Nonparametric Inference for Extreme CoVaR and CoES

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.