Hidden Dependence and Aggregate Tail Risk

2026-06-29

AI Summary

This study addresses the challenge of accurately assessing tail risk under arbitrary non-decreasing aggregation functions when marginal distributions are fixed but the dependence structure is uncertain. To this end, it introduces a novel “hidden dependence” paradigm, constructing worst-case loss vectors driven by risk concentrations and joint tail events within a distributionally robust optimization framework. The analysis reveals how perturbations to a reference Gaussian dependence structure can significantly amplify tail risk. Theoretical results demonstrate that even infinitesimal deviations from the nominal dependence may lead to substantially higher capital requirements. Moreover, the work establishes worst-case risk bounds that coincide with those in the unconstrained setting and quantifies the extent of tail risk underestimation due to dependence uncertainty in credit risk applications.

Abstract

We study risk aggregation problems for arbitrary non-decreasing aggregation functions and tail risk measures under dependence uncertainty in a distributionally robust setting. To this end, we introduce the notion of hidden dependence for random vectors, which is built on the concepts of risk concentration and common tail events developed in Wang and Zitikis (2020). We show that, starting from a tail event $A$ of the aggregate loss for an arbitrary random vector $Y$, one can construct a random vector with hidden dependence that dominates $Y$ on the tail event $A$. We then focus on the case in which model uncertainty is described by small perturbations of the distribution of a random vector with respect to a suitable probability distance without changing the marginals. We show that these perturbations of the reference distribution are compatible with hidden dependence and thus lead to the same worst-case risk bounds as in the unconstrained case for arbitrary $γ$-tail risk measures with a suitable level $γ$. Finally, we apply our results in a credit risk context and quantify the potential underestimation of portfolio risk arising from uncertainty in the dependence structure. In particular, we show that even small deviations from a reference Gaussian dependence model can, in principle, justify dramatic increases in capital requirements.

Problem

• hidden dependence
• aggregate tail risk
• dependence uncertainty
• risk aggregation
• distributionally robust

Innovation

• hidden dependence
• tail risk measures
• risk aggregation
• distributional robustness
• dependence uncertainty