Expected shortfall (ES, also known as CVaR) is the most important coherent risk measure in finance, insurance, risk management, and engineering. Recently, Wang and Zitikis (2021) put forward four economic axioms for portfolio risk assessment and provide the first economic axiomatic foundation for the family of ES. In particular, the axiom of no reward for concentration (NRC) is arguably quite strong, which imposes an additive form of the risk measure on portfolios with a certain dependence structure. We move away from the axiom of NRC by introducing the notion of concentration aversion, which does not impose any specific form of the risk measure. It turns out that risk measures with concentration aversion are functions of ES and the expectation. Together with the other three standard axioms of monotonicity, translation invariance and lower semicontinuity, concentration aversion uniquely characterizes the family of ES. In addition, we establish an axiomatic foundation for the problem of mean-ES portfolio selection and new explicit formulas for convex and consistent risk measures. Finally, we provide an economic justification for concentration aversion via a few axioms on the attitude of a regulator towards dependence structures.

中文翻译：

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风险集中度和平均预期缺口标准

预期缺口（ES，也称为 CVaR）是金融、保险、风险管理和工程领域最重要的连贯风险度量。最近，Wang和Zitikis（2021）提出了投资组合风险评估的四个经济公理，为ES家族提供了第一个经济公理基础。特别是，集中无回报公理（NRC）可以说是相当强的，它对具有一定依赖性结构的投资组合施加了附加形式的风险度量。我们通过引入集中厌恶的概念来摆脱 NRC 的公理，它没有强加任何特定形式的风险措施。事实证明，具有集中厌恶的风险测度是 ES 和期望的函数。与其他三个标准公理（单调性、平移不变性和下半连续性）一起，集中厌恶是 ES 家族的独特特征。此外，我们为均值-ES投资组合选择问题建立了公理基础，并为凸且一致的风险度量建立了新的显式公式。最后，我们通过关于监管者对依赖结构的态度的一些公理，为集中厌恶提供了经济理由。