Estimations of Extreme CoVaR and CoES under Asymptotic Independence

TL;DR

The paper addresses estimating extreme CoVaR and CoES when $X$ and $Y$ are asymptotically independent but positively associated, a regime common in finance. It develops two extrapolation frameworks: (i) a VaR-based approach leveraging an adjustment factor $\xi_{1-k/n}$ and tail-index estimators, and (ii) an intermediate-CoVaR/CoES approach using direct tail estimates and scaling; both yield asymptotic normality and joint estimators for CoVaR and CoES. Theoretical results rely on heavy-tailed $X$ with EVI $\gamma_1$, tail independence parameter $\eta\in(1/2,1)$, and a differentiable tail copula $C$, linking extreme CoVaR to VaR and to CoES via $\mathrm{CoES}_{X|Y}(\tau)/\mathrm{CoVaR}_{X|Y}(\tau) \to 1/(1-\gamma_1)$. Empirical validation via Monte Carlo simulations and a real-data analysis on 12 S&P500 stocks shows the proposed estimators perform well in finite samples, with CoVaR and CoES escalating at more extreme tails and the two extrapolation methods yielding closely aligned results in practice. These findings provide a practical toolkit for systemic risk assessment when tail dependence is weak but positively correlated, expanding the applicability of extreme CoVaR/CoES estimation beyond true tail dependence scenarios.

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.

Figures (8)

• Figure 1: Boxplots of the ratios between $\widetilde{\mathrm{CoVaR}}_{X|Y}^{(i)}(\tau'_n)$ ($i=1,2$), $\widetilde{\mathrm{CoES}}_{X|Y}^{(i)}(\tau'_n)$ ($i=1,2,3$) and their true values for Model 1 with $n \in \{500, 1000, 2000, 5000\}$ and $\tau'_n \in \{0.99, 0.999\}$.
• Figure 2: Boxplots of the ratios between $\widetilde{\mathrm{CoVaR}}_{X|Y}^{(i)}(\tau'_n)$ ($i=1,2$), $\widetilde{\mathrm{CoES}}_{X|Y}^{(i)}(\tau'_n)$ ($i=1,2,3$) and their true values for Model 2 with $n \in \{500, 1000, 2000, 5000\}$ and $\tau'_n \in \{0.99, 0.999\}$.
• Figure 3: Boxplots of the ratios between $\widetilde{\mathrm{CoVaR}}_{X|Y}^{(i)}(\tau'_n)$ ($i=1,2$), $\widetilde{\mathrm{CoES}}_{X|Y}^{(i)}(\tau'_n)$ ($i=1,2,3$) and their true values for Model 3 with $n \in \{500, 1000, 2000, 5000\}$ and $\tau'_n \in \{0.99, 0.999\}$.
• Figure 4: Time series plots of weekly losses for the 12 constituent stocks.
• Figure 5: The estimations for $\gamma_1$ (blue lines) and $\eta$ (red lines) against $k_1$ and $k_2$ for 12 individual losses conditional on S&P500 Index loss.

Theorems & Definitions (8)

• Proposition 2.1
• Remark 2.1
• Lemma 2.1
• Remark 2.2
• Proposition 2.2
• Theorem 2.1
• Proposition 2.3
• Theorem 2.2