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