THE CAPITAL-ASSET-PRICING MODEL AND ARBITRAGE PRICING THEORY - A UNIFICATION

We present a model of a financial market in which naive diversificatio n, based simply on portfolio size and obtained as a consequence of the law of large numbers, is distinguished from efficient diversification , based on mean-variance analysis. This distinction yields a valuation formula involving only the essential risk embodied in an asset's retu rn, where the overall risk can be decomposed into a systematic and an unsystematic part, as in the arbitrage pricing theory; and the systema tic component further decomposed into an essential and an inessential part, as in the capital-asset-pricing model. The two theories are thus unified, and their individual asset-pricing formulas shown to be equi valent to the pervasive economic principle of no arbitrage. The factor s in the model are endogenously chosen by a procedure analogous to the Karhunen-Loeve expansion of continuous time stochastic processes; it has an optimality property justifying the use of a relatively small nu mber of them to describe the underlying correlational structures. Our idealized limit model is based on a continuum of assets indexed by a h yperfinite Loeb measure space, and it is asymptotically implementable in a setting with a large hut finite number of assets. Because the dif ficulties in the formulation of the law of large numbers with a standa rd continuum of random variables are well known, the model uncovers so me basic phenomena not amenable to classical methods, and whose approx imate counterparts are not already, or even readily, apparent in the a symptotic setting.