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.

翻译：尾部风险价值（TVaR）是一种被广泛采用的风险度量指标，在保险领域的学术研究与行业实践中均发挥着至关重要的作用。在数据应用中，由于其简单性和非参数特性，TVaR 常采用经验方法进行估计。监管机构已明确提倡将经验 TVaR 作为计算 TVaR 的标准方法。然而，已有文献指出，经验 TVaR 估计量存在负偏差，这可能导致在有限样本应用中系统性低估风险。本文旨在从两个维度深化对经验 TVaR 估计量偏差的理解：其大小以及驱动偏差严重程度的关键分布性和结构性决定因素。为此，我们基于渐近展开推导了偏差的领头项近似。与领头项近似相关的闭式表达式使我们能够获得关于支配经验 TVaR 估计量偏差的结构特性的解析见解。为了考虑领头项近似与真实偏差之间的差异，我们进一步推导了偏差的一个显式上界。我们通过仿真验证了所提出的偏差分析框架，并利用真实数据证明了其实际相关性。