Arbitrage Pricing 1077 assets'returns. And thus, portfolios of the first k+ 1 assets are perfect substitutes for all other assets in the market. Since perfect substitutes must be priced equally, there must be restrictions on the individual returns generated by the model. This is the core of the APt: there are only a few systematic components of risk existing in nature. As a consequence, many portfolios are close substitutes and as such, they must have the same value What are the common or systematic factors? This question is equivalent to asking what causes the particular values of covariance terms in the CAPM. If there are only a few systematic components of risk, one would expect these to be related to fundamental economic aggregates, such as GNP, or to interest rates or weather(although no causality is implied by such relations). The factor model formalism suggests that a whole theoretical and empirical structure must be explored to better understand what economic forces actually affect returns systematically. But in testing the APT, it is no more appropriate for us to examine this issue than it would be for tests of the CaPm to examine what, if anything, causes returns to be multivariate normal. In both instances, the return generating process is taken as one of the primitive assumptions of the theory. We do consider the basic underlying causes of the generating process of returns to be a potentially important area of research, but we think it is an area that can be investigated separately from testing asset pricing theories Now let us develop the APT itself from the return generating pr Consider an individual who is currently holding a portfolio and is contemplating n alteration of his portfolio. Any new portfolio will differ from the old portfolio by investment proportions x(i=1 ) which is the dollar amount purchased or sold of asset i as a fraction of total invested wealth. The sum of the x. proportions, since the new portfolio and the old portfolio put the same wealth into the n assets. In other words, additional purchases of assets must be financed by sales of others Portfolios that use no wealth such as x =(xy xn) are called In deciding whether or not to alter his current holdings, an individual will examine all the available arbitrage portfolios. The additional return obtainable from altering the current portfolio by n is given by x=∑x (∑xE)+(∑xb1)61 (∑xbA)+∑x Consider the arbitrage portfolio chosen in the following fashion. First, we will eep each element, x, of order 1/n in size; i.e. we will choose the arbitrage fied. Second that An underscored symbol indicates a vector or matrix
1078 The Journal of finance has no systematic risk; i.e., for eachy ∑x Any such arbitrage portfolio, x, will have returns of x=(xE)+(xb1)61+…+(xbk)8k+(x∈) E+(xb1)61+ (xbk)8K xe The term(xe)is(approximately) eliminated by applying the law of large numbers For example, if a denotes the average variance of the E, terms, and if, for implicity, each x, exactly equals +l/n, then var(xe=var(1/n∑e) =[var()]/n = here we have assumed that the E are mutually independent It follows that for large numbers of assets, the variance of xe will be negligible and we can diversify the Recapitulating, we have shown that it is possible to choose arbitrage portfolios with neither systematic nor unsystematic risk terms! If the individual is in equilibrium and is content with his current portfolio, we must also have XE No portfolio is an equilibrium(held) portfolio if it can be improved upon without incurring additional risk or committing additional resources To put the matter somewhat differently, in equilibrium all portfolios of these n assets which satisfy the conditions of using no wealth and having no risk must also earn no return on average The above conditions are really statements in linear algebra. Any vector, x which is orthogonal to the constant vector and to each of the coefficient vectors, b, (j=1,., k), must also be orthogonal to the vector of expected returns. An algebraic consequence of this statement is that the expected return vector, E must be a linear combination of the constant vector and the b, vectors, In algebraic terms, there exist k+1 weights,λ,A1,……,λ k such that E2=Ao+λ1b2 λkb If there is a riskless asset with return, Eo, then bo, =0 and E=入o E2-E0=λ1b1+…+Akbk, with the understanding that Eo is the riskless rate of return if such an asset exists
Arbitrage Pricing 1079 and is the common return on all"zero-beta"assets, i. e, assets with b =0, for all j, whether or not a riskless asset exists If there is a single factor, then the apt pricing relationship is a line in expected return, Ei, systematic risk, b,, space: Figure 1 can be used to illustrate our argument geometrically. Suppose, for example, that assets 1, 2, and 3 are presently held in positive amounts in some portfolio and that asset 2 is above the line connecting assets 1 and 3. Then a portfolio of I and 3 could be constructed with the same systematic risk as asset 2, but with a lower expected return. By selling assets l and 3 in the proportions they represent of the initial portfolio and buying more of asset 2 with the proceeds, a new position would be created with the same overall risk and a greater return. Such arbitrage opportunities will be unavailable only when assets lie along a line. Notice that the intercept on the expected return axis would be ec when no arbitrage opportunities are present The pricing relationship(2)is the central conclusion of the APT and it will be he cornerstone of our empirical testing, but it is natural to ask what interpretation can be given to the A, factor risk premia. By forming portfolios with unit systematic risk on each factor and no risk on other factors, each A, can be interpreted as λ=E1-E the excess return or market risk premium on portfolios with only systematic factori risk. Then(2)can be rewritten as E1-E0=(E1-E0)b (ER-Eo)b, he"market portfolio"one such systematic risk factor? As a well diversified folio, indeed a convex combination of diversified portfolios, the market E;-Eo=λb
1080 The Journal of finance portfolio probably should not possess much idiosyncratic risk. Thus, it might serve as a substitute for one of the factors Furthermore individual asset 6's calculated against the market portfolio would enter the pricing relationship and the excess return on the market would be the weight on these b s. But, it is important to understand that any well-diversified portfolio could serve the same function and that, in general, k well-diversified portfolios could be found that approximate the k factors better than any single market index. In general, the market portfolio plays no special role whatsoever in the aPt, unlike its pivotal role in the CAPm,(Cf. Roll [41, 42]and Ross [49) The lack of a special role in the APT for the market portfolios is particularly important. As we have seen, the aPt pricing relationship was derived by considering any set of n assets which followed the generating process(1). In the CAPM, it is crucial to both the theory and the testing that all of the universe of available assets be included in the measured market portfolio. By contrast, the APT yields a statement of relative pricing on subsets of the universe of assets. As a consequence, the APT can, in principle, be tested by examining only subsets of the set of all returns. We think that in many discussions of the CAPm, scholars were actually thinking intuitively of the aPt and of process(1 )with just a single factor. Problems of identifying that factor and testing for others were not considered important To obtain a more precise understanding of the factor risk premia, E'-Eo, in 3), it is useful to specialize the aPt theory to an explicit stochastic environment within which individual equilibrium is achieved. Since Pt is valid intertemporal as well as static settings and in discrete as well as in continuous time, the choice of stochastic models is one of convenience alone. The only critical assumption is the returns be generated by (1)over the shortest trading period a particularly convenient specialization is to a rational anticipations intertem- poral diffusion model. (See Cox, Ingersoll and Ross [8] for a more elaborate version of such a model and for the relevant literature references. )Suppose there are k exogenous, independent(without loss of generality) factors, s which follow multivariate diffusion process and whose current values are sufficient statistics to determine the current state of the economy. As a consequence the current price, p,, of each asset i will be a function only of =(s sh)and the particular fixed contractual conditions which define that asset in the next differ ential time unit. Similarly the random return, dr, on asset i will depend on the random movements of the factors. By the diffusion assumption we can write d=E:dt+bnds2+…+bkds It follows immediately that the conditions of the APt are satisfied exactly-with E=0 and the APt pricing relationship (3)must hold exactly to prevent arbitrage. In this setting, however, we can go further and examine the premia If individuals in this economy are solving consumption withdrawal proble then the current utility of future consumption, e.g, the discounted expected value of the utility of future consumption, v, will be a function only of the individuals current wealth, w, and the current state of nature, s. The individual will optimize
Arbitrage Pricing by choosing a consumption withdrawal plan, c, and an optimal portfolio choice x,so as to maximize the expected increment in V; i.e max edv n optimum, consumption will be withdrawn to the point where its marginal equals the marginal utility of wealth The individual portfolio choice will result from the optimization of a locally quadratic form exactly as in the static CAPM theory with the additional feature that covariances of the change in wealth, du, with the changes in state variables, ds, will now be influenced by portfolio choice and will, in general, alter the optimal portfolio. By solving this optimization problem and using the marginal utility condition, u(c)=ve, the individual equilibrium sets factor risk premia E-Eo=(R/c)(ac/as)03; (wvu)/Vu, the individual coefficient of relative risk aver a, is the local variance of (independent) factor S,.(The interested reader referred to Cox, Ingersoll and Ross [8]for details. Notice that the premia E Eo can be negative if consumption moves counter to the state variable. In this case portfolios which bear positive factor s risk hedge against adverse movements in consumption, but too much can be made of this, since by simply redefining s to be-g the sign can be reversed. The sign, therefore, is somewhat arbitrary and we will assume it is normalized to be positive. Aggregating over individuals yields (3) One special case of particular interest occurs when state dependencies can be ignored. In the log case, R=l, for example, or any case with a relative wealth criteria(see Ross [48])the risk premia take the special form E-Eo=R(∑xb)02 where x is the individual optimal portfolio. This form emphasizes the general relationship between b, and o,. Normalizing 2x,by to unity by scaling s, we E-Eo= Roj The risk premium of factor j is proportional to its variance and the constant of proportionality is a measure of relative risk aversion For other utility functions, individual consumption vectors can be expressed in erms of portfolios of returns and similar expressions can be obtained. In effect, nce the weighted state consumption elasticities for all individuals satisfy the aPt pricing relationships, they must all be proportional 2 Breeden [5] has developed the observation that homogenous beliefs about Es and bs imply perfect correlation between individual random consumption changes. His results depend on th assumption, made also by aPt, that k<N