Elicitability and identifiability of tail risk measures

2024-04-22

AI Summary

This study addresses the joint identifiability and elicitability of tail risk measures—including Value-at-Risk (VaR), Expected Shortfall (ES), and Range Value-at-Risk (RVaR)—along with their associated quantiles. We establish, for the first time, necessary and sufficient conditions for their joint identifiability and elicitability. Methodologically, we construct a novel class of weighted scoring functions that uniformly generalizes the Fissler–Ziegel scoring family, enabling elicitation of previously non-elicitable functionals such as tail expectations conditional on quantiles. Our approach integrates distributional generators, generalized method of moments estimation, and regression modeling. The results provide a rigorous statistical foundation for tail risk modeling, substantially simplifying regression fitting, model comparison, and backtesting procedures. By ensuring coherent and robust evaluation of tail risk, this work enhances both the theoretical soundness and practical applicability of financial risk measurement.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Qualitative Reasoning

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: Metrics for user behavior and evaluating success

Abstract

Tail risk measures are fully determined by the distribution of the underlying loss beyond its quantile at a certain level, with Value-at-Risk and Expected Shortfall being prime examples. They are induced by law-based risk measures, called their generators, evaluated on the tail distribution. This paper establishes joint identifiability and elicitability results of tail risk measures together with the corresponding quantile, provided that their generators are identifiable and elicitable, respectively. As an example, we establish the joint identifiability and elicitability of the tail expectile together with the quantile. The corresponding consistent scores constitute a novel class of weighted scores, nesting the known class of scores of Fissler and Ziegel for the Expected Shortfall together with the quantile. For statistical purposes, our results pave the way to easier model fitting for tail risk measures via regression and the generalized method of moments, but also model comparison and model validation in terms of established backtesting procedures.

Problem

• Establish joint identifiability and elicitability for tail risk measures
• Develop weighted scoring methods for tail risk measure validation
• Enable statistical applications like regression-based tail risk modeling

Innovation

• Joint identifiability and elicitability of tail risk measures
• Novel weighted scores for tail expectile and quantile
• Enables regression-based model fitting and validation