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not provide theoretical reasons why this is so.

market index are influenced by agentsâ€™ 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 individualâ€™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

agentâ€™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 individualsâ€™ 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 â€˜unrestricted coefficientsâ€™

In matrix notation

E,R,+1 - rr = h ( n â€™ 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 , â‚¬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(66â€™)

+

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 ( â‚¬ â‚¬ â€™ ) .The regression c

recomputed imposing the restrictions that the nij equal their appropriate E (

will worsen the â€˜fitâ€™ of the equations but if the restrictions are statistically acc

â€˜fitâ€™ shouldnâ€™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 (â€˜identifiesâ€™) 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 â€˜Asset Sharesâ€™ Form of the CAPM

The above restriction ll = E(&&â€™) undoubtedly provides a â€˜strong testâ€™ 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 â€˜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(&&â€™) 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 â€˜nationalâ€™ 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 , 03jâ€™ 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 â€˜fitsâ€™ 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 â€˜modellingâ€™ 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)â€™ who use the government debt of s

as the â€˜complete setâ€™ of assets in the portfolio, assume Zjr is either â€˜surp

prices or surprises in the US money supply (the data series for these â€˜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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