标题： 使用CoVaR衡量和分析边际系统性风险贡献：一种Copula方法 显示英文标题

标题： Measuring and Analysing Marginal Systemic Risk Contribution using CoVaR: A Copula Approach

摘要： 本文致力于量化和分析给定单一金融机构i对金融系统s风险的边际风险贡献。我们的工作扩展了Adrian和Brunnermeier提出的CoVaR概念，作为衡量边际系统性风险贡献的工具。 我们首先给出CoVaR_{\alpha }^{s|L^i=l}的数学定义。 我们的定义通过将CoVaR_{\alpha }^{s|L^i=l}表示为状态l和给定概率水平\alpha 的函数，分别相对于i和s，改进了CoVaR概念。 基于Copula理论，我们通过它们的概率解释和定义（条件概率）将CoVaR_{\alpha }^{s|L^i=l}与Copula的偏导数联系起来。 使用这种方法，我们为一大类（边缘）分布和依赖结构（线性和非线性）提供了计算CoVaR_{\alpha }^{s|L^i=l}的闭合公式。我们的公式允许更好地分析系统性风险，使用CoVaR，因为它允许根据i和s的损失的边缘分布以及L^i和L^s之间的copula来定义CoVaR_{\alpha }^{s|L^i=l}。我们在系统性风险贡献的量化和分析背景下讨论了这一结果的影响。一些数学内容使得可能例如我们将分析L^i、L^s和C对i的风险贡献的边缘效应。 显示英文摘要

摘要： This paper is devoted to the quantification and analysis of marginal risk contribution of a given single financial institution i to the risk of a financial system s. Our work expands on the CoVaR concept proposed by Adrian and Brunnermeier as a tool for the measurement of marginal systemic risk contribution. We first give a mathematical definition of CoVaR_{\alpha}^{s|L^i=l}. Our definition improves the CoVaR concept by expressing CoVaR_{\alpha}^{s|L^i=l} as a function of a state l and of a given probability level \alpha relative to i and s respectively. Based on Copula theory we connect CoVaR_{\alpha}^{s|L^i=l} to the partial derivatives of Copula through their probabilistic interpretation and definitions (Conditional Probability). Using this we provide a closed formula for the calculation of CoVaR_{\alpha}^{s|L^i=l} for a large class of (marginal) distributions and dependence structures (linear and non-linear). Our formula allows a better analysis of systemic risk using CoVaR in the sense that it allows to define CoVaR_{\alpha}^{s|L^i=l} depending on the marginal distributions of the losses of i and s respectively and the copula between L^i and L^s. We discuss the implications of this in the context of the quantification and analysis of systemic risk contributions. %some mathematical This makes possible the For example we will analyse the marginal effects of L^i, L^s and C of the risk contribution of i.