The two popular systemic risk measures CoVaR (Conditional Value-at-Risk) and CoES (Conditional Expected Shortfall) have recently been receiving growing attention on applications in economics and finance. In this paper, we study the estimations of extreme CoVaR and CoES when the two random variables are asymptotic independent but positively associated. We propose two types of extrapolative approaches: the first relies on intermediate VaR and extrapolates it to extreme CoVaR/CoES via an adjustment factor; the second directly extrapolates the estimated intermediate CoVaR/CoES to the extreme tails. All estimators, including both intermediate and extreme ones, are shown to be asymptotically normal. Finally, we explore the empirical performances of our methods through conducting a series of Monte Carlo simulations and a real data analysis on S&P500 Index with 12 constituent stock data.

翻译：两种常用的系统性风险度量指标CoVaR（条件风险价值）与CoES（条件期望缺口）近年来在经济学与金融学领域的应用日益受到关注。本文研究当两个随机变量呈渐近独立但正相关时，极端CoVaR与CoES的估计问题。我们提出两类外推方法：第一类基于中间分位的VaR，通过调整因子将其外推至极端CoVaR/CoES；第二类直接将估计的中间CoVaR/CoES外推至极端尾部。所有估计量（包括中间估计量与极端估计量）均被证明具有渐近正态性。最后，我们通过一系列蒙特卡洛模拟实验，以及对标普500指数及其12只成分股数据的实证分析，检验了所提方法的实际表现。