Probability equivalent level for CoVaR and VaR

For a given risk, the well-known classical definition of Value-at-Risk (VaR) does not take into account possible interactions with other observable risks. For this reason, conditional VaRs that capture contagion effects and tail dependence among risks, such as the Co-Value-at-Risk (CoVaR), have been defined and studied in recent literature. In this paper we study conditions that guarantee, in the bivariate setting, the ordering between VaR and CoVaR, allowing to understand which, among the two measures, is more or less conservative than the other. By doing this, we introduce the notion of Probability Equivalent Level of CoVaR-VaR (PELCoV), which is the VaR value of the observable variable for which VaR and CoVaR coincide, and we study some of its properties such as uniqueness and boundedness. In particular, we show that its properties are entirely explained by the copula that describes the dependence between risks, and we provide a list of copulas for which PELCoV is explicitly available, and for which it is or not bounded. A practical applicative example is also presented.