[We present a definition of factor structure that is less restrictive than the one typically used in arbitrage pricing models. Our factor structure restrictions build on the following intuitive distinctions between factor variance and idiosyncratic variance: (i) A well-diversified portfolio contains only factor variance. (ii) If a portfolio is uncorrelated with the well-diversified portfolios, then it contains only idiosyncratic variance; so if a sequence of such portfolios becomes well-diversified, the limiting variance should be zero. Our factor structure restrictions imply Ross' [5] arbitrage pricing formula. We obtain upper and lower bounds on the approximation error in that formula; these bounds may be useful in empirical work. They imply that arbitrage pricing is exact if and only if there is a risky, well-diversified portfolio on the mean-variance frontier. If all mean-variance efficient portfolios are well-diversified, then the well-diversified portfolios provide mutual fund separation. Our factor structure restrictions are satisfied (with K factors) if and only if the covariance matrix of asset returns has only K unbounded eigenvalues as the number of assets increases.]

We examine the implications of arbitrage in a market with many assets. The absence of arbitrage opportunities implies that the linear functionals that give the mean and cost of a portfolio are continuous; hence there exist unique portfolios that represent these functionals. These portfolios span the mean-variance efficient set. We resolve the question of when a market with many assets permits so much diversification that risk-free investment opportunities are available. Ross 112, 141 showed that if there is a factor structure, then the mean returns are approximately linear functions of factor loadings. We define an approximate factor structure and show that this weaker restriction is sufficient for Ross' result. If the covariance matrix of the asset returns has only K unbounded eigenvalues, then there is an approximate factor structure and it is unique. The corresponding K eigenvectors converge and play the role of factor loadings. Hence only a principal component analysis is needed in empirical work.