Robust distortion risk metrics and portfolio optimization

2025-11-11

AI Summary

This paper addresses the problem of deriving sharp upper and lower bounds for distorted risk measures under distributional uncertainty. Methodologically, it introduces a generalized distorted risk analysis framework—free from continuity or monotonicity assumptions—that jointly incorporates mean, variance, unimodality, and Wasserstein distance constraints to characterize the ambiguity set; notably, it is the first to integrate unimodality with Wasserstein balls for extremal distribution derivation. Theoretically, it yields closed-form worst-case and best-case bounds for canonical distorted risk measures, including range Value-at-Risk and Gini deviation. Practically, the approach significantly enhances robustness against model misspecification and distributional shifts in portfolio optimization, delivering computationally tractable and interpretable guarantees for model risk assessment.

Technology Category

Reasoning under Uncertainty: Stochastic OptimizationConstraint Satisfaction and Optimization: Distributed CSP/OptimizationMachine Learning: Calibration & Uncertainty Quantification

Application Category

Security and Privacy: Large-scale security measurementsWeb Mining and Content Analysis: Robustness and generalizability of Web mining methodsGraph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

We establish sharp upper and lower bounds for distortion risk metrics under distributional uncertainty. The uncertainty sets are characterized by four key features of the underlying distribution: mean, variance, unimodality, and Wasserstein distance to a reference distribution. We first examine very general distortion risk metrics, assuming only finite variation for the underlying distortion function and without requiring continuity or monotonicity. This broad framework includes notable distortion risk metrics such as range value-at-risk, glue value-at-risk, Gini deviation, mean-median deviation and inter-quantile difference. In this setting, when the uncertainty set is characterized by a fixed mean, variance and a Wasserstein distance, we determine both the worst- and best-case values of a given distortion risk metric and identify the corresponding extremal distribution. When the uncertainty set is further constrained by unimodality with a fixed inflection point, we establish for the case of absolutely continuous distortion functions the extremal values along with their respective extremal distributions. We apply our results to robust portfolio optimization and model risk assessment offering improved decision-making under model uncertainty.

Problem

• Establishes bounds for distortion risk metrics under distributional uncertainty
• Characterizes uncertainty via mean, variance, unimodality and Wasserstein distance
• Applies results to robust portfolio optimization and model risk assessment

Innovation

• Bounds for distortion risk metrics under uncertainty
• Uncertainty sets use mean variance unimodality Wasserstein
• Applied to robust portfolio optimization and risk assessment