We provide sharp analytical upper and lower bounds for value‐at‐risk (VaR) and sharp bounds for expected shortfall (ES) of portfolios of any dimension subject to default risk. To do so, the main methodological contribution of the paper consists in analytically finding the convex hull generators for the class of exchangeable Bernoulli variables with given mean and for the class of exchangeable Bernoulli variables with given mean and correlation in any dimension. Using these analytical results, we first describe all possible dependence structures for default, in the class of finite sequences of exchangeable Bernoulli random variables. We then measure how model risk affects VaR and ES.

中文翻译：

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在信用风险中模拟风险

我们为存在违约风险的任何维度的投资组合都提供了清晰的风险价值上限（VaR）上限和下限，以及预期投资组合缺口（ES）的清晰上限。为此，本文的主要方法论贡献在于，针对具有给定均值的可交换伯努利变量类别以及具有给定均值和任意维数的相关性的可交换伯努利变量类别，分析找到凸包生成器。使用这些分析结果，我们首先在可交换伯努利随机变量的有限序列类中描述了所有可能的违约依赖结构。然后，我们测量模型风险如何影响VaR和ES。