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

TL;DR

The paper addresses tail-risk evaluation under distributional uncertainty and shows that standard DRO using Wasserstein or polynomial φ-divergence sets can overinflate tail risk, especially under heavy-tailed or light-tailednominal laws. By combining extreme-value theory with carefully calibrated tail-extrapolation, the authors design Rate-Preserving EVT DRO (RPEV-DRO), which preserves the nominal tail growth rate uniformly over ambiguity radii and intermediate tail levels, and extends to multivariate losses. They provide a data-driven implementation with consistent tail-index estimation and prove both rate-preserving and consistency properties. Empirical results across synthetic experiments and real datasets (univariate insurance, delta hedging, fire losses, and Fama-French factors) demonstrate that RPEV-DRO avoids severe tail inflation while maintaining protective tail risk estimates, offering a practical, robust tool for tail-risk management.

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 $φ$-divergence neighborhoods. We show that robustification can alter the nominal tail asymptotic scaling as the tail level $β\to0$, 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 $β\to0$, 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.

Figures (11)

• Figure 1: Worst-case CVaR evaluation under Wasserstein uncertainty
• Figure 2: Worst-case CVaR evaluation with $\chi^2$-divergence ambiguity set
• Figure 3: Common parameters: $\delta=0.1$, $\beta_0 =\min\{0.1, \beta^{0.5}\}$. Solid black line is the true CVaR. At $\beta=10^{-2}$, in Figure \ref{['fig:tailoed_ht']}, RPEV-DRO is $1.1\, C_{1-\beta}(Q)$, $\chi^2$-DRO with EVT nominal is $1.9\,C_{1-\beta}(Q)$ and the Gaussian nominal $\chi^2$-DRO is $0.3\, C_{1-\beta}(Q)$. Similarly, in Figure \ref{['fig:tailored_lt']} RPEV-DRO is $1.2 \,C_{1-\beta}(Q)$, $\chi^2$-DRO with EVT nominal is $1.4\,C_{1-\beta}(Q)$ and the Gaussian nominal $\chi^2$-DRO is $0.7\, C_{1-\beta}(Q)$.
• Figure 4: Common parameters: $n=500$, $\beta_0 = \min\{0.1, \ \beta^{0.5}\}$, $\delta = 0.1$, ${\tt reps}=100$. Markers on each graph denote actual risk evaluations.
• Figure 5: Parameters: For illustration, we choose $\delta=0.05$, $\varepsilon=0.1$, ${\tt reps}=100$. Markers on each graph denote actual risk evaluations. Here $n=500$.

Theorems & Definitions (35)

• Definition 1
• Lemma 2.1
• Lemma 2.2
• Theorem 3.1
• Remark 1
• Theorem 3.2: Characterizing the Worst Case Distribution
• Remark 2
• Proposition 4.1: Contents of divergence ball
• Lemma 4.1: Lower Bound for Robust Risk
• Theorem 4.1: Worst-Case Risk under $\phi-$Divergence Ambiguity