On the Number of Factors in the Arbitrage Pricing Model

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. ARBITRAGE PRICING THEORY (APT) as originally developed by Ross [22, 23] is built on the twin assumptions of no arbitrage opportunities in the capital market and a linear relationship between actual returns and K common factors. The principal hypothesis of the model is that expected returns should be linearly related to the weights of the common factors in the assumed linear process. In every test of the model, the focus has been to use factor analysis to extract K factors from sample covariance matrices and then to test the hypothesis by regressing returns or average returns against the factor loadings of the common factors. Most previous tests have either assumed that the number of factors is equal to some predetermined number or have estimated K simultaneously with estimation of the factor loadings. Thus, existing tests of the APT are joint tests of the APT pricing result and an assumed or estimated number of factors. The question of how many factors are important has not been addressed separately from the pricing tests because the APT pricing result is the significant prediction of a theory which has little to say about the value of K. Nevertheless, crosssection linear models have been a central part of financial economics, and it is clearly important to know at least the approximate number of factors which will be significant in such regressions, particularly given the computational difficulties of large-scale factor analysis. Furthermore, Dhrymes, Friend, and Gultekin [6] report that the number of factors needed for standard chi-squared tests increases * University at Buffalo, SUNY. I am in debt to Donald Bosshardt, Gary Chamberlain, Thomas Ho, Richard Roll, Lemma Senbet, Mark Zmijewski, and the participants in finance workshops at the Universities of Michigan, Wisconsin, and Buffalo for their extensive comments. Nevertheless, I am solely responsible for what remains. This paper would not have been started without the unselfish cooperation of Jay Leavitt, Director of Academic Computing, University at Buffalo, SUNY, who designed the eigenvalue algorithm which I used. I am also grateful for partial financial support and lavish computer funds from the School of Management, and for the expert programming assistance provided by Ravi Shukla.