ON THE UTILITY THEORETIC FOUNDATIONS OF MEAN‐VARIANCE ANALYSIS

Abstract

THE CHOICE MODEL in which alternatives are ordered in terms of the mean and variance of their return has been utilized in a variety of fields with the best developed, both theoretically and empirically, being that of portfolio selection. Since the pioneering works of Markowitz [22] and Tobin [32], a series of results has been obtained regarding the utility theoretic foundations of this model. While these results are widely known, they continue to be the source of discussion.' For example, Markowitz demonstrated that if the ordering of alternatives is to satisfy the von Neumann-Morgenstern (NM) [35] axioms of rational behavior, only a quadratic (NM) utility function is consistent with an ordinal expected utility function that depends solely on the mean and variance of the return. Consequently, even if the return for each alternative has a normal distribution, the mean-variance framework cannot be used to rank alternatives consistently with the NM axioms unless a quadratic NM utility function is specified. The implications of this restriction are disturbing not only because of the undesirable properties of a quadratic utility function but also because, for example, the indifference curves in the mean-standard deviation plane are concentric circles with the center on the mean axis. Furthermore, a quadratic utility function leads to a rather disquieting result in portfolio theory, since it implies that in equilibrium each investor holds an equal percentage of every security (Mossin [23, p. 69]). Also, Ekern and Wilson [11, p. 179] have shown that in a mean-variance portfolio model all shareholders prefer that a firm characterized by stochastic constant returns operate so that its market value is zero. Another disquieting result is that of Borch [7] who demonstrates that if preferences satisfy a monotonicity condition, an indifference curve in the ( ,a)-plane consists of a single point. A number of authors, including Samuelson [27] [29] and Tsiang [33], have argued that mean-variance analysis may be viewed as an approximation to a more general choice model. The appropriateness of such an approximation has been considered by Borch [8], Bierwag [6], Levy [20], and Tsiang [34], for example, and will not be considered further here. The purpose of this paper is instead to consider the foundations of mean-variance analysis and the consistency of the ordering of alternatives according to means and variances with the NM axioms. More specifically, given alternatives a, and a2 and their corresponding random returns X, and X2 with distribution functions F,(x) and F2(x), respectively, preferences satisfying the NM axioms imply the existence of a measurable, continuous utility function