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

Abstract

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.

Linear Conditional Expectation, Return Distributions, and Capital Asset Pricing Theories

“…Recently, Owen and Rabinovitch (1983) show that the class of elliptical distributions extends the domain of the separation results obtained by Ross (1978) and others. They also extend the results of the CAPM to elliptical distributions that are not necessarily normal or stable.…”

Handbook of Computational and Numerical Methods in Finance

“…The joint sub-Gaussian a-stable family is an elliptical family. Hence, as argued by Owen and Rabinovitch (1983), we can extend the classic mean variance analysis to a mean dispersion one. The resulting efficient frontier is formally the same as Markowitz-Tobin's mean-variance one, but, instead of the variance as a risk parameter, we have to consider the scale parameter of the stable distributions.…”

Financial Applications of the Mahalanobis Distance

“…For the definition and properties of elliptical distributions see Cambanis, Jacquier, Kritzman and Lowry (1981), Mitchell and Krzanowski (1985), and for their connection to portfolio theory see Owen and Rabinovitch (1983 . This is necessary as sample mean and variance react quite strongly to outliers, which in turn has a significant impact on portfolios estimated using those parameters (Bodnar, 2009 …”