A characterization of the distributions that imply mean—Variance utility functions☆

An Introduction to Probability Theory and Its Applications. Volume II By William Feller. Pp. xviii, 626. 90s. 1966. (Wiley)

It is shown that the class of elliptical distributions extend the Tobin [14] separation theorem, Bawa's [2] rules of ordering uncertain prospects, Ross's [12] mutual fund separation theorems, and the results of the CAPM to non-normal distributions, which are not necessarily stable. Further, the mean-covariance matrix framework is generalized to a mean-characteristic matrix framework in which the characteristic matrix is the basis for a spread or risk measure, and a generalized equilibrium pricing equation is arrived at. The implications to empirical testing of the CAPM and modeling the empirical distribution of speculative prices are discussed. THE PURPOSE OF THIS paper is to describe the class of elliptical distributions and delineate their relevance to portfolio theory and its empirical applications. The class of elliptical distributions contains the multivariate normal (multinormal, henceforth) distribution as a special case; as well as many non-normal multivariate distributions including the multivariate Cauchy, the multivariate exponential, a multivariate elliptical analog of Student's t-distribution (the multivariate t, henceforth1), and non-normal variance mixtures of multinormal distributions. We show that some of the general characteristics of all members in the class of elliptical distributions make them admissible candidates for generalizing the theory of portfolio choice and equilibrium in capital asset markets, as well as attractive alternatives for use in empirical studies. More specifically, elliptical distributions are characterized by two parameters and extend the Tobin's [14] separation theorem, Bawa's [2] rules of ordering uncertain prospects, Ross's [13] mutual fund separation theorems, and the results of the CAPM to non-normal distributions, which are not necessarily stable. The definitions and properties of elliptical distributions are in Section I. The main results are in Section II and the importance of elliptical distributions in empirical works is discussed in Section III.

On the Class of Elliptical Distributions and their Applications to the Theory of Portfolio Choice

Mutual fund separation in financial theory—The separating distributions

Measuring Mutual Fund Performance Using Lower Partial Moment

Consider the following assertion: If X1, ..., X, are random variables with finite second moments and if all non-trivial linear combinations a1X, +... + a,X, have the same distribution except for location and scale, then that distribution must be normal. True or false? The assertion would be true if any pair of the variables were assumed to be stochastically independent, but it is false in general, even for variables assumed to be uncorrelated. In Section I, we demonstrate that the assertion is false via counter-examples. A recent commentary in this Review by Borch [1], Feldstein [4], and Tobin [13] indicates that the truth of the above assertion has gone unquestioned by some eminent economic theorists. Nevertheless, the generality of Tobin's mean-variance approach to risk analysis is not much enhanced by our revelation; the numerous theoretical criticisms of the mean-variance approach (e.g. Borch [1] and Feldstein [4]) still obtain. As a practical matter, our counter-examples merely extend the class of approximating distributions available to portfolio analysts. A more intriguing ramification of our counter-examples is the possibility of a viable alternative to the stable Paretian model of stochastic price fluctuation [7], [8], [2], [3], [10]. The stable Paretian model follows from the assumption of independent stable increments in the log-price process. In Section II, we construct an alternative model with uncorrelated (but dependent) stable increments.

R. A. Agnew

Papers

Counter-examples to an Assertion concerning the Normal Distribution and a New Stochastic Price Fluctuation Model