Portfolio Selection in a Log-Stable Market

Abstract

The Stable (or Pareto-Levy) distribution has been of considerable interest in describing the behavior of security prices ever since the important work by Mandelbrot [6], [7] and Fama [1]. The aforementioned contributions focused on the empirical hypothesis that security price data are better fitted by theoretical distributions with infinite variance rather than finite variance. Specifically, in the case of Stable distributions, the “characteristic exponent” is less than two, and the data are not adequately fitted by a normal distribution. Remarkably, however, although almost the entire body of literature addressing empirical questions with respect to the distribution of security prices investigages the behavior of the (natural) logarithm of security price relatives, to this author's knowledge no paper exists which analyzes the portfolio choice implications of the assumption that the logarithm of the asset returns has a symmetric Stable distribution with infinite variance. Thus, in Fama [2], Samuelson [10], and Ziemba [12], where the problem of selecting an optimal portfolio in Stable markets is the object of concern, one finds that it is assumed that the price-relatives (returns) have a Stable distribution; this rather than the logarithm of the price relatives. And it should be noted that none of these authors suggests that the untransformed price-relatives are better-fitted by a symmetric Stable distribution as compared with the logarithm of the price-relatives.