Estimations of Extreme CoVaR and CoES under Asymptotic Independence

Abstract

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.