The paper provides an intertemporal version of the capital asset pricing model (CAPM) of Sharpe and Lintner. Although we allow for general changes in the investment opportunity set and for general risk-averse preferences, there are conditions under which two mutual funds are sufficient to generate all optimal portfolios. In particular, we require that the Riesz claim, which represents the date O pricing functional for the marketed claims, should lie in a scalar Brownian information set. Then we obtain an instantaneous counterpart to the CAPM pricing formula: a linear relationship between the conditional mean returns on the securities and conditional covariances with the return on the market portfolio. Our use of option pricing techniques requires continuous trading but does not require continuous consumption. In addition, we consider a large economy with a factor structure, as in Ross' arbitrage pricing theory. The dividends are assumed to have an approximate factor structure, with the factor components lying in the information set generated by an N-dimensional Brownian motion, and with the covariance matrices of the idiosyncratic components having uniformly bounded eigenvalues. We obtain an N-factor version of the pricing formula and relate the factors to the gains processes {price change plus accumulated dividends) for well-diversified portfolios. An approximate factor structure for dividends implies an approximate factor structure for the gains processes of the securities. Furthermore, the assumption_that per.capita supply is well diversified can motivate our condition that the Riesz claim lies in an N-dimensional Brownian information set.

Research Methods/Statistical Methods

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.