(100 − 90)
Approximate YTM = 10
(100 + 2 × 90)
3
Therefore
The cost of debt ( k d ) = 0.225(1 - 0.40) = 0.135= 13.5%
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Activity 9.1
1. Suppose ABC Company ltd can issue debt with a yield of 12%. If the company's
marginal tax rate is 40%, what is its cost of debt?
2. A debenture has a face value of $100 and is currently trading at $90. The
debenture has 10 years to maturity. What is the current yield of the debenture if
the coupon rate is 20%? Hence calculate the cost of the debenture given a tax rate
of 40%.
Since dividends paid on preferred stock are not deductible as an expense for the issuer's tax
purposes, the cost of preferred stock is not adjusted for taxes-- dividends paid on this stock are
paid out of after-tax dollars.
Activity 9.2
Preference stocks are trading at 104 cents per share and are paying a yearly dividend of 24
cents. What is the yield on the preference stock?
Each method relies on different assumptions regarding the cost of equity; each produces
different estimates of the cost of common equity.
i. Zero-growth Model
where
Example 9.3
A company has ordinary shares that are trading at 80cents ex-dividend and expects to pay a
dividend of 20cents forever. What is the cost of equity ( k e ) ?
Solution
20c
Ke = = 0.25 = 25%
80c
PV = Div1 ( re − g )
where
Div1 is next period’s dividends,
g is the growth rate of dividends per year, and
PV is the current stock price per share.
Rearranging this equation to solve instead for k e
Div1
Ke = +g
P
Div1
We see that the cost of common stock is the sum of next period's dividend yield, , plus the
P
growth rate of dividends:
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Example 9.4
Consider a firm that has just paid a dividend of 20cents per share, with a cum dividend price of
200cents. The dividends are expected to grow at a rate of 10% annually forever. What is the
cost equity?
Solution:
The DVM makes some sense regarding the relation between the cost of equity and the dividend
payments. The greater the current dividend yield, the greater the cost of equity, and the greater
the growth in dividends, the greater the cost of equity. An important point to consider is that
when a firm raises new funds it incurs substantial costs in the form of floatation, which are fees
charged by investment banking firms for their services in assisting in selling the bonds in the
primary market, which tends to reduce the net amount that is received by the firm and, the
constant growth model becomes:
Div1
ke =
PV (1 − f )
where
Flotation costs are fees charged by investment banking firms for their services in assisting in
selling the bonds in the primary market. These costs reduce the total proceeds received by the
firm since the fees are paid from the bond funds.
Example 9.5
The firm is expected to pay a dividend of 30c, given that the current share price is 300c, the growth rate is
12% and the floatation costs are 6% of the share price. Calculate the effective cost ( k e ) of an ordinary
share.
Solution
30c
ke = + 0.12 = 0.2264 = 22.64%
300c(1 − 0.06)
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a) Earnings Model
Some analysts use earnings instead of dividends in the no-growth model when
computing the cost of equity using the following formula:
EPS
ke =
PV
where
Example 9.6
A firm has a current earnings per share of 60cents, given that the current share price is
180cents, what is the cost of equity?
Solution
60c 1
ke = = 33 %
180c 3
b) Risk-based Models
We now consider models that are based on the measurement of risk namely:
Example 9.6
A firm has debentures with a yield of 15% and the premium required by ordinary shareholders
over the debenture yield is 6%. What is the cost of equity?
Solution
k e = 15% + 6% = 21%
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ii. Capital Asset Pricing Model
The Capital Asset Pricing Model (CAPM) assumes an investor holds a diversified portfolio,
that is, a collection of investments whose returns are not in synch with one another. The
returns on the assets in a diversified portfolio do not move in the same direction or at the
same time or by the same amount. The result is that the only risk left in the portfolio, as a
whole is the risk related to movements in the market as a whole (market risk).
If investors hold diversified portfolios, the only risk they have is market risk. Investors are
risk averse, meaning they don't like risk. And if they are going to take on risk they want to
be compensated for it. Investors, who only bear market risk, need only be compensated for
market risk.
If we assume all shareholders hold diversified portfolios, the risk that is relevant in the
valuing a particular investment is the market risk of that investment. It is this market risk
that determines the investment's price. The greater the market risk, the greater the
compensation (implying a higher yield) for bearing this risk. And the greater the yield, the
lower the present value of the asset because expected future cash flows are discounted at a
higher rate that reflects the higher risk.
The cost of common stock is the sum of the investor's compensation for the time value of
money and the investor's compensation for the market risk of the stock:
Cost of common stock = Compensation for the time value of money + Compensation for
market risk
Let us represent the compensation for the time value of money as the expected risk-free rate of
interest, rf . If a particular common stock has market risk that is the same as the risk of the
market as a whole, then the compensation for that stock's market risk is the market risk premium.
The market's risk premium is the difference between the expected return on the market, rm , and
the expected risk-free rate, rf:
re = r f + β ( rm − r f )
where rf is the expected risk free rate of interest, β is a measure of the firm's stock return to
changes in the market's return (beta), and rm is the expected return on the market.
Example 9.7
The Zha ltd's common stock has an estimated beta of 1.6. If the expected risk-free rate of interest
is 4% and the expected return on the market is 12%, what is the cost of common stock for Zha
ltd?
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Solution
Given:
rf = 4%
rm = 12%
β = 1.6
re = rf + β ( rm − rf )
re = 4% + 1.6(12% − 4%) = 16.8%
Activity 9.3
1. ABC company limited's last dividend which was paid yesterday was $3.00. The
company has maintained a constant payout ratio of 50% during the past 7 years.
Seven years ago its earnings per share was $1.75. The firm's beta coefficient is 1.3.
The required rate of return on an average stock in the market is 15%, and the risk free
rate is 8%. ABC's A-rated bonds are yielding 12%, and its current stock price is $35.
What is the most reasonable estimate of ABC ltd's cost of retained earnings?
2. A firm has shares that are trading at 95cents ex-dividend and expects to pay a
dividend of 22 cents. Calculate the required rate of return of equity.
3. X ltd has a cum-dividend price of 310 cents and a dividend of 40 cents has just been
paid. The dividends are expected to grow at a rate of 10% per annum annually.
Calculate the cost of equity.
If we assume that the firm maintains the same capital structure (the mix of debt, preferred
stock, and common stock) throughout time, our task is simple. We just figure out the
proportions of capital the firm has at present. If we look at the firm's balance sheet, we can
calculate the book value of its debt, its preferred stock, and its common stock. With these
three book values, we can calculate the proportion of debt, preferred stock, and common
stock that the firm has presently. We could even look at these proportions over time to get a
better idea of the typical mix of debt, preferred stock and common stock.
But are book values going to tell us what we want to know? Probably not. What we are trying
to determine is the mix of capital that the firm considers appropriate. It is reasonable to
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assume that the financial manager recognises that the book values of capital are historical
measures and looks instead at the market values of capital. Therefore, we must obtain the
market value of debt, preferred stock, and common stock.
If the securities represented in a firm's capital are publicly traded, that is, listed on exchanges or
traded in the over-the-counter market, we can obtain market values. If some capital is privately
placed, such as an entire debt issue that was bought by an insurance company or not actively
traded, our job is tougher but not impossible. For example, if we know the interest, maturity
value, and maturity of a bond that is not traded and the yield on similar risk bonds, we can get a
rough estimate of the market value of that bond even though it is not traded.
Once we determine the market value of debt, preferred stock, and common stock, we calculate
the sum of the market values of each, and then figure out what proportion of this sum each
source of capital represents. But the mix of debt, preferred stock, and common stock that a firm
has now may not be the mix it intends to use in the future. So while we may use the present
capital structure as an approximation of the future, we really are interested in the analysis of the
firm and the resulting decision regarding its capital structure in the future.
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Example 9.8
The following information relates to company X ltd:
The long-term debt and the current liabilities have a before tax cost of 26% and 24%
respectively. Xltd's preference stock dividend rate is 20%, the beta of the firm's stock is 1.30,
the average market return is 34% and the risk-free rate is 16%. Calculate WACC using book
values and market values when the tax rate is 35%.
Solution
The Cost of Each Source of Financing
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Calculation of WACC using book values
Long term $180 000 180 000/630 000 0.2857 16.9% 4.82833%
liabilities
Long-term $270 000 270 000/1 639 500 0.1647 16.9% 2.7834%
liabilities
Common $1 200 000 1 200 000/1 639 500 0.7319 39.4% 28.8369%
stock
Example 9.9
A firm has a capital structure made up of $20m worth of equity and $10m worth of debt. The
dividend expected to be paid in the next period is 15c, the after-tax cost of debt is 15%, the
current market price is 80c and the growth rate is 12%.
Given that the cost of equity is 30%, and that the firm is contemplating to raise $10m in new
equity at 60c with floatation costs of 4c per share, calculate WACC and the cost of new equity
and MWACC.
Solution
1 2
a) WACC = (15% × ) + (30% × ) = 25%
3 3
15c
b) Cost of new equity = + 0.12 = 38.79%
60c − 4c
Target weights, which can also be based on either book or market values, reflect the firm’s
desired capital structure proportions. Firms using target weights establish such proportions on
the basis of the “optimal” capital structure they wish to achieve.
When one considers the somewhat approximate nature of the calculation of weighted average
cost of capital, the choice of weights may not be critical. However, from a strictly theoretical
point of view, the preferred weighting scheme is target market value proportions.
(a) the state of the financial markets, including stock prices in general and the level of
interest rates;
(b) investors’ aversion to risk and thus the market risk premium; and
(c) tax rates as set by governments.
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Activity 9.4
1. Identify some problems that occur when estimating the cost of capital for a privately
held firm. What are some solutions to these problems?
2. Longstreet Communications Inc. (LCI) has the following capital structure, which it
considers to be optimal: debt = 25%, preferred stock = 15%, and common stock =
60%. LCI’s tax rate is 40%, and investors expect earnings and dividends to grow at
a constant rate of 6% in the future. LCI paid a dividend of $3.70 per share last year
(D0), and its stock currently sells at a price of $60 per share. Ten-year Treasury
bonds yield 6%, the market risk premium is 5%, and LCI’s beta is 1.3. The
following terms would apply to new security offerings.
• Preferred stock: New preferred stock could be sold to the public at a price of
$100 per share, with a dividend of $9. Flotation costs of $5 per share would be
incurred.
• Debt: Debt could be sold at an interest rate of 9%.
• Common stock: New common equity will be raised only by retaining earnings.
a. Find the component costs of debt, preferred stock, and common stock.
b. What is the WACC?
3. Shi Importer’s balance sheet shows $300 million in debt, $50 million in preferred
stock, and $250 million in total common equity. Shi’s tax rate is 40%, rd = 6%, rps =
5.8%, and re = 12%. If Shi has a target capital structure of 30% debt, 5% preferred
stock, and 65% common stock, what is its WACC?
4. Dillon Labs has asked its financial manager to measure the cost of each specific type
of capital as well as the weighted average cost of capital. The weighted average cost
is to be measured by using the following weights: 40% long-term debt, 10%
preferred stock, and 50% common stock equity (retained earnings, new common
stock, or both). The firm’s tax rate is 40%.
Debt: The firm can sell for $980 a 10-year, $1,000-par-value bond paying annual
interest at a 10% coupon rate. A flotation cost of 3% of the par value is required in
addition to the discount of $20 per bond.
Preferred stock: Eight percent (annual dividend) preferred stock having a par value
of $100 can be sold for $65. An additional fee of $2 per share must be paid to the
underwriters.
Common stock: The firm’s common stock is currently selling for $50 per share.
The dividend expected to be paid at the end of the coming year (2013) is $4. Its
dividend payments, which have been approximately 60% of earnings per share in
each of the past 5 years, were as shown in the following table.
Year Dividend
2012 $3.75
2011 3.50
2010 3.30
2009 3.15
2008 2.85
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It is expected that to attract buyers, new common stock must be underpriced $5 per
share, and the firm must also pay $3 per share in flotation costs. Dividend payments
are expected to continue at 60% of earnings. (Assume that rr = rs ).
a. Calculate the after-tax cost of debt.
b. Calculate the cost of preferred stock.
c. Calculate the cost of common stock.
d. Calculate the WACC for Dillon Labs.
6. Y. ltd's present capital structure, which is also its target capital structure, calls for 50%
debt and 50% equity. The firm has only one potential project, an expansion program
with a 10.2% rate of return and a cost of $20m but which is completely divisible;
that is the company can invest any amount up to $20m. The company expects to
retain $3m of earnings next year. It can raise up to $5m in new debt at a before tax
cost of 8%, and all debt after the first $5m will have a cost of 10%. The cost of
retained earnings is 12%; the company can sell any amount of new common stock
desired at a cost of new equity of 15%. The firm's marginal tax rate is 40%. What is
the company's optimal capital budget?
7. Z ltd is planning next year's budget and projects net income of $10 500 and a payout
ratio of 40%. The company's earnings and dividends are growing at a constant rate
of 5%. The last dividend was 90cents and the current equilibrium stock price is 859
cents. The company can raise up to $10 000 of debt at a 12% before tax cost, the next
$10 000 will cost 14% and all debt after $20 000 will cost 16%. If the company
issues new common stock, a 10% floatation cost will be incurred on the first $16 000
issued, while flotation costs will be 20% on all new stock issued after the first $16
000. The company's optimal structure is composed of 40% debt, 60% equity, and the
marginal tax rate is 40%. Z ltd has the following independent, indivisible and
equally risky investment opportunities:
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9.9 Summary
In this unit we discussed that the cost of capital is the marginal cost of raising additional funds.
This cost is important in our investment decision making because we ultimately want to
compare the cost of funds with the benefits from investing these funds. The cost of capital is
determined in three steps:
(1) determine what proportions of each source of capital we intend to use;
(2) calculate the cost of each source of capital; and
(3) put the cost and the proportions together to determine the weighted average cost of capital.
The required rate of return on debt is the yield demanded by investors to compensate them for
the time value of money and the risk they bear in lending their money. The cost of debt to the
firm differs from this required rate of return due to flotation costs and the tax benefit from the
deductibility of interest expense. The required rate of return on preferred stock is the yield
demanded by investors and differs from the firm's cost of preferred stock because of the costs
of issuing additional shares (the flotation costs). The required rate of return on common stock is
more difficult to estimate than the cost of debt or preferred stock because of the nature of the
return on stock: dividends are neither guaranteed nor fixed in amount, and part of the return is
from the change in the value of the stock.
181
References
Block, S.B. and Hirt, G.A. (1994). Foundations of Financial Management. (Sixth Edition). Boston:
Irwin.
Ehrhardt, M.C. and Brigham, E.F. (2011). Financial Management, Theory and Practice.
(Thirteenth Edition). Mason: South-Western Cengage Learning.
Gitman, L. and Zutter, C.J. (2012). Principles of Managerial Finance. (Thirteenth Edition). Boston:
Pearson Education, Inc.
Kwesu, I., Nyatanga, E. and Zhanje, S. (2002). Business Statistics. Harare: ZOU.
Levy, H. and Sarnat, M. (1991). Capital Investments and Financial Decisions. (Fourth Edition). New
York: Prentice Hall.
McLaney, E..J. (2000). Business Finance: Theory and Practice. New York: Prentice Hall.
182
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