Corporate Fi Nance

8/3/2019 Corporate Fi Nance

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Undergraduate study in

Economics, Management,

Finance and the Social Sciences

This subject guide is for a Level 3 course (also known as a ‘300 course’) offered aspart of the University of London International Programmes in Economics, Management,Finance and the Social Sciences. This is equivalent to Level 6 within the Framework forHigher Education Qualifications in England, Wales and Northern Ireland (FHEQ).

For more information about the University of London International Programmesundergraduate study in Economics, Management, Finance and the Social Sciences, see:www.londoninternational.ac.uk/ current_students/ programme_resources/ lse/ index.shtml

Corporate finance

P. Frantz, R. Payne, J. Favilukis

FN3092, 2790092

2011



This guide was prepared for the University of London International Programmes by:

Dr. P. Frantz, Lecturer in Accountancy and Finance, The London School of Economics andPoliti cal Science

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

Dr. J. Favilukis, Lecturer, 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 int o 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.

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Contents

i

Contents

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

Aims of the course ............................ ................................. ................................ ............ 1Learning outcomes ................................. ................................ ................................. ...... 1

Syllabus ................................ ................................. ................................ ........................ 2

Essential reading .............................. ................................ ................................. ............ 3

Further reading ................................ ................................. ................................ ............. 3

Online study resources ............................ ................................ ................................. ...... 5

Subject guide structure and use ............................... ................................ ...................... 6

Examination advice................................................ ................................ ........................ 7

Glossary of abbreviations used in this subject guide .......................................... ............. 8

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

projects ................................................................................................................... 9Aim ............................... ................................. ................................ .............................. 9

Learning outcomes ................................. ................................ ................................. ...... 9

Essential reading .............................. ................................ ................................. ............ 9

Further reading ................................ ................................. ................................ ............. 9

Overview .............................. ................................. ................................ ...................... 10

Introduction ............................... ................................. ................................ ................ 10

Fisher separation and opt imal decision-making ................................................ ............ 10

Fisher separation and project evaluation .............................. ................................ ........ 13

The time value of money .............................. ................................. ............................... 14

The net present value rule ............................ ................................. ............................... 15Other project appraisal techniques ................................. ................................ .............. 17

Using present value techniques to value stocks and bonds ..... ..... ...... ..... ...... ..... ...... ..... 21

A reminder of your learning outcomes............................ ................................. ............. 23

Key terms ............................. ................................. ................................ ...................... 23

Sample examination questions ............................ ................................ ......................... 23

Chapter 2 : Risk and return: mean–variance analysis and the CAPM........ ......... ... 25

Aim of the chapter............................ ................................ ................................. .......... 25

Learning outcomes ................................. ................................ ................................. .... 25

Essential reading .............................. ................................ ................................. .......... 25

Further reading ................................ ................................. ................................ ........... 25

Introduction ............................... ................................. ................................ ................ 25

Statistical characteristics of portfolios ............................. ................................. ............. 26

Diversification.............................. ................................ ................................ ................ 28

Mean–variance analysis ................................ ................................ ............................... 30

The capital asset pricing model ...................................................... .............................. 34

The Roll critique and empirical t ests of the CAPM ............................. ............................ 37

A reminder of your learning outcomes............................ ................................. ............. 40

Key terms ............................. ................................. ................................ ...................... 40

Sample examination questions ............................ ................................ ......................... 40

Solutions to activities .............................. ................................ ................................. .... 41

Chapter 3: Factor models ........ ......... ........ ......... ........ ......... ........ ......... ......... ........ 43

Aim of the chapter............................ ................................ ................................. .......... 43

Learning outcomes ................................. ................................ ................................. .... 43



92 Corporate finance

ii

Essential reading ............................. ................................ ................................. ........... 43

Further reading ............................... ................................. ................................ ............ 43

Overview ............................. ................................. ................................ ....................... 43

Introduction .............................. ................................ ................................. ................. 44

Single-factor models .............................. ................................ ................................. ..... 44

Mult i-factor models ............................... ................................ ................................ ...... 46

Broad-based portfolios and idiosyncratic returns......................................................... .. 47Factor-replicating portfolios .............................. ................................. .......................... 48

The arbitrage pricing theory .............................. ................................ ........................... 50

Mult i-factor models in practice ................................ ................................ ..................... 51

Summary ............................. ................................. ................................ ....................... 52

A reminder of your learning outcomes ................................ ................................. ......... 52

Key terms ............................ ................................. ................................ ....................... 53

Sample examination question ................................. ................................ ..................... 53

Chapter 4: Derivative securities: properties and pricing ........ ........ ......... ......... ... 55

Aim of the chapter................................ ................................. ................................ ...... 55

Learning outcomes ................................ ................................ ................................ ...... 55Essential reading ............................. ................................ ................................. ........... 55

Further reading ............................... ................................. ................................ ............ 55

Overview ............................. ................................. ................................ ....................... 55

Varieties of derivatives ................................ ................................. ................................ 56

Derivative asset payoff profiles ................................ ................................ ..................... 57

Pricing forward contracts ............................. ................................ ................................ 59

Binomial option pricing sett ing .......................................................... .......................... 60

Bounds on option prices and exercise strategies ..... ..... ..... ..... ..... ..... ..... ..... ...... ..... ...... .. 64

Black–Scholes option pricing ....................... ................................. ............................... 66

Put–call parity ................................. ................................ ................................. ........... 68

Pricing interest rate swaps ................................ ................................ ........................... 69

Summary ............................. ................................. ................................ ....................... 69


Key terms ............................ ................................. ................................ ....................... 70

Sample examination questions ................................ ................................. .................... 71

Chapter 5: Efficient markets: theory and empirical evidence ........ ......... ........ ..... 73


Learning outcomes ................................ ................................ ................................ ...... 73


Further reading ............................... ................................. ................................ ............ 73

Overview ............................. ................................. ................................ ....................... 74

Varieties of efficiency ............................. ................................ ................................. ..... 74

Risk adjustments and the joint hypothesis problem .............. ................................. ....... 75

Weak-form efficiency: implications and tests ............................. ................................ ... 76

Weak-form efficiency: empirical results ................................ ................................ ......... 78

Semi-strong-form efficiency: event studies ................................ ................................. ... 81

Semi-strong-form efficiency: empirical evidence ............................................... ............. 83

Strong-form efficiency ............................ ................................ ................................. ..... 83

Long horizon forecastability ............................................... ................................ .......... 83

Summary ............................. ................................. ................................ ....................... 85

A reminder of your learning outcomes ................................ ................................. ......... 85Key terms ............................ ................................. ................................ ....................... 85




Contents

iii

Chapter 6: The choice of corporate capital structure ........ ......... ........ ......... ........ . 89


Learning outcomes ................................ ................................ ................................ ...... 89


Further reading ............................... ................................. ................................ ............ 89

Overview ............................. ................................. ................................ ....................... 89

Basic features of debt and equity ....................................................... .......................... 90The Modigliani–Miller theorem ............................... ................................. .................... 91

Modigliani–Miller and Black–Scholes ............................. ................................ .............. 93

Modigliani–Miller and corporate taxation ............................ ................................. ........ 94

Modigliani–Miller w ith corporate and personal taxation ............................. .................. 97

Summary ............................. ................................. ................................ ....................... 98


Key terms ............................ ................................. ................................ ....................... 99


Chapter 7: Leverage, WACC and the Modigliani-Miller 2nd proposition .... ....... 101

Aim of the chapter................................ ................................. ................................ .... 101Learning outcomes ................................ ................................ ................................ .... 101

Essential reading ............................. ................................ ................................. ......... 101

Further reading ............................... ................................. ................................ .......... 101

Overview ............................. ................................. ................................ ..................... 101

Weighted average cost of capital ............................................ ................................. .. 102

Modigliani and Miller’s 2nd proposition .............................................. ....................... 103

A CAPM perspective .............................. ................................ ................................. ... 107

Summary ............................. ................................. ................................ ..................... 108

Key terms ............................ ................................. ................................ ..................... 108

A reminder of your learning outcomes ................................ ................................. ....... 108

Sample examination questions ................................ ................................. .................. 109

Chapter 8: Asymmetric informat ion, agency costs and capital structure ........ .. 111

Aim of the chapter................................ ................................. ................................ .... 111

Learning outcomes ................................ ................................ ................................ .... 111


Further reading ............................... ................................. ................................ .......... 111

Overview ............................. ................................. ................................ ..................... 112

Capital structure, governance problems and agency costs ...... ..... ...... ..... ...... ..... ..... ..... 112

Agency costs of outside equity and debt ....................... ................................. ............ 112

Agency costs of free cash flows............................... ................................ ................... 118

Firm value and asymmetric information ............................... ................................ ....... 119

Summary ............................. ................................. ................................ ..................... 123

Key terms ............................ ................................. ................................ ..................... 123



Chapter 9: Dividend policy ................................................................................. 127




Further reading ............................... ................................. ................................ .......... 127

Overview ............................. ................................. ................................ ..................... 128

Modigliani–Miller meets dividends ................................ ................................. ............ 128

Prices, dividends and share repurchases ..... ..... ..... ..... ..... ..... ..... ...... ..... ..... ...... ..... ...... . 129




iv

Dividend policy: stylised facts ............................. ................................ ........................ 129

Taxation and clientele theory ............................ ................................. ........................ 131

Asymmetric information and dividends ................................ ................................. ...... 132

Agency costs and dividends ............................... ................................ ........................ 133

Summary ............................. ................................. ................................ ..................... 133


Key terms ............................ ................................. ................................ ..................... 134Sample examination questions ................................ ................................. .................. 134

Chapter 10: Mergers and takeovers ........ ........ ......... ......... ........ ......... ........ ........ 135




Further reading ............................... ................................. ................................ .......... 135

Overview ............................. ................................. ................................ ..................... 136

Merger motivations .............................. ................................. ................................ .... 136

A numerical takeover example ........................................................... ........................ 137

The market for corporate control ............................. ................................ ................... 138The impossibility of efficient takeovers ........................................... ............................ 139

Two ways to get efficient takeovers .............................. ................................. ............. 140

Empirical evidence ................................. ................................ ................................. ... 141

Summary ............................. ................................. ................................ ..................... 143


Key terms ............................ ................................. ................................ ..................... 143


Appendix 1: Perpetuit ies and annuities ......... ......... ........ ......... ........ ......... ........ . 145

Perpetuit ies ............................... ................................ ................................. ............... 145

Annuities ............................ ................................. ................................ ..................... 146Appendix 2: Sample examination paper ........ ......... ........ ......... ........ ......... ........ . 147



Introduction to the subject guide

1


This subject guide for 9 2 Co r p o r a t e fi n a n c e , a Level 3 course offered

on the Economics, Management, Finance and Social Sciences programme,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.

• The first section 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 statisticaltheory 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.

• 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 firmsmay be explained in theoretical terms.

Aims of the course

This course 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 outcomes

At the end of this course, 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)

• know how to use recent extensions of the CAPM, such as the Fama

and French three-factor model, to calculate expected returns on risky securities




2

• 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 the trade-off firms face between tax advantages of debt and

various costs of debt• understand and explain the capital structure theory, and how information

asymmetries affect it

• 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: A minor revision was made 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 the Regulations.

If you are taking this course as part of a BSc degree, courses which must

be passed before this course may be attempted are 2 I n t r o d u c t io n

t o e c o n o m i cs and 5A Mathem at ic s 1 or 5B Mathem at ic s 2 or

174 Ca lcu lus . This course may not be taken with course 59 F ina nc ia l

m a n a g e m e n t .

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; recent extensions of the

factor framework.

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; long-horizonreturn predictability.

Capital structure: the Modigliani–Miller theorem: capital structure irrelevancy;

taxation, bankruptcy costs and capital structure; weighted average cost

of capital; Modigliani-Miller 2nd proposition; 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.




3

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 course due

to its extensive treatment of many of the issues covered and up-to-date

discussions:

Hillier, D., M. Grinblatt and S. Titman Financial Markets and Corporate Strategy. (Boston, Mass.; London: McGraw-Hill, 2008) European edition

[ISBN 978007119027].

At the start of each chapter of this guide, we will indicate the reading that

you need to do from Hillier, Grinblatt and Titman (2008).

Detailed reading references in this subject guide refer to the editions of the

set textbooks listed above. New editions of one or more of these textbooks

may have been published by the time you study this course. You can use

a more recent edition of any of the books; use the detailed chapter and

section headings and the index to identify relevant readings. Also check

the virtual learning environment (VLE) regularly for updated guidance on

readings.

Further reading

Please note that as long as you read the Essential reading you are then free

to read around the subject area in any text, paper or online resource. You

will need to support your learning by reading as widely as possible and by

thinking about how these principles apply in the real world. To help you

read extensively, you have free access to the VLE and University of London

Online Library (see below).

Other useful texts for this course include:

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].

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.

Bhattacharya, S. ‘Imperfect information, dividend policy, and “the bird in the

hand” fallacy’, Bell Journal of Economics 10(1) 1979, pp.259–70.

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.1731–64.




4

Campbell, J. and R. Shiller ‘The dividend-price ratio and expectations of future

dividends and discount ractors’, Review of Financial Studies 1 1988.

Chen, N-F. ‘Some empirical tests of the theory of arbitrage pricing’, The Journal

of Finance 38(5) 1983, pp.1393–414.

Chen, N-F., R. Roll and S. Ross ‘Economic Forces and the Stock Market’, Journal

of Business 59 1986, pp.383–403.

Cochrane, J.H. ‘Explaining the variance of price-dividend ratios’, Review of

Financial Studies 5 1992, pp.243–80.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.

Fama, E. and K. French ‘Common risk factors in the returns on stocks and

bonds’, Journal of Financial Economics 33 1993, pp.3–56.

Fama, E. and J. MacBeth. ‘Risk, return, and equilibrium: empirical tests’,

Journal of Political Economy 91 1973, pp.607–36.

Gibbons, M.R., S.A. Ross, and J. Shanken. ‘A test of the efficiency of a given

portfolio’, Econometrica 57 1989, pp.1121–52.

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 initiationsand 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.

Jegadeesh, N. and S. Titman ‘Returns to buying winners and selling losers’,

Journal of Finance 48 1993, pp.65–91.

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 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 R. Ruback ‘The market for corporate control: the scientific

evidence’, Journal of Financial Economics 11(1–4) 1983, pp.5–50.

Lakonishok, J., A. Shleifer and R. Vishny ‘Contrarian investment, extrapolation,

and risk’, Journal of Finance 49(5) 1994, pp.1541–78.

Lettau, M. and S. Ludvigson ‘Consumption, aggregate wealth, and expected

stock returns’, Journal of Finance 56 2001, pp.815–49.

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

Money and Finance 12(5) 1993, pp.451–74.




5

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.

Miles, J. and J. Ezzell ‘The weighed average cost of capital, perfect capitalmarkets and project life: a clarification’, Journal of Financial and

Quantitative Analysis 15 1980, pp.719–30.

Miller, M. ‘Debt and taxes’, Journal of Finance 32 1977, pp.261 – 75.

Modigliani, F. and M. Miller ‘The cost of capital, corporation finance and the

theory of investment’, American Economic Review (48)3 1958, pp.261–97.

Modigliani, F. and M. Miller ‘Corporate income taxes and the cost of capital: a

correction’, American Economic Review (5)3 1963, pp.433–43.

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.

Shleifer, A. and R. Vishny ‘Managerial entrenchment: the case of management-

specific investment’, Journal of Financial Economics 25, 1989 pp.123–39.

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, R., W. Maksimovic and

W.T. Ziemba (eds) Handbook of Finance. (Amsterdam: Elsevier Science,

1995) [ISBN 9780444890849].

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].

Online study resources

In addition to the subject guide and the Essential reading, it is crucial that

you take advantage of the study resources that are available online for this

course, including the VLE and the Online Library.

You can access the VLE, the Online Library and your University of London

email account via the Student Portal at:

http://my.londoninternational.ac.uk

You should receive your login details in your study pack. If you have not,or you have forgotten your login details, please email uolia.support@

london.ac.uk quoting your student number.




6

The VLE

The VLE, which complements this subject guide, has been designed to

enhance your learning experience, providing additional support and a sense

of community. It forms an important part of your study experience with the

University of London and you should access it regularly.

The VLE provides a range of resources for EMFSS courses:

• Self-testing activities: Doing these allows you to test your ownunderstanding of subject material.

• Electronic study materials: The printed materials that you receive from

the University of London are available to download, including updated

reading lists and references.

• Past examination papers and Examiners’ commentaries: These provide

advice on how each examination question might best be answered.

• A student discussion forum: This is an open space for you to discuss

interests and experiences, seek support from your peers, work

collaboratively to solve problems and discuss subject material.

• Videos: There are recorded academic introductions to the subject,interviews and debates and, for some courses, audio-visual tutorials and

conclusions.

• Recorded lectures: For some courses, where appropriate, the sessions from

previous years’ Study Weekends have been recorded and made available.

• Study skills: Expert advice on preparing for examinations and developing

your digital literacy skills.

• Feedback forms.

Some of these resources are available for certain courses only, but we are

expanding our provision all the time and you should check the VLE regularly

for updates.

Making use of the Online Library

The Online Library contains a huge array of journal articles and other

resources to help you read widely and extensively.

To access the majority of resources via the Online Library you will either need

to use your University of London Student Portal login details, or you will be

required to register and use an Athens login: http://tinyurl.com/ollathens

The easiest way to locate relevant content and journal articles in the Online

Library is to use the S u m m o n search engine.

If you are having trouble finding an article listed in a reading list, try removing any punctuation from the title, such as single quotation marks,

question marks and colons.

For further advice, please see the online help pages:

www.external.shl.lon.ac.uk/summon/about.php

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.• Ch a p t e r 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.




7

• Ch a p t e r 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).

• Ch a p t e r 3 : we present the arbitrage pricing theory, proposed as an

alternative to the CAPM and discuss multifactor models. We study

several recent multifactor models, such as the Fama and French three-

factor model, and observe that they can explain a large fraction of the variation in risky returns.

• Ch a p t e r 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.

• Ch a p t e r 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. We also note that

returns may be predictable even in efficient markets if risk is also

predictable and discuss evidence in support of predictability of longhorizon returns.

• Ch a p t e r 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.

• Ch a p t e r 7 : this chapter is complementary to Chapter 6, however, rather

than looking at values, as in Chapter 6, this chapter analyses discount

rates. We learn that if there are no taxes, while the return on equity gets

riskier as the level of debt increases, the average rate the firm pays to

raise money is unchanged. In the presence of taxes, as debt increases, the

average rate the firm pays to raise money decreases due to tax shields.

• Ch a p t e r 8 : 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.

• Ch a p t e r 9 : here we examine dividend policy. What is the empirical

evidence on the dividend payout behaviour of firms, and theoretically,

how can we understand the empirical facts?

• Ch a p t e r 1 0 : 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 8 and 10 are inherently motivated by

the existence of such problems. See also Hillier, Grinblatt and Titman

(2008) Chapter 18 for a broad overview on governance-related issues.

Examination advice

I m p o r t a n t : the information and advice given here 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 VLE where you

should be advised of any forthcoming changes. You should also carefully




8

check the rubric/instructions on the paper you actually sit and follow

those instructions.

Remember, it is important to check the VLE for:

• up-to-date information on examination and assessment arrangements

for this course

• where available, past examination papers and Examiners’ commentaries

for the course which give advice on how each question might best beanswered.

This course 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 Examiners will attempt to provide a fairly

balanced coverage of the course, 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

CAPM capital asset pricing model

CML capital market line

IRR internal rate of return

MM Modigliani–Miller

NPV net present value



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

9

Chapter 1: Present value calculations

and the valuation of physical investment

projects

Aim

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 outcomes

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

• analyse optimal physical and financial investment in perfect capital

markets setting 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

Hillier, D., M. Grinblatt and S. Titman Financial Markets and Corporate Strategy.

(Boston, Mass.; London: McGraw-Hill, 2008) Chapters 9 (Discounting

and Valuation), 10 (Investing in Risk-Free Projects), 11 (Investing in Risky

Projects).

Further reading

Brealey, R., S. Myers and F. Allen Principles of Corporate Finance. (Boston,

Mass.; London: McGraw-Hill, 2008) Chapters 2 (Present Values), 3 (How to

Calculate Present Values), 5 (The Value of Common Stocks), 6 (Why NPV Leads to Better Investment Decisions) and 7 (Making Investment Decisions

with the NPV Rule).

Copeland, T. and J. Weston Financial Theory and Corporate Policy. (Reading,

Mass.; Wokingham: Addison-Wesley, 2005) Chapters 1 and 2.

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.




10

Overview

In this chapter we present the basics of the present value methodology

for the valuation of investment projects. The chapter develops the

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.

Introduction

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 ordecisions 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 (POF) that looks as given in Figure

1.1. This curve represents one manner in which the firm can transform

its current funds into future income, where c0

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




11

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).

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 (c0) must be identically m, whereas period 1

consumption (c1) is zero. Alternatively, the firm could lend all of its funds.

This leads to c0 being zero and c1 = m (1 + r ). The relationship betweenperiod 0 and period 1 consumption is therefore:

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




13

Fisher separation and project evaluation

Fisher separation can also be used to justify a certain method of project

appraisal. Figure 1.3 shows a suboptimal 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 I 0. 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:

where Π ( I 0) is the date 1 income from the firm’s physical investment.

Maximising this is equivalent to the following maximisation problem:

.

The prior objective is the NPV 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 transactioncosts 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




14

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 ruleto 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. Theanswer 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 fu t u r e v a l u e of $1. We can similarly

illustrate the time value of money by using p r e s e n t va l u e s . 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:

(1.3)

The value for x defined in equation 1.3 is the p r e s e n t va l u e of $1

received in one year’s time. This quantity is also termed the d i s c o u n t e dva lue of the $1.




15

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 p r e s e n t and f u t u r e value concepts are straightforwardly extended

to cover more than one period. Assume an annual compound interest rateof r . The present value of $100 to be received in k year’s time is:

(1.4)

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

hence is:

FV K (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.

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 NPV 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 thepresent values of the payments.

Using the notation given above and again assuming a rate of return of r ,

the NPV can be written as:

. (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.

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.




16

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 f irm 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 t he machine at the end of its l ifetime 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 t he follow ing 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 f lows 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 i t 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 acrossprojects (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.

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 areconsidering 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

2 By this we mean that

taking on any one of the

set of projects precludes

us from accepting any of

the others.




18

the payback period, giving a d i s co u n t e d p a y b a c k r u l e, but such a

modification still wouldn’t solve the first problem we highlighted.

The internal rate of return rule

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:

(1.7)

where C iis 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 h u r d le

r a t e . 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 the NPV calculation. If r* is the hurdle rate used

in project evaluation, then the project represented by the curve on thefigure 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 k th 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.




19

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 that 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




20

The multiples method

An alternative to using forecasts of a firm’s or project’s cash flows to

calculate value, market information can be used to estimate the value.

The multiples method assesses the firm’s value based on the value of a

comparable publically traded firm. For example, consider the firm’s market

value to earnings ratio, this ratio tells us how much a dollar of earnings

contributes to the present value according to the market’s consensus view. For publically traded firms, this ratio is available. The firm we wish

to value may not have a publically available market value, however we

are likely to know its earnings. If we assume that these two firms should

have similar market value to earnings ratios, then we can value the firm

by taking the publically available ratio and multiplying it by the firm’s

earnings.

Common multiples to use are market value to earnings, market value

to EBITDA, market value to cash flow, and market value to book value.

Some firms, especially younger firms, have no earnings or even negative

earnings. In this case it may be better to value the firm as of some future

date in which the firm’s cash flows have stabilised, and then to discount to

today’s value. An alternative is to use more creative multiples, for example

price to patent ratio, price to subscriber ratio, or price to Ph.D. ratio. It is

often better to take an average over several comparable firms to calculate

the multiple. If you believe the firm being valued is better or worse than

the comparable firms, you can shade the multiple down or up, as in the

example below. The multiples method is not an exact science but rather a

convenient way to incorporate market beliefs. It should always be used in

conjunction with another method, such as NPV.

Example

Below are the equity values, debt values, and earnings (in billions) for several large US

retailers. Additionally provided is earnings growth for the past 10 years.

Equity Debt E E (10 yr) %

JCP 17.48 3.81 1.10 7.8

COST 24.08 2.22 1.10 15.5

HD 82.08 12.39 6.01 21.2

WMT ? 47.44 11.88 15.7

TGT 50.14 14.14 2.58 19.2

Walmart’s (WMT’s) equity value is excluded as this is the quant ity we wish to estimate.

We can first calculate the market value of equity to earnings ratio for the average firm

in the industry (excluding Walmart), this is: [(17.48/1.1) + (24.08/1.1) + (82.08/6.01) +

(50.14/2.58)]/4 = 17.72

We now multiply this number by Walmart’s earnings to get Walmart’s equity value

estimate: 17.72* 11.88= 210.49. Walmart’s actual equity value was $192.48 billion.

In the example above we used multiples to value equity, we sometimes

wish to the value of the full business (sometimes called enterprise value),

in this case we would need to use the full business value (for example,

debt plus equity) in the numerator instead of just equity value.

Notice that the debt to equity ratio of Costco (COST) was 9.2% while that

of Target (TGT) was 28.2%. In this example, we have ignored the effectsof leverage (debt in the capital structure), however as we will see in a later

chapter, leverage affects both firm value and the expected return on equity.

Therefore, firms with different leverage ratios that look otherwise similar




21

may have very different value to earnings ratios. We will learn how to

adjust the multiples method for the effects of leverage later.

The multiples method allows us to check whether the value of a

conglomerate is equal to the sum of its parts. To estimate the value of

each business division of a conglomerate we can calculate each division’s

earnings and multiply it by the average value to earnings multiple of stand

alone firms in the same sector. Adding up the value of all divisions givesus an estimated value for the conglomerate, this estimate is on average

12% greater than the traded value of the conglomerate. This is called the

c o n g l o m e r a t e d i s c o u n t . The reasons for the conglomerate discount

are not fully understood. It is possible that conglomerates are a less

efficient form of organisation due to inefficient capital markets. It is also

possible that the multiples method is inappropriate here because single

segment firms are too different from divisions of a conglomerate operating

in the same industry.

The strength of the multiples approach is that it incorporates a lot of

information in a simple way. It does not require assumptions on the

discount rate and growth rate (as is necessary with the NPV approach)

but just uses the consensus estimates from the market. A weakness is

the assumption that the comparable companies are truly similar to the

company one is trying to value; there is no simple way of incorporating

company specific information. However, its strength is also its biggest

weakness. By using market information, we are assuming that the market

is always correct. This approach would lead to the biggest mistakes

in times of biggest money making opportunities: when the market is

overvalued or undervalued.

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.

Using present value techniques to value stocks andbonds

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 tothe 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 Dt+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:

. (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.




22

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 D0

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 p e r p e t u i t y w i t h g r o w t h , and its

present value can be calculated very simply.3 In this setting the price of the

equity share is:

0 . (1.9)

This is the Go r d o n g r o w t h m o d e l 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:

. (1.10)

The first term in 1.10 is the expected dividend yield on the stock, and thesecond 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 equit y 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?

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 d i sc o u n t o r z e r o c o u p o n b o n d is an instrument that promises

to pay the bearer a given sum (known as the p r i n c i p a l) 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 c ou p o n b o n d s . 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 priceof the bond can be written as:4

. (1.11)

3 See Appendix 1.

4 In our notation a coupon rate of 12 per cent, for example,

implies that c = 0.12; the discount rate used here, r

b , is called the

yield to maturity of the bond.




23

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 a perfect capital

markets setting 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.

Key terms

capital market line (CML)

consumption

Fisher separation theorem

Gordon growth model

indifference curve

internal rate of return (IRR) rule

investment policy

net present value (NPV) 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 six 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%)




24

2. Describe two methods of project evaluation other than NPV. Discuss the

weaknesses of these methods when compared to NPV. (10%)

3. The CEO and other top executives of a firm with no nearby commercial

airports make approximately 300 flights per year with an average

cost per flight of $5,000. The firm is considering buying a Gulfstream

jet for $15 million. The jet will reduce the cost of travel to $300,000

(including fuel, maintenance, and other jet-related expenses).The firm expects to be able to resell the jet in five years for $12.5

million. The firm pays a 25% corporate tax on its profits and can offset

its corporate liabilities by using straight line depreciation on its fixed

assets. The opportunity cost of capital is 4%.

a. Should the firm buy this jet if it has sufficient taxable profits in

order to take advantage of all tax shields?

b. Should the firm buy this jet if it does not have sufficient taxable

profits in order to take advantage of new tax shields?

c. Suppose the firm could lease an airplane for the first year, with

an option to extend the lease. Within that year they would find

out whether the local government has decided to build an airport

nearby which would reduce travel costs. How would this change

your calculations?

4. Suppose that you have a £10,000 student loan with a 5 per cent

interest rate. You also have £1,000 in your zero interest checking

account which you do not plan to use in the foreseeable future. You are

considering three strategies: (i) payoff as much of the loan as possible,

(ii) invest the money in a local bank at 3.5 per cent interest, (iii) invest

in the stock market. The expected return on the stock market is 6 per

cent for the foreseeable future. Your personal discount rate is 4 per

cent for risk-free investments. For simplicity assume all investments are

perpetuities.

a. What is the NPV of strategy (i)?

b. What is the NPV of strategy (ii)?

c. What is the NPV of strategy (iii) if you are risk neutral?

d. What is the NPV of strategy (iv) if your subjective market risk

premium is 3 per cent?



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

25

Chapter 2: Risk and return:

mean–variance analysis and the CAPM

Aim of the chapterThe 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 outcomes

At the end of this chapter, and having completed the Essential reading and

activities, you should be able to:• 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 with confidence

• 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 readingHillier, D., M. Grinblatt and S. Titman Financial Markets and Corporate Strategy.

(Boston, Mass.; London: McGraw-Hill, 2008) Chapters 4 (The Mathematics

and Statistics of Portfolios) and 5 (Mean-Variance Analysis and the CAPM).

Further reading

Brealey, R., S. Myers and F. Allen Principles of Corporate Finance. (Boston,

Mass.; London: McGraw-Hill, 2008) Chapters 8 (Introduction to Risk,

Return, and the Opportunity Cost of Capital) and 9 (Risk and Return).

Copeland, T. and J. Weston Financial Theory and Corporate Policy.

(Reading, Mass.; Wokingham: Addison-Wesley, 2005) Chapters 5 and 6.

Roll, R. ‘A critique of the asset pricing theory’s texts. Part 1: on past andpotential testability of the theory’, Journal of Financial Economics 4(2)

1977, pp.129–76.

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.




26

We begin by introducing the basic statistical tools that will be needed

in our analysis, these being e x p e c t e d v a l u e s , v a r i a n c e s and

c o v a r i a n c e s. This leads to an analysis of the statistical characteristics

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 t w o -fu n d s e p a r a t i o n , and this result

underlies the CAPM, 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 e x p e ct e d r e t u r n s and r e t u r n v a r i a n c e s 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 totalasset 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

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

ij. Similarly, we denote

the expected return of the portfolio held as E ( R p) and its variance by σ 2

P .

Finally, we assume that an investor can pick from N different stocks when

forming their portfolio.

Returning to the example of the American investor given above, assume

that the market price of Microsoft shares is 130 and that of BethlehemSteel 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 por t fo l io we igh t s . 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 α

1= 1.

1 These prices are in US

cents.




27

Our American investor now knows the statistical characteristics of the

return on each of the assets held, plus how to calculate the portfolio

weight on each of the assets. What they would really like to know now

is how to construct the return characteristics for the entire portfolio (i.e.

they are concerned about the risk and reward associated with their entire

investment). In order to do this we will need to introduce some basic

properties of expectations, variances and covariances.

Expectations, variances and covariances

Consider two random variables, x and y. The expected values and

variances of these variables are E ( x) , E ( y) , σ 2 x

and σ 2 y. The covariance

between the random variables is σ 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)

σ 2 P = a

2

σ 2 x + b

2

σ 2 y + 2abσ 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 xi, where i runs from 1 to N . Denote their expected values and

variances as E ( xi) and σ 2

i. The covariance between x

iand x

jis σ

ij. Again

we form a linear combination of the random variables, denoted again by

P , using an arbitrary set of constants denoted ai. The expected value and

variance of the random variable P are given by:

(2.3)

. (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 σ 2 x

and

σ 2 y. The covariance between the random variables is σ

xy. The correlation

coefficient is defined as follows:

, (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




29

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 σ 2iwhere i = 1 to N , and

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

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

. (2.10)

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

equation 2.10 involves N terms. Hence, defining the average variance of

the N assets as σ 2 and average covariance across all assets as C , equation

2.10 can be rewritten as:

. (2.11)

Equation 2.11 obviously simplifies to the following:

. (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 equation 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 equation 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 that they bear. This is illustrated diagrammatically in

Figure 2.1.

Figure 2.1




30

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 o n l y 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.

To begin, assume there is no risk-free aset. The investor can hence only

form their 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 σ 2

xand σ 2

y.

The first question the investor wishes to answer is how the characteristicsof 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 that 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 α on asset X and 1 – α on asset Y . Equation

2.6 then implies that the portfolio variance can be written as follows:

σ 2 P

= (ασ x

+ (1 – α )σ y)2. (2.13)

Taking the square root of equation 2.13, it is clear that the portfoliostandard deviation is linear in α . As the portfolio expected return is linear

in α , 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 their portfolio. As previously discussed, in

this scenario, portfolio standard deviation is n o t a linear combination of

σ x

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

that the mean–standard deviation frontier bows towards the y-axis. This




32

when there are N risky assets available. Figure 2.4 depicts the same type of

diagram for the N asset case.

Figure 2.4

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)

σ 2Q

= a2σ 2 P

. (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:

. (2.16)

Using equations 2.14 and 2.15 we find that:

. (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 (CML), and two such

CMLs are shown in Figure 2.5 for two different portfolios of risky assets.




33

Figure 2.5

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 f KZ ), 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 suboptimal CML (labelled CML2) are

shown on Figure 2.6 along with the indifference curves of two investors.

Figure 2.6

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 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.

The key result that is depicted in Figure 2.6 is known as two- fund

s e p a r a t i o n . Any risk-averse investor (regardless of their degree of risk-

aversion) can form their 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

4 That is, choosing

portfolio K places an

investor on a CML wit h

greater expected returns

at each level of return

variance than does any

other.




34

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

Two-fund separation is the result that underlies the CAPM, which is

developed in the next section.

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 atthe optimum is:

(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 ( R I ) = aE ( R

j) + (1 – a) E ( R

K ) (2.19)

σ 1 = [a2σ 2 j + (1 – a)2σ 2 K + 2a(1 – a)σ 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:

. (2.21)

(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:

. (2.23)

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 t ransactions and

immediately receives the

sale pri ce. 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 cash pro fi t.




35

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

that is,

.

(2.24)

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 2.24and rearranging the resulting expression, we arrive at:

(2.25)

Defining β j

= σ jK

/ σ 2 K , equation 2.26 can be rewritten as:

E ( R j) = r

j+ β

j[ E ( R

K ) – r

f ]. (2.26)

Equation 2.26 is the standard β -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 β .6 Assets that have large

values of β will have large expected returns, whereas those with smaller

values of β will have low expected returns with β 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

m a r k e t p o r t f o li o , 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 ( R j) = R

f + β

j [ E (r

M ) – r

f ] , (2.27)

where E ( RM

) is the expected return on the market portfolio, and β 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

6 The risk premium is

de fi ned as the excess of

the expected return on

the tangency portfolio

over the risk-free rate.

7 All investors perceive

the same ef fi cient

set and tangency portfolio due to our

assumption that they

have homogeneous

expectations regarding

asset returns.




36

measure of an asset’s risk is its β , and equation 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 β -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.

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 β . The expected return on an asset with a β of zero is r f , whereas

an asset with a β 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 β 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 β . Hence

an asset’s β measures its relevant risk and, via equation 2.27, determines

equilibrium expected returns.

The key message of the preceding paragraph is that β measures asset risk.

A high β asset is risky as it has high returns when market returns are high.

An asset with a low β tends to have high returns when market returns are

low. Hence a low β 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 ε j

we have the following equation:

r j

= r f

+ β j[r

M – r

f ] + ε

j. (2.28)




37

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

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

σ 2 j

= β 2 jσ 2

M + σ 2

ε. (2.29)

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 affectexpected returns. The first term on the right-hand side of equation 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 port folio expected return and variance if t he correlation between

returns on X and Y is unity.

2. Using the data from Question 1, recalculate the portfolio expected return andvariance, assuming that t he 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 that t he 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 stat istical properties of

the portfolio, assuming the return correlation for P and Q is –0.5.

The Roll critique and empirical tests of the CAPMThe 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 themarket) 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 thecase 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

8 You will fi nd the

solutions to these

activities at the end of

this chapter.

9 See Roll (1977).




38

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

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 judgethe 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 judgement on the CAPM.

The CAPM as a one-factor model

As we saw above, idiosyncratic risk should not matter for pricing of assets

because investors are able to diversify it away. Only common risk matters.

A one-factor model states that all common risk can be summarised by a

single variable, or factor. Specifically, the return on any asset is given by:

Rit

= ai+ b

i*F

t + e

it E [e

it ] = 0 E [ F

t *e

it ]= 0 (2.30)

Note that aiis an asset specific constant, b

iis an asset specific factor

loading, and eit

is an idiosyncratic variable uncorrelated across assets. On

the other hand F t is a factor common to all assets.

We will now see that the CAPM implies a one-factor model with the factor

being the excess market return. Note that for any two random variable

X t = E [ X

t ] + e

t where e

t is independent of E [ X

t ], therefore R

it – R

f = E [ R

it – R

f ]

+ υit

and Rmt

– R f = E [ R

mt – R

f ] +

t where υ and are idiosyncratic.

E [ Rit – Rf ] = β

i*E [ R

mt – Rf ] (2.31)

Rit – R

f – υ

it = β

i*( R

mt – R

f ) – β

i*η

t (2.32)

Rit – R f = β i*( Rmt – R f ) + (υit – β i*ηt ) = β i*( Rmt – R f ) + eit (2.33)Thus we can write the CAPM as a one-factor model where the excess

market return is the factor.

Suppose we were to regress the excess return on asset i on the excess

market return:

Rit

– R f

= Ai+ B

i*( R

mt – R

f ) (2.34)

By definition of a regression, Bi= Cov( R

it – R

f , R

mt – R

f ) /Var ( R

mt – R

f ), which

is equal to the CAPM β for asset i. The CAPM implies that Ai = 0 for each

asset i. This is one way to test the CAPM (or any factor model). This is

referred to as a first stage test of the CAPM: for each asset we run a time

series regression of that asset’s returns on the market excess return. If wefind that many assets have Ainot equal to zero, we would infer that the

CAPM does not work well.

There is also another test of the CAPM, referred to as the second stage.

As opposed to the first stage test, where we ran a time series regression

for each asset, this test will produce a single cross-sectional regression for

all assets. Note that the CAPM implies that assets with higher betas have

higher expected returns, furthermore, the relationship is linear. We can

test this by regressing the average historical return for each asset on the

β for each asset, which we found in the first stage regression. We run the

cross-sectional regression: E [ Ri– R

f ]= G

0+ G

1* β

i

The CAPM implies that G0 is zero and G1 is the average market premium E [ R

m– R

f ].




39

The data are generally not supportive of the CAPM. The relationship

between an asset’s β and its average return is usually positive, as the CAPM

suggests, but typically flatter than it should be, as can be seen in Figure 2.8.

In this figure the β ’ s are plotted against average returns for 17 portfolios

based on industry (such as food, chemicals or transportation). The dotted

line plots β against β * E [ Rm

– R f ], this is the CAPM predicted expected

return. The solid line plots the actual relationship between β and industry

returns, this relationship is positive but flatter than the dotted line. That is

high β stocks have returns that are lower than predicted by the CAPM while

low β stocks have returns that are higher than predicted by the CAPM.

Furthermore, there are certain assets (to be discussed in the next chapter)

that appear to consistently have non-zero Ai

in time series regressions.10

0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

β

E [ R ]

Figure 2.8

One possible explanation for the too flat relationship between β and

average return is measurement error. Suppose we do not observe an asset’s

true β , but rather its true β plus some measurement error which is mean

zero. Then assets with very high observed β are likely to be assets with very

positive measurement error; therefore their true β is below their observed

β , perhaps consistent with the low observed expected return. Similarly,

assets with very low observed β are likely to be assets with very negative

measurement error and therefore their true β is above the observed β .It is also possible that one factor is simply not enough to explain all of the

variation in expected returns. The CAPM implies that the a firm’s loading

on the market (β ) is the only variable that should cause expected returns to

differ. Adding extra explanatory variables to regression 2.34 will not result

in significant coefficients. In the next chapter we will see that loadings on

other factors, including firm size, book-to-market ratios, P/E ratios and

dividend yields have been shown to explain ex-post realised returns.

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

pradigms such as the APT (which we discuss in the next chapter). Analternative 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).

10 See pp.185–86 of

Brealey and Myers

(2008).




40




• 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 expectedreturns and return variances of constituent assets with confidence

• 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 ( β )

capital asset pricing model (CAPM)

correlationcovariance

diversification

expected return

market portfolio

mean–variance analysis

Roll critique

security market line

standard deviation

systematic risk

two-fund separation

unsystematic risk

variance

Sample examination questions

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 ret urn (% ) M arket return (%)

1 8 6

2 24 12

3 28 15

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

4. Assume that the risk-free rate is 5 per cent. What is the expected return

on ABC’s stock? (3%)

5. The risk-free rate is 4 per cent, firm A has a market β of 2 and an

expected return of 16 per cent.

a. What is the expected return on the market according to the CAPM?




41

b. Draw a graph with β on the x-axis and the expected return on the

y-axis. Indicate the risk-free rate, the market, and firm A. What is

the slope of the securities market line?

c. The standard deviation of the market return is 16 per cent and the

standard deviation of the return of firm A is 40 per cent. What is the

standard deviation of A’s idiosyncratic component?

6. You have 50 years of monthly data on short-term treasury rates andportfolios of 10-year bond returns, an aggregate index of US equities,

a mutual fund focusing on tech firms, a mutual fund focusing on

commodities, a mutual fund focusing on manufacturing, and a hedge

fund index. Describe how you would test the CAPM and the results you

would expect to find.

Solutions to activities

1. The expected return on the equally weighted portfolio is 7.5 per cent.

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 per cent, 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


42



Chapter 3: Factor models

45

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 havedenoted this term εiand 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(εi , ε

j) = 0 i ≠ j Cov(ε

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. thecorrelation 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.

Application exercise

Consider an economy in which the risk-free rate of return is 4 per cent and the expected

rate of return on the market index is 9 per cent. The variance of t he return on t he market

index is 20 per cent. Two portfolios A and B have expected return 7 per cent and 10 per

cent, and variance 20 per cent and 50 per cent, respectively.

a. Work out the portfolios’ β coefficients.

According t o the CAPM:

E (r A

) = r F

+ β A [ E (r

M ) – r

F ]

and

E (r B

) = r F

+ β B [ E (r

M ) – r

F ].

Hence:

β A

= [ E (r A

) – r F

] / [ E (r M ) – r

F ] = (7% − 4%)/(9% − 4%) = 0.6

β B

= [ E (r B

) – r F

] / [ E (r M ) – r

F ] = (10% − 4%)/(9% − 4%) = 1.2.

b. The risk of a port folio 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:

r A

= α A

+ β A

r M

+ ε A

r B

= α B

+ β B

r M

+ ε B

with cov(r M , ε

A) = cov(r

M , ε

B) = 0.

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

, has two

components, systematic and idiosyncratic risk:

σ 2 A

= β 2 A σ 2

M + σ 2ε

A.

Similarly:

σ 2 B

= β 2 B σ 2

M + σ 2ε

B.

The proportion of systematic risk for A is hence

β 2 A σ 2

M / σ 2

A= (0.6)2*20%/20% = 36%.

A A




46

The proportion of idiosyncratic risk for A is hence

1 − [ β 2 A σ 2

M / σ 2

A] = 64%.

The proportion of systematic risk for B is hence

β 2 B σ 2

M / σ 2

B = (1.2)2*20%/50% = 58%.

The proportion of idiosyncratic risk for B is hence

1 − [ β 2 B σ 2

M / σ 2

B] = 42%.

Portfolio B is much riskier than portfolio A as the variance of its returns is 50 percent compared with 20 per cent 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 β is 1.2

compared wit h 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?

Cov(r A ,r

B) = Cov(α

A+ β

Ar

M + ε

A , α

B+ β

Br

M + ε

B) = β

A β

Bσ 2

M = 0.6*1.2*20% = 14%.

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 w ith the returns of the market index.

Multi-factor models

A generalisation of the structure presented in equation 3.1 posits k factors

or sources of common variation in stock returns.

r i = α

i+ β

1i F

1+ β

2i F

2+ .... + β

ki F

k + ε

iE (ε

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) = α

i). Each stock has a complement

of factor sensitivities or factor β s, 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:r X

= 0.05 + F 1

– 0.5 F 2

+ ε X

r

Y = 0.03 + 0.75 F

1+ 0.5 F

2+ ε

Y

r z

= 0.04 + 0.25 F 1

– 0.3 F 2

+ ε 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:

α p = (1/3) (0.05 + 0.03 + 0.04) = 0.04

β 1 p

= (1/3) (1 + 0.75 – 0.25) = 0.5




47

β 2 p

= (1/3) (–0.5 + 0.5 – 0.3) = –0.1

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

r p

= 0.04 + 0.5 F 1

– 0.1 F 2

+ ε p

where the final term is the idiosyncratic component in the portfolio return. Note that

the idiosyncratic volatility of the portfolio is ε p = (1/3)(ε

X + ε

Y + ε

z ) smaller than the

idiosyncratic volat ilit ies of port folios X, Y or Z because the idiosyncratic components are

independent.

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

ω X , ω

Y and ω

Z it must be the case that:

ω X

+ 0.75ωY

– 0.25ω Z

= 0.5 (3.3)

–0.05ω X

+ 0.5ωY

– 0.3ω 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:

ω X

+ ωY

+ ω 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 withsingle 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 εiand assume that all idiosyncratic terms have variance

σ 2. The variance of the idiosyncratic element of the portfolio return is

then:

y y .

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 forthe portfolio return is n-1

σ2. 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.

1 In general, if I have

a k-factor model I will need k+ 1 stocks to

do thi s.




49

Using the prior argument, to replicate asset X’s factor sensitivities, we

construct a portfolio with weight β 1X

on the first factor-replicating portfolio,

weight β 2X

on the second factor-replicating portfolio and weight 1 – β 1X

– β 2X

on the risk-free asset. The expected return of the replicating portfolio is

hence:

β 1X

(r f

+ λ1) + β

2X (r

f + λ

2) + (1 – β

1X– β

2X) r

f = r

f + β

1X λ

1+ β

2X λ

2.

(3.6)

Hence, using our factor-replicating portfolios we can write the expectedreturn 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.

Note that equation 3.6 can be used to test the factor model. This is the

second stage test of factor models mentioned in the previous chapter in the

context of the CAPM. Equation 3.6 states that average returns on assets are

higher if those assets have higher factor loadings ( β s); the factors are the

same for all assets. This is a cross-sectional statement as it compares average

returns for different assets. We can regress average returns on assets in

excess of r f

on the historical β s of these assets (here β is the regressor, not

the coefficient). If the factor model performs well then the intercept of thisregression should be close to zero.

The reason this regression is called a second stage regression is because we

must first find β s by running a time series regression for each asset on the

factor mimicking portfolios. These regressions can also be used to test the

factor model, these are called first stage tests. We can use equation 3.6 to

derive this equation as well. Combine equations 3.2 and 3.6 by noting that

the iin equation 3.6 is the expected return on asset i, given by equation

3.2:

r it = (rf + β

1i λ

1 + β

2i λ

2 ) + β

1i F

1t + β

2i F

2t + ε

it (3.7)

r it – r f = β 1i ( λ1 +F 1t )+ β 2i ( λ2 + F 2t ) + εit = β 2t ( λ1 + F 1t ) + β 2i( λ2 + F 2t ) + εit ,(3.8)

where j+F

jt is the excess return on the jth factor-replicating portfolio

(plus some idiosyncratic risk if markets are incomplete). Thus a time series

regression of r it

– r f

on excess factor returns implies that the intercept must

be zero; this must be true for each asset.

A practical question is how close to zero must the intercept be in both the

first and second stages in order for us to accept a model as being ‘close’ to

the data? Consider the first stage which states that every asset must have

a zero intercept. Suppose we found that 15 out of 100 tested assets had

intercepts different from zero at 5 per cent significance. A naïve application

of statistics would suggest rejection of the factor model. However, rejectionis not as clear cut as it might appear.

Suppose you were told that one of the assets with a non-zero intercept was

McDonalds. It would then not be surprising if we also found Burger King to

have a non-zero intercept because the two are likely to be highly correlated

even when controlling for standard factors. The 100 tested assets may not

all be truly independent and we are likely to see highly correlated assets

both be rejected or both not be rejected. If the 15 assets that are rejected

are all highly correlated, while the remaining 85 are not, we should not

reject the model. Gibbons, Ross and Shanken (1989) provide a procedure

to test the intercepts jointly for many assets, some of which are potentially

correlated.Let us now turn to the second stage test which also states that the intercept

(this time in a cross-sectional regression) must be zero. We can check for

the significance of the intercept in the usual way. However, when doing




50

this we are implicitly making an assumption about the cross-sectional

distribution of returns. Fama and MacBeth (1973) suggested an alternative

implementation of the second stage test which avoids making such

assumptions. Instead of running a single regression of average historical

returns on historical β s they suggest running a separate regression each

year; for each year regress the realised returns on β s calculated over some

recent period. As a result for each year there will be a separate estimate of

the intercept. They suggest using the distribution of intercepts to calculate

significance.

The arbitrage pricing theory

Consider an arbitrary asset. The previous subsection 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, r X, must be identical tothe expression arrived at in equation 3.6, that is:

E (r X) = r

f + β

1X λ

1+ β

2X λ

2. (3.9)

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 areformed.

Example

In the previous two-factor example, we determined the expected returns on the two

factor-replicating portfolios. Denoting the expected return on the i th factor-replicating

portfolio by E (r i) we have:

E (r 1) = 8.29% E (r

2) = 1.71% E (r

3) = 5.14%.

Hence, the premiums associated with the two factors are:

λ1

= 8.29 – 5.14 = 3.15%, λ2

= 1.71 – 5.14 = 3.43%.

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

E (r i) = 5.14 + 3.15 β

1i– 3.43 β

2i.

To check that this works, substitute (for example) portfolio C’s factor sensitivities into the

preceding expression. This gives:

E (r C) = 5.14 + 3.15 (0.5) – 3.43 (0.5) = 5%,

and hence, agrees with the expected return implied by the original representat ion 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.




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Activity

Assume that a new well-diversified portfolio, D, is added to our world. This asset has

sensitivit ies 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.

Multi-factor models in practice

As discussed earlier, the CAPM is a one-factor model where the only factor

is the excess market return. Securities with higher loading ( β ) on the

market return should have higher expected returns; nothing else should

matter for expected returns. Furthermore, the α of each security should be

zero.

Eugene Fama and Ken French illustrated the failure of the CAPM by

forming portfolios of securities in a particular way. First, for each security

they calculated the firm’s size (market cap) and its market-to-book ratio(a ratio of the firm’s market value to its book value). They then formed

cut-offs based on size and book-to-market, and assigned firms to one of

five quintiles for each trait. This resulted in 25 different portfolios (i.e.

large size and small book-to-market, small size and medium size book-to-

market, etc.), this is called a double sort. Once a year the portfolios would

be updated to take into account any changes to firm characteristics.

Fama and French showed that portfolios of small firms tended to have

larger returns than portfolios of large firms, portfolios of high book-to-

market (value) firms tended to have larger returns than portfolios of low

book-to-market (growth) firms. Interestingly, these patterns remained even

once controlling for market risk.

Recall that the first stage test of the CAPM implies that for any asset or

portfolio, a regression of that asset’s returns on the market should have

an intercept (α) of zero. Portfolios of small firms and value firms had

positive α implying their returns were higher than predicted by the CAPM,

conversely portfolios of large and growth firms had negative αs implying

their returns were lower than predicted by the CAPM. This is evident in

Table 3.1, which shows CAPM αs for portfolios double sorted on size and

book-to-market.

Growth 2 3 4 Value

Small –0.573 –0.105 0.151 0.362 0.5282 –0.213 0.146 0.295 0.312 0.363

3 –0.136 0.160 0.262 0.291 0.276

4 0.005 0.049 0.156 0.209 0.163

Big –0.014 0.022 0.038 –0.013 –1.020

Table 3.1

Since the CAPM could not adequately explain the cross-section of returns,

Fama and French looked for additional risk factors. Given the performance

of small and value stocks, it was natural to think those two characteristics

were related to risk. They constructed a zero cost portfolio which took a

long position in small stocks and a short position in large stocks and calledit SMB (small minus big). Similarly, they constructed a zero cost portfolio

which took a long position in value stocks and a short position in growth

stocks and called it HML (high minus low).




52

Fama and French augmented the CAPM by these two additional factors,

creating what is known as the Fama and French three-factor model. As

before with the CAPM, multifactor models can be tested by a first stage

time series test, in which each asset’s return is regressed on the factors;

each should be near zero. The Fama and French three-factor model

performed much better than the CAPM on the 25 portfolios defined

above, Fama and French could not statistically reject that the 25 αs

were different from zero. The Fama and French model is commonly

used as a replacement to the CAPM to assess risk as well as managerial

performance.

Narasimhan Jegadeesh and Sheridan Titman found another set of

portfolios whose returns could not be explained by the CAPM or the Fama

and French three-factor model. Jegadeesh and Titman sorted stocks into

portfolios based on their past performance, they held these portfolios for

a year and then reassigned stocks to new portfolios. They found that a

portfolio long in stocks that performed well in the past, and short in stocks

that performed poorly in the past, had positive αs in both CAPM and

three-factor regressions, they called this portfolio MOM (momentum). The

momentum factor was added to the Fama and French three-factor modelby Mark Carhart. This augmented four-factor model does a somewhat

better job than the three-factor model at explaining the cross-section

of expected stock returns, it is also commonly used to assess risk and

managerial performance.

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 APTallows 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 gross domestic product growth are

important in expected return generation. Fama and French (1992, 1995),

show that size and book-to-market factors can help explain the cross-

section of stock returns while other factors, such as momentum, also

appear to be important. Work in this area is still progressing.




• 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-functioningfinancial markets should be arbitrage-free



Notes


Corporate Fi Nance

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