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Asset Pricing in Multiperiod Securities Markets

Gary Chamberlain

Abstract

The intertemporal models developed in this paper have been stimulated by the capital asset pricing model (CAPM) of Sharpe and Lintner, and the role of factor structure in Ross' arbitrage pricing theory (APT). Suppose that some (one-dimensional) stochastic process S provides a sufficient statistic for aggregate consumption. Then a (heuristic) dynamic programming argument shows that S, market wealth, and the wealth derivative of the value function (for any agent) are all locally perfectly correlated. It follows from Merton's work that there is a linear relationship between the local mean return on a security and the local covariance of that return with the return on the market portfolio. The formal development uses martingale representation and martingale projection to obtain an intertemporal CAPM; the history of a scalar Brownian motion plays the role of a sufficient statistic. We motivate the assumption of a sufficient statistic for aggregate consumption by considering a countable set of securities whose payoffs have an approximate factor structure, where the factor components are in the information set generated by an N-dimensional Brownian motion and the idiosyncratic components are weakly correlated. The approximate factor structure on the security payoffs implies that the rates of return have (locally) an approximate factor structure. The role of the market portfolio can now be played by a set of N well-diversified portfolios.