CoVaR under Asymptotic Independence

TL;DR

This paper develops a semi-parametric extreme-value framework for estimating CoVaR under asymptotic independence between the loss components. It links CoVaR_Y|X(p) to VaR_Y(p η_p) via an adjustment factor and uses a parametric tail dependence model to estimate the tail structure, yielding a practical estimator $ ext{CoVaR}_{Y vert X}(p) = ( ext{η}_p^*)^{- ext{γ}} ext{VaR}_Y(p)$. The authors prove consistency and asymptotic normality of the estimator, validated by simulations and an empirical application to US stock data that demonstrates robust, dynamic CoVaR forecasting even when tail dependence is weak or vanishes. The approach provides a conservative and interpretable tool for systemic risk assessment in settings where joint tails exhibit asymptotic independence, with clear procedures for extensions to varying tail levels and alternative CoVaR definitions.

Abstract

Conditional value-at-risk (CoVaR) is one of the most important measures of systemic risk. It is defined as the high quantile conditional on a related variable being extreme, widely used in the field of quantitative risk management. In this work, we develop a semi-parametric methodology to estimate CoVaR for asymptotically independent pairs within the framework of bivariate extreme value theory. We use parametric modelling of the bivariate extremal structure to address data sparsity in the joint tail regions and prove consistency and asymptotic normality of the proposed estimator. The robust performance of the estimator is illustrated via simulation studies. Its application to the US stock returns data produces insightful dynamic CoVaR forecasts.

Figures (1)

• Figure 1: Estimates of $\eta$ as a function of $k_3$ for the eleven institutions.

Theorems & Definitions (6)

• Theorem 1
• Theorem 2
• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4