EVT-Based Rate-Preserving Distributional Robustness for Tail Risk Functionals

TL;DR

This paper develops a new EVT-based approach to robust tail risk estimation that preserves the nominal asymptotic scaling of risk measures like CVaR, effectively mitigating model misspecification effects.

Contribution

It introduces a tail-calibrated ambiguity set that maintains the original tail risk scaling while providing robustness against misspecification, supported by theoretical guarantees and empirical validation.

Findings

Robustification can cause excess risk inflation in tail risk measures.

The proposed tail-calibrated ambiguity set preserves the asymptotic tail risk scaling.

Experiments show the method avoids severe inflation seen with standard ambiguity sets.

Abstract

Risk measures such as Conditional Value-at-Risk (CVaR) focus on extreme losses, where scarce tail data makes model error unavoidable. To hedge misspecification, one evaluates worst-case tail risk over an ambiguity set. Using Extreme Value Theory (EVT), we derive first-order asymptotics for worst-case tail risk for a broad class of tail-risk measures under standard ambiguity sets, including Wasserstein balls and $\phi$-divergence neighborhoods. We show that robustification can alter the nominal tail asymptotic scaling as the tail level $\beta\to0$, leading to excess risk inflation. Motivated by this diagnostic, we propose a tail-calibrated ambiguity design that preserves the nominal tail asymptotic scaling while still guarding against misspecification. Under standard domain of attraction assumptions, we prove that the resulting worst-case risk preserves the baseline first-order scaling as $\beta\to0$, uniformly over key tuning parameters, and that a plug-in implementation based on consistent tail-index estimation inherits these guarantees. Synthetic and real-data experiments show that the proposed design avoids the severe inflation often induced by standard ambiguity sets.