Monte-Carlo Estimation of CoVaR

TL;DR

This paper introduces Monte-Carlo simulation methods for estimating CoVaR, a key measure of systemic financial risk, demonstrating their theoretical properties and practical effectiveness through numerical experiments.

Contribution

Develops new Monte-Carlo batching and importance-sampling estimators for CoVaR with proven consistency, asymptotic normality, and improved convergence rates.

Findings

Batching estimator converges at rate n^{-1/3}

Importance-sampling estimator converges at rate n^{-1/2}

Both estimators perform well in numerical tests

Abstract

${\rm CoVaR}$ 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 optimal rate of convergence of the batching estimator 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 we show that the rate of convergence of the estimator is $n^{-1/2}$. Numerical experiments support our theoretical findings and show that both estimators work well.