Systemic risk measures quantify the potential risk to an individual financial constituent arising from the distress of entire financial system. As a generalization of two widely applied risk measures, Value-at-Risk and Expected Shortfall, the Conditional Value-at-Risk (CoVaR) and Conditional Expected Shortfall (CoES) have recently been receiving growing attention on applications in economics and finance, since they serve as crucial metrics for systemic risk measurement. However, existing approaches confront some challenges in statistical inference and asymptotic theories when estimating CoES, particularly at high risk levels. In this paper, within a framework of upper tail dependence, we propose several extrapolative methods to estimate both extreme CoVaR and CoES nonparametrically via an adjustment factor, which are intimately related to the nonparametric modelling of the tail dependence function. In addition, we study the asymptotic theories of all proposed extrapolative methods based on multivariate extreme value theory. Finally, some simulations and real data analyses are conducted to demonstrate the empirical performances of our methods.

翻译：系统性风险度量旨在量化由整个金融体系困境引发的个体金融成分的潜在风险。作为两种广泛应用的风险度量——风险价值和预期短缺——的推广，条件风险价值与条件预期短缺因其作为系统性风险度量的关键指标，近年来在经济学与金融学中的应用日益受到关注。然而，现有方法在估计条件预期短缺时，尤其是在高风险水平下，面临统计推断与渐近理论方面的挑战。本文在上尾依赖框架内，提出了几种外推方法，通过调整因子非参数地估计极端条件风险价值与条件预期短缺，这些方法与尾部依赖函数的非参数建模密切相关。此外，基于多元极值理论，我们研究了所有提出外推方法的渐近理论。最后，通过模拟与真实数据分析验证了所提方法的实证表现。