corporate finance lse

Corporate financeP. Frantz and R. Payne2790092

2009

Undergraduate study in Economics, Management, Finance and the Social Sciences

This guide was prepared for the University of London External System by:

Dr Pascal Frantz, Lecturer in Accountancy and Finance, The London School of Economics and Political Science

and

R. Payne, former Lecturer in Finance, The London School of Economics and Political Science.

This is one of a series of subject guides published by the University. We regret that due to pressure of work the authors are unable to enter into any correspondence relating to, or aris-ing from, the guide. If you have any comments on this subject guide, favourable or unfavour-able, please use the form at the back of this guide.

This subject guide is for the use of University of London External students registered for programmes in the fields of Economics, Management, Finance and the Social Sciences (as applicable). The programmes currently available in these subject areas are:

Access route

Diploma in Economics

Diploma in Social Sciences

Diplomas for Graduates

BSc Accounting and Finance

BSc Accounting with Law/Law with Accounting

BSc Banking and Finance

BSc Business

BSc Development and Economics

BSc Economics

BSc Economics and Finance

BSc Economics and Management

BSc Geography and Environment

BSc Information Systems and Management

BSc International Relations

BSc Management

BSc Management with Law/Law with Management

BSc Mathematics and Economics

BSc Politics

BSc Politics and International Relations

BSc Sociology

BSc Sociology with Law.

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Published by: University of London Press

© University of London 2009

Printed by: Central Printing Service, University of London, England

Contents

i

Contents

Introduction to the subject guide .......................................................................... 1

Aims of the unit ............................................................................................................. 1Learning objectives ........................................................................................................ 1Syllabus ......................................................................................................................... 2Essential reading ........................................................................................................... 2Further reading .............................................................................................................. 3Subject guide structure and use ..................................................................................... 5Examination structure .................................................................................................... 5Glossary of abbreviations used in this subject guide ....................................................... 6

Chapter 1: Present-value calculations and the valuation of physical investment projects .................................................................................. 7

Aim of the chapter ......................................................................................................... 7Learning objectives ........................................................................................................ 7Essential reading ........................................................................................................... 7Further reading .............................................................................................................. 7Overview ....................................................................................................................... 7Introduction .................................................................................................................. 8Fisher separation and optimal decision-making .............................................................. 8Fisher separation and project evaluation ...................................................................... 11The time value of money .............................................................................................. 12The net present-value rule............................................................................................ 13Other project appraisal techniques ............................................................................... 15Using present-value techniques to value stocks and bonds ........................................... 18A reminder of your learning outcomes .......................................................................... 19Key terms .................................................................................................................... 20Sample examination questions ..................................................................................... 20

Chapter 2: Risk and return: mean–variance analysis and the CAPM.................... 21

Aim of the chapter ....................................................................................................... 21Learning objectives ...................................................................................................... 21Essential reading ......................................................................................................... 21Further reading ............................................................................................................ 21Introduction ................................................................................................................ 21Statistical characteristics of portfolios ........................................................................... 22Diversification .............................................................................................................. 24Mean–variance analysis ............................................................................................... 25The capital asset pricing model .................................................................................... 30The Roll critique and empirical tests of the CAPM ......................................................... 33A reminder of your learning outcomes .......................................................................... 34Key terms .................................................................................................................... 34Sample examination questions ..................................................................................... 35Solutions to activities ................................................................................................... 35

Chapter 3: The arbitrage pricing theory ............................................................... 37

Aim of the chapter ....................................................................................................... 37Learning objectives ...................................................................................................... 37Essential reading ......................................................................................................... 37Further reading ............................................................................................................ 37Overview ..................................................................................................................... 37

92 Corporate finance

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Introduction ................................................................................................................ 37Single-factor models .................................................................................................... 38Multi-factor models ..................................................................................................... 40Broad-based portfolios and idiosyncratic returns........................................................... 41Factor-replicating portfolios ......................................................................................... 41The arbitrage pricing theory ......................................................................................... 42Summary ..................................................................................................................... 43A reminder of your learning outcomes .......................................................................... 44Key terms .................................................................................................................... 44

Chapter 4: Derivative assets: properties and pricing ........................................... 45

Aim of the chapter ....................................................................................................... 45Learning objectives ...................................................................................................... 45Essential reading ......................................................................................................... 45Further reading ............................................................................................................ 45Overview ..................................................................................................................... 45Varieties of derivatives ................................................................................................. 45Derivative asset pay-off profiles ................................................................................... 47Pricing forward contracts ............................................................................................. 49Binomial option pricing setting .................................................................................... 50Bounds on option prices and exercise strategies ........................................................... 53Black–Scholes option pricing ....................................................................................... 55Put–call parity ............................................................................................................. 56Pricing interest rate swaps ........................................................................................... 58Summary ..................................................................................................................... 58A reminder of your learning outcomes .......................................................................... 58Key terms .................................................................................................................... 59Sample examination questions ..................................................................................... 59

Chapter 5: Efficient markets: theory and empirical evidence .............................. 61

Aim of the chapter ....................................................................................................... 61Learning objectives ...................................................................................................... 61Essential reading ......................................................................................................... 61Further reading ............................................................................................................ 61Overview ..................................................................................................................... 62Varieties of efficiency ................................................................................................... 62Risk adjustments and the joint hypothesis problem ...................................................... 63Weak-form efficiency: implications and tests ................................................................ 64Weak-form efficiency: empirical results ......................................................................... 66Semi-strong-form efficiency: event studies .................................................................... 69Semi-strong-form efficiency: empirical evidence ............................................................ 71Strong-form efficiency .................................................................................................. 71Summary ..................................................................................................................... 71A reminder of your learning outcomes .......................................................................... 72Key terms .................................................................................................................... 72Sample examination questions ..................................................................................... 72

Chapter 6: The choice of corporate capital structure ........................................... 73

Aim of the chapter ....................................................................................................... 73Learning objectives ...................................................................................................... 73Essential reading ......................................................................................................... 73Further reading ............................................................................................................ 73Overview ..................................................................................................................... 73

Contents

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Basic features of debt and equity ................................................................................. 74The Modigliani–Miller theorem .................................................................................... 75Modigliani–Miller and Black–Scholes ........................................................................... 77Modigliani–Miller and corporate taxation ..................................................................... 77Modigliani–Miller with corporate and personal taxation ............................................... 79Summary ..................................................................................................................... 81A reminder of your learning outcomes .......................................................................... 81Key terms .................................................................................................................... 81Sample examination questions ..................................................................................... 81

Chapter 7: Asymmetric information, agency costs and capital structure ................................................................................................... 83

Aim of the chapter ....................................................................................................... 83Learning objectives ...................................................................................................... 83Essential reading ......................................................................................................... 83Further reading ............................................................................................................ 83Overview ..................................................................................................................... 84Capital structure, governance problems and agency costs ............................................. 84Agency costs of outside equity and debt ...................................................................... 84Agency costs of free cash flows .................................................................................... 87Firm value and asymmetric information ........................................................................ 88Summary ..................................................................................................................... 92Key terms .................................................................................................................... 92A reminder of your learning outcomes .......................................................................... 93Sample examination questions ..................................................................................... 93

Chapter 8: Dividend policy ................................................................................... 95

Aim of the chapter ....................................................................................................... 95Learning objectives ...................................................................................................... 95Essential reading ......................................................................................................... 95Further reading ............................................................................................................ 95Overview ..................................................................................................................... 96Modigliani–Miller meets dividends ............................................................................... 96Prices, dividends and share repurchases ....................................................................... 97Dividend policy: stylised facts ....................................................................................... 97Taxation and clientele theory ....................................................................................... 99Asymmetric information and dividends ....................................................................... 100Agency costs and dividends ....................................................................................... 101Summary ................................................................................................................... 101A reminder of your learning outcomes ........................................................................ 102Key terms .................................................................................................................. 102Sample examination questions ................................................................................... 102

Chapter 9: Mergers and takeovers ..................................................................... 103

Aim of the chapter ..................................................................................................... 103Learning objectives .................................................................................................... 103Essential reading ....................................................................................................... 103Further reading .......................................................................................................... 103Overview ................................................................................................................... 104Merger motivations ................................................................................................... 104A numerical takeover example ................................................................................... 105The market for corporate control ................................................................................ 106The impossibility of efficient takeovers ....................................................................... 106


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Two ways to get efficient takeovers ............................................................................ 108Empirical evidence ..................................................................................................... 109Summary ................................................................................................................... 110A reminder of your learning outcomes ........................................................................ 111Key terms .................................................................................................................. 111Sample examination questions ................................................................................... 112

Appendix 1: Perpetuities and annuities .................................................................... 113

Perpetuities ............................................................................................................... 113Annuities .................................................................................................................. 114

Appendix 2: Sample examination paper .................................................................. 115

Introduction to the subject guide

1


This subject guide provides you with an introduction to the modern theory of finance. As such, it covers a broad range of topics and aims to give a general background to any student who wishes to do further academic or practical work in finance or accounting after graduation.

The subject matter of the guide can be broken into two main areas.

1. The first section of material covers the valuation and pricing of real and financial assets. This provides you with the methodologies you will need to fairly assess the desirability of investment in physical capital, and price spot and derivative assets. We employ a number of tools in this analysis. The coverage of the risk-return trade-off in financial assets and mean–variance optimisation will require you to apply some basic statistical theory alongside the standard optimisation techniques taught in basic economics courses. Another important part of this section will be the use of absence-of-arbitrage techniques to price financial assets.

2. In the second section, we will examine issues that come under the broad heading of corporate finance. Here we will examine the key decisions made by firms, how they affect firm value and empirical evidence on these issues. The areas involved include the capital structure decision, dividend policy, and mergers and acquisitions. By studying these areas, you should gain an appreciation of optimal financial policy on a firm level, conditions under which an optimal policy actually exists and how the actual financial decisions of firms may be explained in theoretical terms.

Aims of the unit

This unit is aimed at students interested in understanding asset pricing and corporate finance. It provides a theoretical framework used to address issues in project appraisal and financing, the pricing of risk, securities valuation, market efficiency, capital structure and mergers and acquisitions. It provides students with the tools required for further studies in financial intermediation and investments.

Learning objectives

At the end of this unit, and having completed the essential reading and activities, you should be able to:

explain how to value projects, and use the key capital budgeting techniques (NPV and IRR)

understand the mathematics of portfolios and how risk affects the value of the asset in equilibrium under the fundaments asset pricing paradigms (CAPM and APT)

explain the characteristics of derivative assets (forwards, futures and options), and how to use the main pricing techniques (binomial methods in derivatives pricing and the Black–Scholes analysis)

discuss the theoretical framework of informational efficiency in financial markets and evaluate the related empirical evidence

understand and explain the capital structure theory, and how information asymmetries affect it


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understand and explain the relevance, facts and role of the dividend policy

understand how corporate governance can contribute to firm value

discuss why merger and acquisition activities exist, and calculate the related gains and losses.

Syllabus

Note: There has been a minor revision to this syllabus in 2009.

Students may bring into the examination hall their own hand-held electronic calculator. If calculators are used they must satisfy the requirements listed in paragraphs 10.5 to 10.7 of the General Regulations.

If you are taking this unit as part of a BSc degree, units which must be passed before this unit may be attempted are 02 Introduction to economics and either 05a Mathematics 1 or 05b Mathematics 2. This unit may not be taken with unit 59 Financial management.

Project evaluation: Hirschleifer analysis and Fisher separation; the NPV rule and IRR rules of investment appraisal; comparison of NPV and IRR; ‘wrong’ investment appraisal rules: payback and accounting rate of return.

Risk and return – the CAPM and APT: the mathematics of portfolios; mean-variance analysis; two-fund separation and the CAPM; Roll’s critique of the CAPM; factor models; the arbitrage pricing theory.

Derivative assets – characteristics and pricing: definitions: forwards and futures; replication, arbitrage and pricing; a general approach to derivative pricing using binomial methods; options: characteristics and types; bounding and linking option prices; the Black–Scholes analysis.

Efficient markets – theory and empirical evidence: underpinning and definitions of market efficiency; weak-form tests: return predictability; the joint hypothesis problem; semi-strong form tests: the event study methodology and examples; strong form tests: tests for private information.

Capital structure: the Modigliani–Miller theorem: capital structure irrelevancy; taxation, bankruptcy costs and capital structure; the Miller equilibrium; asymmetric information: 1) the under-investment problem, asymmetric information; 2) the risk-shifting problem, asymmetric information; 3) free cash-flow arguments; 4) the pecking order theory; 5) debt overhang.

Dividend theory: the Modigliani–Miller and dividend irrelevancy; Lintner’s fact about dividend policy; dividends, taxes and clienteles; asymmetric information and signalling through dividend policy.

Corporate governance: separation of ownership and control; management incentives; management shareholdings and firm value; corporate governance.

Mergers and acquisitions: motivations for merger activity; calculating the gains and losses from merger/takeover; the free-rider problem and takeover activity.

Essential reading

There are a number of excellent textbooks that cover this area. However, the following text has been chosen as the core text for this unit due to its extensive treatment of many of the issues covered and up-to-date discussions:

Grinblatt, M. and S. Titman Financial Markets and Corporate Strategy. (Boston, Mass.; London: McGraw-Hill, 2002) second edition [ISBN 9780072294330].

At the start of each chapter of this guide, we will indicate the reading that you need to do from Grinblatt and Titman (2002).


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Further reading

As further material, we will also direct you to the relevant chapters in two other texts. You may wish to look at the following two texts that are standard for many undergraduate finance courses:

Brealey, R., S. Myers and F. Allen Principles of Corporate Finance. (Boston, Mass., London: McGraw-Hill, 2008) ninth international edition [ISBN 9780071266758].

Copeland, T., J. Weston and K. Shastri Financial Theory and Corporate Policy. (Reading, Mass.; Wokingham: Addison-Wesley, 2005) fourth edition [ISBN 9780321223531].

For certain topics, we also list journal articles as Further reading. To help you read extensively, all External System students have free access to the University of London Online Library where you will find either the full text or an abstract of many of the journal articles listed in this subject guide. You will need to have a username and password to access this resource and this is the same one you are sent for accessing the University of London Student Portal where you can also find the Online library website at http://my.londonexternal.ac.uk

A full list of all Further reading referred to in the subject guide is presented here for ease of reference.

Journal articles

Asquith, P. and D. Mullins ‘The impact of initiating dividend payments on shareholders’ wealth’, Journal of Business 56(1) 1983, pp.77–96.

Ball, R. and P. Brown ‘An empirical evaluation of accounting income numbers’, Journal of Accounting Research 6(2) 1968, pp.159–78.

Blume, M., J. Crockett and I. Friend ‘Stock Ownership in the United States: Characteristics and Trends’, Survey of Current Business 54(11) 1974, pp.16–40.

Bradley, M., A. Desai and E. Kim ‘Synergistic Gains from Corporate Acquisitions and Their Division Between the Stockholders of Target and Acquiring Firms’, Journal of Financial Economics 21(1) 1988, pp.3–40.

Brock, W., J. Lakonishok and B. LeBaron ‘Simple technical trading rules and stochastic properties of stock returns’, Journal of Finance 47(5) 1992, pp.1731–764.

DeBondt, W. and R. Thaler ‘Does the stock market overreact?’, Journal of

Finance 40(3) 1984, pp.793–805.Fama, E. ‘The behavior of stock market prices’, Journal of Business 38(1) 1965,

pp.34–105.Fama, E. ‘Efficient capital markets: a review of theory and empirical work’,

Journal of Finance 25(2) 1970, pp.383–417.Fama, E. ‘Efficient capital markets: II’, Journal of Finance 46(5) 1991,

pp.1575–617.Fama, E. and K. French ‘Dividend yields and expected stock returns’, Journal of

Financial Economics 22(1) 1988, pp.3–25.French, K. ‘Stock returns and the weekend effect’, Journal of Financial

Economics 8(1) 1980, pp.55–70.Fama, E. and K. French ‘The cross-section of expected stock returns’, Journal of

Finance 47(2) 1992, pp.427–65.Grossman, S. and O. Hart ‘Takeover Bids, the Free-Rider Problem and the

Theory of the Corporation’, Bell Journal of Economics 11(1) 1980, pp.42–64.Healy, P. and K. Palepu ‘Earnings Information Conveyed by Dividend Initiations

and Omissions’, Journal of Financial Economics 21(2) 1988, pp.149–76.Healy, P., K. Palepu and R. Ruback ‘Does Corporate Performance Improve after

Mergers?’, Journal of Financial Economics 31(2) 1992, pp.135–76.


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Jarrell, G. and A. Poulsen ‘Returns to Acquiring Firms in Tender Offers: Evidence from Three Decades’, Financial Management 18(3) 1989, pp.12–19.

Jarrell, G., J. Brickley and J. Netter ‘The Market for Corporate Control: The Empirical Evidence since 1980’, Journal of Economic Perspectives 2(1) 1988, pp.49–68.

Jensen, M. ‘Some anomalous evidence regarding market efficiency’, Journal of

Financial Economics 6(2–3) 1978, pp.95–101.Jensen, M. ‘Agency costs of Free Cash Flow, Corporate Finance, and Takeovers’,

American Economic Review 76(2) 1986, pp.323–29.Jensen, M. and R. Ruback ‘The Market for Corporate Control: The Scientific

Evidence’, Journal of Financial Economics 11(1–4) 1983, pp.5–50.Jensen, M. and W. Meckling ‘Theory of the firm: managerial behaviour, agency

costs and capital structure’, Journal of Financial Economics 3(4) 1976, pp.305–60.

Jensen, M. and W. Meckling ‘Theory of the Firm: Managerial Behaviour, Agency Costs and Capital Structure’, Journal of Financial Economics 3(4) 1976, pp.305–60.

Lakonishok, J., A. Shleifer and R. Vishny ‘Contrarian investment, extrapolation, and risk’, Journal of Finance 49(5) 1994, pp.1541–578.

Levich, R. and L. Thomas ‘The significance of technical trading-rule profits in the foreign exchange market: a bootstrap approach’, Journal of International

Money and Finance 12(5) 1993, pp.451–74.Lintner, J. ‘Distribution of Incomes of Corporations among Dividends, Retained

Earnings and Taxes’ American Economic Review 46(2) 1956, pp.97–113.Lo, A. and C. McKinlay ‘Stock market prices do not follow random walks:

evidence from a simple specification test’, Review of Financial Studies 1(1) 1988, pp.41–66.

Masulis, R. ‘The impact of capital structure change on firm value: some estimates’, Journal of Finance 38(1) 1983, pp.107–26.

Miller, M. ‘Debt and taxes’, Journal of Finance 32, 1977, pp. 261–75.Myers, S. ‘Determinants of corporate borrowing’, Journal of Financial Economics

5(2) 1977, pp.147–75.Myers, S. and N. Majluf ‘Corporate financing and investment decisions when

firms have information that investors do not have’, Journal of Financial

Economics 13(2) 1984, pp.187–221.Poterba, J. and L. Summers ‘Mean reversion in stock prices: evidence and

implications’, Journal of Financial Economics 22(1) 1988, pp.27–59.

Roll, R. ‘A Critique of the Asset Pricing Theory’s Texts. Part 1: On Past and Potential Testability of the Theory’, Journal of Financial Economics 4(2) 1977, pp.129–76.

Ross, S. ‘The Determination of Financial Structure: The Incentive Signalling Approach’, Bell Journal of Economics 8(1) 1977, pp.23–40.

Shleifer, A. and R. Vishny ‘Large Shareholders and Corporate Control,’ Journal of Political Economy 94(3) 1986, pp.461–88.

Travlos, N. ‘Corporate Takeover Bids, Methods of Payment, and Bidding Firms’ Stock Returns’, Journal of Finance 42(4) 1990, pp.943–63.

Warner, J. ‘Bankruptcy Costs: Some Evidence’, Journal of Finance 32(2) 1977, pp.337–47.

Books

Allen, F. and R. Michaely ‘Dividend Policy’ in Jarrow, Maksimovic and Ziemba (eds) Handbook of Finance. (Elsevier Science, 1995). [No ISBN available].

Haugen, R. and J. Lakonishok The incredible January effect. (Homewood, Ill.: Dow Jones-Irwin, 1988) [ISBN 9781556230424].

Ravenscraft, D. and F. Scherer Mergers, Selloffs, and Economic Efficiency. (Washington D.C.: Brookings Institution, 1987) [ISBN 9780815773481].


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Subject guide structure and use

You should note that, as indicated above, the study of the relevant chapter should be complemented by at least the essential reading given at the chapter head.

The content of the subject guide is as follows.

Chapter 1: here we focus on the evaluation of real investment projects using the net present-value technique and provide a comparison of NPV with alternative forms of project evaluation.

Chapter 2: we look at the basics of risk and return of primitive financial assets and mean–variance optimisation. We go on to derive and discuss the capital asset pricing model (CAPM).

Chapter 3: we present the arbitrage pricing theory, proposed as an alternative to the CAPM for the calculation of expected returns on financial assets.

Chapter 4: here we look at derivative assets. We begin with the nature of forward, future, option and swap contracts, then move on to pricing derivative assets via absence-of-arbitrage arguments. We also include a description of binomial option pricing models and end with the Black–Scholes analysis.

Chapter 5: in this chapter, we examine the efficiency of financial markets. We present the concepts underlying market efficiency and discuss the empirical evidence on efficient markets.

Chapter 6: here we turn to corporate finance issues, treating the decision over a corporation’s capital structure. The essential issue is what levels of debt and equity finance should be chosen in order to maximise firm value.

Chapter 7: we look at more advanced issues in capital structure theory and focus on the use of capital structure to mitigate governance problems known as agency costs and how capital structure and financial decisions are affected by asymmetric information.

Chapter 8: here we examine dividend policy. What is the empirical evidence on the dividend pay-out behaviour of firms, and theoretically, how can we understand the empirical facts?

Chapter 9: we look at mergers and acquisitions, and ask what motivates firms to merge or acquire, what are the potential gains from this activity, and how can this be theoretically treated? We also explore how hostile acquisitions may serve as a discipline device to mitigate governance problems.

There is no specific chapter about corporate governance, but the agency related topics of Chapters 7 and 9 are inherently motivated by the existence of such problems. See also Grinblatt and Titman (2002) Chapter 18 for a broad overview on governance-related issues.

Examination structure

Important: the information and advice given in the following section are based on the examination structure used at the time this guide was written. Please note that subject guides may be used for several years. Because of this, we strongly advise you to always check both the current Regulations for relevant information about the examination, and the current Examiners’ commentaries where you should be advised of any forthcoming changes. You should also carefully check the rubric/instructions on the paper you actually sit and follow those instructions.


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This unit will be evaluated solely on the basis of a three-hour examination. You will have to answer four out of a choice of eight questions. Although the Examiner will attempt to provide a fairly balanced coverage of the unit, there is no guarantee that all of the topics covered in this guide will appear in the examination. Examination questions may contain both numerical and discursive elements. Finally, each question will carry equal weight in marking and, in allocating your examination time, you should pay attention to the breakdown of marks associated with the different parts of each question.

Glossary of abbreviations used in this subject guide

APT arbitrage pricing theory

ARR accounting rate of return

B–S Black –Scholes

CAPM capital asset pricing model

CML capital market line

EMH efficient markets hypothesis

IRR internal rate of return

M&A mergers and acquisitions

M–M Modigliani–Miller

NPV net present value

OTC over the counter

RWM random walk model

SML security market line

Chapter 1: Present-value calculations and the valuation of physical investment projects

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Chapter 1: Present-value calculations

and the valuation of physical investment

projects

Aim of the chapter

The aim of this chapter is to introduce the Fisher separation theorem, which is the basis for using the net present value (NPV) for project evaluation purposes. With this aim in mind, we discuss the optimality of the NPV criterion and compare this criterion with alternative project evaluation criteria.

Learning objectives

At the end of this chapter, and having completed the essential reading and activities, you should be able to:

analyse optimal physical and financial investment in perfect capital markets and derive the Fisher separation result

justify the use of the NPV rules via Fisher separation

compute present and future values of cash-flow streams and appraise projects using the NPV rule

evaluate the NPV rule in relation to other commonly used evaluation criteria

value stocks and bonds via NPV.

Essential reading

Grinblatt, M. and S. Titman Financial Markets and Corporate Strategy. (Boston, Mass.; London: McGraw-Hill, 2002) Chapter 10.

Further reading

Brealey, R., S. Myers and F. Allen Principles of Corporate Finance. (Boston, Mass.; London: McGraw-Hill, 2008) Chapters 2, 3, 5, 6 and 7.

Copeland, T. and J. Weston Financial Theory and Corporate Policy. (Reading, Mass.; Wokingham: Addison-Wesley, 2005) Chapters 1 and 2.


Overview

In this chapter we present the basics of the present-value methodology for the valuation of investment projects. The chapter develops the net present-value (NPV) technique before presenting a comparison with the other project evaluation criteria that are common in practice. We will also discuss the optimality of NPV and give a number of extensive examples.


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Introduction

Let us begin by defining how we are going to think about a firm in this chapter. For the purposes of this chapter, we will consider a firm to be a package of investment projects. The key question, therefore, is how do the firm’s shareholders or managers decide on which investment projects to undertake and which to discard? Developing the tools that should be used for project evaluation is the emphasis of this chapter.

It may seem, at this point, that our definition of the firm is rather limited. It is clear that, in only examining the investment operations of the firm, we are ignoring a number of potentially important firm characteristics. In particular, we have made no reference to the financial structure or decisions of the firm (i.e. its capital structure, borrowing or lending activities, or dividend policy). The first part of this chapter presents what is known as the Fisher separation theorem. What follows is a statement of the theorem. This theorem allows us to say the following: under certain conditions (which will be presented in the following section), the shareholders can delegate to the management the task of choosing which projects to undertake (i.e. determining the optimal package of investment projects), whereas they themselves determine the optimal financial decisions. Hence, the theory implies that the investment and financing choices can be completely disconnected from each other and justifies our limited definition of the firm for the time being.

Fisher separation and optimal decision-making

Consider the following scenario. A firm exists for two periods (imaginatively named period 0 and period 1). The firm has current funds of m and, without any investment, will receive no money in period 1. Investments can be of two forms. The firm can invest in a number of physical investment projects, each of which costs a certain amount of cash in period 0 and delivers a known return in period 1. The second type of investment is financial in nature and permits the firm to borrow or lend unlimited amounts at rate of interest r. Finally the firm is assumed to have a standard utility function in its period 0 and period 1 consumption. (By consumption we mean the use of any funds available to the firm net of any costs of investment.)

Let us first examine the set of physical investments available. The firm will logically rank these investments in terms of their return, and this will yield a production opportunity frontier that looks as given in Figure 1.1 (and is labelled POF). This curve represents one manner in which the firm can transform its current funds into future income, where c

0 is period 0

consumption, and c1 is period 1 consumption. Using the assumed utility

function for the firm, we can also plot an indifference map on the same diagram to find the optimal physical investment plan of a given firm. The optimal investment policies of two different firms are shown in Figure 1.1.

It is clear from Figure 1.1 that the specifics of the utility function of the firm will impact upon the firm’s physical investment policy. The implication of this is that the shareholders of a firm (i.e. those whose utility function matters in forming optimal investment policy) must dictate to the managers of the firm the point to which it invests. However, until now we have ignored the fact that the firm has an alternative method for investment (i.e. using the capital market).


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Figure 1.1

The financial investment allows firms to borrow or lend unlimited amounts at rate r. Assuming that the firm undertakes no physical investment, we can define the firm’s consumption opportunities quite easily. Assume the firm neither borrows nor lends. This implies that current consumption (c

0) must be identically m, whereas period 1

consumption (c1) is zero. Alternatively the firm could lend all of its funds. This leads to c

0 being zero and c1 = (1 + r) m. The relationship between

period 0 and period 1 consumption is therefore given as below:

c1 = (1 + r)(m – c0 ). (1.1)

This implies that the curve which represents capital market investments is a straight line with slope –(1+r). This curve is labeled CML on Figure 1.2. Again, we have on Figure 1.2 plotted the optimal financial investments for two different sets of preferences (assuming that no physical investment is undertaken).

Figure 1.2

Now we can proceed to analyse optimal decision-making when firms invest in both financial and physical assets. Assume the firm is at the beginning of period 0 and trying to decide on its investment plan. It is clear that, to maximise firm value, the projects undertaken should be those


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with the greatest return. Knowing that the return on financial investment is always (1+r), the firm will first invest in all physical investment projects with returns greater than (1+r ). These are those projects on the production possibility frontier (PPF) between points m and I on Figure 1.3.1 Projects above I on the PPF have returns that are dominated by the return from financial investment.

Hence the firm physically invests up to point I. Note that, at this point, we have not mentioned the firm’s preferences over period 0 and period 1 consumption. Hence, the decision to physically invest to I will be taken by all firms regardless of the preferences of their owners. Preferences come into play when we consider what financial investments should be undertaken.

The firm’s physical investment policy takes it to point I, from where it can borrow or lend on the capital market. Borrowing will move the firm to the south-east along a line starting at I and with slope –(1+r); lending will take the firm north-west along a similarly sloped line. Two possible optima are shown on Figure 1.3. The optimum at point X is that for a firm whose owners prefer period 1 consumption relative to period 0 consumption (and have hence lent on the capital market), whereas a firm locating at Y has borrowed, as its owners prefer date 0 to date 1 consumption.

Figure 1.3 demonstrates the key insight of Fisher separation. All firms, regardless of preferences, will have the same optimal physical investment policy, investing to the point where the PPF and capital market line are tangent. Preferences then dictate the firm’s borrowing or lending policy and shift the optimum along the capital market line. The implication of this is that, as it is physical investment that alters firm value, all agents (i.e. regardless of preferences) agree on the physical investment policy that will maximise firm value. More specifically, the shareholders of the firm can delegate choice of investment policy to a manager whose preferences may differ from their own, while controlling financial investment policy in order to suit their preferences.

Figure 1.3

1 The absolute value of

the slope of the PPF can

be equated with the

return on physical

investment. For all points

below I on the PPF, this

slope exceeds that of

the capital market line

set of desirable physical

investment projects.


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Fisher separation and project evaluation

Fisher separation can also be used to justify a certain method of project appraisal. Figure 1.3 shows a sub-optimal physical investment decision (I’) and the capital market line that borrowing and lending from point I’ would trace out. Clearly this capital market line always lies below that achieved through the optimal physical investment policy. Hence, one could say that optimal physical investment should maximise the horizontal intercept of the capital market line on which the firm ends up. Let us, then, assume a firm that decides to invest a dollar amount of I0. Given that the firm has date 0 income of m and no date 1 income, aside from that accruing from physical investment, the horizontal intercept of the capital market line upon which the firm has located is:

!(I0)m – I0 + r1 +where (I0) is the date 1 income from the firm’s physical investment. Maximising this is equivalent to the following maximisation problem:

!(I0)max r1 + – I0I0

.

The prior objective is the net present-value rule for project appraisal. It says that an optimal physical investment policy maximises the difference between investment proceeds divided by one plus the interest rate and the investment cost. Here, the term ‘optimal’ is being defined as that which leads to maximisation of shareholder utility. We will discuss the NPV rule more fully (and for cases involving more than one time period) later in this chapter.

The assumption of perfect capital markets is vital for our Fisher separation results to hold. We have assumed that borrowing and lending occur at the same rate and are unrestricted in amount and that there are no transaction costs associated with the use of the capital market. However, in practical situations, these conditions are unlikely to be met. A particular example is given in Figure 1.4. Here we have assumed that the rate at which borrowing occurs is greater than the rate of interest paid on lending (as the real world would dictate). Figure 1.3 shows that there are now two points at which the capital market lines and the production opportunities frontier are tangential. This then implies that agents with different preferences will choose differing physical investment decisions and, therefore, Fisher separation breaks down.

Figure 1.4


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Agents with strong preferences for future consumption will physically invest to point X and then financially invest to an optimum on the capital market lending line (CML). Those with strong preferences for current consumption physically invest to point Y and borrow (along CML’). Finally, a set of agents may exist who value current and future consumption similarly, and these will optimise by locating directly on the PPF and not using the capital market at all. An example of an optimum of this type is point Z on Figure 1.4.

The time value of money

In the preceding section we demonstrated the Fisher separation theorem and the manner in which physical and financial investment decisions can be disconnected. The major implication of this theorem is that the set of desirable physical investment projects does not depend on the preferences of individuals. In the following sections we shall focus on the way in which individual physical investment projects should be evaluated. Our key methodology for this will be the NPV rule, mentioned in the preceding section. In the following sections we will show you how to apply the rule to situations involving more than one period and with time-varying cash flows.

To begin, let us consider a straightforward question. Is $1 received today worth the same as $1 received in one year’s time? A naïve response to this question would assert that $1 is $1 regardless of when it is received, and hence the answer to the question would be yes. A more careful consideration of the question brings the opposite response however. Let’s assume I receive $1 now. If I also assume that there is a risk-free asset in which I can invest my dollar (e.g. a bank account), then in one year’s time I will receive $(1+r), assuming I invest. Here, r is the rate of return on the safe investment. Hence $1 received today is worth $(1+r) in one year. The answer to the question is therefore no. A dollar received today is worth more than a dollar received in one year or at any time in the future.

The above argument characterises the time value of money. Funds are more valuable the earlier they are received. In the previous paragraph we illustrated this by calculating the future value of $1. We can similarly illustrate the time value of money by using present values. Assume I am to receive $1 in one year’s time and further assume that the borrowing and lending rate is r. How much is this dollar worth in today’s terms? To answer this second question, put yourself in the position of a bank. Knowing that someone is certain to receive $1 in one year, what is the maximum amount you would lend him or her now? If I, as a bank, were to lend someone money for one year, at the end of the year I would require repayment of the loan plus interest (at rate r). Hence if I loaned the individual $x I would require a repayment of $x(1+r). This implies that the maximum amount I should be willing to lend is implicitly defined by the following equation:

$x(1 + r) = $1 (1.2)

such that:

xr

$ 11

=+

. (1.3)

The value for x defined in equation 1.3 is the present value of $1 received in one year’s time. This quantity is also termed the discounted value of the $1.


13

You can see the present and future value concepts pictured in Figure 1.2. If you recall, Figure 1.2 just plots the CML for a given level of initial funds (m) assuming no funds are to be received in the future. The future value of this amount of money is simply the vertical intercept of the CML (i.e. m(1+r)), and obviously the present value of m(1+r) is just m.

The present and future value concepts are straightforwardly extended to cover more than one period. Assume an annual compound interest rate of r. The present value of $100 to be received in k year’s time is:

100(PVK r

( )1001=+ )K

(1.4)

whereas the future value of $100 received today and evaluated k years hence is:

FVK (100) = 100(1 + r)K. (1.5)

Activity

Below, there are a few applications of the present and future value concepts. You should attempt to verify that you can replicate the calculations given below.

Assume a compound borrowing and lending rate of 10 per cent annually.

a. The present value of $2,000 to be received in three years time is $1,502.63.

b. The present value of $500 to be received in five years time is $310.46.

c. The future value of $6,000 evaluated four years hence is $8,784.60.

d. The future value of $250 evaluated 10 years hence is $648.44.

The net present-value rule

In the previous section we demonstrated that the value of funds depends critically on the time those funds are received. If received immediately, cash is more valuable than if it is to be received in the future.

The net present-value rule was introduced in simple form in the section on Fisher separation. In its more general form, it uses the discounting techniques provided in the previous section in order to generate a method of evaluating investment projects. Consider a hypothetical physical investment project, which has an immediate cost of I. The project generates cash flows to the firm in each of the next k years, equal to C

k.

In words, all that the NPV rule does is to compute the present value of all receipts or payments. This allows direct comparisons of monetary values, as all are evaluated at the same point in time. The NPV of the project is then just the sum of the present values of receipts, less the sum of the present values of the payments.

Using the notation given above and again assuming a rate of return of r, the NPV can be written as:

(NPVCir

Ii

k

11= !

= +!) i . (1.6)

Note that the cash flows to the project can be positive and negative, implying that the notation employed is flexible enough to embody both cash inflows and outflows after initiation.


14

Once we have calculated the NPV, what should we do? Clearly, if the NPV is positive, it implies that the present value of receipts exceeds the present value of payments. Hence, the project generates revenues that outweigh its costs and should therefore be accepted. If the NPV is negative the project should be rejected, and if it is zero the firm will be indifferent between accepting and rejecting the project.

This gives a very straightforward method for project evaluation. Compute the NPV of the project (which is a simple calculation), and if it is greater than zero, the project is acceptable.

Example

Consider a manufacturing firm, which is contemplating the purchase of a new piece of plant. The rate of interest relevant to the firm is 10 per cent. The purchase price is £1,000. If purchased, the machine will last for three years and in each year generate extra revenue equivalent to £750. The resale value of the machine at the end of its lifetime is zero. The NPV of this project is:

NPV = 750 + 750 + 750 – 1000 = 865.14.

(1.1)3 (1.1)2 (1.1)1

As the NPV of the project exceeds zero, it should be accepted.

In order to familiarise yourself with NPV calculations, attempt the following activities by calculating the NPV of each project and assessing its desirability.

Activity

Assume an interest rate of 5 per cent. Compute the NPV of each of the following projects, and state whether each project should be accepted or not.

Project A has an immediate cost of $5,000, generates $1,000 for each of the next six years and zero thereafter.

Project B costs £1,000 immediately, generates cash flows of £600 in year 1, £300 in year 2 and £300 in year 3.

Project C costs ¥10,000 and generates ¥6,000 in year 1. Over the following years, the cash flows decline by ¥2,000 each year, until the cash flow reaches zero.

Project D costs £1,500 immediately. In year 1 it generates £1,000. In year 2 there is a further cost of £2,000. In years 3, 4 and 5 the project generates revenues of £1,500 per annum.

Up to this point we have just considered single projects in isolation, assuming that our funds were enough to cover the costs involved. What happens, first of all, if the members of a set of projects are mutually exclusive?2 The answer is simple. Pick the project that has the greatest NPV. Second, what should we do if we have limited funds? It may be the case that we are faced with a pool of projects, all of which have positive NPVs, but we only have access to an amount of money that is less than the total investment cost of the entire project pool. Here we can rely on another nice feature of the NPV technique. NPVs are additive across projects (i.e. the NPV of taking on projects A and B is identical to the NPV of A plus the NPV of B). The reason for this should be obvious from the manner in which NPVs are calculated. Hence, in this scenario, we should calculate all project combinations that are feasible (i.e. the total investment in these projects can be financed with our current funds). Then calculate the NPV of each combination by summing the NPVs of its constituents, and finally choose the combination that yields the greatest total NPV.

2 By this we mean that

taking on any one of the

set of projects precludes

us from accepting any of

the others.


15

Finally, we should devote some time to discussion of the ‘interest rate’ we have used to discount future cash flows. Until now we have just referred to r as the rate at which one can borrow or lend funds. A more precise definition of r is that r is the opportunity cost of capital. If we are considering the use of the NPV rule within the context of a firm, we have to recognise that the firm has several sources of capital, and the cost of each of these should be taken into account when evaluating the firm’s overall cost of capital. The firm can raise funds via equity issues and debt issues, and it is likely that the costs of these two types of funds will differ. Later on in this chapter and in those that follow, we will present techniques by which the firm can compute the overall cost of capital for its enterprise.

Other project appraisal techniques

The NPV methodology for project appraisal is by no means the only technique used by firms to decide on their physical investment policy. It is however the optimal technique for corporate management to use if they wish to maximise expected shareholder wealth. This result is obvious from our Fisher separation analysis. In this section we talk about a couple of NPV’s competitors, the payback and internal rate of return (IRR) rules, which are sometimes used in practice.

The payback rule

Payback is a particularly simple criterion for deciding on the desirability of an investment project. The firm chooses a fixed payback period, for example, three years. If a project generates enough cash in the first three years of its existence to repay the initial investment outlay, then it is desirable, and if it doesn’t generate enough cash to cover the outlay, it should be rejected. Take the cash-flow stream given in the following table as an example.

Year 0 1 2 3 4

Cash flow –1,000 250 250 250 500

Table 1.1

A firm that has chosen a payback period of three years and is faced with the project shown in Table 1.1 will reject it as the cash flow in years 1 to 3 (750) doesn’t cover the initial outlay of 1,000. Note, however, that if the firm used a payback period of four years, the project would be acceptable, as the total cash flow to the project would be 1,250, which exceeds the outlay. Hence, it’s clear that the crucial choice by management is of the payback period.

We can also use the preceding example to illustrate the weaknesses of payback. First, assume the firm has a payback period of three years. Then, as previously mentioned, the project in Table 1.1 will not be accepted. However, assume also that, instead of being 500, the project cash flow in year 4 is 500,000. Clearly, one would want to revise one’s opinion on the desirability of the project, but the payback rule still says you should reject it. Payback is flawed, as a portion of the cash-flow stream (that realised after the payback period is up) is always ignored in project evaluation.

The second weakness of payback should be obvious, given our earlier discussion of NPV. Payback ignores the time value of money. Sticking with the example in Table 1.1, assume a firm has a payback period of four years. Then the project as given should be accepted (as total cash flow of 1,250 exceeds investment outlay of 1,000). But what’s the NPV of this project?


16

If we assume, for example, a required rate of return of 10 per cent, then the NPV can be shown to be negative. (In fact the NPV is –36.78. As a self-assessment activity, show that this is the case.) Hence application of the payback rule tells us to accept a project that would decrease expected shareholder wealth (as shown by application of the NPV rule). This flaw could be eliminated by discounting project cash flows that accrue within the payback period, giving a discounted payback rule, but such a modification still wouldn’t solve the first problem we highlighted.

The internal rate of return criterion

The IRR rule can be viewed as a variant on the apparatus we used in the NPV formulation. The IRR of a project is the rate of return that solves the following equation:

0=(Cir

Ii

k

11!= +

!) i (1.7)

where Ci is the project cash flow in year i, and I is the initial (i.e. year 0)

investment outlay. Comparison of equation 1.7 with 1.6 shows that the project IRR is the discount rate that would set the project NPV to zero. Once the IRR has been calculated, the project is evaluated by comparing the IRR to a predetermined required rate of return known as a hurdle rate. If the IRR exceeds the hurdle rate, then the project is acceptable, and if the IRR is less than the hurdle rate it should be rejected. A graphical analysis of this is presented in Figure 1.5, which plots project NPV against the rate of return used in NPV calculation. If r* is the hurdle rate used in project evaluation, then the project represented by the curve on the figure is acceptable as the IRR exceeds r*. Clearly, if r* is also the correct required rate of return, which would be used in NPV calculations, then application of the IRR and NPV rules to assessment of the project in Figure 1.5 gives identical results (as at rate r* the NPV exceeds zero).

Figure 1.5

Calculation of the IRR need not be straightforward. Rearranging equation 1.7 shows us that the IRR is a solution to a kth order polynomial in r. In general, the solution must be found by some iterative process, for example, a (progressively finer) grid search method. This also points to a first weakness of the IRR approach; as the solution to a polynomial, the IRR may not be unique. Several different rates of return might satisfy equation 1.7; in this case, which one should be used as the IRR? Figure 1.6 gives a graphical example of this case.


17

Figure 1.6

The graphical approach can also be used to illustrate another weakness of the IRR rule. Consider a firm that is faced with a choice between two mutually exclusive investment projects (A and B). The locus of NPV-rate of return pairings for each of these projects is given on Figure 1.7.

The first thing to note from the figure is that the IRR of project A exceeds that of B. Also, both IRRs exceed the hurdle rate, r*. Hence, both projects are acceptable but, using the IRR rule, one would choose project A as its IRR is greatest. However, if we assume the hurdle rate is the true opportunity cost of capital (which should be employed in an NPV calculation), then Figure 1.7 indicates that the NPV of project B exceeds that of project A. Hence, in the evaluation of mutually exclusive projects, use of the IRR rule may lead to choices that do not maximise expected shareholder wealth.

Figure 1.7

The lesson of this section is therefore as follows. The most commonly used alternative project evaluation criteria to the NPV rule can lead to poor decisions being made under some circumstances. By contrast, NPV performs well under all circumstances and thus should be employed.


18

Using present-value techniques to value stocks and

bonds

To end this chapter, we will discuss very briefly how to value common stocks and bonds through the application of present-value techniques.

Stocks

Consider holding a common equity share from a given corporation. To what does this equity share entitle the holder? Aside from issues such as voting rights, the share simply delivers a stream of future dividends to the holder. Assume that we are currently at time t, that the corporation is infinitely long-lived (such that the stream of dividends goes on forever) and that we denote the dividend to be paid at time t+i by D

t+i. Also

assume that dividends are paid annually. Denoting the required annual rate of return on this equity share to be r

e, then a present value argument

would dictate that the share price (P) should be defined by the following formula:

P(Dt+irei

!

11= "

= + ) i. (1.8)

Note that in the above representation we have assumed that there is no dividend paid at the current time (i.e. the summation does not start at zero). In plain terms, what equation 1.8 says is that an equity share is worth only the discounted stream of annual dividends that it delivers.

A simplification of the preceding formula is available when we assume that the dividend paid grows at constant percentage rate g per annum. Then, assuming that a dividend of D

0 has just been paid, the future stream of

dividends will be D0(1+g), D

0(1+g)2, D

0(1+g)3 and so on. This type of cash-

flow stream is known as a perpetuity with growth, and its present value can be calculated very simply.3 In this setting the price of the equity share is:

+1

reD0 g( )

P = – g . (1.9)

This is the Gordon growth model of equity valuation. As is obvious from the preceding discussion, it is only valid if you can assert that dividends grow at a constant rate.

Note also that if you have the share price, dividend just paid and an estimate of dividend growth, you can rearrange equation 1.9 to give the required rate of return on the stock – that is:

re+1D0 g( )P

= + g. (1.10)

The first term in 1.10 is the expected dividend yield on the stock, and the second is expected dividend growth. Hence, with empirical estimates of the previous two quantities, we can easily calculate the required rate of return on any equity share.

Activity

Attempt the following questions:

1. An investor is considering buying a certain equity share. The stock has just paid a dividend of £0.50, and both the investor and the market expect the future dividend to be precisely at this level forever. The required rate of return on similar equities is 8 per cent. What price should the investor be prepared to pay for a single equity share?

3 See Appendix 1.


19

2. A stock has just paid a dividend of $0.25. Dividends are expected to grow at a constant annual rate of 5 per cent. The required rate of return on the share is 10 per cent. Calculate the price of the stock.

3. A single share of XYZ Corporation is priced at $25. Dividends are expected to grow at a rate of 8 per cent, and the dividend just paid was $0.50. What is the required rate of return on the stock?

Bonds

In principle, bonds are just as easy to value.

A discount or zero coupon bond is an instrument that promises to pay the bearer a given sum (known as the principal) at the end of the instrument’s lifetime. For example, a simple five-year discount bond might pay the bearer $1,000 after five years have elapsed.

Slightly more complex instruments are coupon bonds. These not only repay the principal at the end of the term but in the interim entitle the bearer to coupon payments that are a specified percentage of the principal. Assuming annual coupon payments, a three-year bond with principal of £100 and coupon rate of 8 per cent will give annual payments of £8, £8 and £108 in years 1, 2 and 3.

In more general terms, assuming the coupon rate is c, the principal is P and the required annual rate of return on this type of bond is r

b, the price

of the bond can be written as:4

p 1( )+c(cPrbi

k

11!= +

+–1

(1+=PB ) i ) krb. (1.11)

Note that it is straightforward to value discount bonds in this framework by setting c to zero.

Activity

Using the previous formula, value a seven-year bond with principal $1,000, annual coupon rate of 5 per cent and required annual rate of return of 12 per cent. (Hint: the use of a set of annuity tables might help.)

A reminder of your learning outcomes

Having completed this chapter, and the essential reading and activities, you should be able to:

analyse optimal physical and financial investment in perfect capital markets and derive the Fisher separation result

justify the use of the NPV rules via Fisher separation

compute present and future values of cash-flow streams and appraise projects using the NPV rule

evaluate the NPV rule in relation to other commonly used evaluation criteria

value stocks and bonds via NPV.

4 In our notation a

coupon rate of 12

per cent, for example,

implies that c = 0.12;

the discount rate used

here, rb, is called the

yield to maturity of the

bond.


20

Key terms

capital market line (CML)

consumption

Fisher separation theorem

Gordon growth model

indifference curve

internal rate of return (IRR) criterion

investment policy

net present value rule

payback rule

production opportunity frontier (POF)

production possibility frontier (PPF)

time value of money

utility function

Sample examination questions

1. The Toyundai Motor Company has the opportunity to invest in new production line equipment, which would have a working lifetime of 10 years. The new equipment would generate the following increases in Toyundai’s net cash flows.

In the first year of usage the new plant would decrease costs by $200,000. For the following 6 years the cost saving would fall at a rate of 5 per cent per annum. In the remaining years of the equipment’s lifetime, the annual cost saving would be $140,000. Assuming that the cost of the equipment is $1,000,000 and that Toyundai’s cost of capital is 10 per cent, calculate the NPV of the project. Should Toyundai take on the investment? (15%)

2. Describe two methods of project evaluation other than NPV. Discuss the weaknesses of these methods when compared to NPV. (10%)

Chapter 2: Risk and return: mean–variance analysis and the CAPM

21

Chapter 2: Risk and return:

mean–variance analysis and the CAPM

Aim of the chapter

The aim of this chapter is to derive the capital asset pricing model (CAPM) enabling us to price financial assets. In order to do so, we introduce the mean–variance analysis setting, in which investors care solely about financial assets’ expected returns and variances of returns, as well as the statistical tools enabling us to calculate portfolios’ expected returns and variances of returns.

Learning objectives


discuss concepts such as a portfolio’s expected return and variance as well as the covariance and correlation between portfolios’ returns

calculate portfolio expected return and variance from the expected returns and return variances of constituent assets

describe the effects of diversification on portfolio characteristics

derive the CAPM using mean–variance analysis

describe some theoretical and practical limitations of the CAPM.

Essential reading

Grinblatt, M. and S. Titman Financial Markets and Corporate Strategy. (Boston, Mass.; London: McGraw-Hill, 2002) Chapters 4 and 5.

Further reading

Brealey, R., S. Myers and F. Allen Principles of Corporate Finance. (Boston, Mass.; London: McGraw-Hill, 2008) Chapters 8 and 9.

Copeland, T. and J. Weston Financial Theory and Corporate Policy. (Reading, Mass.; Wokingham: Addison-Wesley, 2005) Chapters 5 and 6.


Introduction

In Chapter 1 we examined the use of present-value techniques in the evaluation of physical investment projects and in the valuation of primitive financial assets (i.e. stocks and bonds). A key input into NPV calculations is the rate of return used in the construction of the discount factor but, thus far, we have said little regarding where this rate of return comes from. Our objective in this chapter is to demonstrate how the risk of a given security or project impacts on the rate of return required from it and hence affects the value assigned to that asset in equilibrium.

We begin by introducing the basic statistical tools that will be needed in our analysis, these being expected values, variances and covariances. This leads to an analysis of the statistical characteristics


22

of portfolios of financial assets and ultimately to a presentation of the standard mean–variance optimisation problem. The key result of mean–variance analysis is known as two-fund separation, and this result underlies the capital asset pricing model, which we will present next.

Statistical characteristics of portfolios

A portfolio is a collection of different assets held by a given investor. For example, an American investor may hold 100 Microsoft shares and 650 shares of Bethlehem Steel and therefore holds a portfolio comprising two assets. The objective of this section is to arrive at the statistical characteristics of the return on the entire portfolio, given the statistical features of each of the constituent assets. The key statistical measures used are expected returns and return variances or standard deviations. The expected return on a given asset can be thought of as the reward gained from holding it, whereas the return variance is a measure of total asset risk.

Let us define notation. First, we should clarify the way in which we are thinking about asset returns. The return on an asset is assumed to be a random variable with known distributional characteristics. Each individual asset is assumed to have an expected return of E(r

j) and return variance

2j. Assets i and j are assumed to have covariance

ij . Similarly, we denote

the expected return of the portfolio held as E(Rp) and its variance by 2

P.

Finally, we assume that an investor can pick from N different stocks when forming his portfolio.

Returning to the example of the American investor given above, assume that the market price of Microsoft shares is 130 and that of Bethlehem Steel is 10.1 Hence, given the numbers of each share held, the total value of this investor’s portfolio is $195. We further assume that the expected returns on Microsoft and Bethlehem Steel are 10 per cent and 16 per cent respectively, whereas their variances are 0.25 and 0.49.

We are now in a position to define the share of the entire portfolio value that is contributed by each individual stockholding. These are referred to as portfolio weights. The portfolio weight of Bethlehem Steel, for example, is simply the value of the Bethlehem Steel holding divided by $195 (i.e. 1/3 or approximately 33.3 per cent). Hence our US investor allocates 1/3 of every dollar invested to Bethlehem Steel stock.

Activity

Calculate the portfolio weight for Microsoft, using the method presented above.

From the calculations undertaken it is clear that the sum of portfolio weights must be unity. Each portfolio weight represents the share of total portfolio value contributed by a given asset. Obviously, aggregating these shares across all assets held will give a result of unity. Hence, extending the notation presented above, we denote the portfolio weight on asset i by ai, and the preceding argument implies that 1= 1.

Our American investor now knows the statistical characteristics of the return on each of the assets she holds, plus how to calculate the portfolio weight on each of the assets. What she would really like to know now is how to construct the return characteristics for the entire portfolio (i.e. she’s concerned about the risk and reward associated with her entire investment). In order to do this we will need to introduce some basic properties of expectations, variances and covariances.

1 These prices are in US

cents.


23

Expectations, variances and covariances

Consider two random variables, x and y. The expected values and variances of these variables are E(x), E(y), 2

x and 2

y. The covariance

between the random variables is xy.

Form an arbitrary linear combination of these two random variables and denote it P (i.e. P = ax + by, where a and b are constants). We wish to know the expected return and variance of the new random variable P. These are calculated as follows:

E(P) = aE(x) + bE(y) (2.1)

2P = a

2 2x + b

2 2y + 2ab

xy. (2.2)

The preceding results are readily extended to the case where more than two random variables are linearly combined. Consider N random variables denoted x

i, where i runs from 1 to N. Denote their expected values and

variances as E(xi) and 2

i. The covariance between x

i and x

j is

ij. Again

we form a linear combination of the random variables, denoted again by P, using an arbitrary set of constants denoted a

i. The expected value and

variance of the random variable P are given by:

Ei

N

1=!

=

a1 x1( )E(P) (2.3)

= +i j!!i

N

1!=

"P2 " i2ai2 ai aj " ij. (2.4)

Given that the returns on individual assets are assumed to be random variables with known distributional characteristics, the statistical results given above allow us to calculate portfolio returns and variances very simply.

In addition to the data on Microsoft and Bethlehem Steel provided earlier, we also need to know the covariance between Microsoft and Bethlehem Steel returns in order to determine the statistical characteristics of portfolios of these two assets. However, rather than using covariances, we shall work throughout the rest of this analysis with correlation coefficients. The relationship between correlations and covariances is given below.

Covariances and correlations

Assume two random variables, x and y, with variances denoted by 2x and

2y. The covariance between the random variables is

xy. The correlation

coefficient is defined as follows:

! =!xy

xy !y!x, (2.5)

that is, the correlation between the two random variables is simply the covariance, divided by the product of the respective standard deviations. Clearly, knowledge of the correlation and the variances of the two random variables allows one to retrieve the covariance between the two random variables.

If we again define a linear combination of the two random variables, P, using arbitrary constants a and b, the expression for the variance of the linear combination can be rewritten using the correlation as follows:

2

p = a

2 2

x + b

2 2

y + 2ab

xy x y. (2.6)

This is a straightforward substitution of equation 2.5 into equation 2.2.

Now we are in a position to calculate the characteristics of our American investor’s portfolio. Let us take the simplest possible case first and assume that the returns are uncorrelated (i.e.

xy = 0). Recalling that the portfolio

weights on Microsoft and Bethlehem Steel are 23 and 13 respectively, we


24

can use equations 2.1 and 2.6 to derive the expected return and variance of the investor’s portfolio. These calculations yield:

Rp)E( 23= (0.1) + 1

3 (0.16) = 0.12 = 12 % (2.7)

23 = 16.6 %( (2 (0.25) 1

3( (2+ (0.49)!P2 = . (2.8)

Hence, as we would expect, the expected portfolio return lies between the returns on the individual assets. The portfolio variance, however, is actually less than that on the return of either of the component assets (i.e. the risk associated with the portfolio is lower than the risks associated with either individual asset). This result is one that should be kept in mind and is the focus of the next section.

Now let’s change our assumption regarding the correlation between the two asset returns. Assume now that

xy = 0.5. Obviously, the expected

portfolio return won’t change (as equation 2.1 doesn’t involve the correlation or covariance at all). The portfolio variance now becomes:

23= ( (2 (0.25) 1

3( (2+ (0.49) 13( (+ 2 2

3( (! 0.5 !0.5!0.7 = 24.3%!P2 . (2.9)

The portfolio variance has obviously increased, although it is still less than the return variances of either component assets.

Activity

Assume that xy = – 0.5. Calculate the portfolio return variance in this case, using the

data on portfolio weights and asset return variances given above.

Now, given the expected returns, return variances and covariances for any set of assets, we should be able to calculate the expected return and variance of any portfolio created from those assets. At the end of this chapter, you will find activities that require you to do precisely this, along with solutions to some of these activities.

Diversification

A point that we noted from the calculations of expected portfolio returns and variances above was that, in all of our calculations, the variance of the portfolio return was lower than that on any individual component’s asset return.2 Hence, it seems as though, by forming bundles of assets, we can eliminate risk. This is true and is known as diversification: through holding portfolios of assets, we can reduce the risk associated with our position.

Why is this the case? The key is that, in our prior analysis and in real stock return data, the correlations between returns are less than perfect. If two returns are imperfectly correlated it implies that when returns on the first are above average, those on the second need not be above average. Hence, to an extent, the returns on such assets will tend to cancel each other out, implying that the return variance for a portfolio of these stocks will be smaller than the corresponding weighted average of the individual asset variances.

To illustrate this point in a general setting, consider the following scenario. An investor holds a portfolio consisting of N stocks, with each stock having the same portfolio weight (i.e. each stock has portfolio weight N-1). Denote the return variances for the individual assets by 2

i where i = 1 to N, and

the covariance between returns on assets i and j by ij. Using equation 2.4,

the variance of the investor’s portfolio return can be written as:

1=i j!!i

N

1!=

1+"P2 N 2 "i2 N 2"i j. (2.10)

2 Note that this result

does not hold in general

(i.e. it may be the case

that the return variance

of a portfolio exceeds

the return variance of

one of the component

assets).


25

Examining the second term of equation 2.10, the existence of N component assets implies that the summation for all i not equal to j involves N(N – 1) terms. Obviously the summation in the first term of 2.10 involves N terms. Hence, defining the average variance of the N assets as –2 and average covariance across all assets as C, 2.10 can be rewritten as:

N!P2 = N 2 + N ( (N – 1 C! 2N 2 . (2.11)

Equation 2.11 obviously simplifies to the following:

1 + C1N1 –( (!P2 = N 2 ! 2 . (2.12)

Now we ask the following question. How does the portfolio variance change as the number of assets combined in the portfolio increases towards infinity (i.e. N ). It is clear from 2.12 that, as the number of assets held increases, the first term will shrink towards zero. Also, as N increases the second term in 2.12 tends towards C. Together, these observations imply the following.

1. The portfolio variance falls as the number of assets held increases.

2. The limiting portfolio return variance is simply the average covariance between asset returns: this average covariance can be thought of as the risk of the market as a whole, with the influence of individual asset return variances disappearing in the limit.

The moral of the preceding statistical story is clear. Holding portfolios consisting of greater and greater numbers of assets allows an investor to reduce the risk he or she bears. This is illustrated diagrammatically in Figure 2.1.

Figure 2.1

Mean–variance analysis

In the preceding two sections, we have demonstrated two important facts:

1. The expected return on a portfolio of assets is a linear combination of the expected returns on the component assets.

2. An investor holding a diversified portfolio gains through the reduction in portfolio variance, when asset returns are not perfectly correlated.

In this section, we use these facts to characterise the optimal holding of risky assets for a risk-averse agent. Our fundamental assumption is that all agents have preferences that only involve their expected portfolio return and return variance. Utility is assumed to be increasing in the former and decreasing in the latter. For illustrative purposes we begin using the assumption that only two risky assets are available. The results presented, however, generalise to the N asset case.


26

To begin, assume there is no risk-free aset. The investor can hence only form his or her portfolio from risky assets named X and Y. These assets have expected returns of E(R

x) and E(R

y) and return variances of 2

x and 2

y.

The first question the investor wishes to answer is how the characteristics of a portfolio of these assets (i.e. portfolio expected return and variance) change as the portfolio weights on the assets change. Given equation 2.6, the answer to this question is obviously dependent on the correlation between the returns on the two assets.

First assume the assets are perfectly correlated and, further, assume asset X has lower expected returns and return variance than asset Y. We form a portfolio with weights on asset X and 1 on asset Y. Equation 2.6 then implies that the portfolio variance can be written as follows:

2

P = (

x + (1 )

y)2. (2.13)

Taking the square root of equation 2.13, it is clear that the portfolio standard deviation is linear in . As the portfolio expected return is linear in , the locus of expected return–standard deviation combinations is a straight line. This is shown in Figure 2.2.Figure 2.2

If the correlation between returns is less than unity, however, the investor can benefit from diversifying his portfolio. As previously discussed, in this scenario, portfolio standard deviation is not a linear combination of

x

and y. The reduction of portfolio risk through diversification will imply

that the mean–standard deviation frontier bows towards the y-axis. This is also shown on Figure 2.2. The final curve on Figure 2.2 represents the case where returns are perfectly negatively correlated. In this situation, a portfolio can be constructed, which has zero standard deviation.

Activities

1. Assuming asset returns are perfectly negatively correlated, use equation 2.6 to find the portfolio weights that give a portfolio with zero standard deviation. (Hint: write down 2.6 with the correlation set to minus one and a and b . Then minimise portfolio variance with respect to .)

2. Assume that the returns on Microsoft and Bethlehem Steel have correlation of 0.5. Using the data provided earlier in the chapter, construct the mean–variance frontier for portfolios of these two assets. Start with a portfolio consisting only of Microsoft stock and then increase the portfolio weight on Bethlehem Steel by 0.1 repeatedly, until the portfolio consists of Bethlehem Steel stock only.


27

From here on we will assume that return correlation is between plus and minus one. The expected return–standard deviation locus for this case is redrawn in Figure 2.3. In the absence of a risk-free asset, this locus is named the mean–variance frontier. As our investor’s preferences are increasing in expected return and decreasing in standard deviation, it is clear that his or her optimal portfolio will always lie on the frontier and to the right of the point labelled V. This point represents the minimum-variance portfolio. He or she will always choose a frontier portfolio at or to the right of V, as these portfolios maximise expected return for a given portfolio standard deviation. In the absence of a risk-free asset, this set of portfolios is called the efficient set.

Figure 2.3

We can now, given a set of preferences for the investor, find his or her optimal portfolio. The condition characterising the optimum is that an investor’s indifference curve must be tangent to the mean–variance frontier.3 Two such optima are identified on Figure 2.3 at R and S. The investor locating at equilibrium point R is relatively risk-averse (i.e. his or her indifference curves are quite steep), whereas the equilibrium at S is that for a less risk-averse individual (with correspondingly flatter indifference curves). Figure 2.3 also shows sub-optimal indifference curves for each set of preferences.

Figure 2.4

3 In technical terms the

optimum is characterised

by the marginal rate of

substitution being equal

to the marginal rate of

transformation (i.e. the

slope of the indifference

curve equals the slope of

the frontier).


28

Hence, as Figure 2.3 demonstrates, in a world of two risky assets and no risk-free asset, the optimal portfolio of risky assets held by an investor depends on his or her preferences towards risk and return. The same is true when there are N risky assets available. Figure 2.4 depicts the same type of diagram for the N asset case.

Note that the mean–variance frontier is of the same shape as that in Figure 2.3. However, unlike the two-asset case, the interior of the frontier now consists of feasible but inefficient portfolios (i.e. those that do not maximise expected return for given portfolio risk). The mean–variance frontier now consists of those portfolios that minimise risk for a given expected return, whereas those portfolios on the efficient set (i.e. on the frontier but to the right of V) additionally maximise expected return for a given level of risk.

We now reintroduce a risk-free asset to the analysis (i.e. we assume the existence of an asset with return r

f and zero return–standard deviation).

A key question to address at this juncture is as follows. Assume that we form a portfolio consisting of the risk-free asset and an arbitrary combination of risky assets. How do the expected return and return–standard deviation of this portfolio alter as we vary the weights on the risk-free asset and the risky assets respectively?

Denote our arbitrary risky portfolio by P. We combine P with the risk-free asset using weights 1 – a and a to form a new portfolio Q. The expected return and variance of Q are given by:

E(RQ) = (1 – a)r

f + aE(R

P) = r

f + a[E(R

P) – r

f ] (2.14)

2Q = a

2 2P . (2.15)

In order to analyse the variation in the risk and expected return of the portfolio Q with respect to changes in the portfolio weights, we construct the following expression:

dE(RQ) dE(RQ)

/da/da

d!Q= d!Q

. (2.16)

Using equations 2.14 and 2.15 we find that:

dE(RQ) E(Rp) – rfd!Q

= !p. (2.17)

As this slope is independent of a, the risk–return profile of the portfolio Q is linear. This is known as the capital market line, and two such CMLs are shown in Figure 2.5 for two different portfolios of risky assets.

We now have all the components required to describe the optimal portfolio choice of an investor faced with N risky assets and a risk-free investment. Figure 2.6 replots the feasible set of risky asset portfolios. The key question to answer is, what portfolio of risky assets should an investor hold? Using the analysis from Figure 2.5, it is clear that the optimal choice of risky asset portfolio is at K. Combining K with the risk-free asset places an investor on a capital market line (labelled r

fKZ), which dominates in utility

terms the CML generated by the choice of any other feasible portfolio of risky assets.4 The optimal portfolio choice and a sub-optimal CML (labelled CML

2) are shown on Figure 2.6 along with the indifference curves of two

investors.

Recall that we previously defined the efficient set as the group of portfolios that both minimised risk for a given level of expected return and

4 That is, choosing

portfolio K places an

investor on a CML with

greater expected returns

at each level of return

variance than does any

other.


29

maximised expected return for a given level of risk. With the introduction of the risk-free asset, the efficient set is exactly the optimal CML.

Figure 2.5

The key result that is depicted in Figure 2.6 is known as two-fund separation. Any risk-averse investor (regardless of his or her degree of risk-aversion) can form his or her optimal portfolio by combining two mutual funds. The first of these is the tangency portfolio of risky assets, labelled K, and the second is the risk-free asset. All that the degree of risk-aversion dictates is the portfolio weights placed on each of the two funds. The investor with the optimum depicted at X on Figure 2.6, for example, is relatively risk-averse and has placed positive portfolio weights on both the risk-free asset and K. An investor locating at Y, however, is less risk-averse and has sold the risk-free asset short in order to invest more in K.5

Figure 2.6

Two-fund separation is the result that underlies the capital asset pricing model (CAPM), which is developed in the next section.

5 A short sale is the sale

of an asset that one

does not actually own.

One borrows the asset

in order to complete the

transactions and imme-

diately receives the sale

price. Subsequently, one

uses the proceeds from

the sale to repurchase

a unit of the asset, and

deliver it to the creditor.

If the price of the asset

has dropped in the

interim, one makes a


30

The capital asset pricing model

To begin our derivation of the CAPM, we present the assumptions that underlie the analysis. These assumptions formalise those implicit in the preceding section.

Investors maximise utility defined over expected return and return variance.

Unlimited amounts may be borrowed or loaned at the risk-free rate.

Investors have homogenous expectations regarding future asset returns.

Asset markets are perfect and frictionless (e.g. no taxes on sales or purchases, no transaction costs and no short sales restrictions).

We next need to extend slightly our analysis of the previous section in order to derive the familiar form of the CAPM.

A mathematical characterisation of mean–variance optimisation

Consider Figure 2.6, which graphically identifies the optimal portfolio of risky assets (K), held by an arbitrary risk-averse investor. The key condition for optimality is that the capital market line and the mean–variance frontier are tangent. The following equations give a mathematical description of this optimality condition.

From equation 2.17, we know that the slope of the capital market line at the optimum is:

E(RK) – rf

!K. (2.18)

We also need the slope of the mean–variance frontier at the point of tangency. To derive this, consider a position (called I) with portfolio weight a in an arbitrary portfolio of risky assets (called j) and (1 – a) in the optimal portfolio K. The expected return and standard deviation of this position are:

E(RI) = aE(R

j) + (1 – a)E(R

K) (2.19)

1 = [a2 2

j + (1 – a)2 2

K + 2a(1 – a)

jK]0.5. (2.20)

Using the same method as shown in equation 2.16 to derive the risk–return trade-off at the point represented by portfolio I, we get:

dE(R1) E(Rj) – E(RK)da = . (2.21)

d!1da = 0.5[a2! j

2+(1–a)2!K2+2a(1–a)! jK]-0.5 (2a! j

2–2(1–a)!K2+2(1–2a)! jK).

(2.22)

The slope of the mean–variance frontier at K will be the ratio of 2.21 to 2.22 in the limit as a 0. Note that equation 2.21 does not depend on a. Taking the limit of equation 2.22 as a 0 we get:

1 (! jK – !K2)!K

. (2.23)

The slope of the mean–variance frontier at K is the ratio of 2.21 to 2.23, that is,

!K [E (R j) – E (R K) ]

!JK – !K2

. (2.24)


31

The optimum in Figure 2.6 equates the slope of the mean–variance frontier at K with the slope of the CML. Hence, equating 2.18 and 2.24 and rearranging the resulting expression, we arrive at:

E(Rj ) = rf + !jK

!K2 [E(RK ) – rf ]. (2.25)

Defining j jK

2K, equation 2.26 can be rewritten as:

E(Rj) = r

j j[E(R

K) – r

f ]. (2.26)

Equation 2.26 is the standard representation of the mean–variance optimisation problem. The equation translates as follows: the expected return on a given asset (or portfolio of assets) is equal to the risk-free rate plus a risk premium multiplied by the asset’s 6 Assets that have large values of will have large expected returns, whereas those with smaller values of will have low expected returns with defined as the ratio of the covariance of an asset’s returns with those on the market to the variance of the market return.

Equilibrium and the CAPM

Equation 2.26 is simply derived from mean–variance analysis, and as yet we have said nothing regarding equilibrium in asset markets. Capital market equilibrium requires that the demand for risky securities be identical to their supply. The supply of risky assets is summarised in the market portfolio, which is defined below.

Definition

The market portfolio is the portfolio comprising all assets, where the weights used in the construction of the portfolio are calculated as the market capitalisation of each asset divided by the sum of market capitalisations across all assets.

Two-fund separation gives us the fundamental result that all investors hold efficient portfolios and, further, that all investors hold risky securities in the same proportions (i.e. those proportions dictated by the tangency portfolio (K)).7 For demand to be equal to supply in capital markets, it must be the case that the market portfolio is constructed with identical portfolio weights. The implication of this is simple: the market portfolio and the tangency portfolio are identical. This allows us to express the CAPM in the following form.

The capital asset pricing model

Under the prior assumptions, the following relationship holds for all expected portfolio returns:

E(Rj ) = R

f j [E(r

M ) – r

f ], (2.27)

where E(RM ) is the expected return on the market portfolio, and

j is the

covariance of the returns on asset j with those on the market divided by the variance of the market return.

Equation 2.27 gives the equilibrium relationship between risk and return under the CAPM assumptions. In the CAPM framework, the relevant measure of an asset’s risk is its , and 2.27 implies that expected returns increase linearly with risk.

To clarify the source of the CAPM equation, note that the identification of the tangency portfolio and the linear representation are implied by mean–variance analysis. The CAPM then imposes equilibrium on capital markets and identifies the market portfolio as identical to the tangency portfolio.

6 The risk premium is

the expected return on

the tangency portfolio

over the risk-free rate.

7 All investors perceive

and tangency portfolio

due to our assumption

that they have homo-

geneous expectations

regarding asset returns.


32

The security market line

Given equation 2.27, the equilibrium relationship between risk and return has a very simple graphical depiction. In equilibrium expected returns are linear in . The expected return on an asset with a of zero is r

f , whereas

an asset with a of unity has an expected return identical to that on the market. Plotting this relationship, known as the security market line, we get Figure 2.7.

Comparison of Figures 2.6 and 2.7 implies that, in equilibrium, two assets with identical expected returns must have identical s, although their return variances can differ. The reason that their variances can differ is that a proportion of asset return variance can be eliminated through diversification. Agents should not be rewarded for bearing such risk, and hence, diversifiable risk will not affect expected returns. Undiversifiable risk is that which is driven by variation in the return on the market as a whole, and an asset’s exposure to such risk is summarised by . Hence an asset’s measures its relevant risk and, via equation 2.27, determines equilibrium expected returns.

The key message of the preceding paragraph is that measures asset risk. A high asset is risky as it has high returns when market returns are high. An asset with a low tends to have high returns when market returns are low. Hence a low asset, when included in one’s portfolio, can provide insurance against low market returns and hence is low risk.

Figure 2.7

Systematic and unsystematic risk

To mathematically illustrate the sources of asset risk we can use the CAPM equation to decompose the variance of a given asset. Equation 2.27 gives the equilibrium expected return for asset j. Actual returns on asset j will follow a similar relationship but will also include a random error term. Denoting this error by

j we have the following equation:

rj = r

f j [rM – r

f ] +

j. (2.28)

The variance of the risk-free return is zero by definition. Assuming that j

is fixed we can represent the variance of asset j as:

2j

2j

2M

2. (2.29)


33

The final term on the right-hand side of equation 2.29 is the variance of the error term and represents diversifiable risk. This source of risk is also known as unsystematic and idiosyncratic risk. As emphasised previously, this risk is unrelated to market fluctuations and, therefore, does not affect expected returns. The first term on the right-hand side of 2.29 represents undiversifiable risk, also known as systematic risk. This is risk that cannot be escaped and hence increases equilibrium expected returns.

Activities8

1. An investor forms a portfolio of two assets, X and Y. These assets have expected returns of 9 per cent and 6 per cent and standard deviations of 0.8 and 0.6 respectively. Assuming that the investor places a portfolio weight of 0.5 on each asset, calculate the portfolio expected return and variance if the correlation between returns on X and Y is unity.

2. Using the data from Question 1, recalculate the portfolio expected return and variance, assuming that the correlation between returns is 0.5.

3. An investor forms a portfolio from two assets, P and Q, using portfolio weights of one-third and two-thirds respectively. The expected returns on P and Q are 5 per cent and 7 per cent, and their respective return standard deviations are 0.4 and 0.5. Assuming the return correlation is zero, calculate the expected return and variance of the investor’s portfolio.

4. Assuming identical data to that in Question 3, recalculate the statistical properties of the portfolio, assuming the return correlation for P and Q is –0.5.

The Roll critique and empirical tests of the CAPM

The final topic we touch on in this chapter is the empirical validity of the CAPM. The model of equilibrium expected returns that we have developed in the preceding sections of this chapter is obviously not guaranteed to hold in practice, and hence, rather than just blindly accepting its output, we should examine how it holds up when applied to real data. However, this task brings us face-to-face with a problem first pointed out by Richard Roll and hence known as the Roll critique.9

The statement of the CAPM is identical to the proposition that the market portfolio is mean–variance efficient. Hence, Roll pointed out that empirical tests of the CAPM should seek to examine whether this is indeed the case. However, he also noted that the market portfolio (or the return on the market) is not observable to an econometrician, who wishes to conduct a test. Empirical researchers generally use a broad-based equity index such as the FTSE-100, S&P-500 or Nikkei 250 to proxy the market. But the true market portfolio will contain other financial assets (such as bonds and stocks not included in such indices) as well as non-financial assets such as real estate, durable goods and even human capital. Hence, the validity of tests of the CAPM depend critically on the quality of the proxy used for the market portfolio.

Based on the above, Roll’s critique is simply that, due to the fact that the market portfolio is not observable, the CAPM is not testable. We can understand this through the following arguments. First, it might be the case that the market portfolio is efficient (and hence the CAPM is valid), but our chosen proxy for the market is not efficient, and hence our empirical test rejects the CAPM. Second, our proxy for the market might be efficient whereas the market portfolio itself is not. In this case our test

8

solutions to these

activities at the end of

this chapter.

9 See Roll (1977).


34

will falsely indicate that the CAPM is valid. Put simply, the fact that we can’t guarantee the quality of our proxy for the market implies that we can’t place any faith in the results that tests based upon it generate, and hence it’s impossible to test the CAPM.

The Roll critique is clearly damaging in that it implies that we can’t judge the predictions of the CAPM against reality and trust the results. However, many researchers have disregarded the prior discussion and estimated the empirical counterpart of equation 2.27. From these estimates such researchers pass judgment on the CAPM.

One prediction of the CAPM upon which some have focused is that the expected excess return of a given portfolio over the risk-free rate (the risk premium) is linearly related to with the slope of this relationship given by the market risk premium. Historically, however, when one plots actual excess portfolio returns against their estimated s one finds that the line traced out is flatter than one would expect from the theory (i.e. low portfolios earn greater expected returns than the CAPM predicts, and high portfolios earn less). This is clearly evidence against the CAPM.10

Another prediction of the CAPM, which is also empirically tested, is that is the only factor that should cause expected returns to differ (i.e. no

other variable should explain expected returns once we’ve accounted for the effects of ). Again, the evidence from this line of attack is not good for the CAPM. Among other factors, firm size, book-to-market ratios, P/E ratios and dividend yields have all been shown to explain ex-post realised returns (after accounting for ).

Amalgamating the above evidence implies that, if you are willing to disregard the Roll critique, you should probably conclude that the CAPM does not hold. This has led certain authors to investigate other asset-pricing paradigms such as the APT (which we discuss in the next chapter). An alternative viewpoint would be to argue that such results tell us little or nothing about the validity of the CAPM due to the insight of Roll (1977).



discuss concepts such as a portfolio’s expected return and variance as well as the covariance and correlation between portfolios’ returns

calculate portfolio expected return and variance from the expected returns and return variances of constituent assets

describe the effects of diversification on portfolio characteristics

derive the CAPM using mean–variance analysis

describe some theoretical and practical limitations of the CAPM.

Key terms

beta ( )

capital asset pricing model (CAPM)

correlation

covariance

diversification

expected return

10 See pp.185–86 of

Brealey and Myers

(2008).


35

market portfolio

mean–variance analysis

Roll critique

security market line

standard deviation

systematic risk

two-fund separation

unsystematic risk

variance


1. Detail the assumptions that underlie the CAPM and provide a derivation of the CAPM equation. Support your derivation with graphical evidence. (15%)

2. The returns on ABC stock and on the market portfolio in three consecutive years are given in the following table:

Year ABC return Market return

1 8% 6%

2 24% 12%

3 28% 15%

Showing all your workings, compute the for ABC’s equity. (7%)

3. Assume that the risk-free rate is 5 per cent. What is the expected return on ABC’s stock? (3%)

Solutions to activities

1. The expected return on the equally weighted portfolio is 7.5%. The portfolio return variance is 0.49, and hence the portfolio return standard deviation is 0.7.

2. Obviously, the expected return is the same as in Question 1. With correlation of 0.5, the portfolio return variance is 0.37.

3. The expected return on the portfolio is 6.33%, and the portfolio has a return variance of 0.1289.

4. When the correlation changes to –0.5, the portfolio return variance drops to 0.0844. The expected return on the portfolio doesn’t change from that calculated in Question 3.

Notes


36

Chapter 3: The arbitrage pricing theory

37


Aim of the chapter

The aim of this chapter is to derive arbitrage pricing theory, an alternative to the capital asset pricing model, enabling us to price financial assets.

Learning objectives


understand single-factor and multi-factor model representations

derive factor-replicating portfolios from a set of asset returns

understand the notion of arbitrage strategies and that well-functioning financial markets should be arbitrage-free

derive arbitrage pricing theory and calculate expected returns using the pricing formula.

Essential reading

Grinblatt, M. and S. Titman Financial Markets and Corporate Strategy. (Boston, Mass.; London: McGraw-Hill, 2002) Chapter 6.

Further reading

Brealey, R., S. Myers and F. Allen Principles of Corporate Finance. (Boston, Mass.; London: McGraw-Hill, 2008) Chapter 9.

Chen, N-F. ‘Some Empirical Tests of the Theory of Arbitrage Pricing’, The

Journal of Finance 38(5) 1983, pp.1393–1414.Chen, N-F., R. Roll and S. Ross ‘Economic Forces and the Stock Market’, Journal

of Business 59, 1986, pp.383–403.Copeland, T., J. Weston and K. Shastri Financial Theory and Corporate Policy.

(Reading, Mass.; Wokingham: Addison-Wesley, 2005) Chapter 6.

Overview

The arbitrage pricing theory is an alternative paradigm used to calculate equilibrium expected returns on financial assets. As its name suggests, it rests on the notion that well-functioning financial markets should be arbitrage-free. This, using a factor model of asset returns, implies restrictions on the relationships between asset returns and generates an equilibrium pricing relationship.

Introduction

The arbitrage pricing theory (APT) developed in this chapter gives an alternative to the CAPM as a method to compute the expected returns on stocks. The basis for the APT is a factor model of stock returns, and we will define and discuss these models first. From there we will demonstrate how to derive expected returns using the idea that the returns on stocks, which are exposed to a common set of factors, must be mutually consistent, given each stock’s sensitivity to each factor.


38

To give structure to what we mean by ‘mutually consistent’, we need to define the notion of an arbitrage. An arbitrage strategy can be one of two types.

It could involve investment in a set of assets (both buying and selling) that yields an immediate, positive cash inflow (i.e. the receipts from our sales exceed the cost of our purchases) and, further, is guaranteed not to make a loss tomorrow. Faced by an investment strategy with this pay-off structure, any investor who prefers more to less wealth would try to invest on an infinite scale.

It could be an investment strategy that is costless today but guarantees positive future returns. This is akin to receiving something for nothing, and again, sensible investors would capitalise on the possibility by investing as much as possible.

The idea that underpins the APT is that investment situations, such as those described above, should not be permitted in well-functioning financial markets. Then, if financial markets do not permit the existence of arbitrage strategies, this places restrictions on the relationships between the expected returns on assets given the factor structure underlying returns. We will explain further in later sections of this chapter.

Single-factor models

Before using the notion of absence of arbitrage to provide pricing relations, we need a basis for the generation of stock returns. Within the context of the APT, this basis is given by the assumption that the population of stock returns is generated by a factor model. The simplest factor model, given below, is a one-factor model:

ri =

i +

i i E(

i) = 0. (3.1)

In equation 3.1, the returns on stock i are related to two main components:

1. The first of these is a component that involves the factor F. This factor is posited to affect all stock returns, although with differing sensitivities. The sensitivity of stock i’s return to F is

i. Stocks that have small values

for this parameter will react only slightly as F changes, whereas when i

is large, variations in F cause very large movements in the return on stock i. As a concrete example, think of F as the return on a market index (e.g. the S&P-500 or the FTSE-100), the variations in which cause variations in individual stock returns. Hence, this term causes movements in individual stock returns that are related. If two stocks have positive sensitivities to the factor, both will tend to move in the same direction.

2. The second term in the factor model is a random shock to returns, which is assumed to be uncorrelated across different stocks. We have denoted this term

i and call it the idiosyncratic return component for stock i. An

important property of the idiosyncratic component is that it is also assumed to be uncorrelated with F, the common factor in stock returns. In statistical terms we can write the conditions on the idiosyncratic component as follows: Cov(

i j) = 0 i j Cov(

i, F) = 0 i

An example of such an idiosyncratic stock return might be the unexpected departure of a firm’s CEO or an unexpected legal action brought against the company in question.

The partition of returns implied by equation 3.1 implies that all common variation in stock returns is generated by movements in F (i.e. the correlation between the returns on stocks i and j derives solely from F). As the idiosyncratic components are uncorrelated across assets they do not bring about covariation in stock price movements.


39

Application exercise

Consider an economy in which the risk-free rate of return is 4% and the expected rate of return on the market index is 9%. The variance of the return on the market index is 20%. Two portfolios A and B have expected return 7% and 10%, and variance 20% and 50%, respectively.

a) Work out the portfolios’ beta coefficients.

According to the CAPM:

E(rA) = rF A [E(rM) – rF]

and

E(rB) = rF B [E(rM) – rF].

Hence:

A = [E(rA) – rF]/[E(rM) – rF

B = [E(rB) – rF]/[E(rM) – rFb) The risk of a portfolio can be decomposed into market risk and idiosyncratic

risk. What are the proportions of market risk and idiosyncratic risk for the two portfolios A and B?

From the market model:

rA A A rM A

rB B B rM B

with cov(rM A) = cov(rM B) = 0.

It hence follows that the variance of portfolio A’s returns, 2A, has two

components, systematic and idiosyncratic risk:2A

2A

2M

2A.

Similarly:2B

2B

2M

2B.

The proportion of systematic risk for A is hence

2A

2M

2A

2

The proportion of idiosyncratic risk for A is hence

2A

2M

2A] = 64%.

The proportion of systematic risk for B is hence 2B

2M

2B = (1.2)

2*20%/50% = 58%.

The proportion of idiosyncratic risk for B is hence

2B

2M

2B

Portfolio B is much riskier than portfolio A as the variance of its returns is 50% compared with 20% for A. The main reason why it is riskier is that it is much more sensitive to the return of the market index than portfolio A as its beta is 1.2 compared with 0.6 for portfolio A.

c) Assume the two portfolios have uncorrelated idiosyncratic risk. What is the covariance between the returns on the two portfolios?

The returns of portfolios A and B are hence (positively) correlated even though their idiosyncratic return components are not. These returns are positively correlated because they are positively correlated with the returns of the market index.

Cov(rA,rB A A rM A B B rM B A B 2M


40

Multi-factor models

A generalisation of the structure presented in equation 3.1 posits k factors or sources of common variation in stock returns.

ri =

i + 1iF1 + 2iF2 + .... + kiFk + i E(

i) = 0. (3.2)

Again the idiosyncratic component is assumed uncorrelated across stocks and with all of the factors. Further, we’ll assume that each of the factors has a mean of zero. These factors can be thought of as representing news on economic conditions, financial conditions or political events. Note that this assumption implies that the expected return on asset i is just given by the constant in equation 3.2 (i.e. E(r

i)

i). Each stock has a complement

of factor sensitivities or factor betas, which determine how sensitive the return on the stock in question is to variations in each of the factors.

A pertinent question to ask at this point is how do we determine the return on a portfolio of assets given the k-factor structure assumed? The answer is surprisingly simple: the factor sensitivities for a portfolio of assets are calculable as the portfolio weighted averages of the individual factor sensitivities. The following example will demonstrate the point.

Example

The returns on stocks X, Y, and Z are determined by the following two-factor model:

rX = 0.05 + F1 – 0.5F2 +

X

rY = 0.03 + 0.75 F1 + 0.5F2 +

Y

rz

F1 – 0.3F2 + z

Given the factor sensitivities in the prior three equations, we wish to derive the factor structure followed by an equally weighted portfolio of the three assets (i.e. a portfolio with one-third of the weights on each of the assets). Following the result mentioned above, all we need to do is form a weighted average of the stock sensitivities on the individual assets. Subscripting the coefficients for the equally weighted portfolio with a p we have:

p = 13

1p = 13 (1 + 0.75 – 0.25) = 0.5

2p = 13 (–0.5 + 0.5 –0.3) = –0.1;;

and hence; the factor representation for the portfolio return can be written as:

rp

F1 – 0.1F2 + pwhere the final term is the idiosyncratic component in the portfolio return.

Activity

Using the data given in the previous example, compute the return representation for a portfolio of assets X, Y and Z with portfolio weights –0.25, 0.5 and 0.75.

An important implication of the result is the following. Assume a two-factor model, and also assume that we are given the factor representations for three stocks. I can construct a portfolio of these three assets, which has any desired set of factor sensitivities through appropriate choice of the portfolio weights.1 What underlies this result? Well, to illustrate let’s use the data from the prior example. Assume I wish to construct a portfolio with a sensitivity of 0.5 on the first factor and a sensitivity of 1 on the second factor. Denoting the portfolio weights on the individual assets by

X,

Y and

Z it must be the case that:

1 In general, if I have

a k-factor model I will

need k+1 stocks to

do this.


41

X + 0.75

Y – 0.25

Z = 0.5 (3.3)

–0.05X + 0.5

Y – 0.3

Z = 1. (3.4)

Finally, it must also be the case that the portfolio weights add up to unity, so we must also satisfy the following equation:

X +

Y +

Z = 1.

Equations 3.3, 3.4 and 3.5 are three equations in three unknowns, and we can find values for the portfolio weights which satisfy all three simultaneously. This illustrates the fact that (as the portfolio factor sensitivities were arbitrarily set at 0.5 and 1) we can derive any constellation of factor sensitivities. A particularly interesting case is when the portfolio is sensitive to one of the factors only. We call this a factor-replicating portfolio and discuss it below.

Broad-based portfolios and idiosyncratic returns

In what follows we will assume that the basic securities that we’re going to work with are themselves broad-based portfolios. The reason for this is that it allows us to lose the idiosyncratic risk terms associated with single stocks. Why is this the case? Well, consider the idiosyncratic risk term for an equally weighted portfolio of 100 stocks. Call the ith idiosyncratic term

i and assume that all idiosyncratic terms have variance 2. The variance

of the idiosyncratic element of the portfolio return is then:

Var (!P) = Var (!i

i

100

1"

= 100) = 110000 Var (

i

100

1"

=!i) = 100

10000 = # 2 = #2

100.

Note that, under these assumptions the variance of the idiosyncratic portfolio return is only one-hundredth of the variance of any individual asset’s idiosyncratic return. In a general case, where one forms an equally weighted portfolio of n assets, the variance of the idiosyncratic term for the portfolio return is n-1 2. This is a diversification result just like those we used in Chapter 2. The fact that the idiosyncratic returns are uncorrelated with one another means that their influence tends to disappear when one groups assets into large portfolios.

Factor-replicating portfolios

An important application of the technology developed previously in this chapter is the construction of factor-replicating portfolio. A factor-replicating portfolio is a portfolio with unit exposure to one factor and zero exposure to all others. For example, the portfolio replicating factor 1 in model 3.2 would have 1 = 1 and

j = 0 for all j = 2 to k.

Activity

Assume that stock returns are generated by a two-factor model. The returns on three well-diversified portfolios, A, B and C, are given by the following representations:

rA = 0.10 F1 – 0.5F2

rB = 0.08 + 2F1 + F2

rC = 0.05 + 0.5F1 + 0.5F2.

Determine the portfolio weights you need to place on A, B and C in order to construct the two factor-replicating portfolios plus a portfolio which has zero exposure to both factors. What are the expected returns of the factor-replicating portfolios and what is the expected return of the risk-free portfolio?


42

The question to ask at this point is: why bother constructing factor-replicating portfolios? The reason is as follows. Suppose I want to build a portfolio that has identical factor exposures to a given asset, X. Assume a two-factor world and that asset X has exposure of 0.75 to factor 1 and –0.3 to factor 2. Assume also that I know the two factor-replicating portfolios.

Building a portfolio with the same factor exposures as X is now simple. Construct a new portfolio, Y, which has portfolio weight 0.75 on the replicating portfolio for the first factor, portfolio weight –0.3 on the replicating portfolio for the second factor and the rest of the portfolio weight (i.e. a weight of 1 – 0.75 + 0.3 = 0.55) on the risk-free asset. Via the results on the factor representations of a portfolio of assets and the definition of a factor-replicating portfolio it is easy to see that Y is guaranteed to have identical factor exposures to X.

The replication in the preceding paragraph forms the basis for the APT. For absence of arbitrage we require all assets with identical factor exposures to earn the same return. If they did not, then we would have the chance to make unlimited amounts of money. For example, assume that the expected return on the replicating portfolio Y was greater than that on asset X. Then I should short X and buy Y. The risk exposures of the two portfolios are identical and hence risks cancel out and I am left with an excess return that is riskless (i.e. an arbitrage gain).

In order to progress, let us introduce some notation. Denote the risk-free rate with r

f. Denote the expected return on the ith factor-replicating

portfolio with rf + i such that

i is the risk premium associated with the

ith factor. Again, for simplicity, assume that the world is generated by a two-factor model, and assume that I wish to replicate asset X, which has sensitivity 1X to the first factor and 2X to the second factor. Finally, we will assume that the primary securities being worked with are well-diversified portfolios themselves. Hence, we will ignore any idiosyncratic risk in this derivation.

Using the prior argument, to replicate asset X’s factor sensitivities, we construct a portfolio with weight 1X on the first factor-replicating portfolio, weight 2X on the second factor-replicating portfolio and weight 1 – 1X – 2X on the risk-free asset. The expected return of the replicating portfolio is hence:

1X (rf + 1) + 2X (rf + 2) + (1 – 1X – 2X) rf = rf + 1X 1+ 2X 2. (3.6)

Hence, using our factor-replicating portfolios we can write the expected return on a portfolio which replicates X’s factor exposures as the risk-free rate plus each factor exposure multiplied by the risk premium on the relevant factor-replicating portfolio.

The arbitrage pricing theory

Consider an arbitrary asset. The previous sub-section tells us that it’s simple to replicate this asset’s risk (i.e. its factor exposures) using factor-replicating portfolios. The key to the APT is that absence of arbitrage requires that such a pair of portfolios must have identical expected returns in a financial market equilibrium. If they did not, it would be possible to make unlimited amounts of money without incurring any risk.

This implies that the expected return on asset X, rX, must be identical to

the expression arrived at in equation 3.6, that is:

E(rX) = rf + 1X 1+ 2X 2. (3.7)


43

Equation 3.7 is the statement of the APT. The expected return on a financial asset can be written as the risk-free rate plus sum of the asset’s factor sensitivities multiplied by the factor-risk premiums (which are invariant across assets). If such an expression does not hold at all times, arbitrage opportunities exist. Note the assumptions that are required to achieve this result. First, we require that asset returns are generated by a two-factor (or in general k-factor) model. Second, we assume that arbitrage opportunities cannot exist. Lastly, we assume that enough assets are available such that firm-specific risk washes away when portfolios are formed.

Example

In the previous two-factor example, we determined the expected returns on the two factor-replicating portfolios. Denoting the expected return on the ith factor-replicating portfolio by E(r

i) we have:

E(r1

E(r2) = 1.71% E(r

3

Hence, the premiums associated with the two factors are:

1 2

This implies that the expected return on any asset in this world can be written as:

E(ri 1i 2i.

To check that this works, substitute portfolio C’s (for example) factor sensitivities into the preceding expression. This gives:

E(rC

and hence, agrees with the expected return implied by the original representation for asset C. Check that the expected returns on assets A and B also come out correctly.

To analyse an arbitrage opportunity that might arise in markets, attempt the following Activity.

Activity

Assume that a new well-diversified portfolio, D, is added to our world. This asset has sensitivities of 3 and –1 to the two factors and an expected return of 15 per cent.

Using the equilibrium expected return equation given above, derive the equilibrium expected return on an asset with identical factor exposures to D. Is there now an arbitrage opportunity available? If so, dictate a strategy that could be employed to exploit the arbitrage opportunity.

Summary

The APT gives us a straightforward, alternative view of the world from the CAPM. The CAPM implies that the only factor that is important in generating expected returns is the market return and, further, that expected stock returns are linear in the return on the market. The APT allows there to be k sources of systematic risk in the economy. Some may reflect macroeconomic factors, like inflation, and interest rate risk, whereas others may reflect characteristics specific to a firm’s industry or sector.

Empirical research has indicated that some of the well-known empirical problems with the CAPM are driven by the fact that the APT is really the proper model of expected return generation. Chen (1983), for example, argues that the size effect found in CAPM studies disappears in a multi-factor setting. Chen, Roll and Ross (1986) argue that factors representing default spreads, yield spreads and GDP growth are important in expected return generation. Work in this area is still progressing.


44



understand single-factor and multi-factor model representations

derive factor-replicating portfolios from a set of asset returns

understand the notion of arbitrage strategies and that well-functioning financial markets should be arbitrage-free

derive arbitrage pricing theory and calculate expected returns using the pricing formula.

Key terms

arbitrage pricing theory

factor-replicating portfolio

factor sensitivity

multi-factor model

single-factor model

Sample examination question

1. Assume that stock returns are generated by a two-factor model. The returns on three well-diversified portfolios, A, B and C, are given by the following representations:

rA = 0.10 + F1

rB = 0.08 + 2F1 – F2

rC = 0.05 – 0.5F1 + 0.5F2

a) Discuss what the factor representations above imply for the variation and comovement in the three stock returns. Show how the returns of the stocks should be correlated between themselves.

b) Find the portfolio weights that one must place on stocks A, B and C to construct pure tracking portfolios for the two factors (i.e. portfolios in which the loading on the relevant factor is +1 and the loadings on all other factors are 0).

c) If one was to introduce a new portfolio, D, with loadings of +1 on both of the factors, what would the expected return on D have to be to rule out arbitrage?

d) Explain the concepts of idiosyncratic risk and factor risk in the APT. What role does diversification play in the APT?

Chapter 4: Derivative assets: properties and pricing

45

Chapter 4: Derivative assets: properties

and pricing

Aim of the chapter

The aim of this chapter is to introduce and price derivatives. As in the previous chapter on APT, the valuation of derivatives relies on a no riskless arbitrage argument.

Learning objectives


discuss the main features of the most widely traded derivative securities

describe the pay-off profiles of such assets

understand absence-of-arbitrage pricing of forwards, futures and swaps

construct bounds on option prices and relationships between put and call prices

price options in a binomial framework using the portfolio replicating and the risk-neutral valuation methods.

Essential reading

Grinblatt, M. and S. Titman Financial Markets and Corporate Strategy. (Boston, Mass.; London: McGraw-Hill, 2002) Chapters 7 and 8.

Further reading

Brealey, R., S. Myers and F. Allen Principles of Corporate Finance. (Boston, Mass.; London: McGraw-Hill, 2008) Chapters 21, 22 and 23.

Copeland, T., J. Weston and K. Shastri Financial Theory and Corporate Policy. (Reading, Mass.; Wokingham: Addison-Wesley, 2005) Chapters 8 and 9.

Overview

A derivative asset is one whose pay-off depends entirely on the value of another asset, usually called the underlying asset. In the last 20 years, traded volume in these assets has increased tremendously. Derivatives are widely used for hedging purposes by financial institutions and are also used for speculative purposes. In this chapter we discuss the most commonly traded types of derivative. We go on to introduce the underlying principles of derivative pricing. We devote the final section of the chapter to a more detailed description of the features and pricing of options.

Varieties of derivatives

Forwards and futures

Perhaps the oldest type of derivative asset is the simple forward contract. A forward is an agreement between two parties (called A and B) and has the following features.

1. Party A agrees to supply party B with a specified amount of a specified asset, k periods in the future.


46

2. In return, party B agrees to pay party A $F (the forward price) when the goods are received.

Party A is said to hold a short position in the contract and party B a long position.

Hence, the forward is just an agreement made today to undertake a given transaction at some specified future date, known as the settlement date. Currency and commodities are often traded using forwards, the advantage of such transactions being that they allow an agent to remove any price uncertainty regarding a transaction that must be undertaken in the future.

Example

Assume that party B is American and that in three months he must pay ¥250,000 for a Japanese machine he has purchased. Party B enters into a contract to buy yen three months forward. Party A (the agent who is to supply the yen) specifies that the cost of ¥100 will be $1.20. The total price that party B must pay in three months is therefore $3,000.

Closely related to forward contracts are futures contracts. In fact, futures are refined versions of forwards. Although forwards are generally bilaterally negotiated between two parties directly, futures are standardised forward contracts that are exchange traded. The contracts give precise specifications for the quality and quantity of the assets to be exchanged. The major difference between futures and forwards is in the exchange of monies involved. With a forward, the agent who is long pays the entire forward price at the settlement date. Futures positions, however, are marked to market. This occurs on a daily basis and means that any increases/decreases in the value of the future are received/paid by the party who is long day by day. At the settlement date, the current spot price of the asset is transferred from the agent who is long to the agent who is short.1

Futures are traded on exchanges such as the London International Financial Futures and Options Exchange (LIFFE), the Chicago Board of Trade (CBOT) and the Chicago Mercantile Exchange (CME). Contracts with very high volumes include those on government bonds, interest rates and stock indices.

Options

The option is a less straightforward type of derivative. Although the forward or future contract implies an obligation to trade once the contract is entered into, the option (as its name suggests) gives the agent who is long a right but not an obligation to buy or sell a given asset at a pre-specified price. This price is known as the exercise price and is specified in the option contract. Just as with the forward, another factor specified in the contract is the date on which the exchange is to take place. If, on the maturity date, the holder of an option decides to buy or sell in line with the terms of the contract, he or she is said to have exercised his or her right. A big difference between options and forwards is that, with an option, the agent who is long must pay a price (or premium) at the outset. This is essentially a price paid by the holder for the exercise choice he or she faces at maturity.

Options to buy the specified asset are called call options. Options to sell are called puts. Another distinction is made on the timing of the exercise decision. With European options, the right can only be exercised on the

1 See pp.236–40 in

Grinblatt and Titman

(2002).


47

maturity date itself. With American options, in contrast, the option can be exercised on any date at or before maturity. American options are traded far more frequently than their European counterpart, but for reasons of simplicity, we will focus on the European variety.

Example

A 12-month European call option on IBM has exercise price $45. It gives me the right to purchase IBM stock in one year at a cost of $45 per share. In line with the prior discussion, I am under no obligation to buy at $45 such that, if the market price were less than this amount, I could choose not to exercise and buy in the market instead.

Swaps

Swaps are another type of derivative, which do exactly what their name says. Two counterparties agree to exchange (or swap) periodic interest payments on a given notional amount of money (the notional principal) for a given length of time.

A very common type of swap involves an exchange of interest payments based on a market-determined floating rate (such as the London InterBank Offer Rate (LIBOR)) for those calculated on a fixed-rate basis. Another frequently traded variety of swap involves the exchange of interest payments in different currencies. For example, fixed sterling interest payments may be exchanged for fixed dollar interest payments.2

Derivative asset pay-off profiles

For now we are going to concentrate on forwards and options. As mentioned above, futures are closely related to forwards, and their pricing is based on the technique presented below. The relationship between forwards and swaps will be made clear later.

Before getting on to the principles of derivative pricing, let us take a look at the pay-off profiles of the basic forward and option contracts. The pay-off profile of a long forward position is shown in Figure 4.1. In the figure, F is the price agreed upon in the forward contract, and S is the spot price of the asset at the settlement date. Note that the pay-off profile is linear, positive for values of S greater than F and negative when S is less than F.

Figure 4.1

2 The notional principal

is not exchanged in an

interest rate swap (they

would net out anyway)

but are generally

exchanged in currency

swaps.


48

Understanding the forward pay-off is simple. If the spot price for the asset at maturity exceeds the forward price, then the party that is long has gained by entering into the forward (i.e. he’s got the asset for a lower price than it would have cost if bought in the spot market). If the spot price at maturity is lower than the forward price, then the long pay-off is negative, as it would have been cheaper for the long party to buy the asset in the spot market rather than entering into the forward. Obviously, the pay-off of a short forward position is the negative of that shown in Figure 4.1.

Let’s now consider the pay-off to a holder of a European call option. This is given in Figure 4.2 where the option’s exercise price is labelled X. Remember that a call option gives the holder the right but not the obligation to purchase the asset. What occurs when the price of the spot asset at maturity exceeds the exercise price of the option? Well it is cheaper to buy the asset using the option than in the spot market; hence the option is exercised, and the holder makes a gain of the spot price less the exercise price. When the spot price is lower than the exercise price, then the holder would find it cheaper to buy the asset at spot and hence does not exercise the option. The pay-off to the holder is then zero.

Figure 4.2

The pay-off to the holder of a European put is given in Figure 4.3. As the put gives the holder the right to sell the underlying asset, the holder gains when the exercise price exceeds the spot price and has a zero pay-off when the spot price at maturity is greater than or equal to the exercise price.

Figure 4.3


49

Each option must have one agent who is long and one who is short, with the pay-offs to the long position given in Figures 4.2 and 4.3. An agent who is short is said to have written the option, and his or her pay-offs are the negative of those given above. Note that an agent with a long option position never has a negative pay-off, whereas an agent who has written an option never has a positive pay-off at maturity. The option price, paid at the outset by the agent who is long to that who is short, is the compensation to the writer of the option for holding a position that exposes him or her to weakly negative cash flows.

The key to pricing options, and other derivative assets, is constructing a portfolio of assets that is priced in the market and that has a pay-off structure identical to that of the derivative. As the derivative and replicating portfolio have identical pay-offs, absence-of-arbitrage arguments imply that the cost of these portfolios must be identical. The no-arbitrage price of the derivative is hence just the initial investment cost needed to set up the replicating portfolio.

Pricing forward contracts

In the case of a forward contract, the derivation of the no-arbitrage price is quite simple.3 Assume the current spot asset price is S0 and that the one-period, riskless rate of interest is r. We wish to value a k period forward contract. It is easily verified that the k-period forward price (F

k) is given by

the following expression:

Fk = S0(1 + r)

k. (4.1)

Why is this the case? Well, consider the following pair of investment strategies.

The first is simply a long position in the forward contract. This costs nothing at the present time and yields S

k – F

k at maturity.

The second strategy involves buying a unit of the asset at spot and borrowing F

k(1+r)–k at the risk-free rate for k periods. The k period pay-

off of this strategy is also Sk – F

k, and its net current cost is S0 – Fk(1+r)

–k.

The pay-offs of the two strategies are identical. This implies that the two investments should have identical costs. As the cost of investment in the forward is zero, this implies that the following condition must hold:

S0 – Fk(1+r)–k = 0. (4.2)

Rearranging equation 4.2 we derive the no-arbitrage price for the k period forward contract, which is precisely that given in equation 4.1.

Activity

The current value of a share in Robotronics is $12.50.

1. The one-year riskless rate is 6 per cent. What are the prices of three- and five-year forward contracts on Robotronics stock?

2. Three-year forward contracts are currently being sold for $16 in the market. Outline an investment strategy that could take advantage of the opportunities this presents.

Some of the most active forward markets are those for foreign currency. The forward pricing analysis above, however, is suited only for assets valued in the domestic currency (e.g. individual stocks or stock indices). To illustrate the pricing of currency forwards, consider the following

3 Given the similarities

discussed previously,

we can also use the

derived forward price to

approximate the price of

a futures contract.


50

analysis. A domestic investor (assumed to be located in the UK such that the domestic currency is £) is assumed to face a spot exchange rate of S and a k period forward rate of F

k. These rates are constructed as the

domestic currency price of one unit of foreign currency (i.e. the spot rate implies an exchange rate of £S for $1). The one-period domestic interest rate is denoted r and its foreign counterpart r

f .

Again, let us compare two investment strategies that can be undertaken assuming an investor currently holds £S. The first involves depositing this cash in a domestic risk-free account for k periods. This yields £ S(1+r)k at the maturity date of the investment. Alternatively, the investor could swap his or her sterling for dollars at the spot exchange rate and invest the funds at the US rate. As his or her £S is equivalent to $1 at the spot exchange rate, this investment yields $ (1+r

f)k in k periods. The investor can then sell

the proceeds for sterling using a forward contract yielding £ Fk(1+r

f)k.

Note that both of these investments are riskless, assuming that the interest rates are known and fixed and given that the spot and forward exchange rates are known at the current date. Further, both investments cost £S. This implies that the pay-offs from the two strategies should be identical. Equating these returns we get:

S(1 + r)k = Fk(1 + r

f )k. (4.3)

Rearranging equation 4.3, we get the no-arbitrage k period currency forward price:

1 + r1 + rf

k

Fk = S ( (. (4.4)

Note the simple generalisation of equation 4.1 implicit in equation 4.4. The gross interest rate in equation 4.1 is just replaced by the ratio of domestic to foreign rates in equation 4.4. In the international finance literature, the currency forward rate expression in 4.4 is known as the covered interest rate parity relationship.

Activity

The current spot exchange rate is £0.64 = $1. The riskless rate in the UK is currently 6 per cent and that in the US is 4 per cent. Using equation 4.4, derive the implied five- and 10-year forward exchange rates.

Binomial option pricing setting

Pricing options is far less straightforward than pricing forwards. To begin, however, we introduce a binomial setting, in which the pricing of options turns out to be surprisingly straightforward.

In order to make things as simple as possible, let us consider a binomial setting in which all derivatives last only for one period (starting today and ending tomorrow). Let us denote the current price of the underlying asset by S

0. Let us assume that uncertainty in this world is represented by the

price of the underlying asset, taking one of two values tomorrow.4 If the state of the world is good, the price of the asset will rise tomorrow to S

H,

with SH = (1+u)S

0 and u>0. In contrast, if the state of the world is bad, the

price of the underlying asset will decrease to SL,, with S

L = (1–d)S

0 and d>0.

Let us now consider a one-period derivative asset. If the state of the world is good tomorrow, then the derivative will pay K

H, and if the state of the

world is bad tomorrow the derivative will pay KL. Finally we assume that

the one-period risk-free interest rate is rf (i.e. a safe bond costing one unit

of currency pays 1+ rf units of currency tomorrow).

4 This is where the term binomial

comes from in the name of our

method.


51

In order to price this derivate asset, we will consider two different methods:

the portfolio replicating method

the risk-neutral valuation method.

The portfolio replicating method

The portfolio replicating method prices the derivative asset using absence-of-arbitrage arguments. First, this necessitates constructing a portfolio, containing the underlying asset and the risk-free asset, that has identical pay-offs to the derivative. Assume we purchase a units of the underlying asset and b units of the risk-free asset.

If the state of the world tomorrow is good then the value of our portfolio will be:

aSH + b(1 + r

f ), (4.5)

when the pay-off of the derivative is KH. If the state tomorrow is bad the

portfolio is worth:

aSL + b(1 + r

f ), (4.6)

and the derivative is worth KL. Note that equating the value of the

portfolio with the pay-off of the derivative in each state of the world gives us two equations in two unknowns (a and b). These unknowns are our initial holdings of the underlying and the risk-free asset. Solving the two equations gives us precisely the portfolio weights we need to use to replicate the option pay-off in both states of nature. This yields:

a =KH – KLSH – SL

. (4.7)

and

b =SLKH – SHKL

(1 + rf )(SL – SH). (4.8)

We now know how to construct a portfolio, which has a pay-off profile that replicates that of the derivative (i.e. regardless of the state of the world, the portfolio and the derivative have the same value). If two assets have identical pay-offs then absence-of-arbitrage arguments tell us that the price/cost of the two assets must be identical. The cost of the replicating portfolio is aS

0 + b. It hence follows that:

K0 = aS0 + b. (4.9)

A practical example of how this technique might work for a European call option is given below.

Example

A one-period European call option on ABC stock has an exercise price of 120. The current price of ABC stock is 100, and if things go well, the price in the following period will be 150. If things go badly over the coming period, the future price will be 90. The risk-free rate is 10 per cent. What is the no-arbitrage price of this option?

First we need to know the option pay-offs. In the bad state it pays zero, as the underlying price is less than the exercise price. In the good state it pays the excess of the underlying price over the exercise price (i.e. 30).

Next we construct the replicating portfolio. Using equations 4.3 and 4.4, the quantities of the underlying and risk-free asset we must buy are 0.5 and –40.91 (i.e. we buy half a unit of stock and short 40.91 units of the risk-free asset).5 This portfolio replicates the option pay-off, and therefore the option price is given by the cost of constructing the portfolio. The call price (c) is hence:

c =

5You should check

all these calculations

and further check that

the portfolio we’ve

constructed does indeed

replicate the option

pay-off.


52

Activity

Using the stock price data from the previous example, price a European put option on ABC stock with a strike price of 100.

The Risk-neutral valuation method

Using the portfolio replicating method, we find that the current price of the derivative asset, relative to the current price of the underlying asset, does not depend on the probability that the state of nature will be good (or bad) tomorrow. Neither does it depend on investor risk preferences. The reason for this is that information about probabilities or risk aversion is already captured by the current price of the underlying asset on which we base our valuation of the derivative asset. The fact that the no-arbitrage price of the derivative asset in relation to the price of the underlying asset is the same, regardless of risk preferences, serves as a basis for a neat trick also known as the risk valuation method.

The risk-neutral valuation method is a procedure involving the following steps.

1. Identifying the risk-neutral probabilities, that is, the probabilities which are consistent with investors being risk-neutral. These probabilities are the probabilities for which the current price of the underlying asset is the present value of tomorrow’s asset prices, with the discount rate being equal to the risk-free rate.

2. Calculating the current price of the derivative asset as the present value of tomorrow’s derivative values using the risk-neutral probabilities derived in the previous step and the risk-free rate as the discount rate.

Step 1: Obtaining risk-neutral probabilities

Let us denote the risk-neutral probability that the state of nature will be good tomorrow by q. It hence follows that:

S0 =qSH + (1 – q)SL

1 + rf . (4.10)

Equivalently, the risk-neutral probability q is given by the following identity:

q =S0(1 + rf ) – SL

SH – SL

rf + du + d

= . (4.11)

Step 2: Calculating the current price of the derivative asset

The current price of the derivative asset can be expressed as the present value of tomorrow’s derivative values using the risk-neutral probabilities in equation 4.11 and the risk-free rate as the discount rate:

K0 =qKH + (1 – q)KL

1 + rf . (4.12)

After substituting q from equation 4.11, we obtain:

K0 =(d + rf )KH + (u – rf )KL

(1 + rf )(u + d). (4.13)

Activity

Using the risk-neutral valuation method, price both a European call option and a European put option on the ABC stock (introduced in the previous example) with a strike price of 100.


53

Activity

Show that the current price of the derivative obtained from the portfolio replicating method in equation 4.9 is the same as the one obtained from the risk-neutral valuation method in equation 4.13.

Comments on the binomial option pricing setting

The risk-neutral valuation method is very efficient at pricing multiple derivative assets on the same underlying asset as the same risk-neutral probabilities can be used to price all the derivatives. In these circumstances, the portfolio replicating method is more tedious to use as the replicating portfolio will typically be different for each derivative asset.

The assumptions we have made above may seem very restrictive. We have restricted tomorrow’s price to take one of two values and assumed that derivatives last only for one period. Extending the above model to more than one period is straightforward, and this allows longer maturity instruments to be priced. Also, we can shrink the length of time that we have referred to as one period. It could represent one day, one hour or one minute if we wanted. A binomial model for hourly prices, for example, may be thought more reasonable than a binomial model for annual prices. Then, using a multi-period derivative valuation we could price a one-month option from a binomial model of hourly stock returns. The binomial structure is not as restrictive as you might think.

Bounds on option prices and exercise strategies

The binomial model allows us to derive option prices under certain assumptions on the behaviour of the price of the underlying asset. In this section we present some arguments that place bounds on European option prices and can be made without specification of a model for the underlying price. In order to link up with the following section (on Black–Scholes prices), we will present our arguments using a continuously compounded risk-free rate, r. We assume unlimited borrowing and lending at this rate along with our standard frictionless market assumptions of no transaction costs and taxes. Finally, we also assume that the underlying asset pays out no cash during the option lifetime (such that the option can’t be written on dividend paying stock or coupon bonds, for example).

Upper bounds on European option prices

A call option is the right (but not the obligation) to purchase a unit of a specified asset for price X. It should be obvious to you then that the option can never be worth more than the stock. Hence, denoting the call option price by c we have:

. (4.14)

As a European put gives the holder the right to sell a given quantity of an asset for X, the put can never be worth more than X. Denoting the put price by p we then have:6

. (4.15)

Further, if the put is European, we know that the value at maturity is at most X. If there are T periods to maturity, a present-value argument then implies that:

–rT. (4.16)

6 Clearly both this and

the previous argument

hold for American

options as well as

European options.


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Lower bounds on European option prices

No-arbitrage arguments can be simply employed to develop lower bounds for European puts and calls. A lower bound for a European call option price is given by:

–rT (4.17)

where X is the exercise price, and there are T periods to maturity. To show this, consider the following argument. Assume I hold two portfolios. Portfolio A consists of a European call option struck at price X, plus cash of the amount Xe–rT. Portfolio B consists of the underlying stock.

Assume I invest the cash from portfolio A at the risk-free rate. This implies that, when the option in portfolio A matures, I have cash worth X. If at maturity the underlying price (S

T) exceeds the exercise price, then I

exercise the call option using my cash, and the portfolio is worth ST. If at

maturity the underlying price is less than X, I do not exercise the option, and hence my portfolio is worth X. The value of portfolio A at maturity can be written as:

max(ST,X).

At the maturity date the value of portfolio B is always just ST. Hence,

portfolio A is always worth at least the same as portfolio B, and sometimes (when exercise is not optimal) it is worth more. Reflecting this and to prevent arbitrage, the price of buying portfolio A must exceed the cost of portfolio B. This reasoning implies:

c + Xe–rT > S c > S – Xe

–rT. (4.18)

Also, an option must have positive value since, at the very worst, it is not exercised as it is out of the money. This implies that 4.18 can be generalised to:

[0,S – Xe–rT]. (4.19)

A similar argument to the above can be used to establish a lower bound on the price of a European put. It’s easy to show that:

p > Xe–rT – S. (4.20)

To demonstrate this, consider two more portfolios. Portfolio 1 consists of a European put and a unit of the underlying stock, and portfolio 2 consists of Xe–rT in cash.

At the date at which the put matures, portfolio 1 is worth either X (if it’s profitable to exercise the put, and hence you sell the unit of the underlying for X) or S

T (when exercise isn’t optimal and you’re left with the stock, as

the put expires with zero value). We can then write the value of portfolio 1 as:

max(X,ST).

Portfolio 2 is always worth X at the date when the put matures and is hence weakly dominated in pay-off terms by portfolio 1. Therefore, to prevent arbitrage, portfolio 1 should cost more to set up than portfolio 2, implying:

p + S > Xe–rT p > Xe

–rT – S. (4.21)

Finally, again we know that the worst that can happen for a put option is for it to expire, worth nothing. This implies that its value must exceed zero in all circumstances. Thus:

[0,Xe–rT – S]. (4.22)


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Combining upper and lower bounds

A combination of the upper and lower bounds derived in the preceding two sections can be formed graphically. This gives a set of permissible (in the sense of not admitting arbitrage) put and call prices. As an example, Figure 4.4 shows the permissible call price region (it is the shaded area of the diagram).

Figure 4.4

Black–Scholes option pricing

Our previous pricing analysis was predicated on the assumption that stock prices are well-represented by a discrete time, binomial model. In 1974, Fischer Black and Myron Scholes presented an option pricing formula, based on a continuous time process for the stock price. This analysis gave exact prices for European puts and calls using a continuous time version of the replication strategy followed in our binomial methodology. Unfortunately, derivation of their pricing formula is beyond the scope of the current presentation. However, due to its wide use in the financial markets and the intuition it brings regarding the determinants of option prices, we will describe the pricing formula below.

Assume we wish to price a European call on a stock that never pays dividends. The current price of the stock is S, the exercise price of the option under consideration is denoted X, and the option is to have a maturity of T periods. The continuously compounded risk-free rate is denoted r. One final parameter is needed to calculate the B–S price of the call option. This is the instantaneous volatility of the stock price, and we denote this parameter . It is the standard deviation of the change in the logarithm of the stock price.

The famous B–S formula for the price of a European call option is given below:

c = SN(d1) – Xe–rT N(d2) (4.23)

where

d1= 1n (S / Xe!rT ) + !"T!"T 2

(4.24)

d2= d1 – !"T (4.25)

and N(.) represents the cumulative normal distribution function.7

7 The values of the

cumulative standard

normal distribution

function can be found

in tables in the back

of any good statistical

textbook.


56

Example

The current price of Glaxo Wellcome shares is £2.88. An investor writes a two-year call option on Glaxo with exercise price £3.00. If the annualised, continuously compounded interest rate is 8 per cent, and the volatility of Glaxo’s stock price is 25 per cent, what is the B–S option price?

First we need to derive the values d1 and d

2 defined as above. Using 4.24 and 4.25

these are 0.5139 and 0.1603. The values of the cumulative normal distribution function at 0.5139 and 0.1603 are 0.696 and 0.564. Then, plugging all the available data into equation 4.23 yields a call price of £0.5644.

What does equation 4.23 tell us about the determinants of call prices? Well, there are clearly a number of influences on the price of an option, and these are summarised below.

The effect of the current stock price: the B–S equation tells us that call option prices increase as the current spot asset price increases. This is pretty unsurprising as a higher underlying price implies that the option gives one a claim on a more valuable asset.

The effect of the exercise price: again, as you would expect, higher exercise prices imply lower option prices. The reason for this is clear: a higher exercise price implies lower pay-offs from the option at all underlying prices at maturity.

The effect of volatility: Figure 4.2 gives the pay-off function of a European call option. Note that, although extremely good outcomes (underlying price very high) are rewarded highly, extremely bad outcomes are not penalised due to the kink in the option pay-off function. This would imply that an increase in the likelihood of extreme outcomes should increase option prices, as large pay-offs are increased in likelihood. The B–S formula verifies this intuition, as it shows that call prices increase with volatility, and increased volatility implies a more diverse spread of future underlying price outcomes.

The effect of time to maturity: call option prices increase with time to maturity for similar reasons that they increase with volatility. As the horizon over which the option is written increases, the relevant future underlying price distribution becomes more spread-out, implying increased option prices. Furthermore, as the time to maturity increases, the present value of the exercise that one must pay falls, reinforcing the first effect.

The effect of riskless interest rates: call option prices rise when the risk-free rate rises. This is due to the same effect as above, in that the discounted value of the exercise price to be paid falls when rates rise.

Put–call parity

The B–S formula gives us a closed-form solution for the price of a European call option under certain assumptions on the underlying asset price process. However, as yet, we have said nothing about the pricing of put options. Fortunately, a simple arbitrage relationship involving put and call options allows us to do this. This relationship is known as put–call parity. In what follows we assume the options have the same strike price (X), time to maturity (T) and are written on the same underlying stock.

Consider an investment consisting of a long position in the underlying asset and a put option, called portfolio A. The cost of this position is S0 + p. A second portfolio, denoted B, comprises a long position in a call option and lending Xe–rT. Hence the cost of this position is c + Xe–rT.


57

What are the possible pay-offs of these positions at maturity? Given the pay-off structure on the put shown in Figure 4.3, the pay-off on portfolio A can be written as follows:

max[X – ST,0] + S

T = max[X,S

T]. (4.26)

Similarly, the pay-off on portfolio B can be written as:

max[0,ST – X] + X = max[X,S

T]. (4.27)

Comparison of equations 4.26 and 4.27 implies that the two portfolios always pay identical amounts. Hence, using no-arbitrage arguments, portfolios A and B must cost the same amount. Equating their costs we have:

S + p = c + Xe–rT. (4.28)

Equation 4.28 is the put–call parity relationship. Given the price of a call, the value of the underlying asset and knowledge of the riskless rate, we can deduce the price of a put. Similarly, given the put price, we can deduce the price of a call with similar features.

Example

A call option on BAC stock, with an exercise price of £3.75, costs £0.25 and expires in three years. The current price of BAC stock is £2.00. Assuming the continuously compounded (annual) risk-free rate to be 10 per cent, calculate the price of a put option with three years to expiry and exercise price of £3.75.

From equation 4.28 we have:

p = c + Xe–rT – S.

Plugging in the data we’re given yields:

p = 0.25 + 3.75e–0.1(3) – 2 = 1.03.

Hence, the no-arbitrage put price is £1.03.

Substitution of the B–S call pricing equation gives a closed-form solution for the put price. This equation allows us to deduce the effects of changing the B–S parameters on put prices.

The effect of underlying price: for the opposite reason to that given for the call, put prices drop as underlying prices increase.

The effect of the exercise price: similarly, put prices rise as exercise prices rise.

The effect of volatility: put options and call options are affected in identical ways by volatility. Hence, as volatility increases, put prices rise.

The effects of time to maturity: increased time to maturity will lead to a greater dispersion in underlying prices at maturity, and hence put prices should be pushed higher. However, as the holder of a put receives the exercise price, discounting at higher rates makes puts less valuable. The combined effect is ambiguous.

The effect of the risk-free rate: puts are less valuable as interest rates rise, due to a greater degree of discounting of the cash received.

Activity

ABC corporation’s shares currently sell at $17.50 each. The volatility of ABC stock is 15 per cent. Given a risk-free rate of 7 per cent, price a European call with strike price of $15 and time to maturity 5 years. Use put-call parity to price a put with similar specifications. What are the no-arbitrage prices of the call and the put if the risk-free rate rises to 10 per cent?


58

Pricing interest rate swaps

Recall the definition of an interest rate swap given earlier in the chapter. Agent A contracts to give fixed interest payments (on a given principal) to agent B. In return agent B agrees to deliver to agent A interest payments (on the same principal) based on an agreed floating exchange rate. The frequency and duration of these interest payments are also agreed in advance. A very common choice of floating interest rate used in such contracts is the London Interbank Offer Rate (LIBOR).

An example of such an agreement is as follows. Agent A agrees to pay agent B payments on a $1m principal at a fixed 8 per cent rate. Agent B agrees to pay interest payments of LIBOR plus 0.25 per cent. These payments are to be made annually for the next 10 years.

Note that, from the previous example, the payments made by agent A at every date till maturity are known and fixed (i.e. 8 per cent of $1 m). His or her receipts, however, are uncertain. He or she gains a 0.25 per cent premium above an ex-ante uncertain interest rate. Consider, for example, the transaction at the second payment date. Agent A pays $50,000 and receives LIBOR + 0.25 per cent. This looks identical to the cash flows from a forward contract. Indeed, we can regard the transaction at every payment date as a forward transaction. Hence the swap in entirety can be considered a package of forwards. Using the forward pricing equations given above, the swap is simply priced.

In the situation where interest payments in different currencies are exchanged, the situation is slightly more murky, but the same basic principle maintains. Swaps are just packages of forward contracts and can be priced as such.

Summary

This chapter has treated the nature and pricing of the most important and heavily traded derivative securities. We have looked at the basic specifications of forward, futures, option and swap contracts and what these specifications imply for the pay-off functions of long and short positions. Further, we have looked at methods that can be used to price these securities. The basis of pricing is absence of arbitrage in all cases. We looked most deeply at option contracts, detailing the relationships between put, and call prices, and bounds on option prices, and finally we examined the continuous time option pricing formula of Black and Scholes.

Although we’ve covered a lot of material here, the continual evolution and innovation of derivatives markets and assets means that we missed much more than we’ve treated. However, the basic features of derivatives pricing that we’ve looked at can be extended to new and more complex securities.



discuss the main features of the most widely traded derivative securities

describe the pay-off profiles of such assets

understand absence-of-arbitrage pricing of forwards, futures and swaps

construct bounds on option prices and relationships between put and call prices

price options in a binomial framework using the portfolio replicating and the risk-neutral valuation methods.


59

Key terms

American option

binomial method

Black–Scholes

call option

covered interest rate parity relationship

derivative

European option

exercise price

forward contract

futures contracts

long position

marked-to-market

notional pricing

put option

risk-neutral method

settlement date

short position time to maturity

underlying price


1. Describe the main features of forward and futures contracts. How do forward and futures contracts differ? Derive the no-arbitrage price of a forward contract. (10%)

2. Describe the main features of options contracts. Show how to price a standard European call option using a single-period binomial model. (10%)

3. British Telecom shares are currently trading at 312p. Historically, the (annualised) volatility of BT shares has been 20 per cent. Compute the Black–Scholes price of a European call on BT equity, assuming a strike price of 350p and time to maturity of six months. Assume that the risk-free rate is 5 per cent. (5%)

Notes


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