2 Robert Alan Hill
Portfolio Theory & Financial Analyses
Download free eBooks at bookboon.com
3 Portfolio Theory & Financial Analyses
1st edition© 2010 Robert Alan Hill &
bookboon.com
ISBN 9788776816056Download free eBooks at bookboon.com
4 Contents
Contents
About the Author
8 Part I: An Introduction
91 An Overview
10 Introduction
101.1 ˜e Development of Finance
101.2 E˚cient Capital Markets
121.3 ˜e Role of MeanVariance E˚ciency
141.4 ˜e Background to Modern Portfolio ˜eory
171.5 Summary and Conclusions
181.6 Selected References
20 Part II: ˜e Portfolio Decision
212 Risk and Portfolio Analysis
22 Introduction
222.1 MeanVariance Analyses: Markowitz E˚ciency
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5 Contents
2.2 ˜e Combined Risk of Two Investments
262.3 ˜e Correlation between Two Investments
302.4 Summary and Conclusions
332.5 Selected References
333 ˜e Optimum Portfolio
34 Introduction
343.1 ˜e Mathematics of Portfolio Risk
343.2 Risk Minimisation and the TwoAsset Portfolio
383.3 ˜e Minimum Variance of a TwoAsset Portfolio
403.4 ˜e MultiAsset Portfolio
423.5 ˜e Optimum Portfolio
453.6 Summary and Conclusions
483.7 Selected References
514 ˜e Market Portfolio
52 Introduction
524.1 ˜e Market Portfolio and Tobin™s ˜eorem
534.2 ˜e CML and Quantitative Analyses
574.3 Systematic and Unsystematic Risk
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6 Contents
4.4 Summary and Conclusions
634.5 Selected References
64 Part III: Models Of Capital Asset Pricing
655 ˜e Beta Factor
66 Introduction
665.1 Beta, Systemic Risk and the Characteristic Line
695.2 ˜e Mathematical Derivation of Beta
735.3 ˜e Security Market Line
745.4 Summary and Conclusions
775.5 Selected References
786
˜e Capital Asset Pricing Model (Capm)
79 Introduction
796.1 ˜e CAPM Assumptions
806.2 ˜e Mathematical Derivation of the CAPM
816.3 ˜e Relationship between the CAPM and SML
846.4 Criticism of the CAPM
866.4 Summary and Conclusions
916.5 Selected References
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7 Contents
7
Capital Budgeting, Capital Structure Andthe Capm
93 Introduction
937.1 Capital Budgeting and the CAPM
937.2 ˜e Estimation of Project Betas
957.3 Capital Gearing and the Beta Factor
1007.4 Capital Gearing and the CAPM
1037.5 ModiglianiMiller and the CAPM
1057.5 Summary and Conclusions
1087.6 Selected References
109 Part IV: Modern Portfolio ˜eory
1108
Arbitrage Pricing ˜eory and Beyond
111 Introduction
1118.1 Portfolio ˜eory and the CAPM
1128.2 Arbitrage Pricing ˜eory (APT)
1138.3 Summary and Conclusions
1158.5 Selected References
1189 Appendix for Chapter 1
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8 About the Author
About the Author
With an eclectic record of University teaching, research, publication, consultancy and curricula
development, underpinned by running a successful business, Alan has been a member of national
academic validation bodies and held senior external examinerships and lectureships at both undergraduate
and postgraduate level in the UK and abroad.
With increasing demand for global elearning, his attention is now focussed on the free provision of a
˛nancial textbook series, underpinned by a critique of contemporary capital market theory in volatile
markets, published by bookboon.com.
To contact Alan, please visit Robert Alan Hill at
www.linkedin.com
.Download free eBooks at bookboon.com
9 Part I:
An Introduction
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10 An Overview
1 An Overview
Introduction
Once a company issues shares (common stock) and receives the proceeds, it has no
direct
involvement
with their subsequent transactions on the capital market, or the price at which they are traded. ˜ese
are matters for negotiation between existing shareholders and prospective investors, based on their own
˛nancial agenda.
As a basis for negotiation, however, the company plays a pivotal
agency
role through its implementation of
investment˛nancing strategies designed to maximise pro˛ts and shareholder wealth. What management
do to satisfy these objectives and how the market reacts are ultimately determined by the law of supply
and demand. If corporate returns exceed market expectations, share price should rise (and vice versa).
But in a world where ownership is divorced from control, characterised by economic and geopolitical
events that are also beyond management™s control, this invites a question.
How do companies determine an optimum portfolio of investment strategies that
satisfy a multiplicity of shareholders with di˜erent wealth aspirations, who may also
hold their own diverse portfolio of investments?
1.1 The Development of Finance
As long ago as 1930, Irving Fisher™s
Separation ˜eorem
provided corporate management with a lifeline
based on what is now termed Agency ˜eory.
He acknowledged implicitly that whenever ownership is divorced from control, direct communication
between management (
agents
) and shareholders (
principals
) let alone other stakeholders, concerning the
likely pro˛tability and risk of every corporate investment and ˛nancing decision is obviously impractical.
If management were to implement optimum strategies that satisfy each shareholder, the company would
also require prior knowledge of every investor™s stock of wealth, dividend preferences and riskreturn
responses to their strategies.
According to Fisher, what management therefore, require is a model of
aggregate
shareholder behaviour.
A theoretical abstraction of the real world based on simplifying assumptions, which provides them with
a methodology to communicate a diversity of corporate wealth maximising decisions.
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11 An Overview
To set the scene, he therefore assumed (not unreasonably) that all investor behaviour (including that of
management) is
rational
and
risk averse
. ˜ey prefer high returns to low returns but less risk to more
risk. However, risk aversion does not imply that rational investors will not take a chance, or prevent
companies from retaining earnings to gamble on their behalf. To accept a higher risk they simply require
a commensurately higher return, which Fisher then benchmarked.
Management™s minimum rate of return on incremental projects ˚nanced by retained
earnings should equal the return that existing shareholders, or prospective investors,
can earn on investments of equivalent risk elsewhere.
He also acknowledged that a company™s acceptance of projects internally ˛nanced by retentions, rather
than the capital market, also denies shareholders the opportunity to bene˛t from current dividend
payments. Without these, individuals may be forced to sell part (or all) of their shareholding, or
alternatively borrow at the market rate of interest to ˛nance their own preferences for consumption
(income) or investment elsewhere.
To circumvent these problems Fisher assumed that if capital markets are
perfect
with no barriers to
trade and a free ˝ow of information (more of which later) a ˛rm™s
investment
decisions can not only be
independent
of its shareholders™
˚nancial
decisions but can also satisfy their wealth maximisation criteria.
In Fisher™s
perfect world:
Wealth maximising ˛rms should determine optimum
investment
decisions by
˚nancing
projects based on their
opportunity
cost of capital.
˜e opportunity
cost
equals the
return
that existing shareholders, or prospective investors,
can earn on investments of equivalent risk elsewhere.
Corporate projects that earn rates of return less than the opportunity cost of capital should
be rejected by management. ˜ose that yield equal or superior returns should be accepted.
Corporate earnings should therefore be distributed to shareholders as dividends, or retained
to fund new capital investment, depending on the relationship between project pro˛tability
and capital cost.
In response to rational managerial dividendretention policies, the ˛nal consumption
investment decisions of rational shareholders are then determined independently according
to their personal preferences.
In perfect markets, individual shareholders can always borrow (lend) money at the market
rate of interest, or buy (sell) their holdings in order to transfer cash from one period to
another, or one ˛rm to another, to satisfy their income needs or to optimise their stock of
wealth.
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12 An Overview
Activity 1
Based on Fisher™s Separation Theorem, share price should rise, fall, or remain stable
depending on the interrelationship between a company™s project returns and the
shareholders desired rate of return. Why is this?
For detailed background to this question and the characteristics of perfect markets you
might care to download ﬁStrategic Financial Managementﬂ (both the text and exercises)
from
bookboon.com
and look through their ˚rst chapters.
1.2 E˛cient Capital Markets
According to Fisher, in perfect capital markets where ownership is divorced from control, the separation
of corporate dividendretention decisions and shareholder consumptioninvestment decisions is not
problematical. If management select projects using the shareholders™ desired rate of return as a cuto˙
rate for investment, then at worst corporate wealth should stay the same. And once this information is
communicated to the outside world, share price should not fall.
Of course, the Separation ˜eorem is an abstraction of the real world; a model with questionable
assumptions. Investors do not always behave rationally (some speculate) and capital markets are not
perfect. Barriers to trade do exist, information is not always freely available and not everybody can
borrow or lend at the same rate. But instead of asking whether these assumptions are divorced from
reality, the relevant question is whether the model provides a sturdy framework upon which to build.
Certainly, theorists and analysts believed that it did, if Fisher™s impact on the subsequent development
of ˛nance theory and its applications are considered. So much so, that despite the recent global ˛nancial
meltdown (or more importantly, because the events which caused it became public knowledge) it is still
a basic tenet of ˛nance taught by Business Schools and promoted by other vested interests worldwide
(including governments, ˛nancial institutions, corporate spin doctors, the press, media and ˛nancial
websites) that:
Capital markets may not be
perfect
but are still reasonably
e˜cient with regard to
how analysts
process
information concerning corporate activity and how this changes
market values once it is conveyed to investors.
An e˚cient market is one where:
Information is universally available to all investors at a low cost.
Current security prices (debt as well as equity) re˝ect all relevant information.
Security prices only change when new information becomes available.
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13 An Overview
Based on the pioneering research of Eugene Fama (1965) which he formalised as the ﬁe˚cient market
hypothesisﬂ (EMH) it is also widely agreed that
information processing e˛ciency
can take
three forms
based on
two types
of analyses.
˜e weak form
states that
current
prices are determined solely by a
technical
analysis of
past
prices.
Technical analysts (or
chartists
) study historical price movements looking for cyclical patterns or trends
likely to repeat themselves. ˜eir research ensures that signi˛cant movements in current prices relative to
their history become widely and quickly known to investors as a basis for immediate trading decisions.
Current prices will then move accordingly.
˜e semistrong form
postulates that current prices not only re˝ect price history, but all
public
information.
And this is where
fundamental analysis
comes into play. Unlike chartists,
fundamentalists
study a company
and its business based on historical records, plus its current and future performance (pro˛tability,
dividends, investment potential, managerial expertise and so on) relative to its competitive position, the
state of the economy and global factors.
In weakform markets, fundamentalists, who make investment decisions on the expectations of individual
˛rms, should therefore be able to ﬁoutguessﬂ chartists and pro˛t to the extent that such information is
not assimilated into past prices (a phenomenon particularly applicable to companies whose ˛nancial
securities are infrequently traded). However, if the semistrong form is true, fundamentalists can no
longer gain from their research.
˜e strong form
declares that current prices fully re˝ect
all information
, which not only includes all
publically available information but also
insider
knowledge. As a consequence, unless they are lucky,
even the most privileged investors cannot pro˛t in the long term from trading ˛nancial securities before
their price changes. In the presence of strong form e˚ciency the market price of any ˛nancial security
should represent its intrinsic (true) value based on anticipated returns and their degree of risk.
So, as the EMH strengthens, speculative pro˚t opportunities weaken. Competition
among large numbers of increasingly wellinformed market participants drives security
prices to a consensus value, which re˝ects the best possible forecast of a company™s
uncertain future prospects.
Which strength of the EMH best describes the capital market and whether investors can ever ﬁbeat the
marketﬂ need not concern us here. ˜e point is that whatever levels of e˚ciency the market exhibits
(weak, semi strong or strong):
Current prices re˝ect all the relevant information used by that market (price history, public
data and insider information, respectively).
Current prices only change when new information becomes available.
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14 An Overview
It follows, therefore that prices must follow a ﬁrandom walkﬂ to the extent that new information is
independent
of the last piece of information, which they have already absorbed.
And it this phenomenon that has the most important consequences for how management
model their strategic investment˛nancing decisions to maximise shareholder wealth
Activity 2
Before we continue, you might ˚nd it useful to review the Chapter so far and brie˝y
summarise the main points..
1.3 The Role of MeanVariance E˛ciency
We began the Chapter with an idealised picture of investors (including management) who are rational
and riskaverse and formally analyse one course of action in relation to another. What concerns them
is not only pro˛tability but also the likelihood of it arising; a
riskreturn
tradeo˙ with which they feel
comfortable and that may also be unique.
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15 An Overview
˜us, in a sophisticated mixed market economy where ownership is divorced from control, it follows that
the objective of strategic ˛nancial management should be to implement optimum investment˛nancing
decisions using riskadjusted wealth maximising criteria, which satisfy a multiplicity of shareholders
(who may already hold a diverse portfolio
of investments) by placing them all in an equal, optimum
˛nancial position.
No easy task!
But remember, we have not only assumed that investors are rational but that capital markets are also
reasonably e˚cient at processing information. And this greatly simpli˛es matters for management.
Because today™s price is
independent
of yesterday™s price, e˚cient markets have
no memory
and individual
security price movements are
random
. Moreover, investors who comprise the market are so large in
number that no one individual has a comparative advantage. In the short run, ﬁyou win some, you lose
someﬂ but long term, investment is a
fair game
for all, what is termed a ﬁmartingaleﬂ. As a consequence,
management can now a˙ord to take a
linear
view of investor behaviour (as new information replaces
old information) and model its own plans accordingly.
What rational market participants require from companies is a diversi˚ed investment
portfolio that delivers a maximum return at minimum risk.
What management need to satisfy this objective are investment˚nancing strategies
that maximise corporate wealth, validated by simple
linear models that statistically
quantify the market™s riskreturn
tradeo˚.
Like Fisher™s Separation ˜eorem, the concept of linearity o˙ers management a lifeline because in
e˛cient
capital markets, rational investors (including management) can now assess anticipated investment returns
(ri) by reference to their probability of occurrence, (p
i) using classical statistical theory.
If the returns from investments are assumed to be
random
, it follows that their
expected return
(R) is the
expected monetary value (EMV) of a symmetrical,
normal
distribution (the familiar ﬁbell shaped curveﬂ
sketched overleaf). Risk is de˛ned as the
variance
(or dispersion) of individual returns: the greater the
variability, the greater the risk.
Unlike the mean, the statistical measure of dispersion used by the market or management to assess
risk is partly a matter of convenience. ˜e
variance
(VAR) or its square root, the
standard deviation
( = VAR) is used.
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16 An Overview
When considering the
proportion
of risk due to some factor, the variance (VAR =
2) is su˚cient.
However, because the standard deviation (
) of a normal distribution is measured in the same units
as (R) the expected value (whereas the variance (
2) only summates the squared deviations around the
mean) it is more convenient as an
absolute
measure of risk.
Moreover, the standard deviation (
) possesses another attractive statistical property. Using con˛dence
limits drawn from a Table of
z statistics, it is possible to establish the
percentage probabilities
that a
random variable lies within
one, two or three standard deviations above, below
or around
its expected
value, also illustrated below.
Figure 1.1:
The Symmetrical Normal Distribution, Area under the Curveand Con˚dence Limits
Armed with this statistical information, investors and management can then accept or reject investments
according to the degree of con˛dence they wish to attach to the likelihood (risk) of their desired
returns. Using decision rules based upon their optimum criteria for
meanvariance e˛ciency,
this implies
management and investors should pursue:
Maximum expected return (R) for a given level of risk, (s).
Minimum risk (s) for a given expected return (R).
˜us, our conclusion is that if modern capital market theory is based on the following three assumptions:
(i) Rational investors,
(ii) E˚cient markets,
(iii) Random walks.
˜e normative wealth maximisation objective of strategic ˛nancial management requires the optimum
selection of a portfolio of investment projects, which maximises their expected return (R) commensurate
with a degree of risk (s) acceptable to existing shareholders and potential investors.
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17 An Overview
Activity 3
If you are not familiar with the application of classical statistical formulae to ˚nancial theory, read
Chapter Four of ﬁStrategic Financial Managementﬂ (both the text and exercises) downloadable from
bookboon.com
.Each chapter focuses upon the two essential characteristics of investment, namely expected return
and risk. The calculation of their corresponding statistical parameters, the mean of a distribution
and its standard deviation (the square root of the variance) applied to investor utility should then be
familiar.
We can then apply simple mathematical notation: (r
i, pi, R, VAR,
and U) to develop a more complex
series of ideas throughout the remainder of this text.
1.4 The Background to Modern Portfolio Theory
From our preceding discussion, rational investors in reasonably e˚cient markets can assess the likely
pro˛tability of
individual
corporate investments by a statistical weighting of their expected returns, based
on a
normal
distribution (the familiar bellshaped curve).
Rationalrisk averse investors expect either a
maximum
return for a
given
level of risk, or a
given
return for
minimum
risk.
Risk is measured by the standard deviation of returns and the overall expected return is
measured by its weighted probabilistic average.
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18 An Overview
Using meanvariance e˚ciency criteria, investors then have
three
options when managing a
portfolio
of
investments depending on the performance of its individual components.
(i) To trade (buy or sell),
(ii) To hold (do nothing),
(iii) To substitute (for example, shares for loan stock).
However, it is important to note that what any individual chooses to do with their portfolio constituents
cannot be resolved by
statistical
analyses alone. Ultimately, their behaviour depends on how they
interpret an investment™s riskreturn trade o˙, which is measured by their
utility curve
. ˜is calibrates the
individual™s
current
perception of risk concerning uncertain
future
gains and losses. ˜eoretically, these
curves are simple to calibrate, but less so in practice. Risk attitudes not only di˙er from one investor to
another and may be unique but can also vary markedly over time. For the moment, su˚ce it to say that
there is no
universally
correct decision to trade, hold, or substitute one constituent relative to another
within a ˛nancial investment portfolio.
Review Activity
1. Having read the fourth chapters of the following series from
bookboon.com
recommended in Activity 3:
Strategic Financial Management (SFM),
Strategic Financial Management; Exercises (SFME).
In
SFM
: pay particular attention to Section 4.5 onwards, which explains the
relationship between
meanvariance
analyses, theconcept of
investor utility
and the application of
certainty
equivalent
analysis to investment appraisal.
In
SFME:
work through Exercise 4.1.
2. Next download the free companion text to this ebook:
Portfolio Theory and Financial Analyses; Exercises (PTFAE), 2010.
3. Finally, read Chapter One of
PTFAE.
It will test your understanding so far. The exercises and solutions are presented logically
as a guide to further study and are easy to follow. Throughout the remainder of the
book, each chapter™s exercises and equations also follow the same structure of this
text. So throughout, you should be able to complement and reinforce your theoretical
knowledge of modern portfolio theory (MPT) at
your own
pace.
1.5 Summary and Conclusions
Based on our Review Activity, there are two interrelated questions that we have not yet answered
concerning any wealth maximising investor™s riskreturn trade o˙, irrespective of their behavioural
attitude towards risk.
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19 An Overview
What if investors don™t want ﬁto put all their eggs in one basketﬂ and wish to diversify
beyond a
single asset portfolio?
How do ˚nancial management, acting on their behalf, incorporate the
relative
risk
return tradeo˜ between a
prospective
project and the ˚rm™s
existing asset portfolio into
a quantitative model that still maximises wealth?
To answer these questions, throughout the remainder of this text and its exercise book, we shall analyse
the evolution of Modern Portfolio ˜eory (MPT).
Statistical calculations for the expected riskreturn pro˛le of a
twoasset
investment portfolio will be
explained. Based upon the meanvariance e˚ciency criteria of Harry Markowitz (1952) we shall begin with:
˜e riskreducing e˙ects of a diverse twoasset portfolio,
˜e optimum twoasset portfolio that minimises risk, with individual returns that are
perfectly (negatively) correlated.
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20 An Overview
We shall then extend our analysis to
multiasset
portfolio optimisation, where John Tobin (1958)
developed the
capital market line
(CML) to show how the introduction of riskfree investments de˛ne
a ﬁfrontierﬂ of e˚cient portfolios, which further reduces risk. We discover, however, that as the size of
a portfolio™s constituents increase, the mathematical calculation of the variance is soon dominated by
covariance terms, which makes its computation unwieldy.
Fortunately, the problem is not insoluble. Ingenious, subsequent developments, such as the
speci˚c
capital asset pricing model (CAPM) formulated by Sharpe (1963) Lintner (1965) and Mossin (1966), the
optionpricing model of Black and Scholes (1973) and
general
arbitrage pricing theory (APT) developed
by Ross (1976), all circumvent the statistical problems encountered by Markowitz.
By dividing
total
risk between
diversi˚able
(unsystematic) risk and
undiversi˚able
(systematic or market)
risk, what is now termed Modern Portfolio ˜eory (MPT) explains how rational, risk averse investors and
companies can price securities, or projects, as a basis for pro˛table portfolio trading and investment decisions.
For example, a pro˛table trade is accomplished by buying (selling) an undervalued (overvalued) security
relative to an appropriate stock market index of
systematic
risk (say the FTSE All Share).˜is is measured by
the
beta
factor of the individual security relative to the market portfolio. As we shall also discover it is possible
for companies to de˛ne project betas for project appraisal that measure the systematic risk of speci˛c projects.
So, there is much ground to cover. Meanwhile, you should ˛nd the diagram in the Appendix provides
a useful roadmap for your future studies.
1.6 Selected References
1. Jensen, M.C. and Meckling, W.H., ﬁ˜eory of the Firm: Managerial Behaviour, Agency
Costs and Ownership Structureﬂ,
Journal of Financial Economics
, 3, October 1976.
2. Fisher, I.,
˜e ˜eory of Interest
, Macmillan (London), 1930.
3. Fama, E.F., ﬁ˜e Behaviour of Stock Market Pricesﬂ,
Journal of Business
, Vol. 38, 1965.
4. Markowitz, H.M., ﬁPortfolio Selectionﬂ,
Journal of Finance
, Vol. 13, No. 1, 1952.
5. Tobin, J., ﬁLiquidity Preferences as Behaviour Towards Riskﬂ,
Review of Economic Studies
, February 1958.
6. Sharpe, W., ﬁA Simpli˛ed Model for Portfolio Analysisﬂ,
Management Science
, Vol. 9, No. 2,
January 1963.
7. Lintner, J., ﬁ˜e valuation of risk assets and the selection of risk investments in stock
portfolios and capital budgetsﬂ,
Review of Economic Statistics
, Vol. 47, No. 1, December, 1965.
8. Mossin, J., ﬁEquilibrium in a capital asset marketﬂ,
Econometrica
, Vol. 34, 1966.
9. Hill, R.A.,
bookboon.com
Strategic Financial Management, 2009.
Strategic Financial Management; Exercises, 2009.
Portfolio ˜eory and Financial Analyses; Exercises, 2010.
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21 Part II:
The Portfolio Decision
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22 Risk and Portfolio Analysis
2 Risk and Portfolio Analysis
Introduction
We have observed that
meanvariance e˛ciency
analyses, premised on investor rationality (maximum
return) and risk aversion (minimum variability), are not always su˚cient criteria for investment appraisal.
Even if investments are considered in isolation, wealth maximising acceptreject decisions depend upon
an individual™s perception of the riskiness of its expected future returns, measured by their personal
utility curve
, which may be unique.
Your reading of the following material from the
bookboon.com
companion texts, recommended for
Activity 3 and the Review Activity in the previous chapter, con˛rms this.
Strategic Financial Management
(SFM
): Chapter Four, Section 4.5 onwards,
SFM
; Exercises (SFME)
: Chapter Four, Exercise 4.1,
SFM
: Portfolio ˜eory and Analyses; Exercises (PTAE)
: Chapter One.
Any con˝ict between meanvariance e˚ciency and the
concept of investor utility can only be resolved
through the application of
certainty equivalent
analysis to investment appraisal. ˜e ultimate test of
statistical
meanvariance analysis depends upon
behavioural
risk attitudes.
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23 Risk and Portfolio Analysis
So far, so good, but there is now another complex question to answer in relation to the search for future
wealth maximising investment opportunities:
Even if there is only
one new investment on the horizon, including a
choice
that is either
mutually exclusive,
or if capital is rationed
, (i.e. the acceptance of one precludes the acceptance of others).
How do individuals, or companies and ˚nancial institutions that make decisions on their behalf,
incorporate the
relative
riskreturn tradeo˜ between a
prospective
investment and an
existing asset portfolio into a quantitative model that still maximises wealth?
2.1 MeanVariance Analyses: Markowitz E˛ciency
Way back in 1952 without the aid of computer technology, H.M. Markowitz explained why rational
investors who seek an
e˛cient
portfolio
(one which minimises risk without impairing return, or
maximises return for a given level of risk) by introducing new (or o˙loading existing) investments,
cannot rely on meanvariance criteria alone.
Even before
behavioural
attitudes are calibrated, Harry Markowitz identi˚ed a
third statistical
characteristic concerning the riskreturn relationship between individual investments (or in
management™s case, capital projects) which justi˚es their inclusion within an
existing asset portfolio to
maximise wealth.
To understand Markowitz™ train of thought; let us begin by illustrating his simple
two asset
case,
namely
the construction of an
optimum
portfolio that comprises two investments. Mathematically, we shall
de˛ne their expected returns as R
i(A) and R
i(B) respectively, because their size depends upon which
one of two future economic ﬁstates of the worldﬂ occur. ˜ese we shall de˛ne as S
1 and S
2 with an equal
probability of occurrence. If S
1 prevails, R
1(A) > R1(B). Conversely, given S
2, then R
2(A) < R2(B). ˜e
numerical data is summarised as follows:
Return\State
S1S2Ri(A)
20%10%Ri(B)10%20%Activity 1
The overall expected return R(A) for investment A (its mean value) is obviously 15 per cent (the weighted
average of its expected returns, where the weights are the probability of each state of the world
occurring. Its risk (range of possible outcomes) is between 10 to 20 per cent. The same values also apply
to B.
Meanvariance analysis therefore informs us that because R(A) = R(B) and
(A) =
(B), we should all be
indi˚erent
to either investment. Depending on your behavioural attitude towards risk, one is perceived
to be as good (or bad) as the other. So, either it doesn™t matter which one you accept, or alternatively you
would reject both.
 Perhaps you can con˚rm this from your reading for earlier Activities?
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24 Risk and Portfolio Analysis
However, the question Markowitz posed is whether there is an
alternative
strategy to the exclusive
selection of either investment or their wholesale rejection? And because their respective returns do not
move in
unison
(when one is good, the other is bad, depending on the state of the world) his answer
was yes.
By not ﬁputting all your eggs in one basketﬂ, there is a
third
option that in our example produces an
optimum
portfolio
i.e. one with the
same
overall return as its constituents but with
zero
risk.
If we
diversify
investment and
combine
A and B in a
portfolio
(P) with half our funds in each, then the
overall portfolio return R(P) = 0.5R(A) + 0.5R(B) still equals the 15 per cent mean return for A and
B, whichever state of the world materialises. Statistically, however, our new portfolio not only has the
same return, R(P) = R(A) = R(B) but the risk of its constituents,
(A) = (B), is also eliminated entirely.
Portfolio risk;
(P) = 0. Perhaps you can con˛rm this?
Activity 2
As we shall discover, the previous example illustrates an
ideal portfolio scenario, based upon your
entire knowledge of investment appraisal under conditions of risk and uncertainty explained in
the SFM
texts referred to earlier. So, let us summarise their main points
An
uncertain
investment is one with a
plurality
of cash ˝ows whose probabilities are
non quanti˛able.
A risky
investment is one with a
plurality
of cash ˝ows to which we attach
subjective
probabilities.
Expected returns are assumed to be characterised by a normal distribution (i.e. they are
random variables).
The probability density function of returns is de˚ned by the meanvariance of their
distribution.
An e˛cient choice between individual investments maximises the discounted return of
their anticipated cash ˝ows and minimises the standard deviation of the return.
So, without recourse to further statistical analysis, (more of which later) but using your
knowledge of investment appraisal:
Can you de˚ne the objective of portfolio theory and using our previous numerical example,
brie˝y explain what Markowitz adds to our understanding of meanvariance analyses through
the e˛cient diversi˚cation of investments?
For a given overall return, the objective of e˚cient portfolio diversi˛cation is to determine an overall
standard deviation (level of risk) that is lower than any of its individual portfolio constituents.
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25 Risk and Portfolio Analysis
According to Markowitz, three signi˛cant points arise from our simple illustration with one important
conclusion that we shall develop throughout the text.
1) We can combine risky investments into a less risky, even riskfree, portfolio by ﬁnot putting
all our eggs in one basketﬂ; a policy that Markowitz termed
e˛cient
diversi˛cation, and
subsequent theorists and analysts now term
Markowitz e˛ciency
(praise indeed).
2) A portfolio of investments may be preferred to all or some of its constituents, irrespective
of investor risk attitudes. In our previous example, no rational investor would hold either
investment exclusively, because diversi˛cation can maintain the
same
return for
less
risk.
3) Analysed in isolation, the riskreturn pro˛les of individual investments
are insu˚cient
criteria by which to assess their true value. Returning to our example, A and B initially seem
to be equally valued. Yet, an investor with a substantial holding in A would ˛nd that moving
funds into B is an attractive proposition (and
vice versa
) because of the
inverse
relationship
between the
timing
of their respective riskreturn pro˛les, de˛ned by likely states of the
world. When one is good, the other is bad and
vice versa
.According to Markowitz, risk may be
minimised,
if not eliminated
entirely without compromising
overall return through the diversi˚cation and selection of an
optimum
combination of investments,
which de˚nes an e˜cient asset portfolio.
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26 Risk and Portfolio Analysis
2.2 The Combined Risk of Two Investments
So, in general terms, how do we derive (model) an optimum, e˚cient diversi˛ed portfolio of investments?
To begin with, let us develop the ﬁtwo asset caseﬂ where a company have funds to invest in two pro˛table
projects, A and B. One proportion
x is invested in A and (1
x) is invested in B.
We know from Activity 1 that the
expected return from a portfolio
R(P) is simply a weighted average of
the expected returns from two projects, R(A) and R(B), where the weights are the proportional funds
invested in each. Mathematically, this is given by:
(1) R(P) = x R(A) + (1 Œ x) R(B)
But, what about the likelihood (probability) of the portfolio return R(P) occurring?
Markowitz de˛nes the
proportionate
risk of a twoasset investment as the
portfolio variance:
(2) VAR(P) =
x2 VAR(A) + (1
x) 2 VAR(B) + 2
x(1x) COV(A, B)
Percentage
risk is then measured by the
portfolio standard deviation
(i.e. the square root of the variance):
(3) (P) = VAR (P) =
[ x2 VAR(A) + (1
x) 2 VAR (B) + 2
x(1x) COV(A, B)]
Unlike the risk of a
single
random variable, the variance (or standard deviation) of a
twoasset
portfolio
exhibits
three
separable characteristics:
1) ˜e risk of the constituent investments measured by their respective variances,
2) ˜e squared proportion of available funds invested in each,
3) ˜e relationship between the constituents measured by twice the
covariance
.˜e
covariance
represents the variability of the combined returns of individual investments around their
mean. So, if A and B represent two investments, the degree to which their returns (r
i A and r
i B) vary
together is de˛ned as:
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27 Risk and Portfolio Analysis
For each observation i, we multiply three terms together: the deviation of r
i(A) from its mean R(A), the
deviation of r
i(B) from its mean R(B) and the probability of occurrence p
i. We then add the results for
each observation.
Returning to Equations (2) and (3), the covariance enters into our portfolio risk calculation
twice
and
is
weighted
because the
proportional
returns on A vary with B and
vice versa
.Depending on the state of the world, the logic of the covariance itself is equally simple.
If the returns from two investments are
independent
there is no observable relationship
between the variables and knowledge of one is of no use for predicting the other. ˜e
variance of the two investments combined will equal the sum of the individual variances,
i.e. the covariance is
zero
. If returns are
dependent
a relationship exists between the two and the covariance can take
on either a positive or negative value that a˙ects portfolio risk.
1) When each paired deviation around the mean is negative, their product is positive and so
too, is the covariance.
2) When each paired deviation is positive, the covariance is still positive.
3) When one of the paired deviations is negative their covariance is negative.
˜us, in a state of the world where individual returns are
independent
and whatever happens to one
a˙ects the other to opposite e˙ect, we can reduce risk by diversi˛cation without impairing overall return.
Under condition (iii) the portfolio variance will obviously be less than the sum of its constituent variances.
Less obvious, is that when returns are
dependent
, risk reduction is still possible.
To demonstrate the application of the statistical formulae for a twoasset portfolio let us consider an
equal investment in two corporate capital projects (A and B) with an equal probability of producing the
following paired cash returns.
Pi A B % %0.5 8 14 0.5 12 6Download free eBooks at bookboon.com
28 Risk and Portfolio Analysis
We already know that the expected return on each investment is calculated as follows:
R(A) = (0.5 × 8) + (0.5 × 12) = 10%
R(B) = (0.5 × 14) + (0.5 × 6) = 10%
Using Equation (1), the
portfolio return
is then given by:
R(P) = (0.5 × 10) + (0.5 × 10) = 10%
Since the portfolio return equals the expected returns of its constituents, the question management must
now ask is whether the decision to place funds in both projects in equal proportions, rather than A or
B exclusively, reduces risk?
To answer this question, let us ˛rst calculate the variance of A, then the variance of B and ˛nally, the
covariance of A and B. ˜e data is summarised in Table 2.1 below.
With a negative covariance value of minus 8, combining the projects in equal proportions can obviously
reduce risk. ˜e question is by how much?
Table 2.1:
The Variances of Two Investments and their Covariance
Using Equation (2), let us now calculate the portfolio variance:
VAR(P) = (0.5
2 × 4) + (0.52 × 16) + (2 × 0.5) (0.5 × 8) = 1And ˛nally, the
percentage
risk given by Equation (3), the portfolio standard deviation:
(P) = VAR(P) =
1 = 1%Download free eBooks at bookboon.com
29 Risk and Portfolio Analysis
Activity 3
Unlike our original example, which underpinned Activities 1 and 2, the current statistics reveal
that this portfolio is not
riskless
(i.e. the percentage risk represented by the standard deviation
is not zero
). But given that our investment criteria remain the same (either
minimise , given
R; or maximise R given
) the next question to consider is how the portfolio™s riskreturn pro˚le
compares with those for the individual projects. In other words is diversi˚cation bene˚cial to the
company?
If we compare the standard deviations for the portfolio, investment A and investment B with their
respective expected returns, the following relationships emerge.
(P) < (A) < (B); given
R(P) = R(A) = R(B)˜ese con˛rm that our decision to place funds in both projects in equal proportions, rather than either
A or B exclusively, is the correct one. You can verify this by deriving the standard deviations for the
portfolio and each project from the variances in the Table 2.1.
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30 Risk and Portfolio Analysis
2.3 The Correlation between Two Investments
Because the covariance is an
absolute
measure of the correspondence between the movements of two
random variables, its interpretation is oˆen di˚cult. Not all paired deviations need be negative for
diversi˛cation to produce a degree of risk reduction. If we have small or large negative or positive values
for individual pairs, the covariance may also assume small or large values either way. So, in our previous
example, COV(A, B) = minus 8. But what does this mean exactly?
Fortunately, we need not answer this question? According to Markowitz, the statistic for the
linear
correlation coe˛cient
can be substituted into the third covariance term of our equation for portfolio
risk to simplify its interpretation. With regard to the mathematics, beginning with the variance for a
two asset portfolio:
(2) VAR(P) =
x2 VAR(A) + (1
x)2 VAR(B) + 2
x (1x) COV(A,B)
Let us de˛ne the correlation coe˚cient.
Now rearrange terms to rede˛ne the covariance.
(6) COV(A,B) = COR(A,B)
A B
Clearly, the portfolio variance can now be measured by the substitution of Equation (6) for the covariance
term in Equation (2).
(7) VAR(P) =
x2 VAR(A) + (1
x)2 VAR(B) + 2
x (1x) COR(A,B)
A B˜e standard deviation of the portfolio then equals the square root of Equation (7):
(8) (P) = VAR (P) =
[x2 VAR(A) + (1
x) 2 VAR (B) + 2
x(1x) COR(A,B)
A B]Download free eBooks at bookboon.com
31 Risk and Portfolio Analysis
Activity 4
So far, so good; we have proved mathematically that the correlation coe˛cient can replace the
covariance in the equations for portfolio risk.
But, given your knowledge of statistics, can you now explain why Markowitz thought this was a
signi˚cant contribution to portfolio analysis?
Like the standard deviation, the correlation coe˚cient is a
relative
measure of variability with a convenient
property. Unlike the covariance, which is an
absolute
measure, it has only
limited values between
+1 and
1. ˜is arises because the coe˚cient is calculated by taking the covariance of returns and dividing by
the product (multiplication) of the individual standard deviations that comprise the portfolio. Which is
why, for two investments (A and B) we have de˛ned:
The correlation coe˛cient therefore measures the extent to which two investments vary together
as a proportion
of their respective standard deviations. So, if two investments are
perfectly
and linearly related, they deviate by
constant proportionality
.Of course, the interpretation of the correlation coe˚cient still conforms to the logic behind the covariance,
but with the advantage of limited values.
If returns are
independent
, i.e. no relationship exists between two variables; their correlation
will be zero (although, as we shall discover later, risk can still be reduced by diversi˛cation).
If returns are
dependent
:1) A perfect, positive correlation of +1 means that whatever a˙ects one variable will equally
a˙ect the other. Diversi˛ed riskreduction is
not possible.
2) A perfect negative correlation of 1 means that an
e˛cient
portfolio can be constructed,
with
zero
variance exhibiting
minimum
risk. One investment will produce a return above its
expected return; the other will produce an equivalent return below its expected value and
vice versa
.3) Between +1 and Œ1, the correlation coe˚cient is determined by the proximity of direct and
inverse relationships between individual returns So, in terms of risk reduction, even a low
positive correlation can be bene˛cial to investors, depending on the allocation of total funds
at their disposal.
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32 Risk and Portfolio Analysis
Providing the correlation coe˚cient between returns is less than +1, all investors (including management)
can pro˛tably diversify their portfolio of investments. Without compromising the overall return, relative
portfolio risk measured by the standard deviation will be less than the weighted average standard deviation
of the portfolio™s constituents.
Review Activity
Using the statistics generated by Activity 3, con˚rm that the substitution of the correlation
coe˛cient for the covariance into our revised equations for the portfolio variance and
standard deviation does not change their values, or our original investment decision?
Let us begin with a summary of the previous meanvariance data for the twoasset portfolio:
R(P) = 0.5 R(A) + 0.5 R(B)
VAR(P)
VAR(A)
VAR(B)
COV(A,B)
10%1416(8)˜e correlation coe˚cient is given by:
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33 Risk and Portfolio Analysis
Substituting this value into our revised equations for the portfolio variance and standard deviation
respectively, we can now con˛rm our initial calculations for Activity 3.
(7) VAR(P) =
x2 VAR(A) + (1
x)2 VAR(B) + 2
x (1x)COR(A,B)
A B= (0.52 × 4) + (0.52 × 16) + {2 × 0.5(10.5) × 1(2 × 4)}= 1(8) (P) = VAR(P) =
1.0= 1.00 %˜us, the company™s original
portfolio
decision to place an equal proportions of funds in both investments,
rather than either A or B
exclusively
, still applies. ˜is is also con˛rmed by a summary of the following
interrelationships between the riskreturn pro˛les of the portfolio and its constituents, which are
identical to our previous Activity.
2.4 Summary and Conclusions
It should be clear from our previous analyses that the risk of a
twoasset
portfolio is a function of its
covariability of returns. Risk is at a
maximum
when the correlation coe˚cient between two investments
is +1 and at a
minimum
when the correlation coe˚cient equals 1. For the vast majority of cases where
the correlation coe˚cient is between the two, it also follows that there will be a
proportionate
reduction in
risk, relative to return. Overall portfolio risk will be less than the weighted average risks of its constituents.
So, investors can still pro˛t by diversi˛cation because:
2.5 Selected References
1. Hill, R.A.,
bookboon.com
Strategic Financial Management,
2009. Strategic Financial Management; Exercises,
2009. Portfolio ˜eory and Financial Analyses; Exercises,
2010.2. Markowitz, H.M., ﬁPortfolio Selectionﬂ,
˜e Journal of Finance,
Vol. 13, No. 1, March 1952.
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34 The Optimum Portfolio
3 The Optimum Portfolio
Introduction
In an e˚cient capital market where the random returns from two investments are normally distributed
(symmetrical) we have explained how rational (risk averse) investors and companies who seek an
optimal portfolio can maximise their utility preferences by
e˛cient
diversi˛cation. Any combination of
investments produces a tradeo˙ between the two statistical parameters that de˛ne a normal distribution;
their expected return and standard deviation (risk) associated with the
covariability
of individual returns.
According to Markowitz (1952) this is best measured by the
correlation coe˛cient
such that:
E˜cient
diversi˚ed portfolios are those which
maximise return for a
given
level of risk,
or minimise risk for a
given
level of return for di˜erent correlation coe˛cients.
˜e purpose of this chapter is to prove that when the correlation coe˚cient is at a minimum and
portfolio risk is minimised we can derive an
optimum portfolio
of investments that maximises there
overall expected return.
3.1 The Mathematics of Portfolio Risk
You recall from Chapter Two (both the ˜eory and Exercises texts) that substituting the
relative
linear
correlation coe˚cient for the
absolute
covariance term into a twoasset portfolio™s standard deviation
simpli˛es the wealth maximisation analysis of the riskreturn tradeo˙ between the covariability of
returns. Whenever the coe˚cient falls below one, there will be a
proportionate
reduction in portfolio
risk, relative to return, by diversifying investment.
For example, given the familiar equations for the return, variance, correlation coe˚cient and standard
deviation of a twoasset portfolio:
(1) R(P) = x R(A) +(1x) R(B)
(2) VAR(P) =
x2 VAR(A) + (1
x)2 VAR(B) + 2
x (1x) COV(A,B)
(5) COR(A,B) =
COV(A,B)
A B(8) (P) = VAR(P) =
[ x2 VAR(A) + (1
x) 2 VAR(B) + 2
x(1x) COR(A,B)
A B]Download free eBooks at bookboon.com
35 The Optimum Portfolio
Harry Markowitz (
op. cit.
) proved mathematically that:
However, he also illustrated that if the returns from two investments exhibit
perfect positive, zero
, or
perfect negative
correlation, then portfolio risk measured by the standard deviation using Equation (8)
can be simpli˛ed further.
To understand why, let us return to the original term for the portfolio variance:
(2) VAR(P) =
x2 VAR(A) + (1
x)2 VAR(B) + 2
x (1x) COV(A,B)
Because the correlation coe˚cient is given by:
(5) COR(A,B) =
COV(A,B)
A BDownload free eBooks at bookboon.com
36 The Optimum Portfolio
We can rearrange its terms, just as we did in Chapter Two, to rede˛ne the covariance:
(6) COV(A,B) = COR(A,B)
A B
˜e portfolio variance can now be measured by the substitution of Equation (6) for the covariance term
in Equation (2), so that.
(7) VAR(P) =
x2 VAR(A) + (1
x)2 VAR(B) + 2
x (1x) COR(A,B)
A B˜e standard deviation of the portfolio then equals the square root of Equation (7):
(8) (P) = VAR(P) =
[x2 VAR(A) + (1
x) 2 VAR(B) + 2
x(1x) COR(A,B)
A B]Armed with this information, we can now con˛rm that:
If the returns from two investments exhibit perfect, positive correlation, portfolio risk is
simply the weighted average of its constituent™s risks and at a maximum.
(P) = x (A) + (1x)
(B)If the correlation coe˚cient for two investments is positive and COR(A,B) also equals plus one, then
the correlation term can disappear from the portfolio risk equations without a˙ecting their values. ˜e
portfolio variance can be rewritten as follows:
(9) VAR(P) =
x2 VAR(A) + (1 Œ
x)2 VAR(B) + 2
x (1x) (A) (B)Simplifying, this is equivalent to:
(10) VAR(P) = [
x (A) + (1x) (B)]2And because this is a
perfect square
, our probabilistic estimate for the risk of a twoasset portfolio
measured by the standard deviation given by Equation (8) is equivalent to:
(11) (P) = VAR(P) =
x (A) + (1x)(B)To summarise:
Whenever COR(A, B) = +1 (perfect positive) the portfolio variance VAR(P) and its square
root, the standard deviation
(P), simplify to the weighted average of the respective
statistics, based on the probabilistic returns for the individual investments.
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37 The Optimum Portfolio
But this is not all
. ˜e substitution of Equation (6) into the expression for portfolio variance has two
further convenient properties. Given:
(6) COV(A,B) = COR(A,B)
A BIf the relationship between two investments is
independent
and exhibits
zero
correlation, the portfolio
variance given by Equation (7) simpli˛es to:
(12) VAR(P) =
x2 VAR(A) + (1
x)2 VAR(B)
And its corresponding standard deviation also simpli˛es:
(13) (P) = [x2 VAR(A) + (1
x)2 VAR(B)]
Similarly, with
perfect inverse
correlation we can deconstruct our basic equations to simplify the algebra.
Activity 1
When the correlation coe˛cient for two investments is perfect positive and equals one, the
correlation term disappears from equations for portfolio risk without a˜ecting their values. The
portfolio variance VAR(P)) and its square root, the standard deviation
(P), simplify to the
weighted
average
of the respective statistics.
Can you manipulate the previous equations to prove that if COR(A,B) equals
minus
one (perfect
negative) they still correspond to a weighted average, like their perfect positive counterpart, but
with one fundamental di˜erence? Whenever COR(A, B) = +1 (perfect positive) the portfolio variance
VAR(P) and its square root, the standard deviation
(P), simplify to the weighted average of the
respective statistics, based on the probabilistic returns for the individual investments.
Let us begin again with the familiar equation for portfolio variance.
(7) VAR(P) =
x2 VAR(A) + (1
x)2 VAR(B) + 2
x (1x) COR(A,B)
A BIf the correlation coe˚cient for two investments is negative and COR(A,B) also equals minus one, then
the coe˚cient can disappear from the equation™s third right hand term without a˙ecting its value. It can
be rewritten as follows with only a change of sign (positive to negative):
(14) VAR(P) =
x2 VAR(A) + (1
x)2 VAR(B) Œ 2
x (1x) (A) (B)Simplifying, this is equivalent to:
(15) VAR(P) = [
x (A) Œ (1x) (B)]2Download free eBooks at bookboon.com
38 The Optimum Portfolio
And because this is a
perfect square
, our probabilistic estimate for the risk of a twoasset portfolio
measured by the standard deviation is equivalent to:
(16) (P) = x (A) Œ (1x) (B)The only di˜erence between the formulae for the risk of a twoasset portfolio
where the correlation coe˛cient is at either limit (+1 or Œ1) is simply a matter of
sign (positive or negative) in the right hand term for
(P).3.2 Risk Minimisation and the TwoAsset Portfolio
When investment returns exhibit perfect positive correlation a portfolio™s risk is at a maximum, de˛ned
by the weighted average of its constituents. As the correlation coe˚cient falls there is a proportionate
reduction in portfolio risk relative to this weighted average. So, if we diversify investments; risk is
minimised when the correlation coe˚cient is minus one.
To illustrate this general proposition, Figure 3.1 roughly sketches the various
twoasset portfolios that are possible if corporate management combine two
investments, A and B, in various proportions for di˜erent correlation coe˛cients.
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39 The Optimum Portfolio
Speci˛cally, the
diagonal
line A (+1) B; the
curve
A (E) B and the
ﬁdoglegﬂ
A (1) B are the focus of
all possible riskreturn combinations when our correlation coe˚cients equal plus one, zero and minus
one, respectively.
˜us, if project returns are perfectly, positively correlated we can construct a portfolio with any riskreturn
pro˛le that lies along the
horizontal
line, A (+1) B, by varying the proportion of funds placed in each
project. Investing 100 percent in A produces a minimum return but minimises risk. If management put
all their funds in B, the reverse holds. Between the two extremes, having decided to place say twothirds
of funds in Project A, and the balance in Project B, we ˛nd that the portfolio lies one third along A (+1) B
at point +1.
Figure 3.1:
The Two Asset RiskReturn Pro˚le and the Correlation Coe˛cient
Similarly, if the two returns exhibit perfect negative correlation, we could construct any portfolio that
lies along the line A (1) B. However, because the correlation coe˚cient equals minus one, the line is no
longer straight but a
dogleg
that also touches the vertical axis where
(P) equals zero. As a consequence,
our choice now di˙ers on two counts.
It is possible to construct a
riskfree
portfolio.
No rational, risk averse investor would be interested in those portfolios which o˙er a
lower
expected return for the
same
risk.
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40 The Optimum Portfolio
As you can observe from Figure 3.1, the investment proportions lying along the line 1 to B o˙er
higher returns for a given level of risk relative to those lying between 1 and A. Using the terminology
of Markowitz based on meanvariance criteria; the ˛rst portfolio set is
e˛cient
and acceptable whilst
the second is
ine˛cient
and irrelevant. ˜e line 1 to B, therefore, de˛nes the
e˛ciency frontier
for a
twoasset portfolio.
Where the two lines meet on the vertical axis (point 1 on our diagram) the portfolio standard deviation
is zero. As the
horizontal
line (1, 0, +1) indicates, this
riskless
portfolio also conforms to our decision
to place twothirds of funds in Project A and one third in Project B.
Finally, in most cases where the correlation coe˚cient lies somewhere between its extreme value, every
possible twoasset combination always lies along a
curve
. Figure 3.1 illustrates the riskreturn tradeo˙
assuming that the portfolio correlation coe˚cient is
zero
. Once again, because the data set is not perfect
positive (less than +1) it turns back on itself. So, only a proportion of portfolios are e˚cient; namely
those lying along the EB frontier. ˜e remainder, EA, is of no interest whatsoever. You should also
note that whilst risk is not eliminated entirely, it could still be
minimised
by constructing the appropriate
portfolio, namely point E on our curve.
3.3 The Minimum Variance of a TwoAsset Portfolio
Investors trade ˛nancial securities to earn a return in the form of dividends and capital gains. Companies
invest in projects to generate net cash in˝ows on behalf of their shareholders. Returns might be higher
or lower than anticipated and their variability is the cause of investment risk. Investors and companies
can reduce risk by diversifying their portfolio of investments. ˜e preceding analysis explains why risk
minimisation represents an
objective
standard against which investors and management compare their
variance of returns as they move from one portfolio to another.
To prove this proposition, you will have observed from Figure 3.1 that the decision to place twothirds
of our funds in Project A and onethird in Project B falls between E and A when COR(A,B) = 0. ˜is
is de˛ned by point 0 along the horizontal line (1, 0, +1).
Because portfolio risk is minimised at point E, with a higher return above and to the leˆ in our diagram,
the decision is clearly
suboptimal
. At one extreme, speculative investors or companies would place all
their money in Project B at point B hoping to maximise their return (completely oblivious to risk). At the
other, the most riskaverse among them would seek out the proportionate investment in A and B which
corresponds to E. Between the two, a higher expected return could also be achieved for any degree of risk
given by the curve EA. ˜us, all investors would move up to the e˚ciency frontier EB and depending
upon their risk attitudes choose an appropriate combination of investments above and to the right of E.
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41 The Optimum Portfolio
However, without a graph, let alone data to fall back on, this raises another fundamental question.
How do investors and companies mathematically model an optimum portfolio
with minimum variance from ˚rst principles?
According to Markowitz (
op. cit
) the mathematical derivation of a twoasset portfolio with
minimum
risk is quite straightforward.
Where a proportion of funds
x is invested in Project A and (1
x) in Project B, the portfolio variance can
be de˛ned by the familiar equation:
(7) VAR(P) =
x2 VAR(A) + (1
x)2 VAR(B) + 2
x (1x) COR(A,B)
A B˜e value of
x, for which Equation (7) is at a
minimum
, is given by
di˝erentiating
VAR(P) with respect
to
x and setting
VAR(P) /
x = 0, such that:
(17) x = VAR(B) Œ COR(A,B)
(A) (B) VAR(A) + VAR(B) Œ 2 COR(A,B)
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42 The Optimum Portfolio
Since all the variables in the equation for minimum variance are now known, the riskreturn tradeo˙ can
be solved. Moreover, if the correlation coe˚cient equals
minus one
, risky investments can be combined
into a
riskless
portfolio by solving the following equation when the standard deviation is
zero
.(18) (P) = [x2 VAR(A) + (1
x) 2 VAR (B) + 2
x(1x) COR(A,B)
A B] = 0Because this is a
quadratic
in one unknown (
x) it also follows that to
eliminate
portfolio risk when
COR(A,B) = 1, the proportion of funds (
x) invested in Project A should be:
(19) x = 1  (A) (A) + (B)Activity 2
Algebraically, mathematically and statistically, we have covered a lot of ground since
Chapter Two. So, the previous section, like those before it, is illustrated by the numerical
application of data to theory in the
bookboon
companion text.
Portfolio Theory and Financial Analyses; Exercises (PTFAE):
Chapter Three, 2010.
You might ˚nd it useful at this point in our analysis to crossreference the appropriate
Exercises (3.1 and 3.2) before we continue?
3.4 The MultiAsset Portfolio
In e˚cient capital markets where the returns from
two
investments are normally distributed (symmetrical)
we have explained how rational (risk averse) investors and companies who require an optimal portfolio
can maximise their utility preferences by diversi˛cation. Any combination of investments produce a
tradeo˙ between the statistical parameters that de˛ne a normal distribution; the expected return and
standard deviation (risk) associated with the covariance of individual returns.
E˜cient
diversi˚ed portfolios are those which
maximise return for a
given
level of risk,
or minimise risk for a
given
level of return for di˜erent correlation coe˛cients.
However, most investors, or companies and ˛nancial managers (whether they control capital projects or
˛nancial services (such as insurance premiums, pension funds or investment trusts) may be responsible
for numerous investments. It is important, therefore, that we extend our analysis to portfolios with more
than two constituents.
˜eoretically, this is not a problem. According to Markowitz (
op cit.
) if individual returns, standard
deviations and the covariance for each pair of returns are known, the portfolio return R(P), portfolio
variance VAR(P) and a probabilistic estimate of portfolio risk measured by the standard deviation s(P),
can be calculated.
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43 The Optimum Portfolio
For a
multiasset
portfolio where the number of assets equals n and x
i represents the proportion of funds
invested in each, such that:
We can de˛ne the portfolio return and variance as follows
˜e covariance term, COVij determines the degree to which variations in the return to one investment, i,
can serve to o˙set the variability of another, j. ˜e standard deviation is then derived in the usual manner.
(22) (P) = VAR(P)
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44 The Optimum Portfolio
Assuming we now wish to
minimise
portfolio risk for any given portfolio return; our ˛nancial objective
is equally straightforward:
(23) MIN: (P), Given R(P) = K (constant)
˜is mathematical function combines Equation (22) which is to be
minimised
, with a constraint obtained
by setting Equation (20) for the portfolio return equal to a
constant
(K):Figure 3.2 illustrates all the di˙erent riskreturn combinations that are available from a hypothetical
multi investment scenario.
Figure 3.2:
The Portfolio E˛ciency Frontier: The MultiAsset Case
˜e ˛rst point to note is that when an investment comprises a large number of assets instead of two, the
possible portfolios now lie within on area, rather than along a line or curve. ˜e area is constructed by
plotting (in˛nitely) many lines or curves similar to those in Figure 3.1.
However, like a twoasset portfolio, rational, riskaverse investors or companies are not interested in all
these possibilities, but only those that lie along the upper boundary between F and F
1. ˜e portfolios that
lie along this frontier are e˚cient because each produces the highest expected return for its given level
of risk. To the right and below, alternative portfolios yield inferior results. To the leˆ, no possibilities
exist. ˜us, an optimum portfolio for any investor can still be determined at an appropriate point on
the e˚ciency frontier providing the individual™s attitude toward risk is known.
So how is this calibrated?
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45 The Optimum Portfolio
3.5 The Optimum Portfolio
We have already observed that the
calculation
of statistical means and standard deviations is separate
from their behavioural
interpretation
, which can create anomalies. For example, a particular problem we
encountered within the context of investment appraisal was the ﬁriskreturn paradoxﬂ where one project
o˙ers a
lower return for less risk
, whilst the other o˙ers a
higher return for greater risk
. Here, investor
rationality (maximum return) and risk aversion (minimum variability) may be
insu˛cient
behavioural
criteria for project selection. Similarly, with portfolio analysis:
If two di˜erent portfolios lie on the e˛ciency frontier, it is impossible to choose
between them without information on investor risk attitudes.
One solution is for the investor or company to consider a value for the portfolio™s expected return R(P),
say R(p
i) depicted schematically in Figure 3.3.
Figure 3.3:
The MultiAsset E˛ciency Frontier and Investor Choice
All R(p
i), (pi) combinations for di˙erent portfolio mixes are then represented by points along the
horizontal line R(p
i) Œ R(pi)1 for which R(P) = R(p
i). ˜e leˆmost point on this line, F then yields the
portfolio investment mix that satis˛es Equation (23) for our objective function:
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46 The Optimum Portfolio
By repeating the exercise for all other possible values of R(P) and obtaining every e˚cient value of R(p
i) we can then trace the entire opportunity locus, FF
1.˜e investor or company then subjectively select the
investment combination yielding a maximum return, subject to the constraint imposed by the degree of
risk they are willing to accept, say P* corresponding to R(P*) and s(P*) in the diagram.
Review Activity
As an optimisation procedure, the preceding model is theoretically sound. However, without today™s
computer technology and programming expertise, its practical application was a lengthy, repetitive
process based on trial and error, when ˚rst developed in the 1950s. What investors and companies
needed was a portfolio selection technique that actually incorporated their risk preferences into their
analyses. Fortunately, there was a lifeline.
As we explained in the Summary and Conclusions of Chapter Two™s Exercise text, (
PTFAE
) rational risk
averse investors, or companies, with a
twoasset
portfolio will always be willing to accept higher risk
for a larger return, but
only up to a point
. Their precise cuto˜ rate is de˚ned by an
indi˚erence
curve
that calibrates their risk attitude, based on the concept of
expected utility.
We can apply this analysis to a
multiasset
portfolio of investments. However, before we develop the
mathematics, perhaps you might care to look back at Chapter Two (
PTFAE
)and the simple twoasset
scenario before we continue.
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47 The Optimum Portfolio
In Chapter Two (
PTFAE
) we discovered that if an investor™s or company™s objective is to minimise the
standard deviation of expected returns this can be determined by reference to a their utility
indi˝erence
curve,
which calibrates attitudes toward risk and return. Applied to portfolio analysis, the mathematical
equation for any curve of indi˙erence between portfolio risk and portfolio return for a rational investor
can be written:
(24) VAR(P) =
+ R(P)Graphically, the value of
indicates the
steepness
of the curve and
indicates the
horizontal intercept
. ˜us, the objective of the Markowitz portfolio model is to
minimise
. If we rewrite Equation (24), for
any indi˙erence curve that relates to a portfolio containing n assets, this objective function is given by:
(25) MIN: = VAR (P) Œ
l R(P)For all possible values of
0, where R(P) = K(constant), subject to the nonnegativity constraints:
i 0, i = 1, 2, 3 –nAnd the essential requirement that sources of funds equals uses and
xi be proportions expressed
mathematically as:
n
xi = 1i = 1Any portfolio that satis˛es Equation (25) is
e˛cient
because no other asset combination will have a lower
degree of risk for the requisite expected return.
An optimum portfolio for an individual investor is plotted in Figure 3.4. ˜e e˚ciency frontier F Œ F
1 of
risky portfolios still reveals that, to the right and below, alternative investments yield inferior results. To
the leˆ, no possibilities exist. However, we no longer determine an optimum portfolio for the investor
by trial and error.
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48 The Optimum Portfolio
Figure 3.4:
the Determination of an Optimum Portfolio: The MultiAsset Case
˜e
optimum portfolio
is at
the
point
where one of the curves for their equation of indi˙erence (risk
return pro˛le) is
tangential
to the frontier of e˚cient portfolios (point E on the curve FF
1). ˜is portfolio
is optimal because it provides the best combination of risk and return to suit their preferences.
3.6 Summary and Conclusions
We have observed that the objective function of multiasset portfolio analysis is represented by the
following indi˙erence equation.
˜is provides investors and companies with a standard, against which they can compare their preferred
riskreturn pro˛le for any e˚cient portfolio.
However, its interpretation, like other portfolio equations throughout the Chapter assumes that the
e˚ciency frontier has been correctly de˛ned. Unfortunately, this in itself is no easy task.
Based upon the pioneering work of Markowitz (op. cit.) we explained how a rational and riskaverse
investor, or company, in an e˚cient capital market (characterised by a normal distribution of returns)
who require an optimal portfolio of investments can maximise utility, having regard to the relationship
between the expected returns and their dispersion (risk) associated with the covariance of returns within
a portfolio.
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49 The Optimum Portfolio
Any combination of investments produces a tradeo˙ between the two statistical parameters; expected
return and standard deviation (risk) associated with the covariability of individual returns. And according
to Markowitz, this statistical analysis can be simpli˛ed.
E˛cient diversi˚ed portfolios are those which maximise return for a given level of risk,
or minimise risk for a given level of return for di˜erent correlation coe˛cients.
˜e Markowitz portfolio selection model is theoretically sound. Unfortunately, even if we substitute the
correlation coe˚cient into the covariance term of the portfolio variance, without the aid of computer
soˆware, the mathematical complexity of the variancecovariance matrix calculations associated with a
multiasset portfolio limits its applicability.
˜e
constraints
of Equation (25) are
linear
functions of the
n variables
xi, whilst the
objective function
is
an equation of the
second degree
in these variables. Consequently, methods of
quadratic
programming,
rather than a simple
linear
programming calculation, must be employed by investors to
minimise
VAR(P)
for various values of R (P) = K.
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50 The Optimum Portfolio
Once portfolio analysis extends beyond the twoasset case, the data requirements become increasingly
formidable. If the covariance is used as a measure of the variability of returns, not only do we require
estimates for the expected return and the variance for each asset in the portfolio but also estimates for
the correlation matrix between the returns on all assets.
For example, if management invest equally in three projects, A, B and C, each deviation from the
portfolio™s expected return is given by:
[1/3 riA Œ R(A)] + [1/3 riB Œ R(B)] + [1/3 riC Œ R(C)]If the deviations are now squared to calculate the variance, the proportion 1/3 becomes (1/3)
2, so that:
VAR(P) = VAR[1/3 (A)+1/3 (B)+1/3 (C)]
= (1/3)2 (the sum of three variance terms, plus the sum of six covariances).
For a twenty asset portfolio:
VAR(P) = (1/20)
2 (sum of twenty variance, plus the sum of 380 covariances).
As a
general rule
, if there are
xi = n projects, we ˛nd that:
(26) VAR (P) = (1/
n)2 (sum of
n variance terms, plus the sum of
n (n1) covariances.
In the covariance matrix (
xi – xn), xi is paired in turn with each of the other projects (
x2 – xn) making
(n1) pairs in total. Similarly, (
n 1) pairs can be formed involving
x2 with each other
xi and so forth,
through to
xn making
n (n1) permutations in total.
Of course,
half of these pairs will be duplicates
. ˜e set
x1, x2 is identical with
x2, x1. ˜e
n asset case
therefore requires only
1/2 (n
2 n)
distinct covariance ˛gures altogether, which represents a substantial
data saving in relation to Equation (26). Nevertheless, the decisionmaker™s task is still daunting, as the
number of investments for inclusion in a portfolio increases.
Not surprising, therefore, that without today™s computer technology, a search began throughout the late
1950s and early 1960s for simpler mathematical and statistical measures of Markowitz portfolio risk and
optimum asset selection, as the rest of our text will reveal.
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51 The Optimum Portfolio
3.7 Selected References
1. Markowitz, H.M., ﬁPortfolio Selectionﬂ,
˜e Journal of Finance,
Vol. 13, No. 1, March 1952.
2. Hill, R.A.,
bookboon.com
Strategic Financial Management,
2009. Strategic Financial Management; Exercises,
2009. Portfolio ˜eory and Financial Analyses;
Exercises, 2010.
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52 The Market Portfolio
4 The Market Portfolio
Introduction
˜e objective of e˚cient portfolio diversi˛cation is to achieve an overall standard deviation lower than
that of its component parts without compromising overall return.
In an ideal world portfolio theory should enable:
Investors (private or institutional) who play the stock market to model
the e˜ects of adding new securities to their existing spread.
Companies to assess the extent to which the pattern of returns from
new projects a˜ects the risk of their existing operations.
For example, suppose there is a
perfect positive correlation
between two securities that comprise the
market, or two products that comprise a ˛rm™s total investment. In other words, high and low returns
always move sympathetically. It would pay the investor, or company, to place all their funds in whichever
investment yields the highest return at the time. However, if there is
perfect inverse correlation
, where
high returns on one investment are always associated with low returns on the other and
vice versa
, or
there is
random (zero) correlation
between the returns, then it can be shown statistically that overall risk
reduction can be achieved through diversi˛cation.
According to Markowitz (1952), if the correlation coe˛cient between any number
of investments is less then one (perfect positive), the total risk of a portfolio
measured by its standard deviation is lower than the weighted average of its
constituent parts, with the greatest reduction reserved for a correlation coe˛cient
of minus one (perfect inverse).
˜us, if the standard deviation of an individual investment is higher than that for a portfolio in which
it is held, it would appear that some of the standard deviation must have been diversi˛ed away through
correlation with other portfolio constituents, leaving a residual risk component associated with other
factors.
Indeed, as we shall discover later, the reduction in
total
risk only relates to the
speci˚c
risk associated with
microeconomic
factors, which are unique to individual sectors, companies, or projects. A proportion of
total
risk, termed
market
risk, based on
macroeconomic
factors correlated with the market is inescapable.
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53 The Market Portfolio
˜e distinguishing features of speci˛c and market risk had important consequences for the development
of Markowitz e˚ciency and the emergence of Modern portfolio ˜eory (MPT) during the 1960™s. For
the moment, su˚ce it to say that whilst market risk is not diversi˛able, theoretically, speci˛c risk can be
eliminated entirely if all rational investors diversify until they hold the
market portfolio
, which re˝ects
the riskreturn characteristics for every available ˛nancial security. In practice, this strategy is obviously
unrealistic. But as we shall also discover later, studies have shown that with less than thirty diversi˛ed
constituents it is feasible to reach a position where a portfolio™s standard deviation is close to that for
the market portfolio.
Of course, without today™s computer technology and sophisticated soˆware, there are still problems, as we
observed in previous Chapters (
PTFA
and
PTFAE
). ˜e signi˛cance of covariance terms in the Markowitz
variance calculation are so unwieldy for a welldiversi˛ed risky portfolio that for most investors, with
a global capital market to choose from, it is untenable. Even if we substitute the correlation coe˚cient
into the covariance of the portfolio variance, the mathematical complexity of the variancecovariance
matrix calculations for a risky multiasset portfolio still limits its applicability. So, is there an alternative?
4.1 The Market Portfolio and Tobin™s Theorem
We have already explained that if an individual or company objective is to minimize the standard deviation
of an investment™s expected return, this could be determined by reference to indi˙erence curves, which
calibrate attitudes toward risk and return. In Chapter ˜ree (
PTFA
) and the summary of Chapter Two
(PTFAE
) we graphed an equation of
indi˝erence
between portfolio risk and portfolio return for any
rational investor relative to their optimum portfolio.
Figure 4.1:
the Determination of an Optimum Portfolio: The MultiAsset Case
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54 The Market Portfolio
Diagrammatically, you will recall that the
optimum
portfolio is determined at the
point
where one of the
investor™s indi˙erence curves (riskreturn pro˛le) is
tangential
to the frontier of e˚cient portfolios. ˜is
portfolio (point E on the curve FF
1 in Figure 4.1) is optimal because it provides the best combination
of risk and return to suit their preferences.
However, apart from the computational di˚culty of deriving optimum portfolios using variance
covariance matrix calculations (think 1950™s theory without twenty˛rst century computer technology
soˆware) this policy prescription only concerns
wholly
risky portfolios.
But what if
riskfree
investments (such as government stocks) are included in portfolios? Presumably,
investors who are
totally
riskaverse would opt for a
riskless
selection of ˛nancial and government
securities, including cash. ˜ose who require an element of
liquidity
would construct a
mixed
portfolio
that combines risk and riskfree investments to satisfy their needs.
˜us, what we require is a more sophisticated model than that initially o˙ered by Markowitz, whereby
the returns on new investments (riskfree or otherwise) can be compared with the risk of the market
portfolio.
Fortunately, John Tobin (1958) developed such a model, built on Markowitz e˚ciency and the
perfect
capital market assumptions that underpin the Separation ˜eorem of Irving Fisher (1930) (with which
you should be familiar).
Tobin demonstrates that in a perfect market where risky ˛nancial securities are traded with the option
to lend or borrow at a riskfree rate, using riskfree assets, such as government securities.
Investors and companies need not calculate a multiplicity of covariance terms. All
they require is the covariance of a new investment™s return with the overall return
on the e˜cient market portfolio.
To understand what is now termed Tobin™s
Separation ˜eorem
, suppose every stock market participant
invests in all the market™s risky securities, with their expenditure in each proportionate to the market™s
total capitalisation. Every investor™s risky portfolio would now correspond to the market portfolio with
a market return and market standard deviation, which we shall denote as M, r
m and s
m, respectively.
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55 The Market Portfolio
Tobin maintains that in
perfect
capital markets that are
e˛cient
, such an investment strategy is completely
rational. In
equilibrium
, security prices will re˝ect their ﬁtrueﬂ intrinsic value. In other words, they
provide a return commensurate with a degree of risk that justi˛es their inclusion in the market portfolio.
Obviously, if a security™s return does not compensate for risk; rational investors will want to sell their
holding. But with no takers, price must fall and the yield will rise until the riskreturn tradeo˙ once
again merits the security™s inclusion in the market portfolio. Conversely, excess returns will lead to buying
pressure that raises price and depresses yield as the security moves back into equilibrium.
˜is phenomenon is portrayed in Figure 4.2, where M represents the 100 per cent risky market portfolio,
which lies along the e˚ciency frontier of all risky investment opportunities given by the curve FF
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56 The Market Portfolio
Figure 4.2:
The Capital Market Line
Now, assume that all market participants can not only choose risky investments with the return r
m in the market portfolio M. ˜ey also have the option of investing in
riskfree
securities (such as shortŒterm
government stocks) at a riskfree rate, r
f. According to their aversion to risk and their desire for liquidity,
we can now separate their preferences, (hence the term Separation ˜eorem). Investors may now opt
for a
riskless
portfolio, or a
mixed
portfolio, which comprises any preferred combination of risk and
riskfree securities.
Diagrammatically
, investors can combine the market portfolio with riskfree investments to create a
portfolio between r
f and M in Figure 4.2. If a line is drawn from the riskfree return r
f on the vertical
axis of our diagram to the point of tangency with the e˚ciency frontier at point M, it is obvious that
part of the original frontier (FF
1) is now
ine˛cient
.Below M, a higher return can be achieved for the same level of risk by combining the market
portfolio with riskfree assets. Since rf denotes a riskless portfolio, the line rf M represents
increasing proportions of portfolio M combined with a reducing balance of investment at the
riskfree rate.
Of course, as Fisher ˛rst
explained way back in 1930, if capital markets are perfect (where borrowing and
lending rates are equal) there is nothing to prevent individuals from borrowing at the riskfree rate to
build up their investment portfolios. Tobin therefore adapted this concept to show that if investors could
borrow at a riskfree rate and invest more in portfolio M using borrowed funds, they could construct a
portfolio beyond M in Figure 4.2.
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57 The Market Portfolio
To show this, the line r
f to M has been extended to point P and beyond to CML. ˜e e˙ect eliminates
the remainder of our original e˚ciency frontier. Any initial e˚cient portfolios lying along the curve
MF1 are no longer desirable. With borrowing (leverage) there are always better portfolios with higher
returns for the same risk. ˜e line (r
f MCML) in Figure 4.2 is a new portfolio ﬁe˚ciency frontierﬂ for
all investors, termed the
Capital Market Line
(CML).Activity 1
To illustrate the purpose of the CML, let us assume that historically an investment
company has
passively
held a market portfolio (M) of risky assets. This fund tracks the
London FTSE 100 (Footsie) on behalf of its clients.
However, with increasing global uncertainty the company now wishes to manage their
portfolio
actively
, introducing riskfree investments into the mix and even borrowing
funds if necessary.
Using Figure 4.2 for reference, brie˝y explain how the company™s new strategy would
rede˚ne its optimum portfolio (or portfolios) if it is willing to borrow up to point P?
˜e portfolio lendingborrowing line (r
f MP) in Figure 4.2 is the new e˚ciency frontier (CML) for all
the company™s portfolio constituents. Portfolios lying along the CML between r
f and M are constructed by
placing a proportion of their available funds in the market portfolio and the residual in riskfree assets.
To establish a portfolio lying halfway up the line r
f M, the company should divide funds equally between
the two.
Portfolios lying along the CML beyond M (for example, P in the diagram) are constructed by placing
all their funds in M, plus an amount borrowed at the riskfree rate (r
f). ˜e amount borrowed would
equal the ratio of the line r
f Œ M: MP.
4.2 The CML and Quantitative Analyses
We have observed diagrammatically that if capital markets are e˚cient, all rational investors would
ideally hold the market portfolio (M) irrespective of their risk attitudes. By ˛nding the point of tangency
between the e˚ciency frontier (FF
1) and the capital market line (CML) then borrowing or lending at
the riskfree rate (r
f) it is also possible for individual investors to achieve a desired balance between risk
and return elsewhere on the CML.
Obviously, portfolios whose riskreturn characteristics place it below the CML are ine˚cient and could
be improved by altering their composition. It is also possible that an investor might ﬁbeat the marketﬂ (if
only by luck rather than judgement) so that the portfolio™s riskreturn pro˛le would lie above the CML,
making it ﬁsupere˚cientﬂ. However, if markets are e˚cient without access to insider information (as
portfolio theorists assume) then this will be a temporary phenomenon.
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58 The Market Portfolio
Like the work of Fisher and Markowitz before him, Tobin™s theorem is another landmark in the
development of ˛nancial theory, which you ought to read at source. At the very least you need to be
able to manipulate the following statistical equations, which we shall apply to an Exercise in Chapter
Four of our companion text (
PTFAE
).Portfolio Risk
So, let us begin by
rede˚ning
our general portfolio risk formula based on Equation (7) for the standard
deviation (which you ˛rst encountered in Chapter Two). Combining the market variance of returns
(2m) with the variance of riskfree investments (
2f):(27) p = [x22m + (1x)2 2f + 2x (1x) m f COR
(m,f)]˜e ˛rst point to note is that because the variability of riskfree returns is obviously zero, their variance
(2f) and standard deviation (
f ) equals zero. ˜e second and third terms of Equation (27), which de˛ne
the variance of the riskfree investment and the correlation coe˚cient, disappear completely. ˜us,
Equation (27) for the portfolio™s standard deviation simpli˛es to:
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59 The Market Portfolio
Rearranging the terms of Equation (28) with only one unknown and simplifying, we can also determine
the proportion of funds (
x) invested in the market portfolio. Given any investor™s preferred portfolio and
the market standard deviation of returns (
p and
m):(28) x = p /mPortfolio Return
In Chapter Two we de˛ned the expected return for a twoasset portfolio R(P) as the weighted average of
expected returns from two investments or projects, R(A) and R(B), where the weights are the proportional
funds invested in each. Mathematically, this is given by:
(1) R(P) = x R(A) + (1 Œ x) R(B)
˜e equation can be adapted to calculate the expected return (r
p) for any portfolio that includes a
combination of risky and riskfree investments, whose returns are (r
m) and (r
f) respectively.
(29) rp = x rm + (1x) rf˜e Market Price of Risk or Risk Premium
Because the CML is a
simple linear regression
line, its slope (ˇ
m) is a
constant
, measured by:
(30) ˇm = (rm Œ rf ) / m˜e expected return for any portfolio on the CML (r
p ) can also be expressed as:
(31) rp = rf + [(rm Œ rf ) / m] pGiven r
f (the riskfree rate of return) which is the
intercept
illustrated in Figure 4.2 (where
p equals zero)
rm is still the market portfolio return and
m and
p de˛ne market risk and the risk of the particular
portfolio, respectively.
˜e
constant
slope of the CML (ˇ
m) de˛ned by Equation (30)
is called the
market price of risk
. It represents
the
incremental
return (r
m Œ rf) obtained by investing in the market portfolio (M) divided by market risk
(m). In e˙ect it is the
risk premium
added to the riskfree rate (sketched in Figure 4.2) to establish the
total return for any particular portfolio™s riskreturn trade o˙.
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60 The Market Portfolio
For example, with a risk premium ˇ
m de˛ned by Equation (30), the incremental return from a portfolio
bearing risk (
p) in relation to market risk (
m ) is given by:
ˇm (p Œ m)˜is can be con˛rmed if we were to compare a particular portfolio return with that for the market
portfolio. ˜e di˙erence between the two (r
p Œ rm) equals the market price of risk (ˇ
m) times the spread
(p Œ m).To summarise
, the expected return for any e˛cient portfolio lying on the CML comprising
the market portfolio, plus either borrowing or lending at the risk free rate can be
expressed by simplifying Equations (30) and (31), so that:
(32) rp = rf + ˙m p In other words, the expected return of an e˛cient portfolio (rp) equals the riskfree rate
of return (r
f) plus a risk premium (˙
m.p).This premium re˝ects the market™s riskreturn
tradeo˜ (˙
m) combined with the portfolio™s own risk (
p).4.3 Systematic and Unsystematic Risk
˜e objective of portfolio diversi˛cation is the selection of investment opportunities that reduce
total
portfolio risk without compromising
overall
return. ˜e preceding analysis based on Markowitz e˚ciency
and Tobin™s Separation ˜eorem in perfect capital markets indicates that:
If the standard deviation (risk) of an individual investment is higher than that of the
portfolio in which it is held, then part of the standard deviation must have been
diversi˚ed away through correlation with other portfolio constituents.
A high level of diversi˛cation results in rational investors holding the market portfolio, which they will
do in combination with lending or borrowing at the riskfree rate. ˜is leaves only the element of risk
that is correlated with the market as a whole. In other words portfolio risk equals market risk, which is
undiversi˛able
p Œ mTo clarify this point for future analysis, Figure 4.3 summarises the relationship between total risk and
its component parts where.
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61 The Market Portfolio
Total risk
is split between:
Systematic
or market risk
, so called because it is endemic throughout the system (market)
and is undiversi˛able. It relates to general economic factors that a˙ect all ˛rms and ˛nancial
securities, and explains why share prices tend to move in sympathy.
Unsystematic risk,
sometimes termed
speci˚c
, residual
, or
unique risk
, relates to speci˛c
(unique) economic factors, which impact upon individual industries, companies, securities
and projects. It can be eliminated entirely through e˚cient diversi˛cation.
In terms of our earlier analysis, systematic risk measures the extent to which an investment™s return moves
sympathetically (systematically) with all the ˛nancial securities that comprise the market portfolio (the
system
). It describes a particular portfolio™s inherent sensitivity to global political and macroeconomic
volatility. ˜e best recent example, of course, is the 2007 ˛nancial meltdown and subsequent economic
recession. Because individual companies or investors have no control over such events, they require a
rate of return commensurate with their relative systematic risk. ˜e greater this risk, the higher the rate
of return required by those with widely diversi˛ed portfolios that re˝ect movements in the market as
a whole.
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62 The Market Portfolio
Figure 4.3:
The Interrelationship of Risk Concepts
In contrast, unsystematic risk relates to an individual security™s price or even a project and is independent
of market risk. Applied to individual companies, it is caused by microeconomic factors such as the level
of pro˛tability, product innovation and the quality of ˛nancial management. Because it is completely
diversi˛able (variations in returns cancel out over time) unsystematic risk carries no market premium.
˜us, all the risk in a fully diversi˛ed portfolio is market or systematic risk.
You may have encountered systematic risk elsewhere in your studies under other names. For example,
Figure 4.3 reveals that systemati
c risk comprises a company™s
business risk
and possibly
˚nancial risk.
Certainly if you have read the author™s other
SFM
texts, you will recall that business risk re˝ects the
unavoidable variability of project returns according to the nature of the investment (
investment policy
). ˜is may be higher or lower than that for other projects, or the market as a whole. Systematic risk
may also re˝ect a premium for ˛nancial risk, which arises from the proportion of debt to equity in a
˛rm™s capital structure (gearing) and the amount of dividends paid in relation to the level of retained
earnings, (
˚nancial policy
). Of course, there is considerable empirical support for the view that ˛nancial
risk is irrelevant based on the seminal work of ModiglianiMiller (1958 and 1961) explained in
SFM
. Irrespective of whether ˛nancial policies matter, for the moment all we need say is that for allequity
˛rms with full dividend distribution policies, there is an academic consensus that business risk equals
systematic (market) risk and is not diversi˛able.
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63 The Market Portfolio
Review Activity
Given our analysis of Markowitz e˛ciency and the Separation of Tobin, brie˝y
summarise the implications for optimum portfolio management?
4.4 Summary and Conclusions
Markowitz
, explains how investors or companies can reduce risk but maintain their return by holding
more than one investment providing their returns are not positively correlated. ˜is implies that all
rational investors will diversify their risky investments into a portfolio.
Tobin
illustrates how the introduction of riskfree investments further reduces portfolio risk, using the
CML to de˛ne a new frontier of e˚cient portfolios.
Consequently, all investors are capable of eliminating unsystematic risk by expanding their investment
portfolios until they re˝ect the market portfolio.
Based on numerous studies, Figure 4.4 highlights the empirical fact that up to 95 percent of unsystematic
risk can be diversi˛ed away by randomly increasing the number of investments in a portfolio to about
thirty. With one investment, portfolio risk is represented by the sum of unsystematic and systematic risk,
i.e. the investment™s
total risk
as measured by its standard deviation. When the portfolio constituents
reach double ˛gures virtually all the risk associated with holding that portfolio becomes systematic or
market risk. See Fisher and Lorie (1970) for one of the earliest and best reviews of the phenomenon.
Figure 4.4:
Portfolio Risk and Diversi˚cation
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64 The Market Portfolio
It should therefore come as no surprise that without todays computer technology and soˆware to solve
their problems:
Academic and ˚nancial analysts of the 1960™s, requiring a much simpler model than that
o˜ered by Markowitz to enable them to diversify e˛ciently, were quick to appreciate the
work of Tobin and the utility of the relationship between the systematic risk of either a
project, a ˚nancial security, or a portfolio and their returns.
4.5 Selected References
1. Markowitz, H.M., ﬁPortfolio Selectionﬂ,
˜e Journal of Finance,
Vol. 13, No. 1, March 1952.
2. Tobin, J., ﬁLiquidity Preferences as Behaviour Towards Riskﬂ,
Review of Economic Studies
, February 1958.
3. Fisher, I.,
˜e ˜eory of Interest,
Macmillan (London), 1930.
4. Modigliani, F. and Miller, M.H., ﬁ˜e Cost of Capital, Corporation Finance and the ˜eory
of Investmentﬂ,
American Economic Review,
Vol. XLVIII, No. 4, September 1958.
5. Miller, M.H. and Modigliani, F., ﬁDividend Policy, Growth and the Valuation of Sharesﬂ,
Journal of Business of the University of Chicago,
Vol. 34, No. 4, October 1961.
6. Fisher, L. and Lorie, J., ﬁSome Studies of Variability of Returns on Investment in Common
Stocksﬂ,
Journal of Business,
April 1970.
7. Hill, R.A.,
bookboon.com
Strategic Financial Management,
2009. Strategic Financial Management; Exercises,
2009. Portfolio ˜eory and Financial Analyses; Exercises
, 2010.Download free eBooks at bookboon.com
65 Part III:
Models Of Capital Asset Pricing
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66 The Beta Factor
5 The Beta Factor
Introduction
In an ideal world, the portfolio theory of Markowitz (1952) should provide management with a practical
model for measuring the extent to which the pattern of returns from a new project a˙ects the risk of a
˛rm™s existing operations. For those playing the stock market, portfolio analysis should also reveal the
e˙ects of adding new securities to an existing spread. ˜e objective of e˚cient portfolio diversi˛cation
is to achieve an overall standard deviation lower than that of its component parts without compromising
overall return.
Unfortunately, as we observed in Part Two, the calculation of the covariance terms in the risk (variance)
equation becomes unwieldy as the number of portfolio constituents increase. So much so, that without
today™s computer technology and soˆware, the operational utility of the basic model is severely limited.
Academic contemporaries of Markowitz therefore sought alternative ways to measure investment risk
˜is began with the realisation that the
total risk
of an investment (the standard deviation of its returns)
within a diversi˛ed portfolio can be divided into
systematic
and
unsystematic
risk. You will recall that the
latter can be eliminated entirely by e˚cient diversi˛cation. ˜e other (also termed
market
risk) cannot.
It therefore a˙ects the overall risk of the portfolio in which the investment is included.
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67 The Beta Factor
Since all rational investors (including management) interested in wealth maximisation should be
concerned with individual security (or project) risk relative to the stock market as a whole, portfolio
analysts were quick to appreciate the importance of systematic (market) risk. According to Tobin (1958)
it represents the only risk that they will pay a premium to avoid.
Using this information and the assumptions of perfect markets with opportunities for riskfree
investment, the required return on a risky investment was therefore rede˛ned as the riskfree return,
plus a premium for risk. ˜is premium is not determined by the total risk of the investment, but only
by its systematic (market) risk.
Of course, the systematic risk of an individual ˛nancial security (a company™s share, say) might be higher
or lower than the overall risk of the market within which it is listed. Likewise, the systematic risk for
some projects may di˙er from others within an individual company. And this is where the theoretical
development of the beta factor (b) and the Capital Asset Pricing Model (CAPM) ˛t into portfolio analysis.
We shall begin Part ˜ree by de˛ning the relationship between an individual investment™s systematic risk
and market risk measured by (b
j) its
beta
factor (or coe˚cient). Using our
earlier notation
and continuing
with the
equation numbering
from previous Chapters, which ended with Equation (32):
(33) j = COV(j,m)
VAR(m)
˜is factor equals the covariance of an investment™s return, relative to the market portfolio, divided by
the variance of that portfolio.
As we shall discover, beta factors exhibit the following characteristics:
˜e market as a whole has a b = 1
A riskfree security has a b = 0
A security with systematic risk below the market average has a b < 1
A security with systematic risk above the market average has a b > 1
A security with systematic risk equal to the market average has a b = 1
The signi˚cance of a security™s
value for the purpose of stock market investment is
quite straightforward. If overall returns are expected to fall (
a bear
market) it is worth
buying securities with low
values because they are expected to fall less than the market.
Conversely, if returns are expected to rise generally (a
bull scenario) it is worth buying
securities with high
values because they should rise faster than the market.
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68 The Beta Factor
Ideally, beta factors should re˝ect
expectations
about the
future
responsiveness of security (or project)
returns to corresponding changes in the market. However, without this information, we shall explain
how individual returns can be compared with the market by plotting a
linear
regression line through
historical
data.
Armed with an operational measure for the market price of risk (
), in Chapter Six we shall explain
the rationale for the Capital Asset Pricing Model (CAPM) as an alternative to Markowitz theory for
constructing e˚cient portfolios.
For any investment with a beta of
j, its expected return is given by the CAPM
equation
:(34) rj = rf + (rm Œ rf) jSimilarly, because all the characteristics of systematic betas apply to a
portfolio
, as well as an
individual
security, any portfolio return (r
p) with a portfolio beta (
p) can be de˛ned as:
(35) rp = rf + (rm Œ rf) pFor a given a level of systematic risk, the CAPM determines the expected rate of return for any investment
relative to its beta value. ˜is equals the riskfree rate of interest,
plus
the product of a market risk
premium and the investment™s beta coe˚cient. For example, the mean return on equity that provides
adequate compensation for holding a share is the value obtained by incorporating the appropriate equity
beta into the CAPM equation.
The CAPM can be used to estimate the expected return on a security, portfolio, or project,
by investors, or management, who desire to eliminate unsystematic risk through e˛cient
diversi˚cation and assess the required return for a given level of nondiversi˚able,
systematic (market) risk. As a consequence, they can tailor their portfolio of investments to
suit their individual riskreturn (utility) pro˚les.
Finally, in Chapter Six we shall validate the CAPM by reviewing the balance of empirical evidence for
its application within the context of capital markets.
In Chapter Seven we shall then focus on the CAPM™s operational relevance for strategic ˛nancial
management within a corporate capital budgeting framework, characterised by capital gearing. And as
we shall explain, the stock market CAPM can be modi˛ed to derive a project discount rate based on
the systematic risk of an individual investment. Moreover, it can be used to compare di˙erent projects
across di˙erent risk classes.
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69 The Beta Factor
At the end of Part ˜ree, by crossreferencing this text and its companion Exercises (underpinned by
SFM
and
SFME
material from
Bookboon
) you should therefore be able to con˛rm that:
The CAPM not only represents a viable alternative to managerial investment appraisal
techniques using NPV wealth maximisation, meanvariance analysis, expected utility models
and the WACC concept. It also establishes a mathematical connection with the seminal
leverage theories of Modigliani and Miller (MM 1958 and 1961).
5.1 Beta, Systemic Risk and the Characteristic Line
Suppose the price of a share selected for inclusion in a portfolio happens to increase when the equity
market rises. Of prime concern to investors is the extent to which the share™s total price increased because
of unsystematic (speci˛c) risk, which is diversi˛able, rather than systematic (market) risk that is not.
A practical solution to the problem is to isolate systemic risk by comparing past trends between individual
share price movements with movements in the market as a whole, using an appropriate allshare stock
market index.
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70 The Beta Factor
So, we could plot a ﬁscatterﬂ diagram that correlates percentage movements for:
˜e selected share price, on the vertical axis,
Overall market prices using a relevant index on the horizontal axis.
˜e ﬁspreadﬂ of observations equals unsystematic risk. Our line of ﬁbest ˛tﬂ represents systematic risk
determined by
regressing
historical share prices against the overall market over the time period. Using
the statistical method of
least squares
, this linear regression is termed the share™s
Characteristic Line.
Figure 5.1:
The Relationship between Security Prices and Market Movements The Characteristic Line
As Figure 5.1 reveals, the
vertical intercept
of the regression line, termed the
alpha factor
(ˇ) measures
the average percentage movement in share price if there is no movement in the market. It represents
the amount by which an individual share price is greater or less than the market™s systemic risk would
lead us to expect. A positive alpha indicates that a share has outperformed the market and
vice versa
.˜e
slope
of our regression line in relation to the horizontal axis is the
beta factor
(b) measured by the
share™s covariance with the market (rather than individual securities) divided by the variance of the
market. ˜is calibrates the
volatility
of an individual share price relative to market movements, (more of
which later). For the moment, su˚ce it to say that the steeper the Characteristic Line the more volatile
the share™s performance and the higher its systematic risk.
Moreover, if the slope of the Characteristic
Line is very steep, b will be greater than 1.0. ˜e security™s performance is volatile and the systematic risk
is high. If we performed a similar analysis for another security, the line might be very shallow. In this
case, the security will have a low degree of systematic risk. It is far less volatile than the market portfolio
and b will be less than 1.0. Needless to say, when b equals 1.0 then a security™s price has ﬁtrackedﬂ the
market as a whole and exhibits
zero
volatility.
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71 The Beta Factor
˜e beta factor has two further convenient statistical properties applicable to investors generally and
management in particular.
First, it is a far simpler, computational proxy for the covariance (relative risk) in our original Markowitz
portfolio model. Instead of generating numerous new covariance terms, when portfolio constituents
(securitiesprojects) increase with diversi˛cation, all we require is the covariance on the additional
investment relative to the e˚cient market portfolio.
Second, the Characteristic Line applies to investment
returns
, as well as
prices
. All risky investments with
a market price must have an expected return associated with risk, which justify their inclusion within
the market portfolio that all risky investors are willing to hold.
Activity 1
If you read di˜erent ˚nancial texts, the presentation of the Characteristic Line is a
common source of confusion. Authors often de˚ne the axes di˜erently, sometimes
with prices and sometimes returns.
Consider Figure 5.2, where
returns
have been substituted for the
prices
of Figure 5.1.
Does this a˜ect our linear interpretation of alpha and beta?
Figure 5.2:
The Relationship between Security Returns and Market Returns The Characteristic Line
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72 The Beta Factor
˜e substitution of returns for prices in the regression doesn™t a˙ect our interpretation of the graph,
because returns obviously determine prices.
˜e horizontal intercept (ˇ) now measures the extent to which
returns
on an investment are
greater or less than those for the market portfolio.
˜e steeper the slope of the Characteristic Line, then the more volatile the return, the higher
the systematic risk (b)
and
vice versa
.We began by graphing the security prices of risky investments and total market capitalisation using a
stock market index because it serves to remind us that the development of Capital Market ˜eory initially
arose from portfolio theory as a pricing model. However, because theorists discovered that returns (like
prices) can also be correlated to the market, with important consequences for internal management
decision making, as well as stock market investment, many modern texts focus on returns and skip
pricing theory altogether.
Henceforth, we too, shall place increasing emphasis on returns to set the scene for Chapter Seven. ˜ere
our ultimate concern will relate to strategic ˛nancial management and an optimum project selection
process derived from models of capital asset pricing using b factors for individual companies that provide
the highest expected return in terms of investor attitudes to the risk involved.
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73 The Beta Factor
5.2 The Mathematical Derivation of Beta
So far, we have only explained a beta factor (b) by reference to a
graphical
relationship between the
pricing or return of an individual security™s risk and overall market risk. Let us now derive
mathematical
formulae for b by adapting our
earlier notation
and continuing with the
equation numbering
from
previous Chapters. ˜is ended with Equation (32) and began with Equation (33) in our Introduction
the present one.
Suppose an individual was to place all their investment funds in all the ˛nancial securities that comprise
the global stock market in proportion to the individual value of each constituent relative to the market™s
total value.
˜e market portfolio has a variance of VAR(m) and the covariance of an individual security j with
the market average is COV(j,m). So, the relative risk (the security™s beta) denoted by
j is given by our
earlier equation:
(33) j = COV(j,m)
VAR(m)
Alternatively, we know from Chapter Two that given the relationship between the covariance and the
linear
correlation coe˚cient, the covariance term in Equation (33) can be rewritten as:
COV (
j,m) = COR (
j,m) . j mSo, we can also de˛ne a theoretical value for beta as follows:
j = COR(
j,m) . j m2(m)And simplifying, (allowing for the equation numbering in our Introduction to this Chapter):
(36) j = COR(j m)
j (m)If information on the variance or standard deviation and covariance or correlation coe˚cient is readily
available, the calculation of beta is extremely straightforward using either equation. Ideally, we should
determine b using
forecast
data (in order to appraise
future
investments). In its absence, however, we
can derive an
estimator
using leastsquares regression. ˜is plots a security™s
historical
periodic
return
against the corresponding return for the appropriate market index. For example, an ordinary share™s
return r
t (common stock) is given by:
rt = Increase in the period™s exdiv value per share + the dividend per share paid
Share value at the beginning of the period
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74 The Beta Factor
Obviously it needs to be adjusted for events such as bonus or rights issues and any capital reorganisation
reconstruction. Fortunately, because of their ease of calculation,
estimators are published regularly
by the ˛nancial services industry for stock exchange listings worldwide. A particularly ˛ne example is
the London Business School Risk Management Service (LBSRMS) that supplies details of equity betas,
which are also geared up (leveraged) according to the ˛rm™s capital structure (more of which later in
Chapter Seven).
Given the universal, freely available publication of beta factors, considerable empirical research on their
behaviour has been undertaken over a long period of time. So much so, that as a measure of systematic
risk they are now known to exhibit another extremely convenient property (which also explains their
popularity within the investment community).
Although alpha risk varies considerably over time, numerous studies (beginning with Black, Jensen and
Scholes in 1972) have continually shown that beta values are more stable. ˜ey move only slowly and
display a near
straightline
relationship with their returns. ˜e longer the period analysed, the better. ˜e
more data analysed, the better. ˜us, betas are invaluable for e˚cient portfolio selection. Investors can
tailor a portfolio to their speci˛c riskreturn (utility) requirements, aiming to hold
aggressive
stocks with
a b in excess of one while the market is rising, and less than one (
defensive
) when the market is falling.
Activity 2
Explain the investment implications of a beta factor of 1.15 and a beta factor that
is less than the market portfolio
A beta of 1.15 implies that if the underlying market with a beta factor of one were to rise by 10 per cent,
then the stock may be expected to rise by 11.5 per cent. Conversely, a security with a beta of less than one
would not be as responsive to market movements. In this situation, smaller systemic risk would mean
that investors would be satis˛ed with a return that is below the market average. ˜e market portfolio
has a beta of one precisely because the covariance of the market portfolio with itself is identical to the
variance of the market portfolio. Needless to say, a riskfree investment has a beta of zero because its
covariance with the market is zero.
5.3 The Security Market Line
Let us pause for thought:
Total
risk comprises unsystematic and systematic risk.
Unsystematic
risk, unique to each company, can be eliminated by portfolio diversi˛cation.
Systematic
risk is undiversi˛able and depends on the market as a whole.
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75 The Beta Factor
˜ese distinctions between total, unsystematic and systematic risk are vital to our understanding of the
development of Modern Portfolio ˜eory (MPT). Not only do they validate beta factors as a measure
of the only risk that investors will pay a premium to avoid. As we shall discover, they also explain the
rationale for the Capital Asset Pricing Model (CAPM) whereby investors can assess the portfolio returns
that satisfy their riskreturn requirements. So, before we consider the CAPM in detail, let us contrast
systemic beta analysis with basic portfolio theory that only considers total risk.
˜e linear relationship between
total
portfolio risk and expected returns, the
Capital Market Line
(CML) based on Markowitz e˚ciency and Tobin™s ˜eorem, graphed in Chapter Four does not hold for
individual
risky investments. Conversely, all the characteristics of systemic beta risk apply to portfolios
and
individual securities. ˜e beta of a portfolio is simply the weighted average of the beta factors of
its constituents.
˜is new relationship becomes clear if we reconstruct the CML (Figures 4.2 and 4.1 from Chapter Four
of our ˜eory and Exercise texts, respectively) to form what is termed the
Security Market Line
(SML).
As Figure 5.3 illustrates, the expected return is still calibrated on the vertical axis but the SML substitutes
systemic risk (
) for total risk (
p) on the horizontal axis of our earlier CML diagrams.
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76 The Beta Factor
Once beta factors are calculated (not a problem) the SML provides a universal measure of risk that still
adheres to
Markowitz e˛ciency
and his criteria for portfolio selection, namely:
Maximise return for a given level of risk
Minimise risk for a given level of return
Like the CML, the SML still con˛rms that the
optimum
portfolio is the
market
portfolio. Because the
return on a portfolio (or security) depends on whether it follows market prices as a whole, the closer
the correlation between a portfolio (security) and the market index, then the greater will be its expected
return. Finally, the SML predicts that both portfolios and securities with higher beta values will have
higher returns and
vice versa
.Figure 5.3:
The Security Market Line
As Figure 5.3 illustrates, the expected riskrate return of r
m from a balanced market portfolio (M) will
correspond to a beta value of one, since the portfolio cannot be more or less risky than the market as a
whole. ˜e expected return on riskfree investment (r
f) obviously exhibits a beta value of zero.
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77 The Beta Factor
Portfolio A (or anywhere on the line r
f M) represents a
lending
portfolio with a mixture of risk and
riskfree securities. Portfolio B is a
borrowing
or leveraged portfolio, because beyond (M) additional
securities are purchased by borrowing at the riskfree rate of interest.
Review Activity
Given your knowledge of perfect capital markets, Fisher™s Separation Theorem, stock market
e˛ciency, meanvariance analysis, utility theory, Markowitz e˛ciency and Tobin™s Capital
Market Line (CML):
Brie˝y summarise what the Security Market Line (SML) o˜ers rational, riskaverse individuals
seeking a welldiversi˚ed portfolio of investments?
5.4 Summary and Conclusions
˜roughout our analyses (including the background
SFM
and
SFME
texts) we have observed how rational,
riskaverse individuals and companies operating in perfect markets with no ﬁbarriers to tradeﬂ can rank
individual
investments by interpreting their expected returns and standard deviations using the concept
of expected utility to calibrate their riskreturn attitudes. In this book (and its Exercise companion) we
began with the same meanvariance e˚ciency criteria to derive optimum
portfolio
investments that can
reduce risk (standard deviation) without impairing return. In Part Two this culminated with Tobin™s
˜eorem and the CML that incorporates borrowing and lending opportunities to de˛ne optimum
ﬁe˚cientﬂ portfolio investment opportunities.
Unfortunately, the CML only calibrates total risk (
p) not all of which is diversi˛able. Fortunately, the
SML o˙ers investors a lifeline, by discriminating between nonsystemic and systemic risk. ˜e latter is
de˛ned by a beta factor that measures relative (systematic) risk, which explains how rational investors
with di˙erent utility (riskreturn) requirements can choose an optimum portfolio by borrowing or
lending at the riskfree rate.
We shall return to this topic in Chapter Six when risk is related to the expected return from an investment
or portfolio using the CAPM.
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78 The Beta Factor
5.5 Selected References
1. Markowitz, H.M., ﬁPortfolio Selectionﬂ,
˜e Journal of Finance,
Vol. 13, No. 1, March 1952.
2. Tobin, J., ﬁLiquidity Preferences as Behaviour Towards Riskﬂ,
Review of Economic Studies
, February 1958.
3. Fisher, I.,
˜e ˜eory of Interest,
Macmillan (London), 1930.
4. Modigliani, F. and Miller, M.H., ﬁ˜e Cost of Capital, Corporation Finance and the ˜eory
of Investmentﬂ,
American Economic Review,
Vol. XLVIII, No. 4, September 1958.
5. Miller, M.H. and Modigliani, F., ﬁDividend Policy, Growth and the Valuation of Sharesﬂ,
Journal of Business of the University of Chicago,
Vol. 34, No. 4, October 1961.
6. Black, F., Jensen, M.L. and Scholes, M., ﬁ˜e Capital Asset Pricing Model: Some Empirical
Testsﬂ, reprinted in Jensen, M.L. ed,
Studies in the ˜eory of Capital Markets,
Praeger (New
York), 1972.
7. Hill, R.A.,
bookboon.com
Strategic Financial Management,
2009. Strategic Financial Management; Exercises,
2009. Portfolio ˜eory and Financial Analyses
; Exercises, 2010.
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79 The Capital Asset Pricing Model (Capm)
6
The Capital Asset Pricing Model
(Capm)
Introduction
Basic portfolio theory de˛nes the expected return from a risky investment in general terms as the risk
free return, plus a premium for risk. However, we have observed that this premium is determined not
by the overall risk of the investment but only by its systematic (market) risk.
(36) j = COR(j m)
j (m)Using the geometry of the Security Market Line (SML) that determines the market risk premium (b),
numerous academics, notably Sharpe (1963) followed by Lintner (1965), Treynor (1965) and Mossin
(1966) were quick to develop (quite independently) the
Capital Asset Pricing Model
(CAPM) as a logical
extension to basic portfolio theory.
Today, the CAPM is regarded by many as a superior model of security price behaviour to others based
on wealth maximisation criteria with which you should be familiar. For example, unlike the dividend
and earnings share valuation models of Gordon (1962) and Modigliani and Miller (1961) covered in
our
SFM
and
SFME
texts, the CAPM explicitly identi˛es the risk associated with an ordinary share
(common stock) as well as the future returns it is expected to generate. Moreover, the CAPM can also
express investment returns in two forms
For individual securities:
(34) rj = rf + (rm Œ rf) jAnd because systemic betas apply to a
portfolio
, as well as an
individual
investment:
(35) rp = rf + (rm Œ rf) pDownload free eBooks at bookboon.com
80 The Capital Asset Pricing Model (Capm)
For a given a level of systematic risk, the CAPM determines the expected rate of return for any investment
(security, project, or portfolio) relative to its beta value de˛ned by the SML (a market index). As we
shall discover, it also establishes whether individual securities, projects (or their portfolios) are under
or overpriced relative to the market, (hence its name).˜e CAPM can therefore be used by investors or
management, who desire to eliminate unsystematic risk through e˚cient diversi˛cation and assess the
required return for a given level of nondiversi˛able, systematic (market) risk. As a consequence, they
can tailor their portfolio of investments to suit their individual risk return (utility) pro˛les.
6.1 The CAPM Assumptions
The CAPM is a
singleperiod
model, which means that all investors make the same decision
over the same time horizon. Expected returns arise from expectations over the same period.
The CAPM is a
singleindex
model because systemic risk is prescribed entirely by
one factor; the
beta factor.
The CAPM is de˚ned by random variables that are normally distributed, characterised by mean
expected returns and covariances, upon which all investors agree.
Markowitz meanvariance e˛ciency criteria based on perfect markets still determine the
optimum portfolio (P).
MAX: R(P), given
(P) MIN: (P), given R(P).
All investments are in˚nitely divisible.
All investors are rational and risk averse.
All investors are price takers, since no individual, ˚rm or ˚nancial institution is large
enough to distort prevailing market values.
All investors can borrowlend without restriction at the riskfree market rate of interest.
Transaction costs are zero and the tax system is neutral.
There is a perfect capital market where all information is available and costless.
Table 6.1:
The CAPM Assumptions
˜e application of the CAPM and beta factors is straight forward as far as stock market tactics are
concerned. ˜e model assumes that investors have three options when managing a portfolio:
1) To trade,
2) To hold,
3) To substitute, (
i.e. securities for property, property for cash, cash for gold
etc.
).Download free eBooks at bookboon.com
81 The Capital Asset Pricing Model (Capm)
A pro˛table trade is accomplished by buying (selling), undervalued (overvalued) securities relative
to an appropriate measure of systematic risk, a global stock market index such as the FT/ S&P World
Index. If the market is ﬁbullishﬂ and prices are expected to rise generally, it is worth buying securities
with high b values because they can be expected to rise faster than the market. Conversely, if markets
are ﬁbearishﬂ and expected to fall, then securities with low beta factors are more attractive because they
can be expected to fall less than prices overall.
To validate the CAPM, however, there are other assumptions (many of which should be familiar) that
we will question later. For the moment, they are simply listed in Table 6.1 without comment to develop
our analysis.
6.2 The Mathematical Derivation of the CAPM
Given the perfect market assumptions of the single periodindex CAPM, consider an investor who initially
places nearly all their funds in a portfolio re˝ecting the composition of the market. ˜ey subsequently
invest the balance in security
j. Using sequential numbering from previous equations, let us de˛ne R(P)
the expected return on the revised portfolio as the weighted average of the expected returns of the
individual components. ˜is is given by adapting Equation (1) the basic formula for portfolio return
from Chapter Two (remember?).
(37) R(P) = x rj + (1x) rmDownload free eBooks at bookboon.com
82 The Capital Asset Pricing Model (Capm)
Where:
x = an extremely small proportion,
rj = expected rate of return on security
j,rm = expected rate of return on the market portfolio.
Subject to the original model™s nonnegativity constraints and requirements that sources of funds equal
uses, the portfolio variance is also based on Equation (2) from Chapter Two:
(38) VAR(P) =
x2 VAR (r
j) + (1x)2 VAR(r
m) + 2x (1x) COV(r
j,rm)˜e portfolio will be e˚cient if it has the lowest degree of risk for the highest expected return, given by
the objective functions:
MAX: R(P), given VAR(P)
MIN: VAR(P), given R(P)
But note what has happened. By introducing security
j into the market portfolio, the investor has altered
the riskreturn characteristics of their original portfolio. According to Sharpe and others, the
marginal
return per unit of risk
is derived by:
1) Di˝erentiating
R(P) with respect to the investment in security
j; R(P)/ x,2) Di˝erentiating
VAR(P) with respect to the investment in security
j; VAR(P)/
x.3) Solving
as x 0Since (iii) above simpli˛es to
R(P)/ VAR(P) as
x tends to zero,
the incremental return per unit of risk
is therefore given by:
(39)
for
x 0However, you will recall from our explanation of the SML that an investor can either borrow or lend
at the riskfree rate of interest (r
f) with a beta value of zero. So, by incorporating a riskfree investment
or a liability (if
x is negative) the incremental rate of return given by Equation (39) is established by
substituting rj = r
f and
j = 0 into the equation such that:
(40)
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83 The Capital Asset Pricing Model (Capm)
In a perfectly competitive capital market, the
incremental
riskreturn tradeo˙ must be the same for all
investors. So, Equations (39) and (40) are identical:
(41) Now, multiplying both sides of Equation (41) by the denominator on the leˆ hand side and rearranging
terms, Sharpe™s
one
period,
single
factor Capital Asset Pricing Model (CAPM) for individual investments
(explained earlier) is con˛rmed as follows:
(34) rj = rf + ( rm Œ rf) jAnd because systematic betas apply to a
portfolio
, as well as an
individual
investment we can de˛ne R(P)
using our earlier notation
(35) rp = rf + ( rm Œ rf) pRemember, the CAPM is a
one period
model because the independent variables, r
f, rm and b
j are assumed
to remain constant over the time horizon. It is also a
single factor
model because systematic risk is
prescribed entirely by the beta factor.
Equation (34) represents the expected rate of return on security
j, which comprises a risk free return
plus a premium for accepting market risk (the market rate minus the risk free rate), assuming that all
correctly priced securities will lie on the SML. ˜e market portfolio o˙ers a premium (r
m Œ r
f) j over the
riskfree rate, r
f , which may di˙er from the
jth security™s risk premium measured by the beta factor
j.˜us, Sharpe™s CAPM (like the others mentioned earlier, Lintner
et. al
.) enables an investor to
establish whether individual securities (or portfolios) are under or overpriced, since the linear
relationship between their expected rates of return and beta factors (systematic risk) can be compared
with the SML (the market index).
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84 The Capital Asset Pricing Model (Capm)
6.3 The Relationship between the CAPM and SML
Figure 6.1:
The CAPM and SML
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85 The Capital Asset Pricing Model (Capm)
Activity 1
Take a look at Figure 6.1. This is a reproduction of Figure 5.3, the Security Market Line
(SML) explained in Chapter Five. At one extreme we have the expected return on riskfree
investment (r
f) with a beta value of zero. At the other, portfolio B is a
borrowing
or leveraged
portfolio with a beta of 1.5, which contains securities purchased by borrowing at the riskfree
rate of interest. However, superimposed on the new graph are other beta values associated
with expected returns, one of which is de˚ned by the point X.
Explain its portfolio implications for rational, riskaverse investors.
Suppose we are considering investing in the security denoted by X on the graph with an expected return
of 8 per cent and a beta coe˚cient of 0.5. We can see that the return is too low for the risk involved
and that the security is overpriced because X is located below the SML. Consequently, rational investors
wishing to sell their holdings would need to drop their price and increase the return (yield) until it
impinges upon the SML at point A.
Given the slope of the SML de˛ned by a risk free rate of 6 per cent and a market return of 16 per cent
from a risky balanced portfolio, Figure 6.1 illustrates why the new
equilibrium
rate of return A with a
beta value of 0.5 should be 11%. You can con˛rm this using the CAPM model:
(34) rj = rf + ( rm Œ rf) jwhere the expected return equals the riskfree rate, plus the market rate minus the riskfree rate, multiplied
by the beta factor.
11% = 6% + (16% Œ 6%).0.5It is also clear from Figure 6.1 why investing in a security such as Y is bene˛cial. Stocks above the line
will be in great demand, so they will rise in price causing a fall in yield.
From our examination of the data we can therefore draw the following conclusions.
In theoretical e˛cient capital markets in equilibrium that assimilate all information
concerning a security into its price, all securities (or portfolios) will lie on the SML.
Individual investors need not conform to the market portfolio. They need only determine
how much systematic risk they wish to assume, leaving market forces to ensure that any
security can be expected to yield the appropriate return for its beta.
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86 The Capital Asset Pricing Model (Capm)
6.4 Criticism of the CAPM
Like much else in modern ˛nancial theory, critics of the CAPM maintain that its assumptions are so
restrictive as to invalidate its conclusions, notably investor rationality, perfect markets and linearity.
Moreover, the CAPM is only a singleperiod model, based on estimates for the riskfree rate, market
return and beta factor, which are all said to be di˚cult to determine in practice. Finally, the CAPM also
assumes that investors will hold a well diversi˛ed portfolio. It therefore ignores unsystematic risk, which
may be of vital importance to investors who do not. However, as we have emphasised elsewhere in our
studies, the relevant question is whether a model works, despite its limitations?
Although there is evidence by Black (1993) to suggest that the CAPM does not work accurately for
investments with very high or low betas, overstating the required return for the former and understating
the required return for the latter (suggesting compensation for unsystematic risk) most tests validate the
CAPM for a broad spectrum of beta values.
˜e betareturn characteristics of individual securities also hold for portfolios. In fact, the beta of a
portfolio seems more stable because ˝uctuations among its constituents tend to cancel each other out.
Way back in 1972, Black, Jensen and Scholes analysed
the New York Stock Exchange over a 35 year
period by dividing the listing into 10 portfolios, the ˛rst comprising constituents with the lowest beta
factors and so on. Based on time series tests and crosssectional analyses they found that the intercept
term was not equal to the riskfree rate of interest, r
f , (which they approximated by 30 day Treasury
bills). However, their study revealed an almost
straightline
relationship between a portfolio™s beta and
its average return.
Critics still maintained that beta will only be stable if a company™s systematic risk remains the same
because it continues in the same line of business. However, subsequent studies using historical data to
establish the stability of beta over time con˛rmed that if beta factors are calculated from past observable
returns this problem can be resolved.
˜e longer the period analysed, the better.
˜e more data, the better, which suggests the use of a
sector
beta, rather than a
company
beta.
As an alternative to the basic CAPM, Black (1972) also tested a
twofactor
model, which assumed that
investors couldn™t borrow at a risk free rate but at a rate, r
z, de˛ned as the return on a portfolio with a
beta value of zero. ˜is is equivalent to a portfolio whose covariance with the market portfolio™s rate of
return is zero.
(42) rj = rz + (rm Œ rz) jDownload free eBooks at bookboon.com
87 The Capital Asset Pricing Model (Capm)
˜e Black twofactor model con˛rmed the study by Black, Jensen and Scholes (
op.cit.
) and that a zero
beta portfolio with an expected return, r
z exceeds the risk free rate of interests, r
f .Despite further modi˛cations to the original model, which need not detain us here, (multifactors,
multiperiods) the CAPM in its traditional guise continues to attract criticism, particularly in relation
to its fundamental assumptions.
For example, even if we accept that all investors can borrow or lend at the riskfree rate, it does not follow
that r
f describes a riskfree investment in
real
terms. Future in˝ation rates are neither predetermined,
nor a˙ect individuals equally.
Marginal adjustments to a portfolio™s constituents may also be prohibited by substantial transaction costs
that outweigh their future bene˛ts.
˜e ˛scal system can also be
biased
with di˙erential tax rates on income and capital gains. So much so,
that di˙erent investors will construct or subscribe to portfolios that minimise their personal tax liability
(a clientele
e˙ect).
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88 The Capital Asset Pricing Model (Capm)
And what if the stock market is
ine˛cient
? As we have discussed at great length in this study and
elsewhere in our
SFM
companion texts, investors can not only pro˛t from legitimate data by paying for
the privilege. With access to insider information, which may even anticipate global events (such as the
1987 crash, millennium dot. com. ˛asco and 2007 meltdown) perhaps they can also destabilise markets.
Conversely, even if we assume that the market is
e˛cient
, it has not always responded to signi˛cant
changes in information, ranging from patterns of dividend distribution, takeover activity and government
policies through to global geopolitical events. Why else do even professional
active
managed portfolio
funds periodically underperform relative to the market index? ˜e only way to ﬁbeatﬂ the market, or so
the argument goes, is either through pure speculation or insider information. Otherwise, adopt a
passive
policy of ﬁbuy and holdﬂ to track the market portfolio and hope for the best
Other forces are also at work to invalidate the CAPM. You will recall that the model implies that the
optimum portfolio is the market portfolio, which lies on the Security Market Line (SML) with a beta
factor of one. Individual securities and portfolios with di˙erent levels of risk (betas) can be priced because
their expected rate of return and beta can be compared with the SML. In equilibrium, all securities will
lie on the line, because those above or below are either under or over priced in relation to their expected
return. ˜us, market demand, or the lack of it, will elicit either a rise or fall on price, until the return
matches that of the market.
However, we have a problem, namely how to de˛ne the market. It is frequently forgotten that the CAPM
is a
linear
mode
l based on
partial
equilibrium analysis
that subscribes to the ModiglianiMiller (MM)
law
of one price
. Based on their arbitrage process, (1958 and 1961) explained in our
SFM
companion texts,
you will recall that two similar assets must be valued equally. In other words, two portfolio constituents
that contribute the same amount of risk to the overall portfolio are
close substitutes
. So, they should
exhibit the same return. But what if an asset has no close substitute, such as the market itself? How do
we establish whether the market is under or overvalued?
As Roll (1977) ˛rst noted, most CAPM tests may be invalid because all stock exchange indices are only a
partial
measure of the
true
global market portfolio. Explained simply, by de˛nition the market portfolio
should include every security worldwide.
To prove the point, Roll demonstrated that a change in the surrogate for the American stock market
from the Standard and Poor 500 to the Wilshire 5000 could radically alter a security™s expected return
as predicted by the CAPM. Furthermore, if betas and returns derived from a stock market listing were
unrelated, the securities might still be priced correctly relative to the global market portfolio. Conversely,
even if the listing was e˚cient (shares with high betas did exhibit high returns) there is no obvious reason
for assuming that each constituent™s return is only a˙ected by global systematic risk.
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89 The Capital Asset Pricing Model (Capm)
A further criticism of the CAPM is that however one de˛nes the capital market, movements up and down
are dominated by price changes in the securities of larger companies, Yet as Fama and French (1992)
˛rst observed, it is to these companies that institutional portfolio fund managers (active or passive)
are attracted, though they may underperform relative to smaller companies. Explained simply, fund
managers with perhaps billions to spend are hostages to fortune, even in a ﬁbullﬂ scenario. ˜ey have
neither the time, nor research budgets to scrutinise innumerable companies ﬁneglectedﬂ by the market
with small capitalisations based on little information.
Turning to ﬁbearﬂ markets characterised by rising systematic risk, multinational portfolio fund managers
still have little room for manoeuvre. According to Hill and Meredith (1994):
˜e ˛rst option is to liquidate all or part of a portfolio. However, if the whole portfolio were sold it
could be di˚cult to dispose of a large fund quickly and e˚ciently without a˙ecting the market. Unlike
a private investor, total disposal may also be against the fund™s trust deed. If only part of the portfolio
was liquidated there is the further question of which securities to sell.
˜e second option is to reduce all holdings, to be followed by subsequent reinvestment when the market
bottoms out. However, the fall in prices may have to be in excess of 2 per cent to cover transaction and
commission costs,).
Clearly, both alternatives may be untenable and impose signi˛cant constraints upon the opportunities
to control risk. Indeed, those sceptical of portfolio management generally and the CAPM in particular,
regard successful investment as a matter of luck rather than judgement, insider information, or unlikely
economic circumstances where all prices move in unison.
Review Activity
Assuming the riskfree rate and expected return on the market portfolio for Muse plc are 10
per cent and 18 per cent respectively:
(1) Use the CAPM to calculate the expected returns on stocks with the following beta values:
= 0, 0.5, 1.0, 1.5 (2) How would each stock ˚t into the investment plans for an actively managed portfolio?
(1) Using the data and Equation (34) to derive the expected returns, the CAPM reveals that if:
= 0, 0.5, 1.0 or 1.5
rj =10+(18 Œ10) = 10%, 14%, 18% and 22%, respectively
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90 The Capital Asset Pricing Model (Capm)
(2) ˜e investment plans for an actively portfolio can be explained as follows.
With a beta value greater than one, a stock™s expected return should ﬁbeatﬂ the market and
vice versa.
A beta of one produces a return equal to the market return and a beta value of zero produces an expected
return equal to the riskfree rate.
˜us, we can classify investment into three broad categories of risk for the purpose of ﬁactiveﬂ portfolio
management:
> 1.0 = Aggressive
< 1.0 = Defensive
= 1.0 = Neutral
A portfolio manager™s interest in each category of beta factor concerns the likely impact of changes in a
market index on the share™s expected return. Aggressive shares can be expected to outperform the market
in either direction. If the return on the index is expected to rise, the returns on high beta shares will rise
faster. Conversely, if the market is expected to fall, then their returns will fall faster. Defensive shares
with beta values lower than one will obviously underperform relative to the market in each direction.
Neutral shares will tend to shadow it.
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91 The Capital Asset Pricing Model (Capm)
Hence, rather than adopt a passive policy of ﬁbuy and holdﬂ by constructing a
tracker
fund representative
of a stock market index, ﬁactiveﬂ portfolio managers will wish to pursue:
An
aggressive
investment strategy by moving into
high
beta shares when stock market returns are expected
to rise (a bull market).
A defensive
strategy based on
low
beta shares and even riskfree assets with zero betas, when the market
is about to fall (a bear market).
6.4 Summary and Conclusions
If the capital market is so unpredictable that it is impossible for investors to beat it using the CAPM,
it is important to remember that the operational usefulness of alternative meanvariance analyses and
expected utility models explained at the very beginning of this text are also severely limited in their
application. ˜is is why the investment community turned to Markowitz portfolio theory and the Sharpe
CAPM for inspiration. And why others re˛ned these models into a coherent body of work now termed
Modern Portfolio ˜eory (MPT) to facilitate the e˚cient diversi˛cation of investment.
Since the new millennium, despite the volatility of ˛nancial markets and their tendency to crash (or
perhaps because of it) the portfolio objectives of investors remain the same:
To eliminate unsystematic risk and to establish the optimum relationship between
the systematic risk of a ˚nancial security, project, or portfolio, and their respective
returns; a tradeo˜ with which investors feels comfortable.
So to conclude our studies, what does the
single period
model CAPM based on Markowitz e˚ciency contribute
to Strategic Financial Management within the context of their
multiperiod
investment, dividend and ˛nancing
decisions, which previous models considered throughout this text and
SFM
have failed to deliver?
6.5 Selected References
1. Sharpe, W., ﬁA Simpli˛ed Model for Portfolio Analysisﬂ,
Management Science,
Vol. 9, No. 2,
January 1963.
2. Lintner, J., ﬁ˜e Valuation of Risk Assets and the Selection of Risk Investments in Stock
Portfolios and Capital Budgetsﬂ,
Review of Economic Statistics
, Vol. 47, No. 1, December 1965.
3. Treynor, J.L., ﬁHow to Rate Management of Investment Fundsﬂ,
Harvard Business Review
, JanuaryŒFebruary 1965.
4. Mossin, J., ﬁEquilibrium in a Capital Asset Marketﬂ,
Econometrica,
Vol. 34, 1966.
5. Gordon, M. J., ˜e Investment, Financing and Valuation of a Corporation, Irwin, 1962.
6. Miller, M.H. and Modigliani, F., ﬁDividend policy, growth and the valuation of sharesﬂ,
˜e
Journal of Business of the University of Chicago
, Vol. XXXIV, No. 4 October 1961.
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92 The Capital Asset Pricing Model (Capm)
7. Black, F., ﬁBeta and Returnﬂ,
Journal of Portfolio Management
Vol. 20, Fall 1993.
8. Black, F., Jensen, M.L. and Scholes, M., ﬁ˜e Capital Asset Pricing Model: Some Empirical
Testsﬂ, reprinted in Jensen, M.L. ed,
Studies in the ˜eory of Capital Markets,
Praeger (New
York), 1972.
9. Black, F., ﬁCapital Market Equilibrium with Restricted Borrowingﬂ,
Journal of Business,
Vol. 45, July 1972.
10. Modigliani, F. and Miller, M.H., ﬁ˜e Cost of Capital, Corporation Finance and the ˜eory
of Investmentﬂ,
American Economic Review,
Vol. XLVIII, No.4, September 1958.
11. Roll, R., ﬁA Critique of the Asset Pricing ˜eory™s Testsﬂ,
Journal of Financial Economics,
Vol. 4, March 1977.
12. Fama, E.F. and French, KR., ﬁ˜e CrossSection of Expected Stock Returnsﬂ,
Journal of
Finance,
Vol. 47, No. 3, June 1992.
13. Hill, R.A., and Meredith, S., ﬁInsurance Institutions and Fund Management: A UK
Perspectiveﬂ,
Journal of Applied Accounting Research,
Vol. 1, No. 1994.
14. Fisher, I.,
˜e ˜eory of Interest,
Macmillan (London), 1930.
15. Markowitz, H.M., ﬁPortfolio Selectionﬂ,
˜e Journal of Finance,
Vol. 13, No. 1, March 1952.
16. Hill, R.A.,
bookboon.com
Strategic Financial Management,
2009. Strategic Financial Management; Exercises,
2009. Portfolio ˜eory and Financial Analyses; Exercises
, 2010.Download free eBooks at bookboon.com
93 Capital Budgeting, Capital Structure Andthe Capm
7
Capital Budgeting, Capital
Structure Andthe Capm
Introduction
So far, our study of Markowitz e˚ciency, beta factors and the CAPM has concentrated on the stock
market™s analyses of security prices and expected returns by ˛nancial institutions and private individuals.
˜is is logical because it re˝ects the rationale behind the chronological development of Modern Portfolio
˜eory (MPT). But what about the impact of MPT on individual companies and their appraisal of capital
projects upon which all investors absolutely depend? If management wish to maximise shareholder
wealth, then surely a new project™s expected return and systematic risk relative to the company™s existing
investment portfolio and stock market behaviour, like that for any ˛nancial security, is a vitally important
consideration.
In this Chapter we shall explore the corporate applications of the CAPM by strategic ˛nancial
management, namely:
˜e derivation of a discount rate for the appraisal of capital investment projects on the basis
of their systematic risk.
How the CAPM can be used to match discount rates to the systematic risk of projects that
di˙er from the current business risk of a ˛rm.
Because the model can be applied to projects ˛nanced by debt as well as equity, we shall also establish a
mathematical connection between the CAPM and the ModiglianiMiller (MM) theory of capital gearing
based on their ﬁlaw of one priceﬂ covered in our SFM companion texts.
7.1 Capital Budgeting and the CAPM
As an alternative to calculating a ˛rm™s weighted average cost of capital (WACC) explained in the
SFM
texts, the theoretical derivation of a project discount rate using the CAPM and its application to NPV
maximisation is quite straightforward. A riskadjusted discount rate for the
jth project is simply the
riskfree rate added to the product of the market premium and the
project
beta, given by the following
expression for the familiar CAPM equation:
(45) rj = rf + (rm Œ rf) j˜e project beta (
j) measures the
systematic
risk of a speci˛c project (more of which later). For the
moment, su˚ce it to say that in many textbooks the project beta is also termed an
asset
beta denoted by
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94 Capital Budgeting, Capital Structure Andthe Capm
We then derive the expected NPV by discounting the average net annual cash ˝ows at the riskadjusted
rate from which the initial cost of the investment is subtracted, using a mathematical formulation that
you ˛rst encountered in Part Two of the
SFM
texts.
Individual
projects are acceptable if:
NPV 0Collectively
, projects that satisfy this criterion can also be ranked for selection according to the size of
their NPV. Given:
NPVA > NPVB > –NPVN we prefer project A.
So far, so good; but remember that CAPM project discount rates are still based on a number of
simplifying assumptions. Apart from adhering to the traditional concept of perfect capital markets
(Fisher™s Separation ˜eorem) and meanvariance analysis (Markowitz e˚ciency) the CAPM is only a
singleperiod
model, whereas most projects are
multiperiod
problems.
According to the CAPM, all investors face the same set of investment opportunities,
have the same expectations about the future and make decisions within one time
horizon. Any new investment made
now
will be realised
then, next year (say) and a new
decision made.
Given the assumptions of perfect markets characterised by random cash ˝ow distributions, there is no
theoretical objection to using a
singleperiod
model to generate an NPV discount rate for the evaluation
of a ˛rm™s
multiperiod
investment plans. ˜e only constraints are that the riskfree rate of interest, the
average market rate of return and the beta factor associated with a particular investment are
constant
throughout its life.
Unfortunately, in reality the riskfree rate, the market rate and beta are rarely constant. However the
problem is not insoluble. We just substitute
periodic
riskadjusted discount rates (now dated r
j t) for a
constant r
j into Equation (46) for each future ﬁstate of the worldﬂ, even if only one of the variables in
Equation (45) changes. It should also be noted that the phenomenon of multiple discount rates combined
with di˙erent economic circumstances is not unique to the CAPM. As we ˛rst observed in Part Two of
SFM
, it is common throughout NPV analyses.
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95 Capital Budgeting, Capital Structure Andthe Capm
On ˛rst acquaintance, it would therefore appear that the application of a CAPM return to capital budgeting
decisions provides corporate ˛nancial management with a practical alternative to the WACC approach.
A particular weakness of WACC is that it de˛nes a single discount rate applicable to
all
projects, based
on the assumptions that their acceptance doesn™t change the company™s risk or capital structure and is
marginal
to existing activities. In contrast, the CAPM rate varies from project to project, according to
the systematic risk of each investment proposal. However, the CAPM still poses a number of problems
that must be resolved if it is to be applied successfully, notably how to derive an appropriate
project
beta
factor and how to measure the impact of
capital gearing
on its calculation.
For these reasons, we shall defer a comprehensive numerical example of investment appraisal and the
CAPM until you read the Exercises associated with this chapter, by which time we will have covered
the issues involved.
7.2 The Estimation of Project Betas
For simplicity throughout previous chapters we have used a
general
beta factor (
) applicable to the
overall
systemic risk of portfolios, securities and projects. But now our analysis is becoming more
focussed,
precise
notation and de˛nitions are necessary to
discriminate
between systemic
business
and
˚nancial
risk. Table 7.1 summarises the beta measures that we shall be using for future reference and
also highlights a number of problems.
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96 Capital Budgeting, Capital Structure Andthe Capm
= total
systematic
risk, which relates portfolio, security and project risk to
market
risk.
j = the business risk of a speci˚c project (
project
risk) for investment appraisal.
E = the published equity beta for a company that incorporates business risk and systematic
˛nancial risk if the ˚rm is geared.
A = the overall business risk of a ˚rm™s
assets (projects). It also equals a company™s
deleveraged
published beta (E) which measures business risk
free
from ˚nancial risk.
D = the beta value of debt (which obviously equals zero if it is riskfree).
EU and EG are the respective equity betas for similar allshare and geared companies.
Table 7.1:
Beta Factor De˚nitions
When an allequity company is considering a new project with the same level of risk as its current
portfolio of investments, total systematic risk
equals
business risk, such that:
= j = E = A = EUWhen a company is funded by a combination of debt and equity, this series of equalities must be modi˛ed
to incorporate a
premium
for systematic
˚nancial
risk. As we shall discover, the equity beta (
E) will be
a geared
beta re˝ecting business risk
plus
˛nancial risk, which measures shareholder exposure to debt
in their ˛rm™s capital structure. ˜us, the equity beta of an allshare company is always lower than that
for a geared ˛rm with the same business risk.
EU < EGIrrespective of a gearing problem, Table 7.1 reveals a further weakness of the CAPM. A company™s asset
beta (
A) should produce a discount rate that is appropriate for evaluating projects with the same overall
risk as the company itself. But what if a new project does not re˝ect the average risk of the company™s
assets? ˜en the use of
A is no more likely to produce a correct investment decision than the use of a
WACC calculation.
To illustrate the point, Figure 7.1 graphs the Security Market Line (SML) to show the required return
on a project for di˙erent beta factors, with a company™s WACC. ˜e use of the overall cost of capital to
evaluate projects whose risk di˙ers from the company™s average will be suboptimal where the IRR of
the project is in either of the two shaded sections. To calculate the correct CAPM discount rate using
Equation (45), we must determine the project beta.
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97 Capital Budgeting, Capital Structure Andthe Capm
Figure 7.1:
The SML, WACC and Project Betas
˜e company™s average beta, shown in the diagram, provides a measure of risk for the ˛rm™s overall
returns compared with that of the
market
. However, management™s investment decision is whether or not
to invest in a
project
. So, like the WACC, if the project involves diversi˛cation away from the ˛rm™s core
activities, we must use a beta coe˚cient appropriate to that class of investment. ˜e situation is similar
to a stock market investor considering whether to purchase the shares of the
company
. ˜e individual
would need to evaluate the share™s return by using the
market
beta in the CAPM.
Even if diversi˛cation is not contemplated, the project™s beta factor may not conform to the
average
for the ˛rm™s assets. For example, the investment proposal may exhibit high
operational gearing (
the
proportion of ˛xed to variable costs) in which case the project™s beta will exceed the average for
existing operations.
A serious con˝ict (the
agency
problem) can also arise for those companies producing few products,
or worse still a single product, particularly if management approach their capital budgeting decisions
based on selfinterest and shorttermism, rather than shareholder preferences. Shareholders with well
diversi˛ed corporate holdings who dominate such companies may prefer to see projects with high risk
(high beta coe˚cients) to balance their own portfolios. Such a strategy may carry the very real threat of
bankruptcy but in the event may have very little impact their overall returns. For corporate management,
the ˛rm™s employees and its suppliers, however, the policy may be economic suicide.
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98 Capital Budgeting, Capital Structure Andthe Capm
Fortunately, if a beta is required to validate the CAPM for project appraisal, help is at hand. Management
can obtain factors for companies operating in similar areas to the proposed project by subscribing to the
many commercial services that regularly publish beta coe˚cients for a large number of companies, world
wide. ˜eir listings also include stock exchange classi˛cations for
industry
betas. ˜ese are calculated
by taking the market average for quoted companies in the same industry. Research reveals that the
measurement errors of individual betas cancel out when industry betas are used. Moreover, the larger
the number of comparable beta constituents, the more reliable the industry factor.
So, if management wish to obtain an estimate for a project™s beta, it can identify the industry
in which the project falls, and use that industry™s beta as the project™s beta. This approach is
particularly suitable for highly
diversi˛ed
and divisionalised companies because their WACC or
market beta would be of little relevance as a discount rate for its divisional operations.
As an alternative to stock market data, management can also estimate a project™s beta from ˛rst principles
by calculating its
Fvalue
.The Fvalue of a project is rather like a beta factor in that it measures the variability of a
project™s performance,
relative
to the performance of an entity for which a beta value exists.
The entity could be the industry in which the project falls, the ˚rm undertaking the project, or
a division within the ˚rm that is responsible for the project.
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99 Capital Budgeting, Capital Structure Andthe Capm
A project™s Fvalue is de˛ned as follows:
(47) F = Percentage change in the project™s performance
Percentage change in the ﬁentity™sﬂ performance
As a result, we can obtain an estimate of a project™s beta through
one of three routes:
Activity 1
Let us suppose that a company™s divisional management is considering a capital project,
whose performance may be a˜ected 15 per cent either way, depending on whether the
division™s overall performance rises or falls by 10 per cent. In other words, the project™s
pro˚tability is expected to be more volatile than that of the division because of speci˚c
economic factors.
Calculate the project™s Fvalue and estimate the project™s beta coe˛cient given the
division™s beta factor is 0.80.
Using Equation (47) we can calculate the Fvalue as follows:
F = 15% /10% = 1.5If the divisional beta value is 0.80, then the project beta (
project
) can be estimated as follows:
(% change in the project™s performance / % change in the division™s performance) ×
division
project
= 1.5 × 0.80 = 1.2Download free eBooks at bookboon.com
100 Capital Budgeting, Capital Structure Andthe Capm
7.3 Capital Gearing and the Beta Factor
˜e CAPM de˛nes an individual investment™s risk relative to a welldiversi˛ed portfolio as
systematic
risk.
Measured by the beta coe˚cient, it is the only risk a company or an investor will pay a premium
to avoid. You will recall from Chapter Four (Figure 4.3) that it can be subdivided into:
Business risk
that arises from the variability of a ˛rm™s earnings caused by market forces,
Financial risk
associated with dividend policies and capital gearing, both of which may
amplify business risk
Without getting enmeshed in dividend policies, we shall accept the 1961 MM hypothesis that they are
irrelevant
. Based on their ﬁlaw of one priceﬂ (covered in the
SFM
texts and for which there is considerable
empirical support)
˚nancial
risk should not matter in an allequity company. Applied to the CAPM, the
systematic
risk of investors (who are all shareholders) can be de˛ned by the
business
risk of the ˛rm™s
underlying asset investments.
˜e
equity
beta of an unlevered (allequity) ˛rm equals an
asset
beta, which measures the business risk
of all its investments relative to the market for ordinary shares (common stock). Using earlier notation:
E U = A˜e CAPM return on project (r
j) is then de˛ned by:
(48) rj = rf + (rm Œ rf) AIf there is no debt in the ˛rm™s capital structure, the company™s asset (equity) beta equals the
weighted
average
of its individual project betas (b
i) based on the market value of equity.
(49) A = wi i = EUBut what about companies who decide to fund future investments by gearing up, or the vast majority
who already employ debt ˛nance?
To make rational decisions, it would appear that management now require an asset beta to measure a ˛rm™s
business risk that an ungeared equity beta can no longer provide. For example, an allequity company
may be considering a takeover that will be ˛nanced entirely by debt. To assess the acquisition™s viability,
management will now need to calculate their overall CAPM return on investment using an asset beta
that re˝ects a
leveraged
˛nancial mix of ˛xed interest on debt and dividends on shares.
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101 Capital Budgeting, Capital Structure Andthe Capm
Later in this chapter we shall resolve the dilemma using the predictions of MM™s capital structure
hypothesis (1958). Based on their law of one price, whereby similar ˛rms with the same risk characteristics
(except capital gearing) cannot sell at di˙erent prices, it con˛rms their dividend hypothesis, namely
that ˛nancial policy is irrelevant. First, however, let us develop the CAPM, to illustrate the relationship
between an asset beta and the equity and debt coe˚cients for a geared company.
When a ˛rm is ˛nanced by a debtequity mix, its earnings stream and associated risk is divided between
the ˛rm™s shareholders and providers of corporate debt. ˜e proportion of risk re˝ects the market values
of debt and equity respectively, de˛ned by the
debtequity ratio
. So, the equity beta will be a
geared
equity
beta. It not only incorporates business risk. It also determines shareholders™ exposure to ˛nancial risk
de˛ned by the proportion of contractual, ˛xed interest securities in the capital structure. For this reason
the equity beta of an unlevered company is always lower than the beta of a levered company.
Given a geared equity beta (
E) and debt beta (
D), the asset beta (
A) for a company™s investment in
risky capital projects can be expressed as a weighted average of the two:
(50) A = E G [VE / (VE + VD)] + D [VD / (VE + VD)]Download free eBooks at bookboon.com
102 Capital Budgeting, Capital Structure Andthe Capm
Where:
VE and V
D are the
market
values of equity and debt, respectively,
VE plus
VD de˛ne the ˛rm™s total market value (V).
Activity 2
A ˚rm with respective market values of •60m and •30m for equity and debt has an
equity beta of 1.5. The debt beta is zero.
(1) Use Equation (50) to calculate the asset beta (
A). (2) Explain a simpli˚ed mathematical structure of the calculation.
(1) ˜e asset beta (
A) calculation
(50) A = E G [VE / (VE + VD)] + D [VD /(VE + VD)] = 1.5 [60/(60 +30)] + 0 [30/(60 +30)] = 1.0(2) ˜e mathematical structure of
A.When a company is ˛nanced by debt and equity, management need to derive an asset beta using the
weighted average
of its geared equity and debt components. ˜e market values of debt and equity provide
the weightings for the calculation. Note, however, that because the market risk of debt (
D) was set to
zero, the right hand side of Equation (50) disappears.
˜is is not unusual. As explained in
SFM
, debt has priority over equity™s share of pro˛ts and the sale of
assets in the event of liquidation. ˜us, debt is more secure and if it is riskfree, there is no variance. So
if D equals zero, our previous equation for an asset beta reduces to:
(51) A = EG [VE / (VE + VD)]For example, if a company has an equity beta of 1.20, a debtequity ratio of 40 per cent and we assume
that debt is riskfree, the asset beta is given by:
A = 1.20 [100 / (100 + 40)]= 0.86Perhaps you also recall from
SFM
that debt is also a
tax deductible
expense in many economies. If we
incorporate this ˛scal adjustment into the previous equations (where t is the tax rate) we can rede˛ne the
mathematical relationship between the asset beta and its geared equity and debt counterparts as follows.
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103 Capital Budgeting, Capital Structure Andthe Capm
(52) A = E G {VE / [VE + VD(1t)]} + D {[VD(1t) / (VE + VD(1t))]}(53) A = E G {VE / [VE + VD(1t)]} if debt is riskfree
Despite the tax e˙ect, our methodology for deriving a company™s asset beta still reveals a
universal
feature
of the CAPM that ˛nancial management can usefully adopt to assess individual projects.
Whenever risky investments are combined, the asset beta of the resultant portfolio is a
weighted
average
of the debt and equity betas.
Activity 3
Consider a company with a current asset beta of 0.90. Itaccepts a project with a beta of
0.5 that is equivalent to 10 per cent of its corporate value after acceptance.
Con˚rm that:
1. The new (
expost
)beta coe˛cient of the company equals 0.86.
2. The
new project
reduces the original (
exante
) risk of the ˚rm™s
existing portfolio.
7.4 Capital Gearing and the CAPM
˜e CAPM de˛nes a project™s discount rate as a return equal to the riskfree rate of interest, plus the
product of the market premium and the project™s asset beta (a risk premium) to compensate for systematic
(business) risk. However, we now know that the ˛nancial risk associated with capital gearing can also
a˙ect beta factors. So, the discount rate derived from the CAPM for investment appraisal must also be
a˙ected, but how?
Let us ˛rst consider a company funded entirely by equity that is considering a new project with the same
level of risk as its existing activities. ˜e ˛rm™s equity beta (
EU) can be used as the project™s asset beta
(A) because the shareholders™ return (K
e) equals the company™s return (r
j) on a new project of equivalent
risk. So, the project return that provides adequate compensation for holding shares in the company is
the equity return (K
e) obtained by substituting the appropriate equity beta (
E) into the familiar CAPM
formula.
(54) Ke = rj = rf + ( rm Œ rf) EUDownload free eBooks at bookboon.com
104 Capital Budgeting, Capital Structure Andthe Capm
˜e CAPM therefore o˙ers management an important alternative to the derivation of project discount
rates that use the traditional dividend and earnings valuation models explained in the
SFM
texts. In
an
unlevered
(allequity) ˛rm, the
shareholders™
return (K
e) de˛nes the
company™s
cost of capital (K
U) as
follows:
(55) KU = Ke = rj = rf + ( rm Œ rf) EU˜e question we must now ask is whether Equation (55) has any parallel if the ˛rm is geared.
˜e short answer is yes. Rather than use traditional dividend, earnings and interest models to derive a
WACC (explained in
SFM
) we can substitute an appropriately
geared
asset beta for an
allequity
beta
into the CAPM to estimate the overall return on debt and equity capital for project appraisal.
(56) KG = rj = rf + ( rm Œ rf) ADownload free eBooks at bookboon.com
105 Capital Budgeting, Capital Structure Andthe Capm
7.5 ModiglianiMiller and the CAPM
Without debt in it capital structure, a company™s asset beta equals its equity beta for projects of equivalent
risk. However, according to MM™s theory of capital structure (
op. cit.
) based on their ﬁlaw of one priceﬂ
and the
arbitrage
process, companies that are identical in every respect apart from their gearing should
also have the same asset betas. Because their business risk is the same, the factors are not in˝uenced by
methods of ˛nancing. To summarise MM™s position
An ungeared company™s asset beta equals its equity beta.
A geared company™s asset beta is lower than its equity beta.
Irrespective of gearing, the asset beta for any company equals the equity beta of an
ungeared company with the same business risk.
The asset beta (equity beta) of an unlevered company can be used to evaluate
projects in the same risk class without considering their ˚nance.
j = A = EU < E GYou will recall from your studies that MM™s capital theory (like their dividend irrelevancy hypothesis)
depends on perfect market assumptions. However, because these assumptions also underpin much else
in ˛nance (including the CAPM) for the moment we shall accept them. To illustrate the MM relationship
between the beta factors of allequity and geared companies with the same systemic business risk, let us
begin with the following equation using our familiar notation in a taxless world.
(57) A = E U = E G [VE / (VE + VD)] + D [VD / (VE + VD)]If we now rearrange terms, divide through by V
E and solve for
EG, the mathematical relationship between
the geared and ungeared equity betas can be expressed as follows:
(58) EG = E U + (EU Œ D) VD / VE˜is equation reveals that the equity beta in a geared company equals the equity beta for an allshare
company in the same class of business risk,
plus
a premium for systemic ˛nancial risk. ˜e premium
represents the di˙erence between the allequity beta and debt beta multiplied by the debtequity ratio.
However, the important point is that the increase in the equity beta measured by the risk premium is
exactly o˙set by a lower debt factor as the ˛rm gears up leaving the asset beta una˙ected. In other words,
irrespective of leverage, the asset betas of the two ˛rms are still identical and equal the equity beta of
the ungeared ˛rm.
A = EU < E GDownload free eBooks at bookboon.com
106 Capital Budgeting, Capital Structure Andthe Capm
For those of you familiar with MM™s capital structure hypothesis, the parallels are striking. According
to MM, the expected return on equity for a geared ˛rm (K
eG) relative to the return (K
eU) for an allshare
˛rm in a taxless world equals:
(59) KeG = KeU + ( KeU Œ Kd ) VD / VE.˜is states that the return for a geared ˛rm equals an allequity return for the same class of business
risk,
plus
a ˛nancial risk premium de˛ned by the di˙erence between the allequity return and the cost of
debt multiplied by the debtequity ratio. ˜e premium compensates shareholders for increasing exposure
to ˛nancial risk as a ˛rm gears up. As we observed in
SFM
, however, because the cheaper cost of debt
exactly o˙sets rising equity yields, the overall cost of capital (WACC) is una˙ected. So, irrespective of
leverage, all ˛rms with the same business risk can use the cost of equity for an allshare ˛rm as a project
discount rate before considering methods of ˛nancing.
Turning to a world of taxation, where debt is a
taxdeductible
expense with a tax rate (t), we can rede˛ne
the equity beta of a geared company from Equation (58) as follows:
(60) EG = E U + [(EU Œ D) (1t) VD / VE ]And if debt is riskfree with
zero
variance, so that
D is zero, the formula simpli˛es to:
(61) EG = E U + [(EU (1t) VD / VE ]Review Activity
To illustrate the union between MM and the CAPM, consider a leveraged company in an
economy where interest is tax deductible at a 20 per cent corporate rate. 20 million ordinary
shares are authorised and issued at a current market value of £2.00 each (
exdiv
). The equity
beta is 1.5. Debt capital comprises £10 million, irredeemable 10 per cent loan stock, currently
trading at par value.
Calculate the company™s asset beta and brie˝y explain the result.
Since the equity beta for an
ungeared
company equals the asset beta for any company in the same risk
class, we can use Equation (61) to solve for
EU and hence
A as follows.
First, de˛ne the market values of equity and debt:
VE = £2.00 × 20 million = £40 million
VD = £10 million
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107 Capital Budgeting, Capital Structure Andthe Capm
Next, de˛ne the geared equity beta of 1.5 assuming that debt sold at par is riskfree (
D = 0).EG = 1.5 = EU + [EU (10.2) (10/40)]= EU {1 + [(10.2) (10/40)]}Finally, rearrange terms to solve for
EU and
A.A = EU = 1.5/1.2 = 1.25˜e result is to be expected. ˜e asset beta should be smaller than the geared equity beta (i.e. 1.25 <
1.5) since the systemic risk associated with the asset investment is only one component of the total risk
associated with the shares. ˜e asset beta measures business risk, whereas the geared beta measures
business and ˛nancial risk
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108 Capital Budgeting, Capital Structure Andthe Capm
7.5 Summary and Conclusions
If management use the CAPM rather than a WACC to obtain a riskadjusted discount rate for project
appraisal, they need to resolve the following questions
Question:
Is the business risk of a project equivalent to that for the company?
Answer:
YES
NO
Solution:
Use the company™s current
Use an equity beta for similar
equity beta
companies with similar projects
Question:
Is the chosen equity beta a˚ected by capital gearing?
Answer:
YES
NO
Solution:
Deleverage ﬁungearﬂ the
Use an equity beta equivalent to an
equity beta to derive an
asset beta if it is not a˙ected by gearing
asset beta
Having obtained an appropriate asset beta, the project discount rate may then be calculated using the
CAPM formula.
(62) rj = rf + ( rm Œ rf) bAAccording to MM™s capital structure theory, the asset betas of companies, or
projects, in the same class of business risk are identical irrespective of leverage.
Higher equity betas are o˜set by lower debt betas, just as higher equity yields
o˜set cheaper ˚nancing, as a ˚rm gears up
Even in a taxed world, it is possible to establish a connection between MM and the CAPM. With tax,
the MM cost of equity for a geared ˛rm is given by:
(63) KeG = KeU + [(KeU Œ Kd) (1t) VD / VE]According to the CAPM, the equity costs for an ungeared and geared ˛rm are given by:
(64) KeU = rj = rf + (rm Œ rf) bEU(65) KeG = rj = rf + (rm Œ rf) bEGWhere:
bA = bEU < bEGDownload free eBooks at bookboon.com
109 Capital Budgeting, Capital Structure Andthe Capm
If we assume that the company™s pretax cost of debt (K
d) in Equation (63) equals the riskfree rate (r
f) in Equations (64) and (65) we can write r
f for K
d in Equation (63). If we now substitute Equations (64)
and (65) into Equation (63) rearrange terms and simplify the result, we can con˛rm our earlier equation
for a geared equity beta:
(61) bEG = bE U + [bEU (1t) VD / VE ] = bEU {1 + [(1t) (10/40)]}For an application of this formula and the derivation of the cost of equity using the CAPM see Exercise
7.2 in the companion text.
7.6 Selected References
1. Modigliani, F. and Miller, M.H., ﬁ˜e Cost of Capital, Corporation Finance and the ˜eory
of Investmentﬂ,
American Economic Review,
Vol. XLVIII, No. 4, September 1958.
2. Miller, M.H. and Modigliani, F., ﬁDividend Policy, Growth and the Valuation of Sharesﬂ,
Journal of Business of the University of Chicago,
Vol. 34, No. 4, October 1961.
3. Hill, R.A.,
bookboon.com
 Strategic Financial Management,
2009. 
Strategic Financial Management; Exercises,
2009. 
Portfolio ˜eory and Financial Analyses; Exercises
, 2010.Download free eBooks at bookboon.com
110 Part IV:
Modern Portfolio Theory
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111 Arbitrage Pricing Theory and Beyond
8
Arbitrage Pricing Theory and
Beyond
Introduction
Previous chapters have presented a series of mathematical models representing a body of work termed
Modern Portfolio ˜eory (MPT) available to ˛nancial management when making strategic investment
decisions. MPT was originally developed for use by investors in securities, primarily fund managers and
professional analysts with the time, resources and expertise to implement the models and interpret their
˛ndings. Today, anybody with access to a computer, the appropriate soˆware and a reasonable ˛nancial
education can
model
quite complex tasks. Ultimately, however, it is people who should
interpret
the results
and not the computer. One lesson to be learnt from the1987 stock market crash is the catastrophic e˙ect
of automated trading. Another from the 2007 meltdown and ongoing ˛nancial crises is that computer
driven models can be so complex that hardly anybody understands what is going on anymore.
Like all ˛nancial theories, MPT should therefore be a guide to human action and not a substitute. And
while the bene˛ts of IT cannot be overstressed, you should always understand the ˛nancial model that
underpins the computer program you are running. So, let us review the original purpose of MPT, notably
the CAPM and then outline its subsequent development, notably
Arbitrage Pricing ˜eory
(APT).Download free eBooks at bookboon.com
112 Arbitrage Pricing Theory and Beyond
8.1 Portfolio Theory and the CAPM
You will recall that portfolio theory was initially developed by Harry Markowitz in the early 1950s to
explain how rational investors in perfect markets can minimise the risk of investment without comprising
return by diversifying and building up an e˚cient portfolio of investments. ˜e risk of each portfolio
is measured by the variability of possible returns about the mean measured by the standard deviation.
Investor riskreturn attitudes can be expressed by indi˙erence curves.
In 1958, John Tobin explained how the introduction of riskfree investments into Markowitz™ theory
further reduces the risk of a portfolio. According to Tobin, the Capital Market Line (CML) de˛nes a
new ﬁe˚cient frontierﬂ of investments for all investors.
Applied to project appraisal, Markowitz theory reveals that an individual project™s risk is not as important
as its e˙ect on the portfolio™s overall risk. So, whenever management evaluate a risky project they must
correlate the individual project risk with that for the existing portfolio it will join to assess its suitability.
Without the bene˛t of today™s computer technology, the mathematical complexity of the Markowitz
model arising from its covariance calculations prompted other theorists to develop alternative approaches
to e˚cient portfolio diversi˛cation. In the early 1960s by common consensus, the CAPM emerged as
a means whereby investors in ˛nancial securities were able to reduce their total risk by constructing
portfolios that discriminate between systematic (market risk) and unsystematic (speci˛c) risk.
˜e CAPM (usually associated with its prime advocate William Sharpe) states that the return on a security
or portfolio depends on whether their prices follow prices in the market as a whole by reference to a
suitable index, such as the FTSE 100. ˜e closer the correlation between the price of either an individual
security or a portfolio and this market proxy (measured by the beta factor) the greater will be their
expected returns. ˜us, if an investor knows the beta factor (relative risk) of a security or portfolio, their
returns can be predicted with accuracy. Pro˛table trading of portfolios is then accomplished by buying
(selling) undervalued (overvalued) securities relative to their systematic or market risk.
˜e CAPM also states that rational investors would choose to hold a portfolio that comprises the stock
market as a whole. By de˛nition, the market portfolio has a beta of one and is the most ﬁe˚cientﬂ in the
sense that no other combination of securities would provide a higher return for the same risk. You will
recall that it is a benchmark by which the CAPM establishes the Security Market Line (SML) in order
to compare other beta factors and returns. From this linear relationship, rational investors can ascertain
whether individual shares are underpriced or overpriced and determine other e˚cient portfolios that
balance their personal preference for risk and return.
According to the CAPM:
Any security with the same risk as the market will have a beta of 1.0; half as risky it will have a beta of
0.5; twice as risky it will have a beta of two.
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113 Arbitrage Pricing Theory and Beyond
˜e required rate of return given by the CAPM formula is composed of the return on riskfree investments,
plus
a risk premium measured by the di˙erence between the market return and the risk free rate multiplied
by an appropriate beta factor. For example, using Equation (45) for an investment with a beta of b
j:rj = rf + (rm Œ rf) bjIf we use the CAPM for project appraisal, rather than stock market analysis, the procedure remains the
same. Essentially, we are substituting an investment project for a security into a company™s portfolio of
investments, rather than a market portfolio. Risk relates to the cost of capital and management™s objective
is to obtain a discount rate to appraise individual projects.
8.2 Arbitrage Pricing Theory (APT)
So far, so good, but if we consider the purpose for which the CAPM was originally intended, namely stock
market investment, it has limitations. As we observed in Chapter Six, even the most actively managed,
institutional portfolio funds periodically underperform relative to the market as a whole.
Leaving aside the questionable assumptions that investors are rational, markets are e˚cient and prices
perform a ﬁrandom walkﬂ (dealt with in our Introduction and elsewhere) one early explanation of the
variable performance of portfolios, institutional or otherwise, was provided by Roll™s critique of the
CAPM (1977).Download free eBooks at bookboon.com
114 Arbitrage Pricing Theory and Beyond
According to Roll, it is not only impossible for the most discerning investor to establish the composition
of the
true
market portfolio, but there is also no reason to assume that a security™s expected return is
only a˙ected by systematic risk. In the same year, Firth (1977) also observed that if the stock market is
so e˚cient at assimilating all relevant information into security prices, it is impossible to claim that it is
either e˚cient or ine˚cient, since by de˛nition there is no alternative measurement criterion.
Such criticisms are important, not because they invalidate the CAPM (most empirical tests support it).
But because they gave credibility to an alternative approach to portfolio asset management and security
price determination based on stock market e˚ciency presented by Ross (1976). ˜is is termed
Arbitrage
Pricing ˜eory
(APT).Unlike the CAPM, which prices securities in relation to a
global market portfolio
, the APT possesses the
advantage of pricing of securities
in relation to each other
. ˜e
single
index
(beta factor) CAPM focuses
upon an assumed
speci˚c
linear relationship between betas and expected returns (systemic risk plotted
by the SML). ˜e APT is a
general
model that
subdivides
systematic risk into smaller components, which
need not be speci˛ed in advance. ˜ese de˛ne the
Arbitrage Pricing Plane
(APP). Any macroeconomic
factors, including market sentiment, which impact upon investor returns may be incorporated into
the APP (or ignored, if inconsequential.) For example, an unexpected change in the rate of in˝ation
(purchasing power risk) might a˙ect the price of securities generally. ˜e advantage of the APT, however,
is that it can be used to eliminate this risk speci˛cally, such as a pension fund portfolio™s requirement
that it should be immune to in˝ation.
Statistical tests on the model, including those of Roll and Ross (1980), established that a
four factor
linear version of the APT is a more accurate predictor of security and portfolio returns than the
single
factor
(index) CAPM. Speci˛cally, their APT states that the expected return is directly proportional to
its sensitivity to the following:
1) Interest rates,
2) In˝ation
3) Industrial productivity,
4) Investor risk attitudes.
˜e return equation for a fourfactor APP conforms to the following
simple linear
relationship for the
expected return on the
j th security in a portfolio:
(66) rj = a + b1 (r1) + b2(r2) + b3(r3) + b4 (r4)Download free eBooks at bookboon.com
115 Arbitrage Pricing Theory and Beyond
Where:
rj = expected rate of return on security
j,ri = expected return on factor i, (i = 1,2,3,4),
a = intercept,
bi = slope of r
i.˜e expected risk premium on the
jth security is de˛ned as the di˙erence between its expected return (r
j) and the riskfree fate of interest (r
f) associated with each factor™s return (r
i) and the security™s sensitivity
to each of these factors (b
i). ˜e fourfactor equation is given by:
(67) (rj Œ rf) = b1 (r1 Œ rf) + b2 (r2 Œ rf) + b3 (r3 Œ rf) + b4 (r4 Œ rf)Like the
speci˚c
CAPM, the
general
APT is still a
linear
model. ˜eoretically, it assumes that unsystematic
(unique) risk can be eliminated in a welldiversi˛ed portfolio, leaving only the portfolio™s sensitivity to
unexpected changes in
macroeconomic
factors. Subsequent studies, such as Chen, Roll and Ross (1986)
therefore focused upon identifying further signi˛cant factors and why the sensitivity of returns on a
particular share to each factor will vary. However, the work of Dhrymes, Friend and Gultekin (1984)
had already suggested that this line of research may be redundant. ˜eir study concluded that as the
number of portfolio constituents increases, a greater number of factors must be incorporated into the
model. ˜us, at the limit, the APT could be equivalent to the CAPM, which de˛nes risk in terms of a
single
overarching
microeconomic factor
relative to the return on the market portfolio.
For one of the ˛rst comprehensive reviews of the APT, which explains why even today it is not fully
developed and its application has been less successful than the CAPM, you should read Elton, Gruber
and Mei (1994). A more recent perspective on the APT is provided by Huberman and Wang (2005).
8.3 Summary and Conclusions
By now you appreciate that ˛nancial analysis is not an exact science and the theories upon which it is
based may even be ﬁbadﬂ science. ˜e fundamental problem is that real world economic decisions are
characterised by uncertainty. By de˛nition
uncertainty is nonquanti˚able
. Yet, rather than bury their
heads in sand, academics continue to defend ˛nancial models, such as the CAPM based on
simplifying
assumptions
that
rationalise
a search for investment opportunities in the
chaotic
world we inhabit. See
Fama and French (2003).
New mathematical theories and statistical models of investor
irrationality
and market
ine˛ciency,
characterised by
nonrandom
walks are being crystallised. ˜ese
postmodern
ﬁQuantsﬂ reject the
assumptions of a normal distribution of returns. See Peters (1991) for a comprehensive exposition.
Scienti˛c ﬁcatastrophe theoryﬂ is also being applied to stock market analysis to explain why ﬁbullﬂ markets
crash
without warning
. See Varian (2007).
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116 Arbitrage Pricing Theory and Beyond
Academics and ˛nancial analysts are also returning to twentiethcentury economic theorists for
inspiration, from John Maynard Keynes to the
behaviouralists
who dispensed with the assumption that
we can maximise anything.
Today™s proponents of behaviouralism, such as Montier (2002) reject the
neoclassical
economic pro˛t
motive and the wealth maximisation objectives of twentiethcentury ˛nance ˜ey believe that ˛nance
is a blend of economics and psychology that determines how investor attitudes can determine ˛nancial
decisions. Explained simply, investors do not appreciate what motivates them to make one choice, rather
than another. Behavioural Finance therefore seeks to explain why individuals, companies, or institutions
make mistakes and how to avoid them.
Su˚ce it to say, that much of the ﬁnewﬂ
Quants is so complex as to confuse most ˛nancial analysts,
let alone individuals who wish to beat the market (think the millennium dot.com ˛asco and the 2007
meltdown). Likewise, the ﬁnewﬂ behavioural ˛nance (just like the ﬁnewﬂ behavioural economics of
the1960s) seems to prefer ﬁa sledgehammer to crack a walnutﬂ (see Hill 1990).
As a parting shot, let us therefore return to
˚rst principles
and
common sense
with a guide to your future
studies or investment plans, which places Modern Portfolio ˜eory in a human context.
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117 Arbitrage Pricing Theory and Beyond
Ignore forecasts
: Evidence suggests that predictions are invariably wrong. ˜e behavioural trait to avoid is
known as
anchoring
, whereby you latch on to uncertain data that is hopelessly wrong. Develop a strategy
that does not depend on them.
Information Overload
: ˜e ˛nancial services industry believes that to ﬁbeatﬂ the market they need to know
more than everybody else. But empirical studies reveal too much information leads to overcon˛dence,
rather than accuracy. So concentrate on an investment™s ﬁkeyﬂ elements.
Overcon˚dence
: Most investors overestimate their skills. Prepare a plan based on your riskreturn pro˛le
and ability. ˜en stick to it.
Denial
: Investors are more attracted to good news, rather than bad. Prior to the millennium, the market
did not want the dot.com boom to fail. So, any information suggesting that technoshares were overvalued
was ignored. ˜e lesson is not to be complacent.
Overreaction
: Investors become optimistic in a rising (bull) market and pessimistic in a falling (bear)
market. When a signi˛cant proportion of investors believe that the market will rise or fall it may be a
signal that the opposite will happen.
Crowd Behaviour
: People feel safer herded together, which is why investors mimic the behaviour of
others and buy fashionable securities and funds. Speculative investors turn this to their own advantage
by acquiring stocks that are cheap and unfashionable.
Selective Memory
: Most investors tend to forget failure but remember success. To beat the market and
keep ahead of the crowd, keep a record of your decisions (good or bad) and learn from your mistakes.
Ignore Current Market Sentiment and Noise
: Today, most investors are doing the opposite. ˜e average
holding period for a share on the New York Stock Exchange is eleven months, compared with eight
years in the 1950s
.Go for longterm investment:
Over time, most shareholder returns come from dividends.
But remember
the
expected return from a stock is equal to the dividend yield,
plus
any dividend growth,
plus
any changes
in valuation that occur. ˜e strategy to adopt is ﬁvalue investingﬂ, where you buy stocks that are cheap
with high dividend yields.
To summarise:
Shortterm gain equals longterm pain:
According to Patrick Hosking (2010) the global ˛nancial crisis,
which has cost somewhere between one and ˛ve times the entire world™s ˛nancial output, started with
reckless bankers lending to poor Americans. Since 2007, other contributory factors have also been
suggested for the meltdown. Central banks ignored rising asset prices, governments talked up a global
economic boom and ˛nancial regulators still adhered to e˚cient market theory by using a light touch.
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118 Arbitrage Pricing Theory and Beyond
However, these are merely the consequences of a more fundamental problem, motivated by greed, referred
to throughout our analyses (including the
SFM
texts), namely:
The principalagency con˝ict associated with inappropriate shortterm, managerial reward
structures that arise from a bonus culture and lack of corporate governance, ˚rst explained by
Jensen and Meckling (1976).
As Hosking observes, these ˝awed incentives still exist today not just in banks, but also within ˛nancial
institutions and companies whose management (agents) hold shares on behalf of their owners (principles).
Management rarely accept responsibility if things go wrong, but always accept rewards, even if their
strategies have no lasting value. ˜us, managerial shorttermism rarely coincides with the longterm
income and capital aspirations of their shareholders.
If proof be needed that professional portfolio management has lost its way, let us conclude with two telling
UK statistics from the
London School of Economics™ Centre for the Study of Capital Market Disfunctionality.
In real terms, pension fund returns grew by an average of 4.1 per cent per annum between 1963
and2009, but by only 1.1 per cent a year in the last 10 years.
Unless corporate management are held personally responsible for their bonuses long aˆer their receipt
(perhaps a decade) it is therefore di˚cult to see how the rational objectives of e˚cient portfolio theory
can ever match the rational expectations of a portfolio™s clientele.
8.5 Selected References
1. Markowitz, H.M., ﬁPortfolio Selectionﬂ,
˜e Journal of Finance,
Vol. 13, No. 1, March 1952.
2.
Sharpe, W., ﬁA Simpli˛ed Model for Portfolio Analysisﬂ,
Management Science,
Vol. 9, No. 2,
January 1963.
3. Roll, R., ﬁA Critique of the Asset Pricing ˜eory Testsﬂ,
Journal of Financial Economics
, Vol. 4, March 1977.
4. Firth, M.,
˜e Valuation of Shares and the E˛cient Markets ˜eory
, Macmillan (London) 1977.
5. Ross, SA., ﬁArbitrage ˜eory of Capital Asset Pricingﬂ,
Journal of Economic ˜eory
, Vol. 13,
December 1976.
6.
Roll, R. and Ross, S.A., ﬁAn Empirical Investigation of the Arbitrage Pricing ˜eoryﬂ,
Journal
of Finance
, Vol. 35, No. 5, December 1980.
7. Chen, N.F., Roll, R. and Ross, S.A., ﬁEconomic Forces and the Stock Marketﬂ,
Journal of
Business,
Vol. 59, July 1986.
8. Dhrymes, P.J., Friend, I. and Gultekin, N.B., ﬁA Critical Reexamination of the Empirical
Evidence on the Arbitrage Pricing ˜eoryﬂ,
Journal of Finance,
Vol. 39, No. 3, June 1984.
9. Elton, E.J., Gruber, M.J. and Mei, J., ﬁCost of Capital Using Arbitrage Pricing ˜eory: A Case
Study of Nine New York Utilitiesﬂ,
Financial Markets, Institutions and Instrumentsﬂ,
Vol. 3,
August 1994.
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119 Arbitrage Pricing Theory and Beyond
10.
Huberman, G. and Wang, Z., ﬁArbitrage Pricing ˜eoryﬂ,
Federal Reserve Bank of New York
, Sta˙ Report, No. 216, August 2005.
11. Fama, E.F. and French, K.,ﬂ ˜e Capital Asset Pricing Model: ˜eory and Evidenceﬂ Working
Paper No.550,
Center for Research in Security Price s(CRSP),
University of Chicago, 2003.
12. Peters, E.E.,
Chaos and Order in Capital Markets
, Wiley (New York), 1991.
13. Varian, Hal R., ﬁCatastrophe ˜eory and the Business Cycleﬂ,
Economic Enquiry
, Vol. 17,
Issue 1, September 2007.
14. Montier, J.,
Behavioural Finance: Insights into Irrational Minds and Markets
, Wiley
(International), 2002.
15.
Hill, R.A., ﬁ˜eories of the Firm and Corporate Objectivesﬂ,
ACCA
, April 1990.
16.
Jenson, M.C. and Meckling, W.H., ﬁ˜e ˜eory of the Firm: Managerial Behaviour, Agency
Costs and Ownership Structureﬂ,
Journal of Financial Economics,
3, October 1976.
17. Hosking, P., ﬁShortterm gain, longterm gainﬂ,
˜e Times,
July 17, 2010.
thetimes.co.uk
18. Hill, R.A.,
bookboon.com

Strategic Financial Management,
2009. 
Strategic Financial Management; Exercises,
2009. 
Portfolio ˜eory and Financial Analyses
; Exercises, 2010.
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120 Appendix for Chapter 1
9 Appendix for Chapter 1
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