Robust Empirical Optimization is Almost the Same As Mean-Variance Optimization

We formulate a distributionally robust optimization problem where the empirical distribution plays the role of the nominal model, the decision maker optimizes against a worst-case alternative, and the deviation of the alternative distribution from the nominal is controlled by a $\phi$-divergence penalty in the objective, and show that a large class of these problems are essentially equivalent to an in-sample mean-variance problem. Intuitively, controlling the variance reduces the sensitivity of the expected reward to perturbations in the tail of the in-sample reward distribution, which reduces the sensitivity of its out-of-sample performance to errors in the nominal model. We also quantify the relationship between the ``level of robustness" in the robust optimization model and the mean and variance of the objective and we show that while a ``small amount of robustness" reduces the expected reward, the reduction in the variance is an order of magnitude larger. As a practical application of our theory, we introduce the notion of a robust mean-variance frontier and show how it can be used to balance out-of-sample performance and robustness when calibrating the robust model. To illustrate the usefulness of such frontiers we consider a real world portfolio optimization application