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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




X


Standard
Deviation

Figure 2.2 Efficient Frontier and Correlation.
any point in time) and hence only one risk-return locus and corresponding xi v
risk-return combination is part of the feasible set or opportunity set for every

More than Two Securities
If we allow the investor to distribute his wealth between any two securities A
and C or A and C we obtain ( p p ,u p )combinations given by curves I, 11, I11 r
in Figure 2.3. If we now allow the agent to invest in all three securities, that
the proportions x1, x2, x3 (with Cixi = 1) then the ( p p ,u p )locus is a cur
This demonstrates that holding more securities reduces portfolio risk for any
of expected return (i.e. the three-security portfolio at Y is preferred to portfolio
(securities B and C only) in Figure 2.3 and X also dominates Y and 2). It f
any agent adopting the mean-variance criterion will wish to move from points
Z to points at X.
The slope of IV is a measure of how the agent can trade-off expected ret
risk by altering the proportions Xi held in the three assets. By altering the c
of his portfolio from M to Q he can obtain an increase in expected return (p
taking on an amount of additional risk ( 0 2 - 01).
Note that the dashed portion of the curve IV indicates a mean-variance
portfolio. An investor would never choose portfolio L rather than M becau
lower expected return for a given level of portfolio risk than does portfolio M
M is said to dominate portfolio L on the mean-variance criterion.

Limited Investment and Portfolio Variance
In general the agent may reduce op for any given p p by including additiona
his portfolio (particularly those that have negative covariances with the exis
already held). In fact portfolio variance (0;) falls very quickly as one in




Figure 2.3 Efficient Frontier.
5 Numbers of
20
10
Stocks

Figure 2.4 Portfolio Combinations.

number of stocks held from 1 to 10, and thereafter the reduction in portfol
is quite small (Figure 2.4). This, coupled with the brokerage fees and inform
of monitoring a large number of stocks, may explain why individuals tend t
only a relatively small number of stocks. Individuals may also obtain the
diversification by investing in mutual funds (unit trusts) and pension funds
institutions use funds from a large number of individuals to invest in a very
of financial assets and each individual then owns a proportion of this ˜large p

2.23 The Efficient Frontier
Consider now the case of N assets. When we vary the proportions Xi (i = 1 ,
form portfolios it is obvious that there is potentially a large number of such
We can form 2,3, . . . to N asset portfolios. We can also form portfolios consis
same number of assets but in different proportions. The set of every possible
given by the convex ˜egg™ of Figure 2.5.
If we apply the mean-variance dominance criterion then all of the points in
of the porlcfolio opportunity set (e.g. PI, Figure 2.5) are dominated by th
P2,
curve AB since the latter has a lower variance for a given expected return. Po
curve AB also dominate those on BC, so the curve AB represents the propo
the eficient set of portfolios and is referred to as the eficient frontier.
We now turn to the problem of how the investor calculates the Xj values th
the efficient frontier. The investor faces a known set of n expected returns an
of and n(n - 1)/2 covariances O i j (or correlation coefficients p i j ) and the fo
the expected return and variance of the portfolio are:
-
Staendafd
Deviation
Figure 2.5

or




We assume our investor wishes to choose the proportions invested in each ass
is concerned about expected return and risk. Risk is measured by the standard d
returns on the portfolio (up). efficientfiontier shows all the combinations
The
which minimises risk u p for a given level of p,.
The investor™s budget constraint is C x i = 1, that is all his wealth is placed
of risky assets. (For the moment he is not allowed to borrow or lend money i
asset.) Short sales X i < 0 are permitted.
A stylised way of representing how the agent seeks to map out the efficien
as follows:
1. Choose an arbitrary ˜target™ return on the portfolio pZ, (e.g. p p = 10 perc
2. Arbitrarily choose the proportions of wealth to invest in each asset
1,2, . . . n) such that p p is achieved (using equation (2.15)).
3. Work out the variance or standard deviation of this portfolio (a,)˜ with th
of ( X i ) l using equation (2.17).
4. Repeat (2) and (3) with a new set of ( ˜ i ) 2if ( a p ) 2 < (op)lthen discard t
in favour of ( X i ) 2 (and vice versa).
5 . Repeat (2)-(4) until you obtain that set of asset proportions x (with Cx;
;
meets the target rate of return p i and yields the minimum portfolio varian
(up)*. assets held in the proportions x; is an ejfficientportfolio and
The
point in ( p p ,op) space - point A in Figure 2.6.
6 . Choose another arbitrary ˜target™ rate of return p y (= 9 percent say) and
above to obtain the new efficient portfolio with proportions xi** and minimu
( , * - point B.
a)*
Figure 2.6

We could repeat this exercise for a wide range of values for alternative expe
rates p , and hence trace out the curve XABCD. However, only the upper po
curve, that is XABC, yields the set of efficient portfolios and this is the eficie
It is worth noting at this stage that the general solution to the above problem
usually does) involve some xf being negative as well as positive. A positive x
stocks that have been purchased and included in the portfolio (i.e. stocks h
Negative xf represent stocks held ˜short™, that is stocks that are owned by so
(e.g. a broker) that the investor borrows and then sells in the market. He th
a negative proportion held in these stocks (i.e. he must return the shares to
at some point in the future). He uses the proceeds from these short sales to a
holding of other stocks.

Interim Summary
1. Our investor, given expected returns pi and the variances a, and covarian
p i j ) on all assets, has constructed the efficient frontier. There is only on
frontier for a given set of pi, a;,ajj, ( p i j ) .
2. He has therefore chosen the optimal proportions xi* which satisfy the budge
Cxf = 1 and minimise the risk of the portfolio a for any given level o
,
return on the portfolio p p .
3. He has repeated this procedure and calculated the minimum value of a fo,
of expected return p p and hence mapped out the (p,, a) points which co
,
efficient frontier.
4. Each point on the efficient frontier corresponds to a different set of optim
tions x : , x z , x;, . . . in which the stocks are held.

Points 1-4 constitute the first ˜decision™ the investor makes in applying th
separation theorem - we now turn to the second part of the decision process

2.2.4 Borrowing and Lending: The ™hansformation Line
Our agent can now be allowed to borrow or lend an asset which has the same
the holding period and yields a ˜certain™ and hence risk-free rate of interest,
(ii) invest less than his total wealth in the risky assets and use the remainde
the risk-free rate,
(iii) invest more than his total wealth in the risky assets by borrowing the
funds at the risk-free rate. In this case he is said to hold a leveredportf

The transformation line is a relationship between expected return and risk on
portfolio. This specific portfolio consists of (i) a riskless asset and (ii) a portfo
assets.
The transformation line holds for any portfolio consisting of these two as
turns out that the relationship between expected return and risk (measured by th
deviation of the ˜new™ portfolio) is linear. Suppose we construct a portfolio
consisting of one risky asset with expected return ER1 and variance 0 and :
asset. Then we can show that the relationship between the return on this new
and its standard deviation is
pk = a -k buk
where ˜a™ and ˜b™ are constants and PI( = expected return on the new portf
standard deviation on the new portfolio. Similarly we can create another new
˜N™ consisting of (i) a set of q risky assets held in proportions xi (i = 1,2, .
together constitute our one risky portfolio and (ii) the risk-free asset. Again w
pN = 60 -k 6 I O N
To derive the equation of the transformation line let us assume the individual ha
already chosen a particular combination of proportions (i.e. the x i ) of q r
(stocks) with actual return R, expected return p˜ and variance 0 Note that
.
:
not be optimal proportions but can take any values. Now he is considering what
of his wealth to put in this one portfolio of q assets and how much to borr
at the riskless rate. He is therefore considering a ˜new™ portfolio, namely co
of the risk-free asset and his ˜bundle™ of risky assets. If he invests a proportio
own wealth in the risk-free asset, then he invests (1 - y ) in the risky ˜bund
the actual return and expected return on his new portfolio as RN and p ˜resp ,
+
RN = yr (1 - y ) R
+
PN = yr (1 - y)pR
where (R, p ˜ is) the (actual, expected) return on the risky ˜bundle™ of his po
in stocks. When y = 1 all wealth is invested in the risk-free asset and p˜ = r
y = 0 all wealth is invested in stocks and p˜ = p ˜For y < 0 the agent borro
.
at the risk-free rate r to invest in the risky portfolio. For example, when y =
initial wealth = $100, the individual borrows $50 (at an interest rate r ) but in
in stocks (i.e. a levered position).
Since r is known and fixed over the holding period then the standard devia
˜new™ portfolio depends only on the standard deviation of the risky portfolio
OR. From (2.19) and (2.20) we have
2
- y)2E(R - pR)2
= E(& - pN)2 = (1
ON
and (2.22) are both definitional but it is useful to rearrange them into a sing
in terms of mean and standard deviation ( p ˜ N ) of the ˜new™ portfolio. From
O,

(1 - Y ) = O N / O R
Y = 1- (ON/OR)
Substituting for y and (1 - y) from (2.23) and (2.24) in (2.20) gives the iden



where SO = r and 61 = ( p - OR. Thus for any portfolio consisting of two
˜
of which is a risky asset (portfolio) and the other is a risk-free asset, the r
between the expected return on this new portfolio p˜ and its standard error O
with slope given by 81 and intercept = r . Equation (2.25) is, of course, an ide
is no behaviour involved. ( p - r ) is always positive since otherwise no one w
˜
the set of risky assets.
When a portfolio consists only of n risky assets, then as we have seen th
opportunity set in return-standard deviation space is curved (see Figure 2.6)
the opportunity set for a two-asset portfolio consisting of a risk-free asset and
risky portfolio is a positive straight line. This should not be unduly confusin
portfolios considered in the two cases are different and in the case of the ˜effic
curve is derived under an optimising condition and is not just a rearrangeme
identities.
Equation (2.25) says that p˜ increases with ( O N / G R ) . This arises because f
an increase in O N / O R simply implies an increase in the proportion of wea
the risky asset (i.e. 1 - y) and since ER > r this raises the expected return o
portfolio p ˜Similarly for a given ( O N / O R )= (1 - y) (see equation (2.23)), a
.
in the expected excess return on the risky asset ( p - r ) increases the overa
˜
return p ˜This is simply because here, the investor holds a fixed proportion
.
the risky asset but the excess return on the latter is higher.
We can see from (2.25) that when all wealth is held in the set of risky
0 and hence ON = OR and this is designated the 100 percent equity portfolio
Figure 2.7). When all wealth is invested in the risk-free asset y = 1 and ,UN
O N / O R = 0 from (2.33)). At points between r and X, the individual holds so
initial wealth in the risk-free asset and some in the equity portfolio. At points
individual holds a levered portfolio (i.e. he borrows some funds at a rate r and
all his own wealth to invest in equities).


DERIVATION OF THE CAPM
23
.
231 The Optimal Portfolio
..
The transformation line gives us the risk-return relationship for any portfolio
of a combination of investment in the risk-free asset and any ˜bundle™ of sto
All wealth in risky
assets
I
1
I


risk free asset I
I
I
I
I
I

+
bN
OR
Standard
Deviation

Figure 2.7 The Transformation Line.

is no behavioural or optimisation by agents behind the derivation of the tran
line: it is an identity. At each point on a given transformation line the agen
risky assets in the same fixed proportions x i . Suppose point X (Figure 2.7) r
combination of x; = 20 percent, 25 percent and 55 percent in the three risk
of firms ˜alpha™, ˜beta™ and ˜gamma™. Then points Q, L and Z also represen
proportions of the risky assets. The only ˜quantity™ that varies along the tran
line is the proportion held in the one risky bundle of assets relative to that
risk-free asset.
The investor can borrow or lend and be anywhere along the transformati
(Exactly where he ends up along rZ depends on his preferences for risk versus r
we shall see this consideration does not enter the analysis until much later.) Fo
point Q in Figure (2.8) might represent 40 percent in the riskless asset and 60
the bundle of risky securities. Hence an investor with $100 would at point Q




Expected
Return




r




Standard
Deviation

Figure 2.8 Portfolio Choice.
because at any point on rZ™ the investor has a greater expected return for any
of risk compared with points on rZ. In fact because rZ™ is tangent to the effici
it provides the investor with the best possible set of opportunities. Point M
a ˜bundle™ of stocks held in certain fixed proportions. As M is on the effici
the proportions X i held in risky assets are optimal (i.e. the x: referred to e
investor can be anywhere along rZ™, but M is always a fixed bundle of stock
fixed proportions of stocks) held by all investors. Hence point M is known as
portfolio and rZ˜ is known as the capital market line (CML). The CML is the
transformation line which is tangential to the efficient frontier.
Investor™s preferences only determine at which point along the CML, rZ™,
vidual investor ends up. For example, an investor with little or no risk aver
end up at a point like K where he borrows money (at r) to augment his own
he then invests all of these funds in the bundle of securities represented by
still holds all his risky stocks in the fixed proportions xi*).

Separation Principle
Thus the investor makes two separate decisions

( 9 He uses his knowledge of expected returns, variances and covariances t
the efficient set of stocks represented by the efficient frontier YML (F
He then determines point M as the point of tangency of the line from
efficient frontier. All this is accomplished without any recourse to the i
preferences. All investors, regardless of preferences (but with the same
expected returns, etc.) will ˜home in™ on the portfolio proportions (xi*) o
securities represented by M. All investors hold the marketportfolio or mor
all investors hold their risky assets in the same proportions as their rel
in the market. Thus if the value of ICI shares constitutes 10 percent of
market valuation then each investor holds 10 percent of his own risky p
ICI shares.
(ii) The investor now determines how he will combine the market portfol
assets with the riskless asset. This decision does depend on his subje
return preferences. At a point to the left of M the individual investor is
risk averse and holds a percentage of his wealth in the market portfolio (i
optimal proportions xi*) and a percentage in the risk-free asset. If the
investor is less risk averse then he ends up to the right of M such as
levered portfolio (i.e. he borrows to increase his holdings of the market p
excess of his own initial wealth). At M the individual puts all his own w
the market portfolio and neither borrows nor lends at the risk-free rate.

The CML, rZ™, which is tangential at M, the market portfolio, must have the
by (2.25), that is:
in the market portfolio (point A, Figure 2.8). A less risk averse investor w
borrowing in order to invest more in the risky assets than allowed by his in
(point K, Figure 2.8). However, one thing all investors have in common is that
portfolio of risky assets for all investors lies on the CML and for each invest
slope of CML = ( p m - r)/Om
= slope of the indifference curve
The slope of the CML is often referred to as the ˜market price of risk™. The s
indifference curve is referred to as the marginal rate of substitution, MRS, sin
rate at which the individual will ˜trade off more return for more risk.
All investors portfolios lie on the CML and therefore they all face the sa
price of risk. From (2.27) and Figure (2.8) it is clear that for both investors a
the market price of risk equals the MRS. The latter measures the individlclal™
trade off or ˜taste™ between risk and return. Hence in equilibrium, all individ
the same trade-off between risk and return.
The derivation of the efficient frontier and the market portfolio have been
in terms of the standard deviation being used as a measure of risk. When risk i
in terms of the variance of the portfolio then
2
km = ( P m - r)/gm
is also frequently referred to as ˜the market price of risk™. Since a m and U: are co
very similar, this need not cause undue confusion. (See Roll (1977) for a di
the differences in the representation of the CAPM when risk is measured in
different ways.)

Market Equilibrium
In order that the efficient frontier be the same for all investors they must h
= ERi, U,? a
geneous expectations about the underlying market variables
course, this does not mean that they have the same degree of risk aversion.) H
homogeneous expectations all investors hold all the risky assets in the propor
by point M, the market portfolio. The assumption of homogeneous expectation
in producing a market equilibrium where all risky assets are willingly held in
proportions x: given by M or in other words, in producing market clearing. Fo
if the price of shares of ˜alpha™ is temporarily low and those of delta shares t
high, then all investors will wish to hold more alpha shares and less delta s
price of the former rises and the latter falls as investors with homogeneous e
buy alpha and sell delta shares.

In What Proportions are the Assets Held?
When we allow borrowing and lending we know that the individual will hold
risky assets in the optimal proportions represented by the point M. He holds
each individual to be on his own highest indifference curve.)
But a problem remains. How can we calculate the risky asset proportions xf
by point M? So far we have only shown how to calculate each set of xf for ea
the efficient frontier. We have not demonstrated how the proportions xf for th
point M are derived. This can, in fact, be quite a technically complicated p
illustrate, note from Figure 2.8 that for any transformation line:
[+I
tan@= ER, -


where ˜ p ™ represents any risky portfolio, and as we have seen ER, and opdepen
well as the known values of Pj and Ojj for the risky assets). Hence to achiev
equation (2.29) can be maximised with respect to xi, subject to the budget
Cxi = 1 and this yields the optimum proportions xf. Some of the x; may b
zero, indicating short selling of assets. If short sales are not allowed then the
constraint, xf 3 0 for all i, is required and to find the optimal xf requires a sol
quadratic programming techniques.

23.2 Determining Equilibrium Returns
Let us now use the ideas developed above to derive the CAPM equation re
the equilibrium return on each security. We do so using a mixture of graphic
(Figure 2.9) and simple algebra (see Appendix 2.1 for a more formal deriva
slope of the CML is constant and represents the market price of risk which i
for all investors.
Pm -r
slope of CML = -
am

We now undertake a thought experiment whereby we ˜move™ from M (which c
assets in fixed proportions) and create an artificial portfolio by investing some o

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:



and we can use this heuristically to derive a variant of the CAPM and the S
there is no risk-free asset. A convenient point to focus on is where the SM
vertical axis (i.e. at the point where = 0). The intercept in equation (3.2)




I .C




-
Pi

Figure 3.1 Security Market Line.
The SML holds for the return on the market portfolio ER"' and noting that, for
portfolio = 1, we have
+ b(1)
ER" = E R or b = ERm - E R
Hence the expected return on any security (on the SML) may be written
+ (ERm- E R ) p j
ERi = ERZ
where ERZ is the rate of return on any portfolio that has a zero-beta coeffi
respect to any other portfolio that is mean-variance efficient).
A more rigorous proof (see Levy-Sarnat, 1984) derives the above equilibri
equation in a similar fashion to that of the standard CAPM. The only differe
with no borrowing and lending, investors choose the x to minimise portfol
:
0; subject to:



(budget constraint: no borrowing/lending)
e x j =1
i= 1

ERP = CxjERj (a given level of expected return)
whereas in the standard CAPM the budget constraint and the definition of th
return are slightly different (see Appendix 2.1) because of the presence of
asset.
We can represent our zero-beta portfolio in terms of our usual graphic
using the efficient frontier (Figure 3.2). Portfolio Z is constructed by drawing
to the efficient frontier at M, and where it cuts the horizontal axis at ER" we
a horizontal line to Z. All risky portfolios on the SML obey equation (3.9
portfolio Z. At point X, ERj = ERZ by construction, hence from (3.5) it follo
B for portfolio Z must be zero (given that ERm # ERZ by construction). Henc
cov(RZ,R")
=o
=
Bz
var(Rm)




r
I
orn oi

Figure 32 Zero-beta CAPM.
.
b purchase M, hence reaching M,




bi

Figure 3.3 Asset Choice: No Risk Free Asset.

By construction, a zero-beta portfolio has zero covariance with the market port
we can measure cov(RZ,Rm)from a sample of data this allows us to ˜choose™
portfolio. We simply find any portfolio whose return is not correlated with
portfolio. Note that all portfolios along ZZ˜ are zero-beta portfolios but Z
portfolio which has minimum variance (within this particular set of portfol
also be shown that Z is always an inefficient portfolio (i.e. lies on the segmen
efficient frontier).
Since we chose the portfolio M on the efficient frontier quite arbitrarily
possible to construct an infinite number of combinations of various Ms with
sponding zero-beta counterparts. Hence we lose a key property found in th
CAPM, namely that all investors choose the same mix of risky assets, regardl
preferences. This is a more realistic outcome since we know that individua
different mixes of the risky assets. The equilibrium return on asset i could e
be represented by (3.5) or by an alternative combination of portfolios M* and
+
ERj = ERZ* (EP* - EF* p ˜
)
Of course, both equations (3.5) and (3.6) must yield the same expected return
This result is in contrast to the standard CAPM where the combination (r,M
unique opportunity set. In addition, in the zero-beta CAPM the line XX™ does no
the opportunity set available to investors.
Given any two mutual funds M and their corresponding orthogonal risky
then all investors can (without borrowing or lending) reach their optimum p
combining these two mutual fund portfolios.
Thus a separation property also applies for the zero-beta CAPM. Investors
the efficient portfolio M and its inefficient counterpart Z, then in the second
investor mixes the two portfolios in proportions determined by his individual p
For example, the investor with indifference curve 11, will short portfolio Z an
proceeds and his own resources in portfolio M thus reaching point M1. The in
indifference curve I2 reaches point M2 by taking a long position in both p
and Z.
The zero-beta CAPM provides an alternative model of equilibrium returns
dard CAPM and we investigate its empirical validity in later chapters. The m
of the zero-beta model are as follows:
that the return on this portfolio ERZ = ZxiERi is
(i) uncorrelated with the risky portfolio M,
(ii) the minimum variance portfolio (in the set of portfolios given by (i))
The combination of portfolios M and Z is not unique. Nevertheless the
0

return on any asset i (or portfolio of assets) is a linear function of ERZ an
is given by equation (3.5).

3.1.2 Different Lending and Borrowing Rates
Next, consider what is in fact a rather realistic case for many individual invest
that the risk-free borrowing rate r˜ exceeds the risk-free lending rate r t . The i
for the CAPM of this assumption are similar to the no-borrowing case, in that a
no longer hold the same portfolio of risky assets in equilibrium.
In Figure 3.4, if an individual investor is a lender, his optimal portfol
anywhere along the straight line segment rtL. If he is a borrower, then t
segment is BC. LL' and ˜ B are not feasible. Finally, if he neither borrows no
B
optimum portfolio lies at any point along the curved section LMB. The mark
is (by definition) a weighted average of the portfolios at L, B and all the portf
the curved segment LMB, held by the set of investors. In Figure 3.4 M rep
market portfolio. We can always construct a zero-beta portfolio for those w
borrow nor lend and for such an unlevered portfolio the equilibrium return o
given by:
+
ERj = E R (ER"' - ERZ)Pi
For portfolios held by lenders we have
+ (ER"' - Q )&t
ER, = rt
where P q ˜ the beta of the portfolio or security q relative to the lender'
is
unlevered portfolio at L:
PqL = COV(Rq9


L'




r



Qi

Different Borrowing and Lending Rates.
Figure 3.4
however, differ among individuals depending on the point where their indiffere
are tangent along LMB. Similarly all borrowers hold risky assets in the same
as at B but the levered portfolio of any individual can be anywhere along B-
For large institutional investors it may be the case that the lending rate i
different from the borrowing rate and that changes in both are dominated by
the return on market portfolio (or in the over time, see Part 6). In this case t
CAPM may provide a reasonable approximation for institutional investors wh
be large players in the market.

3.1.3 Non-Marketable Assets
Some risky assets are not easily marketable yet the standard CAPM, in princi
only to the choice between the complete set of all risky assets. For exam
capital, namely an individual's future lifetime income, cannot be sold because
illegal. Some assets such as one's house may not for psychological reasons o
considered marketable. When some assets are not marketable the CAPM can b
and results in the equilibrium return on asset i being determined by
+ /?;(ER" - r )
ER; = r
where


and VN = value of all non-marketable assets, V m = value of marketable asset
one-period rate of return on non-marketable assets.
Our 'new' beta, denoted pi", consists in part of the 'standard-beta'
cov(R;, R m ) / o : ) but also incorporates a covariance term between the ret
(observed) marketable assets and that on the (unobserved) non-marketa
cov(Ri, R˜ 1.
In empirical work one has to make a somewhat arbitrary choice as to w
tutes the portfolio of marketable assets (e.g. an all-items stock or bond index)
anomalies arising from the standard CAPM may be due to time variation in c
In Part 6 we make an attempt at modelling the time variation in the covari
'omitted assets' and hence attempt to improve the empirical performance of t

3.1.4 Taxes and Ransactions Costs
Investors pay tax on dividends received and may also be subject to capital gains
securities are sold. Hence investors who face different tax rates will have dif
tax efficient frontiers. Under such circumstances the standard CAPM has to b
so that the equilibrium return on asset i (or portfolio i ) is given by an equa
form (see Litzenberger and Ramaswamy (1979))
+ (ER" +
- r)pi f(6;, 6 m , t)
ERj = r
where 6; = dividend yield of the ith stock, 6, = dividend yield of the mark
and t = weighted average of various tax rates.
number of stocks in their portfolio. Most small investors (usually individuals
a few stocks, say five or less. If the number of stocks held is limited by t
costs then there is no simple relationship for equilibrium returns (Levy, 1978
individuals can avoid very high transactions costs by purchasing one or more m
and so the ˜high transactions cost™ variant of the CAPM may not be particular

3.1.5 Heterogeneous Expectations
Investors may have different subjective expectations of expected returns, va
covariances. This could not be the case under rational expectations where all in
assumed to know the true probability distribution (model) of the stochastic re
points in time. Hence the assumption of heterogeneous expectations is a viola
assumption that all investors have rational expectations. Under heterogeneous e
each investor will have his own subjective efficient frontier and hence each inve
different proportion of risky assets (xi) in his optimal unlevered portfolio: the
theorem no longer holds. In order to guarantee that the market clears the
the same amount of ˜buy™ and ˜sell™ orders and some investors may be sel
short while others with different expectations are holding stock i . For each i
problem is the standard CAPM one of minimising U / , subject to his budge
and a given level of expected return (see section 2.3 and Appendix 2.1). Th
optimum portfolio of the riskless asset and the bundle of risky assets X! (for i =
assets) where the X! differ for different investors.
In general, when we aggregate over all investors (k = 1, 2, . . . P) so that the
each asset clears, we obtain a complex expression for the expected return on
which is a complex weighted average of investors™ subjective preferences (of
return) and the C J ; ˜ .In general the marginal rate of substitution depends on t
wealth of the individual. Hence in general, equilibrium returns and asset pri
on wealth, which itself depends on prices, so there is no ˜closed form™ or expli
in the heterogeneous expectations case.
We can obtain a solution in the heterogeneous expectations case if we
utility function so that the marginal rate of substitution between expected retu
(variance) is not a function of wealth. Lintner (1971) assumed a negative
utility function in wealth which implies a constant absolute risk aversion par
Even in this case equilibrium returns, although independent of wealth, still d
complex weighted average of individuals™ subjective expectations of aikj and i
risk aversion parameters, ck.
In general, the heterogeneous expectations version of the CAPM is largely
However, as will be seen later, one can introduce different expectations in a mo
fashion. In Chapter 8, we assume that rational traders (or ˜smart money™) obey t
CAPM while there are some other traders who have different expectations an
different model of returns, based on the view that stock prices are influenced b
fashions™. This group of ˜noise traders™ then react with the smart money and
of agent may influence equilibrium returns.
returns measured in real terms:
ER; = ER?* + (ER" - E R Z * ) ˜ ;
Under certain restrictive assumptions, the above equation can be transpos
equation involving only nominal returns (Friend et a1 (1976)).



where a = ratio of nominal risky assets to total nominal value of all assets (
= covariance
= covariance of Ri with inflation (n)and
non-risky),
inflation (n).
If inflation is uncorrelated with the returns on the market portfolio or on
gin = = 0 and (3.15) reduces to the standard CAPM.However, in gene
see from equation (3.15) that the equilibrium return on asset i is far more co
in the standard one-period CAPM of Chapter 2.
The main conclusions to emerge from relaxing some of the restrictive assu
the standard CAPM are as follows.
Assuming that no riskless borrowing or lending opportunities are possib
seriously undermine the basic results of the standard CAPM. The equilibriu
return on asset i depends linearly on a weighted average of the returns o
portfolios. One is an arbitrary efficient portfolio M (say), and the other
1
risky portfolio that has a zero covariance with M1, and is known as th
portfolio. A form of the separation theorem still holds.
Introducing taxes or inflation uncertainty yields an equilibrium returns
which is reasonably tractable. However, assuming either heterogeneous e
by rational mean-variance investors or assuming agents hold only a limited
stocks, produces acute problems which results in complex and non-tractabl
for equilibrium returns.
The variants on the standard CAPM may well appeal to some on grounds of
realism'. However, this is of limited use if the resulting equations are s
as to be virtually inestimable, for whatever reason. Ultimately, the criter
simplicity versus fruitfulness. Nevertheless, examining a variety of models
does alert one to possible deficiencies and hence may provide some insig
empirical failures of the standard CAPM that may occur.


3 2 A SIMPLE MEAN-VARIANCE MODEL OF
.
ASSET DEMANDS
In this section we derive a simple model of asset demands based on the me
criterion, which often goes under the name of Tobin's risk aversion model,
discuss the relationship between the mean-variance (MV) model and the CAPM
involving only the expected return p˜ and the variance of the portfolio 0 .H
;
do not wish to delve into the complexities of this link (see Cuthbertson (1991)
Roley (1983) and Courakis (1989)) and sidestep the issue somewhat by assert
individual wishes to maximise a function depending only on p˜ and 0;:

U = U ( P N , 0;) U > 0, U2 > 0, w 1 1 ,
1 <0
U221

and in particular that the explicit maximand can be approximated by:



where c is a constant representing the degree of risk aversion. In the simpl
of the MV model the individual is faced with a choice between a single ris
and a bundle of risky assets which we can consider as a single risky asset
therefore only two assets. The riskless asset carries a known certain return, r, an
asset has an actual return R, expected return P R and variance a2.This set-up
that discussed under the heading of the ˜transformation line™ in section 2.2.4
what we want to do here is to concentrate on how this problem can yield a
to determine the optimal amount of the riskless and risky asset held as a func
relative rates of return and the riskiness of the portfolio. We shall find that the
held in the risky asset xT is given by:



Equation (3.18) is Tobin™s (1958) mean-variance model of asset demands. I
any risky portfolio and not necessarily for just the market portfolio.

Derivation of Asset Demands
Repeating the algebra in the derivation of the transformation line gives the expe
and standard deviation for the two-asset portfolio:
+ (1- y)PR
PN = y r
= (1 - y)GR
ON

where y = proportion of wealth held in the safe asset. The transformation l
arrangement of (3.19) and (3.20):
]0 -
p N = r + [ + PR N


which is a straight line in ( p ˜ N ) space with intercept r and slope (
O,
(Figure 3.5).
As we see below equation (3.20) is reinterpreted as the budget constraint f
vidual. By superimposing the indifference curves 11 and I2 in Figure 3.4 the
in attempting to maximize utility, will attempt to reach the highest indiffere
Figure 3.5 Simple Mean-Variance Model.

However, his choices are limited by his budget constraint. If all wealth is
risk-free asset then (1 - y) = 0 and from (3.20), ON = 0 and hence p˜ = r , u
Hence we are at point A. If all wealth is held in risky assets y = 0 and f
ON = OR which when substituted in (3.21) gives p˜ = p ˜ At point D the
.
incurs maximum risk OR and a maximum expected return on the whole portf
combination of the risk-free and risky assets is represented by points on the li
hence AD can be interpreted as the budget constraint. Given expected retur
hence the budget constraint AD, the individual attains the highest indifferen
point B which involves a diversified portfolio of the risky and risk-free ass
a nutshell, is the basis of mean-variance models of asset choice and in Tob
article the risk-free asset was taken to be money and the risky assets, bonds.
If the expected return on the risky assets p˜ increases or its perceived r
falls, then the budget line pivots about point A and moves upwards (e.g. li
In Figure 3.5 this results in a new equilibrium at E where the individual hol
the risky asset in his portfolio. This geometric result is consistent with the as
function in (3.18).
The asset demand function can be derived algebraically in a few steps. U
(3.19) and (3.20), maximising utility is equivalent to choosing y in order to m
* = yr + (1 - y)PR - (c/2)(1 - y)”O;
+ c(1 - y)oi = 0
a*/ay = (pR - r )
Therefore the optimal proportion of the holding of the risky asset x: = (1 -



Hence the demand for the risky asset increases with the expected excess retu
and is inversely related to the degree of riskiness 0;.Equation (3.24) is of
models of asset demands and will be used in Chapter 8 on noise trader mode
In a rather simplistic way the MV model can be turned into a model of


.&
returns. If the supply of the risky assets xt is exogenous then using xs = x an
:
(3.24) we obtain
(pR - r ) =
assets xi which may be considered as one asset. However, the MV model ca
alised to the choice between a set of risky assets and the risk-free asset: this i
in Chapter 18. The observant reader might also note that in this general me
framework the decision problem appears to be identical to that for the CAP
choosing the optimum proportions of the risky assets and the riskless ass
This is in fact correct, except that the mean-variance asset demand appro
representative individual who has subjective preferences for the rate at wh
substitute additional risk for additional return. This is encapsulated in a spe
tional form for the utility function and the associated indifference curves. T
avoids this problem by splitting the decision process into two independent de
separation theorem). In the CAPM it is only in the second stage of the decis
that the preferences of the individual influence his choices. This is because
stage decision we assume all investors wish to minimise the risk of the por
not the utility from risk), for any given level of expected return. The corr
between the CAPM and the mean-variance model of asset demands is discus
in Chapter 18.


3.3 PERFORMANCE MEASURES
The CAPM predicts that the excess return on any stock adjusted for the r
stock pi, should be the same for all stocks (and all portfolios). Algebraically t
expressed as:
(ERi - r)/Bj = (ERj - r ) / p j = . . .
Equation (3.26) applies, of course, under the somewhat restrictive assumpt
standard CAPM which include
all agents have homogeneous expectations
0

agents maximise expected return relative to the standard deviation of the
0

agents can borrow or lend unlimited amounts at the riskless rate
0

the market is in equilibrium at all times
0


In the real world, however, it is possible that over short periods the marke
equilibrium and profitable opportunities arise. This is more likely to be the ca
agents have divergent expectations, or if they take time to learn about a new en
which affects the returns on stocks of a particular company, or if there are so
who base their investment decisions on what they perceive are ˜trends™ in t
Future chapters will show that, if there is a large enough group of agents w
trends™ then the rational agents (smart money) will have to take account of the m
in stock prices produced by these ˜trend followers™ or noise traders (as they are o
in the academic literature).
It would be useful if we could assess actual investment performance ag
overall index of performance. Such an index would have to measure the act
rank alternative portfolios accordingly.
One area where a performance index would be useful is in ranking the perf
specific mutual funds. A mutual fund allows the managers of that fund to inve
portfolio of securities. For the moment, put to one side the difference in transa
of the individual versus the mutual fund manager (who buys and sells in larg
and may reap economies of scale). What one might wish to know is whether
fund manager provides a ˜better return™ than is provided by a random selectio
by the individual. It may also be the case that whatever performance index i
may itself be useful in predicting the fiture performance of a particular mutu
Any chief executive or group manager of a mutual fund will wish to kno
his subordinates are investing wisely. Again, a performance index ought to be
us whether some investment fund managers are either doing better than othe
or better than a policy of mere random selection of stocks. The indices we w
are known as the Sharpe, Treynor and Jensen performance indices.

3.3.1 Sharpe™s Performance Index (S)
The index suggested by Sharpe is a reward to variability ratio and is define
folio i as:


where ERj = expected return on portfolio i , a = variance of portfolio i and r
j
rate.
Sharpe™s index measures the slope of the transformation line and can be ca
any portfolio using historic data. Over a run of (say) quarterly periods we ca
the average ex-post return and the standard deviation on the portfolio, held by
fund manager. We can also do the same for a portfolio of stocks which has bee
selected. The mutual fund manager will be outperforming the random selecti
provided his value of Si is greater than that given by the randomly selecte
The reason for this is that the mutual fund portfolio will then have a transfor
with a higher slope than that given by the randomly selected portfolio. It follo
investor could end up with a higher level of expected utility if he mixed the
the mutual fund manager with the riskless asset.
Underlying the use of Sharpe™s performance index are the following assum

(i) Investors hold only one risky portfolio (either the mutual fund or a random
portfolio) together with the risk-free asset.
(ii) Investors are risk averse and rates of return are normally distributed
assumption is required if we are to use the mean-variance framework).

3.3.2 Weynor™s Performance Index (T)
When discussing Sharpe™s performance index it was assumed that individua
hold only the riskless asset and a single portfolio of risky assets. Treynor™s p
Pi

It is therefore a measure of excess return per unit of risk but this time the risk i
by the beta of the portfolio. The Treynor index comes directly from the CA
may be written as



Under the CAPM the value of Ti should be the same for all portfolios of secu
the market is in equilibrium. It follows that if the mutual fund manager in
portfolio where the value of Ti exceeds the excess return on the market portfo
be earning an abnormal return relative to that given by the CAPM.
We can calculate the sample value of the excess return on the market portfol
the right-hand side of equation (3.29). We can also estimate the Pi for any give
using a time series regression of (Ri - r)f on (RM- r)t. Given the fund manage
excess return (ERi - r ) we can compute all the elements of (3.29). The fun
outperforms the market if his particular portfolio (denoted i) has a value of
exceeds the expected excess return on the market portfolio. Values of Ti can
rank individual investment manager™s portfolios. There are difficulties in inter
Treynor index if Pi < 0, but this is uncommon in practice.

3.3.3 Jensen™s Performance Index (J)
Jensen™s performance index also assumes that investors can hold either the m
denoted i or a well-diversified portfolio such as the market portfolio. Jensen
given by the intercept J i in the following regression:


To run the regression we need time series data on the expected excess ret
market portfolio and the expected excess return on the portfolio i adopted by
fund manager to obtain estimates for Ji and P i . In practice, one replaces th
return variables by the actual return (by invoking the rational expectations a
(see Chapter 5)). It is immediately apparent from equation (3.30) that if Ji =
have the standard CAPM. Hence the mutual fund denoted i earns a return in
that given by the CAPM if J i is greater than zero. For J i less than zero,
fund manager has underperformed relative to the risk adjusted rate of return gi
CAPM. Hence Jensen™s index actually measures the abnormal return on the p
the mutual fund manager (i.e. a return in excess of that given by the CAPM).

Comparing Treynor ™s and Jensen”s Pegormance
We can rearrange equation (3.30) as follows:
It may be shown, however, that when ranking two mutual funds, say X and
indices they can give different inferences. That is to say a higher value of Ti
over fund X may be consistent with a value of Jj for fund Y which is les
for fund X. Hence the relative performance of two mutual funds depends on
chosen.
In section 6.1 the above indices will be used to rank alternative portfolio
a particular ˜passive™ and a particular ˜active™ investment strategy. This al
ascertain whether the active strategy outperforms a buy-and-hold strategy.

Roll™s Critique and Performance Measures
Roll™s critique, which concerns the estimation of the CAPM using a samp
indicates that in any dataset the following relationship will always hold


= sample mean of the return on portfolio i and
where Ri
Em = sample mean of the return on the market portfolio.
There is an exact linear relationship in any sample of data between the mea
portfolio i and that portfolio™s beta, if the market portfolio is correctly measu
if the CAPM were a correct description of investor behaviour then Treynor™s in
always be equal to the sample excess return on the market portfolio and Jens
would always be zero. It follows that if the measured Treynor or Jensen indice
than suggested by Roll then that simply means that we have incorrectly me
market portfolio.
Faced with Roll™s critique we can therefore only recommend the use of th
mance indices on the basis of a fairly ad-hoc argument. Because of transac
and information costs, investors may not fully diversify their portfolio and h
risky assets in the market portfolio. On the other hand, most investors do n
their holdings to a single security: they hold a diversified, but not the fully
portfolio given by the market portfolio of the CAPM.Thus the appropriate
risk for them is neither the (correctly measured) Bj of the portfolio nor the ow
0; on their own individual portfolio.
It may well be the case that something like the S&P index is a good app

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