Portfolio Theory: A Step Toward Its Practical Application

The modern theory of portfolio analysis dates from the pioneering work of Markowitz.' Of paramount importance to the practical application of this theory is the ability to assess accurately the future performance of portfolios of risky assets. With accurate assessments, an investor can optimize his choice of a portfolio on the two dimensions of risk and expected return. In spite of the need of obtaining accurate assessments of future performance in applying portfolio analysis to a practical situation, there has been virtually no statistical analysis of the empirical properties of explicit methods of assessing future performance. This paper proposes two different methods of assessing future performance or, more technically, of assessing predictive or subjective distributions of future returns. These two different methods will be compared by evaluating the accuracy of the predictive distributions derived under these two alternatives, using as a criterion the degree of approximation between these predictive distributions and the underlying distributions generating the future returns. After reviewing some of the more important theoretical and empirical properties of symmetric, stable distributions in section II, section III develops the first method which assesses predictive distributions of the returns for all the assets individually and then aggregates these distributions into predictive distributions for portfolios. This procedure is implicit in Sharpe's well-known diagonal algorithm for calculating the efficient set.2 This paper shows that this first method of aggregation requires certain empirical assumptions about the predictive distributions for the individual assets. After sections IV and V describe the sample and the statistical design, section VI uses this first method to assess predictive distributions for a sample of portfolios of common stocks using historical data and compares these predictive distributions to the generating distributions of the future returns. The closeness of the approximation of these predictive distributions to the future generating distributions varies considerably with the particular empirical assumptions used in the process of aggregation. The second method to be presented in section VII assesses the predictive distributions of future returns for portfolios directly rather than first assessing the distributions for the individual assets. This second or alternative method still requires some of the assumptions needed by the first method. Neverthe* The author is deeply indebted to Professors Eugene Fama, Lawrence Fisher, Merton Miller, and Harry Roberts for their very helpful advice. In addition, the comments of Professors Robert Keeley, Benjamin King, Michael Jensen, and Myron Scholes were appreciated.