The Geometric Mean and Stochastic Dominance

Abstract

TWO APPROACHES TO THE choice among risky alternatives that have been developed independently over the past twenty years are the geometric mean criterion and the stochastic dominance decision models. Both can be justified by the expected utility hypothesis, with the geometric mean criterion following as a result of the assumption that the decision-maker has a logarithmic utility function, and the stochastic dominance models requiring the less restrictive assumption of the signs of the first few derivatives of the decision-maker's utility function. In this paper the mathematical relationship between the geometric mean and the integrals of the probability density functions used in stochastic dominance test is derived for a very general class of distributions, and the principal result of the mathematical comparison is that a geometric mean ranking is a necessary condition for a stochastic dominance ranking. Stochastic dominance showed great promise in the early years of its development as a method for constructing a theory of security prices because of the limited set of assumptions necessary regarding investor preferences. The evolution of a stochastic dominance-based theory of security prices has been stalled because of the frequency with which a no-ranking result occurs, and the lack of efficient algorithms to construct optimum portfolios. Algorithms to construct optimum geometric mean portfolios have been developed and, as a result of the derivations of this paper, those algorithms can be used in the exploration of the optimum stochastic dominance portfolio. Latane [9] justified the geometric mean criterion on the basis of maximization of terminal wealth in a many-period reinvestment process, the optimal growth approach, and also pointed out that the criterion maximizes expected utility if the Bernoulli [1] logarithmic utility function is assumed. The growth optimal portfolio has been studied extensively since then, with an example being Markowitz's [16] derivation of the asymptotic characteristics of the model when smaller numbers of periods are used. The concept of stochastic dominance has been used for many years, but modern development of the approach is usually credited to Quirk and Saposnik [17], Hadar and Russell [4], Hanoch and Levy [7], and Whitmore [19]. In the case of continuous distributions of returns stochastic dominance ranking models use repeated integrals of the probability density function f(x) to order alternatives. In this paper those repeated integrals are