The capital-asset-pricing model and arbitrage pricing theory: a unification

We present a model of a financial market in which naive diversification, 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 return, where the overall risk can be decomposed into a systematic and an unsystematic part, as in the arbitrage pricing theory; and the systematic 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 equivalent to the pervasive economic principle of no arbitrage. The factors in the model are endogenously chosen by a procedure analogous to the Karhunen-Loéve expansion of continuous time stochastic processes; it has an optimality property justifying the use of a relatively small number of them to describe the underlying correlational structures. Our idealized limit model is based on a continuum of assets indexed by a hyperfinite Loeb measure space, and it is asymptotically implementable in a setting with a large but finite number of assets. Because the difficulties in the formulation of the law of large numbers with a standard continuum of random variables are well known, the model uncovers some basic phenomena not amenable to classical methods, and whose approximate counterparts are not already, or even readily, apparent in the asymptotic setting.