<< . .

. 7
( : 51)



. . >>




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

<< . .

. 7
( : 51)



. . >>