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

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.