Robust Portfolio Choice with Learning in the Framework of Regret: Single-Period Case

In this paper, we formulate a single-period portfolio choice problem with parameter uncertainty in the framework of relative regret. Relative regret evaluates a portfolio by comparing its return to a family of benchmarks, where the benchmarks are the wealths of fictitious investors who invest optimally given knowledge of the model parameters, and is a natural objective when there is concern about parameter uncertainty or model ambiguity. The optimal relative regret portfolio is the one that performs well in relation to all the benchmarks over the family of possible parameter values. We analyze this problem using convex duality and show that it is equivalent to a Bayesian problem, where the Lagrange multipliers play the role of the prior distribution, and the learning model involves Bayesian updating of these Lagrange multipliers/prior. This Bayesian problem is unusual in that the prior distribution is endogenously chosen by solving the dual optimization problem for the Lagrange multipliers, and the objective function involves the family of benchmarks from the relative regret problem. These results show that regret is a natural means by which robust decision making and learning can be combined. This paper was accepted by Dimitris Bertsimas, optimization.