Corporate finance
P. Frantz, R. Payne, J. Favilukis
FN3092, 2790092
2011
Undergraduate study in
Economics, Management,
Finance and the Social Sciences
This subject guide is for a 300 course offered as part of the University of London
International Programmes in Economics, Management, Finance and the Social Sciences.
This is equivalent to Level 6 within the Framework for Higher Education Qualifications in
England, Wales and Northern Ireland (FHEQ).
For more information about the University of London International Programmes
undergraduate study in Economics, Management, Finance and the Social Sciences, see:
www.londoninternational.ac.uk
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 and
Political 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 into any correspondence relating to, or arising from, the guide. If you have any comments on this subject guide, favourable or unfavourable, please use the form at the back of this guide.
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Contents
Contents
Introduction to the subject guide .......................................................................... 1
Aims of the course......................................................................................................... 1
Learning 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 ................................................................................................................... 9
Aim .............................................................................................................................. 9
Learning outcomes ........................................................................................................ 9
Essential reading ........................................................................................................... 9
Further reading.............................................................................................................. 9
Overview ..................................................................................................................... 10
Introduction ................................................................................................................ 10
Fisher separation and optimal decision-making ............................................................ 10
Fisher separation and project evaluation ...................................................................... 13
The time value of money .............................................................................................. 14
The net present value rule ............................................................................................ 15
Other 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 tests 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
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92 Corporate finance
Essential reading ......................................................................................................... 43
Further reading............................................................................................................ 43
Overview ..................................................................................................................... 43
Introduction ................................................................................................................ 44
Single-factor models .................................................................................................... 44
Multi-factor models ..................................................................................................... 46
Broad-based portfolios and idiosyncratic returns........................................................... 47
Factor-replicating portfolios ......................................................................................... 48
The arbitrage pricing theory ......................................................................................... 50
Multi-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 ...................................................................................................... 55
Essential reading ......................................................................................................... 55
Further reading............................................................................................................ 55
Overview ..................................................................................................................... 55
Varieties of derivatives ................................................................................................. 56
Derivative asset payoff profiles ..................................................................................... 57
Pricing forward contracts ............................................................................................. 59
Binomial option pricing setting .................................................................................... 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
A reminder of your learning outcomes.......................................................................... 70
Key terms .................................................................................................................... 70
Sample examination questions ..................................................................................... 71
Chapter 5: Efficient markets: theory and empirical evidence .............................. 73
Aim of the chapter....................................................................................................... 73
Learning outcomes ...................................................................................................... 73
Essential reading ......................................................................................................... 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.......................................................................... 85
Key terms .................................................................................................................... 85
Sample examination questions ..................................................................................... 86
ii
Contents
Chapter 6: The choice of corporate capital structure ........................................... 89
Aim of the chapter....................................................................................................... 89
Learning outcomes ...................................................................................................... 89
Essential reading ......................................................................................................... 89
Further reading............................................................................................................ 89
Overview ..................................................................................................................... 89
Basic features of debt and equity ................................................................................. 90
The Modigliani–Miller theorem .................................................................................... 91
Modigliani–Miller and Black–Scholes ........................................................................... 93
Modigliani–Miller and corporate taxation..................................................................... 94
Modigliani–Miller with corporate and personal taxation ............................................... 97
Summary ..................................................................................................................... 98
A reminder of your learning outcomes.......................................................................... 99
Key terms .................................................................................................................... 99
Sample examination questions ..................................................................................... 99
Chapter 7: Leverage, WACC and the Modigliani-Miller 2nd proposition ........... 101
Aim of the chapter..................................................................................................... 101
Learning 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 information, agency costs and capital structure .......... 111
Aim of the chapter..................................................................................................... 111
Learning outcomes .................................................................................................... 111
Essential reading ....................................................................................................... 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
A reminder of your learning outcomes........................................................................ 124
Sample examination questions ................................................................................... 124
Chapter 9: Dividend policy ................................................................................. 127
Aim of the chapter..................................................................................................... 127
Learning outcomes .................................................................................................... 127
Essential reading ....................................................................................................... 127
Further reading.......................................................................................................... 127
Overview ................................................................................................................... 128
Modigliani–Miller meets dividends ............................................................................. 128
Prices, dividends and share repurchases ..................................................................... 129
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92 Corporate finance
Dividend policy: stylised facts ..................................................................................... 129
Taxation and clientele theory ..................................................................................... 131
Asymmetric information and dividends ....................................................................... 132
Agency costs and dividends ....................................................................................... 133
Summary ................................................................................................................... 133
A reminder of your learning outcomes........................................................................ 134
Key terms .................................................................................................................. 134
Sample examination questions ................................................................................... 134
Chapter 10: Mergers and takeovers ................................................................... 135
Aim of the chapter..................................................................................................... 135
Learning outcomes .................................................................................................... 135
Essential reading ....................................................................................................... 135
Further reading.......................................................................................................... 135
Overview ................................................................................................................... 136
Merger motivations ................................................................................................... 136
A numerical takeover example ................................................................................... 137
The market for corporate control ................................................................................ 138
The impossibility of efficient takeovers ....................................................................... 139
Two ways to get efficient takeovers ............................................................................ 140
Empirical evidence ..................................................................................................... 141
Summary ................................................................................................................... 143
A reminder of your learning outcomes........................................................................ 143
Key terms .................................................................................................................. 143
Sample examination questions ................................................................................... 144
Appendix 1: Perpetuities and annuities ............................................................. 145
Perpetuities ............................................................................................................... 145
Annuities .................................................................................................................. 146
Appendix 2: Sample examination paper ............................................................ 147
iv
Introduction to the subject guide
Introduction to the subject guide
This subject guide for 92 Corporate finance, a ‘300’ 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 statistical
theory alongside the standard optimisation techniques taught in basic
economics courses. Another important part of this section will be the
use of absence-of-arbitrage techniques to price financial assets.
• In the second section, we will examine issues that come under the
broad heading of corporate finance. Here we will examine the key
decisions made by firms, how they affect firm value and empirical
evidence on these issues. The areas involved include the capital
structure decision, dividend policy, and mergers and acquisitions.
By studying these areas, you should gain an appreciation of optimal
financial policy on a firm level, conditions under which an optimal
policy actually exists and how the actual financial decisions of firms
may be explained in theoretical terms.
Aims of the 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
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92 Corporate finance
• 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 Introduction to
economics and 5A Mathematics 1 or 5B Mathematics 2 or 174
Calculus.
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; meanvariance 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-horizon
return 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.
2
Introduction to the subject guide
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.
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92 Corporate finance
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 initiations
and omissions’, Journal of Financial Economics 21(2) 1988, pp.149–76.
Healy, P., K. Palepu and R. Ruback ‘Does corporate performance improve after
mergers?’, Journal of Financial Economics 31(2) 1992, pp.135–76.
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 in
the foreign exchange market: a bootstrap approach’, Journal of International
Money and Finance 12(5) 1993, pp.451–74.
4
Introduction to the subject guide
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 capital
markets 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 managementspecific 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.
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92 Corporate finance
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 own
understanding 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 Summon 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.
• Chapter 1: here we focus on the evaluation of real investment projects
using the net present value technique and provide a comparison of NPV
with alternative forms of project evaluation.
6
Introduction to the subject guide
• Chapter 2: we look at the basics of risk and return of primitive
financial assets and mean–variance optimisation. We go on to derive
and discuss the capital asset pricing model (CAPM).
• Chapter 3: we present the arbitrage pricing theory, proposed as an
alternative to the CAPM and discuss multifactor models. We study
several recent multifactor models, such as the Fama and French threefactor model, and observe that they can explain a large fraction of the
variation in risky returns.
• Chapter 4: here we look at derivative assets. We begin with the
nature of forward, future, option and swap contracts, then move on to
pricing derivative assets via absence-of-arbitrage arguments. We also
include a description of binomial option pricing models and end with
the Black–Scholes analysis.
• Chapter 5: in this chapter, we examine the efficiency of financial
markets. We present the concepts underlying market efficiency and
discuss the empirical evidence on efficient markets. 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 long
horizon returns.
• Chapter 6: here we turn to corporate finance issues, treating the decision
over a corporation’s capital structure. The essential issue is what levels of
debt and equity finance should be chosen in order to maximise firm value.
• Chapter 7: 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.
• Chapter 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.
• Chapter 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?
• Chapter 10: 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
Important: 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
7
92 Corporate finance
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 be
answered.
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
8
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
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 and
activities, 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.
9
92 Corporate finance
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 or
decisions of the firm (i.e. its capital structure, borrowing or lending
activities, or dividend policy). The first part of this chapter presents what
is known as the Fisher separation theorem. What follows is a statement
of the theorem. This theorem allows us to say the following: under
certain conditions (which will be presented in the following section), the
shareholders can delegate to the management the task of choosing which
projects to undertake (i.e. determining the optimal package of investment
projects), whereas they themselves determine the optimal financial
decisions. Hence, the theory implies that the investment and financing
choices can be completely disconnected from each other and justifies our
limited definition of the firm for the time being.
Fisher separation and optimal decision-making
Consider the following scenario. A firm exists for two periods
(imaginatively named period 0 and period 1). The firm has current funds
of m and, without any investment, will receive no money in period 1.
Investments can be of two forms. The firm can invest in a number of
physical investment projects, each of which costs a certain amount of cash
in period 0 and delivers a known return in period 1. The second type of
investment is financial in nature and permits the firm to borrow or lend
unlimited amounts at rate of interest r. Finally the firm is assumed to have
a standard utility function in its period 0 and period 1 consumption. (By
consumption we mean the use of any funds available to the firm net of any
costs of investment.)
Let us first examine the set of physical investments available. The firm
will logically rank these investments in terms of their return, and this will
yield a production opportunity frontier (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
10
Chapter 1: Present value calculations and the valuation of physical investment projects
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 between
period 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
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92 Corporate finance
Now we can proceed to analyse optimal decision-making when firms
invest in both financial and physical assets. Assume that the firm is at the
beginning of period 0 and trying to decide on its investment plan. It is
clear that, to maximise firm value, the projects undertaken should be those
with the greatest return. Knowing that the return on financial investment
is always (1+r), the firm will first invest in all physical investment projects
with returns greater than (1+r ). These are those projects on the production
possibility frontier (PPF) between points m and I on Figure 1.3.1 Projects
above I on the PPF have returns that are dominated by the return from
financial investment.
Hence, the firm physically invests up to point I. Note that, at this point,
we have not mentioned the firm’s preferences over period 0 and period
1 consumption. Hence, the decision to physically invest to I will be taken
by all firms regardless of the preferences of their owners. Preferences
come into play when we consider what financial investments should be
undertaken.
The firm’s physical investment policy takes it to point I, from where it can
borrow or lend on the capital market. Borrowing will move the firm to
the south-east along a line starting at I and with slope –(1+r); lending will
take the firm north-west along a similarly sloped line. Two possible optima
are shown on Figure 1.3. The optimum at point X is that for a firm whose
owners prefer period 1 consumption relative to period 0 consumption (and
have hence lent on the capital market), whereas a firm locating at Y has
borrowed, as its owners prefer date 0 to date 1 consumption.
Figure 1.3 demonstrates the key insight of Fisher separation. All firms,
regardless of preferences, will have the same optimal physical investment
policy, investing to the point where the PPF and capital market line are
tangent. Preferences then dictate the firm’s borrowing or lending policy
and shift the optimum along the capital market line. The implication of
this is that, as it is physical investment that alters firm value, all agents
(i.e. regardless of preferences) agree on the physical investment policy that
will maximise firm value. More specifically, the shareholders of the firm
can delegate choice of investment policy to a manager whose preferences
may differ from their own, while controlling financial investment policy in
order to suit their preferences.
Figure 1.3
12
1
The absolute value of
the slope of the PPF can
be equated with the
return on physical
investment. For all points
below I on the PPF, this
slope exceeds that of
the capital market line
and hence defines the
set of desirable physical
investment projects.
Chapter 1: Present value calculations and the valuation of physical investment projects
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 I0. Given that
the firm has date 0 income of m and no date 1 income, aside from that
accruing from physical investment, the horizontal intercept of the capital
market line upon which the firm has located is:
where Π(I0) 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 transaction
costs associated with the use of the capital market. However, in practical
situations, these conditions are unlikely to be met. A particular example
is given in Figure 1.4. Here we have assumed that the rate at which
borrowing occurs is greater than the rate of interest paid on lending (as
the real world would dictate). Figure 1.3 shows that there are now two
points at which the capital market lines and the production opportunities
frontier are tangential. This then implies that agents with different
preferences will choose differing physical investment decisions and,
therefore, Fisher separation breaks down.
Figure 1.4
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92 Corporate finance
Agents with strong preferences for future consumption will physically
invest to point X and then financially invest to an optimum on the
capital market lending line (CML). Those with strong preferences for
current consumption physically invest to point Y and borrow (along
CML’). Finally, a set of agents may exist who value current and future
consumption similarly, and these will optimise by locating directly on the
PPF and not using the capital market at all. An example of an optimum of
this type is point Z on Figure 1.4.
The time value of money
In the preceding section we demonstrated the Fisher separation theorem
and the manner in which physical and financial investment decisions can
be disconnected. The major implication of this theorem is that the set of
desirable physical investment projects does not depend on the preferences
of individuals. In the following sections we shall focus on the way in
which individual physical investment projects should be evaluated. Our
key methodology for this will be the NPV rule, mentioned in the preceding
section. In the following sections we will show you how to apply the rule
to situations involving more than one period and with time-varying cash
flows.
To begin, let us consider a straightforward question. Is $1 received today
worth the same as $1 received in one year’s time? A naïve response to
this question would assert that $1 is $1 regardless of when it is received,
and hence the answer to the question would be yes. A more careful
consideration of the question brings the opposite response however. Let’s
assume I receive $1 now. If I also assume that there is a risk-free asset in
which I can invest my dollar (e.g. a bank account), then in one year’s time
I will receive $(1+r), assuming I invest. Here, r is the rate of return on the
safe investment. Hence $1 received today is worth $(1+r) in one year. The
answer to the question is therefore no. A dollar received today is worth
more than a dollar received in one year or at any time in the future.
The above argument characterises the time value of money. Funds are
more valuable the earlier they are received. In the previous paragraph we
illustrated this by calculating the future value of $1. We can similarly
illustrate the time value of money by using present values. Assume I
am to receive $1 in one year’s time and further assume that the borrowing
and lending rate is r. How much is this dollar worth in today’s terms?
To answer this second question, put yourself in the position of a bank.
Knowing that someone is certain to receive $1 in one year, what is the
maximum amount you would lend him or her now? If I, as a bank, were to
lend someone money for one year, at the end of the year I would require
repayment of the loan plus interest (at rate r). Hence if I loaned the
individual $x, I would require a repayment of $x(1+r). This implies that the
maximum amount I should be willing to lend is implicitly defined by the
following equation:
$x(1+r) = $1
(1.2)
such that:
(1.3)
The value for x defined in equation 1.3 is the present value of $1
received in one year’s time. This quantity is also termed the discounted
value of the $1.
14
Chapter 1: Present value calculations and the valuation of physical investment projects
You can see the present and future value concepts pictured in Figure 1.2.
If you recall, Figure 1.2 just plots the CML for a given level of initial funds
(m) assuming no funds are to be received in the future. The future value
of this amount of money is simply the vertical intercept of the CML (i.e.
m(1+r)), and obviously the present value of m(1+r) is just m.
The present and future value concepts are straightforwardly extended
to cover more than one period. Assume an annual compound interest rate
of r. The present value of $100 to be received in k year’s time is:
(1.4)
whereas the future value of $100 received today and evaluated k years
hence is:
FVK (100) = 100(1 + r)K.
(1.5)
Activity
Below, there are a few applications of the present and future value concepts. You should
attempt to verify that you can replicate the calculations.
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 Ck.
In words, all that the NPV rule does is to compute the present value of all
receipts or payments. This allows direct comparisons of monetary values,
as all are evaluated at the same point in time. The NPV of the project is
then just the sum of the present values of receipts, less the sum of the
present values of the payments.
Using the notation given above and again assuming a rate of return of r,
the NPV can be written as:
.
(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.
15
92 Corporate finance
This gives a very straightforward method for project evaluation. Compute
the NPV of the project (which is a simple calculation), and if it is greater
than zero, the project is acceptable.
Example
Consider a manufacturing firm, which is contemplating the purchase of a new piece of
plant. The rate of interest relevant to the firm is 10 per cent. The purchase price is £1,000.
If purchased, the machine will last for three years and in each year generate extra revenue
equivalent to £750. The resale value of the machine at the end of its lifetime is zero. The
NPV of this project is:
NPV = 750 + 750 + 750 – 1000 = 865.14.
(1.1)3 (1.1)2 (1.1)1
As the NPV of the project exceeds zero, it should be accepted.
In order to familiarise yourself with NPV calculations, attempt the following
activities by calculating the NPV of each project and assessing its desirability.
Activity
Assume an interest rate of 5 per cent. Compute the NPV of each of the following projects,
and state whether each project should be accepted or not.
• Project A has an immediate cost of $5,000, generates $1,000 for each of the next six
years and zero thereafter.
• Project B costs £1,000 immediately, generates cash flows of £600 in year 1,
£300 in year 2 and £300 in year 3.
• Project C costs ¥10,000 and generates ¥6,000 in year 1. Over the following years, the
cash flows decline by ¥2,000 each year, until the cash flow reaches zero.
• Project D costs £1,500 immediately. In year 1 it generates £1,000. In year 2 there is a
further cost of £2,000. In years 3, 4 and 5 the project generates revenues of £1,500
per annum.
Up to this point we have just considered single projects in isolation,
assuming that our funds were enough to cover the costs involved. What
happens, first of all, if the members of a set of projects are mutually
exclusive?2 The answer is simple. Pick the project that has the greatest
NPV. Second, what should we do if we have limited funds? It may be the
case that we are faced with a pool of projects, all of which have positive
NPVs, but we only have access to an amount of money that is less than the
total investment cost of the entire project pool. Here we can rely on
another nice feature of the NPV technique. NPVs are additive across
projects (i.e. the NPV of taking on projects A and B is identical to the NPV
of A plus the NPV of B). The reason for this should be obvious from the
manner in which NPVs are calculated. Hence, in this scenario, we should
calculate all project combinations that are feasible (i.e. the total investment
in these projects can be financed with our current funds). Then calculate
the NPV of each combination by summing the NPVs of its constituents, and
finally choose the combination that yields the greatest total NPV.
Finally, we should devote some time to discussion of the ‘interest rate’
we have used to discount future cash flows. Until now we have just
referred to r as the rate at which one can borrow or lend funds. A more
precise definition of r is that r is the opportunity cost of capital. If we are
considering the use of the NPV rule within the context of a firm, we have
to recognise that the firm has several sources of capital, and the cost of
each of these should be taken into account when evaluating the firm’s
16
2
By this we mean that
taking on any one of the
set of projects precludes
us from accepting any of
the others.
Chapter 1: Present value calculations and the valuation of physical investment projects
overall cost of capital. The firm can raise funds via equity issues and
debt issues, and it is likely that the costs of these two types of funds will
differ. Later on in this chapter and in those that follow, we will present
techniques by which the firm can compute the overall cost of capital for its
enterprise.
Other project appraisal techniques
The NPV methodology for project appraisal is by no means the only
technique used by firms to decide on their physical investment policy. It is,
however, the optimal technique for corporate management to use if they
wish to maximise expected shareholder wealth. This result is obvious from
our Fisher separation analysis. In this section we talk about three of NPV’s
competitors, the payback rule, the internal rate of return (IRR) rule,
and the multiples method, which are sometimes used in practice.
The payback rule
Payback is a particularly simple criterion for deciding on the desirability
of an investment project. The firm chooses a fixed payback period, for
example, three years. If a project generates enough cash in the first three
years of its existence to repay the initial investment outlay, then it is
desirable, and if it doesn’t generate enough cash to cover the outlay, it
should be rejected. Take the cash-flow stream given in the following table as
an example.
Year
Cash flow
0
1
2
3
4
–1,000
250
250
250
500
Table 1.1
A firm that has chosen a payback period of three years and is faced with
the project shown in Table 1.1 will reject it as the cash flow in years 1 to
3 (750) doesn’t cover the initial outlay of 1,000. Note, however, that if the
firm used a payback period of four years, the project would be acceptable,
as the total cash flow to the project would be 1,250, which exceeds the
outlay. Hence, it’s clear that the crucial choice by management is of the
payback period.
We can also use the preceding example to illustrate the weaknesses
of payback. First, assume that the firm has a payback period of three
years. Then, as previously mentioned, the project in Table 1.1 will not be
accepted. However, assume also that, instead of being 500, the project
cash flow in year 4 is 500,000. Clearly, one would want to revise one’s
opinion on the desirability of the project, but the payback rule still says
you should reject it. Payback is flawed, as a portion of the cash-flow
stream (that realised after the payback period is up) is always ignored in
project evaluation.
The second weakness of payback should be obvious, given our earlier
discussion of NPV. Payback ignores the time value of money. Sticking with
the example in Table 1.1, assume a firm has a payback period of four years.
Then the project as given should be accepted (as total cash flow of 1,250
exceeds investment outlay of 1,000). But what’s the NPV of this project?
If we assume, for example, a required rate of return of 10 per cent, then
the NPV can be shown to be negative. (In fact the NPV is –36.78. As a
self-assessment activity, show that this is the case.) Hence application of
the payback rule tells us to accept a project that would decrease expected
shareholder wealth (as shown by application of the NPV rule). This flaw
could be eliminated by discounting project cash flows that accrue within
17
92 Corporate finance
the payback period, giving a discounted payback rule, but such a
modification still wouldn’t solve the first problem we highlighted.
The internal rate of return 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 Ci is the project cash flow in year i, and I is the initial (i.e. year 0)
investment outlay. Comparison of equation 1.7 with 1.6 shows that the
project IRR is the discount rate that would set the project NPV to zero.
Once the IRR has been calculated, the project is evaluated by comparing
the IRR to a predetermined required rate of return known as a hurdle
rate. If the IRR exceeds the hurdle rate, then the project is acceptable,
and if the IRR is less than the hurdle rate it should be rejected. A graphical
analysis of this is presented in Figure 1.5, which plots project NPV against
the rate of return used in the NPV calculation. If r* is the hurdle rate used
in project evaluation, then the project represented by the curve on the
figure is acceptable as the IRR exceeds r*. Clearly, if r* is also the correct
required rate of return, which would be used in NPV calculations, then
application of the IRR and NPV rules to assessment of the project in Figure
1.5 gives identical results (as at rate r* the NPV exceeds zero).
Figure 1.5
Calculation of the IRR need not be straightforward. Rearranging equation
1.7 shows us that the IRR is a solution to a kth order polynomial in r.
In general, the solution must be found by some iterative process, for
example, a (progressively finer) grid search method. This also points to
a first weakness of the IRR approach; as the solution to a polynomial,
the IRR may not be unique. Several different rates of return might satisfy
equation 1.7; in this case, which one should be used as the IRR? Figure 1.6
gives a graphical example of this case.
18
Chapter 1: Present value calculations and the valuation of physical investment projects
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
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92 Corporate finance
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.
E (10 yr) %
Equity
Debt
E
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 quantity 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 effects
of 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
20
Chapter 1: Present value calculations and the valuation of physical investment projects
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 gives
us an estimated value for the conglomerate, this estimate is on average
12% greater than the traded value of the conglomerate. This is called the
conglomerate discount. 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 and
bonds
To end this chapter, we will discuss very briefly how to value common
stocks and bonds through the application of present value techniques.
Stocks
Consider holding a common equity share from a given corporation. To
what does this equity share entitle the holder? Aside from issues such as
voting rights, the share simply delivers a stream of future dividends to
the holder. Assume that we are currently at time t, that the corporation is
infinitely long-lived (such that the stream of dividends goes on forever)
and that we denote the dividend to be paid at time t+i by Dt+i. Also
assume that dividends are paid annually. Denoting the required annual
rate of return on this equity share to be re, 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.
21
92 Corporate finance
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), D0(1+g)2, D0(1+g)3 and so on. This type of
cash-flow stream is known as a perpetuity with growth, and its
present value can be calculated very simply.3 In this setting the price of the
equity share is:
0
.
3
See Appendix 1.
(1.9)
This is the Gordon growth model of equity valuation. As is obvious
from the preceding discussion, it is only valid if you can assert that
dividends grow at a constant rate.
Note also that if you have the share price, dividend just paid and an
estimate of dividend growth, you can rearrange equation 1.9 to give the
required rate of return on the stock – that is:
.
(1.10)
The first term in 1.10 is the expected dividend yield on the stock, and the
second is expected dividend growth. Hence, with empirical estimates of
the previous two quantities, we can easily calculate the required rate of
return on any equity share.
Activity
Attempt the following questions:
1. An investor is considering buying a certain equity share. The stock has just paid a
dividend of £0.50, and both the investor and the market expect the future dividend to
be precisely at this level forever. The required rate of return on similar equities is 8 per
cent. What price should the investor be prepared to pay for a single equity share?
2. A stock has just paid a dividend of $0.25. Dividends are expected to grow at
a constant annual rate of 5 per cent. The required rate of return on the share
is 10 per cent. Calculate the price of the stock.
3. A single share of XYZ Corporation is priced at $25. Dividends are expected
to grow at a rate of 8 per cent, and the dividend just paid was $0.50. What is
the required rate of return on the stock?
Bonds
In principle, bonds are just as easy to value.
• A discount or zero coupon bond is an instrument that promises
to pay the bearer a given sum (known as the principal) at the end of
the instrument’s lifetime. For example, a simple five-year discount bond
might pay the bearer $1,000 after five years have elapsed.
• Slightly more complex instruments are coupon bonds. These not
only repay the principal at the end of the term but in the interim entitle
the bearer to coupon payments that are a specified percentage of
the principal. Assuming annual coupon payments, a three-year bond
with principal of £100 and coupon rate of 8 per cent will give annual
payments of £8, £8 and £108 in years 1, 2 and 3.
In more general terms, assuming the coupon rate is c, the principal is P
and the required annual rate of return on this type of bond is rb, the price
of the bond can be written as:4
.
22
(1.11)
4
In our notation a
coupon rate of 12
per cent, for example,
implies that c = 0.12;
the discount rate used
here, rb , is called the
yield to maturity of the
bond.
Chapter 1: Present value calculations and the valuation of physical investment projects
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%)
23
92 Corporate finance
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?
24
Chapter 2: Risk and return: mean–variance analysis and the CAPM
Chapter 2: Risk and return:
mean–variance analysis and the CAPM
Aim of the chapter
The aim of this chapter is to derive the capital asset pricing model (CAPM)
enabling us to price financial assets. In order to do so, we introduce the
mean–variance analysis setting, in which investors care solely about
financial assets’ expected returns and variances of returns, as well as the
statistical tools enabling us to calculate portfolios’ expected returns and
variances of returns.
Learning 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 reading
Hillier, 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 and
potential 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.
25
92 Corporate finance
We begin by introducing the basic statistical tools that will be needed
in our analysis, these being expected values, variances and
covariances. This leads to an analysis of the statistical characteristics
of portfolios of financial assets and ultimately to a presentation of the
standard mean–variance optimisation problem. The key result of mean–
variance analysis is known as two-fund separation, and this result
underlies the 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 expected returns and return variances or standard deviations.
The expected return on a given asset can be thought of as the reward
gained from holding it, whereas the return variance is a measure of total
asset risk.
Let us define notation. First, we should clarify the way in which we are
thinking about asset returns. The return on an asset is assumed to be a
random variable with known distributional characteristics. Each individual
asset is assumed to have an expected return of E(rj) and return variance
σ2j. Assets i and j are assumed to have covariance σij . Similarly, we denote
the expected return of the portfolio held as E(Rp) and its variance by σ2P.
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 Bethlehem
Steel is 10.1 Hence, given the numbers of each share held, the total value
of this investor’s portfolio is $195. We further assume that the expected
returns on Microsoft and Bethlehem Steel are 10 per cent and 16 per cent
respectively, whereas their variances are 0.25 and 0.49.
We are now in a position to define the share of the entire portfolio value
that is contributed by each individual stockholding. These are referred
to as portfolio weights. The portfolio weight of Bethlehem Steel, for
example, is simply the value of the Bethlehem Steel holding divided by
$195 (i.e. 1/3 or approximately 33.3 per cent). Hence our US investor
allocates 1/3 of every dollar invested to Bethlehem Steel stock.
Activity
Calculate the portfolio weight for Microsoft, using the method presented above.
From the calculations undertaken it is clear that the sum of portfolio
weights must be unity. Each portfolio weight represents the share of total
portfolio value contributed by a given asset. Obviously, aggregating these
shares across all assets held will give a result of unity. Hence, extending
the notation presented above, we denote the portfolio weight on asset i by
ai, and the preceding argument implies that α1= 1.
26
1
These prices are in US
cents.
Chapter 2: Risk and return: mean–variance analysis and the CAPM
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), σ2x and σ2y. The covariance
between the random variables is σxy.
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 P = a σ x + b σ y + 2abσxy.
(2.2)
2
2 2
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 σ2i. The covariance between xi and xj is σij. 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 σ2x and
σ2y. The covariance between the random variables is σxy. 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
27
92 Corporate finance
linear combination can be rewritten using the correlation as follows:
σ2p = a2σ2x + b2σ2y + 2abxyσxσy.
(2.6)
This is a straightforward substitution of equation 2.5 into equation 2.2.
Now we are in a position to calculate the characteristics of our American
investor’s portfolio. Let us take the simplest possible case first and assume
that the returns are uncorrelated (i.e. xy = 0). Recalling that the portfolio
weights on Microsoft and Bethlehem Steel are 2/3 and 1/3 respectively, we
can use equations 2.1 and 2.6 to derive the expected return and variance
of the investor’s portfolio. These calculations yield:
(2.7)
.
(2.8)
Hence, as we would anticipate, the expected portfolio return lies between
the returns on the individual assets. The portfolio variance, however, is
actually less than that on the return of either of the component assets (i.e.
the risk associated with the portfolio is lower than the risks associated
with either individual asset). This result is one that should be kept in mind
and is the focus of the next section.
Now let’s change our assumption regarding the correlation between the
two asset returns. Assume now that xy = 0.5. Obviously, the expected
portfolio return won’t change (as equation 2.1 doesn’t involve the
correlation or covariance at all). The portfolio variance now becomes:
.
(2.9)
The portfolio variance has obviously increased, although it is still less than
the return variances of either component assets.
Activity
Assume that xy = – 0.5. Calculate the portfolio return variance in this case, using the
data on portfolio weights and asset return variances given above.
Now, given the expected returns, return variances and covariances for
any set of assets, we should be able to calculate the expected return and
variance of any portfolio created from those assets. At the end of this
chapter, you will find activities that require you to do precisely this, along
with solutions to some of these activities.
Diversification
A point that we noted from the calculations of expected portfolio returns
and variances above was that, in all of our calculations, the variance of the
portfolio return was lower than that on any individual component’s asset
return.2 Hence, it seems as though, by forming bundles of assets, we can
eliminate risk. This is true and is known as diversification: through holding
portfolios of assets, we can reduce the risk associated with our position.
Why is this the case? The key is that, in our prior analysis and in real stock
return data, the correlations between returns are less than perfect. If two
returns are imperfectly correlated it implies that when returns on the first are
above average, those on the second need not be above average. Hence, to an
extent, the returns on such assets will tend to cancel each other out, implying
that the return variance for a portfolio of these stocks will be smaller than
the corresponding weighted average of the individual asset variances.
28
2
Note that this result
does not hold in general
(i.e. it may be the case
that the return variance
of a portfolio exceeds
the return variance of
one of the component
assets).
Chapter 2: Risk and return: mean–variance analysis and the CAPM
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 σ2i where i = 1 to N, and
the covariance between returns on assets i and j by σij. 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 σ2 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
29
92 Corporate finance
Mean–variance analysis
In the preceding two sections, we have demonstrated two important facts:
1. The expected return on a portfolio of assets is a linear combination of
the expected returns on the component assets.
2. An investor holding a diversified portfolio gains through the reduction
in portfolio variance, when asset returns are not perfectly correlated.
In this section, we use these facts to characterise the optimal holding of
risky assets for a risk-averse agent. Our fundamental assumption is that all
agents have preferences that only involve their expected portfolio return
and return variance. Utility is assumed to be increasing in the former
and decreasing in the latter. For illustrative purposes we begin using the
assumption that only two risky assets are available. The results presented,
however, generalise to the N asset case.
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(Rx) and E(Ry) and return variances of σ2x and σ2y.
The first question the investor wishes to answer is how the characteristics
of a portfolio of these assets (i.e. portfolio expected return and variance)
change as the portfolio weights on the assets change. Given equation 2.6,
the answer to this question is obviously dependent on the correlation
between the returns on the two assets.
First assume 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:
σ2P = (ασx + (1 – α)σy)2.
(2.13)
Taking the square root of equation 2.13, it is clear that the portfolio
standard 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 not a linear combination of
σx and σy. The reduction of portfolio risk through diversification will imply
that the mean–standard deviation frontier bows towards the y-axis. This
30
Chapter 2: Risk and return: mean–variance analysis and the CAPM
is also shown on Figure 2.2. The final curve on Figure 2.2 represents the
case where returns are perfectly negatively correlated. In this situation, a
portfolio can be constructed, which has zero standard deviation.
Activities
1. Assuming asset returns are perfectly negatively correlated, use equation 2.6 to find
the portfolio weights that give a portfolio with zero standard deviation. (Hint: write
down 2.6 with the correlation set to minus one and a =  and b = 1 – . Then
minimise portfolio variance with respect to .)
2. Assume that the returns on Microsoft and Bethlehem Steel have a correlation of 0.5.
Using the data provided earlier in the chapter, construct the mean–variance frontier
for portfolios of these two assets. Start with a portfolio consisting only of Microsoft
stock and then increase the portfolio weight on Bethlehem Steel by 0.1 repeatedly,
until the portfolio consists of Bethlehem Steel stock only.
From here on we will assume that return correlation is between plus and
minus one. The expected return–standard deviation locus for this case
is redrawn in Figure 2.3. In the absence of a risk-free asset, this locus is
named the mean–variance frontier. As our investor’s preferences are
increasing in expected return and decreasing in standard deviation, it
is clear that their optimal portfolio will always lie on the frontier and to
the right of the point labelled V. This point represents the minimumvariance portfolio. They will always choose a frontier portfolio at or to
the right of V, as these portfolios maximise expected return for a given
portfolio standard deviation. In the absence of a risk-free asset, this set of
portfolios is called the efficient set.
Figure 2.3
We can now, given a set of preferences for the investor, find their optimal
portfolio. The condition characterising the optimum is that an investor’s
indifference curve must be tangent to the mean–variance frontier.3 Two
such optima are identified on Figure 2.3 at R and S. The investor locating
at equilibrium point R is relatively risk-averse (i.e. their indifference curves
are quite steep), whereas the equilibrium at S is that for a less risk-averse
individual (with correspondingly flatter indifference curves). Figure 2.3
also shows suboptimal indifference curves for each set of preferences.
Hence, as Figure 2.3 demonstrates, in a world of two risky assets and no
risk-free asset, the optimal portfolio of risky assets held by an investor
depends on their preferences towards risk and return. The same is true
3
In technical terms, the
optimum is characterised
by the marginal rate of
substitution being equal
to the marginal rate of
transformation (i.e. the
slope of the indifference
curve equals the slope of
the frontier).
31
92 Corporate finance
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 rf 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)rf + aE(RP) = rf + a[E(RP) – rf ]
(2.14)
σ2Q = a2σ2P .
(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.
32
Chapter 2: Risk and return: mean–variance analysis and the CAPM
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 rf 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.
4
That is, choosing
portfolio K places an
investor on a CML with
greater expected returns
at each level of return
variance than does any
other.
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 riskfree asset, the efficient set is exactly the optimal CML.
The key result that is depicted in Figure 2.6 is known as two-fund
separation. Any risk-averse investor (regardless of their degree of riskaversion) 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
33
92 Corporate finance
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 at
the 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(RI) = aE(Rj) + (1 – a)E(RK)
(2.19)
σ1 = [a2σ 2j + (1 – a)2σ 2K + 2a(1 – a)σjK]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:
.
34
(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 transactions and
immediately receives the
sale price. Subsequently,
one uses the proceeds
from the sale to
repurchase a unit of the
asset, and deliver it to
the creditor. If the price
of the asset has dropped
in the interim, one
makes a cash profit.
Chapter 2: Risk and return: mean–variance analysis and the CAPM
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 2.24
and rearranging the resulting expression, we arrive at:
(2.25)
Defining βj = σjK / σ2K, equation 2.26 can be rewritten as:
E(Rj) = rj + βj[E(RK) – rf ].
(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.
6
The risk premium is
defined as the excess of
the expected return on
the tangency portfolio
over the risk-free rate.
Equilibrium and the CAPM
Equation 2.26 is simply derived from mean–variance analysis, and as
yet we have said nothing regarding equilibrium in asset markets. Capital
market equilibrium requires that the demand for risky securities be
identical to their supply. The supply of risky assets is summarised in the
market portfolio, which is defined below.
Definition
The market portfolio is the portfolio comprising all assets, where the
weights used in the construction of the portfolio are calculated as
the market capitalisation of each asset divided by the sum of market
capitalisations across all assets.
Two-fund separation gives us the fundamental result that all investors
hold efficient portfolios and, further, that all investors hold risky securities
in the same proportions (i.e. those proportions dictated by the tangency
portfolio (K)).7 For demand to be equal to supply in capital markets, it
must be the case that the market portfolio is constructed with identical
portfolio weights. The implication of this is simple: the market portfolio
and the tangency portfolio are identical. This allows us to express the
CAPM in the following form.
The capital asset pricing model
7
All investors perceive
the same efficient
set and tangency
portfolio due to our
assumption that they
have homogeneous
expectations regarding
asset returns.
Under the prior assumptions, the following relationship holds for all
expected portfolio returns:
E(Rj ) = Rf + βj [E(rM ) – rf ],
(2.27)
where E(RM ) is the expected return on the market portfolio, and βj 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
35
92 Corporate finance
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 rf , 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 βs, 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 εj we have the following equation:
rj = rf + βj [rM – rf ] + εj.
36
(2.28)
Chapter 2: Risk and return: mean–variance analysis and the CAPM
The variance of the risk-free return is zero by definition. Assuming that βj
is fixed we can represent the variance of asset j as:
σ2j = β2jσ2M + σ2ε.
(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 affect
expected 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 portfolio expected return and variance if the correlation between
returns on X and Y is unity.
8
You will find the
solutions to these
activities at the end of
this chapter.
2. Using the data from Question 1, recalculate the portfolio expected return and
variance, assuming that the correlation between returns is 0.5.
3. An investor forms a portfolio from two assets, P and Q, using portfolio weights of
one-third and two-thirds respectively. The expected returns on P and Q are
5 per cent and 7 per cent, and their respective return standard deviations are 0.4 and
0.5. Assuming that the return correlation is zero, calculate the expected return and
variance of the investor’s portfolio.
4. Assuming identical data to that in Question 3, recalculate the statistical properties of
the portfolio, assuming the return correlation for P and Q is –0.5.
The Roll critique and empirical tests of the CAPM
The final topic we touch on in this chapter is the empirical validity of the
CAPM. The model of equilibrium expected returns that we have developed
in the preceding sections of this chapter is obviously not guaranteed to
hold in practice and, hence, rather than just blindly accepting its output,
we should examine how it holds up when applied to real data. However,
this task brings us face-to-face with a problem first pointed out by Richard
Roll and hence known as the Roll critique.9
9
See Roll (1977).
The statement of the CAPM is identical to the proposition that the market
portfolio is mean–variance efficient. Hence, Roll pointed out that empirical
tests of the CAPM should seek to examine whether this is indeed the case.
However, he also noted that the market portfolio (or the return on the
market) is not observable to an econometrician, who wishes to conduct a
test. Empirical researchers generally use a broad-based equity index such
as the FTSE-100, S&P-500 or Nikkei 250 to proxy the market. But the true
market portfolio will contain other financial assets (such as bonds and
stocks not included in such indices) as well as non-financial assets such as
real estate, durable goods and even human capital. Hence, the validity of
tests of the CAPM depend critically on the quality of the proxy used for the
market portfolio.
Based on the above, Roll’s critique is simply that, due to the fact that
the market portfolio is not observable, the CAPM is not testable. We can
understand this through the following arguments. First, it might be the
case that the market portfolio is efficient (and hence the CAPM is valid),
but our chosen proxy for the market is not efficient, and hence our
37
92 Corporate finance
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 judge
the predictions of the CAPM against reality and trust the results. However,
many researchers have disregarded the prior discussion and estimated
the empirical counterpart of equation 2.27. From these estimates, such
researchers pass 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 + bi*Ft + eit
E[eit ] = 0
E[Ft*eit ]= 0
(2.30)
Note that ai is an asset specific constant, bi is an asset specific factor
loading, and eit is an idiosyncratic variable uncorrelated across assets. On
the other hand Ft 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
Xt = E[Xt] + et where et is independent of E[Xt], therefore Rit – Rf = E[Rit – Rf ]
+ υit and Rmt – Rf = E[Rmt – Rf ] + t where υ and  are idiosyncratic.
E[Rit– Rf ] = βi*E[Rmt– Rf ]
(2.31)
Rit – Rf – υit = βi*(Rmt – Rf) – βi*ηt
(2.32)
Rit – Rf =βi*(Rmt – Rf ) + (υit – βi*ηt) = βi*(Rmt – Rf ) + 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 – Rf = Ai + Bi*(Rmt – Rf )
(2.34)
By definition of a regression, Bi = Cov(Rit – Rf , Rmt – Rf )/Var(Rmt – Rf ), 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 we
find that many assets have Ai not 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 – Rf ]= G0+ G1*βi
The CAPM implies that G0 is zero and G1 is the average market premium
E[Rm – Rf ].
38
Chapter 2: Risk and return: mean–variance analysis and the CAPM
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 – Rf ], 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
0.9
10
See pp.185–86 of
Brealey and Myers
(2008).
0.85
0.8
0.75
E[R]
0.7
0.65
0.6
0.55
0.5
0.45
0.4
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
β
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). An
alternative viewpoint would be to argue that such results tell us little or
nothing about the validity of the CAPM due to the insight of Roll (1977).
39
92 Corporate finance
A reminder of your learning outcomes
Having completed this chapter, and 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.
Key terms
beta (β)
capital asset pricing model (CAPM)
correlation
covariance
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 return (%)
Market 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?
40
Chapter 2: Risk and return: mean–variance analysis and the CAPM
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 and
portfolios 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.
41
92 Corporate finance
Notes
42
Chapter 3: Factor models
Chapter 3: Factor models
Aim of the chapter
The aim of this chapter is to derive arbitrage pricing theory, an alternative
to the capital asset pricing model, enabling us to price financial assets.
Learning outcomes
By the end of this chapter, and having completed the Essential reading and
activities, you should be able to:
• understand single-factor and multi-factor model representations
• derive factor-replicating portfolios from a set of asset returns
• understand the notion of arbitrage strategies and that well-functioning
financial markets should be arbitrage-free
• derive arbitrage pricing theory and calculate expected returns using the
pricing formulas
• know how to test multifactor models.
Essential reading
Hillier, D., M. Grinblatt and S. Titman Financial Markets and Corporate Strategy.
(Boston, Mass.; London: McGraw-Hill, 2008) Chapter 6 (Factor Models and
the APT).
Further reading
Brealey, R., S. Myers and F. Allen Principles of Corporate Finance. (Boston,
Mass.; London: McGraw-Hill, 2008) Chapter 9 (Risk and Return).
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.
Copeland, T., J. Weston and K. Shastri Financial Theory and Corporate Policy.
(Reading, Mass.; Wokingham: Addison-Wesley, 2005) Chapter 6.
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.
Jegadeesh, N. and S. Titman ‘Returns to buying winners and selling losers’,
Journal of Finance 48 1993.
Overview
Empirically, expected returns appear to depend on several factors. For
this reason, multifactor models, such as the Fama and French three-factor
model are commonly used in practice to calculate expected returns. The
arbitrage pricing theory gives a theoretical basis for using such models.
As its name suggests, it rests on the notion that well-functioning financial
markets should be arbitrage-free. This, using a factor model of asset
43
92 Corporate finance
returns, implies restrictions on the relationship between asset returns and
generates and equilibrium pricing relationship.
Introduction
As we saw in the previous chapter, the CAPM was not sufficient to explain
the cross-section of expected asset returns. The CAPM was a one-factor
model and we can improve on the CAPM by including additional factors.
However, the CAPM was derived from micro-economic foundations, why
should additional factors matter for risk?
The arbitrage pricing theory (APT) gives an alternative to the CAPM as a
method to compute expected returns on stocks. The basis for the APT is a
factor model of stock returns, and we will define and discuss these models
first. From there we will demonstrate how to derive expected returns using
the idea that the returns on stocks, which are exposed to a common set of
factors, must be mutually consistent, given each stock’s sensitivity to each
factor.
To give structure to what we mean by ‘mutually consistent’, we need to
define the notion of an arbitrage. An arbitrage strategy is a strategy that
delivers non-negative returns in all states of the world, and strictly positive
returns in at least one state of the world. For example, a strategy that
yields an immediate, positive cash inflow and, further, is guaranteed not to
make a loss tomorrow. Faced with an investment strategy with this payoff
structure, any investor who prefers more to less would try to invest on an
infinite scale.
The idea that underpins the APT is that investment situations, such
as those described above, should not be permitted in well-functioning
financial markets. Then, if financial markets do not permit the existence
of arbitrage strategies, this places restrictions on the relationships between
the expected returns on assets given the factor structure underlying
returns.
Although the APT gives justification for why there may be multiple factors,
it does not identify specific factors. Factors should proxy for risk and may
be identified from economic fundamentals (such as the CAPM), or from
empirical observation. Eugene Fama and Ken French identified three
factors that do a relatively good job at explaining much of the variation
in expected stock returns. We will learn about their model, as well as
improvements on it, at the end of the chapter.
Single-factor models
Before using the notion of absence of arbitrage to provide pricing
relations, we need a basis for the generation of stock returns. Within
the context of the APT, this basis is given by the assumption that the
population of stock returns is generated by a factor model. The simplest
factor model, given below, is a one-factor model:
ri = αi + βi F + εi
E(εi) = 0.
(3.1)
In equation 3.1, the returns on stock i are related to two main components:
1. The first of these is a component that involves the factor F. This
factor is posited to affect all stock returns, although with differing
sensitivities. The sensitivity of stock i’s return to F is βi. Stocks that
have small values for this parameter will react only slightly as F
changes, whereas when βi is large, variations in F cause very large
movements in the return on stock i. As a concrete example, think of F
44
Chapter 3: Factor models
as the return on a market index (e.g. the S&P-500 or the FTSE-100),
the variations in which cause variations in individual stock returns.
Hence, this term causes movements in individual stock returns that are
related. If two stocks have positive sensitivities to the factor, both will
tend to move in the same direction.
2. The second term in the factor model is a random shock to returns,
which is assumed to be uncorrelated across different stocks. We have
denoted this term εi and call it the idiosyncratic return component for
stock i. An important property of the idiosyncratic component is that
it is also assumed to be uncorrelated with F, the common factor in
stock returns. In statistical terms we can write the conditions on the
idiosyncratic component as follows:
Cov(εi, εj) = 0 i ≠ j Cov(εi, F) = 0
i
A
A
An example of such an idiosyncratic stock return might be the unexpected
departure of a firm’s CEO or an unexpected legal action brought against
the company in question.
The partition of returns implied by equation 3.1 implies that all common
variation in stock returns is generated by movements in F (i.e. the
correlation between the returns on stocks i and j derives solely from F). As
the idiosyncratic components are uncorrelated across assets they do not
bring about covariation in stock price movements.
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 the return on the 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 to the CAPM:
E(rA) = rF + βA [E(rM) – rF ]
and
E(rB) = rF + βB [E(rM) – rF ].
Hence:
βA = [E(rA) – rF]/[E(rM) – rF ] = (7% − 4%)/(9% − 4%) = 0.6
βB = [E(rB) – rF]/[E(rM) – rF ] = (10% − 4%)/(9% − 4%) = 1.2.
b. The risk of a portfolio can be decomposed into market risk and idiosyncratic risk.
What are the proportions of market risk and idiosyncratic risk for the two portfolios
A and B?
From the market model:
rA = αA + βA rM + εA
rB = αB + βB rM + εB
with cov(rM , εA) = cov(rM , εB) = 0.
It hence follows that the variance of portfolio A’s returns, σ2A, has two
components, systematic and idiosyncratic risk:
σ2A = β2A σ2M + σ2εA.
Similarly:
σ2B = β2B σ2M + σ2εB.
The proportion of systematic risk for A is hence
β2A σ2M / σ2A = (0.6)2*20%/20% = 36%.
45
92 Corporate finance
The proportion of idiosyncratic risk for A is hence
1 − [β2A σ2M / σ2A] = 64%.
The proportion of systematic risk for B is hence
β2B σ2M / σ2B = (1.2)2*20%/50% = 58%.
The proportion of idiosyncratic risk for B is hence
1 − [β2B σ2M / σ2B] = 42%.
Portfolio B is much riskier than portfolio A as the variance of its returns is 50 per
cent 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 with 0.6 for portfolio A.
c. Assume the two portfolios have uncorrelated idiosyncratic risk. What is the
covariance between the returns on the two portfolios?
Cov(rA,rB) = Cov(αA +βA rM + εA, αB +βB rM + εB) = βA βB σ2M = 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 with 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.
ri = αi + β1iF1 + β2iF2 + .... + βkiFk + εi
E(εi) = 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(ri) = αi). Each stock has a complement
of factor sensitivities or factor βs, which determine how sensitive the
return on the stock in question is to variations in each of the factors.
A pertinent question to ask at this point is how do we determine the return
on a portfolio of assets given the k-factor structure assumed? The answer
is surprisingly simple: the factor sensitivities for a portfolio of assets are
calculable as the portfolio weighted averages of the individual factor
sensitivities. The following example will demonstrate the point.
Example
The returns on stocks X, Y, and Z are determined by the following two-factor model:
rX = 0.05 + F1 – 0.5F2 + εX
rY = 0.03 + 0.75 F1 + 0.5F2 + εY
rz = 0.04 + 0.25 F1 – 0.3F2 + εz
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:
αp = (1/3) (0.05 + 0.03 + 0.04) = 0.04
β1p = (1/3) (1 + 0.75 – 0.25) = 0.5
46
Chapter 3: Factor models
β2p = (1/3) (–0.5 + 0.5 – 0.3) = –0.1
and hence; the factor representation for the portfolio return can be written as:
rp = 0.04 + 0.5F1 – 0.1F2 + εp
where the final term is the idiosyncratic component in the portfolio return. Note that
the idiosyncratic volatility of the portfolio is εp = (1/3)(εX + εY + εz) smaller than the
idiosyncratic volatilities of portfolios 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 twofactor 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
ωX, ωY and ωZ it must be the case that:
ωX + 0.75ωY – 0.25ωZ = 0.5
(3.3)
–0.05ωX + 0.5ωY – 0.3ωZ = 1.
(3.4)
1
In general, if I have
a k-factor model I will
need k+1 stocks to
do this.
Finally, it must also be the case that the portfolio weights add up to unity,
so we must also satisfy the following equation:
ωX + ωY + ωZ = 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 factorreplicating portfolio and discuss it below.
Broad-based portfolios and idiosyncratic returns
In what follows we will assume that the basic securities that we’re going
to work with are themselves broad-based portfolios. The reason for this
is that it allows us to lose the idiosyncratic risk terms associated with
single stocks. Why is this the case? Well, consider the idiosyncratic risk
term for an equally weighted portfolio of 100 stocks. Call the ith
idiosyncratic term εi and assume that all idiosyncratic terms have variance
σ2. 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 for
the portfolio return is n-1σ2. 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.
47
92 Corporate finance
Factor-replicating portfolios
An important application of the technology developed previously in this
chapter is the construction of a factor-replicating portfolio. A factorreplicating portfolio is a portfolio with unit exposure to one factor and
zero exposure to all others. For example, the portfolio replicating factor
1 in model 3.2 would have β1 = 1 and βj = 0 for all j = 2 to k. We will use
factor-replicating portfolios to show that a factor structure for asset returns
implies a β pricing model. In such a model, expected returns depend only
on βs, or risk loadings.
Activity
Assume that stock returns are generated by a two-factor model. The returns on three
well-diversified portfolios, A, B and C, are given by the following representations:
rA = 0.10 + F1 – 0.5F2
rB = 0.08 + 2F1 + F2
rC = 0.05 + 0.5F1 + 0.5F2.
Determine the portfolio weights you need to place on A, B and C in order to construct
the two factor-replicating portfolios plus a portfolio which has zero exposure to both
factors. What are the expected returns of the factor-replicating portfolios and what is the
expected return of the risk-free portfolio?
The question to ask at this point is: why bother constructing factorreplicating portfolios? The reason is as follows. Suppose I want to build
a portfolio that has identical factor exposures to a given asset, X. Assume
a two-factor world and that asset X has exposure of 0.75 to factor 1
and –0.3 to factor 2. Assume also that I know the two factor-replicating
portfolios.
Building a portfolio with the same factor exposures as X is now simple.
Construct a new portfolio, Y, which has portfolio weight 0.75 on the
replicating portfolio for the first factor, portfolio weight –0.3 on the
replicating portfolio for the second factor and the rest of the portfolio
weight (i.e. a weight of 1 – 0.75 + 0.3 = 0.55) on the risk-free asset. Via
the results on the factor representations of a portfolio of assets and
the definition of a factor-replicating portfolio it is easy to see that Y is
guaranteed to have identical factor exposures to X.
The replication in the preceding paragraph forms the basis for the APT. For
absence of arbitrage we require all assets with identical factor exposures
to earn the same return. If they did not, then we would have the chance to
make unlimited amounts of money. For example, assume that the expected
return on the replicating portfolio Y was greater than that on asset X.
Then I should short X and buy Y. The risk exposures of the two portfolios
are identical and hence risks cancel out and I am left with an excess return
that is riskless (i.e. an arbitrage gain).
In order to progress, let us introduce some notation. Denote the riskfree rate with rf. Denote the expected return on the ith factor-replicating
portfolio with rf + λi such that λi is the risk premium associated with the
ith factor. Again, for simplicity, assume that the world is generated by a
two-factor model, and assume that I wish to replicate asset X, which has
sensitivity β1X to the first factor and β2X to the second factor. Finally, we will
assume that the primary securities being worked with are well-diversified
portfolios themselves. Hence, we will ignore any idiosyncratic risk in this
derivation.
48
Chapter 3: Factor models
Using the prior argument, to replicate asset X’s factor sensitivities, we
construct a portfolio with weight β1X on the first factor-replicating portfolio,
weight β2X on the second factor-replicating portfolio and weight 1 – β1X – β2X
on the risk-free asset. The expected return of the replicating portfolio is
hence:
β1X (rf + λ1) + β2X (rf + λ2) + (1 – β1X – β2X) rf = rf + β1X λ1+ β2X λ2.
(3.6)
Hence, using our factor-replicating portfolios we can write the expected
return on a portfolio which replicates X’s factor exposures as the riskfree 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 (βs); 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 rf on the historical βs of these assets (here β is the regressor, not
the coefficient). If the factor model performs well then the intercept of this
regression should be close to zero.
The reason this regression is called a second stage regression is because we
must first find βs 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 i in equation 3.6 is the expected return on asset i, given by equation
3.2:
rit= (rf + β1i λ1 + β2i λ2 ) + β1i F1t + β2i F2t + εit
(3.7)
rit – rf = β1i (λ1 +F1t )+β2i (λ2 + F2t ) + εit = β2t(λ1 + F1t ) + β2i(λ2 + F2t ) + εit ,
(3.8)
where j+Fjt is the excess return on the jth factor-replicating portfolio
(plus some idiosyncratic risk if markets are incomplete). Thus a time series
regression of rit – rf 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, rejection
is 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
49
92 Corporate finance
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 βs they suggest running a separate regression each
year; for each year regress the realised returns on βs 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 factorreplicating portfolios. The key to the APT is that absence of arbitrage
requires that such a pair of portfolios must have identical expected returns
in a financial market equilibrium. If they did not, it would be possible to
make unlimited amounts of money without incurring any risk.
This implies that the expected return on asset X, rX, must be identical to
the expression arrived at in equation 3.6, that is:
(3.9)
E(rX) = rf + β1X λ1+ β2X λ2.
Equation 3.7 is the statement of the APT. The expected return on a
financial asset can be written as the risk-free rate plus sum of the asset’s
factor sensitivities multiplied by the factor-risk premiums (which are
invariant across assets). If such an expression does not hold at all times,
arbitrage opportunities exist. Note the assumptions that are required
to achieve this result. First, we require that asset returns are generated
by a two-factor (or in general k-factor) model. Second, we assume that
arbitrage opportunities cannot exist. Lastly, we assume that enough assets
are available such that firm-specific risk washes away when portfolios are
formed.
Example
In the previous two-factor example, we determined the expected returns on the two
factor-replicating portfolios. Denoting the expected return on the i th factor-replicating
portfolio by E(ri) we have:
E(r1) = 8.29%
E(r2) = 1.71%
E(r3) = 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(ri) = 5.14 + 3.15β1i – 3.43β2i .
To check that this works, substitute (for example) portfolio C’s factor sensitivities into the
preceding expression. This gives:
E(rC) = 5.14 + 3.15 (0.5) – 3.43 (0.5) = 5%,
and hence, agrees with the expected return implied by the original representation for
asset C. Check that the expected returns on assets A and B also come out correctly.
To analyse an arbitrage opportunity that might arise in markets, attempt
the following activity.
50
Chapter 3: Factor models
Activity
Assume that a new well-diversified portfolio, D, is added to our world. This asset has
sensitivities of 3 and –1 to the two factors and an expected return of 15 per cent.
Using the equilibrium expected return equation given above, derive the equilibrium
expected return on an asset with identical factor exposures to D. Is there now an
arbitrage opportunity available? If so, dictate a strategy that could be employed to exploit
the arbitrage opportunity.
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-tomarket, 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-tomarket (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
0.528
2
–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 called
it 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).
51
92 Corporate finance
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 model
by 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 APT
allows there to be k sources of systematic risk in the economy. Some
may reflect macroeconomic factors, like inflation, and interest rate risk,
whereas others may reflect characteristics specific to a firm’s industry or
sector.
Empirical research has indicated that some of the well-known empirical
problems with the CAPM are driven by the fact that the APT is really the
proper model of expected return generation. Chen (1983), for example,
argues that the size effect found in CAPM studies disappears in a multifactor 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 crosssection of stock returns while other factors, such as momentum, also
appear to be important. Work in this area is still progressing.
A reminder of your learning outcomes
Having completed this chapter, and the Essential reading and activities,
you should be able to:
• understand single-factor and multi-factor model representations
• derive factor-replicating portfolios from a set of asset returns
• understand the notion of arbitrage strategies and that well-functioning
financial markets should be arbitrage-free
52
Chapter 3: Factor models
• derive arbitrage pricing theory and calculate expected returns using the
pricing formulas
• know how to test multifactor models.
Key terms
arbitrage pricing theory
factor-replicating portfolio
factor sensitivity
multi-factor model
single-factor model
Sample examination question
1. Assume that stock returns are generated by a two-factor model. The
returns on three well-diversified portfolios, A, B and C, are given by the
following representations:
rA = 0.10 + F1
rB = 0.08 + 2F1 – F2
rC = 0.05 – 0.5F1 + 0.5F2
a. Discuss what the factor representations above imply for the
variation and comovement in the three stock returns. Show how the
returns of the stocks should be correlated between themselves.
b. Find the portfolio weights that one must place on stocks A, B and
C to construct pure tracking portfolios for the two factors (i.e.
portfolios in which the loading on the relevant factor is +1 and the
loadings on all other factors are 0).
c. If one was to introduce a new portfolio, D, with loadings of +1 on
both of the factors, what would the expected return on D have to be
to rule out arbitrage?
d. Explain the concepts of idiosyncratic risk and factor risk in the APT.
What role does diversification play in the APT?
2. Explain the first and second stage tests of factor models. Discuss how
you would look for significance.
3. Explain how Fama and French form their portfolios and factors. What
does it mean for a factor model to work well? What is Fama and
French’s explanation for why their factor model works well?
53
92 Corporate finance
Notes
54
Chapter 4: Derivative securities: properties and pricing
Chapter 4: Derivative securities:
properties and pricing
Aim of the chapter
The aim of this chapter is to introduce and price derivatives. As in
the previous chapter on APT, the valuation of derivatives relies on the
impossibility of riskless arbitrage.
Learning outcomes
At the end of this chapter, and having completed the Essential reading and
activities, you should be able to:
• discuss the main features of the most widely traded derivative securities
• describe the payoff profiles of such assets
• understand the absence-of-arbitrage pricing of forwards, futures and
swaps
• construct bounds on option prices and relationships between put and
call prices
• price options in a binomial framework using the portfolio replicating
and the risk-neutral valuation.
Essential reading
Hillier, D., M. Grinblatt and S. Titman Financial Markets and Corporate
Strategy. (Boston, Mass.; London: McGraw-Hill, 2008) Chapters 7 (Pricing
Derivatives) and 8 (Options Part III).
Further reading
Brealey, R., S. Myers and F. Allen Principles of Corporate Finance. (Boston,
Mass.; London: McGraw-Hill, 2008) Chapters 21 (Understanding Options),
22 (Valuing Options) and 23 (Real Options).
Copeland, T., J. Weston and K. Shastri Financial Theory and Corporate Policy.
(Reading, Mass.; Wokingham: Addison-Wesley, 2005) Chapters 8 and 9.
Overview
A derivative asset is one whose payoff depends entirely on the value of
another asset, usually called the underlying asset. In the last 20 years,
traded volume in these assets has increased tremendously. Derivatives
are widely used for hedging purposes by financial institutions and are
also used for speculative purposes. In this chapter we discuss the most
commonly traded types of derivative. We go on to introduce the underlying
principles of derivative pricing. We devote the final section of the chapter
to a more detailed description of the features and pricing of options.
55
92 Corporate finance
Varieties of derivatives
Forwards and futures
Perhaps the oldest type of derivative asset is the simple forward
contract. A forward is an agreement between two parties (called A and
B) and has the following features:
• Party A agrees to supply party B with a specified amount of a specified
asset, k periods in the future.
• In return, party B agrees to pay party A $F (the forward price) when
the goods are received.
• Party A is said to hold a short position in the contract and party B a
long position.
Hence, the forward is just an agreement made today to undertake a given
transaction at some specified future date, known as the settlement
date. Currency and commodities are often traded using forwards, the
advantage of such transactions being that they allow an agent to remove
any price uncertainty regarding a transaction that must be undertaken in
the future.
Example
Assume that party B is American and that in three months he must pay ¥250,000 for
a Japanese machine he has purchased. Party B enters into a contract to buy yen threemonths forward. Party A (the agent who is to supply the yen) specifies that the cost of
¥100 will be $1.20. The total price that party B must pay in three months is therefore
$3,000.
Closely related to forward contracts are futures contracts. In fact, futures
are refined versions of forwards. Although forwards are generally bilaterally
negotiated between two parties directly, futures are standardised forward
contracts that are exchange traded. The contracts give precise specifications
for the quality and quantity of the assets to be exchanged. The major
difference between futures and forwards is in the exchange of monies
involved. With a forward, the agent who is long pays the entire forward
price at the settlement date. Futures positions, however, are marked to
market. This occurs on a daily basis and means that any increases/
decreases in the value of the future are received/paid by the party who is
long day by day. At the settlement date, the current spot price of the asset is
transferred from the agent who is long to the agent who is short.1
Futures are traded on exchanges such as the London International
Financial Futures and Options Exchange (LIFFE), the Chicago Board of
Trade (CBOT) and the Chicago Mercantile Exchange (CME). Contracts
with very high volumes include those on government bonds, interest rates
and stock indices.
Options
The option is a less straightforward type of derivative. Although the
forward or future contract implies an obligation to trade once the contract
is entered into, the option (as its name suggests) gives the agent who is
long a right but not an obligation to buy or sell a given asset at a prespecified price. This price is known as the exercise price and is specified in
the option contract. Just as with the forward, another factor specified in
the contract is the date on which the exchange is to take place. If, on the
maturity date, the holder of an option decides to buy or sell in line with
56
1
See pp.236–40 in
Hillier, Grinblatt and
Titman (2008).
Chapter 4: Derivative securities: properties and pricing
the terms of the contract, they are said to have exercised their right. A
big difference between options and forwards is that, with an option, the
agent who is long must pay a price (or premium) at the outset. This is
essentially a price paid by the holder for the exercise choice they face at
maturity.
Options to buy the specified asset are called call options. Options to sell
are called puts. Another distinction is made on the timing of the exercise
decision. With European options, the right can only be exercised on the
maturity date itself. With American options, in contrast, the option can be
exercised on any date at or before maturity. American options are traded
far more frequently than their European counterpart, but for reasons of
simplicity, we will focus on the European variety.
Example
A 12-month European call option on IBM has exercise price $45. It gives me the right
to purchase IBM stock in one year at a cost of $45 per share. In line with the prior
discussion, I am under no obligation to buy at $45 such that, if the market price were less
than this amount, I could choose not to exercise and buy in the market instead.
Swaps
Swaps are another type of derivative, which do exactly what their name
says. Two counterparties agree to exchange (or swap) periodic interest
payments on a given notional amount of money (the notional principal)
for a given length of time.
A very common type of swap involves an exchange of interest payments
based on a market-determined floating rate (such as the London InterBank
Offer Rate (LIBOR)) for those calculated on a fixed-rate basis. Another
frequently traded variety of swap involves the exchange of interest
payments in different currencies. For example, fixed sterling interest
payments may be exchanged for fixed dollar interest payments.2
Derivative asset payoff profiles
For now we are going to concentrate on forwards and options. As
mentioned above, futures are closely related to forwards, and their pricing
is based on the technique presented below. The relationship between
forwards and swaps will be made clear later.
2
The notional principal
is not exchanged in an
interest rate swap (they
would net out anyway)
but are generally
exchanged in currency
swaps.
Before getting on to the principles of derivative pricing, let us take a look
at the payoff profiles of the basic forward and option contracts. The payoff
profile of a long forward position is shown in Figure 4.1. In the figure, F
is the price agreed upon in the forward contract, and S is the spot price
of the asset at the settlement date. Note that the payoff profile is linear,
positive for values of S greater than F and negative when S is less than F.
Understanding the forward payoff is simple. If the spot price for the asset
at maturity exceeds the forward price, then the party that is long has
gained by entering into the forward (i.e. they have got the asset for a
lower price than it would have cost if bought in the spot market). If the
spot price at maturity is lower than the forward price, then the long payoff
is negative, as it would have been cheaper for the long party to buy the
asset in the spot market rather than entering into the forward. Obviously,
the payoff of a short forward position is the negative of that shown in
Figure 4.1.
57
92 Corporate finance
St – F
Payoff
F
St
–F
Figure 4.1
Let’s now consider the payoff to a holder of a European call option.
This is given in Figure 4.2 where the option’s exercise price is labelled
X. Remember that a call option gives the holder the right but not the
obligation to purchase the asset. What occurs when the price of the
spot asset at maturity exceeds the exercise price of the option? Well it is
cheaper to buy the asset using the option than in the spot market; hence
the option is exercised, and the holder makes a gain of the spot price less
the exercise price. When the spot price is lower than the exercise price,
then the holder would find it cheaper to buy the asset at spot and hence
does not exercise the option. The payoff to the holder is then zero.
Payoff
[St – X]
0
X
St
Figure 4.2
The payoff to the holder of a European put is given in Figure 4.3. As the
put gives the holder the right to sell the underlying asset, the holder gains
when the exercise price exceeds the spot price and has a zero payoff when
the spot price at maturity is greater than or equal to the exercise price.
Each option must have one agent who is long and one who is short, with
the payoffs to the long position given in Figures 4.2 and 4.3. An agent
who is short is said to have written the option, and their payoffs are the
negative of those given above. Note that an agent with a long option
position never has a negative payoff, whereas an agent who has written an
option never has a positive payoff at maturity. The option price, paid at the
outset by the agent who is long to that who is short, is the compensation
to the writer of the option for holding a position that exposes them to
weakly negative cash flows.
58
Chapter 4: Derivative securities: properties and pricing
Payoff
X
[X – St ]+
0
X
St
Figure 4.3
The key to pricing options, and other derivative assets, is constructing
a portfolio of assets that is priced in the market and that has a payoff
structure identical to that of the derivative. As the derivative and
replicating portfolio have identical payoffs, absence-of-arbitrage arguments
imply that the cost of these portfolios must be identical. The no-arbitrage
price of the derivative is hence just the initial investment cost needed to
set up the replicating portfolio.
Pricing forward contracts
In the case of a forward contract, the derivation of the no-arbitrage price
is quite simple.3 Assume that the current spot asset price is S0 and that the
one-period, riskless rate of interest is r. We wish to value a k-period
forward contract. It is easily verified that the k-period forward price (Fk)
is given by the following expression:
Fk = S0(1+r)k.
(4.1)
3
Given the similarities
discussed previously,
we can also use the
derived forward price to
approximate the price of
a futures contract.
Why is this the case? Well, consider the following pair of investment
strategies.
• The first is simply a long position in the forward contract. This costs
nothing at the present time and yields Sk – Fk at maturity.
• The second strategy involves buying a unit of the asset at spot and
borrowing Fk(1+r)–k at the risk-free rate for k-periods. The k-period
payoff of this strategy is also Sk – Fk, and its net current cost is
S0 – Fk(1+r)–k.
The payoffs of the two strategies are identical. This implies that the two
investments should have identical costs. As the cost of investment in the
forward is zero, this implies that the following condition must hold:
S0 – Fk(1+r)–k = 0.
(4.2)
Rearranging equation 4.2 we derive the no-arbitrage price for the k-period
forward contract, which is precisely that given in equation 4.1.
Activity
The current value of a share in Robotronics is $12.50.
1. The one-year riskless rate is 6 per cent. What are the prices of three- and five-year
forward contracts on Robotronics stock?
2. Three-year forward contracts are currently being sold for $16 in the market.
Outline an investment strategy that could take advantage of the opportunities
this presents.
59
92 Corporate finance
Some of the most active forward markets are those for foreign currency.
The forward pricing analysis above, however, is suited only for assets
valued in the domestic currency (e.g. individual stocks or stock indices).
To illustrate the pricing of currency forwards, consider the following
analysis. A domestic investor (assumed to be located in the UK such that
the domestic currency is £) is assumed to face a spot exchange rate of
S and a k-period forward rate of Fk. These rates are constructed as the
domestic currency price of one unit of foreign currency (i.e. the spot rate
implies an exchange rate of £S for $1). The one-period domestic interest
rate is denoted r and its foreign counterpart rf .
Again, let us compare two investment strategies that can be undertaken
assuming an investor currently holds £S. The first involves depositing this
cash in a domestic risk-free account for k-periods. This yields £S(1+r)k at
the maturity date of the investment. Alternatively, the investor could swap
their sterling for dollars at the spot exchange rate and invest the funds
at the US rate. As their £S is equivalent to $1 at the spot exchange rate,
this investment yields $(1+rf )k in k-periods. The investor can then sell the
proceeds for sterling using a forward contract yielding £Fk(1+rf )k.
Note that both of these investments are riskless, assuming that the interest
rates are known and fixed and given that the spot and forward exchange
rates are known at the current date. Further, both investments cost £S.
This implies that the payoffs from the two strategies should be identical.
Equating these returns we get:
S(1 + r)k = Fk(1 + rf )k.
(4.3)
Rearranging equation 4.3, we get the no-arbitrage k-period currency
forward price:
.
(4.4)
Note the simple generalisation of equation 4.1 implicit in equation 4.4.
The gross interest rate in equation 4.1 is just replaced by the ratio of
domestic to foreign rates in equation 4.4. In the international finance
literature, the currency forward rate expression in 4.4 is known as the
covered interest rate parity relationship.
Activity
The current spot exchange rate is £0.64 = $1. The riskless rate in the UK is currently 6
per cent and that in the USA is 4 per cent. Using equation 4.4, derive the implied fiveand 10-year forward exchange rates.
Binomial option pricing setting
Pricing options is far less straightforward than pricing forwards. To begin,
however, we introduce a binomial setting, in which the pricing of options
turns out to be surprisingly straightforward.
In order to make things as simple as possible, let us consider a binomial
setting in which all derivatives last only for one period (starting today and
ending tomorrow). Let us denote the current price of the underlying asset
by S0. Let us assume that uncertainty in this world is represented by the
price of the underlying asset, taking one of two values tomorrow.4 If the
state of the world is good, the price of the asset will rise tomorrow to SH,
with SH = (1+u)S0 and u > 0. In contrast, if the state of the world is bad, the
price of the underlying asset will decrease to SL, with SL = (1 – d)S0 and
d > 0.
60
4
This is where the term
‘binomial’ comes from
in the name of our
method.
Chapter 4: Derivative securities: properties and pricing
Let us now consider a one-period derivative asset. If the state of the world
is good tomorrow, then the derivative will pay KH, and if the state of the
world is bad tomorrow the derivative will pay KL. Finally, we assume that
the one-period risk-free interest rate is rf (i.e. a safe bond costing one unit
of currency pays 1+ rf units of currency tomorrow). In order to price this
derivate asset, we will consider two different methods:
• the portfolio replicating method
• the risk-neutral valuation method.
The portfolio replicating method
This method prices the derivative asset using absence-of-arbitrage
arguments. First, this necessitates constructing a portfolio, containing the
underlying asset and the risk-free asset, that has identical payoffs to the
derivative. Assume we purchase a units of the underlying asset and b units
of the risk-free asset.
If the state of the world tomorrow is good then the value of our portfolio
will be:
aSH + b(1 + rf ),
(4.5)
when the payoff of the derivative is KH. If the state tomorrow is bad the
portfolio is worth:
aSL + b(1 + rf ),
(4.6)
and the derivative is worth KL. Note that equating the value of the
portfolio with the payoff of the derivative in each state of the world gives
us two equations in two unknowns (a and b). These unknowns are our
initial holdings of the underlying and the risk-free asset. Solving the
two equations gives us precisely the portfolio weights we need to use to
replicate the option payoff in both states of nature. This yields:
.
(4.7)
and
.
(4.8)
We now know how to construct a portfolio, which has a payoff profile that
replicates that of the derivative (i.e. regardless of the state of the world,
the portfolio and the derivative have the same value). If two assets have
identical payoffs then absence-of-arbitrage arguments tell us that the
price/cost of the two assets must be identical. The cost of the replicating
portfolio is aS0 + b. It hence follows that:
K0 = aS0 + b.
(4.9)
A practical example of how this technique might work for a European call
option is given below.
Example
A one-period European call option on ABC stock has an exercise price of 120. The current
price of ABC stock is 100 and, if things go well, the price in the following period will be
150. If things go badly over the coming period, the future price will be 90. The risk-free
rate is 10 per cent. What is the no-arbitrage price of this option?
First, we need to know the option payoffs. In the bad state it pays zero, as the underlying
price is less than the exercise price. In the good state it pays the excess of the underlying
price over the exercise price (i.e. 30).
61
92 Corporate finance
Next we construct the replicating portfolio. Using equations 4.3 and 4.4, the quantities of
the underlying and risk-free asset we must buy are 0.5 and –40.91 (i.e. we buy half a unit
of stock and short 40.91 units of the risk-free asset).5 This portfolio replicates the option
payoff, and therefore the option price is given by the cost of constructing the portfolio.
The call price (c) is hence:
c = 0.5(100) – 40.91 = 9.09.
Activity
Using the stock price data from the previous example, price a European put option on
ABC stock with a strike price of 100.
The risk-neutral valuation method
Using the portfolio replicating method, we find that the current price of
the derivative asset, relative to the current price of the underlying asset,
does not depend on the probability that the state of nature will be good
(or bad) tomorrow. Neither does it depend on investor risk preferences.
The reason for this is that information about probabilities or risk aversion
is already captured by the current price of the underlying asset on
which we base our valuation of the derivative asset. The fact that the
no-arbitrage price of the derivative asset in relation to the price of the
underlying asset is the same, regardless of risk preferences, serves as a
basis for a neat trick also known as the risk valuation method.
The risk-neutral valuation method is a procedure involving the following
steps.
1. Identifying the risk-neutral probabilities, that is, the probabilities which
are consistent with investors being risk-neutral. These probabilities are
the probabilities for which the current price of the underlying asset
is the present value of tomorrow’s asset prices, with the discount rate
being equal to the risk-free rate.
2. Calculating the current price of the derivative asset as the present value
of tomorrow’s derivative values using the risk-neutral probabilities
derived in the previous step and the risk-free rate as the discount rate.
Step 1: Obtaining risk-neutral probabilities
Let us denote the risk-neutral probability that the state of nature will be
good tomorrow by q. It hence follows that:
.
(4.10)
Equivalently, the risk-neutral probability q is given by the following
identity:
.
(4.11)
Step 2: Calculating the current price of the derivative asset
The current price of the derivative asset can be expressed as the present
value of tomorrow’s derivative values using the risk-neutral probabilities in
equation 4.11 and the risk-free rate as the discount rate:
.
(4.12)
After substituting q from equation 4.11, we obtain:
.
62
(4.13)
5
You should check
all these calculations
and further check that
the portfolio we’ve
constructed does indeed
replicate the option
payoff.
Chapter 4: Derivative securities: properties and pricing
Activity
Using the risk-neutral valuation method, price both a European call option and a European
put option on the ABC stock (introduced in the previous example) with a strike price of 100.
Activity
Show that the current price of the derivative obtained from the portfolio replicating
method in equation 4.9 is the same as the one obtained from the risk-neutral valuation
method in equation 4.13.
Comments on the binomial option pricing setting
The risk-neutral valuation method is very efficient at pricing multiple
derivative assets on the same underlying asset as the same risk-neutral
probabilities can be used to price all the derivatives. In these circumstances,
the portfolio replicating method is more tedious to use as the replicating
portfolio will typically be different for each derivative asset.
The assumptions we have made above may seem very restrictive. We
have restricted tomorrow’s price to take one of two values and assumed
that derivatives last only for one period. Extending the above model to
more than one period is straightforward, and this allows longer maturity
instruments to be priced. Also, we can shrink the length of time that we
have referred to as one period. It could represent one day, one hour or one
minute if we wanted. A binomial model for hourly prices, for example,
may be thought more reasonable than a binomial model for annual prices.
Then, using a multi-period derivative valuation we could price a onemonth option from a binomial model of hourly stock returns. The binomial
structure is not as restrictive as you might think.
Example
In this example we will see how to extend the binomial approach to a more realistic multiperiod problem. We will also see that sometimes it is best to exercise an American put
option early.
Consider an underlying security which is worth 100 today and will either increase by 25
per cent or decrease by 20 per cent in value in six months. In the following six months,
it will again either increase in value by 25 per cent or decrease in value by 20 per cent.
There is a risk-free asset with a 1 per cent semi-annual return. We will first price a one-year
European put on this security with a strike of 105, we will then price an American put with
the same strike but the option to exercise at the six-month interval.
It is often best to draw a tree diagram of the payout for the put and the underlying
security, as in Figure 4.4. After six months the underlying security is worth either 100*1.25
= 125 (node 1.1) or 100*.8 = 80 (node 1.2). If in node 1, it will subsequently either
increase to 125*1.1 = 156.25 (node 1.1.1) or decrease to 125*.8 = 100 (node 1.1.2). If
in node 2, it will subsequently either increase to 80*1.1 = 100 (node 1.2.1) or decrease
to 80*.8 = 64 (node 1.2.2). Note that nodes 1.1.2 and 1.2.1 have the same payoff but
different histories; the probability of having this ‘medium’ payoff of 100 is higher than
either of the ‘extreme’ payoffs of 156.25 or 64.
At maturity the put option pays max(0,105 – 156.25) = 0 in node 1.1.1, max(0,105 –
100) = 5 in nodes 1.1.2 and 1.2.1, and max(0,105 – 64) = 41 in node 1.2.2.
We can replicate its payoff at each node to calculate the price of the option using
equations 4.7, 4.8, and 4.9.
In node 1.1, K0 = 125, KH = 0, KL = 5, SH = 156.25, SL = 100. Equation 4.7 gives
a = –0.089. Equation 4.8 gives b = 13.75. Equation 4.9 gives 2.64 as the option’s price
in node 1.1.
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92 Corporate finance
In node 1.2, K0 = 80, KH = 5, KL = 41, SH = 100, SL = 64. Equation 4.7 gives
a = –1. Equation 4.8 gives b = –103.96. Equation 4.9 gives 23.96 as the option’s price
in node 1.1.
In node 1, K0 = 100, KH = 2.64, KL = 23.96, SH = 125, SL = 80. Equation 4.7 gives a
= –0.4738. Equation 4.8 gives b = 61.252. Equation 4.9 gives 13.872 as the option’s
price in node 1.1.
Turning to the American put, note that the calculation in nodes 1.1 and 1.2 remains
identical. That is, if we want to create a replicating portfolio starting in 1.1, it would be
exactly the same. However, because this is an American put, we also have the option of
exercising it at six months rather than waiting for expiry. If we are in node 1.1 exercising
today gives us max(0,105 – 125) = 0 which is less than the option value and we
would not exercise. If we are in node 1.2 exercising today gives us max(0,105 – 80) =
25 which is greater than the 23.96 that the European put is worth. Thus at node 1.2,
we would always exercise and the option is worth 25. We must now redo the node 1
calculation:
In node 1, K0 = 100, KH = 2.64, KL = 25, SH = 125, SL = 80. Equation 4.7 gives
a = –0.4969. Equation 4.8 gives b = 64.11. Equation 4.9 gives 14.42 as the option’s
price in node 1.1.
Note that the American put is more valuable because it has an option to exercise early,
and there are some states of the world in which we would choose to exercise that option.
The value of the option is the difference between the two prices: 14.42 – 13.87 = 0.55
The Black–Scholes formula discussed below is derived as the limit of adding more
intermediate steps in a binomial calculation. Instead of splitting up the year into two
six-month intervals, we can split it up into four three-month intervals, 12 one-month
intervals, and so on. As the number of intervals gets very large, the option price converges
to the Black–Scholes price.
Figure 4.4
Bounds on option prices and exercise strategies
The binomial model allows us to derive option prices under certain
assumptions on the behaviour of the price of the underlying asset. In this
section we present some arguments that place bounds on European option
prices and can be made without specification of a model for the underlying
price. In order to link up with the following section (on Black–Scholes
prices), we will present our arguments using a continuously compounded
risk-free rate, r. We assume unlimited borrowing and lending at this rate
along with our standard frictionless market assumptions of no transaction
costs and taxes. Finally, we also assume that the underlying asset pays out
no cash during the option lifetime (such that the option can’t be written on
dividend paying stock or coupon bonds, for example).
64
Chapter 4: Derivative securities: properties and pricing
Upper bounds on European option prices
A call option is the right (but not the obligation) to purchase a unit of a
specified asset for price X. It should be obvious to you then that the option
can never be worth more than the stock. Hence, denoting the call option
price by c we have:
c ≤ S.
(4.14)
As a European put gives the holder the right to sell a given quantity of an
asset for X, the put can never be worth more than X. Denoting the put
price by p we then have6:
p ≤ X.
(4.15)
Further, if the put is European, we know that the value at maturity is at
most X. If there are T periods to maturity, a present value argument then
implies that:
p ≤ Xe–rT.
6
Clearly, both this and
the previous argument
hold for American
options as well as
European options.
(4.16)
Lower bounds on European option prices
No-arbitrage arguments can be simply employed to develop lower bounds
for European puts and calls. A lower bound for a European call option
price is given by:
c ≥ S – Xe–rT
(4.17)
where X is the exercise price, and there are T periods to maturity. To show
this, consider the following argument. Assume I hold two portfolios.
Portfolio A consists of a European call option struck at price X, plus cash of
the amount Xe–rT. Portfolio B consists of the underlying stock.
Assume I invest the cash from portfolio A at the risk-free rate. This implies
that, when the option in portfolio A matures, I have cash worth X. If
at maturity the underlying price (ST) exceeds the exercise price, then I
exercise the call option using my cash, and the portfolio is worth ST. If at
maturity the underlying price is less than X, I do not exercise the option,
and hence my portfolio is worth X. The value of portfolio A at maturity can
be written as:
max(ST,X).
At the maturity date the value of portfolio B is always just ST. Hence,
portfolio A is always worth at least the same as portfolio B, and sometimes
(when exercise is not optimal) it is worth more. Reflecting this and to
prevent arbitrage, the price of buying portfolio A must exceed the cost of
portfolio B. This reasoning implies:
c + Xe–rT > S  c > S – Xe–rT.
(4.18)
Also, an option must have positive value since, at the very worst, it is
not exercised as it is out of the money. This implies that 4.18 can be
generalised to:
c ≥ max[0,S – Xe–rT].
(4.19)
A similar argument to the above can be used to establish a lower bound on
the price of a European put. It’s easy to show that:
p > Xe–rT – S.
(4.20)
To demonstrate this, consider two more portfolios. Portfolio 1 consists of a
European put and a unit of the underlying stock, and portfolio 2 consists
of Xe–rT in cash.
At the date at which the put matures, portfolio 1 is worth either X (if it’s
profitable to exercise the put, and hence you sell the unit of the underlying
65
92 Corporate finance
for X) or ST (when exercise isn’t optimal and you’re left with the stock, as
the put expires with zero value). We can then write the value of portfolio
1 as:
max(X,ST).
Portfolio 2 is always worth X at the date when the put matures and is
hence weakly dominated in payoff terms by portfolio 1. Therefore, to
prevent arbitrage, portfolio 1 should cost more to set up than portfolio 2,
implying:
p + S > Xe–rT  p > Xe–rT – S.
(4.21)
Finally, again we know that the worst that can happen for a put option is
for it to expire, worth nothing. This implies that its value must exceed zero
in all circumstances. Thus:
p ≥ max[0, Xe–rT –S].
(4.22)
Combining upper and lower bounds
A combination of the upper and lower bounds derived in the preceding
two sections can be formed graphically. This gives a set of permissible (in
the sense of not admitting arbitrage) put and call prices. As an example,
Figure 4.5 shows the permissible call price region (it is the shaded area of
the diagram).
Figure 4.5
Black–Scholes option pricing
Our previous pricing analysis was predicated on the assumption that
stock prices are well-represented by a discrete time, the binomial model.
In 1974, Fischer Black and Myron Scholes presented an option pricing
formula, based on a continuous time process for the stock price. This
analysis gave exact prices for European puts and calls using a continuous
time version of the replication strategy followed in our binomial
methodology. Unfortunately, derivation of their pricing formula is beyond
the scope of the current presentation. However, due to its wide use in the
financial markets and the intuition it brings regarding the determinants of
option prices, we will describe the pricing formula below.
Assume we wish to price a European call on a stock that never pays
dividends. The current price of the stock is S, the exercise price of the
option under consideration is denoted X, and the option is to have a
maturity of T periods. The continuously compounded risk-free rate is
denoted r. One final parameter is needed to calculate the Black–Scholes
66
Chapter 4: Derivative securities: properties and pricing
price of the call option. This is the instantaneous volatility of the stock
price, and we denote this parameter . It is the standard deviation of the
change in the logarithm of the stock price.
The famous Black–Scholes formula for the price of a European call option
is given below:
c = SN(d1) – Xe–rT N(d2)
(4.23)
where
(4.24)
(4.25)
and N(.) represents the cumulative normal distribution function.7
7
The values of the
cumulative standard
normal distribution
function can be found
in tables in the back
of any good statistical
textbook.
Example
The current price of Glaxo Wellcome share is £2.88. An investor writes a two-year call
option on Glaxo with exercise price £3.00. If the annualised, continuously compounded
interest rate is 8 per cent, and the volatility of Glaxo’s stock price is 25 per cent, what is
the Black–Scholes option price?
First, we need to derive the values d1 and d2 as defined above. Using equations 4.24
and 4.25 these are 0.5139 and 0.1603. The values of the cumulative normal distribution
function at 0.5139 and 0.1603 are 0.696 and 0.564. Then, plugging all the available
data into equation 4.23 yields a call price of £0.5644.
What does equation 4.23 tell us about the determinants of call prices?
Well, there are clearly a number of influences on the price of an option,
and these are summarised below.
• The effect of the current stock price: the Black–Scholes equation
tells us that call option prices increase as the current spot asset price
increases. This is pretty unsurprising as a higher underlying price
implies that the option gives one a claim on a more valuable asset.
• The effect of the exercise price: again, as you would expect,
higher exercise prices imply lower option prices. The reason for this is
clear: a higher exercise price implies lower payoffs from the option at
all underlying prices at maturity.
• The effect of volatility: Figure 4.2 gives the payoff function of a
European call option. Note that, although extremely good outcomes
(underlying price very high) are rewarded highly, extremely bad
outcomes are not penalised due to the kink in the option payoff
function. This would imply that an increase in the likelihood of extreme
outcomes should increase option prices, as large payoffs are increased
in likelihood. The Black–Scholes formula verifies this intuition, as it
shows that call prices increase with volatility, and increased volatility
implies a more diverse spread of future underlying price outcomes.
• The effect of time to maturity: call option prices increase with
time to maturity for similar reasons that they increase with volatility.
As the horizon over which the option is written increases, the relevant
future underlying price distribution becomes more spread-out, implying
increased option prices. Furthermore, as the time to maturity increases,
the present value of the exercise that one must pay falls, reinforcing the
first effect.
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92 Corporate finance
• The effect of riskless interest rates: call option prices rise when
the risk-free rate rises. This is due to the same effect as above, in that
the discounted value of the exercise price to be paid falls when rates
rise.
Put–call parity
The Black–Scholes formula gives us a closed-form solution for the price of
a European call option under certain assumptions on the underlying asset
price process. However, as yet, we have said nothing about the pricing of
put options. Fortunately, a simple arbitrage relationship involving put and
call options allows us to do this. This relationship is known as put–call
parity. In what follows we assume the options have the same strike price
(X), time to maturity (T) and are written on the same underlying stock.
Consider an investment consisting of a long position in the underlying
asset and a put option, called portfolio A. The cost of this position is
S0 + p. A second portfolio, denoted B, comprises a long position in a call
option and lending Xe–rT. Hence the cost (c) of this position is c + Xe–rT.
What are the possible payoffs of these positions at maturity? Given the
payoff structure on the put shown in Figure 4.3, the payoff on portfolio A
can be written as follows:
max[X – ST,0] + ST = max[X,ST].
(4.26)
Similarly, the payoff on portfolio B can be written as:
max[0,ST – X] + X = max[X,ST].
(4.27)
Comparison of equations 4.26 and 4.27 implies that the two portfolios
always pay identical amounts. Hence, using no-arbitrage arguments,
portfolios A and B must cost the same amount. Equating their costs we
have:
S + p = c + Xe–rT.
(4.28)
Equation 4.28 is the put–call parity relationship. Given the price of a call,
the value of the underlying asset and knowledge of the riskless rate, we
can deduce the price of a put. Similarly, given the put price, we can deduce
the price of a call with similar features.
Example
A call option on BAC stock, with an exercise price of £3.75, costs £0.25 and expires
in three years. The current price of BAC stock is £2.00. Assuming the continuously
compounded (annual) risk-free rate to be 10 per cent, calculate the price of a put option
with three years to expiry and exercise price of £3.75.
From equation 4.28 we have:
p = c + Xe–rT – S.
Plugging in the data we’re given yields:
p = 0.25 + 3.75e–0.1(3) – 2 = 1.03.
Hence, the no-arbitrage put price is £1.03.
Substitution of the Black–Scholes call pricing equation gives a closed-form
solution for the put price. This equation allows us to deduce the effects of
changing the Black–Scholes parameters on put prices.
• The effect of underlying price: for the opposite reason to that
given for the call, put prices drop as underlying prices increase.
68
Chapter 4: Derivative securities: properties and pricing
• The effect of the exercise price: similarly, put prices rise as
exercise prices rise.
• The effect of volatility: put options and call options are affected in
identical ways by volatility. Hence, as volatility increases, put prices rise.
• The effects of time to maturity: increased time to maturity will
lead to a greater dispersion in underlying prices at maturity, and hence
put prices should be pushed higher. However, as the holder of a put
receives the exercise price, discounting at higher rates makes puts less
valuable. The combined effect is ambiguous.
• The effect of the risk-free rate: puts are less valuable as interest
rates rise, due to a greater degree of discounting of the cash received.
Activity
ABC corporation’s shares currently sell at $17.50 each. The volatility of ABC stock is 15
per cent. Given a risk-free rate of 7 per cent, price a European call with strike price of
$15 and time to maturity five years. Use put-call parity to price a put with similar
specifications. What are the no-arbitrage prices of the call and the put if the risk-free
rate rises to 10 per cent?
Pricing interest rate swaps
Recall the definition of an interest rate swap given earlier in the chapter.
Agent A contracts to give fixed interest payments (on a given principal) to
agent B. In return, agent B agrees to deliver to agent A interest payments
(on the same principal) based on an agreed floating exchange rate. The
frequency and duration of these interest payments are also agreed in
advance. A very common choice of floating interest rate used in such
contracts is the LIBOR.
An example of such an agreement is as follows. Agent A agrees to pay
agent B payments on a $1m principal at a fixed 8-per cent rate. Agent
B agrees to pay interest payments of LIBOR plus 0.25 per cent. These
payments are to be made annually for the next 10 years.
Note that, from the previous example, the payments made by agent A at
every date till maturity are known and fixed (i.e. 8 per cent of $1m). Their
receipts, however, are uncertain. They gain a 0.25 per cent premium above
an ex-ante uncertain interest rate. Consider, for example, the transaction
at the second payment date. Agent A pays $50,000 and receives LIBOR
+ 0.25 per cent. This looks identical to the cash flows from a forward
contract. Indeed, we can regard the transaction at every payment date
as a forward transaction. Hence the swap in entirety can be considered a
package of forwards. Using the forward pricing equations given above, the
swap is simply priced.
In the situation where interest payments in different currencies are
exchanged, the situation is slightly more complicated, but the same basic
principle maintains. Swaps are just packages of forward contracts and can
be priced as such.
Summary
This chapter has treated the nature and pricing of the most important
and heavily traded derivative securities. We have looked at the basic
specifications of forward, futures, option and swap contracts and what
these specifications imply for the payoff functions of long and short
positions. Further, we have looked at methods that can be used to price
69
92 Corporate finance
these securities. The basis of pricing is absence of arbitrage in all cases. We
looked most deeply at option contracts, detailing the relationships between
put, and call prices, and bounds on option prices, and finally we examined
the continuous time option pricing formula of Black and Scholes.
Although we’ve covered a lot of material here, the continual evolution and
innovation of derivatives markets and assets means that we missed much
more than we’ve treated. However, the basic features of derivatives pricing
that we’ve looked at can be extended to new and more complex securities.
A reminder of your learning outcomes
Having completed this chapter, and the Essential reading and activities,
you should be able to:
• discuss the main features of the most widely traded derivative securities
• describe the payoff profiles of such assets
• understand the absence-of-arbitrage pricing of forwards, futures and
swaps
• construct bounds on option prices and relationships between put and
call prices
• price options in a binomial framework using the portfolio replicating
and the risk-neutral valuation.
Key terms
American option
binomial method
Black–Scholes
call option
covered interest rate parity relationship
derivative
European option
exercise price
forward contract
futures contracts
long position
marked-to-market
notional pricing
put option
risk-neutral method
settlement date
short position time to maturity
underlying price
70
Chapter 4: Derivative securities: properties and pricing
Sample examination questions
1. Describe the main features of forward and futures contracts. How do
forward and futures contracts differ? Derive the no-arbitrage price of a
forward contract. (10%)
2. Describe the main features of options contracts. Show how to price a
standard European call option using a single-period binomial model.
(10%)
3. British Telecom shares are currently trading at 312p. Historically, the
(annualised) volatility of BT shares has been 20 per cent. Compute the
Black–Scholes price of a European call on BT equity, assuming a strike
price of 350p and time to maturity of six months. Assume that the riskfree rate is 5 per cent. (5%)
4. The S&P-500 ETF is trading at 1,260 today. In one year the price will
either grow by 15 per cent if there is an expansion, or fall by 15 per
cent if there is a recession. There are no dividends. A one-year zerocoupon bond purchased today has a 1 per cent interest rate.
a. What is the price of a European call option with strike price 1,100?
1,260? Describe the portfolio which would exactly replicate the first
of these securities.
b. What is the price of a European put option with strike price 1,100?
1,260?
c. What is the price of a quasi-American put with a strike price of
1,260? Assume that every six-months the S&P-500 ETF either goes
up by 10 per cent or down by 10 per cent and that the six-month
interest rate is 1 per cent (Note that the annual standard deviation
of the underlying is 0.1*√2=14.1%, which is nearly the same as
before).
71
92 Corporate finance
Notes
72
Chapter 5: Efficient markets: theory and empirical evidence
Chapter 5: Efficient markets: theory and
empirical evidence
Aim of the chapter
The aim of this chapter is to introduce the notions underlying
informational efficiency and provide a summary of some of the main
empirical tests of financial market efficiency.
Learning outcomes
By the end of this chapter, and having completed the Essential reading and
activities, you should be able to:
• understand the concept of market efficiency
• distinguish among varieties of efficiency
• understand the methodologies used to test for market efficiency
• explain the joint hypothesis problem
• present empirical evidence on varieties of market efficiency.
Essential reading
Hillier, D., M. Grinblatt and S. Titman Financial Markets and Corporate Strategy.
(Boston, Mass.; London: McGraw-Hill, 2008) no specific chapters.
Further reading
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.
Brealey, R., S. Myers and F. Allen Principles of Corporate Finance. (Boston,
Mass., London: McGraw-Hill, 2008) ninth edition, Chapter 14 (Efficient
Markets and Behavioral Finance).
Brock, W., J. Lakonishok and B. LeBaron ‘Simple technical trading rules and
stochastic properties of stock returns’, Journal of Finance 47(5) 1992,
pp.1731–764.
Campbell, J. and R. Shiller ‘The dividend-price ratio and expectations of future
dividends and discount factors’, Review of Financial Studies 1 1988.
Cochrane, J.H. ‘Explaining the variance of price-dividend ratios’, Review of
Financial Studies 5 1992, pp.243–80.
Copeland, T. and J. Weston Financial Theory and Corporate Policy. (Reading,
Mass., Wokingham: Addison-Wesley, 2005) Chapters 10 and 11.
DeBondt, W. and R. Thaler ‘Does the stock market overreact?’, Journal of
Finance 40(3) 1985, 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.
73
92 Corporate finance
Fama, E. and K. French ‘The cross-section of expected stock returns’, Journal of
Finance 47(2) 1992, pp.427–65.
French, K. ‘Stock returns and the weekend effect’, Journal of Financial
Economics 8(1) 1980, pp.55–70.
Haugen, R. and J. Lakonishok The Incredible January Effect. (Homewood, Ill.:
Dow Jones-Irwin, 1988).
Jensen, M. ‘Some anomalous evidence regarding market efficiency’, Journal of
Financial Economics 6(2–3) 1978, pp.95–101.
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.
Levich, R. and L. Thomas ‘The significance of technical trading-rule profits in
the foreign exchange market: a bootstrap approach’, Journal of International
Money and Finance 12(5) 1993, pp.451–74.
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.
Poterba, J. and L. Summers ‘Mean reversion in stock prices: evidence and
implications’, Journal of Financial Economics 22(1) 1988, pp.27–59.
Overview
In this chapter, we define and explore empirical evidence for informational
efficiency in financial markets. We begin by defining varieties of efficiency.
We then examine tests of weak-form efficiency and semi-strong-form
efficiency, and then move briefly on to strong-form efficiency tests.
We examine the issues surrounding a single set of questions, which are
of interest to finance academics and practitioners alike. Do marketdetermined financial asset prices reflect all information relevant to that
asset? Do stock prices speedily and accurately react to data on corporate
earnings and dividends? Do foreign exchange rates move quickly to adjust
to interest rate movements and capital flows?
These issues are of interest to finance practitioners, as violations of
efficiency will lead to situations where markets display unexploited profit
opportunities. Finance academics, on the other hand, have generated
reams of studies testing the efficient markets hypothesis. In this chapter
we provide an introduction to the notions underlying informational
efficiency and a summary of some of the empirical tests of financial market
efficiency.
Varieties of efficiency
The basic definition of market efficiency, which we will use in our
discussion, is as follows1:
A market is said to be efficient with respect to a given information set  if no agent
can make economic profit through the use of a trading rule based on . Economic
profit is defined as the level of return after costs are adjusted appropriately for risk.
To paraphrase the above; if I am aware of a piece of information relevant
to a given asset and the market for that asset is efficient, then I cannot
exploit my information and earn a positive net risk-adjusted return. This
seems like a fairly straightforward concept. However, in order for our
definition to be useful in an empirical context, we must specify broad
information sets observable to econometricians, which can be used along
with statistical analysis to test the efficiency of financial markets.
74
1
This definition is based
on that contained in
Jensen (1978).
Chapter 5: Efficient markets: theory and empirical evidence
Throughout the rest of this chapter we will use the definitions of market
efficiency employed in a famous survey of such issues by Fama (1991).
Fama works with three varieties of market efficiency which are repeated
below.
• Weak-form efficiency: a market is said to be weak-form efficient if
prices fully reflect all historical information. Such historical information
will include past prices (and returns) plus past data on the financial
characteristics of firms and information on macroeconomic conditions.
• Semi-strong-form efficiency: a market is said to be semi-strongform efficient if price fully, accurately and speedily reflects all new
public information releases. Further, price must reflect all past public
information.
• Strong-form efficiency: a market is strong-form efficient if prices
reflect all information, both public and private.
Note the scopes of the information sets used in the prior definitions.
That for strong form is the largest, containing any relevant information
whether known only to a few insiders (e.g. company directors who know
that a takeover is just around the corner) or to everyone. The next largest
information set is associated with the semi-strong form, containing all
information in the public domain. Examples of such information would
be corporate price-to-earnings ratios, past dividends, interest rates
and inflation rates. Finally, the most restricted information set is that
associated with the weak form (i.e. past data only).
The vast majority of academic empirical work on market efficiency has
concentrated on the first two varieties of efficiency. This is not to say
that the final definition is less important, just that it’s more difficult to
test. However, if one rejects either weak- or semi-strong-form efficiency,
then a rejection of strong-form efficiency is automatic. In the following
sections we will follow the empirical finance literature and concentrate
on the weak and semi-strong forms. We present the implications of each
for models of financial asset prices, the relationship between prices and
information announcements and, finally, the results from empirical studies
of efficiency.
Risk adjustments and the joint hypothesis problem
Definition 1 characterises markets as efficient with respect to some
information if one can’t make positive risk-adjusted returns by trading
on that information. This clearly implies that, if one wishes to test
informational efficiency, one needs a technique for calculating riskadjusted returns.
The way in which this is generally done is as follows. The lesson of
Chapters 2 and 3 was that riskier assets earn higher expected returns
(whether we’re in a CAPM world and risk comes from the market portfolio
or an APT world with multiple-risk factors). Hence, we first find a model
that allows us to estimate the expected returns on an asset. This model
may be the CAPM, the APT or a less theory-motivated choice, such as
the return on a broad stock index. A fairly popular choice of model for
generating expected asset returns is the market model, which just
estimates the expected return of stock i through a regression of stock i’s
actual returns on those of the market. An even more naïve method for
generating expected returns that has been employed is to simply assume
that they are constant. Another commonly used model is the Fama and
French three-factor model, whose factors are the market return, a portfolio
75
92 Corporate finance
long small stocks and short large stocks, and a portfolio long value stocks
and short growth stocks.
This then gives us a time-series of expected returns for stock i. Riskadjusted or abnormal or excess returns are then just calculated as the
difference between the actual returns on stock i and expected returns,that
is,
rtX = rt – E(rt)
(5.1)
where rtx is the excess return and rt the actual stock return at time t. The
efficient markets hypothesis is concerned with the ability to make excess
returns based on a certain information set. Hence, the object of our
attention when testing market efficiency is the excess return derived in
equation 5.1. Throughout the rest of this chapter, we will discuss tests of
market efficiency and, unless explicitly stated otherwise, use of the word
return will mean excess return.
First, there is one further important point to be made at this juncture.
Empirical researchers do not know the true model that generates expected
returns in the economy. Hence, their choice of expected return-generating
mechanism, used to adjust actual returns, may be wrong. This implies
that abnormal returns may be incorrectly measured. These (inaccurate)
abnormal returns are then used in tests of market efficiency. Let’s assume
that the tests indicate that abnormal returns can be earned on the basis
of a given piece of information. We would then conclude that markets are
not efficient with respect to this information. However, it might be the
case that markets are actually efficient and that our use of an incorrect
risk-adjustment technique is driving the result that abnormal returns can
be gained.
Therefore, we are left in a position where we are not sure whether markets
are inefficient or our model of expected returns is wrong. This is known as
the joint hypothesis problem associated with testing market efficiency. The
null hypothesis of any test of efficiency is comprised of two components:
• informational efficiency
• the accuracy of one’s model for expected returns.
As the true model of expected returns is unknown, a rejection of this null
hypothesis cannot be immediately taken as evidence that markets are not
efficient. The existence of the joint hypothesis problem should be kept at
the forefront of your mind when discussing empirical results on efficiency.
Weak-form efficiency: implications and tests
Recalling the above, in a weak-form efficient market, prices should fully
reflect historical data. What implications does this have for processes to be
followed by asset prices?
A first, very straightforward, implication is that current and past asset
returns should have no predictive power for future returns on that asset.
Another way of saying this is that you cannot form a trading rule based on
current and historical returns, as this allows you to make more than a fair
return (where the fair return is determined by the risk of the investment).
Yet another way of saying this uses statistical notation. The inability of
current and past returns to forecast the level of future returns can be
written as:
E(rt+1 |rt , rt–1 , rt–2 , rt–3 , …) = 0,
76
(5.2)
Chapter 5: Efficient markets: theory and empirical evidence
that is, the expectation of next period’s return conditional on the entire
history of returns is zero. An implication of this statement is that returns
are uncorrelated with their own past values. This can be written as:
A
Cov(rt , rt–s) = 0, s > 0.
(5.3)
Tests of weak-form efficiency or return predictability can be based upon
5.3. Take a time-series of stock returns and compute the autocorrelations
of returns.2 Weak-form efficiency implies that all autocorrelations of
returns should be statistically indistinguishable from zero.3 Otherwise,
current or past returns have a systematic relationship with future returns
and can hence be used in prediction.
The random walk model
A popular model for asset prices is based on 5.2. This is the random walk
model (RWM) of stock prices and it is given in equation 5.4. Denoting the
log of the stock price by P we have,
A
Pt = Pt–1 + εt, E(εt ) = 0, Cov(εt , εs ) = 0, t t ≠ s.
(5.4)
Equation 5.4 says that the change in price from time t–1 to t is a mean
zero, serially uncorrelated innovation, εt. We can think of this innovation
as representing new public information arriving at market during period t.
As it represents new information that is equally likely to be good or bad, it
has zero mean. Further, new information is by definition unpredictable,
such that εt is uncorrelated with its own past values. Hence, past price
changes carry no information about current or future price changes.4
Note that the stock price return is just the first difference of the log stock
price (i.e. rt = Pt – Pt-1) and equation 5.4 then implies that rt = εt. Via the
properties of the innovation, εt, it is clear that returns have zero mean and
are uncorrelated over time in line with equation 5.3. Hence, tests of return
autocorrelation can be viewed as tests of the random walk model.5
In the preceding discussion, we concentrated on predicting future returns
using the history of returns only. Weak-form efficiency would also be
violated, however, if any information available at time t or before allowed
us to forecast returns. As a result, researchers have run regressions of the
following type in order to assess weak-form efficiency:
rt+1 = α + βXt + ut, E(ut) = 0, Var(ut) = σ2.
(5.5)
Here, Xt is the forecasting variable for returns and ut is a regression error
term. Weak-form efficiency would imply that the coefficient β in equation
5.5 should be statistically indistinguishable from zero reflecting the
inability of Xt to forecast returns.
Calendar effects
A last group of studies that we will treat in empirical analysis of weak-form
efficiency is that looking for calendar effects in stock returns. A calendar
effect is defined as a pattern in stock returns related to either the day of
the week, the week of the month or the month of the year. An example
of such an effect would be the idea that stock returns were consistently
greater on Wednesdays than on other days of the week. Alternatively, a
researcher might examine whether stock returns are lower in the first
week of every month relative to all other weeks of the month. Tests of this
type fit into the statistical testing framework developed around equation
5.5. In the case of calendar effects, Xt would be defined as a dummy
variable (or set of dummy variables) that picks out the desired calendar
effect. Using the Wednesday effect example mentioned above, Xt would be
defined to take the value 1 if rt+1 was realised on a Wednesday and zero
2
The autocorrelation at
displacement s is simply
the autocovariance at
displacement s divided
by the sample variance
of returns where the
autocovariance at
displacement s is given
by 5.3.
3
A statistical test can be
formed by using the
result that the
asymptotic variance
of an estimated
autocorrelation is T–1
where T is the number
of return observations in
the sample.
4
Note that, while past
price changes can’t be
used to forecast current
price changes, past
prices give non-zero
forecasts of current and
future prices. Indeed,
if the price at time t
is Pt then the optimal
forecast of all future
prices is Pt also. This
can be checked from
equation 5.4.
5
Other tests of the
usefulness of past
returns for prediction of
future returns include
those which examine
whether the sign of
returns is predictable.
These tests are based
on the likelihood of
observing sequences of
positive and negative
returns over time.
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92 Corporate finance
otherwise. Hence, the regression 5.5 then picks out the systematic effect
on stock returns of the fact that the day is Wednesday.
Weak-form efficiency: empirical results
The amount of academic time and effort devoted to testing weak-form
efficiency over the past 30 years is staggering. This can be seen to reflect
the importance academics place on the informational efficiency of financial
markets. A selective review of some of this research is given below.
Tests of return autocorrelation
There is a large literature that examines the autocorrelations of returns on
individual stocks and portfolios. Results from such studies vary with the
frequency over which returns are calculated. Here we mention just a few.
When looking at daily and weekly returns, researchers have generally
found that returns are positively autocorrelated. Examples of such papers
are Fama (1965) and Lo and MacKinlay (1988). Interestingly, Lo and
MacKinlay show that the strength of the autocorrelation is dependent on
the size of the stock in question (where size might be measured by market
capitalisation, for example.) Portfolios of small stocks tend to have much
higher positive autocorrelation than returns on large stocks. One reason
put forward to explain this is that infrequent and non-synchronous trading
of small stocks will generate positive portfolio return autocorrelation,
even when individual stock returns are uncorrelated over time. Hence,
it is not obvious that the return predictability implied by short-term
autocorrelation evidence reflects informational inefficiency.
Autocorrelation evidence is reversed when one looks at very long horizons
though. Fama and French (1988) and Poterba and Summers (1988) both
show that portfolio returns measures over three to five years demonstrate
negative autocorrelation. This would seem to indicate that stocks that
have increased in price over the five years up to today should tend to fall
in price in the five years from today and hence to indicate informational
inefficiency. However, it might be the case that such long swings in prices
(which generate mean reversion in long horizon returns) reflect mean
reversion in expected returns over time, which is not picked up by our
expected return generating model (i.e. this result may be a manifestation
of the joint hypothesis problem).
Calendar effects
One of the most famous empirical findings in finance is the so-called
‘incredible January effect’. This result is that stock/portfolio returns are
statistically positive and greater in January than in any other month of
the year. Again, this result is most pronounced for small stocks. Hence, it
would seem that a trading rule that indicated that one should buy (small)
stocks at the end of December and sell them at the end of January would
make money. Potential explanations for the January effect include:
• taxation impacts
• year-end effects
• effects from the remuneration packages of fund managers.
None of these seems completely plausible. Another point to note is that the
January effect seems to be an international phenomenon.
The existence of the January effect is puzzling to economists because of
the following logic. Assume that all agents in the economy observe that a
trading strategy consisting of being in the market in January only makes
78
Chapter 5: Efficient markets: theory and empirical evidence
excess returns. Then, all agents would follow such a strategy. However,
the impact of this would be that stock prices would be bid up at the end
of December due to buying pressure. Similarly stock prices at the end of
January would drop due to extra sales of equity. This would tend to erode
the abnormal return that could be earned in January until, ultimately, it
was zero. Hence, the actions of rational agents should eliminate these
types of effects. The continued existence of the January effect is therefore
extremely puzzling.
Other calendar effects that have been uncovered include:
• day of the week effects (French (1980))
• holiday effects (Haugen and Lakonishok (1988)).
These are, however, not as well known as the January effect and are less
consistent. All in all, the calendar-effects literature gives strong indications
of market inefficiencies. It is difficult to invent stories that suggest there
should be calendar effects in expected returns (so we can’t turn to the
joint hypothesis problem as a way out) and we are left with the possibility
that profits are available on this basis.
Impact of other variables on stock returns
Lakonishok, Shleifer and Vishny (1994) investigate whether it is possible
or not to beat the market by choosing shares whose price is low relative
to fundamentals such as earnings, dividends, the book value of equity, or
cash-flows. In order to do so, they allocate stocks to 10 different portfolios
according to the magnitude of prices relative to a given fundamental, the
portfolio consisting of the stocks with the lowest prices to fundamental
being referred to as the value portfolio and the portfolio consisting of the
stocks with the highest prices to fundamental being referred to as the
glamour portfolio. They track the performance of these portfolios over five
years following the allocation of the stocks to the portfolios. Their findings
are as follows:
• The lower the portfolio’s average price to fundamental (at the time of
the allocation of the stocks to the portfolios), the higher the portfolio’s
subsequent average return.
• The value portfolio furthermore outperforms the glamour portfolio by
about 10 to 11 per cent per annum (or equivalently 8.5 to 9 per cent
after adjustments for size) over a period of five years.
• The excess returns of value over glamour stocks have persisted over the
1968–90 period.
The empirical evidence reported by Lakonishok, Shleifer and Vishny
in the context of fundamental analysis is furthermore consistent with
that reported by DeBondt and Thaler (1985) in the context of technical
analysis. DeBondt and Thaler allocate stocks to portfolios on the basis of
past performance as measured by excess returns in prior years. Portfolios
of previous ‘losers’ are found to subsequently outperform previous
‘winners’: over the three years following the allocation of the stocks to the
portfolios, the losing stocks earn about 25 per cent more than the winners,
even though the latter are significantly more risky. Furthermore, the
subsequent excess returns tend to take place in January.
For the empirical evidence reported by Lakonishok, Shleifer and Vishny to
be consistent with market efficiency,6 the value portfolio must be riskier
than the glamour portfolio. Using conventional measures of risk, such as a
measure of systematic risk (β) or the standard deviation of portfolio
6
In efficient markets,
higher subsequent
returns on average
can only be explained
through risk: riskier
investment strategies
must generate on
average higher
subsequent returns.
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92 Corporate finance
returns, the value portfolio is found to be quite risky. The difference in risk
between the value and glamour portfolios is, however, not remotely high
enough to justify the observed differences in subsequent average portfolio
returns.
An examination of the value portfolio reveals that value stocks tend to
experience poor performance in previous years, as measured by growth
in sales, earnings or cash flows, resulting in highly negative excess stock
returns. Value stocks also tend to have small market capitalisations. In
contrast, glamour stocks tend to experience high performance in previous
years, resulting in highly positive excess stock returns, and tend to have
large market capitalisations. In a period of two to five years following the
allocation of the stocks to the portfolios, the performance of the glamour
stocks, as measured by growth, tends to deteriorate while the performance
of the value stocks tends to improve to the point where it exceeds many
attributes of the glamour stocks.
By comparing the actual earnings growth rates with the expected earnings
growth rates implicit in stock prices, Lakonishok, Shleifer and Vishny find that
the high expected earnings growth rate of glamour stocks is only validated
for one to two years. Lakonishok, Shleifer and Vishny therefore argue that
their empirical evidence is consistent with investors pursuing naïve strategies
by always treating a well-run company as a good investment, extrapolating
trends and overreacting. This interpretation is furthermore consistent with
evidence from the psychology literature suggesting that as individuals we
tend to rely too much on very recent data when making decisions.
Fama and French (1992) have a different interpretation for the 10 to 11
per cent per annum excess return of value over glamour stocks. Fama and
French recognise that variables like size (market capitalisation), earnings
yield, dividend yield, leverage, and book-to-market are all scaled versions
of a firm’s stock price and are hence correlated.7 When trying to explain
portfolios’ average stock returns (proxying for expected returns), Fama
and French find that size and book-to-market capture the cross-sectional
variation in average stock returns associated with size, earnings yield,
dividend yield, book-to-market, leverage and other fundamentals. Fama
and French therefore argue that a stock’s size and book-to-market proxy
for the firm’s exposure to risks are priced by the capital market.8
According to Fama and French, the reason for value strategies’ superior
returns is that they are fundamentally riskier (higher average returns are
simply a compensation for these risks). The debate about the
interpretation for the 10 to 11 per cent per annum excess return of value
over glamour stocks illustrates again the joint hypothesis problem.
Technical trading rule applications
Finally, we will discuss briefly weak-form tests, which are pretty much
direct examinations of market efficiency according to definition 1.
One of the things that finance academics find most puzzling is finance
practitioners’ reliance on technical trading rules to generate trading
signals. Via the logic used above, if a trading rule actually did generate
profits, then its adoption by the masses would eliminate the gains it had
generated in the past. Hence, technical-trading rules would appear to be
valueless and practitioners’ trust in them is misguided.
However, recently, certain academics have tested this argument by
examining how very simple technical-trading rules would have worked on
historical-data spans. An example of a simple technical rule is the moving
average cross-over.
80
7
The earnings yield and
the dividend yield are
respectively the inverse
of the price-to-earnings
ratio and the inverse of
the price-to-dividend
ratio.
8
According to the CAPM,
b is the only factor that
should cause expected
returns to differ (i.e. no
other variable should
explain expected returns
once we have accounted
for the effects of b). Fama
and French, however,
show that, when allowing
for variations in b that
are unrelated to size,
there is no reliable
relation between b and
average portfolio return.
Chapter 5: Efficient markets: theory and empirical evidence
A moving average of stock prices at length k is an equally weighted
average of the current and past k–1 stock prices. The moving average
cross-over rule compares a long (k high, e.g. 100) and short (k low, e.g. 5)
moving average in order to determine one’s trading position. If the short
moving average cuts the long moving average from below, one should
buy the asset in question. If the short cuts the long from above, then one
should go short in the asset. The reasoning behind this is that, when the
short cuts the long from below, it is seen to signal the start of an upward
trend in prices, and vice versa.
To the surprise of many academics, empirical studies have shown that rules
as simple as that given above generate positive excess returns on average.
A famous study by Brock, Lakonishok and LeBaron (1992) applies such
rules to foreign exchange rate and US stock index data with some success.
Similarly, Levich and Thomas (1993) show that profits are available from
the application of technical rules to currency futures markets. Such results
do not inspire confidence in the weak-form efficiency of financial markets.
Semi-strong-form efficiency: event studies
Semi-strong-form efficiency is concerned with the speed at which new
information is impounded into asset prices. The primary empirical
methodology used for examining semi-strong-form efficiency is the event
study. In this section, we give an overview of the event study methodology.
To illustrate, we consider a hypothetical situation in which we examine
the impact of firms’ earnings announcements on stock prices. As earnings
announcements reflect the financial health of a firm, we would expect
stock prices to rise upon the announcement of better-than-expected
earnings (good news) and fall if earnings are below expectations (bad
news.) As emphasised above, the event study characterises the speed at
which good/bad news is assimilated into prices.
• The first step in conducting an event study is to collect a sample of firms,
all of which have had an earnings announcement within your chosen
interval. It is very important that you know precisely on which day each
firm’s earning announcement was actually made. The reason for this will
become clear below. Also, you must have access to stock prices for these
firms prior to, and after, the date of the earnings announcement.
• If you recall the earlier discussion, we were concerned with the impact of
unexpected earnings on stock prices. This implies that we need a measure
of the market’s ex-ante expectation of earnings for each firm in the sample.
Fortunately, several corporations collect and collate such expectations from
analysts. Using these expectations we can derive the unexpected portion
of each firm’s earnings announcement. From here onwards, we assume
that all of the firms in our sample experienced positive earnings surprises
in order to make things easier to present.9
• You must next decide on the period the event study is going to be
based on. Let’s assume in our earnings announcement study that we
are going to look at a period starting 50 days before the announcement
until 50 days after the announcement. Denoting the actual
announcement date for each firm by date 0, this implies we have stock
returns for each firm dated from –50 to 50. In order to account for risk
(as treated earlier in this chapter), we must also deduct expected
returns from the actual return on each date to get a series of abnormal
returns running from date –50 to date 50.10
• The final step in the event study is to construct average abnormal
returns for each date. The date –50 average abnormal return, for
9
Alternatively, assume
that we have discarded
any firms in the sample
with a negative earnings
surprise.
10
We might use the
CAPM to estimate
expected returns, for
example. Note, however,
that we should estimate
the CAPM relationship
using return data
from the period prior
to the event window
and then extrapolate
the expected return
generation through
the event window. This
ensures that the event
itself doesn’t impinge
upon our estimation of
expected returns.
81
92 Corporate finance
example, is just the sum of the date –50 abnormal return across all
firms divided by the number of firms in the sample. This operation
is repeated for every date in the event window (i.e. for all dates
between –50 and 50). In general, these average abnormal returns are
accumulated from date –50 to 50, and a plot of the cumulative average
abnormal return against the date is formed.
Figure 5.1
If stock markets were efficient with respect to the positive earnings
surprises we are studying, then we would hope to see a cumulative
abnormal return diagram as shown in Figure 5.1. Why is this the case?
Well, on the announcement date (date 0), we see a large increase in
cumulative abnormal returns. This reflects the assimilation of the
unexpected earnings information into prices.11 Note that there is no
systematic increase in the cumulative abnormal return after the
announcement date. If one were to see continued systematic increases in
abnormal returns after the announcement date, this would imply that it
was taking time for the earnings information to be reflected in prices and
hence informational inefficiency. Such a situation is shown in Figure 5.2.
One feature of such a diagram that we haven’t yet mentioned is the
systematic rise in prices prior to the announcement. This can occur for
several reasons.
• The earnings information may be partially leaked prior to the official
announcement, and (in line with informational efficiency) the leaked
information is reflected in price.
• Certain announcements are only made after increased prices (i.e. the
announcement date is chosen by firm management to be just after a
price rise). Stock splits, for example, generally occur after rising stock
prices and hence would demonstrate the pre-event pattern shown in
Figure 5.2.
Figure 5.2
82
11
Of course, if we were
studying a sample
of negative earnings
surprises (i.e. bad news),
then we would hope for
a picture which looked
like the mirror image of
Figure 5.1 in the x-axis.
Chapter 5: Efficient markets: theory and empirical evidence
Semi-strong-form efficiency: empirical evidence
A multitude of announcement types have been studied by academics. A
large number of event studies imply that new information is quickly and
accurately reflected in prices. Many authors find that new information
is quickly and accurately reflected in stock prices (often within 5 to 10
minutes of the announcement). Asquith and Mullins (1983), for instance,
demonstrate that unexpected dividend increases cause stock price rises.
The same authors show that stock issues are bad news in a 1986 study.
Empirical evidence from some types of event studies, however, would
appear to be inconsistent with semi-strong efficiency. For instance, Ball
and Brown (1968) report that stock prices do not fully incorporate new
information embodied in unexpected earnings announcements. Prices of
good news stocks continue to rise after earnings announcements, while
prices of bad news stocks continue to fall. Ball and Brown, hence, provide
evidence of underreaction to earnings announcements: the financial
market requires up to a few months to fully incorporate the information
content of earnings announcements.
Strong-form efficiency
Strong-form efficiency has received the least attention in empirical work
and we will only briefly mention it here. Certain studies examine whether
corporate insiders (e.g. company directors) make gains from trading in
their own company’s stock. Results suggest that insider trades can
generally be used to predict subsequent stock price changes, and hence
such work concludes that markets are not strong-form efficient.12 Other
work shows there to be information in the forecasts of professional
analysts and surveys (for example the Value Line survey). Again, this
would seem to indicate the existence of private information in the hands
of professional or privileged agents.
12
Insiders tend to buy
prior to stock price rises
and sell prior to stock
price drops.
On the other hand, however, work on mutual fund performance shows
that these actively managed portfolios underperform other broad-based
portfolios with similar risk. A recent study on UK funds by Blake and
Timmermann, for example, indicates that, over a 23-year span, funds
underperformed the market by about 2 per cent per annum.
Hence, evidence of private information on stock prospects is also mixed.
Results on mutual fund performance would certainly suggest that fund
managers are no better informed than the average investor, whereas
company directors seem to trade in a way that betrays the fact that they
possess information that markets do not. The latter finding of private
information is strengthened by results on the information content of
analyst forecasts.
Long horizon forecastability
A common misconception about the efficient market hypothesis is that
stock returns should be unpredictable. The efficient market hypothesis
actually says that risk-adjusted returns should be unpredictable, as in
equation 5.1. If the random walk model is the appropriate model for
asset prices, then both stock returns and excess stock returns should be
unpredictable because expected stock returns are constant. However, more
generally, expected stock returns, and therefore realised stock returns may
be predictable.
83
92 Corporate finance
We have already seen that assets that are more risky have higher expected
returns as compensation for that risk. For example, the CAPM implies that
any asset with a higher β must also have a higher expected return. This is a
cross-sectional implication. However, this can also happen in time series. If
certain times are riskier than other times then investors will not be willing
to pay a high price for risky assets, therefore prices of risky assets will be
low and expected returns on risky assets will be high. Thus, if risk is time
varying, then expected returns should also be time varying; if a variable
can describe the quantity of risk, this variable should also predict stock
returns.
Economists do not agree on what exactly is the right benchmark for
calculating abnormal returns; it may be correlation with the market
(CAPM), changes to the growth rate of productivity as suggested by Bansal
and Yaron (2004), changes to the standard of living we are used to as in
Campbell and Cochrane (2000), or a liquidity crunch. However, regardless
of what exactly constitutes risk, risk is likely to be changing through time.
For example, Robert Engle and Clive Granger won the 2003 Nobel Prize
in Economics for improving our understanding of how volatility of asset
returns moves through time.
Time varying risk implies that prices of risky assets should be relatively
low during risky times and that this should also forecast high expected
returns. We can look at ratios of prices relative to fundamentals to check
when prices are low, such ratios include the price dividend ratio, the price
earnings ratio, the wealth to consumption ratio, and the price to rent
ratio when considering housing. Indeed, such ratios are high some times
and low other times, as can be seen in Figure 5.3, which plots the price
dividend ratio for the aggregate US equity market over an 80-year period.
140
120
100
80
60
40
20
1920
1930
1940
1950
1960
1970
1980
1990
2000
2010
Figure 5.3
Consider the price dividend ratio, according to the Gordon growth model,
P/D = 1/(r–g). If the price dividend ratio is low today, it must be either
that expected growth rates are low, or that expected discount rates are
high. John Cochrane showed that variation in price dividend ratios comes
mostly from variation in discount rates rather than growth rates. That is,
price dividend ratios are low when discount rates are high.
Several studies have shown that such ratios do forecast asset returns, but
only at longer horizons. Both the significance of coefficients and the R2
increase as the time period over which returns are calculated increases. For
example, at horizons of three to five years, the combination of the price to
84
Chapter 5: Efficient markets: theory and empirical evidence
dividend ratio and the consumption to wealth ratio can forecast aggregate
stock returns with R2 of nearly 40 per cent.
However, here too, we cannot tell if movements in the aforementioned
ratios and the predictability of returns are due to market inefficiencies or
time varying risk driving expected returns. For example, consider a world
in which public sentiment, independent of fundamentals, can affect stock
prices. That is, in certain times people are very optimistic about stocks
for no fundamental reason, and at other times they are overly pessimistic
about stocks. Then, during times of such optimism, the price earnings ratio
would be high and would forecast low future returns as eventually everyone
would realise that the market is overvalued and sell. Similarly in times of
pessimism the price earnings ratio would be low and forecast high future
returns.
Summary
The evidence given above provides much food for thought. Results from
event studies tend to indicate that markets are close to (if not perfectly)
informationally efficient. Return predictability tests, on the other hand,
indicate some striking departures from weak-form efficiency. Research on
these issues is still progressing. Some more recent event study results (on
initial public offerings and new stock market listings, for example) seem to
be less supportive of efficiency. At the same time, more careful statistical
procedures are indicating that at least some of the weak-form efficiency
rejections may be dubious. Putting together this diverse group of results
with the joint hypothesis problem and the problems in modelling expected
returns means that a definitive answer on market efficiency is difficult
to come by. Indeed, although we have dichotomised results as indicating
‘efficiency’ or ‘inefficiency’, it may be more sensible to talk about degrees of
efficiency and classify certain markets or markets at certain times as being
more efficient than others.
A reminder of your learning outcomes
Having completed this chapter, and the Essential reading and activities, you
should be able to:
• understand the concept of market efficiency
• distinguish among varieties of efficiency
• understand the methodologies used to test for market efficiency
• explain the joint hypothesis problem
• present empirical evidence on varieties of market efficiency.
Key terms
calendar effect
contrarian strategies
economic profit
efficient markets hypothesis
event study
excess return
glamour stocks
market model
85
92 Corporate finance
momentum strategies
moving average cross-over
random walk model
return autocorrelation
semi-strong-form efficiency
strong-form efficiency
value stocks
weak-form efficiency
Sample examination questions
1. Discuss why prices cannot be forecasted in an efficient market. Evaluate
the empirical evidence for and against the weak-form efficiency. (10%)
2. Recent research has shown that a firm’s market capitalisation and book
value relative to its market value explain the cross-section of stock returns
better than β. Is this consistent with stock market efficiency? (10%)
3. Technology stocks are coming to the new issues market at very high price
earnings multiples. Is this consistent with stock market
efficiency? (5%)
4. Consider running a regression of the three-year aggregate stock market
return on the lagged price-to-dividend ratio for the aggregate market.
What sign do you expect the slope coefficient to have? Discuss possible
explanations.
5. Below are hypothetical daily returns for the aggregate stock market, as
well as for securities A to F over one month. The βs of the securities are
provided above their names. The yield on treasury bonds over this period
was so low that it is safe to assume that it is zero. Marked with stars are
days on which these firms announced they would issue common equity
and use it to payoff some of the firm’s outstanding debt. Assume that
the announcement happened in the morning. Conduct an event study.
Describe what you are doing and why you are doing it. Assuming nothing
special happens on any other day, what does the market think about these
equity issuances qualitatively and quantitatively? What does the result
for day –1 say about market efficiency? What does the result for day 0
say about market efficiency? What does the result for day +1 say about
market efficiency?
86
Chapter 5: Efficient markets: theory and empirical evidence
1
0.8
1.2
1.8
2
0.9
Market
A
B
C
D
E
F
1.71
1.56
1.1
2.27
3.35
3.44
1.68
–3.03
–3.04
–2.1
–3.45
–5.3
–5.72
–2.54
0.15
0.3
0.14
*0.14*
0.33
0.47
0.41
–0.28
*–0.11*
0.1
–0.62
–0.81
–0.33
0.04
0.51
0.75
0.25
0.58
0.82
0.98
0.74
0.35
0.27
0.29
0.22
0.29
0.79
0.46
0.11
0.3
–0.03
0.01
0.36
0.42
0.03
–0.8
–0.8
–0.71
–0.61
–1.2
–1.29
–0.49
0.33
0.17
0.43
0.05
0.91
*1.37*
0.01
–0.37
–0.24
*0.23*
–0.44
–0.79
–0.57
–0.14
–0.21
–0.01
0.18
–0.19
–0.04
–0.37
0.1
0.05
–0.25
0.4
–0.18
–0.08
0.27
–0.21
0.25
0.33
0.05
0.41
0.15
0.26
–0.06
0.19
0.14
0.25
0.51
0.89
0.33
0.29
0.82
0.68
0.9
0.89
*2.23*
1.62
0.58
–0.34
–0.37
–0.33
–0.73
–0.03
–0.8
–0.22
0.38
0.39
0.38
0.27
1.02
0.58
0.38
0.2
0.34
0.37
0.27
0.45
0.51
*0.98*
0.41
0.69
0.17
0.54
0.85
0.66
0.66
–0.13
–0.07
–0.13
–0.06
–0.31
–0.15
–0.38
87
92 Corporate finance
Notes
88
Chapter 6: The choice of corporate capital structure
Chapter 6: The choice of corporate
capital structure
Aim of the chapter
The aim of this chapter is to analyse and explain the choices of corporate
capital structures made by firms’ managers. With this aim in mind, we
first introduce a stylised model in which capital structure is irrelevant
(Modigliani–Miller). We then relax some of the assumptions made in this
stylised model in order to explain empirical evidence on firms’ capital
structures.
Learning outcomes
By the end of this chapter, and having completed the Essential reading and
activities, you should be able to:
• outline the main features of risky debt and equity
• derive and discuss the Modigliani–Miller theorem
• draw the link between Modigliani–Miller and Black–Scholes
• analyse the impact of taxes on the Modigliani–Miller propositions.
Essential reading
Hillier, D., M. Grinblatt and S. Titman Financial Markets and Corporate Strategy.
(Boston, Mass.; London: Macmillan, 2008) Chapters 14 (How Taxes Affect
Financing Choices) and 16 (Bankruptcy Costs and Debt–Holder–EquityHolder Conflicts).
Further reading
Brealey, R., S. Myers and F. Allen Principles of Corporate Finance. (Boston,
Mass.; London: McGraw-Hill, 2003) Chapter 19 (How Much Should a Firm
Borrow?).
Copeland, T. and J. Weston Financial Theory and Corporate Policy. (Reading,
Mass.; Wokingham: Addison-Wesley, 2005) Chapter 15.
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.
Warner, J. ‘Bankruptcy costs: some evidence’, Journal of Finance 32(2) 1977,
pp.337–47.
Overview
In most of the preceding chapters of this guide we have examined
the pricing of assets – both physical investment projects and financial
securities. With respect to the latter, we examined the pricing of stocks
and bonds using present value techniques and equilibrium financial asset
pricing via the CAPM and APT.
Thus far, however, we have said nothing about the mix of securities
actually issued by corporations. Should firms aim to use a large proportion
of debt in their financing or, conversely, should they employ equity
financing in the main? In this chapter and the next we examine the
firm’s decision over which types of claim to issue. The most important
89
92 Corporate finance
result we will find is that, under a certain set of assumptions, the firm is
indifferent about the mix of debt and equity that it uses in its financing.
This result is the first Modigliani–Miller theorem (MM1). We go on
to explore deviations from the MM1 assumptions and how this affects
the debt–equity choice through the introduction of taxation effects, costly
bankruptcy and information asymmetries.
Basic features of debt and equity
Before moving into our analysis it is useful to introduce the most basic
securities actually issued by corporations: risky debt and equity.
Corporations hold debt in many forms. They borrow money from banks
through straightforward loan and overdraft facilities, they issue corporate
debt, and they have trade credit with their trading partners. The bonds
issued by firms can have complicated features, such as convertibility, the
ability to be called and differences in seniority. To simplify matters,
however, we will think of corporate debt as being composed of a number
of bonds.1 Each bond entitles the holder to claim a fixed amount of cash
from the firm at a given maturity date. The amount reclaimed is termed
the face value of the debt.
Two important features of corporate debt are as follows.
1. In times of corporate bankruptcy (the cash flow to the firm being less
than the claims upon it), bond-holders have priority over equity-holders
(i.e. they get their share of the cash first).
2. Interest paid to debt claims is deductible from a corporation’s corporate
tax bill.
The latter point will not be used at present but will come in later. The
first of the preceding pair of points implies that corporate debt has the
following payoff function.
Payoff
[Xt , B] –
B
0
B
Xt
Figure 6.1
The horizontal axis of the graph above represents the cash flow to the firm
(X), and the vertical axis shows the payoff to debt assuming the amount
promised to the group of all debt-holders (the face value) is denoted B.
When the cash flow to the firm is less than the face value, the debt-holders
gain the entire amount. For values of the cash flow at and above the face
value, the payoff to debt-holders is constant at B.
90
1
We have already
talked about bond
characteristics and
pricing in Chapter 2.
Chapter 6: The choice of corporate capital structure
The holders of corporate equity receive the residual cash flow accruing to
the firm after payments to debt-holders. However, despite having a claim
that is junior to that of debt-holders, equity-holders elect the board of a
firm and have voting rights over corporate activities and are hence the true
owners of the corporation. Equity also comes in many forms, but we will
focus on the characteristics of common stock.2
Stock-holders receive cash income in the form of dividend payments.
These payments, unlike payments to service debt, are not deductible from
corporation tax obligations. Given the residual nature of the equity claim,
the payoff to equity is as given in Figure 6.2.
2
Other types of equity
include preferred stock
and warrants.
Payoff
[Xt – B, 0]+
0
B
Xt
Figure 6.2
When the firm’s cash flow (X) is at or less than the face value of debt (B),
equity-holders receive nothing. However, they receive every dollar of cash
flow greater than B. This gives the kinked payoff function shown in Figure
6.2, which (anticipating future developments) is of precisely the same
form as that of a European call option.
The Modigliani–Miller theorem
We now know what corporate debt and equity claims look like. One
unanswered question, however, is what mix of debt versus equity should
firms issue? In finance parlance, the ratio of the market value of debt
to that of equity is known as the leverage or gearing ratio. Hence,
the preceding question can be rephrased as follows. What is the optimal
leverage ratio that a firm should aim for? This question was addressed in
the 1950s by Franco Modigliani and Merton Miller. They showed the result
that is the focus of the current section: under given assumptions, firms are
indifferent about their leverage. This is because firms with differing debtto-equity ratios but the same investment policies have identical values, and
hence there is no value to leverage.
The assumptions underlying MM1 are as follows:
• capital markets have no frictions (including no taxes or transactions
costs)
• investors have perfect information and homogeneous expectations
• investors care only about their wealth
• financing decisions do not affect investment outcomes.
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92 Corporate finance
To prove their indifference proposition, Modigliani and Miller used the
notion of absence of arbitrage, which we have already come across in
previous chapters.
Consider two firms. The first is entirely equity-financed, and we call it firm
U. A second firm has an identical set of investment projects but has issued
both debt and equity. We shall refer to the second firm as firm L and assume
it has issued B units of debt that earn interest at rate rd. Finally, assume for
simplicity’s sake that everything in our world lasts for one period only.
Consider an investor who holds a proportion α of firm U’s equity. As this
firm is solely equity-financed, our investor always earns a proportion α of
the firm’s cash flow (X). Assume that the same agent also buys α of firm
L’s equity and α of firm L’s debt. When the cash flow to firm L is less than
the face value of its debt (B) obligations, our investor earns α of the cash
flows through his share of total debt. When cash flow exceeds the face
value of debt, he also gets a payoff on his equity claim.
In Table 6.1 we show the payoff to our investors’ positions in both firms
under two scenarios. The first represents the case where the cash flows to
the two firms are smaller than the face value of firm L’s debt. The second
case is when firm U’s cash flow exceeds firm L’s debt obligations. Note
that, in both cases, the investor earns an identical amount from the two
positions, regardless of the actual cash-flow outcome. Hence, in line with
the absence of arbitrage arguments used in Chapters 3 and 4, the two
positions must be identically priced.
Type of claim
Debt
Payoff from position in U
X < B(1 + rd ) X > B(1 + rd )
0
0
Payoff from position in L
X < B(1 + rd ) X > B(1 + rd )
αX
αB(1 + rd )
Equity
αX
αX
0
α(X–B(1+rd))
Total
αX
αX
αX
αX
Table 6.1
The price of the position in the unlevered firm is just αVU where the
value of the unlevered firm is denoted VU. The value of the position in
the levered firm is αE + αD = α(E+D), where E is the market value of the
levered firm’s equity, and D is the market value of the levered firm’s debt.
Of course, the total value of the levered firm (VL) must be the sum of E and
D. Hence, the price of the levered position is αVL. Equating the price of
levered and unlevered position yields the result that VU = VL, which is the
MM capital structure irrelevance proposition.
The key to the above result is that financing decisions do not affect
investment outcomes. Hence, two firms with identical investment policies
will derive identical returns regardless of their financing. As their
investment proceeds are the same, they should have the same value.3
Another key point is that none of their cash flow goes to anyone outside
those who own debt and equity.
An alternative way to show the MM capital structure irrelevance
proposition is to show that stake-holders in the firm are indifferent to
changes in the firm’s capital structure. The reason for this is that stakeholders can, without cost, undo any changes the firm makes through their
own trading in the firm’s securities.
Consider once more an investor who owns a proportion α of firm L’s
equity. The payoff associated with this position is α(X – B(1+rd)). Firm
L now chooses to repurchase half of its equity (costing E/2) and funds
92
3
You can think of this
result in the following
way: when you slice a
cake, you do not reduce
the size of the cake you
sliced. Debt and equity
are just different slices
of firm cash flow and,
based on the preceding
logic, the value of the
firm (size of the cake)
is independent of the
leverage ratio (way in
which you slice the cake).
Chapter 6: The choice of corporate capital structure
the repurchase with the issue of new debt. Hence, the face value of debt
outstanding becomes B1 = B + E/2. Assuming that none of our investor’s
equity was repurchased, the payoff would be 2α(X – B1(1 + rd)) after the
repurchase. This is obviously different to that prior to the capital structure
change.
However, our investor can restore their original payoff profile using the
following strategy. Sell one-half of the equity stake and use the proceeds to
buy debt. The payoff from the new position is α(X – B1(1+rd)) + α(1 + rd)E/2
= α(X – B(1 + rd)). Hence our investor can, without cost, undo any change
the firm makes in its capital structure. This implies that investors will be
indifferent to such changes, and hence the valuation of a firm will not
depend on the specific debt–equity ratio it chooses (i.e. the MM irrelevance
proposition is valid).
Example
Consider an entrepreneur with a project which requires an initial investment of $100m
and which will have perpetual cash flows of $20m forever or $5m forever with equal
probability. Assume that all investors are risk neutral and require a 10 per cent expected
rate of return. We can show that the entrepreneur is indifferent between raising $100m
with debt, equity, or a mix of debt and equity.
• Debt: the entrepreneur must promise investors a coupon such that in expectations
they receive interest of 100*.1 = $10m every year. Since in the bad state of the world
investors will receive no more than $5m, it must be the case that .5*c + .5*5 = 10
and c = 15. The entrepreneur will receive the remainder: 0 in the bad state of the world
and 20 – 15 = 5 in the good state of the world. In expectation, the present value of
this is .5*5/.1 = $25m.
• Equity: the entrepreneur must promise investors a fraction  of future equity payouts.
In expectation, outside equity investors will receive *(.5*5 + .5*20) = 12.5 each
year. The present value of this is 12.5/.1 = 125. This must equal to the amount they
put in: 100 = 125and = 80 per cent. The entrepreneur receives the remainder of
the equity, (1 – )*12.5 = $2.5m every year. The present value of this is $25m.
• Mix: the entrepreneur raises $50m through debt. She must promise investors a coupon
such that in expectations they receive interest of 50*.1 = $5m every year. Since even in
the bad state of the world the firm can pay $5m, they promise them a coupon of $5m.
The total equity payout is the remainder: 0 in the bad state of the world and 20 – 5 =
$15m in the good state of the world; this is equal to .5*15 = $7.5m in expectation.
The entrepreneur promises equity investors a fraction  of future equity payouts. In
expectation outside equity investors will receive 7.5, per year, or 7.5/.1 = 75
in present value. This must equal to the $50m they have contributed, resulting in
 = 66.7 per cent. The entrepreneur is left with (1 – )*75 = $25m.
The entrepreneur is indifferent to the choice of capital structure because capital structure
does not affect total cash flows produced by the firm.
Modigliani–Miller and Black–Scholes
MM irrelevance tells us that, under the assumptions listed above, firm value
is independent of leverage. Another way to see this is to use the Black–
Scholes option pricing analysis presented in Chapter 4. As we remarked
above, the payoff profile for equity in a levered firm is precisely the same as
that of a call option with exercise price equal to the face value of the firm’s
debt. The payoff profile for debt can be replicated by a risk-free investment
paying B and simultaneously writing a put option struck at B.
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92 Corporate finance
Given put–call parity, the sum of the values of debt and equity (i.e. a
position consisting of a call option less a put option (both struck at B) plus
lending B/(1 + rd)) must be equal to the value of the firm’s assets. This
holds whatever the specific value taken by B, and hence, as the face value
of debt varies firm value is unchanged.4
The Black–Scholes analysis of the MM proposition also gives us a simple
way in which to value debt and equity claims on firms. Knowing the face
value of debt, the maturity of debt, the risk-free rate and the parameters of
the process governing the value of the firm’s assets, we can use the Black–
Scholes equation and put–call parity to gain the market values of debt and
equity. An example is given below.
4
This argument is also
robust to the issue
by the firm of more
complex securities (e.g.
warrants, convertible
debt and subordinated
debt).
Example
Assume a levered firm has current market value of assets equal to $100m. This firm has
issued zero coupon debt with face value $80m, which matures in five years. Assume that
the risk-free rate is 0.05 and that the volatility of the firm asset value process is 0.5.
Using the Black–Scholes option pricing analysis from Chapter 4 and the fact that equity
can be treated as a call option, you should be able to verify that the market value of
equity is:
E = $55.97m.
Obviously then, as the total firm value is $100m, the market value of debt is equal to:
D = $44.03m.
Do the calculations yourself, and make sure you get the correct answer.
Modigliani–Miller and corporate taxation
One of the assumptions underlying MM’s irrelevance proposition is that
there are no frictions in capital markets. One very pertinent and realistic
friction is taxation, however. Firms are taxed on their profits and investors
on their income from dividends, capital gains and interest income.
Incorporating taxation into our analysis will result in the irrelevance of
capital structure breaking down. The reason underlying this problem is
that dividend and interest payments are not treated symmetrically in the
calculation of a firm’s corporation tax bill, and similarly investors are
taxed differentially on their income from interest and from capital gains.
Hence, the choice of firm capital structure will affect the after-tax stream
of payments to all stakeholders and hence change the value of the firm.
To start, consider a world in which investors are not taxed at all on their
personal incomes. However, firm profits are taxed. Interest payments to
debt, however, are made prior to the calculation of the corporation tax
bill, whereas dividend payments must be paid out of after-tax income.
As suggested above, the fact that debt service payments are made out of
pre-tax cash flow and payments to equity out of post-tax cash flow will
cause the breakdown of the irrelevance proposition. Debt is now a more
favourable security to issue than equity.
To illustrate, consider an infinitely lived, levered firm. Assume that the
firm earns net cash flow Xt in period t, and that interest of rdB must be paid
every period. Finally, assume that the probability of defaulting on the debt
is always zero.5 In period t, the following funds are paid to investors in the
firm:
Ct = rd B + (1 – τc )(Xt – rdB) = (1 – τc ) Xt + τcrd B,
(6.1)
where τc is the corporation tax rate. The first term on the right-hand side
of equation 6.1 is precisely the payment made by an unlevered firm with
94
5
For this to hold we
must have Xt > rd B in
every period t.
Chapter 6: The choice of corporate capital structure
cash flow Xt in period t. The second term is the gain made by the levered
firm in saving on its corporation tax bill through using debt in the capital
structure. This is known as the tax shield advantage of debt finance.
As our firm is infinitely lived, its market value is calculated as the present
value of the perpetual stream of payments to investors. Discounting and
adding up the stream of payments represented by the first term on the
right-hand side of equation 6.1 gives us the value of an unlevered firm
(VU), with identical cash flows to our levered firm. Discounting the stream
of payments represented by the second term on the right-hand side of
equation 6.1 gives τcD, where D is the market value of debt. Hence the
value of the levered firm can be written as:
VL = VU + τcD.
(6.2)
The value of a firm increases linearly with the market value of its debt
and, as such, firms should aim to have as high a leverage as possible.
Note that, when the corporation tax rate is zero, the MM proposition is
satisfied once more. In the following section, we show how firm valuation
is affected by the introduction of personal taxes on investor income as well
as taxes on corporate profits.
Example
Consider the same entrepreneur as in the previous example but now living in a world
where corporate taxes are 15 per cent. We can show that the entrepreneur wishes to
raise as much money as possible through debt.
• Debt: the coupon payment offered to creditors is c = $15m, exactly as before. The
entrepreneur will receive the remainder, but must pay taxes on it. This is 0 in the bad
state of the world and (20 – 15)*(1 – .15) = 4.25 in the good state of the world. In
expectation the present value of this is .5*4.25/.1 = $21.25m.
• Equity: the entrepreneur must promise investors a fraction  of future equity
payouts. In expectation, outside equity investors will receive α*(.5*5 + .5*20)
(1 – .85) = 10.625 each year. The present value of this is 10.625α/.1=106.25α.
This must equal to the amount they put in: 100 = 106.25α and α= 94.12%. The
entrepreneur receives the remainder of the equity, (1 – α)*10.625 = $.625m every
year. The present value of this is $6.25m.
• Mix: the coupon payment offered to creditors is $5m, exactly as above. The total
equity payout is the remainder: 0 in the bad state of the world and (20 – 5)*
(1 – .15) = $12.75m in the good state of the world; this is equal to .5*12.75 =
$6.375m in expectation. The entrepreneur promises equity investors a fraction  of
future equity payouts. In expectation outside equity investors will receive 6.375, per
year, or 6.375α/.1 = 63.75α in present value. This must equal to the $50m they have
contributed, resulting in = 78.43%. The entrepreneur is left with (1 – α)*63.75 =
$13.75m.
The entrepreneur is best off raising money with 100 per cent debt, next best off with a
50/50 mix, and worst off raising money with 100 per cent equity.
As noted above, the addition of corporation tax to the MM analysis implies
that firms should choose leverage ratios as large as possible. However, this
is a clearly counterfactual implication. It has been suggested that relaxing
another of MM’s assumptions can reconcile the facts with our analysis. The
assumption that we relax is that bankruptcy is a cost-less process for firms
to undergo.6 MM assume that, if a firm’s cash flow is insufficient to cover
debt service (bankruptcy), then all funds are transferred immediately and
without cost to bond-holders. However, in reality bankruptcy involves
direct costs, such as lawyers’ fees, and indirect costs, such as debt-holder–
equity-holder conflicts in financially distressed firms.
6
For empirical
evidence on the costs
of bankruptcy in US
railroad firms, see
Warner (1977).
95
92 Corporate finance
Figure 6.3
As a result, we once more modify our analysis to allow for the effects of
bankruptcy costs. We assume that firms with higher levels of debt in their
capital structure incur greater costs of financial distress and that, at very
high debt levels, the effect of such costs may outweigh tax shield effects.7
You will find a diagrammatic analysis of this situation in Figure 6.3, which
plots firm value against leverage under three different scenarios. The first
is when corporation tax and bankruptcy costs are both zero. The second
scenario introduces non-zero corporation tax, and the third allows for
non-zero costs of bankruptcy.
Figure 6.3 makes the point quite well. When debt levels become too large,
the costs of financial distress outweigh tax shield gains and imply a finite
optimal leverage ratio. This is in contrast to the case when bankruptcy is
costless as firm value then increases without bound as leverage rises.
Example
Consider the same entrepreneur as in the previous examples who still faces a 15 per
cent corporate tax, but now also a drop of 40 per cent in all future income in case of
bankruptcy. We can show that the entrepreneur wishes to raise money through a mix of
debt and equity because using all equity results in losses of tax shields while too much
debt results in paying bankruptcy costs.
• Debt: the entrepreneur must promise investors a coupon such that in expectations
they receive interest of 100*.1 = $10m every year. In the bad state of the world
the firm is unable to fully pay its creditors and the firm will default. At this point, the
creditors will take over the firm, but 20 per cent is lost to bankruptcy costs so their
annual payout is 5*(1 – .4) = 3. It must be the case that .5*c + .5*3 = 10 and c
= 17. The entrepreneur will receive the remainder, after taxes. This is 0 in the bad
state of the world and (20 – 17)*(1 – .15) = 2.55 in the good state of the world. In
expectation the present value of this is .5*2.55/.1 = $12.75m.
• Equity: the firm cannot be bankrupt since it carries no debt, therefore the solution is
identical to the previous example. The entrepreneur receives $6.25m.
• Mix: note that in the previous example the coupon payment was just low enough
for the firm to not default (in the bad state of the world equity is left with zero but
creditors are fully paid, this is not default). Since no bankruptcy costs are paid, the
solution is identical to the previous example. The entrepreneur receives $13.75m
The entrepreneur is best off raising money by a mix of debt and equity so that they can
take advantage of the tax benefits of debt without having leverage so high as to suffer
bankruptcy costs.
96
7
High debt levels imply
large fixed nominal
payments every period
and hence expose
the firm to financial
distress if cash flows are
unexpectedly low.
Chapter 6: The choice of corporate capital structure
The idea that firm value is maximised by some intermediate leverage
which balances out the tax benefit of debt and the costs of financial
distress is called trade-off theory. However trade-off theory is out of
favour because empirically the costs of bankruptcy appear to be too low
to observe the low amounts of debt firms typically have in their capital
structure. The average leverage ratio for large US firms is 1/3. Estimates of
direct costs have been estimated as 7.5 per cent of market value for small
firms by Ang (1982) but only 2.9 per cent for firms listed on AMEX and
NYSE by Weiss (1990). Indirect costs are likely to be somewhat larger, but
are harder to estimate.
Modigliani–Miller with corporate and personal taxation
Before closing this chapter, we briefly examine how personal taxation
affects the MM analysis when introduced in conjunction with corporate
taxation. For the analysis in this section, we revert to the assumption that
bankruptcy costs are zero.
Consider a world with the following tax structure. Corporate profits are
taxed at τc. Personal income, including that obtained from corporate
interest payments, is taxed at rate τd. Finally, personal income from
holdings of equity is taxed at rate τe. Assume that firms are infinitely
lived, and consider a firm that pays rD B of its gross income at any point as
interest. As interest payments are tax-deductible, the amount of interest
that reaches the firm’s bond-holders’ bank accounts is:
rDB(1 – τd ).
(6.3)
In period t, the firm pays out an amount Xt – rD B to equity-holders. This
amount is taxed twice: first at the corporate level and second at the personal
level. Hence, the net amount that reaches equity-holders’ pockets is:
(Xt – rD B)(1 – τc )(1 – τe ).
(6.4)
Hence, in total, in period t, the firm pays out the following amount:
Ct = (Xt – rD B) (1 – τc)(1 – τe) + rDB(1 – τd ).
(6.5)
This expression can be rearranged to yield the following:
Ct = Xt(1 – τc)(1 – τe) + rD B[(1 – τd) – (1 – τe)(1 – τc )].
(6.6)
Note that the first term in equation 6.6 is precisely the cash-flow stream
accruing to equity-holders in an unlevered firm (with identical cash
flows to the levered firm). Hence, discounting this stream of funds at the
appropriate rate yields a present value of VU. The second term is the extra
money paid out, as the firm has debt in its capital structure. This should be
discounted at the after-tax rate of return on debt (i.e. (1 – τd)rD). The sum
of the present values of these two terms is clearly the value of the levered
firm. Hence we can write:
.
(6.7)
This generalises equation 6.2 to the personal (as well as corporate)
taxation case. Note that equation 6.2 can be retrieved as a special case
of equation 6.7, when both personal tax rates are set to zero. The second
term on the right-hand side of 6.7 is the taxation gain of debt. It is
increasing in the corporate tax rate and the tax rate on equity income and
decreasing in the tax rate on debt income. Note that, if (1 – τc)(1 – τe) >
(1 – τd), then the tax advantage is negative, such that the optimal capital
structure choice is to be all equity. If the preceding inequality is reversed,
though, the tax advantage is clearly positive and, as such, optimal capital
structure involves a firm issuing as much debt as possible.
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92 Corporate finance
The Miller equilibrium
Let us consider again the MM setting with corporate and personal taxes. The
Miller equilibrium is derived in such a setting when investors differ in their
tax rates on personal income. The Miller equilibrium is obtained by stating
that demand for debt must be equal to supply for debt in equilibrium.
Let us denote respectively the (expected) rates of return offered by debt and
equity, gross of personal taxes, but after adjusting for risk premiums, by rD
and rE. In this new setting, firms are willing to issue debt exclusively as long
as, after adjusting for risk premiums, the cost of debt after corporate taxes is
strictly lower than the cost of equity, that is, as long as:
rD (1 – τc) < rE.
(6.8)
Investors are willing to hold debt as long as, after adjusting for risk
premiums, the return after personal income taxes offered by debt is weakly
higher than the return after personal taxes on equity income offered by
equity, that is, as long as:
rD (1 – τd) ≥ rE (1 – τe).
(6.9)
In order to understand the Miller equilibrium, let us first assume that the
pre-tax return on debt, rD, offered by firms is equal to the pre-tax return
on equity, rE. In this case, firms are willing to issue debt which tax-exempt
investors are willing to buy as both inequalities (equations 6.8 and 6.9) are
satisfied. Firms have an incentive to increase leverage and will continue to
replace equity with debt, moving up the demand curve by increasing the
return rD they offer to attract investors with higher personal income tax rates,
until:
rD = [rE (1 – τe)]/(1 – τd) = rE /(1 – τc).
(6.10)
If the rate of return offered on debt is lower than rE /(1 – τc), firms have still
incentives to issue more debt as, at this point, it is still profitable to issue
debt to investors with marginally higher personal income tax rates. In
contrast, if the rate of return offered on debt is higher than rE/(1 – τc), firms
would be better off issuing equity than debt as it is cheaper.
In equilibrium, there is thus no advantage for firms to issue debt as the
equilibrium rate of return offered to debt-holders is such that firms are
indifferent between issuing debt and equity. In equation 6.7, the value of the
levered firm, VL, is equal to the value of the unlevered firm, VU, as:
(1 – τc) (1 – τe) = 1 – τd.
(6.11)
The after-tax Miller’s theory hence implies that there is an equilibrium
aggregate amount of debt outstanding in the economy which is determined
by relative corporate and personal tax rates. The amount of debt issued by
any particular firm is, however, a matter of indifference.
Summary
98
In this chapter we have presented a fundamental analysis of the capital
structure of a firm. Initially we show that, under the MM assumptions,
capital structure does not affect firm value. We then present relaxations of
the MM assumptions and demonstrate how the MM result is altered. With
the introduction of corporate taxation it becomes clear that firm value is
increasing with the level of debt in the capital structure. Also allowing for
costly bankruptcy, we find that an optimal, finite capital structure may result.
When personal taxes and corporate taxes are included, then the prescriptions
for optimal capital structure are unclear. The optimum depends on the
particular constellation of corporate and personal taxation rates.
Chapter 6: The choice of corporate capital structure
In the next chapter we will explore the same relationships but from the
perspective of returns rather than values. In the following chapter we will
examine how conflicts between debt and equity-holder interests will also
imply that the MM result is violated. The analysis presented focuses on
simple cases in which the choices of equity-holders (those who dictate the
firm’s investment policy) are not aligned with the interests of debt-holders.
A reminder of your learning outcomes
Having completed this chapter, and the Essential reading and activities,
you should be able to:
• outline the main features of risky debt and equity
• derive and discuss the Modigliani–Miller theorem
• draw the link between Modigliani–Miller and Black–Scholes
• analyse the impact of taxes on the Modigliani–Miller propositions.
Key terms
bankruptcy costs
Black–Scholes
capital structure
corporate taxes
leverage
Miller equilibrium
Modigliani–Miller irrelevance theorem
personal taxes
tax shield of debt
Sample examination questions
1. What assumptions underlie Modigliani and Miller’s proposition that
firm value should be independent of capital structure? (5%)
2. Using a simple two-period model of an unlevered firm and a levered
firm with B units of riskless debt outstanding, demonstrate the
MM proposition. In the same framework, show that an investor is
indifferent to the firm altering its capital structure. (10%)
3. Demonstrate the impact of corporate and personal taxation on the
relationship between firm value and capital structure using a simple
infinite horizon framework. What would be the optimal capital
structure for firms if the only form of taxation was corporate? (10%)
4. A start-up firm needs $100 million to launch its product. It has already
signed a contract to provide its services to one major customer, this
will result in $5 million in profits annually in perpetuity, starting this
year. There is a 50 per cent chance the firm will sign a contract with a
second customer with expected profits of $15 million in annual profits.
If this deal is not signed, the firm only has $5 million in profits.
The corporate tax is 15 per cent. In case of bankruptcy, 40 per cent of
firm value is lost. Everyone is risk neutral with a 10 per cent discount
rate.
a. Suppose the start-up funds the $100 million through equity. What
share of equity must be offered to outside investors? What is the
present value of the initial investors’ stake.
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92 Corporate finance
b. Suppose the start-up funds the $100 million through debt
(perpetuity). What coupon payment must be offered to creditors?
What is the present value of the initial investors’ stake.
c. Suppose the start-up funds half of the $100 million through debt
and the rest through equity. What coupon payment must be offered
to creditors? What share of equity must be offered to outside
investors? What is the present value of the initial investors’ stake.
What is the best way to finance this project? Comment on trade-off
theory.
d. Suppose there were no bankruptcy costs. What would be the
optimal choice of financing?
5. Firm A pays ¥15 million in the good state and ¥10 million in the bad
state. It is an all equity firm and you own 10 per cent of the equity.
Assume there are no taxes. The price per share is ¥10 with one million
shares outstanding.
a. What is your payout in the good state and in the bad state?
b. The other owners have decided to recapitalise the firm. They raise
¥6 million by selling riskless bonds with a face value ¥7 million.
They use this money to repurchase equity at the market price.
You did not sell any of your shares. How much equity did they
repurchase? What share of equity do you now on? What is your
payout in the good state and in the bad state?
c. Compare the expected return on your investment before and after
the transaction. Why did the expected return change?
d. You are risk averse and do not like the change to your return profile.
Describe what you can do to get your payoff to be just the same as
before the transaction. Comment on what the MM 1st proposition in
relation to this question.
100
Chapter 7: Leverage, WACC and the Modigliani-Miller 2nd proposition
Chapter 7: Leverage, WACC and the
Modigliani-Miller 2nd proposition
Aim of the chapter
The aim of this chapter is to derive relationships between the rate of
return on a firm’s equity, the rate of return on a firm’s debt, and the rate
of return on the firm’s total assets (WACC). We will derive the Modigliani
and Miller 2nd proposition to analyse these relationships in the presence
of corporate taxes.
Learning outcomes
By the end of this chapter, and having completed the Essential reading and
activities, you should be able to:
• write down the relationship between debt, equity, the unlevered return
on the firm, and the levered return on the firm
• understand what happens to equity returns, and the weighted average
cost of capital as leverage increases with and without taxes
• draw a link between Modigliani and Miller’s 1st and 2nd propositions
• find the equity beta of a firm by unlevering and relevering the equity
beta of a comparable firm with different capital structure.
Essential reading
Hillier, D., M. Grinblatt and S. Titman Financial Markets and Corporate Strategy.
(Boston, Mass.; London: McGraw-Hill, 2008) Chapters 13 (Corporate Taxes
and the Impact of Financing on Real Asset Valuation), 14 (How Taxes
Affect Financing Choices) and 15 (How Taxes Affect Dividends and Share
Repurchases).
Further reading
Brealey, R., S. Myers and F. Allen Principles of Corporate Finance. (Boston,
Mass.; London: McGraw-Hill, 2008) Chapters 18 (Does Debt Policy
Matter?) and 20 (Financing and Valuation).
Miles, J. and J. Ezzell ‘The weighed average cost of capital, perfect capital
markets and project life: a clarification’, Journal of Financial and
Quantitative Analysis (15) 1980, pp.719–30.
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.
Overview
In Chapter 1 we learned how to calculate the value of a project by computing
the present value of the project’s future cash flows. This was done by
discounting the cash flows by the appropriate discount rate. In Chapter 2 we
learned that this discount rate depends on the project’s risk. In this chapter
we will see how this discount rate changes as the capital structure of the firm
changes.
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92 Corporate finance
We will see that as the firm increases its leverage, its equity becomes more
risky. The required rate of return on equity therefore increases. However
the overall return on the firm’s assets (WACC) does not change if there
are no corporate taxes. This is analogous to results from the previous
chapter: Modigliani–Miller’s 1st proposition stated that the firm’s value
did not change with leverage when there were no corporate taxes. We will
see that because taxes result in a safe cash flow returned to the firm in
the form of a tax refund, in the presence of corporate taxes the expected
return on the firm’s assets decreases with leverage as the assets become
safer due to increased tax shields. This is also analogous to results from
the previous chapter: as the firm increases leverage, its value increases in
the presence of corporate taxes.
Weighted average cost of capital
Consider two all equity firms, A and B, whose values are VA and VB, and
whose betas are βA and βB. The CAPM implied expected returns on the
two firms are rA = rf + βA (rm – rf) and rB = rf + βB (rm – rf). If these two firms
merged, then the expected return on the merged firm would be:
rAB = (VA /(VA + VB))rA + (VB /(VA + VB))rB,
(7.1)
and the β of the merged firm would be:
βAB = (VA /(VA + VB))βA + (VB / (VA + VB))βB.
(7.2)
The merged return and βs were weighted averages of the individual
returns and βs, with the weights depending on the weight of each firm
within the conglomerate.
The intuition above carries over to debt and equity inside of the firm.
When the firm is financed by a mix of debt and equity, and when there are
no taxes, the average rate of return which a firm pays to raise money is a
weighted average between the cost of debt and the cost of equity. Without
taxes, the weighted average cost of capital (WACC) is given by:
WACC = (E/(B + E))re + (B/(E + B))rd,
(7.3)
where the value of debt and equity are given by B and E, while their
respective returns are rd and re. This is the rate of return which should
discount the total cash flow coming from the firm (that is, the cash flows
to debt and equity) in order to calculate the total value of the firm (that is,
the value of debt plus equity). More generally, the WACC will also account
for taxes because taxes make the cost of borrowing through debt cheaper.
Consider a firm with pre-tax annual cash flows Xt. Its value today is V0
and its value next year, after X1 has been paid out, is V1. If this firm has
outstanding debt with market value B0, then its equity is valued E0 = V0 –
B0. Suppose that the appropriate returns on debt and equity are rd and re
respectively.
Recall from the previous chapter that if this firm has perpetual outstanding
debt with face value B then rd B will be distributed to the creditors in the
form of a dividend, and the rest (Xt – rd B)(1 – C) will be distributed to
equity-holders after corporate taxes. Define the free cash flow (FCF) as the
after-tax cash flow available to be distributed by a similar but all equity
firm. In this case, the firm’s FCF each year is Xt(1 – C). Let us calculate the
discount rate r, which would make the discounted present value of the
FCF equal to V0, the combined value of the debt and equity.
By definition of a return,
V0 = [Xt(1 – τC) + V1]/(1 + r)
102
(7.4)
Chapter 7: Leverage, WACC and the Modigliani-Miller 2nd proposition
which can be rewritten as:
r = (Xt (1 – τC ) + V1 – V0 ) / V0.
(7.5)
We wish to solve for the r in equation 7.5 as a function of the return on
debt, return on equity, and the tax rate.
Note that the expected increase in value between years 0 and 1 is:
(X1 – rd B0 )(1 – τC ) + rd Bv + V1 – V0 = [X1 (1 – τC ) + V1 – V0 ] + τC rd B0
(7.6)
where, on the left-hand side of the equals sign, the first term is the payment
to equity-holders, the second term is the payment to creditors, and the third
term is the value of all assets remaining in the firm. The formulation on the
right of equation 7.6 merely rearranges terms of the left-hand side. Note
that this increase in expected value must be split between the return to
equity-holders and the return to debt-holders:
[X1 (1 – τC ) + V1 – V0]+ τC rd B0 = E0 rd + B0 re,
(7.7)
[X1 (1 – τC) + V1 – V0]/V0 = (E0 rd + (1 – τC ) B0 re ) /V0.
Finally, substitute equation 7.5 for the left-hand side, and note that
V0 = E0 + B0 to find the WACC:
WACC = r = (E0 /(E0+B0 ))re + (1 – τC )(B0 /(E0 + B0))rd
(7.8)
Thus, the WACC is the discount rate at which the FCF needs to be
discounted in order to calculate the firm’s value. The FCF is the cash flow
to a hypothetical all equity firm, while the WACC accounts for the firm’s
leverage.
When corporate taxes are zero, equation 7.8 collapses to 7.3, however in
the presence of taxes, WACC decreases as leverage increases. The intuition
is similar to the MM 1st proposition. For every extra dollar of debt in its
capital structure, the firm receives τC rd back as a tax refund. This tax refund
is a riskless payment, therefore the firm appears less risky and the average
rate of return it pays to raise money decreases. Because of the refund,
effectively, the firm is paying (1 – τC )rd instead of rd to raise money through
debt.
Example
The historic risk-free rate is 4 per cent and the historic market premium is 5 per cent.
Walmart has an equity β of 0.9, implying an expected equity return of re = 4 + 0.9*5 =
8.5% according to the CAPM.
Walmart has AA debt which matures in 2023 and has a yield of 5.9 per cent. Walmart’s
tax rate is 35 per cent so Walmart is paying (1 – τC )rd =(1 – .35)*5.9 = 3.835% to raise
money through debt.
Walmart’s outstanding debt has a value of $22.7 billion. Walmart has 4,269 million shares
outstanding with a price of $55.69 per share, implying an equity market capitalisation of
4.269*55.69 = $237.7 billion. Walmart’s weight of debt in the capital structure is 22.7/
(237.7 + 22.7) = 8.7% and its weight of equity is 237.7/(237.7 + 22.7) = 91.3%.
Walmart’s WACC is 0.087*3.835 + 0.913*8.5 = 8.09%.
Modigliani and Miller’s 2nd proposition
In the previous section we derived the relationship between the return
on the firm’s debt, the return on its equity, and the average cost of capital
for that firm. In this section we will make a distinction between the firm’s
unlevered (or asset return), which is the return this firm would pay to
raise capital if it was an all equity firm, and the firm’s actual cost of capital,
once we account for leverage, this is the WACC from the previous section.
We will also find a relationship between the firm’s equity return and its
unlevered return.
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92 Corporate finance
In the absence of taxes, the MM 2nd proposition states that:
(7.9)
re = ru + (B/E)(ru – rd ),
where B/E is the debt to equity ratio in the firm’s capital structure, re is
the return on the firm’s equity, rd is the return on the firm’s debt, and ru is
the unlevered return, or the return on a hypothetical firm that is financed
by all equity (or unlevered) but otherwise similar to the firm we are
considering.
As leverage increases, the expected return on equity grows because equity
becomes riskier. Equity is riskier because it is a residual payment, it is paid
last after all other claims (such as debt) have been settled. When leverage
is high, equity is only a small portion of the firm, but must take the brunt
of most of the firm’s losses. This makes the equity of a highly levered firm
very risky.
Notice that equation 7.9 is identical to equation 7.3 if we substitute WACC
for ru and rearrange terms. When there are no taxes (or other frictions), as
leverage increases, the equity return becomes riskier and its expectation
grows to compensate investors for that risk. However, the average return
that the firm pays to borrow does not change. This is because although
equity returns grow, equity is a smaller part of the firm and carries less
weight. Thus the firm is borrowing more through debt, which has a lower
rate of return. The weighted average does not change. In the absence
of corporate taxes, the average rate at which the firm raises money, the
WACC, is equal to the rate at which an all equity (or unlevered) firm raises
money, ru. The WACC is independent of capital structure, analogous to
the MM 1st proposition in the absence of taxes. The relationship between
equity, debt, WACC and leverage in the absence of taxes is illustrated
graphically in Figure 7.1.
25
Debt
WACC
Equity
20
E[R]
15
10
5
0
0
0.1
0.2
0.3
0.4
0.5
D/ V
0.6
0.7
0.8
0.9
1
Figure 7.1
We will now derive a more general version of the MM 2nd proposition, in
the presence of taxes. Consider a firm that lives for one period. It has both
debt and equity in its capital structure and its value is V0 = E0 + B0 today
and V1 = E1 + B1 next period. Also note that from the definition of return,
E1 = (1 + re)E0 and B1 = (1 + rd)B0 as there are no intermediate payments.
104
Chapter 7: Leverage, WACC and the Modigliani-Miller 2nd proposition
This firm will have a cash flow X1 which it will distribute to its debt and
equity holders in period 1. Also consider a similar firm that is all equity
owned. This unlevered firm has value V0U today; for this firm B = 0.
Since next period the cash flows will be distributed first to creditors, and
then to equity-holders (after taxes), we can write the value of the firm as
the value of the total distributions:
V1 = (X1 – B1)(1 – τC ) + B1 = X1 (1 – τC ) + τC B1 = V1U + τC B1,
(7.10)
where the first term is the payout to equity-holders and the second term is
the payout to creditors. The last equality uses the fact that the value of the
unlevered firm next period is just equal to its after-tax cash flows.
From the definitions of debt and equity we know that:
V1 = E1 + B1 = (1 + re )E0 + (1 + rd )B0.
(7.11)
Setting equations 7.10 and 7.11 equal to each other and substituting
V1U = (1 + ru)V0U and B1 = (1 + rd )B0 we get the following equation:
(1 + ru)V0U + τC (1 + rd )B0 = (1 + re )E0 + (1 + rd )B0.
(7.12)
Now, we can rearrange the terms of this to solve for the return on equity:
(7.13)
1 + re = (1 + ru)(V0U/E0 ) – (1 – τC )(1 + rd)(B0 /E0).
Finally, we can use the fact that V0U = V0 – CB0 = E0 + B0 – C B0 (this is just
the present value of equation 7.10) to rewrite this as:
(7.14)
re = ru + (1 – τC)(B0 / E0)(ru – rd).
Equation 7.14 is the MM 2nd proposition in the presence of corporate
taxes. When C = 0 this equation becomes identical to equation 7.9.
However when C > 0, the expected return on equity rises by less in
comparison to equation 7.9 as leverage (B/E) increases. This is because
even though extra leverage makes equity more risky for the same
arguments as before, tax shield reduce some of this risk. This can also be
seen by comparing the equity return in Figure 7.1 to that of Figure 7.2
which has the same returns in the presence of taxes.
25
Debt
WACC
Equity
20
E[R]
15
10
5
0
0
0.1
0.2
0.3
0.4
0.5
D/V
0.6
0.7
0.8
0.9
1
Figure 7.2
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92 Corporate finance
The MM 2nd proposition gives us a relationship between the unlevered
return on a firm, and the return on the debt and equity of a similar but
levered firm. The WACC is the average rate of return the firm pays to raise
money, it is defined as a function of the returns on debt and equity. We
can combine the MM 2nd proposition (equation 7.14) with the definition
of WACC (equation 7.8) to find the WACC as a function of the unlevered
return on the firm:
WACC = ru (1 – τC (B0 /V0)).
(7.15)
Activity
Combine equations 7.14 and 7.8 to derive equation 7.15.
We can split up the risk investors of a firm face into two types of risk. The
first is business risk, this depends on the risk of the firm’s underlying assets
and activities. All similar firms should have similar business risk regardless
of capital structure. The second is financial risk, this is additional risk
that the firm faces due to its choice of capital structure. The return on an
unlevered firm ru is based on the firm’s business risk, since this firm has no
leverage. WACC is the return on the levered firm, this combines business
and financial risk. From equation 7.15, it is evident that without taxes
financial risk is irrelevant. The WACC of any firm is equal to the return on
an unlevered firm, regardless of the amount of leverage. This is analogous
to the 1st proposition of MM: the value of any firm is equal to the value of
an unlevered firm, regardless of the amount of leverage. In the presence
of taxes, the WACC decreases as we add leverage because of additional
tax shields. With more leverage, the firm becomes safer, its borrowing rate
decreases (equation 7.15), and its value increases (equation 6.2). The MM
1st and 2nd propositions are flip sides of the same coin.
Example
Consider two firms with the same unlevered (asset) β of 0.5. The risk-free rate is 3 per
cent and the market premium is 6 per cent. The corporate tax rate is 35 per cent.
Firm A has no debt. Current pre-tax earnings are $23 million with no growth prospects.
Firm B has AAA-rated long-term debt with 4 per cent yield to maturity and market value
$50 million. Current pre-tax earnings are $8.75 million with no growth prospects.
What are the WACC, equity return, total firm value, and equity value for each firm?
The unlevered return is ru = 3 +.5*(6 – 3) = 4.5% for both firms.
The FCF of firm A is 23*(1 – .35) = $23.98 million. We use ru = 4.5% as the discount
rate and find an unlevered firm value of VU = 23.98/.045 = $332.2 million.
Since this firm is debt free, its equity value and its total value are the same as the
unlevered value. Again, because this firm is unlevered, its WACC and its equity return are
both equal to ru.
The FCF of firm B is 8.75*(1 – 0.35) = $5.69 million. We use ru = 4.5% as the discount
rate and find an unlevered firm value of VU = 5.69/ 0.045 = $126.4 million.
Using the MM 1st proposition, we can calculate the levered value as the unlevered value
plus the present value of tax shields where the present value of tax shields is given by cB:
V = 126.4 + 0.35*50 = $143.9 million. The equity value is the total firm value minus
the value of the debt: 143.9 – 50 = $93.9 million.
106
Chapter 7: Leverage, WACC and the Modigliani-Miller 2nd proposition
We can use the MM 2nd proposition (7.14) to calculate the return on equity:
re = ru + (1 – τC)(B0 / E0)(ru – rd) = 4.5 + (1 – 0.35)*(50/93.9)*(4.5 – 4) = 4.67%.
We can now calculate the WACC either through equation 7.8 or 7.15. Both give the same
answer. First using equation 7.8:
WACC = (E/(E + B))re + (1 – τC)(B/ (E + B))rd
WACC = (50/143.9)*(1 – 0.35)*4 + (93.9/143.9)*4.67 = 3.95%
Alternately using 7.15:
WACC = ru(1 – τC(B/V)) = 4.5*(1 – 0.35*(50/143.9)) = 3.95%.
A CAPM perspective
So far we have looked at the relationships between returns implied by the
MM 2nd proposition. We can instead look at the relationships between βs.
Recall that the CAPM implies that every security lies on the security market
line. We can write down CAPM equations for the unlevered, equity, and
bond returns.
ru = rf + βu (rm – rf )
(7.16)
re = rf + βe (rm – rf )
(7.17)
rd = rf + βd (rm – rf )
(7.18)
By plugging equation 7.16 into equation 7.14 (MM 2nd proposition) and
then rearranging terms, we can rewrite the return on equity as:
re = rf + [βu+ (1 – τC )(B/E)(βu – βd )](rm – rf ).
(7.19)
This itself is a CAPM equation, by comparing equation 7.19 to 7.17 we can
see that βe must equal to the term in brackets from equation 7.19:
βe = βu + (1 – τC )(B/E)(βu – βd ).
(7.20)
In the special case when the firm’s debt is riskless and therefore βd = 0, this
equation simplifies to:
βe = βu (1 + (1 – τC )(B/E)).
(7.21)
With equation 7.21 we can compare the β of an unlevered firm to the β of a
levered firm. We can also use the equation backwards to find the unlevered
β for a levered firm. Suppose you wish to find the expected equity return
for a firm with no past financial data. It is possible to find a comparable
publically trading firm with the same business risk (for example a firm in
the same industry), however this firm may have different financial risk
(different leverage).
Using historical market information we can find the β of the comparable
firm by running a regression, analogous to equation 2.32. The slope from
this regression is the equity β of the comparable firm. However, due to
different leverage, the β we are looking for may be different from this β.
Using equation 7.21 with the capital structure of the publically traded
firm, we can unlever this β and find the unlevered (asset) β, which is the
same for both firms. We can then again use equation 7.21, this time with
the leverage ratio of the firm whose β we wish to know, to get the desired
equity β.
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92 Corporate finance
Example
Firm A is looking to do an IPO with a debt to value ratio of 0.7. The average equity beta
of similar, publically traded firms is 0.85 and the average debt to value ratio is 0.22. The
tax rate is 35 per cent. What rate of return should we use to discount Firm A’s expected
equity cash flows?
Using equation 7.21 backwards with the capital structure of the comparables, we find
that the unlevered (asset) β of this industry is:
βu = βe/(1 + (1 – τC)(B/E)) = 0.85/(1 + (1 – 0.35)*.22/(1 – 0.22)) = 0.718
Now we can use equation 7.21 forwards, with the target leverage of firm A:
βe = βu(1 + (1 – τC)(B/E)) = 0.718*(1 + (1 – 0.35)* 0.7/(1 – 0.7)) = 1.81
With a 4 per cent historical risk-free rate and a 6 per cent historical market premium, the
required equity return is: 4 + 1.81*6 = 14.86%.
Summary
In this chapter we derived relationships between the return on a firm’s
equity, a firm’s debt and a firm’s total assets. We saw that if there are no
taxes, increasing leverage makes equity riskier and increases expected
returns. However, the return on the firm’s total assets does not change
because more weight is given to the safe debt return. However, in the
presence of taxes, the increase of expected equity returns with leverage
was smaller, due to a tax refund. The return on the firm’s total asset
actually declined with leverage in the presence of taxes, because tax
refunds make the firm safer. This is analogous to firm value rising with
leverage in the presence of taxes, as we saw in the previous chapter.
Key terms
business risk
financial risk
leverage
tax shields
weighted average cost of capital (WACC)
unlevered (asset) return
unlevered β
A reminder of your learning outcomes
Having completed this chapter, and the Essential reading and activities,
you should be able to:
• write down the relationship between debt, equity, the unlevered return
on the firm, and the levered return on the firm
• understand what happens to equity returns, and the weighted average
cost of capital as leverage increases with and without taxes
• draw a link between Modigliani and Miller’s 1st and 2nd propositions
• find the equity beta of a firm by unlevering and relevering the equity
beta of a comparable firm with different capital structure.
108
Chapter 7: Leverage, WACC and the Modigliani-Miller 2nd proposition
Sample examination questions
1. Consider an all equity firm with an equity β of 0.7. The risk-free rate
is 3 per cent and the market risk premium is 6 per cent. The company
is considering a recapitalisation to a debt-to-value ratio of 0.25; at this
ratio the before-tax cost of debt will be 5 per cent. For a tax rate of 35
per cent, what is the WACC at this new level of leverage?
2. Stagnant Inc. is a swimming pool supply company that is currently
unlevered with a P/E ratio of 12. The company has no growth
prospects. The tax rate is 35 per cent.
a. What is Stangant’s cost of capital?
b. Stagnant is considering adopting a new capital structure with 50
per cent debt. It has consulted with a bank which is willing to lend
at a 5 per cent rate. What will be the new return on equity, WACC
and P/E ratio?
3. The earnings for firm A and firm B are given below (year –5 indicates
5 years ago, year 0 indicates this year’s dividend, which has not been
paid out yet but is already known, year +1 indicates the forecast of
next year’s dividend). All numbers are in millions of dollars.
Year
–5
–4
–3
–2
–1
0
+1
+2
A
–11
0
1
2
21
22
23
23
B
5
13
7
4
15
13
3
10
Both firms pay out nearly 100 per cent of their after-tax cash flows
to the owner. A has no debt. B has AAA-rated long-term debt with 4
per cent yield to maturity and market value of 50 million. The asset
(unlevered) β for firms in the same industry as A and B is 0.5. The
corporate tax rate is 35 per cent, assume no personal taxes. The
historical risk-free rate is
3 per cent and the historical return on the stock market is 6 per cent.
a. For each firm calculate the WACC, the firm (enterprise) value, and
the equity value. Give justification for your calculation.
b. What changes to capital structure would you make for firm A? Firm
B?
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92 Corporate finance
Notes
110
Chapter 8: Asymmetric information, agency costs and capital structure
Chapter 8: Asymmetric information,
agency costs and capital structure
Aim of the chapter
The aim of this chapter is to analyse and explain the choices of corporate
capital structures made by firms’ managers through theories involving
agency costs or asymmetries of information.
Learning outcomes
By the end of this chapter, and having completed the Essential reading and
activities, you should be able to:
• understand the concept of agency costs and governance problems in
general
• discuss the intuition behind the agency costs of debt, equity and free
cash-flows
• calculate the agency cost of debt in stylised settings
• discuss the effects of asymmetric information on capital structure
• explain the intuition behind the pecking order theory of finance.
Essential reading
Hillier, D., M. Grinblatt and S. Titman Financial Markets and Corporate Strategy.
(Boston, Mass.; London: McGraw-Hill, 2008) Chapters 16 (Bankruptcy
Costs and Debt–Holder–Equity-Holder Conflicts), 17 (Capital Structure
and Corporate Strategy), 18 (How Managerial Incentives Affect Financial
Decisions) and 19 (The Information Conveyed by Financial Decisions).
Further reading
Brealey, R., S. Myers and F. Allen Principles of Corporate Finance. (Boston,
Mass.; London: McGraw-Hill, 2008) Chapters 13 (Agency Problems,
Management Compensation, and the Measurement of Performance) and 19
(How Much Should a Firm Borrow?).
Copeland, T. and J. Weston Financial Theory and Corporate Policy. (Reading,
Mass.; Wokingham: Addison-Wesley, 2005) Chapter 15.
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.
Masulis, R. ‘The impact of capital structure change on firm value: some
estimates’, Journal of Finance 38(1) 1983, pp.107–26.
Miller, M. ‘Debt and taxes’, Journal of Finance 32, 1977, pp.261–75.
Modigliani, F. and M. Miller ‘The cost of capital, corporate finance and the
theory of investment’, American Economic Review 48(3) 1958, pp.261–97.
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.
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92 Corporate finance
Ross, S. ‘The determination of financial structure: the incentive signalling
approach’, Bell Journal of Economics 8(1) 1977, pp.23–40.
Overview
In the previous chapter we introduced the capital irrelevance proposition
first put forward by Miller and Modigliani (1958). We also explored cases
in which the capital structure of a firm did matter in its valuation due to
relaxations of the MM assumptions (e.g. the introduction of corporation
tax and bankruptcy costs). In this chapter we will focus on two classes of
problem in which MM1 does not hold. In the first, firms are subject to agency
problems between outside stakeholders and insiders (managers). The second
class of problem allows the possibility that insiders to the firm are better
informed about its quality than the market or potential external investors.
Capital structure, governance problems and agency
costs
In most Western corporations, ownership and control are separate, in that
the owners of a firm (the firm’s security-holders) entrust the day-to-day
running of the firm to managers. In general, although owners may have an
idea of what the optimal strategy for the firm is, it is impossible to force
managers to follow this plan. Managers may then behave opportunistically,
taking inflated salaries, investing in pet projects and enjoying other
perquisites (perks). Hence, in such scenarios, managers can corporate
policy to maximise their own utility rather than setting the policy which
would maximise shareholder wealth. This is the agency problem that
arises in modern corporations and was first talked about in relation to
capital structure by Jensen and Meckling (1976).
Agency costs of outside equity and debt
Jensen and Meckling (1976) argue that understanding of two types of
agency cost is important in understanding why firm value is not invariant
to capital structure. The first of these is an agency cost associated with
outside equity.
Assume a firm that is financed solely by equity. A proportion of the
equity is held by the management of the firm, whereas the rest is held
by outsiders to the firm. Jensen and Meckling argue that such a situation
leads to firm values which are lower than that which would obtain if
the manager was the sole owner of the firm. To see why this is the case,
consider the rewards and costs facing the manager/equity-holder.
The manager is the agent who undertakes activities that add value to
the firm. Let’s call these activities ‘effort’. Increased effort supply leads to
greater firm value and vice versa. However, supplying effort is also costly to
the manager (it takes up their time and tires them mentally and physically,
for example). In situations where a proportion α of the firm’s equity is held
by outsiders, the manager bears the entire cost of effort supply but reaps
only a portion (1 – α) of the benefit. Hence, the outside equity-holders gain
from the manager increasing effort but don’t bear any costs. This induces
the manager to supply lower levels of effort for higher values of α (i.e. when
the proportion of profits the manager appropriates is low, their incentive is
to supply little amounts of effort). Hence, firm value is decreased when the
proportion of equity held by outsiders is increased, and MM1 does not hold.
This is the agency cost of outside equity.
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Chapter 8: Asymmetric information, agency costs and capital structure
Jensen and Meckling argue that the agency cost of outside equity is
decreasing in the leverage ratio of the firm (where leverage is the ratio of
debt to equity values). The argument runs as follows: assume that the firm
repurchases some of the equity held by outsiders, funding this with a debt
issue – hence, leverage is increased. Also, the proportion of outstanding
equity held by the manager is now increased. Thus, as his share of the
residual value of the firm is increased, the manager chooses to supply
more effort, leading to increased firm value. Then, as leverage rises,
agency costs of outside equity fall.
Example
In this example we will see that when issuing outside equity, a project’s owner is worse
off because she uses too little effort. On the other hand, when using debt, she uses
optimal effort.
Consider an entrepreneur with a project that next year pays $10 million with probability p
and $20 million with probability 1 – p. This project requires an initial investment of
$11 million.
The entrepreneur can pick the probability of success p to be any number they want
between 0.25 and 0.75. However, choosing a higher p requires effort e, which the
entrepreneur dislikes; e = k*p. In this case k = 4 is the disutility of raising probability of
success by 1 expressed in millions of dollars. In particular, if X is the monetary the utility
function is:
U = E[X] – k*e
The required discount rate is zero and everyone is risk neutral.
Suppose the entrepreneur finances the project with equity by promising a share  of
equity to outside investors in return for them paying the $11 million necessary for the
initial investment. Then the expected payoff is:
E[X] = (1 – )(20p + 10(1 – p)) = (1 – )(10p + 10),
and the utility is:
U = E[X] – e = (1 – )(10p + 10) – k*p = 10*(1 – ) + [10*(1 – ) – k]*p.
Therefore, the entrepreneur will choose p to be as small as possible if 10*(1 – ) – k
< 0. Suppose outside investors believe that the entrepreneur will choose p = 0.75, then
their expected payout is: (0.75*20 + 0.25*10) = 17.5.
This must equal to their initial investment of 11, implying = 62.9%. However, that
implies that 10*(1 – ) – k = 3.71 – k < 0 and the entrepreneur would choose
p = 0.25, therefore this cannot be an equilibrium.
Suppose outside investors believe our investor will choose p = 0.25, then their expected
payout is: (0.25*20 + 0.75*10) = 12.5
This must equal their initial investment of 11, implying  = 88%. Indeed 10*(1 – ) – k
= 1.2 – k < 0, thus the entrepreneur will choose p = 0.25, consistent with the beliefs of
outside equity-holders.
The entrepreneur’s utility is:
U = 10*(1 – ) + [10*(1 – ) – k]*p = 1.5 – k*p = 0.5.
Suppose instead the entrepreneur financed this investment with debt by promising a face
value F to creditors in return for $11 million to cover the initial investment. In this case
the entrepreneur’s equity will always be bankrupt in the bad state of the world and they
will receive zero; in this case creditors receive the full $10 million. In the good state of the
world, the entrepreneur will receive 20 – F. Their utility is:
U =E[X] – e = p(20 – F) – k*p = (20 – F – k]*p.
They will choose p to be as large as possible as long as 20 – F – k > 0.
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92 Corporate finance
Suppose creditors believe that p = 0.25. Then their expected payout is:
p*F + (1 – p)*10 = 0.25F + 7.5
This must equal their initial investment of 11, implying F = 14. However, this implies that
20 – F – k > 0 and the entrepreneur would choose p = 0.75, therefore this cannot be
an equilibrium.
Suppose creditors believe that p = 0.75. Then their expected payout is:
p*F + (1 – p)*10 = 0.75F + 2.5.
This must equal to their initial investment of 11, implying F = 11.33. Indeed, 20 – F – k
> 0 and the entrepreneur chooses p = 0.75, consistent with the beliefs of outside equityholders.
The entrepreneur’s utility is:
U = (20 – F – k)*p = 6.50 – k*p = 3.5.
Note that this is much higher than when the entrepreneur uses equity. In this example
the MM proposition did not hold because one type of security was better than another.
As we increased the proportion of debt used to finance the firm, the entrepreneur chose
to exert more effort and increased value. Increasing leverage reduced the agency cost of
outside equity because it aligned the payoff to the entrepreneur with their cost of effort.
With a fraction  of outside equity, for every dollar of value they took out of the firm due
to decreased effort, the entrepreneur lost only (1 – ) of wealth.
Activity
First, show that in the above example, if the entrepreneur could commit to using the
optimal amount of effort, then they could get maximum utility even when using equity.
Next, show that in the above example if the entrepreneur is less averse to effort, for
example k = 3, then two possible equilibria can arise in the equity financing case. Thus
market beliefs may play an important role.
The second agency cost highlighted by Jensen and Meckling is that
associated with debt finance. It is also known as the asset substitution
or risk-shifting problem associated with debt finance. To illustrate the
problem, consider the following example.
Example
Assume that a firm that is financed by both debt and equity. A manager runs the firm in
the interest of current equity-holders (i.e. the manager sets investment policy in order
to maximise the expected shareholder payoff). The manager is faced with the choice
between two investment projects, A and B. These projects are assumed to have zero cost
and are mutually exclusive. The cash flows to projects A and B are given in Table 8.1.
Cash flow A
State 1
State 2
State 3
40
50
60
Table 8.1
Clearly, both projects have positive expected NPV. Project A has the lowest risk and the
higher expected NPV (50), whereas project B is the riskier and its expected NPV is 45.1
We assume that debt-holders have a claim of 50 that must be repaid out of
the cash flow to the chosen project. Given this debt claim, which project
will the manager choose?
If we first analyse project A, it is obvious that, with a debt obligation of 50,
only in state 3 will equity-holders get any payoff, this payoff being 10. This
implies that the expected payoff from project A to shareholders is 10 ∞ 0.25
= 2.5. The expected payoff to debt-holders from A is equal to (0.25 ∞ 40) +
(0.5 ∞ 50) + (0.25 ∞ 50) = 47.5.
114
1
When we say that
project B is riskier, we
mean that it has higher
cash-flow variance than
project A.
Chapter 8: Asymmetric information, agency costs and capital structure
Moving on to the analysis of project B, again equity-holders only get some
cash in state 3 and their expected payoff is 0.25 ∞ 30 = 7.5. The payoff
to debt-holders from project B is (0.25 ∞ 20) + (0.5 ∞ 40) + (0.25 ∞ 50) =
37.5. Hence, from the equity-holders point of view, project B maximises
expected payoff and, as a result, this will be the project chosen by the
manager. Note that the choice of this project implies that debt-holders are
worse off and firm value lower than in the case where project A is chosen.
When the face value of debt is 50, the firm invests in the project with
the lower expected NPV and higher risk, as this project maximises the
expected return to equity. What would happen if the debt repayment
outstanding were 30 instead of 50? In this case the expected payoffs to
equity-holders are 20 from project A and 17.5 from project B. Therefore,
the manager will choose project A. This choice also implies that debtholders are happy as project A maximises their expected payoff (they get
30 rather than the 27.5 that they would expect to receive if project B were
chosen). Note that, when the face value of debt is lower, the manager
switches and chooses the low-risk, high-expected-return project. This, in
turn, implies that, when face value of debt is lower, firm value is higher.
Example
In this example we will see that when issuing debt, a project’s owner is worse off because
they choose to take on too much risk. On the other hand, when using outside equity, they
choose the optimal amount of risk.
Consider an entrepreneur with a choice of one of two projects. Project A pays $5
million or $15 million with equal probability. Project B pays 0 or $18 million with equal
probability.
Each project requires an initial investment of $3 million. The entrepreneur will have the
freedom to choose the project after they raise financing.
The required discount rate is zero and everyone is risk neutral. There are no taxes or
bankruptcy costs.
Note that the expected value of project A is 0.5*5 + 0.5*15 = 10 while the expected
value of project B is 0.5*0 + 0.5*18 = 9 so project A is better. Project A is also less
volatile; in this example investors are risk neutral but typically they would prefer less
volatile projects.
Consider debt financing. For any face value of debt F shareholders receive the residual
after creditors have been paid. From project A their expected payout is:
0.5*(5 – F) + 0.5*(15 – F) = 10 – F if F < 5
0.5*0 + 0.5*(15 – F) = 7.5 – 0.5F if 5 < F < 15.
From project B their expected payout is:
0.5*(18 – F) = 9 – 0.5F if F < 18.
Comparing these two equations we can see that project B is preferred by equity-holders
for any F > 2, this can also be seen graphically in Figure 8.1. Project B is preferred
because equity-holders have a limited downside but care very much about the upside.
On the other hand, creditors expected payout from project A is:
F if F < 5
0.5*5 + 0.5*F = 2.5 + 0.5F if 5 < F < 15.
From project B their expected payout is:
0.5*F if F < 18.
Comparing these two equations we can see that project A is preferred by creditors for any
F, this can also be seen graphically in Figure 8.1. Project A is preferred because creditors
have no upside, and care only about limiting losses in the downside.
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92 Corporate finance
Since the necessary initial investment is 3, the face value of debt will have to be at least
3. This leads equity-holders to choose project B. Knowing this, creditors will ask for a face
value of debt such that they receive their initial investment back in expectation:
3 = 0.5*F and F = 6.
With this F, the initial entrepreneur’s payout is:
0.5*(18 – 6) = $6 million.
Suppose the entrepreneur could credibly commit to choose project A. In that case
creditors would ask for a smaller face value of debt, F = 3, because even in the bad
scenario, project A will be more than enough to repay the initial investment. The payout
to equity-holders would be:
10 – F = $7 million.
The shareholders would be better off if they could ex-ante commit to invest in A because
A has higher NPV. However, as we saw earlier, with F = 3 they are ex-post better off
choosing B. Since the creditors have no reason to trust them, creditors will assume B will
be chosen and ask for F = 6.
Now consider using outside equity to finance this project. Outside equity-holders
are promised a fraction  of the project and the entrepreneur receives the rest. The
entrepreneur’s payoff from choosing A is:
(1 – )[0.5*5 + 0.5*15] = (1 – α)*10,
and from choosing B it is:
(1 – α)[0.5*0 + 0.5*18] = (1 – α)*9.
Clearly the entrepreneur always chooses A. Knowing this, outside equity-holder will ask
for  such that their expected payoff 10α is equal to their initial investment of 3. This
implies that α= 30% and the entrepreneur’s share is worth (1 – 0.3)*10 = 7. This is just
as good as the commitment case and better than the debt financing case.
In this example the MM proposition did not hold because one type of security was better
than another. Debt financing caused the entrepreneur to choose a very risky project (risk
shift) because their downside was limited. As a result, creditors asked for a very high
interest rate to protect their investment and the entrepreneur was worse off for this.
Equity financing did not face this problem because the entrepreneur was just receiving
a fixed share of total profits, therefore it was in their interest to maximise total profits
both ex-ante and ex-post. Commitment was a possible substitute to equity, but it may be
difficult to implement in a real world situation.
Figure 8.1
Jensen and Meckling argue that the agency costs of debt are increasing in
the level of debt outstanding and hence in corporate leverage. Combining
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Chapter 8: Asymmetric information, agency costs and capital structure
the two agency costs then allows us to argue that an optimal (in the sense
of maximising firm value) capital structure might exist. We contended
that the agency cost of outside equity was decreasing in leverage, whereas
the agency cost of debt increased with leverage. Firm value would be
maximised where total agency costs are minimised, and this leads to the
optimal leverage ratio shown on Figure 8.2.
Figure 8.2
The Myers (1977) debt overhang problem
Another agency cost of debt was pointed out by Myers (1977). Rather than
arguing that debt obligations induce managers to invest in excessively
risky projects, Myers argues that the management of firms with large
levels of debt outstanding will choose to reject some positive NPV projects.
As a result, heavily indebted firms will see reductions in corporate value,
and MM1 is violated. This is known as the debt overhang problem.
To illustrate the previous argument consider the situation depicted in Table
8.2. A given firm is presented with the opportunity to invest in a certain
project at the current time. The payoff of this investment is $20,000 at time
t + 1 regardless of the state of nature, and the cost at time t is $10,000. We
assume, for simplicity, that interest rates are zero such that the investment
has a positive NPV. Further, the firm receives cash flow at time t, which
reflects its past investments. This cash flow is uncertain. As depicted in
Table 8.2, with probability 0.25 it will be $50,000; it will be $80,000 with
probability 0.5 and, finally, with probability 0.25 it will be $120,000.
The firm is run by a manager who acts in the interest of current
shareholders. In the past, the firm issued debt with a face value of
$100,000. This debt must be repaid out of the cash flow to the firm, after
the investment decision has been made and any payoffs realised. Note that,
if the project is accepted by the manager, its cost must be met out of the
pockets of equity-holders.
Probabilities
State 1
State 2
State 3
0.25
0.5
0.25
Cash flow existing assets
50
80
120
Cost new project
10
10
10
Return new project
20
20
20
Table 8.2
When the face value of debt is $100,000, the manager will reject the new
project. Why is this? Note that, in states 1 and 2, the new project pays
$20,000, but this simply goes straight into the pockets of debt-holders
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92 Corporate finance
through the required payment of $100,000. It is only in state 3 that the
$20,000 payoff of the new project accrues to equity-holders. Hence, in this
case the expected net payoff to equity-holders is:
(0.25 ∞ 20) – 10 = –5.
As this is negative, the manager rejects the new project. The implication
of this is that, when debt levels are high, a firm may reject a project with
positive NPV, as little of that project’s payoff accrues to equity-holders.
To confirm this, consider the case in which the required debt payment is
$80,000 rather than $100,000. In this case, the payoff from existing assets
is sufficient to service the debt in both states 2 and 3. Hence, in both these
states the equity-holders reap all of the rewards from the new project,
whereas the new project payoff goes to debt-holders in state 1. Hence, the
expected net return to equity-holders from the new project is:
(0.5 ∞ 20) + (0.25 ∞ 20) – 10 = 5.
As this is positive, the manager will accept the project as it increases
expected shareholder wealth.
Activity
Compute the expected payoff to equity-holders if the required debt repayment is 90. Will
the manager accept or reject the project?
The preceding example illustrates the debt overhang argument. Managers
that run heavily indebted corporations in the interest of equity-holders
may reject positive NPV projects as the cash flows from such projects
accrue mostly to debt-holders, whereas equity-holders bear the costs. The
rejection of such projects implies that firm values are suboptimal.
Agency costs of free cash flows
Although debt may generate agency costs, as discussed in the previous
section, Jensen (1986) argues that debt may also alleviate agency costs
of free cash flows. In this framework, debt is valuable as it motivates
managers to disgorge cash (in the form of interest and principal payments)
rather than investing it at below the cost of capital or wasting it on
organisation inefficiencies.
Jensen argues that growth is associated with increases in managers’
compensation and power. Managers have thus incentives to grow their
firms beyond their optimal size; that is, to engage in ‘empire-building’.
Managers of firms with substantial free cash flow, that is, cash flows
in excess of that required to fund all projects with non-negative NPVs,
are thus tempted to invest it at below the cost of capital or waste it on
organisation inefficiencies rather than return the cash to shareholders
through the payment of dividends or repurchase of shares. The agency cost
of free cash flows is the negative NPV of the investments made at below
the cost of capital. In this context, debt creation, without the retention
of the proceeds of the issue, enables managers to bond their promise to
pay out future cash flows in the form of interest and principal payments.
Although increases in dividends can be reversed, an issue of debt used to
repurchase equity is a credible bond as debt-holders are given the right
to take the firm into bankruptcy court if managers do not respect their
promise to make interest and principal payments. Debt thus reduces the
agency costs of free cash flow by decreasing the cash flow available for
spending at the discretion of managers.
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Chapter 8: Asymmetric information, agency costs and capital structure
Firm value and asymmetric information
The preceding sections emphasised the point that agency problems may
lead to departures from MM1. An alternative reason for such departures is
the presence of information asymmetries between corporate insiders and
outsiders. The role played by asymmetric information is emphasised by
Ross (1977) and Myers and Majluf (1984).
Ross (1977) signalling argument for debt
The crux of Ross’ argument is as follows. Assume firms differ according
to their future cash-flow prospects. High-quality firms expect large future
cash flows, whereas low-quality firms expect cash flows to be small. Firm
quality is not observable to outsiders to the firm. The managers of highquality firms have an incentive to attempt to signal their quality to the
market, as in the absence of signals the market can’t distinguish high- and
low-quality firms and will value them identically.
One way the management can signal is through debt policy. High-quality
firms choose large leverage ratios and lower quality firms choose low
leverage ratios. The market can observe leverage and hence values firms
accordingly (assigning firm values increasing in leverage.) Leverage is a
credible signal, as it is assumed that firm managers are averse (in terms
of their own utility) to bankruptcy. High levels of debt imply a higher
probability of bankruptcy, and only managers in charge of high-quality
firms are willing to expose themselves to this probability.
The preceding intuition can be formalised with the following model,
which is a simplified version of that contained in Ross (1977). Assume a
population of firms, each of which has future cash flow that is uniformly
distributed.2 Firm quality varies, as the upper bound of the cash flow
distribution (call this parameter t) varies across firms (i.e. a high-quality
firm may have cash flow distributed on [0, t1] and a low-quality firm might
have cash flow distributed on [0, t2] where t1 exceeds t2). Managers of firms
know the value of t for their own firms, but the market as a whole does
not.
Managerial utility is increasing in date 0 firm value and date 1 firm value,
but is decreasing in the expected cost of bankruptcy. In line with the prior
argument, managers will try to use debt to signal their quality. However,
non-zero debt levels imply that bankruptcy is possible. Hence, we can
write the managerial optimisation problem as follows:
.
2
If cash flow is
uniformly distributed
on [a, b] it means that
the probability density
of cash flow is constant
from a to b and zero
elsewhere. This implies
that the probability
distribution function of
cash flow is F(x)=(x–a)/
(b–a) for x between a
and b.
(8.1)
where we have assumed firm quality of t, V0(B) is date 0 firm value, L is a
parameter reflecting the cost (in managerial utility terms) of bankruptcy
and γ is a weight parameter. Given that the manager knows the true
t
distribution of firm cash flow, his assessment of date 1 firm value is 2.
Similarly, if a debt level of B is chosen, the manager knows the firm will be
bankrupt with probability
hence
.
and the expected utility cost of bankruptcy is
Assume that the market assigns a firm with debt level B a date 0 value of
f(B). Substituting this into equation 8.1 gives:
(8.2)
To compute the optimal level of debt, we compute the first order condition
of 8.2 with respect to B. After rearrangement this yields:
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92 Corporate finance
.
(8.3)
Finally, we assume that in equilibrium, the market’s beliefs about firm
quality (based on a firm’s debt level) are correct. Hence, we have the
t
condition f (B(t)) = 2 where we have also acknowledged the dependence
of the debt level, B, on firm quality through managerial actions.
Differentiating this condition yields:
f’(B)B’(t) = ½.
(8.4)
Substituting f’(B) from 8.4 into 8.3 yields the following differential
equation:
.
(8.5)
This differential equation has the following general solution:
(8.6)
where c is a constant term. The constant c can be assigned a value through
noting that the lowest quality firm in the population has no incentive
to signal and will hence elect not to have any debt. Denoting the lowest
quality by tc, use of this intuition in 8.6 gives:
.
(8.7)
Substitution of 8.7 in 8.6 gives the final debt rule:
.
(8.8)
Equation 8.8 gives us the key results from the Ross (1977) model. Debt
level (B) is increasing in firm quality (t). Clearly then, firms with higher
debt levels will have greater date 0 market values and MM1 is violated
once more.
In more loose terms, the arguments in Ross (1977) are that debt is a
costly signal (due to the possibility of bankruptcy it entails), and hence its
use implies higher-quality firms. From an empirical standpoint, evidence
that supports this notion can be found in Masulis (1983). This paper
demonstrates that firms which swap debt for equity (hence increasing
leverage) experience positive stock price returns whereas firms swapping
equity for debt experience negative stock returns. The stock price reactions
are interpreted as implying that leverage-increasing transactions are good
news whereas leverage-decreasing transactions are bad news, consistent
with the asymmetric information story.
The Myers–Majluf (1984) pecking order theory of finance
Another study that generates departures from MM1 through information
asymmetries is Myers and Majluf (1984). Although Ross focuses on the
level of the debt–equity ratio as a signal of firm quality, Myers–Majluf
concentrate on the information revealed by security issues. The intuition
behind their arguments is as follows.
We start by assuming a population of firms differing in both the quality
(value) of their assets in place and the quality (NPV) of their investment
projects. Any investment project has to be financed through an issue of
equity. Assume also that the managers of any firm are better informed
about both the quality of their firm’s assets in place and the quality of their
firm’s investment project than are outsiders. Furthermore, assume that
managers act in the interests of their firm’s existing equity-holders.
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Chapter 8: Asymmetric information, agency costs and capital structure
Only managers know whether the equity of their firm is over- or underpriced though, and this creates an opportunity for them to exploit the
market in order for existing shareholders to profit. The existence of
information asymmetries thus implies that the market can misprice
corporate equity: some firms’ equity may be over priced and others will be
under priced.
In this setting, managers may raise equity for two reasons.
• They may wish to invest in a positive NPV investment, which would
result in an increase in the value of the firm’s equity.
• Alternatively, they may wish to issue overpriced equity, which would
result in a transfer of wealth from the new to the old equity-holders.
Given rational expectations, the financial market correctly recognises
both incentives to raise equity. In equilibrium, managers of low-quality
firms (i.e. managers of firms with assets in place whose true worth is low
enough – and are hence overvalued), raise equity in order to take projects
with a small but negative NPV. The benefit to the existing equity-holders
that results from issuing overvalued equity exceeds the cost resulting
from taking the negative NPV project. Similarly, managers of high-quality
firms (i.e. managers of firms with assets in place whose true worth is high
enough – and are hence undervalued), abstain from raising equity and
hence from taking projects with a small but positive NPV. The dilution to
the existing equity-holders that results from issuing undervalued equity
exceeds the benefit resulting from the positive NPV generated by taking
the project. The presence of information asymmetries between managers
and equity-holders hence leads to distortions in investments.
Issue decisions affect prices as they reveal information on firm quality.
Managers are more likely to issue equity when their firm’s assets in place
are overvalued, as opposed to undervalued. On average, equity issues thus
lead to stock price drops. Furthermore, the highest quality firms avoid
issues at all costs.
Generalising the above somewhat, we can fit riskless debt, risky debt and
other securities into our pecking order. Obviously, issuing riskless debt to
finance investments conveys no information to the market, as there is no
possibility of exploitation (as there is no risk). Thus, stock prices should
not react to riskless debt issues and the highest quality firms will issue riskless debt in order to finance any investments. Low-quality firms don’t issue
riskless debt, as they cannot exploit new investors through its issue. Risky
debt comes with a possibility of default and hence could be overpriced if
the market underestimates the probability of default. Issues of risky debt,
therefore, convey some information, but clearly less than issues of equity.
Putting this all together leads to a model in which equity issues cause
stock prices to drop a lot (as the market infers that firms that issue are
very poor quality), risky debt issues cause small price decreases (as fairly
low-quality firms issue risky debt) and riskless debt issues cause no price
impact (as only high-quality firms issue riskless debt). Hence, in a dynamic
sense, Myers–Majluf implies that capital structure decisions do affect firm
values. This is the pecking order theory of finance.
There is a fair amount of empirical evidence that supports the pecking
order theory. First, the event study results on exchange offers detailed
above are consistent with the pecking order theory. Second, event study
evidence on new security issues confirms the theory too. Common stock
issues lead to price impacts of around –3 per cent, for example, whereas
risky debt issues cause small price drops, which are not statistically
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92 Corporate finance
different from zero. Hence, the intuition that underlies the model is
regarded by many as very plausible.
Example
Project Universe Industries (PUI), an all equity firm, currently has 20 million shares
outstanding. The value of the company is the sum of the value of the assets in place and
the NPV of the project. As shown in the following table, both the value of the assets in
place and the NPV from the project crucially depend on the price of oil:
Valuation
Assets
State A (cheap oil)
State B (expensive oil)
Assets in place
£130m
£220m
NPV of the project’s cash flows
£10m
£40m
The positive NPV project requires an initial investment of K = £600m irrespective of
the state of nature. In order to fund its project, PUI must raise £600m in equity. Assume
that managers maximise the wealth of the existing shareholders and that the states are
equally likely.
a. If managers must issue equity prior to knowing the price of oil, how many shares
should the firm issue and at which price will they sell for?
In each state, the post-issue firm value will be equal to the sum of the value of the
assets in place, the NPV of the project, and the capital (K = $600m) contributed
by the new equity-holders. In state A, the post-issue firm value is thus £740m. In
state B, the post-issue firm value is thus £860m. As both states are equally likely,
the expected post-issue firm value is thus £800m (derived as 50%*£740m +
50%*£860m). The fraction of the value of the firm that the new shareholder should
be getting is hence £600m/£800 = 75%. The value of the firm’s equity prior to the
share issue is thus £600m, and the share price is thus £200m/20m = £10. As
ex-post, all the shares have an equal claim, the firm must thus issue 60 million new
shares (derived as £600m/£10).
b. If the manager knew the state of the world before investing, in which state (A or B)
would the manager raise equity and invest in the project? In order to answer this
question, let us assume that the capital can be raised under the terms found in part
a) of this example and that the market does not know the state of the world.
Let us derive the ex-post payoffs to the existing shareholders in each state of nature
when the manager raises equity and invests in the project and when the manager
abstains from raising any equity and does not invest in the project. These payoffs
can be found in the following table:
Payoff to existing shareholders
Do nothing
Issue equity invested in
the project
State A (cheap oil)
State B (expensive oil)
£130m
£220m
(1 – 75%) * £740m
(1 – 75%) * £860m
The manager, when informed about the realisation of the state of nature, will issue
equity and invest in the positive NPV project in state A as (1 − 75%)*£740m =
£185m is strictly higher than £130m and refrain from issuing equity and forego
the positive NPV project in state B as (1 − 75%)*£860m = £215m is strictly lower
than £220m.
The manager of the firm hence abstains from issuing any equity and does not invest
in the strictly positive NPV project in the favourable state of nature. The intuition
behind this result is as follows. Although taking this project would increase the value
of the firm overall as it has a strictly positive NPV, it also leads to a reduction in the
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Chapter 8: Asymmetric information, agency costs and capital structure
wealth of the existing shareholders. The reason for this is that, in the favourable
state of nature, the financial market undervalues both the NPV of the project and
the intrinsic value of the firm’s existing assets. The effect of the dilution of the
existing shareholders, resulting from issuing undervalued shares, turns out to be so
high that the existing shareholders are better off without the project whenever the
project has to be financed through outside equity.
c. Now let us assume that the market knows that managers will make a decision after
observing the state of the world. When managers announce that they will not issue
equity to fund the project, the stock price of the firm may change. How would you
expect it to change? In order to answer this question, let us assume that the firm
does not have any other source of capital to take the project and that the market
does not know the state of the world.
Upon the announcement that equity will not be issued and the investment project
will not be taken, the market updates its estimate of the value of the firm, infers
that state B is obtaining, and hence prices the firm’s stock at £11 per share
(£220m/20m), hence rises by 10 per cent.
Summary
In this chapter we have examined theoretical models (and examples),
which imply that firm value does depend on the financing choices it
makes and on capital structure choices in particular. First, we examined
arguments based on agency costs and then looked at a model of
asymmetric information. The empirical evidence for these models is
mixed. Evidence for agency problems can be found in the specification
of corporate debt contracts, which contain clauses aimed specifically at
preventing debt overhang and asset substitution problems. The previously
discussed evidence on exchange offers is supportive of asymmetric
information models (although it would contradict the implications of a
debt overhang model). Research in these areas still proceeds. The most
recent strand of literature on capital structure builds on the agency cost
approach and examines incomplete contracts as the source of violations of
MM1.
Key terms
agency costs of debt
agency costs of free cash flows
agency costs of outside equity
asset substitution problem
asymmetric information
capital structure
debt-overhang problem
event study
governance problems
overinvestment
pecking order theory
risk-shifting problem
separation of ownership and control
signalling
underinvestment
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92 Corporate finance
A reminder of your learning outcomes
Having completed this chapter, and the Essential reading and activities,
you should be able to:
• understand the concept of agency costs and governance problems in
general
• discuss the intuition behind the agency costs of debt, equity and free
cash-flows
• calculate the agency cost of debt in stylised settings
• discuss the effects of asymmetric information on capital structure
• explain the intuition behind the pecking order theory of finance.
Sample examination questions
1. Explain the debt-overhang problem. (5%)
2. What are the agency costs of equity? Explain. (5%)
3. A firm has £100m in cash on hand and a debt obligation of £100m
due next period. With this cash, it can take one of two projects (A
and B) which cost £100m each. Assume that the firm cannot raise
any additional funds. If the economy is favourable, project A will pay
£120m and project B will pay £101m. If the economy is unfavourable,
project A will pay £60m and project B will pay £101m. Assume
that investors are risk-neutral, there are no taxes or direct costs of
bankruptcy, the risk-free rate of interest is nil, and the probability of
each state of nature obtaining is equal.
a. What is the NPV of each project?
b. Which project will equity-holders want the firm’s manager to take?
c. Show that debt-holders would find it incentive-compatible to cut the
face value of their claim to £82m. (10%)
4. What are the consequences of asymmetries of information between
managers and investors, as in Myers and Majluf, for investments and
the funding of investments? (15%)
5. Consider an entrepreneur who has a project that will cost $20 million
to implement and will produce cash flows of either $3 million or $5
million per year in perpetuity with equal probability. The entrepreneur
does not have the $20 million and must raise it externally. Assume risk
neutrality and a 10 per cent opportunity cost of capital.
a. Calculate the annual cash flow to the entrepreneur and its present
value if they raise the $20 million through perpetual debt.
b. Calculate the annual cash flow to the entrepreneur and its present
value if they raise the initial investment with equity.
c. As CEO of the firm the entrepreneur is able to spend $200,000
per year on a marketing relationship with their favourite celebrity.
This advertising relationship is worth only $150,000 annually
for a net loss of $50,000. However, the CEO receives utility from
the relationship, in particular, they would be willing to spend up
to $30,000 of their own money purely to spend time with this
celebrity. Show that if the entrepreneur uses equity to raise money,
they will engage in the wasteful advertising relationship but if they
use debt, they will not.
124
Chapter 8: Asymmetric information, agency costs and capital structure
d. Suppose the outside investors are aware of the CEO’s penchant for
spending time with celebrities. What share of equity would they
demand? What would be the present value of the entrepreneur’s
total payoff?
6. A firm’s productive assets will be worth either $100 million in a good
state or $10 million in a bad state with equal probability. Additionally,
the firm has $15 million in cash, which it could pay out as a dividend,
and outstanding debt with a face value of $35 million due next year.
The firm also has a project which would require an investment of $15
million this year and produce $22 million with certainty regardless of
the state of the world. Assume risk neutrality and a 10% cost of capital.
a. Do stockholders choose to take this positive NPV project? What is
the present value of the creditors payoff?
b. Suppose creditors suggest to financially restructure by reducing the
face value of debt to 24 if the shareholders promise to use the $15
million to invest. Will the shareholders agree? Will the creditors
prefer to do this?
125
92 Corporate finance
Notes
126
Chapter 9: Dividend policy
Chapter 9: Dividend policy
Aim of the chapter
The aim of this chapter is to analyse and explain the choices of dividend
policies made by firms’ managers. With this aim in mind, we first introduce
a stylised model in which dividend policy is irrelevant (Modigliani–Miller).
We then relax some of the assumptions made in this stylised model in
order to explain empirical evidence on firms’ dividend policies.
Learning outcomes
By the end of this chapter, and having completed the Essential reading and
activities, you should be able to:
• show that dividend policy (and share repurchases) are irrelevant to
firm valuation under the Modigliani–Miller assumptions
• discuss the stylised facts of dividend policy as provided by Lintner
• present the clientele model of dividends
• discuss the effects of asymmetric information and agency costs on
dividend behaviour.
Essential reading
Hillier, D., M. Grinblatt and S. Titman Financial Markets and Corporate Strategy.
(Boston, Mass.; London: Macmillan, 2008) Chapters 15 (How Taxes Affect
Dividends and Share Repurchases) and 19 (The Information Conveyed by
Financial Decisions).
Further reading
Allen, F. and R. Michaely ‘Dividend Policy’ in Jarrow R.A., V. Maksimovic and
W.T. Ziemba (eds) Handbooks in Operational Research and Management
Science: Volume 9: Finance. (Amsterdam: North Holland, 1995).
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.
Brealey, R., S. Myers and F. Allen Principles of Corporate Finance. (Boston,
Mass.; London: McGraw-Hill, 2008) Chapter 17 (Payout Policy).
Copeland, T. and J. Weston Financial theory and corporate policy.
(Reading, Mass; Wokingham: Addison-Wesley, 2005) Chapter 16.
Healy, P. and K. Palepu ‘Earnings information conveyed by dividend initiations
and omissions’, Journal of Financial Economics 21(2) 1988, pp.149–76.
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.
Lintner, J. ‘Distribution of incomes of corporations among dividends, retained
earnings and taxes’, American Economic Review 46(2) 1956, pp.97–113.
Myers, S. ‘Determinants of corporate borrowing’, Journal of Financial Economics
5(2) 1977, pp.147–75.
Ross, S. ‘The determination of financial structure: the incentive signalling
approach’, Bell Journal of Economics 8(1) 1977, pp.23–40.
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92 Corporate finance
Overview
The dividend is a cash payment (usually made on an annual or semiannual basis) to the owners of corporate equity and is the basic financial
inducement for individuals to hold shares. In Chapter 1, when analysing
discounted cash-flow techniques, we demonstrated how to price an
equity share, given knowledge of the future dividend stream that would
accrue to the share. Such an analysis might be undertaken by an investor
in order to assess the ‘value’ of an equity share. The current chapter
analyses dividends from the opposite perspective, that of the manager
of a corporation who must decide on the level of dividends to pay out.
In a similar vein to the analysis of capital structure in Chapters 6 and 7,
the fundamental question we wish to answer is: what dividend policy is
optimal for management in that its adoption results in maximum
firm value?
Modigliani–Miller meets dividends
In Chapter 6 we argued that, under a given set of assumptions, firm value
is independent of capital structure (i.e. the MM theorem was valid). These
assumptions include the following:
• frictionless markets (no taxes or transaction costs)
• symmetric information
• no agency costs
• investment outcomes independent of financing decisions.
The assumptions that give us MM1 actually yield a far more powerful
result than just the irrelevancy of debt policy. They imply that the entire
financial policy followed by a firm is irrelevant for its valuation; all that
matters is the firm’s portfolio of investment projects. Hence, capital
structure, dividend policy and risk management activities (among other
things) are all ineffectual in altering firm value. We have restated the
theorem and application of its logic to dividend policy, below.
Consider a firm that has fixed its investment policy. In each period, it is left
with a net cash flow, which is simply the difference between operating
income and investment costs. A straightforward corporate dividend
policy would just be to pay out this net cash flow to the holders of equity.
However, consider a firm that desires to pay a dividend in excess of its
net cash flow. In order to do this, the firm can raise funds by issuing
new equity. Alternatively, the firm could borrow money which, assuming
perfect capital markets, is a transaction with NPV of zero. Conversely, a
firm wishing to pay a smaller dividend might spend the balance of its net
cash flow on repurchasing equity. The key idea here is that a firm can
choose whatever payout policy it desires, funding the policy through share
issues/repurchases; hence, dividend policy is irrelevant.
From the individual investor’s point of view we can show that dividend
policy is irrelevant too. To do this we can use a similar argument to that
employed in our argument that shareholders are indifferent to capital
structure changes; shareholders are indifferent to dividend policy as,
through appropriate purchases or sales of shares, they can replicate any
dividend policy they wish. Hence, investors will not value a firm paying a
particular dividend policy different to any other firm such that firm
value does not depend on dividends. We will pick up this theme in the
following section.
128
Chapter 9: Dividend policy
Prices, dividends and share repurchases
It is straightforward to show that investors are indifferent to cash received
through dividends or share repurchases. To see this, consider an all-equity
firm, which has a current market value of $100,000. There are 2,000
shares outstanding, such that the current share price is $50. The firm is
due to pay a $10 per share dividend tomorrow. In this scenario (i.e. just
before the payment of a dividend) the current share price of $50 is called
the cum-dividend share price.
First, let’s analyse what would happen to the share price after dividend
payment. The total dividend payment is $10 ∞ 2,000 = $20,000. Hence,
after a dividend payment, the total firm value will be $100,000 – $20,000 =
$80,000. As there are still 2,000 shares outstanding, the share price after
dividend payment is $80,000/2,000 = $40. This is called the ex-dividend
share price. Note the obvious result that the sum of dividend paid and exdividend share price is equal to the cum-dividend share price.
Activity
A firm has current share price of £2.50 and will pay a £0.15 per share dividend
tomorrow. What is the share price immediately after dividend payment?
Consider the cash position of an individual who originally held five shares
in our firm. The value of their shareholding was originally $250. After
the dividend payment, they have cash of $50, and the value of their
shareholding is $200. Hence, the dividend has just altered the composition
of their wealth rather than changing its dollar amount.
What happens if, instead, the firm decides to use the cash it had originally
earmarked for dividend payment for a share repurchase instead? As
mentioned above, the total dividend amount was $20,000. As the
original share price was $50, this implies that the firm can repurchase
$20,000/$50 = 400 shares. As a result, after the share repurchase, there
are 1,600 shares outstanding, and the firm is again worth $80,000 in total.
Therefore, the post-share repurchase share price must be $80,000/1,600
= $50. Note that a share repurchase (at a fair price) does not alter share
prices.
Again, consider the position of our individual who originally owned
five shares. The firm repurchases 400 shares, which is one-fifth of all
equity. Now, assume that one share of this individual’s holding of five
is repurchased. The repurchase thus gives them $50 and, after the
repurchase, their four remaining shares are worth $200 in all. As a result,
in this case also, their $250 invested in equity has been changed into
$50 of cash and $200 still in equity. This is identical to the case where
dividends were paid.
Thus, the individual is indifferent between dividends and share
repurchases. The manner in which the firm chooses to distribute cash does
not matter to them and, as a result, they will not discriminate (in value
terms) between stocks that do and do not pay dividends.
Dividend policy: stylised facts
Our prior discussion led to the conclusion that dividend policy is irrelevant
(i.e. the choice of policy doesn’t affect firm value). However, certain formal
and casual empirical observations point in the opposite direction. In this
section we will provide a brief and selective review of such empirical
research on dividend policy.
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92 Corporate finance
Perhaps the most famous set of results on actual dividend policy was
compiled and presented by John Lintner (1958). Lintner interviewed the
management of a sample of US corporations in order to determine what
lay behind their dividend-setting decisions. His research led to the four
following stylised facts.
1. Managers seem to have a target dividend payout level.
2. This payout level is determined as a proportion of long-run
(i.e. sustainable) earnings of the firm.
3. Managers are more concerned with changes in dividends rather than
the actual level of dividends.
4. Managers prefer not to make dividend changes that might need to
be reversed (e.g. cutting dividends after having raised them in the
previous period).
As the second fact implies, it is not current but long-run earnings
that matter in setting dividends such that dividends can be seen to be
smoothed relative to earnings. These observations led Lintner to develop
the characterisation of dividend behaviour that is given in equation 9.1. It
is a simple partial adjustment model:
ΔDt = λ(αEPSt – Dt–1 ), 0 < α < 1, 0 < λ < 1
(9.1)
where Dt is the time t dividend per share, EPSt is earnings per share at t, α
is the target payout ratio, and λ is the parameter governing the degree of
dividend smoothing. In line with facts 1 and 2, equation 9.1 embodies a
target payout, which is a simple proportion of earnings. Also, the change
in dividends appears on the left-hand side of 9.1 in line with fact 3.
Note that, if λ was equal to one, then the dividend change at time t would
always ensure that dividends were at precisely their target level (i.e. we
would have Dt = α EPSt ). However, for values of λ less than one, dividends
change towards their target level gradually. This reflects the smoothing of
dividends that Lintner’s stylised facts indicate.
The other major source of empirical observations on the effects of dividend
policy has been the event study literature, which has also emphasised the
vast importance of changes in dividends.1 A wide range of studies for
equity from many different countries has demonstrated that dividend cuts
lead to drops in stock price on average, whereas dividend increases on
average lead to stock price rises.2 The interested reader can consult Healy
and Palepu (1988), among other writers.
Clearly then, putting together the empirical evidence from interviews and
event studies yields an impressive case for the relevance of dividend policy.
The results of Lintner (1956) indicate that corporate managers do not
perceive dividend policy as irrelevant. Rather, they seem to follow similar
plans in their payout policy. Further, the event study evidence tells us that
the market interprets unexpected dividend increases as good news for a
stock, whereas unexpected dividend cuts are regarded as bad news.
Hence, we have a case for arguing that the dividend version of the MM
theorem is invalid. However, we have not yet come up with reasons for
why it is invalid. In the following two sections we will explore three sets
of reasons (similar to those put forward to explain the relevancy of capital
structure): namely, the existence of taxation, asymmetric information and
agency costs.
130
1
The event study
was introduced in
Chapter 5 as the basic
testing methodology for
semi-strong-form market
efficiency.
2
No change in dividends
is (as one might expect)
associated with little or
no effect on stock prices
on average.
Chapter 9: Dividend policy
Taxation and clientele theory
An obvious omission from our story of dividend policy irrelevancy is
taxation. Previously we argued that, with no taxes, share-holders should be
indifferent between income in the form of dividends or income from capital
gains. This would still be true if dividends and capital gains were taxed
symmetrically. However, it is generally true that the dividend payments
accruing to individuals are taxed more heavily than capital gains. We
would therefore expect individuals to prefer income in the form of capital
gains. Corporations, on the other hand, are taxed very favourably on
dividend income on the shares of other firms that they hold. Corporations,
therefore, should prefer dividend income to capital gains income. Finally,
some institutions pay no taxes whatsoever. These institutions will not care
whether income is earned as either dividends or capital gains.
The preceding observations on taxes lay the foundations for the clientele
theory of dividends. The notion behind this theory is straightforward.
Given the three groups above, we might expect some stocks to pay high
dividends (with these stocks held by corporations), some stocks to pay
medium dividend levels (and these are held by tax-exempt institutions)
and finally certain firms to pay low dividends (and their shares are held by
individuals). Each type of stock (classified according to dividend levels)
caters to its own ‘clientele’ of investor. A numerical example will yield
further insights.3
Assume an economy populated by risk-neutral agents. Individuals pay
a tax rate of 50 per cent on dividend income and 20 per cent on capital
gains. Corporations pay tax at rate 10 per cent on dividend income and 35
per cent on capital gains. Three types of stock exist in the economy: high,
medium and low payout stocks. Each stock has earnings per share of 100.
Payout policies, stock prices and after-tax payoffs are given in Table 9.1.
3
This example is based
on that given in Allen
and Michaely (1995).
Payout policy
High
Medium
Low
100
50
0
0
50
100
Individuals
50
60
80
Corporations
90
77.5
65
Dividend
Capital gain
After tax payoffs
Institutions
Equilibrium price
100
100
100
1,000
1,000
1,000
Table 9.1
Clearly, given the after-tax payoffs to each group, individuals will hold low
payout stocks, corporations will hold high payout stocks, and institutions
are indifferent. Assume that in equilibrium the total holdings of each
group are as given in Table 9.2.
Payout policy
High
Medium
Low
0
0
320
Corporations
110m
0
0
Institutions
500m
730m
220m
Total
610m
730m
540m
Individuals
Table 9.2
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92 Corporate finance
Note that in Table 9.1 we displayed the equilibrium price of each equity
share as 1,000. Why is this the case? To see this, assume that the price of
low payout stock is 1,050, whereas the price of all other stock is 1,000.
This would imply that high and medium dividend level firms have an
incentive to switch to low dividend policies (to take advantage of the high
share prices). Such actions would increase the supply of low dividend
stocks and hence depress their price.
A reinforcing effect comes from the demand side. The return that
individuals get from holding low payout stock is 80/1,050 = 7.62%.
This exceeds the returns they would gain from holding medium and high
payout stocks (which are 6.5 per cent and 5 per cent respectively), and
hence individuals continue to demand low dividend stocks. Institutions, on
the other hand, only get a return of 9.52 per cent from holding low payout
stock (100/1,050 = 9.52 per cent), whereas they get a return of 10 per
cent on other types of equity. Thus, institutions rationally sell their low
dividend equity. This further depresses the price. It is only when the price
of low dividend stock is 1,000 that equilibrium is reached.
The clientele model leads to the same main result as MM. Firm values (or
stock prices) are unaffected by dividend policy. There are obviously
underlying differences to these theories though. For example, the clientele
theory implies that investors in high tax brackets should hold portfolios
with low dividend yields and vice versa.4
Asymmetric information and dividends
A popular version of the asymmetric information story for the relevance
of dividends is very similar to the reasoning underlying the relevance
of capital structure in Ross (1977). This model argued that debt policy
was relevant as, in a world where firm quality was not observable to the
market, the level of debt chosen by a firm’s management signalled the
quality of the firm. High-quality firms would choose high debt levels (as
they could afford the interest payments without running into cash-flow
problems), whereas poor firms would choose low levels of debt. Hence,
debt acted as an observable signal of firm quality upon which the market
would base its valuation of a firm.
Exactly the same type of logic can be applied to dividend policy. If we
again assume that corporate managers’ objective function is increasing in
expected firm value but decreasing in expected bankruptcy costs then, in
a world where firm quality is not observable to outsiders, dividend policy
can be used as a signal. High-quality firms (i.e. firms with large average
cash flows) can afford to pay large dividends, as they worry less about
bankruptcy than low-quality firms. The latter pay low dividends to avoid
bankruptcy. Interpretation of such signals by investors means that firms
paying high dividends are valued more highly in the market than those
paying low amounts.
In empirical terms, the prior argument would then imply a positive
relationship between dividend levels and firm value. Further, we might
also expect that cuts in dividends would result in share price reductions,
as this might be interpreted as a signal of reductions in a firm’s quality.
Conversely, dividend increases should correlate with share price rises. Such
empirical predictions fit quite nicely with those empirical results discussed
earlier in the chapter.
132
4
The dividend yield on
a stock is the ratio of
dividend payment to
stock price. Evidence for
this prediction is given
in Blume, Crockett and
Friend (1974).
Chapter 9: Dividend policy
Agency costs and dividends
Consider a situation where the ownership and control of corporations
are separated. Organisations are assumed to be controlled by managers,
who can only be imperfectly monitored by owners/shareholders and, as
a result, there is scope for managers to behave opportunistically. In such
situations, our analysis of the results of Jensen and Meckling (1976)
and Myers (1977) indicated that capital structure changes may alter
firm value, such that MM1 was violated. The same situation may imply
that dividend policy affects firm value. Here, we give only the briefest
treatment of this possibility.
Both of the agency cost models of capital structure referenced above
include situations where managers, acting in the interest of equityholders, transfer value away from debt-holders towards those who own
shares.5 Similar activities may be undertaken with dividend policy.
Managers may pay out large levels of dividends (benefiting equityholders), financing these payments by rejecting positive NPV projects or
by increasing debt levels. If debt-holders do not anticipate this behaviour,
the value of debt will be reduced while the value of equity increases. Note
that, in both cases, ‘excessive’ dividend payments will lead to lower firm
values.
5
Asset substitution
and debt overhang
are examples of such
behaviour.
An interesting feature of this argument is that it predicts that dividend
increases should be reflected in higher market values for equity but
lower market values for debt. This contrasts with the implications of the
asymmetric information-based theories, which, as dividend increases are
good news in general, predict that they should lead to increases in the
values of both debt and equity.
From the preceding section we know that dividend increases result
in higher equity values empirically, consistent with both agency- and
information-based theories. However, recent empirical evidence suggests
that, at least for US firms, corporate bond prices drop when dividends
are cut and don’t change significantly when dividend increases are
announced. Such results would seem to indicate that theories of dividend
policy based on asymmetric information are more realistic than those based
on agency costs.
Summary
We started this chapter by arguing that, like capital structure, dividend
policy should not affect firm value. Subsequent to this, however, we
pointed out several sources of real world imperfection that might lead to
optimal dividend policies (in the sense of firm value maximisation). Such
imperfections included taxation, information asymmetries and agency
costs.
We also explored some of the empirical results on dividend policy.
Empirical evidence shows that equity prices tend to rise after unexpected
dividend increases and fall after unexpected dividend cuts (with bond
prices following a similar pattern). This, we argued, seemed most
supportive of dividend models based on asymmetric information.
The dividend puzzle is far from resolved, however. Much research
work remains to be done in the area to clarify our understanding of the
fundamental determinants of corporate dividend policy. Lintner’s stylised
facts and results from event studies have given us a good empirical basis
upon which to construct realistic theories of dividend behaviour, and it is
precisely this task that currently confronts finance theorists.
133
92 Corporate finance
A reminder of your learning outcomes
Having completed this chapter, and the Essential reading and activities,
you should be able to:
• show that dividend policy (and share repurchases) are irrelevant to
firm valuation under the Modigliani–Miller assumptions
• discuss the stylised facts of dividend policy as provided by Lintner
• present the clientele model of dividends
• discuss the effects of asymmetric information and agency costs on
dividend behaviour.
Key terms
agency costs
asymmetric information
capital structure
clientele model
dividend policy
frictionless markets
Lintner’s stylised facts
Modigliani–Miller irrelevance theorem
personal taxes
share repurchases
target dividend payout level
taxes on capital gains
taxes on dividends
Sample examination questions
1. Describe the model of dividend policy formulated by Lintner (1956)
and detail the stylised facts upon which this model is based. (10%)
2. ‘The Modigliani–Miller theorems imply that firms’ dividend policy does
not affect their value in the slightest.’ What assumptions underlie this
statement? Give two scenarios in which the statement is invalid. (15%)
3. For tax reasons it is cheaper to pay equity-holders through share
repurchases than with dividends. Nevertheless, many firms use
dividends to pay their investors. What is the signaling explanation for
this?
134
Chapter 10: Mergers and takeovers
Chapter 10: Mergers and takeovers
Aim of the chapter
The aim of this chapter is to explain why managers of firms are engaging
in mergers and acquisitions. With this aim in mind, we first introduce
a stylised model in which efficient takeovers cannot possibly obtain
(Grossman–Hart). We then introduce institutional mechanisms which
enable takeovers to occur. Finally, we investigate whether or not mergers
and acquisitions create value and provide empirical evidence on returns to
shareholders of bidding and target firms.
Learning outcomes
By the end of this chapter, and having completed the Essential reading,
you should be able to:
• discuss motivations for merger activity
• analyse simple numerical examples of efficient takeover activity
• detail the argument of Grossman–Hart (1980) regarding the
impossibility of efficient takeovers
• present ways in which this analysis can be modefied to permit
takeovers to occur.
Essential reading
Hillier, D., M. Grinblatt and S. Titman Financial Markets and Corporate Strategy.
(Boston, Mass.; London: Macmillan, 2008) Chapter 20 (Mergers and
Acquisitions).
Further reading
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.
Brealey, R., S. Myers and F. Allen Principles of Corporate Finance. (Boston,
Mass.; London: McGraw-Hill, 2008) Chapter 32 (Mergers).
Copeland, T. and J. Weston Financial Theory and Corporate Policy.
(Reading, Mass.; Wokingham: Addison-Wesley, 2004) Chapter 18.
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., K. Palepu and R. Ruback ‘Does corporate performance improve after
mergers?’, Journal of Financial Economics 31(2) 1992, pp.135–76.
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.
Jarrell, G. and A. Poulsen ‘Returns to acquiring firms in tender offers: evidence
from three decades’, Financial Management 18(3) 1989, pp.12–19.
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.
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92 Corporate finance
Jensen, M. and R. Ruback ‘The market for corporate control: the scientific
evidence’, Journal of Financial Economics 11(1–4) 1983, pp.5–50.
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.
Ravenscraft, D. and F. Scherer Mergers, selloffs, and economic efficiency.
(Washington D.C.: Brookings Institution, 1987).
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 managementspecific 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) 1987, pp.943–63.
Overview
The post-Second World War period has seen an unprecedented amount
of corporate activity resulting in the combination of two or more firms
under a single corporate banner and legal status. Such activity comes in
many forms and is initiated for varying reasons. This chapter gives an
introduction to the concepts underlying merger/takeover/acquisition
activity and provides a basic review of the theory of takeover activity, and
supplies empirical evidence on returns to takeovers.
In line with the arguments presented throughout this guide, we argue
that merger activity should be judged in terms of the value it delivers.
Mergers should be undertaken if they are positive NPV transactions. A
mathematical way of stating this is that:
VXY > VX + VY,
(10.1)
that is, the value of the merged firm created from firms X and Y (VXY)
exceeds the sum of pre-merger values of X and Y (i.e. VX + VY). Such value
may come about through the exploitation of scale economies or elimination
of inefficiencies. We will give a classification of merger and acquisition
behaviour based on the source of value in the following section.
Merger motivations
Following Hillier, Grinblatt and Titman (2008), we will split merger and
takeover activity into three distinct sub-groups:
• financial activity
• strategic activity
• conglomerate activity.
1. Financial mergers: these are takeovers or acquisitions that are
initiated to take advantage of corporate inefficiencies that lead to the
under-valuation of firms. This allows an acquiring firm to buy assets
cheaply, implement strategies that increase the value of the acquired
firm and then sell on the acquired assets at a profit (if so desired).
Such activity yields a positive NPV. Opportunities for financial mergers
are likely to come about due to managers of acquired firms following
their own, rather than shareholders’, goals and hence not maximising
firm value. In this way, the market for corporate control is said to exert
discipline on a firm’s management.1 The merger wave of the 1980s may
be thought of as largely comprised of such activity. An active market
for corporate control (in the form of hostile takeovers) is therefore an
important force that mitigates the problems arising from the separation
of ownership and control in modern corporations.
136
1
This is because, if
a takeover occurs,
incumbent management
are likely to lose their
jobs. Hence, assuming
management would prefer
to retain their jobs, the
possibility of takeover
limits managerial scope for
inefficiency.
Chapter 10: Mergers and takeovers
2. Strategic mergers: financial mergers generate value through
eliminating corporate inefficiency induced by bad management.
Strategic mergers yield value through the taking advantage of
economies of scale and scope in production, purchasing and marketing.
Hence, horizontal integration activity undertaken to increase and
exploit market power and to take advantage of scale economies fall
into this category. Also, acquisitions that are vertically integrating may
be thought of as strategic activity due to their yielding lower
production costs or marketing expenses. A recent example of such
activity might be the announced link-ups within the French banking
sector in February 1999.2
3. Conglomerate mergers: certain mergers are clearly not motivated
by scale economies and are not attempts to take advantage of
corporate mismanagement. The most obvious examples of such activity
are between firms in very different industries and these link-ups are
known as conglomerate mergers. This type of activity was very popular
in the 1960s and 1970s (although much of the conglomeration that
occurred in these decades was reversed in the 1980s). Motivations for
conglomerate merger are unclear. Some have stated that the element
of diversification that conglomeration yields adds to value. However,
given that investors can diversify their own portfolios in order to
reduce risk (i.e. they don’t need firms to diversify for them), the idea
that value is added for this reason is flawed. Along similar lines, some
have argued that a gain from conglomeration is derived due to lower
interest rates that conglomerates are charged.3 Again, however, this
argument doesn’t stand up to close scrutiny. One reason why
conglomeration may occur is that it allows firms with large amounts of
cash (who do not want to increase dividends or repurchase equity) to
profitably employ this cash in positive NPV projects.
2
In early February
1999, BNP and Société
Générale announced
plans to merge. Later,
Paribas entered the
fray, announcing that
it would take over the
other two banks.
3
Conglomerates may be
charged lower interest rates
as cash-flow risk is reduced
through precisely the
diversification argument
already mentioned.
A numerical takeover example
Consider two firms, X and Y, that compete in the same product market.
Corporation X currently has one million shares outstanding, each with
value $2. Firm Y has 500,000 shares on offer and share price $10. Firm Y
is contemplating a takeover of corporation X, as it knows that corporation
X is being run inefficiently. Firm Y estimates that, if it takes corporation X
over, it could increase firm X’s net cash flow by $300,000 per year. Assume
that these firms are infinitely lived. The relevant cost of capital for firm X
is 10 per cent.
Given the prior information, it is clear that, if firm Y does take over
corporation X, the increase in X’s value would be the present value of a
perpetuity paying $300,000 each year. This present value is $3m, which
represents the gain from the merger.4 It is clear that, given that the merger
creates value, it is socially desirable. However, the terms by which the
merger actually occurs will dictate the net payoffs to the shareholders of X
and Y. For the merger to occur, both net payoffs must be positive.
4
Make sure you can
derive this PV for
yourself.
Assume, for example, that the merger is to occur by firm Y agreeing to
purchase every share in firm X at a price of $3 per share. This implies that
(as there are one million shares in firm X in issue) X’s shareholders get a
total payout of $3m, which exceeds the value of their initial shareholding
(i.e. $2m). Hence firm X’s shareholders are happy to participate in the
merger, as their payoff is $1m. Firm Y’s shareholders are paying $3m for a
firm which, under their management, will be worth $2m + $3m = $5m.
Hence their gain is also positive at $2m, and they are happy to participate.
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92 Corporate finance
Note that, quite obviously, the sum of the gains to X and Y shareholders is
the total value creation of $3m.
Another way in which this merger could have been financed is if firm Y
offered to issue a certain amount of new shares and gave these to the
shareholders of firm X instead of cash. Consider the following offer as an
example. One new share in firm Y is exchanged for every four existing firm
X shares. Note that this freshly issued equity will be a claim on the value of
the merged enterprise and hence priced as such.
The value of the merged firm will be the sum of the pre-merger values of X
and Y plus the value created of $3m. The pre-merger value of X is $2m
and that of Y is $5m. Hence the total value of the firm after the merger is
$10m. After the merger there are 0.75m shares in issue. This comprises
the original 0.5m shares in firm Y plus the 250,000 new shares issued.5
Hence the share price of the merged enterprise is:
$
.
(10.2)
The original shareholders of Y hold two-thirds of the equity of the merged
enterprise, which has a value $6.67m. The value of their original position
is $5m and hence they gain to the tune of $1.67m. The old X shareholders
own one-third of the equity of the merged enterprise, which is worth
$3.33m. Their gain is hence $1.33m, as the value of firm X pre-merger
was $2m. Both sets of shareholders are winners therefore, and hence the
merger goes ahead. Again, note that the sum of the gains is $3m, the total
value created.
The market for corporate control
As a result of the separation of ownership and control, managers may not
act in the firm owners’ best interest. Managers may:
• exercise insufficient effort
• make extravagant investments (Jensen (1986))
• use entrenchment strategies; that is, take actions that hurt shareholders
in order to secure their position (Shleifer and Vishny (1989))
• increase their private benefits from running the firm by engaging in a
variety of self-dealing behaviour (Jensen and Meckling (1976)).
This moral hazard between firms’ managers and owners may be mitigated
through corporate governance. A firm’s board of directors in principle
monitors managers on behalf of owners. It is furthermore in charge of
managers’ compensation, audits and oversight of risk management.
Moral hazard between firms’ managers and owners may be mitigated
through the market for corporate control. In the market for control,
disciplinary takeovers, which are usually hostile, create value by
substituting efficient teams for entrenched money-wasting managers.
These disciplinary takeovers may be needed to keep managers on their
toes if the board of directors is an ineffective monitor and, more generally,
if corporate governance is failing. This is particularly important for firms
with a disperse mass of small shareholders. However, as we will see in the
following section, free-rider problems make hostile takeovers particularly
difficult when ownership is disperse.
138
5
One new share was
offered for every four old
X shares. As there were
originally one million X
shares outstanding, this
implies 250,000 new Y
shares must be issued.
Chapter 10: Mergers and takeovers
The impossibility of efficient takeovers
In the previous sections, we examined the types of merger activity commonly
seen in reality and the motives for such activity generally given by managers.
In this section, we will introduce you to a simple theoretical model of merger
activity, which yields the result that any efficient takeover bid will fail.6 This
extreme outcome comes from rational shareholders free-riding on the (effort
and) firm value improvement delivered by a takeover raider.
Assumptions
Our assumptions here are as follows:
• the firm is subject to a takeover bid from an external takeover raider
• firm value will improve, if the bid succeeds: the value increase is
common knowledge
6
The model developed
in this section is based
on Grossman and
Hart (1980). Efficient
takeover activity is
defined as activity for
which the increase in
the market value of the
acquired firm exceeds
any associated costs.
• the equity of the target firm is held by many, small shareholders
• the raider incurs administrative takeover costs of c.
Assume that the current firm value is y, and let the firm value if the
takeover were to succeed be y + z. The takeover is efficient as the
following condition holds:
z > c.
(10.3)
The raider must gain at least 50 per cent of the shares to implement the
takeover. Note, however, that as shareholders are assumed to be identical,
if any one shareholder finds it profitable to tender their shares to the
raider then all will. The raider offers a premium p over the current firm
value to equity-holders for their shares. Hence, for the bid to be profitable
for the raider we must have:
z > p + c,
(10.4)
that is, the improvement in firm value must exceed the cost of takeover
and the premium paid to original equity-holders.
Consider the position of a single, small shareholder. As their shareholding
is minor relative to the sum of all equity, they do not consider their
decision to be pivotal. Assume that they believe that the bid will be
successful. Then they will only sell their shares to the raider if:
p > z,
(10.5)
that is, it is only in the shareholder’s interest to tender if the premium they
get outweighs the money they would make by hanging on to their equity
and profiting from the value improvement associated with the takeover.
If the shareholder believes that the takeover bid will fail, then they will
be indifferent between offering their shares to the raider and not offering
them.
Our key result can be derived from a comparison of equations 10.4 and
10.5. They are clearly contradictory, implying that the raider cannot
simultaneously succeed with the bid and make a profit. Hence, profitable
takeover activity cannot occur.
A crucial assumption here is that all shareholders are small in size.
This then implies that none of them perceive themselves to be pivotal
to the success of the takeover bid. This results in all small shareholders
attempting to free-ride on the value improvement offered by the raider
and, ultimately, the bid then fails.
Another way to see the result is as follows. A premium that allows the
raider to make a profit must satisfy the following condition:
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92 Corporate finance
p  (0, z – c).
(10.6)
However, a premium in this region implies that shareholders are better off
not selling to the raider and hanging on to their equity as:
y + p < y + z – c < y + z.
(10.7)
The first term in equation 10.7 is the money they get for selling to the
raider, and the final term is the value of their shareholding if they do not
sell (conditional on the bid being successful).
Two ways to get efficient takeovers
In light of casual and formal empirical evidence, the result of the previous
section seems untenable. Most would argue that at least some of the
takeovers that occur in reality lead to both the raider and the target
shareholders making some money. This section provides two ways in
which we can overturn the results from the previous section.
Dilution
Grossman and Hart (1980) first pointed out the free-riding problem we
discussed in the preceding section. In the same paper they also indicated a
solution to the free-riding problem. This solution was dilution.
Dilution is the ability of a raider to extract value from the target, if they
successfully complete the takeover. This might be done by placing themself
in charge and paying themself an astronomical salary, selling the firm’s
output to another corporation they own at a very low price, and other
diverse means. Hence, if the takeover is successful and the raider dilutes
the firm, the firm’s market value ends up being less than y + z (to use the
notation of the previous section).
To make the prior argument concrete, assume the raider can appropriate
an amount  of firm value if the takeover is successful. Hence, if
shareholders believe the bid will be successful, they will be willing to
tender their shares if offered a premium (over current value) that satisfies
the following condition:
p > z – .
(10.8)
The raider makes money if equation 10.4 holds, and this leads to the
following condition for profitable takeover activity to occur:
z – c > p > z –  |  > c.
(10.9)
The interpretation of equation 10.9 is simple – takeovers can be
profitable if the amount the raider can grab through dilution exceeds the
administrative cost of takeover. Note also that, once they gain control,
the raider need not actually dilute the firm. Merely the threat of dilution
allows the takeover to proceed.
A final issue about dilution that should be addressed is the source of the
raider’s ability to dilute. Grossman and Hart assume that the target firm is
originally a private enterprise. The original owners of the firm then decide
to take the firm public and write provisions that allow dilution into the
corporate charter. These individuals do this in order to ensure that the firm
is efficiently run in future years (i.e. they write in dilution provisions to
allow efficient future takeover activity).
Large shareholders (toehold)
Another scenario in which efficient takeover activity might occur is when
a single shareholder owns a large block of equity. In such a situation we
can think of the large shareholder and the raider synonymously (i.e. it is
140
Chapter 10: Mergers and takeovers
the large shareholder who can possibly implement an efficient takeover).
Sticking with the notation used in the Grossman and Hart (1980) analysis,
assume that the large shareholder originally owns a proportion α of firm
equity (toehold). Assuming no dilution, the condition for shareholders to
tender if they believe the bid will succeed is again:
p > z.
(10.10)
Hence, shareholders require a premium that exceeds the size of the value
improvement. The condition that must hold for the large shareholder to
profit is:
z > (1 – α)p + c,
(10.11)
that is, the value improvement must exceed the cost of takeover, plus the
premium the large shareholder must pay to buy the remaining (1 – α)
of firm equity. Both equations 10.10 and 10.11 are satisfied when the
following condition holds:
αz > c.
(10.12)
Hence, large shareholders can implement efficient takeovers, when the
proportion of the value improvement that accrues to their original holding
exceeds the takeover cost.
Thus our analysis tells us that large shareholders are important in that
their existence allows the free-rider problem to be circumvented. This is
exploited in Shleifer and Vishny (1986) who also relax the assumption
of perfect information. In their analysis, the value improvement is only
known by the large shareholder, and this provides another reason for
the existence of takeover activity in the model. The role of the large
shareholder is emphasised in some of the empirical predictions from their
model. They show, for example, that firm values increase with the size of
the large shareholding. The intuition for this is that a larger shareholding
means more efficient takeover decisions and hence a firm with larger
future values and hence greater current market value.
Empirical evidence
Are mergers and acquisitions value-enhancing? This section reviews
empirical evidence from two types of studies: accounting and event studies.
The first type, accounting studies, examine financial results
(accounting data) to draw inferences about the underlying economic
impact of mergers and acquisitions. These studies tend to investigate
whether acquirers outperform their non-acquirer peers. Alternatively,
these studies compare the performance of the combined firm following
a merger or an acquisition with the performance obtaining prior to the
transaction. Performance tends to be measured by net income, operating
margin, or return on equity or assets.
The second type, event studies, do not directly measure performance.
Instead, these studies attempt to measure the value created by the merger
or acquisition through abnormal stock returns around the announcement
date of a tender offer. Hence, event studies rely on financial markets being
efficient.
Accounting studies
The empirical evidence from accounting studies is mixed. Ravenscraft and
Scherer (1987) investigate more than 5,000 mergers occurring between
1950 and 1975, calculate and compare the post-merger performance of
acquiring firms with that of non-acquiring firms in the same industries,
141
92 Corporate finance
with performance being measured as return on assets, and report that
performance is 1 to 2 per cent less for acquiring firms.
In contrast, Healy, Palepu and Ruback (1992) examine 50 large mergers
between 1979 and 1983 and report improvement in performance of the
combined firms following the mergers, where performance is measured
by sales and profits. Asset productivity is furthermore shown to improve
significantly following acquisitions.
The difference in findings between both accounting studies may be due to
differences in the motivation for mergers and acquisitions. The motivation
for many of the mergers in the 1960s and 1970s (and much fewer in the
1980s) was diversification and there can be efficiency losses associated
with diversification. Accounting studies are, however, vulnerable to
discrepancies introduced by accounting for mergers and acquisitions.
Event studies
Empirical evidence from event studies suggests that shareholders from
target firms gain from takeovers. This should not come as a surprise as
target shareholders require a premium in order to induce them to sell
their shares to the acquiring firm. Jensen and Ruback (1983) report
that target share prices increase, on average, by about 16 to 30 per
cent around the date of the announcement of a tender offer. Empirical
evidence reported by Jarrell, Brickley and Netter (1988) suggests that
these returns increased substantially during the 1980s to an average of
about 53 per cent. Jensen and Ruback (1983) furthermore report that the
average return to shareholders from target firms in negotiated mergers is,
however, only about 10 per cent.
The empirical evidence from event studies on returns to shareholders of
bidding firms tends to be quite mixed: returns to bidders are, on average,
small, time-varying, but may be positive or negative. For instance, Jarrell
and Poulsen (1989) show that the announcement return to bidder in
tender offers dropped from a statistically significant 5 per cent gain in
the 1960s to an insignificant 1 per cent loss in the 1980s. The means of
payment used for the transaction is furthermore shown to have a major
effect on returns to bidders. For instance, Travlos (1987) finds that the
average return on the two days around the announcement of a cash offer
is only marginally different from zero (+0.24 per cent). In contrast, in
acquisitions financed by an exchange of equity, stock prices of bidding
firms fall, on average, by about 1.5 per cent. The means of payment may
hence act as a signal for the quality of the bidder. Consistent with the
pecking order theory reviewed in Chapter 7 (Myers and Majluf (1984)),
bidders offer stock when they believe that their stock is overvalued. A
stock offer may furthermore indicate that the bidder was unable to get any
financial backing from a bank or another financial institution.
Adding the bidder and target returns generates positive returns, implying
that, on average, there is a net gain to shareholders around the time of the
merger or acquisition. For instance, Bradley, Desai and Kim (1988) provide
evidence suggesting that successful tender offers increase the combined
value of the merging firms by an average of 7.4 per cent or $117m (stated
in 1984 dollars). The empirical evidence from event studies hence suggests
that mergers and acquisitions are, on average, value enhancing.
142
Chapter 10: Mergers and takeovers
Summary
In this chapter we have given you an overview of the facts involved in,
and theory surrounding, mergers and takeovers. The main lesson of
this chapter is that mergers that should go ahead (i.e. efficient merger
activity) are those that are positive NPV transactions. See equation 10.1.
Such positive NPV can come from exploitation of economies of scale in
production or sales (strategic mergers), removal of bad management and
elimination of inefficiencies (financial mergers) or possibly through the
purchase of firms in an unrelated industry but with a strong portfolio of
possible investment projects (conglomerate mergers).
We discussed theoretical models indicating that such efficient merger
activity may be blocked in economies without frictions or information
asymmetries. The source of problems here is shareholder free riding.
The prevention of profitable takeovers by free riding is shown to
disappear when allowances are made for dilution, large shareholders and
asymmetric information.
Towards the end of the chapter, we investigate whether mergers and
acquisitions are value-enhancing. Empirical evidence from event studies
suggests that mergers and acquisitions create, on average, joint value.
Most of the value created is, however, appropriated by the shareholders of
target firms.
A reminder of your learning outcomes
Having completed this chapter, and the Essential reading, you should be
able to:
• discuss motivations for merger activity
• analyse simple numerical examples of efficient takeover activity
• detail the argument of Grossman–Hart (1980) regarding the
impossibility of efficient takeovers
• present ways in which this analysis can be modefied to permit
takeovers to occur.
Key terms
asymmetric information
bidders
capital structure
clientele model
conglomerate mergers
corporate governance
dilution
disciplinary takeover
efficient takeovers
event studies
financial mergers
free-riding
frictionless markets
Grossman–Hart model
143
92 Corporate finance
large shareholders
mergers and acquisitions
strategic mergers
targets
takeover premium
toehold
Sample examination questions
1. Present the assumptions behind, and give a derivation of, the
Grossman–Hart analysis, which implies that efficient takeover activity
is impossible. (15%)
2. Describe the dilution solution to the preceding solution as suggested by
Grossman and Hart. (5%)
3. How does the existence of a large shareholder affect the Grossman–
Hart result? (5%)
4. Exporting firm Euro Importing has a market value of €100 million.
There are one million shares outstanding, 20 per cent of them are
controlled by the CEO who is the original founder. The present value of
the firm’s profits is €130 million, however the CEO uses up €30 million
of firm value for pet projects that do not add value to the firm. All
other shares are controlled by dispersed shareholders.
An asset management firm worth €500 million, and which has five
million shares outstanding, is considering acquiring Euro Importing.
a. What is the current price per share of Euro Importing?
b. If the acquirer buys 51 per cent of the shares, it would control the
firm and cancel wasteful perk spending. What is the maximum the
acquirer would be willing to pay for 51 per cent? What if purchasing
51 per cent also involved €1 million in additional fees?
c. The acquirer announces that it will attempt a takeover of Euro
Importing by purchasing shares at the price in (b). Assume €1
million fees as in (b). What happens to the price per share if (i)
the market believes the raid will succeed; (ii) the market believes
the raid will fail. What does a rational investor do if the rest of the
market believes (i)? If the rest of the market believes (ii)? Is there
an inconsistency? What happens to the price per share of the asset
management firm if (i)? If (ii)?
d. Suppose half of the dispersed shareholders believe the acquirer
succeeds and half believe that he will fail. Does the raid succeed?
e. How many shareholders are willing to sell if the offer price is €130?
How many are willing to sell if the offer price is €100? Assume you
can linearly interpolate the probability that a shareholder succeeds
between these two extreme values. What price must be paid for the
raid to succeed? Is it worth it to the acquirer? What if the fees were
€6 million?
f. Suppose that after buying the firm, the acquirer can also use up €30
million on private benefits. At what price would the shareholders
now be willing to sell? Relate this to Grossman and Hart’s solution
to the free rider problem.
g. Explain why current ownership would be willing to outbid the
acquirer.
144
Appendix 1: Perpetuities and annuities
Appendix 1: Perpetuities and annuities
This short appendix gives some formulae that will allow you to compute
the present value of certain types of income stream quickly and easily.
The mathematics behind these formulas is based upon the summation
of convergent geometric progressions, a topic that should be treated in
any basic mathematics text. Throughout our examples we will think of
cash flows as being received on an annual basis (but this is obviously not
critical).
Perpetuities
A perpetuity is an income stream that promises us a payment of a fixed
amount, X, at the end of every year from now until the end of time. Hence,
the income stream is perpetual. Assuming that the appropriate positive
rate for discounting this income stream is r, then the present value of the
income from the perpetuity is given by:
A1
where the sum extends out forever. The summation in A1 is a very simple
progression and has a straightforward closed-form solution, which is:
A2
X
PVP = –r
(i.e. the present value of the income stream associated with a perpetuity is
just the ratio of the fixed payment to the interest rate).
Activity
Calculate the present value of a perpetuity stream that promises a cash payment of
$15,000 per year, assuming that the annual interest rate is 8 per cent.
Growing perpetuities
The preceding example can be generalised to permit the annual cash
payment to grow at a fixed percentage rate. Again, denote the first cash
payment by X, and let g be the annual growth rate of the payment. We
assume that the growth rate of the payment is less than the interest rate, r.
Then the present value of the perpetuity income stream can be written:
A3
Again, it’s simple to calculate the value of this infinite summation
explicitly. It’s just:
A4
Activity
Calculate the present value of a perpetuity stream that promises an initial cash payment of
$15,000 and growth of 5 per cent. Assume that the annual interest rate is 8 per cent.
145
92 Corporate finance
Annuities
The two income streams above are assumed to be infinite in nature.
Quite clearly, however, it is very important to be able to value projects/
assets which have finite lifetimes (in terms of years). An annuity is such
an income stream and promises fixed annual cash payments for the next
T years only. Hence, one can think of an annuity as a kind of truncated
perpetuity (as the perpetuity would go on paying annual cash flows after
the annuity had expired). The present value of an annuity paying £K per
year for the next T years is:
A5
where the interest rate is again denoted r. The term in square brackets in
A5 is known as the annuity factor, and tables of such factors (for various T
and r) are widely available.
The derivation of A5 can be performed using the formula for the present
value of a perpetuity. An annuity can be thought of as the cash-flow
difference between a perpetuity with cash flows beginning in one year and
a perpetuity with cash flows beginning in T+1 years. The present value
of the first stream is just K/r from equation A2. The value of the second
perpetuity is K/r at time T, which yields a present value of K/[r(1+r)T].
Taking the difference between the two present values yields the expression
in A5.
Activity
What is the present value of a 15-year annuity promising an annual payment of
£250,000 assuming that the interest rate is 10 per cent? What is the future value of this
annuity at a 15-year horizon? (Hint: the factor which should be used to calculate the
future value is just (1+r)T.)
146
Appendix 2: Sample examination paper
Appendix 2: Sample examination paper
Important note: This Sample examination paper reflects the
examination and assessment arrangements for this course in the academic
year 2010−2011. The format and structure of the examination may have
changed since the publication of this subject guide. You can find the most
recent examination papers on the VLE where all changes to the format of
the examination are posted.
Time allowed: three hours.
Candidates should answer FOUR of the following EIGHT questions: ONE
from Section A, ONE from Section B and TWO further questions from
either section. All questions carry equal marks.
A calculator may be used when answering questions on this paper and
it must comply in all respects with the specification given with your
Admission Notice. The make and type of machine must be clearly stated
on the front cover of the answer book.
Section A
Answer one question from this section and not more than a further
two questions. You are reminded that four questions in total are to be
attempted with at least one from Section B.
1. a. Derive and explain the Fisher separation result, which implies
that firm owners can delegate choice of investment projects to firm
managers.
(10 marks)
b. Using the Fisher separation analysis, justify the use of the net
present value rule as a project evaluation criterion.
(10 marks)
c. Show how the Fisher separation result breaks down in a world
in which capital markets are not perfect in that the interest rate
charged on borrowed funds exceeds the rate paid on loaned
monies.
(5 marks)
2. a. Stock X has an expected return of 6 per cent and a return variance
of 36 per cent. Stock Y has expected return 12 per cent and return
variance of 81 per cent. An investor forms a portfolio of these two
stocks, placing one-third of his wealth in stock X and the remainder
in stock Y. Showing all of the steps in your calculations, compute
the expected return and return standard deviation of this portfolio,
assuming that returns on the two stocks are perfectly correlated.
Graph the points representing the portfolio and the two stocks in
mean–standard deviation space.
(8 marks)
b. Assume now that the returns on stocks X and Y are uncorrelated.
Recompute the expected return and return standard deviation
of the investor’s portfolio. Plot the point now represented by the
portfolio on the previously constructed graph.
(7 marks)
c. Using the results derived above, discuss the impact of diversification
on the characteristics of investors’ portfolios. Give a mathematical
treatment of the effect of diversification on portfolio variance.
(10 marks)
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92 Corporate finance
3. a. Futura Computers Inc. is a relatively new British IT firm. In the
recent past its equity has had return variance of 35 per cent. Over
the same period the market has had an average return of 8 per
cent and return variance of 25 per cent. The covariance between
Futura’s returns and the market’s return was 40 per cent. Compute
the  for Futura’s stock. What does the level of the  imply for the
relationship between Futura’s stock returns and those on the
market?
(10 marks)
b. The risk-free rate is 5 per cent. Compute the expected return on
Futura stock.
(5 marks)
c. Futura is evaluating a project that would involve the installation of
a new inventory control system. The project would have an effective
lifetime of seven years. Futura’s management estimates that in the
first year of its life the project would increase profits by £54,000.
This figure would increase by 5 per cent per annum until the project
was over. The cost of the project is £325,000. Should Futura invest
in the new inventory control system?
(10 marks)
4. a. A UK exporter knows that he is due to receive a payment of
$650,000 in one year. The current spot exchange rate is $1.6 per
£1. Given annual UK and US interest rates of 5 per cent and
3 per cent, construct the implied one-year forward exchange
rate. Assuming that the exporter hedges exchange rate risk using
a forward contract, how much in sterling will he receive in one
year’s time?
(10 marks)
b. Using absence-of-arbitrage arguments, derive upper and lower
bounds that must hold for the price of a European call option on a
non-dividend paying stock.
(10 marks)
c. Derive the put–call parity condition that links the prices of
European puts, calls and underlying prices. If a stock is priced at
$2.25, and a call with exercise price of $2.75 and time to maturity
of one year has a price of $0.20, derive the price of a put with the
same specifications to the nearest whole cent. Assume that the riskfree rate is 8 per cent.
(5 marks)
Section B
Answer one question from this section and not more than a further
two questions. You are reminded that four questions in total are to be
attempted with at least one from Section A.
5. a. What is the free-rider problem in corporate takeovers? In reality,
how do acquiring firms get around this problem?
(15 marks)
b. Describe briefly two takeover defence strategies. Can they ever
benefit shareholders?
(10 marks)
6. a. Certain authors have recently found evidence of positive
autocorrelation in short-term stock returns and negative
autocorrelation in longer horizon returns. What are the implications
of these findings for weak-form efficiency?
(10 marks)
b. Discuss how one might use information on mutual fund
performance or the predictive accuracy of investment analyst
expectations to evaluate the hypothesis that markets are strongform efficient.
(7 marks)
c. Is event-study evidence of positive abnormal returns prior to stock
splits consistent with semi-strong form efficiency? How might these
abnormal returns be explained?
(8 marks)
148
Appendix 2: Sample examination paper
a. What is the empirical evidence on the impact of dividend
announcements on stock prices?
(5 marks)
b. How do you think that this empirical relationship is affected by
asymmetric information regarding the quality of firms’ investment
projects?
(10 marks)
c. What effect would you expect an increase in the higher rate of
personal taxation to have on the dividend payout decisions of
firms?
(10 marks)
7. Explain the tax trade-off and pecking order theories of corporate capital
structure. Compare and contrast the empirical implications of these
theories.
(25 marks)
END OF PAPER
149
92 Corporate finance
Notes
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