Stochastic Portfolio Theory and Stock Market Equilibrium

Abstract

HARRY MARKOWITZ [1952, 1956, 1959] developed a theory of portfolio selection based on the optimization of a quadratic function subject to linear constraints. His work led to the development of a single period equilibrium model, the SharpeLintner capital asset pricing model (CAPM) (see Sharpe [1964], Lintner [1965a, 1965b]). R. C. Merton [1973] extended the CAPM to a continuous time model using lognormal diffusion processes to represent stock price series and showed that the original conclusions continued to hold virtually without change. In this continuous time model it became possible to observe the dynamic interaction between investors' behavior and the behavior of the stocks. In fact, Rosenberg and Ohlson [1976] showed that his interaction led to internal inconsistencies in the continuous time CAPM. In this paper we analyze long term portfolio performance compatible with equilibrium constraints on global portfolio structure. We present a portfolio theory which is based on Markowitz's theory, but which emphasizes the long term performance of the portfolios in continuous time. The concept of is introduced, a quantity which measures the relative performance of a portfolio compared to that of its component stocks. The equilibrium model we present does not consider questions dealing with optimal strategies for investors, and specifically avoids all normative issues. Rather, it provides constraints on global portfolio structure based on the principle that excess growth is conserved, that the total excess growth in the market at any instant is zero. These constraints generate a distribution of portfolios in the market similar to the energy distribution in thermodynamic equilibrium. This approach provides an alternative to classical supply-demand equilibrium. We adopt some standard conventions here, but most of the results in the paper will carry over to more general settings. The assumptions we make are: (1) Each stock price follows a lognormal diffusion process with constant drift and variance parameters. The covariance parameters between stocks are constant. Stock portfolios can be represented as Ito integrals in the various stock price processes. (2) There are no transaction costs, taxes, or problems with the indivisibility of assets. (3) The number of shares of each corporation remains constant.