Asymptotic Expansion and Bounds for the Bias of Empirical Tail Value-at-Risk

Abstract

Tail Value-at-Risk (TVaR) is a widely adopted risk measure playing a critically important role in both academic research and industry practice in insurance. In data applications, TVaR is often estimated using the empirical method, owing to its simplicity and nonparametric nature. The empirical TVaR has been explicitly advocated by regulatory authorities as a standard approach for computing TVaR. However, prior literature has pointed out that the empirical TVaR estimator is negatively biased, which can lead to a systemic underestimation of risk in finite-sample applications. This paper aims to deepen the understanding of the bias of the empirical TVaR estimator in two dimensions: its magnitude as well as the key distributional and structural determinants driving the severity of the bias. To this end, we derive a leading-term approximation for the bias based on its asymptotic expansion. The closed-form expression associated with the leading-term approximation enables us to obtain analytical insights into the structural properties governing the bias of the empirical TVaR estimator. To account for the discrepancy between the leading-term approximation and the true bias, we further derive an explicit upper bound for the bias. We validate the proposed bias analysis framework via simulations and demonstrate its practical relevance using real data.