1082 The Journal of finance The risk premium can be written in general as E-Eo=∑R where l indexes individual agents, W is the proportion of total wealth held by gent l, Ri is his coefficient of relative risk aversion is the partial elasticity of his consumption with respect to changes in the jth factor, ando, is the variance of the jth factor. Not very much is known about the term in parentheses and, all other things being equal, about all we can conclude is that risk premia should be larger, the larger the own variance of the factor. We would not expect this result to be specialized to the diffusion model and, in general, we would expect, with beta weights appropriately normalized, that factors with larger own variances would have larger associated risk premia Let us return now to the general APt model and aggregate it to a testable arket relationship. The key point in aggregation is to make strong enough assumptions on the homogeneity of individual anticipations to produce a testable theory. To do so with the aPt we need to assume that individuals agree on both he factor coefficients, by, and the expected returns, Er. It now follows that the pricing relationship(2)which holds for each individual holds at the market level as well. Notice that individual, and aggregate risk premia must coincide when here are homogenous beliefs on the expected returns and the factor coefficients As with the CAPM, the purpose of assuming homogenous anticipations is not to facilitate the algebra of aggregation. Rather, it is to take the final step to a testable theory. We can now make the rational anticipations assumption that(1) not only describes the ex ante individual perceptions of the returns process but also that ex post returns are described by the same equation. This fundamental intertemporal rationality assumption permits the ex ante theory to be tested by examining ex post data. In the next section we will discuss the possibilities for empirical testing which derive from this assumption. B. Testing the apT Our empirical tests of the APT will follow a two step procedure. In the first tep, the expected returns and the factor coefficients are estimated from time eries data on individual asset returns. The second step uses these estimates to test the basic cross-sectional pricing conclusion, (2), of the APT. This procedure is analogous to familiar CAPM empirical work in which time series analysis is used to obtain market betas, and cross-sectional regressions are the expected returns, estimated for various time periods, on the estimated betas While flawed in some respects, the two step procedure is free of some major conceptual difficulties in CAPM tests. In particular, the aPt applies to subsets course, developed a complete rational anticipations model in diffusio from this outline that the aPt is compatible with the more specific results of Merton [35], Lucas [31], Cox, Ingersoll, and Ross[], and Ross[48]
Arbitrage Pricing of the universe of assets; this eliminates the need to justify a particular choice of a surrogate for the market portfolio If we assume that returns are generated by(1), then the basic hypothesis we wish to test is the pricing relationship, Ho: There exist non- zero constants,(E,λ1,∵…,λk) such that E4-E0=A1b1+…+λkb1,forl The theory should be tested by its conclusions, not by its assumptions. One should not reject the APt hypothesis that assets were priced as if (2) held by merely observing that returns do not exactly fit a k-factor linear process. The theory says nothing about how close the assumptions must fit. Rejection is justified only if the conclusions are inconsistent with the observed data To estimate the b coefficients, we appeal to the statistical technique of factor analysis. In factor analysis, these coefficients are called factor loadings and they are inferred from the sample covariance matrix, I. From (1), the population variance, V, is decomposed into V=BAB+D covariances and d is the diagonal matrix of own asset variances, oF=/ where B = [bi] is the matrix of factor loadings, A is the matrix of fact From (5),V will be unaltered by any transformation which leaves BAB unaltered. In particular, if G is an orthogonal transformation matrix, GG=I, then V=BAB+D BGG AGGB+D BG)(GAG)(BG)+D e, If B is to be estimated from y, then all transforms BG will be equivalent. For xample, it clearly makes no difference in (1)if the first two factors switch places ore importantly, we could obviously scale up factor j's loadings and scale down factor by the same constant g and since bz,8=gb,(-8, the distributions of returns would be unaltered. To some extent we can eliminate ambiguity restricting the factors to be orthonormal so they are independent and have unit variance. Alternatively, we could maintain the independence of the factors and construct the loadings for each factor to have a particular norm value, e. g, to 4 This is a strongly positive view. Testing the aPt involves testing Ho and not testing the k-fact model. The latter tests may be of interest in their own right just as any examination of the distribution of returns is of interest, but it is irrelevant for the APT. As Friedman [16, pp. 19-20] points out: one would not be inclined to reject the hypothesis that the leaves on a tree arranged themselves so as to aximize the amount of sunlight they received by observing that trees did not have free will Similarly, one should not reject the conclusions derived from firm profit maximization on the basis of sample surveys in which managers claim that they trade off profit for social good