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when we have a risky environment. The issue of how shareholders ensure tha
act in the best interest of the shareholders, by maximising the value of the fi
under the heading of corporate control mechanisms (e.g. mergers, takeovers). Th
of corporate control is not directly covered in this book. Consideration is g
under a risky environment.

( 0 In a risky environment a somewhat different separation principle applies.
each investor when choosing his portfolio of risky marketable assets (
bonds) will hold risky assets in the same proportion as all other investors,
of his preferences of risk versus return. Having undertaken this first stag
each investor then decides how much to borrow or lend in the money
the 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 individual™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 individual™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 ˜simple™ 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 ˜tomorrow™ 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 investor™s choice between his holdings of t
0

asset and the ˜bundle™ 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 ˜basic™ 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 investor™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 individual™s preferences concerning his subjective trade-off between risk
This choice depends only on the individual™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 ˜a shares™, 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 individual™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 ˜neutral stocks™. 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 CAPM™s key predictions, namely that al
should hold stocks in the same optimal proportions x:. Hence the ˜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
position™ in particular stocks may use betas to rank the stocks to include in h
(i.e. he doesn™t obey the assumptions of the CAPM and therefore doesn™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 ˜Performance Index™ (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 ˜A™ (of n securities) and a portfolio ˜B™ (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 ˜inefficient™ 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 ˜riskiness™ 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) ˜the total variance™ 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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