Robust trade-off portfolio selection

Robust portfolio selection explicitly incorporates a model of parameter uncertainty in the problem formulation, and optimizes for the worst-case scenario. We consider robust mean–variance portfolio selection involving a trade-off between the worst-case utility and the worst-case regret, or the largest difference between the best utility achievable under the model and that achieved by a given portfolio. While optimizing for the worst-case utility may yield an overly pessimistic portfolio, optimizing for the worst-case regret may result in a complete loss of robustness; we theoretically demonstrate this point. Robust trade-off portfolio compromises these two extremes, enabling more informative selections. We show that, under a widely used ellipsoidal uncertainty model, the entire optimal trade-off curve can be found via solving a series of semidefinite programs (SDPs). We then extend the model to handle the union of finitely many ellipsoids, and show that trade-off analysis under this quite general uncertainty model also reduces to a series of SDPs. For more general uncertainties, we propose an iterative algorithm based on the cutting-set method. Under the finite-union-of-ellipsoids model, this algorithm offers an alternative to the SDP in exploring the optimal trade-off curve. We illustrate the promises of the trade-off portfolios by using historical stock returns data.