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.

对于给定的风险，众所周知的风险价值 (VaR) 经典定义并未考虑与其他可观察风险之间可能的相互作用。因此，最近的文献中已经定义和研究了捕捉风险之间的传染效应和尾部依赖性的条件 VaR，例如共同风险价值 (CoVaR)。在本文中，我们研究了在双变量环境中保证 VaR 和 CoVaR 之间的排序的条件，从而可以了解这两种衡量标准中哪一个比另一个或多或少保守。通过这样做，我们引入了 CoVaR-VaR 的概率等效水平（PELCoV）的概念，即 VaR 和 CoVaR 重合的可观察变量的 VaR 值，并研究了其一些属性，例如唯一性和有界性。特别是，我们表明其属性完全由描述风险之间依赖性的联结函数来解释，并且我们提供了 PELCoV 明确可用的联结函数列表，以及它是否有界。还提供了一个实际应用示例。