Relative performance measures of portfolio robustness

In this paper, we propose and analyze an alternative measure of “robust performance”. This alternative measure differs from the typical “worst case expected utility” and “worst case meanvariance” formulations in that a (dynamic) portfolio is evaluated not only on the basis of its performance when there is an adversarial opponent (“nature”), but also by its performance relative to a fully informed “benchmark investor” who behaves optimally given complete knowledge of the otherwise ambiguous model. This “relative performance” approach has several important properties: (i) decisions arising from this approach are less pessimistic than the portfolios obtained from the typical “worst case expected utility” and “worst case mean-variance” formulations, and (ii) the problem is computationally tractable for important classes of models such as ambiguous jump-diffusion processes where it reduces to a convex static optimization problem under reasonable choices of the benchmark portfolio. This static problem is interesting in its own right: it can be interpreted as a less pessimistic alternative to the single period “worst case mean-variance” problem. Key words– ambiguity, model uncertainty, relative performance measure, competitive ratio, min-max regret, asset allocation, robust control, convex duality.