[论文解读] On dependence consistency of CoVaR and some other systemic risk measures

Georg Mainik, Eric Schaanning|arXiv (Cornell University)|Jul 14, 2012

Financial Risk and Volatility Modeling参考文献 13被引用 6

一句话总结

本文比较了两种CoVaR的定义——以X = VaRα(X)为条件与以X ≥ VaRα(X)为条件——并证明后者在多元正态、t分布及t边缘的Gumbel copula等关键模型中能确保依赖一致性。主要发现是，仅基于X ≥ VaRα(X)的CoVaR随依赖参数单调递增，而原始定义则不满足此性质，尤其在正态分布情形下，当相关系数超过1/√2时，其值随相关性增加而下降。

ABSTRACT

This paper is dedicated to the consistency of systemic risk measures with respect to stochastic dependence. It compares two alternative notions of Conditional Value-at-Risk (CoVaR) available in the current literature. These notions are both based on the conditional distribution of a random variable Y given a stress event for a random variable X, but they use different types of stress events. We derive representations of these alternative CoVaR notions in terms of copulas, study their general dependence consistency and compare their performance in several stochastic models. Our central finding is that conditioning on X>=VaR_α(X) gives a much better response to dependence between X and Y than conditioning on X=VaR_α(X). We prove general results that relate the dependence consistency of CoVaR using conditioning on X>=VaR_α(X) to well established results on concordance ordering of multivariate distributions or their copulas. These results also apply to some other systemic risk measures, such as the Marginal Expected Shortfall (MES) and the Systemic Impact Index (SII). We provide counterexamples showing that CoVaR based on the stress event X=VaR_α(X) is not dependence consistent. In particular, if (X,Y) is bivariate normal, then CoVaR based on X=VaR_α(X) is not an increasing function of the correlation parameter. Similar issues arise in the bivariate t model and in the model with t margins and a Gumbel copula. In all these cases, CoVaR based on X>=VaR_α(X) is an increasing function of the dependence parameter.

研究动机与目标

• 评估不同条件事件下CoVaR的依赖一致性。
• 比较原始CoVaR（以X = VaRα(X)为条件）与改进版本（以X ≥ VaRα(X)为条件）的差异。
• 评估系统性风险度量如CoES、MES和SII是否继承自CoVaR定义的依赖一致性。
• 提供反例以表明原始CoVaR在依赖参数上不具备单调性，从而削弱其在压力情景下的可靠性。

提出的方法

• 通过椭copula表示CoVaR与CoES，以分析依赖结构。
• 利用二元分布与copula的等级序关系，建立依赖一致性的理论条件。
• 将改进的CoVaR定义（以X ≥ VaRα(X)为条件）应用于三种随机模型：二元正态、二元t分布及t边缘的Gumbel copula。
• 通过解析推导与数值比较，评估CoVaR与CoES随依赖参数变化的单调性。
• 将结果扩展至相关系统性风险度量：边际预期缺口（MES）与系统性影响力指数（SII）。
• 使用反例表明，原始CoVaR（以X = VaRα(X)为条件）在二元正态及其他模型中均不随相关性增加而上升。

实验结果

研究问题

• RQ1 基于X = VaRα(X)的CoVaR是否在多元模型中对依赖增强作出一致响应？
• RQ2 基于X ≥ VaRα(X)的CoVaR在常见copula族中是否随依赖参数单调递增？
• RQ3 条件事件的选择如何影响系统性风险度量的一致性与可靠性？
• RQ4 CoVaR的依赖一致性是否可与copula的等级序关系等既定概念相联系？
• RQ5 如MES与SII等相关系统性风险度量是否继承自改进CoVaR定义的依赖一致性？

主要发现

• 基于X = VaRα(X)的CoVaR不具备依赖一致性；在二元正态模型中，当相关系数ρ > 1/√2时，其值随相关性增加而下降，违背系统性风险的直觉。
• 基于X ≥ VaRα(X)的CoVaR在所有测试模型中（包括二元正态、t分布及t边缘的Gumbel copula）均随依赖参数单调递增。
• 改进CoVaR的依赖一致性在数学上建立于copula的等级序关系之上，提供了坚实的理论基础。
• 原始CoVaR定义在系统性风险最显著时无法识别风险，反例显示其在高相关性情形下失效。
• 改进CoVaR及其扩展（CoES、MES、SII）继承了依赖一致性，支持其在监管压力测试中的应用。
• 以X ≥ VaRα(X)为条件可产生更稳健且可解释的系统性风险度量，相较以X = VaRα(X)为条件的高选择性与乐观情形更具优势。