Deviation Measures in Risk Analysis and Optimization

Abstract

General deviation measures, which include standard deviation as a special case but need not be symmetric with respect to ups and downs, are defined and shown to correspond to risk measures in the sense of Artzner, Delbaen, Eber and Heath when those are applied to the dierence between a random variable and its expectation, instead of to the random variable itself. A property called expectation-boundedness of the risk measure is uncovered as essential for this correspondence. It is shown to be satisfied by conditional value-at-risk and by worst-case risk, as well as various mixtures, although not by ordinary value-at-risk. Interpretations are developed in which inequalities that are “acceptably sure”, relative to a designated acceptance set, replace inequalities that are “almost sure” in the usual sense being violated only with probability zero. Acceptably sure inequalities fix the standard for a particular choice of a deviation measure. This is explored in examples that rely on duality with an associated risk envelope, comprised of alternative probability densities. The role of deviation measures and risk measures in optimization is analyzed, and the possible influence of “acceptably free lunches” is thereby brought out. Optimality conditions based on concepts of convex analysis, but relying on the special features of risk envelopes, are derived in support of a variety of potential applications, such as portfolio optimization and variants of linear regression in statistics.