Elicitability and identifiability of tail risk measures
Tobias Fissler, Fangda Liu, Ruodu Wang, Linxiao Wei
TL;DR
The paper addresses the identifiability and elicitability of tail risk measures by linking each tail-functional ρ to a generator ρ^* via ρ(F) = ρ^*(F_p). It proves that, under suitable regularity, joint identifiability and elicitability of the tail risk measure with its quantile (Q_p) follow from analogous properties of the generator, and vice versa, providing explicit identification and scoring function forms. A key contribution is a novel class of weighted scores that extends the Fissler–Ziegel framework to accommodate forecast-dependent weights, enabling practical regression, GMM, backtesting, and model comparison for tail risks; the approach is illustrated with the tail expectile and generalizes to other tail-functionals such as VaR, ES, and RVaR. The results yield both conditional and unconditional elicitability results, along with left-tail and body-region extensions, offering a coherent strategy for estimation, comparison, and validation of tail-risk measures in financial risk management. The work thus provides a theoretically grounded, implementable pathway for fitting and validating tail-risk models using regression and backtesting while clarifying when and how tail-functionals can be elicitable or identifiable via their generators.
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, Expected Shortfall and Range Value-at-Risk 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.