ROBUST ASSET ALLOCATION WITH BENCHMARKED OBJECTIVES

In this paper, we introduce a new approach for finding robust portfolios when there is model uncertainty. It differs from the usual worst case approach in that a (dynamic) portfolio is evaluated not only by its performance when there is an adversarial opponent ("nature"), but also by its performance relative to a stochastic benchmark. The benchmark corresponds to the wealth of a fictitious benchmark investor who invests optimally given knowledge of the model chosen by nature, so in this regard, our objective has the flavor of min-max regret. This relative performance approach has several important properties: (i) optimal portfolios seek to perform well over the entire range of models and not just the worst case, and hence are less pessimistic than those obtained from the usual worst case approach, (ii) the dynamic problem reduces to a convex static optimization problem under reasonable choices of the benchmark portfolio for important classes of models including ambiguous jump-diffusions, and (iii) this static problem is dual to a Bayesian version of a single period asset allocation problem where the prior on the unknown parameters (for the dual problem) correspond to the Lagrange multipliers in this duality relationship. This dual static problem can be interpreted as a less pessimistic alternative to the single period worst case Markowitz problem. More generally, this duality suggests that learning and robustness are closely related when benchmarked objectives are used.