On the Number of Factors in the Arbitrage Pricing Model

Abstract

ABSTRACT Recent theory has demonstrated that the Arbitrage Pricing Model with K factors critically depends on whether K eigenvalues dominate the covariance matrix of returns as the number of securities grows large. The purpose of this paper is to test whether sample covariance matrices can be characterized as having K large eigenvalues. Using all available data on the 1983 CRSP tapes, we compute sample covariance matrices of returns in sequentially larger portfolios of securities. Analyzing their eigenvalues, we find evidence that one eigenvalue dominates the covariance matrix indicating that a one‐factor model may describe security pricing. We also find that, for values of K larger than one, there is no obvious way to choose the number of factors. Nevertheless, we find that while only the first eigenvalue dominates the matrix, the first five eigenvalues are growing more distinct.

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