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act in the best interest of the shareholders, by maximising the value of the fi

under the heading of corporate control mechanisms (e.g. mergers, takeovers). Th

of corporate control is not directly covered in this book. Consideration is g

under a risky environment.

( 0 In a risky environment a somewhat different separation principle applies.

each investor when choosing his portfolio of risky marketable assets (

bonds) will hold risky assets in the same proportion as all other investors,

of his preferences of risk versus return. Having undertaken this first stag

each investor then decides how much to borrow or lend in the money

the risk-free interest rate. This separation principle is the basis of optim

choice and of the capital asset pricing model (CAPM) which provides

equilibrium asset returns.

(ii) The optimal amount of borrowing and lending in the money market in

case occurs where the individualâ€™s subjective marginal rate of substitutio

+

for current consumption (i.e. (dCl/dCo)u) equals -(1 r), where r

opportunity cost of money. Under uncertainty a parallel condition appli

that the individualâ€™s subjective trade-off between expected return and ri

to the market price of risk.

(iii) Under certainty, the slope of the money market (or present value) line m

price (interest rate) of money, or equivalently the increase in value of $

today in the money market. Under risky investment opportunities the s

so-called capital market line (CML) along which investors can borrow

provides a measure of the market price of risk.

1.4 SUMMARY

In this chapter we have developed some basic tools for analysing in financi

There are many nuances on the topics discussed which have not been elaborat

and in future chapters these omissions will be rectified.

The main conclusions to emerge are:

Market participants generally quote â€˜simpleâ€™ annual interest rates but these

be converted to effective annual (compound) rates or to continuously co

rates.

The concepts of DPV and IRR can be used to analyse physical investme

and provide measures of the return on bills and bonds.

Theoretical models often have either returns and risk or utility as their m

Utility functions and their associated indifference curves can be used to rep

averters, risk lovers and risk neutral investors.

Under certainty, a type of separation principle applies. Managers can cho

ment projects to maximise the value of the firm and disregard investor p

Then investors are able to borrow and lend to allocate consumption betw

and â€˜tomorrowâ€™ in order to maximise their satisfaction (utility).

2. There are two points worth noting at this point. First, the expectations ope

applied to the whole of the RHS expression in (1.30). If qi and Dr+; boare

+

variables then, for example, E,[Dr+l/(l ql)] does not equal ErD,+l

Second, equation (1.30) is expressed in terms of one-period rates 4;. If

(annual) rate applicable between t = 0 and t = 2 on a risky asset, then we

+ +

define (1 r(2))2 (1 q1)(1+ q 2 ) . Then (1.30) is of a similar form to

=

we ignore the expectations operator. These rather subtle distinctions need

the reader at this point and they will become clear later in the text.

3. Note that the function U = U ( F N 0) is very different from either E U

,:

from E[U(W)] where W is end of period wealth.

I &

The Capital Asset

Pricing Model: CAPM

This chapter presents a detailed derivation of the (basic) one-period capital a

model (CAPM). This model, interpreted as a model of equilibrium asset return

used in the finance literature and the concepts which underlie its derivation, su

folio diversification, measures of risk and return, and the concept of the mark

are also fundamental to the analysis of all asset prices. Throughout this chap

consider that the only risky assets are equities (stocks) although strictly the mo

to choices among all risky assets (i.e. stocks, bonds, real estate, etc.).

The CAPM attempts to answer what at first sight appears to be a rather c

of interrelated questions, namely:

Why it is beneficial for agents to hold a diversified portfolio consisting of a

0

risky assets rather than say one single risky asset or a small subset of all th

risky assets?

What determines the expected equilibrium return on each individual risky

0

market, so that all the risky assets are willingly held by investors?

What determines an individual investorâ€™s choice between his holdings of t

0

asset and the â€˜bundleâ€™ of risky assets?

As will be seen in the next chapter, there are a number of models of equilib

returns and there are a number of variants on the â€˜basicâ€™ CAPM. The con

in the derivation of the CAPM are quite numerous and somewhat complex,

useful at the outset to sketch out the main elements of its derivation and draw

basic implications. Section 2.2 carefully sets out the principles that underli

diversification and the efficient set of portfolios. Section 2.3 derives the optim

and the equilibrium returns that this implies.

21 AN OVERVIEW

.

Our world is restricted to one in which agents can choose a set of risky assets (

a risk-free asset (e.g. fixed-term bank deposit or a three-month Treasury bill).

borrow and lend as much as they like at the risk-free rate. We assume agents

returns. Transactions costs and taxes are assumed to be zero.

Consider the reason for holding a diversifiedportfolio consisting of a set of r

Assume for the moment that the funds allocated to the safe asset have already

Putting all your wealth in asset 1, you incur an expected return ER1 and a ri

o , the variance of the returns on this one asset. Similarly holding just asset 2

:

to earn ER2 and incur risk 0;.Let us assume a two-asset world where there is

covariance of returns 0 1 2 < 0. Hence when the return on asset 1 rises that on as

to fall. (This also implies a negative correlation coefficient p12 = 012/0102.) H

diversify and hold both assets, this would seem to reduce the variance of

portfolio (i.e. of asset 1 plus asset 2). To simplify even further suppose that E

and a = 0;.In addition assume that when the return on asset 1 increases by

:

that on asset 2 falls by 1 percent (i.e. returns are perfectly negatively correlated

Under these conditions when you hold half your initial wealth in each of the r

the expected return on the overall portfolio is ER1 = E&. However, diversif

reduced the risk on the portfolio to zero: an above average return on asset 1

matched by an equal below average return on asset 2 (since p = -1). Our exa

course, a special case but in general, even if the covariance of returns is zero

(but not perfectly positively correlated) it still pays to diversify and hold a c

of both assets.

The above simple example also points to the reason why each individual inv

at least hold some of each of all the available stocks in the market, if we al

borrow (or lend) unlimited funds at the risk-free rate r. To demonstrate this p

up a counter example. If one stock were initially not desired by any of the inv

its current price would fall as investors sold it. However, a fall in the current pr

that the expected return over the coming period is higher, ceterisparibus (ass

expected it to pay some dividends in the future). One might therefore see the cu

fall until the expected return increases so that the stock is sufficiently attractiv

The reader may now be summising that the individual investorâ€™s tastes or p

must come into the analysis at some point and he would be correct. Howev

a quite remarkable result, known as the two-fund separation theorem. The

decision can be broken down into two separate decisions. The first decision

the choice of the optimal proportions xi* of risky assets held and this is inde

the individualâ€™s preferences concerning his subjective trade-off between risk

This choice depends only on the individualâ€™s views about the objective marke

namely, expected returns, variances and covariances. Expectations about thes

are assumed to be homogeneous across investors. All individuals therefore hol

proportions of the risky assets (e.g. all investors hold 1/20 of â€˜a sharesâ€™, 1/80 of

etc.) irrespective of their preferences. Hence aggregating, all individuals will

risky assets in the same proportions as in the (aggregate) market portfolio

share of ICI in the total stock market index is 1/20 by value, then all investors

of their own risky asset portfolio, in ICI shares).

It is only after mimicking the market portfolio that the individualâ€™s prefere

the calculation. In the second stage of the decision process the individual de

pays r ) and only invest a small amount of his own wealth in the risky assets i

proportions x:. The converse applies to a less risk averse person who will bo

risk-free rate and use these proceeds (as well as his own initial wealth) to invest

bundle of risky assets in the optimal proportions XI. however, this sec

Note,

which involves the individual's preferences, does not impinge on the relativ

for the risky assets (i.e. the proportions x:). Hence the equilibrium expected

the set of risky assets are independent of individuals' preferences and depe

market variables such as the variances and covariances in the market.

Throughout this and subsequent chapters the following equivalent ways of

expected returns, variances and covariances will be used:

Expected return = pi = ER;

Variance of returns = 0;= var(Ri)

Covariance of returns = 0 ; j = cov(Ri, R j )

Let us turn now to some specific results about equilibrium returns which

the CAPM. The CAPM provides an elegant model of the determinants of t

rium expected return ER; on any individual risky asset in the market. It predi

expected excess return on an individual risky asset (ERi - r ) is directly rel

expected excess return on the market portfolio (ERm- r), with the constant

tionality given by the beta of the individual risky asset:

ERm is the expected return on the market portfolio that is the 'average' expe

from holding all assets in the optimal proportions x:. Since actual returns on

portfolio differ from expected returns, the variance var(Rm) on the market

non-zero. The definition of firm i's beta, namely pi indicates that it depends

(i) the covariance between the return on security i and the marke

cov(Ri, R m ) and

(ii) is inversely related to the variance of the market portfolio, var(Rm).

Loosely speaking, if the ex-post (or actual) returns when averaged appro

ex-ante expected return ERi, then we can think of the CAPM as explaining

return (over say a number of months) on security i .

What does the CAPM tell us about equilibrium returns on individual secu

stock market? First note that (ERm - r ) > 0, otherwise no risk averse agent

covariance with the market portfolio, they will be willingly held as long as th

expected return equal to the risk-free rate (put pi = 0 in (2.1)). Securities that h

positive covariance with the market return (pi > 0) will have to earn a rela

expected return: this is because the addition of such a security to the portfolio d

reduce overallportfolio variance. Conversely any security for which cov(Rj, R

hence pi < 0 will be willingly held even though its expected return is below th

rate (equation (2.1) with pi < 0) because it tends to reduce overall portfolio v

The CAPM also allows one to assess the relative volatility of the expec

on individual stocks on the basis of their pi values (which we assume are

measured). Stocks for which pi = 1 have a return that is expected to move o

with the market portfolio (i.e. ER, = ERm) and are termed â€˜neutral stocksâ€™. If

stock is said to be an aggressive stock since it moves more than changes in th

market return (either up or down) and conversely defensive stocks have pi < 1

investors can use betas to rank the relative safety of various securities. Howeve

should not detract from one of the CAPMâ€™s key predictions, namely that al

should hold stocks in the same optimal proportions x:. Hence the â€˜market por

by all investors will include neutral, aggressive and defensive stocks held in t

proportions x,? predicted by the CAPM. Of course, an investor who wishes

positionâ€™ in particular stocks may use betas to rank the stocks to include in h

(i.e. he doesnâ€™t obey the assumptions of the CAPM and therefore doesnâ€™t attemp

the market portfolio).

The basic concepts of the CAPM can be used to assess the performance o

managers. The CAPM can be applied to any portfolio p of stocks composed

of the assets of the market portfolio. For such a portfolio we can see that PI

r)/a,] is a measure of the excess return (over the risk-free rate) per unit of ris

may loosely be referred to as a â€˜Performance Indexâ€™ (PI). The higher the PI

is the expected return corrected for risk (a,).Thus for two investment manage

whose portfolio has the higher value for PI may be deemed the more succes

ideas are developed further in Chapter 5.

2.2 PORTFOLIO DIVERSIFICATION, EFFICIENT FRO

AND THE TRANSFORMATION LINE

Before we analyse the key features of the CAPM we discuss the mean-varianc

the concept of an efficient portfolio and the gains from diversification in m

portfolio risk. We then consider the relationship between the expected retur

diversified portfolio and the risk of the portfolio, ap.If agents are interested in m

risk for any given level of return, then such efficient portfolios lie along th

frontier which is non-linear in ( p p ,a ) space. We then examine the return-risk r

,

for a specific two-asset portfolio where one asset consists of the amount of

or lending in the safe asset and the other asset is a single portfolio of risky a

gives rise to the transformation line which gives a linear relationship betwee

return and portfolio risk for any two-asset portfolio comprising a risk-free asse

It is assumed that the investor would prefer a higher expected return (ER) ra

lower expected return, but he dislikes risk (i.e. is risk averse). We choose to m

by the variance of the returns var(R) on the portfolio of risky assets. Thus, i

is presented with a portfolio â€˜Aâ€™ (of n securities) and a portfolio â€˜Bâ€™ (of a d

of n securities), then according to the MVC, portfolio A is preferred to portfo

where SD = standard deviation. Of course if, for example, EA(R)> E B ( R )but

varB(R) then we cannot say what portfolio the investor prefers using the MVC

Portfolios that satisfy the MVC are known as the set of eficientportfolio

folio A that has a lower expected return and a higher variance than another po

said to be â€˜inefficientâ€™ and an individual would (in principle) never hold a por

as A if portfolio B is available.

2.2.2 Portfolio Diversification

To demonstrate in a simple fashion the gains to be made from holding a

portfolio of assets, the simple two-(risky) asset model will be used.

Suppose the actual return (over one period) on each of the two assets is R1 a

expected returns p1 = ER1 and p2 = E&. The variance of the returns on ea

is measured by o?(i= 1,2) which is defined as

In addition assume that the correlation coeficient between movements in the

the two assets is p, (-1 < p < 1) where p is defined as

Hence 0 1 2 = cov(R1, R 2 ) is the covariance between the two returns. If p = +

asset returns are perfectly positively (linearly) related and the asset returns al

in the same direction. For p = -1 the converse applies and for p = 0 the a

are not (linearly) related. As we see below, the â€˜riskinessâ€™ of the portfolio co

both asset 1 and asset 2 depends crucially on the sign and size of p. If p = -1

be completely eliminated by holding a specific proportion of initial wealth in b

Even if p is positive (but less than +1) the riskiness of the overall portfolio

(although not to zero) by diversification.

Suppose for the moment that the investor chooses the proportion of his tota

invest in each asset in order to minimise portfolio risk. He is not, at this stag

to borrow or lend or place any of his wealth in a risk-free asset. Should the i

a proportion x1 of his wealth in asset 1 and a proportion x2 = (1 - X I ) in a

actual return on this diversified portfolio (which will not be revealed until

later) is:

+

R p = xiRi ˜ 2 R 2

The expected return on the portfolio (formed at the beginning of the period) is

For the moment we assume the investor is not concerned about expected

equivalently that both assets have the same expected return, so only the v

returns matters to him). Knowing of, ; and p (or 0 1 2 ) the individual has to

0

value of x1 (and hence x2 = 1 - X I ) to minimise the total portfolio risk, 0;.Dif

(2.8) gives

Solving (2.9) for x1 gives

Note that from (2.10) â€˜the total varianceâ€™ will be smallest when p = -1 and la

p = +l.

For illustrative purposes assume a = (0.4)2,0; = p = 0.25 (i.e. pos

t

lation). Then the value of x1 for minimum variance using (2.10) is:

- 0.25(0.4)(0.5) 20

--

x1 =

+ - 2(0.25)(0.4)(0.5)

(0.4)2 31

and substituting this value of x1 in (2.8) gives

0; = 12.1 percent

the two assets returns are perfectly negatively correlated. It follows from this a

an individual asset may be highly risky taken in isolation (i.e. has a high varian

if it has a negative covariance with assets already held in the portfolio, then

be willing to add it to their existing portfolio even if its expected return is rel

since such an asset tends to reduce overall portfolio risk ( : .

0 ) This basic intui

lies behind the explanation of determination of equilibrium' asset returns in th

Even in the case where asset returns are totally uncorrelated then portfol

can be reduced by adding more assets to the portfolio. To see this, note that f

(all of which have pij = 0) the portfolio variance is:

Simplifying further, if all the variances are equal (of = a2)and all the assets

equal proportions ( l / n ) we have

1 1

0; = -(no2) = - 0 2

n2 n

Hence as n + 09 the variance of the portfolio approaches zero. Thus, if u

risks are pooled, total risk is diversified away. The risk attached to each se

which is known as idiosyncratic risk can be completely diversified away. Intu

is inclined to suggest that such idiosyncratic risk should not require any additi

over the risk-free rate. As we shall see this intuition carries through to the CA

Portfolio Expected Return and Portfolio Variance as p Varies

In the above example we neglected the expected return on the portfolio. Clea

uals are interested in expected portfolio return p p as well as the risk of the p

The question now is how p p and a p vary, relative to each other, as the agen

proportion of wealth held in each of the risky assets. Remember that p1, p2,01

(or p ) are fixed and known. As the agent alters x1 (and x2 = 1 - XI) then equ

and (2.8) allow us to calculate the combinations of p p and a p that ensue for

values of x1 (and x2) that we have arbitrarily chosen. (Note that there is no

tion/minimisation problem here, it is a purely arithmetic calculation given the

of p p and ap).A numerical example is given in Table 2.1 for

and is plotted in Figure 2.1.

The above calculations could be repeated using different values for p (b

and -1). In general as p approaches -1 the ( p p ,u p )locus moves closer to

axis as in Figure 2.2, indicating that a greater reduction in portfolio risk is p

any given expected return. (Compare portfolios A and B corresponding to p

p = -0.5.) For p = -1 the curve hits the vertical axis indicating there is som

900 (30.0)

0 20

18 532 (23.1)

115

16 268 (16.4)

215

14 108 (10.4)

315

52 (7.2)

12

415

10 100 (10.0)

1

t

Expanded

Return

'1

12 --

"1

10 -- I

I I

I

I I

1 I

I I

I I

I I

I I

I I

1

I

L

w

I I I

10 30

20 Standard

Deviation

Figure 2.1 Expected Return on Standard Deviation.

t

Expected

Return

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