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Expected
Return




r




Standard
am
Deviation
Figure 29 The Market Portfolio.
.
The portfolio ˜p™ lies along the curve AMB and is tangent at M. It doesn™
efficient frontier since the latter by definition is the minimum variance portfo
given level of expected return. Note also that at M there is no borrowing
Altering xi and moving along MA we are ˜shorting™ security i and investing
100 percent of the funds in portfolio M.
The key element in this derivation is to note that at point M the curves
A M B coincide and since M is the market portfolio xi = 0. To find the sl
efficient frontier at M,we require




where all the derivatives are evaluated at xi = 0. From (2.31) and (2.32):
-
- P i - pm



At xi = 0 (point M) we know op = 0 and hence
,




Substituting in (2.34) and (2.36) in (2.33):




But at M the slope of the efficient frontier (equation (2.37)) equals the slope o
(equation (2.30)):



From (2.38) we obtain the CAPM relationship:


Using alternative symbols:

[cov(Rj, Rm)] (ERm-
+ r)
ERj = r
var(Rm)
=
Pi
var (Rm)
and the CAPM relationship is:

+ pi(ERm - r )
ERi = r
There is one further rearrangement of (2.42) to consider. Substituting for (ER"
(2.28) in (2.24)
+
ERi = r Am COV(Ri, R")
The CAPM therefore gives three equivalent ways represented by (2.40), (2.42)
of expressing the equilibrium required return on any single asset or subset o
the market portfolio.

2.3.3 Beta and Systematic Risk
If we define the extra return on asset i over and above the risk-free rate as a ris

+ rpi
ERi r
E


then the CAPM gives an explicit form for the risk premium


or equivalently:
rpi = A, cov(Ri, Rm)
The CAPM predicts that only the covariance of returns between asset i and
portfolio influences the excess return on asset i. No additional variables s
dividend price ratio, the size of the firm or the earnings price ratio should
expected excess returns. All changes in the portfolio risk of asset i are sum
changes in cov(Ri, Rm).The expected excess return on asset i relative to asset
by Pi/B j since



Given the definition of in (2.41) we see that the market portfolio has a bet
any individual security that has a Pi = 1, then its expected excess return mov
one with the market excess return (ER" - r ) and could be described as a ne
For pi > 1, the stock's expected return moves more than the market return
be described as an aggressive stock, while for Pi < 1 we have a defensive
CAPM only explains the excess rate of return relative to the excess rate of
the market portfolio (equation (2.42)): it is not a model of the absolute pri
individual stocks.
The systematic risk of a portfolio is defined as risk which cannot be diversifi
adding extra securities to the portfolio (this is why it is also known as 'non-div
i= 1 i j

With n assets there are n variance terms and n(n - 1)/2 covariance terms tha
to the variance of the portfolio. The number of covariance terms rises much
the number of assets in the portfolio and the number of variance terms (b
latter increase at the same rate, n ) . To illustrate this dependence on the cova
consider a simplified portfolio where all assets are held in the same proportion
and where all variances and covariances are constant (i.e. a,?= var and 0 i j =
˜var™ and ˜cov™ are constant). Then (2.48) becomes
1
[ i]cov
]
o;=n [ivar] +n(n-1). -cov =-var+ 1--
[n12 n
It follows that as n + 00 the influence of the variance term approaches ze
variance of the portfolio equals the (constant) covariance (cov) of the asset
variance of the individual securities is diversified away. However, the covar
cannot be diversified away and the latter (in a loose sense) give rise to syste
which is represented by the beta of the security.
We can rearrange the definition of the variance of a portfolio as follows:




where we have rewritten a,?as 0 i i . If the Xi are those for the market portfo
equilibrium we can denote the variance as 0 . ;
The contribution of security 2 to the portfolio variance may be interpr
bracketed term in the second line of (2.50) which is then ˜weighted™ by the pr
of security 2 held in the portfolio. The bracketed term contains the covarian
security 2 with all other securities including itself (i.e. the term ˜ 2 0 2 2and each
)
is weighted by the proportion of each asset in the market portfolio. It is easy t
the term in brackets in the second line of (2.50) is the covariance of security
return on the market portfolio Rm:




It is also easy to show that the contribution of security 2 to the risk of the
given by the above expression since & ˜ 2 / a x 2= 2cov(R2, R“). Similarly, we
Now, rearranging the expression for the definition of pi:

cov(R;, Rrn)= p;o;
and substituting (2.53) in (2.52) gives:




The pi of a security therefore measures the relative impact of security i on the
portfolio of stocks, as a proportion of the total variance of the portfolio. A se
/?; = 0 when added to the portfolio has zero additional proportionate influen
variance, whereas /?i < 0 reduces the variance of the portfolio. Of course, the
amount of security i held (i.e. the larger is the absolute value of xi) the more
of /?; on total portfolio variance, ceterisparibus. Since an asset with a small
considerably reduces the overall variance of a risky portfolio, it will be willingly
though the security has a relatively low expected return. All investors are tradi
which they dislike, against expected return, which they like. Assets which red
portfolio risk therefore command relatively low returns but are nevertheless wi
in equilibrium.

2.3.4 The Predictability of Equilibrium Returns
This section outlines how our equilibrium model of returns, namely the CAPM
tent with returns being both variable and predictable. The CAPM applied to
portfolio implies that equilibrium expected (excess) returns are given by:


where subscripts ˜t™ have been added to highlight the fact that these variables w
over time. From (2.55) we see that equilibrium excess returns will vary over tim
the conditional variance of the forecast error of returns is not constant. From a
standpoint the CAPM is silent on whether the conditional variance is time va
the sake of argument suppose it is an empirical fact that periods of turbulen
uncertainty in the stock market are generally followed by further periods of
Similarly, assume that periods of tranquillity are generally followed by further
tranquillity. A simple mathematical way of demonstrating such persistence i
is to assume volatility follows an autoregressive AR(1) process. When vol
the second moment of the distribution) is autoregressive this process is ref
autoregressive conditional heteroscedasticity or ARCH for short:
+
2
= aa, vt


where vf is a zero mean (white noise) error process independent of a The be
:
.
of o;+l at time t is:
E,a,+,
2 =ao;
(i) non-constant
(ii) depend on information available at time t, namely a
:
.

Hence we have an equilibrium model in which expected returns vary and
012.
information at time t , namely The reason expected returns vary with
straightforward. The conditional variance (.YCJ; is the investor™s best guess of ne
systematic risk in the market E,c$+,. In equilibrium such risks are rewarded w
expected return.
The above model may be contrasted with a much simpler hypothesis, n
equilibrium expected returns are constant. Rejection of the latter model, for e
finding that actual returns depend on information R, at time t, or earlier (e.
price ratio), may be because the variables in 52, are correlated with the omitt
+
af which occurs in the ˜true™ model of expected returns (i.e. CAPM ARCH
The above argument about the predictability of returns can be repeated fo
librium excess return on an individual asset


If the covariance term is, in part, predictable from information at time t then e
returns on asset i will be non-constant and predictable. Hence the empirical f
returns are predictable need not necessarily imply that investors are irratio
ignoring potentially profitable opportunities in the market. It is important to b
mind when discussing certain empirical tests of the so-called efficient markets
(EMH) in Chapter 5.


SUMMARY
2.4
The basic one-period CAPM seeks to establish the optimal proportions in w
assets are held. Since the risky assets are all willingly held, the CAPM can a
to establish the determinants of equilibrium returns on all of the individual a
portfolio. The key results from the one-period CAPM are:
All investors hold their risky assets in the same proportions (xf) regardle
0

preferences for risk versus return. These optimal proportions constitute
portfolio.
Investors™ preferences enter in the second stage of the decision process,
0

choice between the fixed bundle of risky securities and the risk-free asset
risk averse is the individual the smaller the proportion of his wealth he w
the bundle of risky assets.
The CAPM implies that in equilibrium the expected excess return on any s
0

asset ERi - r is proportional to the excess return on the market portfolio
The constant of proportionality is the asset™s beta, where pi = cov(Ri, R,)
themselves predictable.


APPENDIX 2.1 DERIVATION OF THE CAPM
The expected return and standard deviation for any portfolio ˜p™ consisting of n risky
risk-free asset are:
1

ER, =
L 1 J
1

In n n
0=
,
i=l j=1
i#j

where xi = proportion of wealth held in asset i. The CAPM is the solution to the
minimising op subject to a given level of expected return ER,. The Lagrangian is:

+ \I, [ER, - C x i E R i + (1 - E x ; ) r ]
C = op

Choosing xi (i = 1,2, . . . n) to minimise C gives a set of first-order conditions (FOC)

[ I
+i)-1/22x + 2 k x j cov(R1, R,)
ac/axl = - *(ER1 - r ) = 0
10;

j=2




Differentiation with respect to \ , gives:
I

n
aC/W =ER, - x x i E R i -
i= 1

Multiplying the first equation in (4)by xl, the second by x2, etc. and summing over a
gives:




xi = 1 gives:
At the point where
= *(ER,,, - r )
a,
,,
derive the CAPM expression for equilibrium returns on each individual share in the p
xi = 1 is:
ith equation in (4) at the point


1I
1L A
+ - IX;C$ + >,
ERi = r COV(R,,
Rj)
xj
UT
r




Substitute for (1/Q) from (8) in (9):
r 1




Note that:




and substituting (11) in (10) we obtain the CAPM expression for the equilibrium exp
on asset i:
+
ER; = r (ER, - r)P,

P, = cov(Ri, R,,,)/oi.
where
3



Modelling Equilibrium Return

The last chapter dealt at some length with the principles behind the simple
CAPM. In this chapter this model is placed in the wider context of alternative
seek to explain equilibrium asset returns. The mean-variance analysis that un
CAPM also allows one to determine the optimal asset proportions (the xy) for th
risk-free asset. This chapter highlights the close relationship between the me
analysis of Chapter 2 and a strand of the monetary economics literature that
the determination of asset demands for a risk averse investor. This mean-vari
of asset demands is elaborated and used in later chapters. In this chapter w
look at the following interrelated set of ideas.
Some of the restrictive assumptions of the basic CAPM model are relaxed
0

general principles of the model still largely apply. Expected returns on
depend on the asset™s beta and on the excess return on market portfolio.
The mean-variance criterion is used to derive a risk aversion model of ass
0

and this model is then compared with the CAPM.
We examine how the CAPM can be used to provide alternative performance
0

to assess the abilities of portfolio managers.
The arbitrage pricing theory APT provides an alternative model of equilibr
0

to the CAPM and we assess its strengths and weaknesses.
We examine the single index model and early empirical tests of the CAPM
0




3.1 EXTENSIONS OF THE CAPM
The standard one-period CAPM is derived under somewhat restrictive assu
is possible to relax some of these assumptions and yet still retain the basic
framework of the CAPM together with its predictions for the determinants of
returns. In any economic model there is often a trade-off between ˜simplicit
of the theory and ˜fruitfulness™ in terms of a good statistical model. The po
the standard CAPM arises in part because it is relatively tractable when i
testing the model (and it does have some empirical validity). Inevitably whe
its restrictive assumptions are relaxed, the resulting models become more co
tractable and for some, less mathematically elegant. In interpreting the empir
from the standard CAPM it is important to be aware of the less restrictive var
Although investors can lend as much as they like at the riskless rate (e.g. by
government bills and bonds), usually they cannot borrow unlimited amounts. I
if the future course of price inflation is uncertain then there is no riskless bo
real terms (riskless lending is still possible in this case, if government issues in
government bonds).
This section reworks the CAPM under the assumption that there is no riskless
or lending (although short sales are still allowed). This gives rise to the so-calle
CAPM where the equilibrium expected return on any asset (or portfolio i ) is

ERi = ERZ + (ER"' - E R ) B j

where ERZ is the expected return on the so-called zero-beta portfolio (see bel
To get some idea of this rather peculiar entity called the zero-beta portfoli
the security market line (SML) of Figure 3.1. The SML is a graph of the expe
on a set of securities against their beta values. From the standard CAPM we
(ER; - r)//3; is the same for all securities (and equals ERm - r). Hence if the
correct all assets should, in equilibrium, lie along the SML.



If a security C existed it would be preferred to security B because it offer
expected return but has the same systematic risk (i.e. value of Pi). Investors w
security B and buy security C thus raising the current price of C until its expe
+
(from t to t 1) equalled that on security B.
In general the equation of the SML is that of a straight line:

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