Monte Carlo Estimation of CoVaR

CoVaR is one of the most important measures of financial systemic risks. It is defined as the risk of a financial portfolio conditional on another financial portfolio being at risk. In this paper we first develop a Monte Carlo simulation-based batching estimator of CoVaR and study its consistency and asymptotic normality. We show that the best rate of convergence that the batching estimator can achieve is n(-1/3), where n is the sample size. We then develop an importance sampling-inspired estimator under the delta-gamma approximations to the portfolio losses and show that the best rate of convergence that the estimator can achieve is n(-1/2). Numerical experiments support our theoretical findings and show that both estimators work well.