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.

Funding: This work was partially supported by the National Natural Science Foundation of China [Projects 72161160340 and 12301601].

Supplemental Material: The computer code and data that support the findings of this study are available within this article’s supplemental material at https://doi.org/10.1287/opre.2023.0211.