Worst-Case Expected Shortfall with Univariate and Bivariate Marginals

Abstract

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 [Formula: see text]-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.

Tree Bounds for Sums of Bernoulli Random Variables: A Linear Optimization Approach

“…Thus, the bounds in the paper can provide robust estimates on probabilities when the conditional independence assumption on the underlying structured graphical models is violated. A similar problem was recently studied by Dhara et al (2020), in which tight tree bounds were proposed for the expectation of a sum of discrete random variables beyond a threshold using linear optimization. In contrast to their work, our focus in this paper is on probability bounds, which requires the use of different proof techniques.…”

On the Impossibility of Statistically Improving Empirical Optimization: A Second-Order Stochastic Dominance Perspective

“…In the DRO literature, the choice of U λ can be categorized roughly into two groups. The first group is based on partial distributional information, such as moment and support (Ghaoui et al 2003, Delage and Ye 2010, Goh and Sim 2010, Wiesemann et al 2014, Hanasusanto et al 2015, shape (Popescu 2005, Van Parys et al 2016, Li et al 2017, Lam and Mottet 2017, Chen et al 2021) and marginal distribution , Doan et al 2015, Dhara et al 2021. This approach has proven useful in robustifying decisions when facing limited distributional information, or when data is scarce, e.g., in the extremal region.…”

Generalization Bounds with Minimal Dependency on Hypothesis Class via Distributionally Robust Optimization

“…The construction can be roughly categorized into two approaches: 1) neighborhood ball using statistical distance, which include most commonly φ-divergence (Ben-Tal et al 2013, Bayraksan and Love 2015, Jiang and Guan 2016, Lam 2016 and Wasserstein distance (Esfahani and Kuhn 2018, Blanchet and Murthy 2019, Gao and Kleywegt 2016, Chen and Paschalidis 2018. 2) partial distributional information including moment (Ghaoui et al 2003, Delage and Ye 2010, Goh and Sim 2010, Wiesemann et al 2014, Hanasusanto et al 2015, distributional shape (Popescu 2005, Van Parys et al 2016, Li et al 2017, Chen et al 2020) and marginal , Doan et al 2015, Dhara et al 2021 constraints.…”