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errors of stock returns are persistent although ARCH models being purely
not provide theoretical reasons why this is so.
market index are influenced by agents™ changing perceptions of risk. Hence
with time varying conditional variances would appear to be an improvem
assumption that equilibrium returns are constant. Nevertheless the CAPM
varying returns does not provide a complete explanation of equilibrium r
the volume of trading may influence expected returns).
It is possible that noise traders as well as smart money influence the expe
on an aggregate stock market index, even after making allowance for ti
risk premia.
7the CAPM 1
This chapter is concerned with two main topics: first, the relationship between
variance model of asset demands and the CAPM, and second, tests of the CAPM
a formulation based on asset shares. The strengths and weaknesses of the latt
are assessed in relation to the tests of the CAPM outlined in the previous c
focus of the empirical work is on testing a set of implied restrictions on the
of the CAPM and on the importance or otherwise of time varying risk premi


18.1 THE MEAN-VARIANCE MODEL
There is a strong tradition in monetary economics of optimising models wh
determine an investor's desired demand for individual assets in a portfolio. T
generalises on the results from the mean-variance model of asset demands
discussed in Section 3.2. The MVMAD (or one version of it) predicts tha
equilibrium demand for a single risky asset (as a share of total wealth) depen
of expected returns and the variances and covariances of the forecast errors
which represent the riskiness of these assets. Hence we have

= p-l E (E,&+' - rr)
;
'
X:

where x; = (n x 1) vector of n risky asset shares, C, = (n x n ) matrix of for
of returns comprising the variance-covariance terms (aij), E,Rr+1 - rr is
vector of expected excess returns, rr = risk-free rate and p = coefficient of r
+
aversion. The demand (share) of the (n l)th risk-free asset is derived from
constraint
n


i=l

The MVMAD (or rather one version of it) assumes that investors choose t
values of asset shares XT in order to maximise a function that depends on expe
risk and a measure of the individual's aversion to risk. The latter sounds very
the objective function in the CAPM. The obvious question is whether these
in the literature are interrelated. The answer is that they are but the corresp
not one-for-one. The MVMAD is based on expected utility theory. One can a
The budget constraint for end-of-period wealth is:




The first term on the RHS of (18.4) is initial wealth, the second term is the retu
the portfolio multiplied by initial wealth and the final term is the receipts from
risk-free asset. If asset returns are normally distributed and we assume a consta
risk aversion utility function then maximising (18.3) subject to (18.4) reduces



where E&'+, = Cxj(ErRj,+l) = expected return on the portfolio, 0;= vari
portfolio and p = risk aversion parameter. There is much debate in the as
mean-variance literature (see, for example, Cuthbertson (1991b) and Courakis
an overview) about whether specific utility functions can give rise to asset d
desired asset shares xi*, which are independent of initial wealth. Another co
is whether the model expressed in terms of maximising end-of-period utility
wealth (or real returns) yields a similar functional form for asset demand func
the problem is conducted in terms of end-of-period nominal wealth (or nomin
The reader will have noted from the CAPM that it is crucial that asset de
equilibrium asset returns are independent of the level of initial wealth. If
returns are not independent of initial wealth then the theory is somewhat ci
equilibrium asset returns then depend on initial wealth which itself depends
asset returns. Hence if the mean-variance model of asset demands is to y
similar to the CAPM it must (at a minimum) yield desired asset demands in d
Aj that are proportional to initial wealth and hence asset shares that are i
of wealth. To choose a particular functional form for the utility function w
such asset demand functions is not straightforward and the current state of th
suggests that this can only be achieved as an approximation and results are on
small changes around the initial point.
We will sidestep these theoretical issues somewhat and assume that, to a
approximation, a version of the mean variance model of asset demands base
and (18.4) does yield asset demands in terms of equilibrium asset shares of the f
Frequently this type of model assumes that the 'constant' p in (18.1) is the co
relative risk aversion. In general p does depend on the level of wealth but the
made is that for small changes in the variables p may be considered to b
constant. Having obtained the mean-variance asset demand functions it is t
assumed that desired asset supplies xs (e.g. for government bonds, equity
bonds and foreign assets) are exugenous. Market equilibrium is given by xs =
xr denote the market equilibrium (which also equals the actual stock of assets o
and inverting (18.1) we have
over all individuals, which will in general depend on the distribution of weal
It is worth noting that the MVMAD only deals with the demand side of
If supply side decisions are governed by a set of economic variables zf (e.g
government bonds is determined by the budget deficit and corporate bonds b
gearing position of firms) then market equilibrium becomes:



Hence in general
(ERt+1 - rt) = P ˜ X " Z t )
and equilibrium expected returns depend not only on {Oi,} but also on a set o
Equation (18.7) is the basis of the so-called structural or portfolio appro
zf.
determination of asset returns which occurs frequently in the monetary econo
ture. In this approach a set of asset demand and asset supply equations are es
then these resulting equations are solved algebraically to yield the reduced form
for expected and actual returns given in (18.7) (Friedman, 1979). Alternatively
economists estimate the reduced form equation (18.7) directly: that is they reg
or relative returns on a whole host of potential economic variables represent
(18.7) (e.g. budget deficit, changes in wealth, inflation).

The CAPM Revisited
At present the reader must be somewhat bewildered since the RHS of (18.6)
be nothing like the CAPM equation which for asset i is:




where E&"+, = expected return on the market portfolio and

h = (EfRzl - rf)/cri = market price of risk

However, (18.6) does reduce to something close to (18.8) as can be seen in th
illustrative case of three risky assets. Writing (18.6) in full:




where o,?, 2 = 021. It follows that
01
0ji
Thus the standard C M M (18.8) and the mean-variance model of asset de
exogenous supply (18.10) are equivalent providing p = A. Now unfortunately
not quite the same thing. However, both p and A are similar in that they are bo
of risk. In broad terms p measures the curvature of the individual™s utility fun
l / A measures the additional return on the market portfolio per unit of marke
is about as far as we can go in terms of drawing on the common elements
approaches.
To summarise, the MVMAD with an exogenous supply of assets yields
for equilibrium asset returns that is very similar to that given by the CAPM
differ in their measure of overall risk. The MVMAD uses a measure of risk
depends on the curvature of the utility function while the CAPM uses a m
which depends on the additional return on the market portfolio to compensate
risk (0;).
Both the mean-variance asset demand approach and the standard CAPM are
problems. The investor is only concerned with (wealth) at the end of the first
all future periods are ignored. However, Lintner (1971) has shown that in a
minimisation problem of the form (18.5) that h is equal to the harmonic m
agent™s coefficient of relative risk aversion.

Summary: MVMAD and the CAPM
We have concentrated on the relationship between the MVMAD which app
monetary economics literature and the CAPM of the finance literature. The m
sions are:
(i) The MVMAD is usually couched in terms of maximising the expect
utility from end-of-period wealth. Under certain restrictive assumption
form of the utility function this may be reduced to a maximand (or m
terms of portfolio variance and portfolio expected return as in the CAP
(ii) Unlike the CAPM, the MVMAD has asset demands that depend on th
erences of individuals and are in general not independent of initial wea
(iii) The MVMAD is usually inverted to give an equation for asset return
CAPM) on the assumption that asset supplies are exogenous. The latter
assumption unlikely to hold in practice.
(iv) Both the CAPM and the MVMAD (with exogenous asset supplies) i
asset returns depend on (a) cov(Ri, Rm)or equivalently (b) Ex,, where
(v) The weakness of the MVMAD is that it requires the researcher to assume
utility function and equilibrium prices depend on individuals™ preferen
distribution of wealth, represented by the average coefficient of risk
across agents in the market.
18.2 TESTS OF THE CAPM USING ASSET SHAR
We now want to explore an alternative method of presenting and testing the CA
is often discussed in the literature. For each asset i we have:


using the definition of cov(R;, Rm)we have:




For illustrative purposes consider again the case of n = 3 assets then (18.15) i
Rlf+l - r
r


or in matrix form
E,R,+l - rr = hEx,
First assume the 0 i j are constant (and note that a ,= 0 j i ) . The CAPM then p
;
the expected excess return on asset i depends only on a weighted average
asset shares Xi, (which are equilibrium/desired asset shares in the CAPM). We
how economists often like to test the implications of a theory by testing res
parameters. Are there any restrictions in the system (18.16)? Writing out (18.




Assume for the moment that h is known so that we can use (Rj,+l - r r ) / h as
variables in (18.18). Suppose we run the unrestricted regressions in (18.18)
six ˜unrestricted coefficients™


In matrix notation
E,R,+1 - rr = h ( n ™ x , )
If the CAPM with constant is true then from (18.18) we expect
aij
is symmetric.

Restrictions on the Variance-Covariance Matrix
Tests of the above restrictions are rarely conducted since there is another mo
set of restrictions for the CAPM implicit in (18.18) which logically subsume
restrictions. If we add the assumption of rational expectations:




where Qr is the information set available at time t. Then from (
variance -covariance matrix of forecast errors is:




Substituting (18.23) in (18.16) we obtain a regression equation in terms of ac
excess returns
+
(R,+1 - r,) = AXx, Er
where 6; = (Elr, c z r , ‚¬2,). However, from the derivation of the theoretical
know that
I = ( O i j } where aij = E[(Ri - ERi)(Rj - E R j ) ]
:
But under RE comparing (18.25) and (18.23) we see that
I: = E(66™)

+
Thus the CAPM RE imposes the restriction that the estimated parameters
regression of the excess returns on the X i [ (i.e. equation (18.19)) should equal
sion estimates of the variance-covariance matrix of regression residuals E
unrestricted set of equations (18.19) can be estimated using SURE or maxim
hood and yield a value for Il which does not equal E ( ‚¬ ‚¬ ™ ) .The regression c
recomputed imposing the restrictions that the nij equal their appropriate E (
will worsen the ˜fit™ of the equations but if the restrictions are statistically acc
˜fit™ shouldn™t worsen appreciably (or the estimates of the l l i j alter in the
regressions). This is the basis of the likelihood ratio test of these restrictions.
The elements nij are determined exclusively by the estimates of the E
therefore the regression also provides (˜identifies™) an estimate of A, the m
of risk. The estimate for A (or p) can be compared with estimates obtained
studies. Also note that because the regression package automatically imposes E
symmetric, then, under the restriction that Il = E(&), Il is also automatically
Difficulties in Estimating the ˜Asset Shares™ Form of the CAPM
The above restriction ll = E(&&™) undoubtedly provides a ˜strong test™ of the
in practice we face the usual difficulties with data, estimation and interpretatio
these are as follows:
The returns Rif+l in the CAPM are holding period returns (e.g. over
and these are not always available, particularly for foreign assets (e.g. s
use monthly Eurocurrency rates to approximate the monthly holding p
on government bonds).
The shares Xit according to the CAPM are equilibrium holdings at m
Often data on Xir for marketable assets (e.g. for government bonds) are on
at the issue price and not at market value.
There is the question of how many assets one should include in the emp
Under the CAPM investors who hold the market portfolio hold all asset
real estate, land, gold, etc.). Clearly, given a finite data set, to include a la
of assets would involve a loss of degrees of freedom and multicollinearit
would probably arise.
If either any important asset shares are excluded or if the asset shar
are measured with error then this may result in biased parameter es
principle the measurement error problem can be mitigated by applying
instrumental variables. A measure of omitted variables bias can be asc
trying different sets of Xit and comparing the sensitivity of results obt
similar vein, omitted variables bias might show up as temporal instab
parameter estimates (Hendry, 1988). Often in empirical studies these ˜r
to the estimation procedure are not undertaken.

Under the assumption of constant aij (the so-called static CAPM) it is invar
that the restrictions ll = E(&&™) rejected. Also as the variability in the Xi
are
rather low relative to the variability in (Rit+l - r f ) ,the fit of these CAPM r
is very poor (Frankel, 1982, Engel and Rodrigues, 1989, Giovannini and Jo
and Thomas and Wickens, 1993). Finally these studies find that h (or p) is oft
(rather than positive) and is extremely poorly determined statistically. Indeed
does not reject h(or p ) = 0 statistically. The CAPM then reduces to the hyp
expected excess returns are zero (or equal a constant)



In the context of international assets with a known own currency nominal
Eurocurrency rates) this amounts to uncovered interest parity (for a = 0). Th
i
expected excess returns are found to depend on asset shares Xir the restricti
parameters llij do not conform to the static CAPM (i.e. constant aij) and th
equations is rather poor.
ARCH model
+
A2
(for all i and j )
o ˜ ˜ ,PO ˜P l & $ ,
=+
Thomas and Wickens (1993) find little evidence of time varying variances and
for a diverse set of assets which include foreign as well as domestic assets. H
a subset of assets, for example only ˜national™ assets or only equities, are incl
CAPM model then ARCH effects are found to be present. Other empirical s
do not test for ARCH effects but assume they may be present and then rec
above model and test the time varying covariance restrictions Il, = E ( E i E j ) f
As far as estimation is concerned the introduction of ARCH effects can
horrendous computational requirements as the number of parameters can be
For example, with seven risky assets each single O i j t + l can in principle de
the other seven lagged values of O i j r for the other assets (and a constant).
first-order ARCH model for asset 1 only we have:




Equation (18.28) for asset 1 includes individual terms for 0 1 1 , 0 1 2 , 0 1 3 , . . ., 01
similar equations are needed for 0 2 j , 03j™ etc. (of course remembering that O i j
higher-order ARCH processes additional lagged terms also need to be estim
is the computational task (and loss of degrees of freedom) that usually only
ARCH processes are used and sometimes only own lagged covariances are e
the ARCH process:
+
2
( O i j ) t + 1 = ˜1 eij<a$)t

Since we are now allowing the C matrix in (18.24) to vary over time (as we
naturally the time varying covariance CAPM ˜fits™ better but again the CAPM
that (schematically) Il, = C, is invariably rejected (e.g. Giovannini and Jo
and Thomas and Wickens (1993). Also it is often the case that the point es
(or p ) is negative and again the null hypothesis that is zero is easily accepted
is a rejection of the CAPM.
An alternative method of ˜modelling™ the time varying variances in th
to assume that the variances O i j are (linearly) related to a small set of mac
variables Zir. Engel and Rodrigues (1989)™ who use the government debt of s
as the ˜complete set™ of assets in the portfolio, assume Zjr is either ˜surp
prices or surprises in the US money supply (the data series for these ˜su
residuals from ARIMA models). They find that both surprises in oil prices an
money supply help to explain the relative rates of return on this international
government bonds. However, the CAPM restriction that C, = Il, is still reje
or p is statistically zero).
There is an additional potential weakness of the above reported tests of
This is that they only consider returns of over one month. Now while in p
horizon returns, for example, for 3, 6 months or even 1-2 years. In this case
ARCH process might also be a better representation than for one-month retur
All the above studies only consider the one-period CAPM model. The c
CAPM and the MVMAD yield expressions for equilibrium returns under the
that agents are concerned only with one-period returns (or next periods wealth
of the MVMAD). Merton (1973) in his intertemporal CAPM has demonstrate
constant preferences (e.g. constant relative risk aversion parameter p) and som
restrictions, the CAPM yields a constant price of risk (A):



for the entire market portfolio. However, in Merton's model the price of
constant for components of the market portfolio (Chou et al, 1989, page 4)
therefore that in an intertemporal CAPM one would not expect the 'price
be constant for returns on subsets of the portfolio. However, the latter is p
assumption made in much empirical work. Put another way, a weakness of
empirical studies is that they assume that the portfolios chosen are good repr
of the market portfolio (i.e. the 'errors in variables' problem is not acute)
section discusses a model which relaxes the assumption that the price of risk
and hence attempts to tackle this potential omitted variables problem.

Time Varying Price of Volatility
The CAPM for asset i may be written


where we shall refer to Qr as the price of volatility and allow it to be time var
term is chosen because the price of risk h is usually assumed to be a constant.)
Chou et a1 (1989) suppose we approach the problem of the potential mism
of the market portfolio by assuming it consists of an observable stock return
unobservable component with return RU.The variances are 0 and o, and the
:
:
between the stock return and the unobservable return is usu. The CAPM fo
portfolio (we omit t subscripts on the oij terms):

+ (1 - xr)0sul
E&+, - rf = Qt[xra,"
w r + (1 - ˜ r ) # I a , 2
=
= utera,"

+
where #' = (osu/o:)r, = xr (1 - xr)B,", and xr is share of equities in the
Or
portfolio. Hence, 3 is the unobserved but possibly time varying CAPM be
/:
the stock portfolio and the unobserved part of the true market portfolio.
equation (18.31) becomes:
+
R;+, - rt = q r 8 p ; Er+l
+
R:+, - r, = aaS Et+,

where a is constant only if

(i) x, = 1, that is the true market portfolio consists only of stocks, or
=
(ii) that is the (conditional) covariances between the 'omitted
(asu), (a:),,
the included assets (stocks) are equal for all time periods, and
(iii) ' U r the price of volatility is constant.
Any time variation found in a, in an equation of the form (18.33) will be
breakdown of one or all of the above assumptions. A number of studies find
varying own variance U: can help to explain movements in own expected ex
but 'the linkage' is unstable (i.e. a is time varying). This is consistent with
variables interpretation of equation (18.33). A purely statistical method of mod
variation in a which is reasonably general is to assume a, follows a random w

+ U,
= at-1

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