ńňđ. 1 
Quantitative Financial Economics
SERIES IN
FINANCIAL ECONOMICS
AND QUANTITATIVEANALYSIS
Stephen Hall, London Business School, UK
Series Editor:
Editorial Board: Robert F. Engle, University of California, USA
John Flemming, European Bank, UK
Lawrence R. Klein, University of Pennsylvania, USA
Helmut Liitkepohl, Humboidt University, Germany
The Economics of Pensions and Variable Retirement Schemes
Oliver Fabel
Applied General Equilibrium Modelling:
Imperfect Competition and European Integration
Dirk Willenbockel
Housing, Financial Markets and the Wider Economy
David Miles
Maximum Entropy Econometrics: Robust Estimation with Limited Data
Amos Golan, George Judge and Douglas Miller
Estimating and Interpreting the Yield Curve
Nicola Anderson, Francis Breedon, Mark Deacon,
Andrew D e r v and Gareth Murphy
Further titles in preparation
Proposals will be welcomed by the Series Editor
L Quantitative Financial Economi
Stocks, Bonds and Foreign Exchange
Keith Cuthbertson
Newcastle upon Tyne University
and
City University Business School
JOHN WILEY & SONS
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Data
Library of Congress Caf˜ginginPublicafion
Cuthbertson, Keith.
Quantitative financial economics : stocks, bonds, and foreign
exchange / Keith Cuthbertson.
cm.  (Series in financial economics and quantitative
p:
analysis)
Includes bibliographical references and index.
ISBN 0471953598 (cloth).  ISBN 0471953601 (pbk.)
1. Investments  Mathematical models. 2. Capital assets pricing
model. 3. Stocks  Mathematical models. 4. Bonds  Mathematical
models. 5 . Foreign exchange  Mathematical models. I. Title.
11. Series.
HG4515.2.C87 1996
95  48355
332.6  dc20
CIP
British Library Cataloguing in Publicafion Data
A catalogue record for this book is available from the British Library
ISBN 0471953598 (Cased) 0471953601 (Paperback)
Typeset in 10/12pt Times Roman by Laser Words, India
Printed and bound in Great Britain by Bookcraft (Bath) Ltd, Avon
This book is printed on acidfree paper responsibly manufactured from sustainable
forestation, for which at least two trees are planted for each one used for paper production.
I
Dedication
To June
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Contents
Series Preface
Introduction
Acknowledgements
Part 1 Returns and Valuation
1 Basic Concepts in Finance
1.1 Returns on Stocks, Bonds and Real Assets
1.2 Utility and Indifference Curves
1.3 Physical Investment Decisions and Optimal Consumption
1.4 Summary
Endnotes
2 The Capital Asset Pricing Model: CAPM
2.1 An Overview
2.2 Portfolio Diversification, Efficient Frontier and
the Transformation Line
2.3 Derivation of the CAPM
2.4 Summary
Appendix 2.1 Derivation of the CAPM
3 Modelling Equilibrium Returns
3.1 Extensions of the CAPM
3.2 A Simple MeanVariance Model of Asset Demands
3.3 Performance Measures
3.4 The Arbitrage Pricing Theory (APT)
3.5 Testing the Single Index Model, the CAPM and the APT
3.6 Summary
4 Valuation Models
4.1 The Rational Valuation Formula (RVF)
4.2 Summary
Endnotes
Further Reading
5.2 Implications of the EMH
5.3 Expectations, Martingales and Fair Game
5.4 Testing the EMH
5.5 Summary
Endnotes
Empirical Evidence on Efficiency in the Stock Market
6.1 Predictability in Stock Returns
6.2 Volatility Tests
6.3 Summary
Endnotes
Appendix 6.1
Rational Bubbles
7.1 Euler Equation and the Rational Valuation Formula
7.2 Tests of Rational Bubbles
7.3 Intrinsic Bubbles
7.4 Summary
Endnotes
Anomalies, Noise Traders and Chaos
8.1 The EMH and Anomalies
8.2 Noise Traders
8.3 Chaos
8.4 Summary
Appendix 8.1
Appendix 8.2
Endnote
Further Reading
Part 3 The Bond Market
9 Bond Prices and the Term Structure of Interest Rates
9.1 Prices, Yields and the RVF
9.2 Theories of the Term Structure
9.3 Summary
Endnotes
10 Empirical Evidence on the Term Structure
10.1 The Behaviour of Rates of Return
10.2 Pure Discount Bonds
10.3 Coupon Paying Bonds: Bond Prices and the Yield to Maturity
10.4 Summary
Appendix 10.1 Is the Long Rate a Martingale?
Appendix 10.2 Forward Rates
Endnotes
Further Reading
11.2 Purchasing Power Parity (PPP)
11.3 Interrelationships between CIP, UIP and PPP
11.4 Summary
Appendix 11.1 PPP and the WagePrice Spiral
12 Testing CIP, UIP and FRU
12.1 Covered Interest Arbitrage
12.2 Uncovered Interest Parity and Forward Rate Unbiasedness
12.3 Forward Rate: Risk Aversion and Rational Expectations
12.4 Exchange Rates and News
12.5 Peso Problems and Noise Traders
12.6 Summary
Appendix 12.1 Derivation of Famaâ€™s Decomposition of the Risk Premium
in the Forward Market
13 The Exchange Rate and Fundamentals
13.1 FlexPrice Monetary Model
13.2 StickyPrice Monetary Model (SPMM)
13.3 Dornbusch Overshooting Model
13.4 Frankel Real Interest Differential Model (RIDM)
13.5 Testing the Models
13.6 Chaos and Fundamentals
13.7 Summary
Further Reading
Part 5 Tests of the EMH using the VAR Methodology
14 The Term Structure and the Bond Market
14.1 Crossequation Restrictions and Informational Efficiency
14.2 The VAR Approach
14.3 Empirical Evidence
14.4 Summary
Endnotes
15 The FOREX Market
15.1 Efficiency in the FOREX Market
15.2 Recent Empirical Results
15.3 Summary
Endnotes
16 Stock Price Volatility
16.1 Theoretical Issues
16.2 Stock Price Volatility and the VAR Methodology
16.3 Empirical Results
16.4 Persistence and Volatility
Appendix 16.1 Returns, Variance Decomposition and Persistence
17 Risk Premia: The Stock Market
17.1 What Influences Stock Market Volatility?
17.2 The Impact of Risk on Stock Returns
17.3 Summary
18 The MeanVariance Model and the CAPM
18.1 The MeanVariance Model
18.2 Tests of the CAPM Using Asset Shares
18.3 Summary
19 Risk Premia and the Bond Market
19.1 Time Varying Risk: Pure Discount Bonds
19.2 Time Varying Risk: LongTerm Bonds
19.3 Interaction Between Stock and Bond Markets
19.4 Summary
Endnotes
Further Reading
Part 7 Econometric Issues in Testing Asset Pricing Models
20 Economic and Statistical Models
20.1 Univariate Time Series
20.2 Multivariate Time Series Models
20.3 Simple ARCH and GARCH Models
20.4 Rational Expectations: Estimation Issues
Further Reading
References
Index
I
Series Preface I
This series aims to publish books which give authoritative accounts of major ne
financial economics and general quantitative analysis. The coverage of the seri
both macro and micro economics and its aim is to be of interest to practi
policymakers as well as the wider academic community.
The development of new techniques and ideas in econometrics has been rap
years and these developments are now being applied to a wide range of areas an
Our hope is that this series will provide a rapid and effective means of com
these ideas to a wide international audience and that in turn this will contri
growth of knowledge, the exchange of scientific information and techniqu
development of cooperation in the field of economics.
St
Imperial College, L
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I
Introduction I
This book has its genesis in a final year undergraduate course in Financia
although parts of it have also been used on postgraduate courses in quantitat
of the behaviour of financial markets. Participants in these courses usually h
what heterogeneous backgrounds: some have a strong basis in standard und
economics, some in applied finance while some are professionals working
institutions. The mathematical and statistical knowledge of the participants in th
is also very mixed. My aim in writing the book is to provide a selfcontaine
introduction to some of the theories and empirical methods used by financial ec
the analysis of speculative assets prices in the stock, bond and foreign exchan
It could be viewed as a selective introduction to some of the recent journa
in this area, with the emphasis on applied work. The content should enable
to grasp that although much of this literature is undoubtedly very innovative
grounded in some fairly basic intuitive ideas. It is my hope that after reading
students and others will feel confident in tackling the original sources.
The book analyses a number of competing models of asset pricing and the me
to test these. The baseline paradigm throughout the book is the efficient market
EMH. If stock prices always fully reflect the expected discounted present valu
dividends (i.e. fundamental value) then the market will allocate funds among
firms, optimally. Of course, even in an efficient market, stock prices may be hig
but such volatility does not (generally) warrant government intervention since
the outcome of informed optimising traders. Volatility may increase risk (of b
for some financial institutions who hold speculative assets, yet this can be m
portfolio diversification and associated capital adequacy requirements.
Part 1begins with some basic definitions and concepts used in the financial
literature and demonstrates the â€˜separation principleâ€™ in the certainty case.
period) Capital Asset Pricing Model (CAPM) and (to a much lesser extent) th
Pricing Theory (APT) provide the baseline models of equilibrium asset retu
two models, presented in Chapters 2 and 3, provide a rich enough menu t
many of the empirical issues that arise in testing the EMH. It is of course
made clear that any test of the EMH is a joint test of an equilibrium returns
rational expectations (RE). Also in Part 1, the theoretical basis of the CAP
variants, including the consumption CAPM), the APT and some early emp
of these models are discussed, and it is concluded with an examination, in
of the relationship between returns and prices. It is demonstrated that any
In Part 2, Chapter 5, the basic assumptions and mathematical formulat
REEMH approach are outlined. One view of the EMH is that equilibri
returns are unpredictable, another slightly different interpretation is that one ca
persistent abnormal profits after taking account of transactions costs and adju
risk. In Chapter 6, an examination is made of a variety of statistical tests wh
establish whether stock returns (over different holding periods) are predictabl
whether one can exploit this predictability to earn â€˜abnormalâ€™ profits. This
by a discussion of the behaviour of stock prices and whether these are
solely by fundamentals or are excessively volatile. When discussing â€˜vola
it is possible to highlight some issues associated with inference in small sa
problems encountered in the presence of nonstationary data. The usefulness
Carlo methods in illuminating some of these problems is also examined. Th
evidence in Part 2 provides the reader with an overview of the difficultie
establishing firm conclusions about competing hypotheses. However, at a m
prima facie case is established that when using fairly simple models, the EM
adequately capture the behaviour of stock prices and returns.
It is well known that stock returns may contain a (rational) bubble whic
dictable, yet this can lead to a discrepancy between the stock price and fundam
Such bubbles are a â€˜selffulfilling prophecyâ€™ which may be generated exogenou
depend on fundamentals such as dividends (i.e. intrinsic bubbles). The intrin
is â€˜anchoredâ€™ to dividends and if dividends are fairly stable then the actual
might not differ too much from its fundamental value. However, the divide
may be subject to â€˜regime changesâ€™ which can act as a catalyst in generatin
in an intrinsic bubble. Periodically collapsing bubbles are also possible: when
is positive, stock prices and fundamentals diverge, but after the â€˜collapseâ€™ the
brought into equality. These issues are addressed in Chapter 7 which also asses
the empirical evidence supports the presence of rational bubbles.
Stock market â€˜anomaliesâ€™ and models of noise trader behaviour are d
Chapter 8, the final chapter in Part 2. The evidence on â€˜anomaliesâ€™ in the stoc
voluminous and students love providing a â€˜listâ€™ of them in examination answ
they are invaluable pieces of evidence, which may be viewed as being comple
the statistical/regressionbasedapproaches, I have chosen to â€˜listâ€™ only a few o
ones, since the analytic content of these studies is usually not difficult for
to follow, in the original sources. Such anomalies highlight the potential im
noise traders, who follow â€˜fads and fashionsâ€™ when investing in speculative a
asset prices are seen to be the outcome of the interaction between â€˜smart mon
and â€˜noise tradersâ€™. The relative importance of these two groups in particular m
at particular times may vary and hence prices may sometimes reflect fundam
and at other times may predominantly reflect fads and fashions.
There are several approaches to modelling noise trader behaviour. For exa
are based on maximising an explicit objective function, while others involve
responses to market signals. As soon as one enters the domain of nonlinear
possibility of chaotic behaviour arises. It is possible for a purely (nonlinear) de
in the near future, relative to those in the more distant future, when pricing sto
are therefore mispriced and physical investment projects with returns over a sh
are erroneously preferred to those with long horizon returns, even though the
a higher expected net present value. Illustrative models which embody the a
are presented in Chapter 8, along with some empirical tests.
Overall, the impression imparted by the theoretical models and empiri
presented in Part 2 is that for the stock market, the EMH under the assumptio
invariant risk premium may not hold, particularly for the post1950s period.
the reader is made aware that such a conclusion is by no means clear cut and
sophisticated tests are to be presented in Parts 5 and 6 of the book. Through
it is deliberately shown how an initial hypothesis and tests of the theory of
the unearthing of further puzzles, which in turn stimulates the search for e
theoretical models or improved data and test procedures. Hence, by the end of
reader should be well versed in the basic theoretical constructs used in anal
prices and in testing hypotheses using a variety of statistical techniques.
Part 3 examines the EMH in the context of the bond market. Chapter 9 o
various hypotheses of the term structure of interest rates applied to spot yield
period yields and the yield to maturity and demonstrates how these are interr
dominant paradigms here are the expectations hypothesis and the liquidity
hypothesis, both of which assume a time invariant term premium. Chapter 10
empirical tests of the competing hypotheses, for the short and long ends of th
spectrum. In addition, cointegration techniques are used to examine the comp
rity spectrum. On balance, the results for the bond market (under a time inv
premium) are found to be in greater conformity with the EMH than are the
the stock market (as reported in Part 2). These differing results for these two
asset markets are reexamined later in the book.
Part 4 examines the FOREX market and in particular the behaviour of spot a
exchange rates. Chapter 11 begins with a brief overview of the relationshi
covered and uncovered interest parity, purchasing power parity and real interest
Chapter 12 is mainly devoted to testing covered and uncovered interest parity a
rate unbiasedness. The degree to which the apparent failure of forward rate un
may be due to a failure either of rational expectations or of risk neutrality is
The difficulty in assessing â€˜efficiencyâ€™ in the presence of the socalled Peso pr
the potential importance of noise traders (or chartists) are both discussed, in the
illustrative empirical results presented. In the final section of Part 4, in Chapter
theories of the behaviour of the spot exchange rate based on â€˜fundamentalsâ€™
flexprice and stickyprice monetary models, are outlined. These monetary mod
pursued at great length since it soon becomes clear from empirical work that
for periods of hyperinflation) these models, based on economic fundamentals, ar
deficient. The final chapter of Part 4 therefore also examines whether the â€˜sty
of the behaviour of the spot rate may be explained by the interaction of no
and smart money and, in one such model, chaotic behaviour is possible. T
conclusion is that the behaviour of spot and forward exchange rates is little u
of market participants (although space constraints prevent a discussion of
models).
In Part 5 the EMH using the VAR methodology is tested. Chapter 14 begi
term structure of interest rates and demonstrates how the VAR equations can
provide a time series for the forecast of (a weighted average of) future chang
term rates of interest, which can then be compared with movements in the
spread, using a variety of metrics. Under the null of the expectations hypothes
yields a set of crossequation parameter restrictions. These restrictions are sho
an intuitive interpretation, namely, that forecast errors are independent of inform
in generating the forecast and that no abnormal profits can be made. Having
the basic principles behind the VAR methodology it is then possible succinc
with its application to the FOREX (Chapter 15) and stock market (Chapter
are two further interesting aspects to the VAR methodology applied to the sto
First, the VAR methodology is useful in establishing links between early emp
that looked at the predictably of oneperiod returns and multiperiod return
that examined volatility tests on stock prices. Second, the link between the per
oneperiod returns and the volatility of stock prices is easily examined withi
framework. Broadly speaking the empirical results based on the VAR approa
that the stock and FOREX markets (under a time invariant risk premium) do n
to the EMH, while for the bond market the results are more in conformity
some puzzles still remain.
Part 6 examines the potential impact of time varying risk premia in the
bond markets. If returns depend on a time varying risk premium which is pers
sharp movements in stock prices may ensue as a result of shocks to such pre
observed price movements may not be â€˜excessivelyâ€™ volatile. An analysis is m
usefulness of the CAPM with time varying variances and covariances which a
by ARCH and GARCH processes. This framework is applied to both the (in
stock and bond markets. There appears to be more support for a time varyin
type) risk premium influencing expected returns in the stock market than i
market. Some unresolved issues are whether such effects are stable over ti
robust to the inclusion of other variables that represent trading conditions (e
in the market).
As the book progresses, the reader should become aware that to establish
particular speculative market is efficient, in the sense that either no excess
profits can be earned or that market price reflects economic fundamentals,
straightforward. It often requires the use of sophisticated statistical tests man
have only recently appeared in the literature. Data on asset prices often exhi
and such â€˜nonstationary â€™ data require analysis using concepts from the li
unit roots and cointegration  otherwise grossly misleading inferences may e
readers will also be aware that, although the existence of time varying risk
always been acknowledged in the theoretical finance literature, it is only re
empirical work has been able to make advances in this area using ARCH an
book. However, I did not want these issues to dominate the book and â€˜crow
economic and behavioural insights. I therefore decided that the best way forw
the heterogeneous background of the potential readership of the book, was to
overview of the purely statistical aspects in a selfcontained section (Part 7)
of the book. This has allowed me to limit my comments on the statistical nu
minimum, in the main body of the text. A prerequisite for understanding Pa
be a final year undergraduate course or a specialist option on an MBA in a
series econometrics.
Naturally, space constraints imply that there are some interesting areas tha
omitted. To have included general equilibrium and other â€˜factor modelsâ€™ of
returns based on continuous time mathematics (and associated econometric p
would have added considerably to the mathematical complexity and length o
While continuous time equilibrium models of the term structure would have
useful comparison to the discrete time approach adopted, I nevertheless felt i
to exclude this material. This also applies to some material I initially wrote on
futures  I could not do justice to these topics without making the book inordi
and there are already some very good specialist, academically oriented texts i
I also do not cover the recent burgeoning theoretical and applied literature
microstructureâ€™ and applications of neural networks to financial markets.
Readership
In order to make the book as selfcontained as possible and noting the often sh
of even some central concepts in the minds of some students, I have included
basic theoretical material at the beginning of the book (e.g. the CAPM and i
the APT and valuation models). As noted above, I have also relegated detaile
issues to a separate chapter. Throughout, I have kept the algebra as simple as p
usually I provide a simple exposition and then build up to the more general
I hope will allow the reader to interpret the algebra in terms of the econom
which lies behind it. Any technically difficult issues or tedious (yet important)
I relegate to footnotes and appendices. The empirical results presented in th
merely illustrative of particular techniques and are not therefore meant to be
In some cases they may not even be representative of â€˜seminal contributionsâ€™,
are thought to be too technically advanced for the intended readership. As
will already have gathered, the empirics is almost exclusively biased towards
analysis using discrete time data.
This book has been organised so that the â€˜average studentâ€™ can move fr
to more complex topics as he/she progresses through the book. Theoretical
constructs are developed to a particular level and then tests of these ideas are
By switching between theory and evidence using progressively more difficu
the reader becomes aware of the limitations of particular approaches and ca
this leads to the further development of the theories and test procedures. Hen
less adventuresome student one could end the course after Part 4. On the o
the advanced student would probably omit the more basic material in Part 1
approach. In a survey article, one often presents a general framework from w
other models may be viewed as special cases. This has the merit of great el
it can often be difficult for the average student to follow, since it requires an
understanding of the general model. My alternative approach, I believe, is to b
on pedagogic grounds but it does have some drawbacks. Most notably, no
possible theoretical approaches and empirical evidence for a particular marke
stocks, bonds or foreign exchange, appear in one single chapter. However, this i
and I can only hope my ordering of the material does not obscure the underlyin
approaches that may be applied to all speculative markets.
The book should appeal to the rising undergraduate final year, core financ
area and to postgraduate courses in financial economics, including electives o
MBA finance courses. It should also provide useful material for those wor
research departments of large financial institutions (e.g. investment banks, pen
and central and commercial banks). The book covers a number of impor
advances in the financial markets area, both theoretical and econometric/empiri
innovative areas that are covered include chaos, rational and intrinsic bubbles
action of noise traders and smart money, shorttermism, anomalies, predict
VAR methodology and time varying risk premia. On the econometrics sid
of nonstationarity, cointegration, rational expectations, ARCH and GARCH
examined. These issues are discussed with empirical examples taken from the
and FOREX markets.
Professional traders, portfolio managers and policymakers will, I hope, fin
of interest because it provides an overview of some of the theoretical mod
explaining the determination of asset prices and returns, together with the
used to assess their empirical validity. The performance of such models provid
input to key policy issues such as capital adequacy proposals (e.g. for securiti
the analysis of mergers and takeovers and other aspects of trading arrangeme
margin requirements and the use of trading halts in stock markets. Also, to
that monetary policy works via changes in interest rates across the maturity sp
changes in the exchange rate, the analysis of the bond and FOREX markets
relevance. At a minimum the book highlights some alternative ways of exa
behaviour of asset prices and demonstrates possible pitfalls in the empirical
these markets.
I remember, from reading books dealing with the development of quantum
that for several years, even decades, there would coexist a number of competing
the behaviour of elementary particles. Great debates would ensue, where often
than lightâ€™ would be generated  although both could be construed as manife
(intellectual) energy. What becomes clear, to the layman at least, is that as one
closer to the â€˜microbehaviourâ€™ of the atom, the more difficult it becomes to
the underlying physical processes at work. These controversies in natural sc
me a little more sanguine about disputes that persist in economics. We know (
think we know) that in a risky and uncertain world our â€˜simpleâ€™ economic mode
not work terribly well. Even more problematic is our lack of data and inability
analysis of speculative asset prices and I hope this is reflected in the material i
It has been said that some write so that other colleagues can better unders
others write so that colleagues know that only they understand. I hope this
achieve the former aim and will convey some of the recent advances in t
of speculative asset prices. In short, I hope it ameliorates the learning proce
stimulates others to go further and earns me a modicum of â€˜holiday moneyâ€™.
if the textbook market were (instantaneously) efficient, there would be no ne
book  it would already be available from a variety of publishers. My expe
success are therefore based on a view that the market for this type of book is no
and is currently subject to favourable fads.
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Acknowledgements
I have had useful discussions and received helpful comments on various cha
many people including: David Ban, George Bulkley , Charles Goodhart, Davi
Eric Girardin, Louis Gallindo, Stephen Hall, Simon Hayes, David Miles, Mich
Dirk Nitzsche, Barham Pesaran, Bob Shiller, Mark Taylor, Dylan Thomas, Ian
Mike Wickens. My thanks to them and naturally any errors and omissions a
me. I also owe a great debt to Brenda Munoz who expertly typed the various
to my colleagues at the University of Newcastle and City University Busine
who provided a conducive working environment.
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PART1
1
I
L I
Returns and Valuation
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1
Basic Concepts in Finance
The aim in this chapter is to quickly run through some of the basic tools of an
in the finance literature. The topics covered are not exhaustive and they are d
a fairly intuitive level. The topics covered include
Compounding, discounted present value DPV, the rate of return on pu
bonds, coupon paying bonds and stocks.
Utility functions, indifference curves, measures of risk aversion, and in
utility .
The use of DPV in determining the optimal level of physical investme
optimal consumption stream for a twoperiod horizon problem.
1.1 RETURNS ON STOCKS, BONDS AND REAL ASS
Much of the theoretical work in finance is conducted in terms of compound rat
or interest rates even though rates of interest quoted in the market use â€˜simp
For example, an interest rate of 5 percent payable every six months will be
simple interest rate of 10 percent per annum in the market. However, if an inv
over two sixmonth bills and the interest rate remained constant, he could ac
a â€˜compoundâ€™ or â€˜trueâ€™ or â€˜effectiveâ€™ annual rate of (1.05)* = 1.1025 or 10.
The effective annual rate of return exceeds the simple rate because in the form
investor earns â€˜interestoninterestâ€™.
We now examine how to calculate the terminal value of an investmen
frequency with which interest rates are compounded alters. Clearly, a quoted
of 10 percent per annum when interest is calculated monthly will amount to
end of the year than if interest accrues only at the end of the year.
Consider an amount $x invested for n years at a rate of R per annum (
expressed as a decimal). If compounding takes place only at the end of th
future value after n years is FV, where
R / m is often referred to as the periodic interest rate. As rn, the frequency of com
increases the rate becomes continuously compounded and it may be shown that
where exp = 2.71828. For example, using (1.2) and (1.3) if the quoted (simp
rate is 10 percent per annum then the value of $100 at the end of one ye
for different values of m is given in Table 1.1. For practical purposes daily co
gives a result very close to continuous compounding (see the last two entries in
We now consider how we can switch between simple interest rates, per
effective annual rates and continuously compounded rates. Suppose an investm
periodic interest rate of 2 percent each quarter. This will usually be quoted in
as 8 percent per annum, that is, as a simple annual rate. The effective ann
exceeds the simple rate because of the payment of interestoninterest. At the
year $x = $100 accrues to
The effective annual rate R f is clearly 8.24 percent since
$100(1+ R f ) = 108.24
The relationship between the quoted simple rate R with payments m times p
the effective annual rate Rf is
]: m
[l + R f ] = [ 1 +
We can use equation (1.6) to move from periodic interest rates to effectiv
vice versa. For example, an interest rate with quarterly payments that would
effective annual rate of 12 percent is given by
34
+
1.12 = [l

R = [(l.l2)lI4 l] 4 = 0.0287 x 4 = (11.48 percent)
'hble 1.1 Compounding Frequency
Compounding Value of $100
Frequency at End of Year ( R = 10%p.a.)
Annually (rn = 1) 110.00
Quarterly (rn = 4) 110.38
Weekly (rn = 52) 110.51
Daily (rn = 365) 110.52
Continuous (n = 1) 110.517
compounded rate, Re. One reason for doing this calculation is that much of th
theory of bond pricing (and the pricing of futures and options) uses co
compounded rates.
Suppose we wish to calculate a value for R, when we know the mperiod ra
the terminal value after n years of an investment of $A must be equal when u
interest rate we have
Aexp(R,.n)=A
and
Also, if we are given the continuously compounded rate R, we can use the abo
to calculate the simple rate R which applies when interest is calculated m time
R = rn[exp(R,/m)  11
We can perhaps best summarise the above array of alternative interest rate
one final illustrative example. Suppose an investment pays a periodic inter
5 percent every six months (m = 2, R/2 = 0.05). In the market, this would
as 10 percent per annum and clearly the 10 percent represents a simple annu
+
investment of $100 would yield lOO(1 (0.1/2))2= $110.25 after one year (u
Clearly the effective annual rate is 10.25 percent per annum. Suppose we wish
the simple annual rate of R = 0.10 to an equivalent continuously compounded
+
(1.9) with rn = 2 we see that this is given by Re = 2  ln(1 0.10/2) = 0.09
percent per annum). Of course, if interest is continuously compounded at an
of 9.758 percent then $100 invested today would accrue to exp(R,n) = $110.2
years' time.
Discounted Present Value (DPV)
Let the annual rate of interest on a completely safe investment over n years
r d n ) .The future value of $x in n years' time with interest calculated annually
It follows that if you were given the opportunity to receive with certainty $
years time then you would be willing to give up $x today. The value today o
payment of FV, in n years time is $x. In more technical language the discoun
value (DPV) of FV, is
DPV = FVn
+
(1 r d n ) ) n
We now make the assumption that the safe interest rate applicable to 1 , 2 , 3
horizons is constant and equal to r . We are assuming that the term structure
Physical Investment Project
Consider a physical investment project such as building a new factory whic
of prospective net receipts (profits) of FVi. Suppose the capital cost of the pro
we assume all accrues today (i.e. at time t = 0) is $ K C . Then the entrepren
invest in the project if
DPV 2 KC
or equivalently if the net present value (NPV) satisfies
NPV = DPV KC 2 0
If NPV = 0 it can be shown that the net receipts (profits) from the investment
just sufficient to pay back both the principal ( $ K C ) and the interest on the l
was taken out to finance the project. If NPV > 0 then there are surplus fund
even after these loan repayments.
As the cost of funds r increases then the NPV falls for any given stream
FVi from the project (Figure 1.1).There is a value of r (= 10 percent in Fig
which the NPV = 0. This value of r is known as the internal rate of return (
investment. Given a stream of net receipts F Vi and the capital cost KC for a p
can always calculate a projectâ€™s IRR.It is that constant value of y for which
FVi
KC=C
+
i=l (1 Y)â€™
An equivalent investment rule to the NPV condition (1.15) is then to in
project ifcl)
IRR(= y) 2 cost of borrowing (= r)
Figure 1.1 NPV and the Discount Rate.
Suppose, however, that 'oneyear money' carries an interest rate of rd'), twoy
costs rd2),etc. Then the DPV is given by
n
i=l
+
where Sj = (1 di))'. rd') are known as spot rates of interest since t
The
rates that apply to money lent over the periods rdl) = 0 to 1 year, rd2) = 0
etc. (expressed at annual compound rates). The relationship between the spot
on default free assets is the subject of the term structure of interest rates. Fo
if rdl) < rd2) c r d 3 ) .. . then the yield curve is said to be upward sloping
formula can also be expressed in real terms. In this case, future receipts FVi a
by the aggregate goods price index and the discount factors are then real rates
In general, physical investment projects are not riskless since the future r
uncertain. There are a number of alternative methods of dealing with uncerta
DPV calculation. Perhaps the simplest method and the one we shall adopt has t
rate Sj consisting of the riskfree spot rate rd') plus a risk premium rpi:
Equation (1.19) is an identity and is not operational until we have a model
premium (e.g. rpj is constant for all i). We examine some alternative model
Chapter 3.
Pure Discount Bonds and Spot Yields
Instead of a physical investment project consider investing in a pure discount
coupon bond). At short maturities, these are usually referred to as bills (e.g
bills). A pure discount bond has a fixed redemption price M1, a known matu
and pays no coupons. The yield on the bill if held to maturity is determined
that it is purchased at a market price Pt below its redemption price M1. For
bill it seems sensible to calculate the yield or interest rate as:
where rsj') is measured as a proportion. However, viewing the problem in term
we see that the oneyear bill promises a future payment of M1 at the end of
exchange for a capital cost paid out today of PI,.Hence the IRR, ylt of the
calculated from
and hence the oneyear spot yield rsf') is simply the IRR of the bill. Applying
principal to a twoyear bill with redemption price M 2 , the annual (compou
rate rsi2) on the bill is the solution to
which implies
rst(2)= [M2/P2,I1'2  1
We now see how we can, in principle, calculate a set of (compound) spo
different maturities from the market prices of pure discount bonds (bills). O
practical issue the reader should note is that, in fact, dealers do not quote the
rates r#)(i = 1,2, 3 , 4 . . .) but the equivalent simple interest rates. For exam
periodic interest rate on a sixmonth bill using (1.20) is 5 percent, then the quot
be 10 percent. However, we can always convert the periodic interest rate to an
compound annual rate of 10.25 percent or, indeed, into a continuously compo
of 9.758, as outlined above.
Coupon Paying Bonds
A level coupon (noncallable) bond pays a fixed coupon $C at known fixe
(which we take to be every year) and has a fixed redemption price Mn payabl
bond matures in year n. For a bond with n years left to maturity let the cur
price be P f " ) .The question is how do we measure the return on the bond
to maturity? The bond is analogous to our physical investment project with
outlay today being Pf"' and the future receipts being $C each year (plus the
price). The internal rate of return on the bond, which is called the yield to m
can be calculated from
+... C + M n
C C
p("' =
+
˜
(1 + R : ) k (1 + R r ) 2 (1 R:)"
The yield to maturity is that constant rate of discount which at a point in ti
the DPV of future payments with the current market price. Since P("',Mn an
known values in the market, equation (1.24) has to be solved to give the qu
rate for the yield to maturity RY. There is a subscript 't' on R because as
Y
price falls, the yield to maturity rises (and vice versa) as a matter of actuaria
in equation (1.24). Although widely used in the market and in the financial pre
some theoretical/conceptual problems in using the yield to maturity as an un
measure of the return on a bond even when it is held to maturity. We deal w
these issues in Part 3.
In the market, coupon payments C are usually paid every six months and
rate from (1.24) is then the periodic sixmonth rate. If this periodic yield
is calculated as say 6 percent, then in the market the quoted yield to matu
pricing and return on bonds.
Aperpetuity is a level coupon bond that is never redeemed by the primary
n + 00). If the coupon is $C per annum and the current market price of the b
then a simple measure of the return Rfm) is the flat yield:
R(@ = c/pj"'
This simple measure is in fact also the yield to maturity for a perpetuity, since
in (1.24) then it reduces to (1.25). It is immediately obvious from (1.25) tha
changes, the percentage change in the price of a perpetuity equals the percent
in the yield to maturity.
Holding Period Return
Much empirical work on stocks deals with the oneperiod holding period re
which is defined as
[
Pr+l  Dr+l
pr p r ] + pt
Hr+l =
The first term is the proportionate capital gain or loss (over one period) and
term is the (proportionate) dividend yield. H t + l can be calculated expost but
viewed from time t, Pr+l and (perhaps) D1+l are uncertain and investors can o
forecast these elements. It also follows that
+ ++
where Ht+j is the one period return between t i and t i 1. Hence, expo
invested in the stock (and all dividend payments are reinvested in the stock) t
payout after n periods is
Beginning with Chapter 4 and throughout the book we will demonstrate how
oneperiod returns H r + l can be directly related to the DPV formula. Much o
empirical work on whether the stock market is efficient centres on trying t
whether oneperiod returns H r + l are predictable. Later empirical work conce
whether the stock price equalled the DPV of future dividends and the most recen
work brings together these two strands in the empirical literature.
With slight modifications the oneperiod holding period return can be defin
asset. For a coupon paying bond with initial maturity of n periods and coupo
of C we have
and is referred to as the (oneperiod) holding period yield (HPY). The first
capital gain on the bond and the second is the coupon (or running) yield. Broadl
stocks
The difficulty with direct application of the DPV concept to stocks is that futur
namely the dividends, are uncertain. Also because the future dividend payment
tain these assets are risky and one therefore might not wish to discount all futur
some constant riskfree interest rate. It can be shown (see Chapter 4) that if t
oneperiod holding period return E,Hf+I equals qt then the fundamental value
can be viewed as the DPV of expected future dividends E,D,+j deflated by the
discount factors (which are likely to embody a risk premium). The fundamen
+
+
In (1.30) qi is the oneperiod return between time period t i  1 and t i(
If there are no systematic profitable opportunities to be made from buying
shares between wellinformed rational traders, then the actual market price o
P, must equal fundamental value V , , that is, the DPV of expected future div
example, if P, < V f then investors should purchase the undervalued stock and
a capital gain as P, rises towards V,.In an efficient market such profitable o
should be immediately eliminated.
Clearly one cannot directly calculate V , to see if it does equal P,because ex
dends (and discount rates) are unobservable. However, in Chapters 6 and 16
methods of overcoming this problem and examine whether the stock market is
the sense that P, = V,. Also if we add some simplifying assumptions to the D
(e.g. future dividends are expected to be constant) then it can be used in a rela
manner to calculate V , and assess whether shares are under or overvalued
to their current market price. Such models are usually referred to as dividen
models (see Elton and Gruber (1987)) and are dealt with in Chapter 4.
1.2 UTILITY AND INDIFFERENCE CURVES
In this section we briefly discuss the concept of utility but only to a level su
reader can follow the subsequent material on portfolio choice.
Economists frequently set up portfolio models where the individual choo
assets in order to maximise either some monetary amount such as profits or
returns on the portfolio or the utility (satisfaction) that such assets yield. For
certain level of wealth will imply a certain level of satisfaction for the indiv
contemplates the goods and services he could purchase with the wealth. If h
doubled his level of satisfaction may not be. Also, for example, if the individua
one bottle of wine per night the additional satisfaction from consuming an
may not be as great as from the first. This is the assumption of diminishin
utility. Utility theory can also be applied to decisions involving uncertain o
fact we can classify investors as â€˜risk avertersâ€™, â€˜risk loversâ€™ or â€˜risk neutralâ€™
the shape of their utility function. Finally, we can also examine how indivi
Suppose W represents the possible outcomes of a football game, namely, win, lo
Suppose an individual attaches probabilities p ( W ) to these outcomes, that
N ( W ) / T where N ( W ) equals the number of wins, losses or draws in the
T = total number of games played. Finally, suppose the individual attaches
levels of satisfaction or utility U to win (= 4 units), lose (= 0 units) and draw
so that U(win) = 4, etc. Then his expected utility from the seasonâ€™s forthcomin
W
Uncertainty and Risk
The first restriction placed on utility functions is that more is always preferred
that Uâ€™( W) > 0 where U â€™ ( W )= aU(W)/aW. Now, consider a simple gamble o
$2 for a â€˜headâ€™ on the toss of a coin and $0 for tails. Given a fair coin th
monetary value of the risky outcome is $1:
+ (1/2)0 = $1
(1/2)2
Suppose it costs the investor $1 to â€˜investâ€™ in the game. The outcome from n
the game (i.e. not investing) is the $1 which is kept. Risk aversion means t
will reject a fair gamble; $1 for certain is preferred to an equal chance of $2
aversion implies that the second derivative of the utility function is negative U
To see this, note that the utility from not investing U(1) must exceed the expe
from investing
â€™ +
U(1) (1/2)U(2) (1/2)U(O)
or
U(1)  U ( 0 ) > U(2)  U(1)
so that the utility function has the concave shape given in Figure 1.2 marked â€˜ri
It is easy to deduce that for a risk lover the utility function is convex while
neutral investor who is just indifferent to the gamble or the certain outcome,
function is linear (i.e. the equality sign applies to equation (1.33)). Hence we
U â€ť ( W ) < 0 risk averse
U â€ť ( W )= 0 risk neutral
U â€ť ( W ) > 0 risk lover
A risk averse investor is also said to have diminishing marginal utility of w
additional unit of wealth adds less to utility the higher the initial level of w
U â€ť ( W ) < 0). The degree of risk aversion is given by the concavity of the utili
in Figure 1.2 and equivalently by the absolute size of Uâ€ť(W).Two measures of
Figure 1.2 Utility Functions.
of risk aversion are commonly used:
R A ( W ) is the ArrowPratt measure of absolute risk aversion, the larger is
greater the degree of risk aversion. R R ( W )is the coefficient of relative risk a
and RR are a measure of how the investorâ€™s risk preferences change with a
wealth. For example, assume an investor with $10000 happens to hold $50
assets. If his wealth were to increase by $10000 and he then put more than $5
into risky assets, he is said to exhibit decreasing absolute risk aversion. (The
of increasing and constant absolute risk aversion are obvioils.)
The natural assumption to make as to whether relative risk aversion is
increasing or constant i s less clear cut. Suppose you have 50 percent of y
(of $lOOQOO) in risky assets. If, when your wealth doubles, you increase the
held in risky assets then you are said to exhibit decreasing relative risk aversi
definitions app!y for constant and increasing relative risk aversion.) Different m
functions give rise to different implications for the form of risk aversion. F
the function
U ( W ) = In W
exhibits diminishing absslute risk aversion and constant relative risk aversio
Certain utility functions allow one to reduce the problem of maximising exp
to a problem involving only the maximisation of a function of expected return
risk of the return (measured by the variance) ch.For example, maximising t
absolute risk aversion utiiity fmction
E[U(W)] = E [ a  bexp(cW)]
risk aversion. Apart from the unobservable 'c' the maximand (1.38) is in t
mean and variance of the return on the portfolio: hence the term meanvarianc
However, the reader should note that in general maximising E U ( W ) cannot be
a maximisation problem in terms of He and afr only and often portfolio mod
at the outset that investors are concerned with the meanvariance maximan
discard any direct link with a specific utility function(3).
Indifference Curves
Although it is only the case under somewhat restrictive circumstances, let us
the utility function in Figure 1.2 for the risk averter can be represented solely
the expected return and the variance of the return on the portfolio. The link b
of period wealth W and investment in a portfolio of assets yielding an expe
+
l is W = (1 n ) W o where W Oequals initial wealth. However, we assume
l
function can be represented as
U1 > 0, U2 < 0, u11, U22 < 0
U = U(n',a;)
The sign of the firstorder partial derivatives ( U l , U2) imply that expected
to utility while more 'risk' reduces utility. The secondorder partial derivativ
diminishing marginal utility to additional expected 'returns' and increasing m
utility with respect to additional risk. The indifference curves for the above util
are shown in Figure 1.3.
""r

0,
2
Figure 1.3 Indifference Curves.
that at higher levels of risk, say at C, the individual requires a higher expe
(Câ€ť to Câ€ťâ€™ > Aâ€ť to Aâ€ťâ€˜) for each additional increment to risk he undertakes,
at A: the individual is â€˜risk averseâ€™.
The indifference curves in riskreturn space will be used when analysin
choice in the oneperiod CAPM in the next chapter and in a simple meanvari
in Chapter 3.
Intertemporal Utility
A number of economic models of individual behaviour assume that inves
utility solely from consumption goods. At any point in time, utility depends po
consumption and exhibits diminishing marginal utility
The utility function therefore has the same slope as the â€˜risk averterâ€™ in Figur
C replacing W). The only other issue is how we deal with consumption which
different points in time. The most general form of such an intertemporal life
function is
U N = u(Cr,C t + l , Cr+2  .  Cr+N)
However, to make the mathematics tractable some restrictions are usually pla
form of U ,the most common being additive separability with a constant sub
of discount 0 < 6 < 1:
It is usually the case that the functional form of U ( C , ) ,U(Cr+I),etc. are take
same and a specific form often used is
where d < 1. The lifetime utility function can be truncated at a finite valu
if N + 09 then the model is said to be an overlapping generations mod
individualâ€™s consumption stream is bequeathed to future generations.
The discount rate used in (1.41) depends on the â€˜tastesâ€™ of the individu
+
present and future consumption. If we define S = 1/(1 d) then d is kn
subjective rate of time preference. It is the rate at which the individual will sw
++
+
time t j for utility at time t j 1 and still keep lifetime utility constant. T
separability in (1.41) implies that the extra utility from say an extra consumptio
10 yearsâ€™ time is independent of the extra utility obtained from an identical c
bundle in any other year (suitably discounted).
L c
C
O
Figure 1.4 Intertemporal Consumption: Indifference Curves.
For the twoperiod case we can draw the indifference curves that foll
simple utility function of the form U = Czl C y ( 0 < CYI,a < 1) and these
2
in Figure 1.4. Point A is on a higher indifference curve than point B since at
vidual has the same level of consumption in period 1, C1 as at B, but at A, h
of consumption in period zero, CO.At point H if you reduce COby xo units t
individual to maintain a constant level of lifetime utility he must be compens
extra units of consumption in period 1, so he is then indifferent between poin
Diminishing marginal utility arises because at F if you take away xo units of
requires yl(> yo) extra units of C1 to compensate him. This is because at F h
with a lower initial level of CO than at H, so each unit of CO he gives up i
more valuable and requires more compensation in terms of extra C1.
The intertemporal indifference curves in Figure 1.4 will be used in
investment decisions under certainty in the next section and again when
the consumption CAPM model of portfolio choice and equilibrium asset ret
uncertainty.
1.3 PHYSICAL INVESTMENT DECISIONS AND OPTI
CONSUMPTION
Under conditions of certainty about future receipts the investment decisio
section 1.1 indicate that managers should rank physical investment projects
either to their net present value (NPV), or internal rate of return (IRR).
projects should be undertaken until the NPV of the last project undertaken e
or equivalently until IRR = r, the riskfree rate of interest. Under these cir
the marginal (last) investment project undertaken earns just enough net retur
to cover the loan interest and repayment of principal. For the economy a
undertaking real investment requires a sacrifice in terms of lost current co
output. Higher real investment implies that labour skills, manhours and mach
t = 0, devoted to producing new machines or increased labour skills, which
output and consumption but only in future periods. The consumption profile
prefer, atthemargin, consumption today rather than tomorrow. How can financ
through facilitating borrowing and lending ensure that entrepreneurs produce
level of physical investment (i.e. which yields high levels of future consump
and also allows individuals to spread their consumption over time accordi
preferences? Do the entrepreneurs have to know the preferences of individual
in order to choose the optimum level of physical investment? How can the
acting as shareholders ensure that the managers of firms undertake the 'correc
investment decisions and can we assume that financial markets (e.g. stock mark
funds are channelled to the most efficient investment projects? These quest
interaction between 'finance' and real investment decisions lie at the heart of
system. The full answer to these questions involves complex issues. Howev
gain some useful insights if we consider a simple two period model of the
decision where all outcomes are certain (i.e. riskless) in real terms (i.e. we a
price inflation). We shall see that under these assumptions a separation princip
If managers ignore the preferences of individuals and simply invest in pr
the NPV = 0 or IRR = r , that is, maximise the value of the firm, then this
given a capital market, allow each consumer to choose his desired consumpt
namely, that which maximises his individual welfare. There is therefore a
process or separation of decisions, yet this still allows consumers to max
welfare by distributing their consumption over time according to their pref
step one, entrepreneurs decide the optimal level of physical investment, disre
preferences of consumers. In step two, consumers borrow or lend in the cap
to rearrange the time profile of their consumption to suit their individual pref
explaining this separation principle we first deal with the production decisio
the consumers' decision before combining these two into the complete model
All output is either consumed or used for physical investment. The entrep
an initial endowment W O at time t = 0. He ranks projects in order of decre
using the riskfree interest rate r as the discount factor. By foregoing consump
obtains resources for his first investment project 10 = W O Cf'.The physical
in that project which has the highest NPV (or IRR) yields consumption output
Cil) (where C','' > C t ) ,see Figure 1.5). The IRR of this project (in terms of co
goods) is:
1 + IRR") = ˜ 1 1 1) 0 1 )
˜(
(
As he devotes more of his initial endowment W Oto other investment projects
NPVs then the internal rate of return (C1/Co) falls, which gives rise to the
opportunity curve with the shape given in Figure 1.5 (compare the slope at A
The first and most productive investment project has a NPV of
and
c (1) WO Consumption
in Period Zero
Figure 1.5 Production Possibility Curve. Note (AA" = BB").
Let us now turn to the financing problem. In the capital market, any two co
streams COand C1 have a present value (PV) given by:
hence
+ r)Co
V ( l + r )  (1
c =P
1
For a given value of PV, this gives a straight line in Figure 1.6 with a slop
+ r). Equation (1.45) is referred to as the money market line since it rep
(1
rate of return on lending and borrowing money in the financial market place.
+
an amount COtoday you will receive C1 = (1 r)Co tomorrow.
Our entrepreneur, with an initial endowment of WO, will continue to invest
assets until the IRR on the nth project just equals the riskfree market interest
IRR'") = r
A
Consumption
in Period One
*
Consumption
in Period Zero
Figure 1.6 Money Market Line.
current consumption of C and consumption at t = 1 of Cl; (Figure 1.6). At a
G
the right of X the slope of the investment opportunity curve (= IRR) exceeds
interest rate (= r ) and at points to the left of X, the opposite applies.
However, the optimal levels of consumption (Cg, C ; ) from the production de
not conform to those preferred by individual consumers. We now leave the
decision and turn exclusively to the consumerâ€™s decision.
Suppose the consumer has income accruing in both periods and this inco
has present value of PV. The consumption possibilities which fully exhaust t
(after two periods) are represented by:
+ C1
PV = CO
(1 +
We now assume that lifetime utility (satisfaction) of the consumer depends on
and there is diminishing marginal utility in both COand C1 (i.e. aU/aCi > 0, a
0, for i = 0, 1). The indifference curves are shown in Figure 1.7. To give up
C the consumer must be compensated with additional units of C1, if he is to m
O
initial level of utility. The consumer wishes to choose CO and C1 to maximi
utility subject to his budget constraint (1.48). Given his endowment PV, h
consumption in the two periods is (C;t*,C;*). general, the optimal production
In
investment plan which yields consumption (C8,C;) will not equal the consume
consumption profile (C;*,C;*).However, the existence of a capital market e
the consumerâ€™s optimal point can be attained. To see this consider Figure 1.8
The entrepreneur has produced a consumption profile (CC;,C;)which max
value of the firm. We can envisage this consumption profile as being paid
t
Consumption
in period One
c;
C; Consumption
in Period Zero
Figure 1.7 Consumersâ€™ Maximisation Problem.
e
C^
O Consumption
in Period Zero
Figure 1.8 Investing 10 and Lending â€˜Lâ€™ in the Capital Market.
owners of the firm in the form of (dividend) income. The present value of
flowâ€™ is PV* where
+
PV* = c;; CT/(l+ r )
This is, of course, the â€˜incomeâ€™ given to our individual consumer as owner o
But, under conditions of certainty, the consumer can â€˜swapâ€™ this amount PV
combination of consumption that satisfies
Given PV* and his indifference curves (i.e. tastes or preferences) in Figure
then borrow or lend in the capital market at the riskless rate r to achieve that co
C:*,Cy* which maximises his own utility function.
Thus, there is a separation of investment and financing (borrowing and len
sions. Optimal borrowing and lending takes place independently of the physical
decision. If the entrepreneur and consumer are the same person(s), the separ
ciple still applies. The investor (as we now call him) first decides how much
initial endowment W Oto invest in physical assets and this decision is indepen
own (subjective) preferences and tastes. This first stage decision is an objecti
tion based on comparing the internal rate of return of his investment project
riskfree interest rate. His second stage decision involves how much to borrow
the capital market to â€˜smooth outâ€™ his desired consumption pattern over time.
decision is based on his preferences or tastes, at the margin, for consumption to
(more) consumption tomorrow.
Much of the rest of this book is concerned with how financing decisions
when we have a risky environment. The issue of how shareholders ensure tha
act in the best interest of the shareholders, by maximising the value of the fi
under the heading of corporate control mechanisms (e.g. mergers, takeovers). Th
of corporate control is not directly covered in this book. Consideration is g
under a risky environment.
( 0 In a risky environment a somewhat different separation principle applies.
each investor when choosing his portfolio of risky marketable assets (
bonds) will hold risky assets in the same proportion as all other investors,
of his preferences of risk versus return. Having undertaken this first stag
each investor then decides how much to borrow or lend in the money
the riskfree interest rate. This separation principle is the basis of optim
choice and of the capital asset pricing model (CAPM) which provides
equilibrium asset returns.
(ii) The optimal amount of borrowing and lending in the money market in
case occurs where the individualâ€™s subjective marginal rate of substitutio
+
for current consumption (i.e. (dCl/dCo)u) equals (1 r), where r
opportunity cost of money. Under uncertainty a parallel condition appli
that the individualâ€™s subjective tradeoff between expected return and ri
to the market price of risk.
(iii) Under certainty, the slope of the money market (or present value) line m
price (interest rate) of money, or equivalently the increase in value of $
today in the money market. Under risky investment opportunities the s
socalled capital market line (CML) along which investors can borrow
provides a measure of the market price of risk.
1.4 SUMMARY
In this chapter we have developed some basic tools for analysing in financi
There are many nuances on the topics discussed which have not been elaborat
and in future chapters these omissions will be rectified.
The main conclusions to emerge are:
Market participants generally quote â€˜simpleâ€™ annual interest rates but these
be converted to effective annual (compound) rates or to continuously co
rates.
The concepts of DPV and IRR can be used to analyse physical investme
and provide measures of the return on bills and bonds.
Theoretical models often have either returns and risk or utility as their m
Utility functions and their associated indifference curves can be used to rep
averters, risk lovers and risk neutral investors.
Under certainty, a type of separation principle applies. Managers can cho
ment projects to maximise the value of the firm and disregard investor p
Then investors are able to borrow and lend to allocate consumption betw
and â€˜tomorrowâ€™ in order to maximise their satisfaction (utility).
2. There are two points worth noting at this point. First, the expectations ope
applied to the whole of the RHS expression in (1.30). If qi and Dr+; boare
+
variables then, for example, E,[Dr+l/(l ql)] does not equal ErD,+l
Second, equation (1.30) is expressed in terms of oneperiod rates 4;. If
(annual) rate applicable between t = 0 and t = 2 on a risky asset, then we
+ +
define (1 r(2))2 (1 q1)(1+ q 2 ) . Then (1.30) is of a similar form to
=
we ignore the expectations operator. These rather subtle distinctions need
the reader at this point and they will become clear later in the text.
3. Note that the function U = U ( F N 0) is very different from either E U
,:
from E[U(W)] where W is end of period wealth.
I &
The Capital Asset
Pricing Model: CAPM
This chapter presents a detailed derivation of the (basic) oneperiod capital a
model (CAPM). This model, interpreted as a model of equilibrium asset return
used in the finance literature and the concepts which underlie its derivation, su
folio diversification, measures of risk and return, and the concept of the mark
are also fundamental to the analysis of all asset prices. Throughout this chap
consider that the only risky assets are equities (stocks) although strictly the mo
to choices among all risky assets (i.e. stocks, bonds, real estate, etc.).
The CAPM attempts to answer what at first sight appears to be a rather c
of interrelated questions, namely:
Why it is beneficial for agents to hold a diversified portfolio consisting of a
0
risky assets rather than say one single risky asset or a small subset of all th
risky assets?
What determines the expected equilibrium return on each individual risky
0
market, so that all the risky assets are willingly held by investors?
What determines an individual investorâ€™s choice between his holdings of t
0
asset and the â€˜bundleâ€™ of risky assets?
As will be seen in the next chapter, there are a number of models of equilib
returns and there are a number of variants on the â€˜basicâ€™ CAPM. The con
in the derivation of the CAPM are quite numerous and somewhat complex,
useful at the outset to sketch out the main elements of its derivation and draw
basic implications. Section 2.2 carefully sets out the principles that underli
diversification and the efficient set of portfolios. Section 2.3 derives the optim
and the equilibrium returns that this implies.
21 AN OVERVIEW
.
Our world is restricted to one in which agents can choose a set of risky assets (
a riskfree asset (e.g. fixedterm bank deposit or a threemonth Treasury bill).
borrow and lend as much as they like at the riskfree rate. We assume agents
returns. Transactions costs and taxes are assumed to be zero.
Consider the reason for holding a diversifiedportfolio consisting of a set of r
Assume for the moment that the funds allocated to the safe asset have already
Putting all your wealth in asset 1, you incur an expected return ER1 and a ri
o , the variance of the returns on this one asset. Similarly holding just asset 2
:
to earn ER2 and incur risk 0;.Let us assume a twoasset world where there is
covariance of returns 0 1 2 < 0. Hence when the return on asset 1 rises that on as
to fall. (This also implies a negative correlation coefficient p12 = 012/0102.) H
diversify and hold both assets, this would seem to reduce the variance of
portfolio (i.e. of asset 1 plus asset 2). To simplify even further suppose that E
and a = 0;.In addition assume that when the return on asset 1 increases by
:
that on asset 2 falls by 1 percent (i.e. returns are perfectly negatively correlated
Under these conditions when you hold half your initial wealth in each of the r
the expected return on the overall portfolio is ER1 = E&. However, diversif
reduced the risk on the portfolio to zero: an above average return on asset 1
matched by an equal below average return on asset 2 (since p = 1). Our exa
course, a special case but in general, even if the covariance of returns is zero
(but not perfectly positively correlated) it still pays to diversify and hold a c
of both assets.
The above simple example also points to the reason why each individual inv
at least hold some of each of all the available stocks in the market, if we al
borrow (or lend) unlimited funds at the riskfree rate r. To demonstrate this p
up a counter example. If one stock were initially not desired by any of the inv
its current price would fall as investors sold it. However, a fall in the current pr
that the expected return over the coming period is higher, ceterisparibus (ass
expected it to pay some dividends in the future). One might therefore see the cu
fall until the expected return increases so that the stock is sufficiently attractiv
The reader may now be summising that the individual investorâ€™s tastes or p
must come into the analysis at some point and he would be correct. Howev
a quite remarkable result, known as the twofund separation theorem. The
decision can be broken down into two separate decisions. The first decision
the choice of the optimal proportions xi* of risky assets held and this is inde
the individualâ€™s preferences concerning his subjective tradeoff between risk
This choice depends only on the individualâ€™s views about the objective marke
namely, expected returns, variances and covariances. Expectations about thes
are assumed to be homogeneous across investors. All individuals therefore hol
proportions of the risky assets (e.g. all investors hold 1/20 of â€˜a sharesâ€™, 1/80 of
etc.) irrespective of their preferences. Hence aggregating, all individuals will
risky assets in the same proportions as in the (aggregate) market portfolio
share of ICI in the total stock market index is 1/20 by value, then all investors
of their own risky asset portfolio, in ICI shares).
It is only after mimicking the market portfolio that the individualâ€™s prefere
the calculation. In the second stage of the decision process the individual de
pays r ) and only invest a small amount of his own wealth in the risky assets i
proportions x:. The converse applies to a less risk averse person who will bo
riskfree rate and use these proceeds (as well as his own initial wealth) to invest
bundle of risky assets in the optimal proportions XI. however, this sec
Note,
which involves the individual's preferences, does not impinge on the relativ
for the risky assets (i.e. the proportions x:). Hence the equilibrium expected
the set of risky assets are independent of individuals' preferences and depe
market variables such as the variances and covariances in the market.
Throughout this and subsequent chapters the following equivalent ways of
expected returns, variances and covariances will be used:
Expected return = pi = ER;
Variance of returns = 0;= var(Ri)
Covariance of returns = 0 ; j = cov(Ri, R j )
Let us turn now to some specific results about equilibrium returns which
the CAPM. The CAPM provides an elegant model of the determinants of t
rium expected return ER; on any individual risky asset in the market. It predi
expected excess return on an individual risky asset (ERi  r ) is directly rel
expected excess return on the market portfolio (ERm r), with the constant
tionality given by the beta of the individual risky asset:
ERm is the expected return on the market portfolio that is the 'average' expe
from holding all assets in the optimal proportions x:. Since actual returns on
portfolio differ from expected returns, the variance var(Rm) on the market
nonzero. The definition of firm i's beta, namely pi indicates that it depends
(i) the covariance between the return on security i and the marke
cov(Ri, R m ) and
(ii) is inversely related to the variance of the market portfolio, var(Rm).
Loosely speaking, if the expost (or actual) returns when averaged appro
exante expected return ERi, then we can think of the CAPM as explaining
return (over say a number of months) on security i .
What does the CAPM tell us about equilibrium returns on individual secu
stock market? First note that (ERm  r ) > 0, otherwise no risk averse agent
covariance with the market portfolio, they will be willingly held as long as th
expected return equal to the riskfree rate (put pi = 0 in (2.1)). Securities that h
positive covariance with the market return (pi > 0) will have to earn a rela
expected return: this is because the addition of such a security to the portfolio d
reduce overallportfolio variance. Conversely any security for which cov(Rj, R
hence pi < 0 will be willingly held even though its expected return is below th
rate (equation (2.1) with pi < 0) because it tends to reduce overall portfolio v
The CAPM also allows one to assess the relative volatility of the expec
on individual stocks on the basis of their pi values (which we assume are
measured). Stocks for which pi = 1 have a return that is expected to move o
with the market portfolio (i.e. ER, = ERm) and are termed â€˜neutral stocksâ€™. If
stock is said to be an aggressive stock since it moves more than changes in th
market return (either up or down) and conversely defensive stocks have pi < 1
investors can use betas to rank the relative safety of various securities. Howeve
should not detract from one of the CAPMâ€™s key predictions, namely that al
should hold stocks in the same optimal proportions x:. Hence the â€˜market por
by all investors will include neutral, aggressive and defensive stocks held in t
proportions x,? predicted by the CAPM. Of course, an investor who wishes
positionâ€™ in particular stocks may use betas to rank the stocks to include in h
(i.e. he doesnâ€™t obey the assumptions of the CAPM and therefore doesnâ€™t attemp
the market portfolio).
The basic concepts of the CAPM can be used to assess the performance o
managers. The CAPM can be applied to any portfolio p of stocks composed
of the assets of the market portfolio. For such a portfolio we can see that PI
r)/a,] is a measure of the excess return (over the riskfree rate) per unit of ris
may loosely be referred to as a â€˜Performance Indexâ€™ (PI). The higher the PI
is the expected return corrected for risk (a,).Thus for two investment manage
whose portfolio has the higher value for PI may be deemed the more succes
ideas are developed further in Chapter 5.
2.2 PORTFOLIO DIVERSIFICATION, EFFICIENT FRO
AND THE TRANSFORMATION LINE
Before we analyse the key features of the CAPM we discuss the meanvarianc
the concept of an efficient portfolio and the gains from diversification in m
portfolio risk. We then consider the relationship between the expected retur
diversified portfolio and the risk of the portfolio, ap.If agents are interested in m
risk for any given level of return, then such efficient portfolios lie along th
frontier which is nonlinear in ( p p ,a ) space. We then examine the returnrisk r
,
for a specific twoasset portfolio where one asset consists of the amount of
or lending in the safe asset and the other asset is a single portfolio of risky a
gives rise to the transformation line which gives a linear relationship betwee
return and portfolio risk for any twoasset portfolio comprising a riskfree asse
It is assumed that the investor would prefer a higher expected return (ER) ra
lower expected return, but he dislikes risk (i.e. is risk averse). We choose to m
by the variance of the returns var(R) on the portfolio of risky assets. Thus, i
is presented with a portfolio â€˜Aâ€™ (of n securities) and a portfolio â€˜Bâ€™ (of a d
of n securities), then according to the MVC, portfolio A is preferred to portfo
where SD = standard deviation. Of course if, for example, EA(R)> E B ( R )but
varB(R) then we cannot say what portfolio the investor prefers using the MVC
Portfolios that satisfy the MVC are known as the set of eficientportfolio
folio A that has a lower expected return and a higher variance than another po
said to be â€˜inefficientâ€™ and an individual would (in principle) never hold a por
as A if portfolio B is available.
2.2.2 Portfolio Diversification
To demonstrate in a simple fashion the gains to be made from holding a
portfolio of assets, the simple two(risky) asset model will be used.
Suppose the actual return (over one period) on each of the two assets is R1 a
expected returns p1 = ER1 and p2 = E&. The variance of the returns on ea
is measured by o?(i= 1,2) which is defined as
In addition assume that the correlation coeficient between movements in the
the two assets is p, (1 < p < 1) where p is defined as
Hence 0 1 2 = cov(R1, R 2 ) is the covariance between the two returns. If p = +
asset returns are perfectly positively (linearly) related and the asset returns al
in the same direction. For p = 1 the converse applies and for p = 0 the a
are not (linearly) related. As we see below, the â€˜riskinessâ€™ of the portfolio co
both asset 1 and asset 2 depends crucially on the sign and size of p. If p = 1
be completely eliminated by holding a specific proportion of initial wealth in b
Even if p is positive (but less than +1) the riskiness of the overall portfolio
(although not to zero) by diversification.
Suppose for the moment that the investor chooses the proportion of his tota
invest in each asset in order to minimise portfolio risk. He is not, at this stag
to borrow or lend or place any of his wealth in a riskfree asset. Should the i
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