Portfolio Analysis with General Deviation Measures

Abstract

Generalized measures of deviation, as substitutes for standard deviation, are considered in a framework like that of classical portfolio theory for coping with the uncertainty inherent in achieving rates of return beyond the risk-free rate. Such measures, associated for example with conditional value-at-risk and its variants, can reflect the different attitudes of different classes of investors. They lead nonetheless to generalized one-fund theorems as well as to covariance relations which resemble those commonly used in capital asset pricing models (CAPM), but have wider interpretations. A more customized version of portfolio optimization is the aim, rather than the idea that a single might arise from market equilibrium and serve the interests of all investors. The results cover discrete distributions along with continuous distributions, and therefore are applicable in particular to financial models involving finitely many future states, whether introduced directly or for purposes of numerical approximation. Through techniques of convex analysis, they deal rigorously with a number of features that have not been given much attention in this subject, such as solution nonuniqueness, or nonexistence, and a potential lack of differentiability of the deviation expression with respect to the portfolio weights. Moreover they address in detail the previously neglected phenomenon that, if the risk-free rate lies above a certain threshold, a master fund of the usual type will fail to exist and need to be replaced by one of an alternative type, representing a net short position instead of a net long position in the risky instruments.