Estimation Risk and Simple Rules for Optimal Portfolio Selection

Abstract

Elton, Gruber and Padberg (EGP) [6, 7] have recently simplified the process of constructing optimal portfolios by developing simple criteria for optimal portfolio selection which do not involve use of mathematical programming. Their simple decision rules permit one to determine easily which securities to include in an optimal portfolio and how much to invest in each. However, in practical applications of theoretical models, sample estimators are usually treated as if they were true values of unknown parameters. As a result, the effect of the standard errors of sample estimators on decision rules are completely ignored. Bawa, Brown and Klein [1] have shown that what is optimal in the absence of estimation risk is not necessarily optimal or even approximately optimal in the presence of estimation risk. Moreover, Brown [4] examined optimal portfolio choice under uncertainty for various portfolio selection procedures-the diffuse Bayes rule, the Markowitz Certainty Equivalent (CE) rule, the aggregation CE rule, and the equal weight rule,1 and found that the diffuse Bayes rule uniformly dominates the Markowitz CE rule in repeated samples for the quadratic utility case. As the sample size increases, the Bayes rule becomes superior to the aggregation CE and the equal weight rules. In addition, the result holds even where the probability distribution of returns is seriously misspecified. Thus, Brown's [4] study has clearly indicated that, without taking estimation risk into account, portfolio selection rules other than the Bayes rule can lead investors to select suboptimal portfolios. This paper shows by using the single index model for the return generating process that the simple decision rules for optimal portfolio selection derived by Elton, Gruber and Padberg [7] are not identical under the Bayesian and the traditional methods of analysis.2 Moreover, in the case where short sales are not