Worst-Case Expected Shortfall with Univariate and Bivariate Marginals

Worst-case bounds on the expected shortfall risk given only limited information on the distribution of the random variables has been studied extensively in the literature. In this paper, we develop a new worst-case bound on the expected shortfall when the univariate marginals are known exactly and additional expert information is available in terms of bivariate marginals. Such expert information allows for one to choose from among the many possible parametric families of bivariate copulas. By considering a neighborhood of distance ρ around the bivariate marginals with the Kullback-Leibler divergence measure, we model the trade-off between conservatism in the worst-case risk measure and confidence in the expert information. Our bound is developed when the only information available on the bivariate marginals forms a tree structure in which case it is efficiently computable using convex optimization. For consistent marginals, ∗Engineering Systems and Design, Singapore University of Technology and Design, 8 Somapah Road, Singapore 487372. Email: [email protected] †Engineering Systems and Design, Singapore University of Technology and Design, 8 Somapah Road, Singapore 487372. Email: [email protected] ‡Engineering Systems and Design, Singapore University of Technology and Design, 8 Somapah Road, Singapore 487372. Email: karthik [email protected] §This research was partly funded by the SUTD-MIT International Design Center grant number IDG31300105 on ‘Optimization for Complex Discrete Choice’ and the MOE Tier 2 grant number MOE2013-T2-2-168 on ‘Distributional Robust Optimization for Consumer Choice in Transportation Systems’.