P roc. Natl. Acad. Sci. USA
Vol. 94, pp. 4229–4232, April 1997
The capital-asset-pricing model and arbitrage pricing
theory: A unification
M. A LI K HAN*
*Department of Economics, Johns Hopkins University, Baltimore, MD 21218; †Department of Mathematics, National University of Singapore, Singapore 119260; and ‡Cowles Foundation, Yale University, New Haven, CT 06520
Communicated by Paul A. Samuelson, Massachusetts Institute of Technology, Cambridge, MA, October 3, 1996 (received for review August 14, 1996)
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expected return of an asset is related to its exposure to each of these factors, and now summarized by a vector of factor
loadings. The reward to the residual component in the return to a particular asset, unsystematic or idiosyncratic risk, can be made arbitrarily small simply by considering portfolios with an arbitrarily large number of assets.
The basic point, however, is that the two theories capture
two different sets of risks and address different aspects of the premium-awarding scheme for taking such risks. The CAPM,
by its emphasis on efficient diversification in the context of a finite number of assets, neglects unsystematic risks in the sense of the APT; whereas the APT, with its explicit focus on
markets with a ‘‘large’’ number of assets, and by its emphasis on naive diversification and on the law of large numbers,
neglects essential risks. The two theories seem to be inherently disjoint. It is surprising, however, that a model which unifies their basic ingredients can nevertheless be found; and moreover, that it is one in which the absence of arbitrage opportunities is not only sufficient, but in contrast to the literature, also necessary for the validity of the APT pricing formula. We present this model here.
It is easy to see why such a unification has not been
considered so far. A natural way to proceed is to work with a limit model of a financial market with a continuum of assets, to identify the ensemble of systematic risks, and within this, the essential risk emanating from a suitably constructed ‘‘market’’ portfolio. Standard methods, however, do not permit any
progress toward a limit model in which nontrivial portfolios with genuinely unsystematic or asset-specific risks can be
included; the difficulties associated with a version of the law of large numbers for a standard continuum of random variables— say the Lebesgue unit interval as an index set—are well
understood [see refs. 10 (theorem 2.2), and 11–13]. An alternative way is to follow the APT literature and work with an increasing sequence of asset markets, but here one has to
overcome at least three obstacles: unsystematic risks are never completely eliminated, exogenous factor structures are not
sufficiently refined to yield orthogonal factor loadings, and pricing formulas are approximate with convergent requirements on infinite series (see refs. 14–16). Under these considerations, it is not evident how to introduce a simple explicit formula for essential risk, or more generally, how to relate important portfolios to the associated factor structures.
Our idealized limit model of asset pricing is based on a
hyperfinite continuum of assets (17, 18). In this setting, we can appeal to a hyperfinite analogue of the Karhunen-Loeve
expansion of continuous time stochastic processes (18–22), and derive factors endogenously from the process of asset
returns. These factors are used to formalize systematic risks and to construct a ‘‘market’’ portfolio for a further specification of essential risk. The valuation formula then shows that the usual claim, based on the APT, that the market only
rewards the holding of systematic risks, is simply not sharp
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