The consumption based asset pricing model

Journal

of Flnanclal

Economics

Y (1981) 103 -108. North-Holland

THE CONSUMPTION

BASED

ASSET

Pubhshlng

PRICING

Company

MODEL

A Note on Potential Tests and Applications Bradford

Received

CORNELL*

April 1980, final version

received September

1980

Breeden’s demonstration that Merton’s multl-beta capital asset prxing model can be collapsed mto a single-beta model where betas are computed with respect to aggregate consumption IS an Important theoretlcal advance Nonetheless, Breeden’s model retams many of the empuxal problems that beset Merton’s earher version In general the consumption betas WIII be nonstationary, so that the state variables must be observable for the model to be estimated.

1. Introduction In a recent issue of this journal, Breeden (1979) presents a model of asset pricing in contmuous time that appears to offer a dramatic simplification of Merton’s (1971, 1973) earlier model. Whereas, Merton’s pricing equation includes the covariance of each asset’s return with both the market portfoho and all the state variables, Breeden’s pricing equation includes only the covariance of the asset’s return with aggregate real consumption. Through Rubinstein (1976) derives the same result from a discrete time model, he had to assume weak aggregation. By moving to continuous time, Breeden is able to dispense with this assumption. In addition to highlighting the unique role of consumption in asset prlcmg, Breeden’s model also appears to solve some difficult empirical problems. First, cstlmation of the model does not seem to require observations on the state variables. Breeden, for instance, states that (p. 266): ‘The fact variable, variables, stationary assets and

that this model involves a single beta relative to a specific rather than many betas measured relative to unspecified may make it easier to test and to implement, given certain assumptions on the joint distributions of rates of return on aggregate consumption.’

This is particularly important because be identifiable, let alone observable. *The author would hke to thank Jr, for helpful comment\

Doug

in general

Brceden,

the state variables

James Ohlson

and the referee, John

may not

B Long,

104

B. Cornell,

Note

on rhc, consumption

bawd

usscf pricing

model

Secondly, the consumption based capital asset pricing model surmounts some of the problems discussed by Roll (1977). Though the market portfolio may be unobservable, data on aggregate consumption are available in the national income accounts. Thus Breeden suggests that (p. 292): ‘The principal virtue of aggregate consumption measures, in comparison with the market proxies used, is that the consumption measures available cover a greater fraction of the true consumption variable than the fraction that the market portfolio measures cover of the true market portfolio.. .’ The thesis of this note is that tests and applications of the consumption based CAPM are not as straightforward as the foregoing quotations apparently imply.’ I will argue, in fact, that Breeden’s model has essentially the same empirical problems that beset Merton’s original model. What has happened is that Breeden has collapsed the state variables into the consumption betas. While this simplifies the pricing equations, it also means that each asset’s consumption beta is a function of the state variables. As a result the consumption betas will not be stationary if the state variables are random. To surmount the non-stationarity problem, the dependence of the consumption betas must be accounted for explicitly. In empirical work, this means that the state variables must be both identifiable and observable. 2. The consumption based CAPM: A quick review To analyze the role of the consumption Merton-Breeden model is required. The model barest bones to highlight the relevant issues. Breeden (1979). To begin, all asset returns, assumed to follow diffusion processes. Thus we

betas, a brief review of the presented here is pared to its The details are contained in a, and state variables, s, are can write,

da=Bo(s,t)dt+~,(s,t)dzp

(A x 1 vector),

ds=&t)dt+o,(s,t)dz,

(S x 1 vector),

where bold italic symbols represent vectors and bold roman ones are matrices. It is assumed that all agents agree on the probability distributions. Investors maximize a time-additive state independent expected utility function of the form,

‘I do not mean to argue that Breeden was unaware goal is simply to clarify an important empirical issue.

of the problems

I discuss in this note. My

B. Cornell, Note on the consumption based subject

to the budget

asset pricrng

105

model

constraint,

dWk=[wk’(p,-r)+r]Wkdt-c’dt+

Wkwk’~,dza,

where ck(t)=individual

k’s consumption

Wk =individual

k’s wealth,

wk =vector

of k’s wealth in each risky security.

of the fraction

The familiar

first-order

at time t,

conditions

are

U;(Ck,t)=J;(Wk,S,t),

(1)

wkWk = (-J”,lJk,,)V,’ (A-- r)-v,‘v,,vs,lJk,,~),

(2)

where the subscriptions

on J, U and later c denote

Jk(Wk,s, t) =individual

k’s derived

V,,

=covariance

matrix

utility

function

partial

derivatives,

and

for wealth,

of asset rates of return

over the interval

(t, r + dr), VLl,

=covariance matrix between s(t + dt) given s(t).

asset returns

Define Tk = - Ut/IJic and note that (1) implies Eq. (2) can then be written as

over (t, t + dt) and

J”,, = IJEc cf and J”,,,, = U~J”,.

wkWk= ( Tk/Ck,)Va;l(& - r) - VnoVas(C~/C~), or equivalently,

wkWkck,V,, = Tk (p, - r) - V,,

c”,V,,wk + Vosct = Tk (p, -

r ).

ca, (3)

The most important step in collapsing Merton’s model to a CAPM based only on consumption, is to see that Ito’s lemma implies that (3) can be drastically simplified because

Substituting

(4) into (3) yields

v,ck= Tk(A - r).

(5)

106

Defining

B. Cornell, Note on Ihe consumption

T” =&

Tk and summing

based assetprrcing model

(5) over k gives

V,,m=Tm(yo-r)~

(6)

Assuming that there exists a security with return, (I,, equal to the growth rate in real aggregate consumption, eq. (6) for that security becomes cov (Lf,, C”, ) = T”(&

- r),

(7)

where /‘c = /Le. Dividing

(7) into (6) gives CL,-

r = B,.,(P,- r),

(8)

where

B,,, = cov (a, u, )/var (a,) = cov (a, c,)/var

(c,).

Eq. (8) is the consumption based version of the continuous time CAPM. The excess return on any security 1s solely a function of that security’s consumption beta and the difference between the expected growth rate in aggregate consumptton and the riskless rate.

3. Testing and applying the model: The stationarity

problem

With the state variables eliminated and the index problem solved, the consumption based CAPM can apparently be tested and applied to problems such as estimating a lit-m’s cost of capita1 in the fashion in which earlier practitioners attempted to use the orrginal CAPM. For these tests and between asset applications to be feasible, however, the joint distribution returns and aggregate consumption must be sufficiently stable to allow estimation of beta from time series data. The problem 1s that except for trivial cases, the joint distribution is unlikely to be stable! This is true even of the underlying distributions for the state variables are stable. Because they are a function of the random state variables, consumption betas will be stochastic. In retrospect this fact seems obvious. The consumptron based CAPM reduces to a simple form precisely because uttlity maximizing agents adjust their consumption in responses to variation in the state vartables. In this manner aggregate consumption comes to serve as a sufficient statistic for the underlying state variables. Accordingly, the covarlance between asset returns

107

B. Cornell, Note on the consumpr~on bused asset prvzing model

and the growth rate of aggregate consumption reflects the level of the state variables at each instant. The problem is best seen by returning to eq. (4), rewritten below for an individual asset [for clarity, the partial derivatives are written out in (4’) rather than using the shorthand notation] cov (a, ck) = (dch/dWk) cov(u, W”) + C (dck/ds) cov(a, s ).

(4’ )

From (4’) it can be seen that cov(a,c,) will be stable only for special cases. One obvious case is non-stochastic state variables so that cov(u, s) =O. Another example is logarithmic utility which implies that dc’/ds=O for all s. In both these cases, Merton (1971) has shown that the single beta version of the CAPM holds with betas calculated with respect to the market portfolio. Finally, it is possible that &(dck/ds) cov(a,s) is constant, or nearly so, even though cov(a,s) are random. In general, of course, the values of the state variables must be known before consumption betas can be estimated. If the state variables were observable, however, then Merton’s multi-beta model could also be estimated. The hope that there are only a few relevant state variables is also called into question by Breeden’s work. Breeden proves a theorem which states that (p. 272): ‘All individuals in the economy, regardless of preferences, their optimal positions by investing in at most S+2 funds.’

may

obtain

As Rosenberg and Olsen (1976) note, this Implies that all relative security prices must lie in an S-t2 dimensional space. Since very few linear dependencies among stock prices have been documented, we know that S must at least be of the same order of magnitude as the number of listed stocks. The existence of a large number of state variables does not necessarily preclude empirical tests of asset pricing models. In the arbitrage pricing model of Ross (1976), for instance, the fact that each security has an independent risk factor means that the dimension of the space in which relative security prices lie in n- 1, where n is the number of securities. Nonetheless, by adding additional assumptions Ross does reach empirically testable conclusions. In conclusion, Breeden’s realization that the continuous time CAPM can be collapsed so that excess returns depend only on a security’s beta with aggregate consumption does not solve the empirical problems that beset Merton’s earlier model. Though a single-beta pricing equation can be derived, the effects of the state variables have simply been impounded in the covariance between consumption and asset returns. This means that in general consumption betas will be functions of the state variables. Though

108

B. Cornell, Note on the consumption

based asset pricing

model

the stationarity of the consumption betas remains an empirical question, theory shows that in most cases consumption betas will be random.

the

References Breeden, D.T., 1979, An mtertemporal asset pricing model with stochastic consumptton and Investment opportumties, Journal of Fmanctal Economics 7, Sept., 265-296. Merton, R.C., 1971, Opttmum consumptton and mvestment rules in a continuous time model, Journal of Economtc Theory, 373-413. Merton, R.C., 1973, An intertemporal capttal asset prtcmg model, Econometrrca 41, Sept., 867887. Roll, R., 1977, A crmque of the asset prtcmg theory’s tests, Journal of Fmanctal Economics 4, March, 129-176. Rosenberg, B. and J.A. Ohlson, 1976, The stationary dtstribution of returns and portfolio separation in caprtal markets: A fundamental contradiction, Journal of Financtal Quantitattve Analysts 11, Sept., 393402. Ross, S.A., 1976, Tlx arbttrage theory of capttal asset pricmg, Journal of Economic Theory 13, Dec., 341-360 Rubinstem, M., 1976, The valuation of uncertam income streams and the prtcing opttons, Bell Journal of Economtcs 7, Autumn, 407425.