The paper deals with Conditional Value at Risk (CoVaR) for copulas with nontrivial tail dependence. We show that both in the standard and the modified settings, the tail dependence function determines the limiting properties of CoVaR as the conditioning event becomes more extreme. The results are illustrated with examples using the extreme value, conic and truncation invariant families of bivariate tail-dependent copulas.

Let ${𝔽}_{}^{}$F be a filtration andτbe a random time. Let ${𝔾}_{}^{}$G be the progressive enlargement of ${𝔽}_{}^{}$F withτ. We study the following formula, called the optional splitting formula: For any ${𝔾}_{}^{}$G-optional processY, there exists an ${𝔽}_{}^{}$F-optional processY′ and a function Y′′ defined on [0,∞] × (ℝ+ × Ω) being $ℬ[0,\infty ]\otimes 𝒪\left({𝔽}_{}^{}\right)$ℬ[0,∞]⊗x1d4aa;(F) measurable, such that $Y=Y^{\text{'}}{1}_{[0,\tau )}^{}+Y^{\text{'}\text{'}}\left(\tau \right){\mathbb{1}}_{[\tau ,\infty )}^{}.$Y=Y′1[0,τ)+Y′′(τ)1[τ,∞). (This formula can also be formulated for multiple random timesτ1,...,τk). We are interested in this formula because of its fundamental role in many...