Worst-Case Expected Shortfall with Univariate and Bivariate Marginals

Computing and minimizing the worst-case bound on the expected shortfall risk of a portfolio given partial information on the distribution of the asset returns is an important problem in risk management. One such bound that been proposed is for the worst-case distribution that is “close” to a reference distribution where closeness in distance among distributions is measured using φ -divergence. In this paper, we advocate the use of such ambiguity sets with a tree structure on the univariate and bivariate marginal distributions. Such an approach has attractive modeling and computational properties. From a modeling perspective, this provides flexibility for risk management applications where there are many more choices for bivariate copulas in comparison with multivariate copulas. Bivariate copulas form the basis of the nested tree structure that is found in vine copulas. Because estimating a vine copula is fairly challenging, our approach provides robust bounds that are valid for the tree structure that is obtained by truncating the vine copula at the top level. The model also provides flexibility in tackling instances when the lower dimensional marginal information is inconsistent that might arise when multiple experts provide information. From a computational perspective, under the assumption of a tree structure on the bivariate marginals, we show that the worst-case expected shortfall is computable in polynomial time in the input size when the distributions are discrete. The corresponding distributionally robust portfolio optimization problem is also solvable in polynomial time. In contrast, under the assumption of independence, the expected shortfall is shown to be #P-hard to compute for discrete distributions. We provide numerical examples with simulated and real data to illustrate the quality of the worst-case bounds in risk management and portfolio optimization and compare it with alternate probabilistic models such as vine copulas and Markov tree distributions.