CoVaR under Asymptotic Independence

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.

💡 Research Summary

This paper introduces a novel semi‑parametric methodology for estimating Conditional Value‑at‑Risk (CoVaR) when the two loss variables, X (institution‑specific loss) and Y (system‑wide loss), exhibit asymptotic independence—a situation increasingly observed in financial applications but largely ignored by existing CoVaR estimators that assume asymptotic dependence. The authors build on the Ledford‑Tawn framework, introducing the tail dependence coefficient η (½ < η < 1) and a homogeneous tail dependence function c(x, y) that captures the residual dependence structure in the joint upper tail.

The CoVaR of interest is defined via the conditional exceedance probability
 P(Y ≥ CoVaR | X ≥ VaR_X(p)) = p,
and is linked to an unconditional VaR of Y through an adjustment factor η_p:
 CoVaR = VaR_Y(p η_p).
Because η_p cannot be directly estimated from sparse joint tail data, the authors propose an asymptotic approximation η*_p obtained by solving
 c(1, η*_p) = p^{2 − 1/η},
where η and c are estimated from the data.

The estimation proceeds in three steps. First, the extreme‑value index γ of Y’s marginal distribution is estimated using the Hill estimator based on the top k₁ order statistics. Second, VaR_Y(p) is extrapolated from the empirical tail using the de Haan–Ferroira (2006) method with k₂ exceedances. Third, the parametric family {c(·,·; θ)} is fitted via an M‑estimator that minimizes a moment condition involving a non‑parametric estimator Q_n(x, y) of the joint tail probability. The minimization yields estimates θ̂ and a scale parameter ζ̂; η̂ is then obtained as a known function η(θ̂). With η̂ and θ̂, η*_p is solved from the implicit equation c(1, η*_p; θ̂) = p^{2 − 1/η̂}. The final CoVaR estimator is
 \widehat{CoVaR} =