[Paper Review] CoVaR under Asymptotic Independence

Zhaowen Wang, Yutao Liu+1 more|arXiv (Cornell University)|Feb 11, 2026

Financial Risk and Volatility Modeling0 citations

TL;DR

This paper develops a semi-parametric estimator for CoVaR when X and Y are asymptotically independent, proves consistency and asymptotic normality, and demonstrates performance via simulations and US stock data.

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.

Motivation & Objective

• Motivate the need for measuring systemic risk when tail dependence is asymptotically independent.
• Propose a semi-parametric estimator for CoVaR under asymptotic independence using bivariate extreme value theory.
• Establish consistency and asymptotic normality of the estimator.
• Assess finite-sample performance through simulations.
• Apply the method to US stock returns to generate dynamic CoVaR forecasts.

Proposed method

• Adopt a bivariate extreme value framework with asymptotic independence characterized by Ledford–Tawn regular variation.
• Define an adjustment factor eta_p* to connect CoVaR to the unconditional VaR via CoVaR_Y|X(p) ≈ VaR_Y(p·eta_p*).
• Estimate the tail index gamma of Y with the Hill estimator and VaR_Y(p) via extrapolation.
• Model the tail dependence function c(x,y;θ) parametrically and estimate θ by a method-of-moments (M-estimator) using a nonparametric proxy Q̂_n.
• Solve c(1, η_p*; θ̂)=p^{2−1/η} to obtain η_p*, then construct CoVaR̂_Y|X(p) = (η_p*)^{−γ̂}·VaR̂_Y(p).
• Provide extensions to CoVaR(p,q), CoVaR^{=}(p), and their estimators under various tail-dependence and conditioning schemes.

(a) AIG

Experimental results

Research questions

• RQ1 How can CoVaR be estimated when X and Y are asymptotically independent?
• RQ2 What semi-parametric framework accurately captures tail dependence via a parametric c(x,y;θ) in the bivariate extreme value setting?
• RQ3 Are the proposed estimators of CoVaR consistent and asymptotically normal under the asymptotic independence regime?
• RQ4 How do the proposed estimators perform relative to naive estimators in finite samples and under model misspecification?
• RQ5 Can the method be extended to CoVaR with different conditioning and/or different tail-dependence definitions?

Key findings

• The proposed estimator of CoVaR under asymptotic independence is more accurate than the naive estimator in simulations (lower bias and variance).
• Consistency holds for the CoVaR estimator under a suite of technical assumptions (Assumptions 1–7).
• Asymptotic normality is established for the CoVaR estimator under additional regularity (Assumptions 8–9).
• Finite-sample performance depends on the tail model and choice of k-parameters; larger k reduces variance, smaller k can reduce bias.
• The application to US stock returns yields dynamic CoVaR forecasts; in simulations, performance degrades under model misspecification (Model 2) but remains improved over the naive approach.
• The method supports extensions to CoVaR(p,q) and CoVaR^{=}(p) with corresponding estimators.

(b) ALL