Conditional Value-at-Risk (CVaR) is a risk measure widely used to quantify the impact of extreme losses. Owing to the lack of representative samples CVaR is sensitive to the tails of the underlying distribution. In order to combat this sensitivity, Distributionally Robust Optimization (DRO), which evaluates the worst-case CVaR measure over a set of plausible data distributions is often deployed. Unfortunately, an improper choice of the DRO formulation can lead to a severe underestimation of tail risk. This paper aims at leveraging extreme value theory to arrive at a DRO formulation which leads to representative worst-case CVaR evaluations in that the above pitfall is avoided while simultaneously, the worst case evaluation is not a gross over-estimate of the true CVaR. We demonstrate theoretically that even when there is paucity of samples in the tail of the distribution, our formulation is readily implementable from data, only requiring calibration of a single scalar parameter. We showcase that our formulation can be easily extended to provide robustness to tail risk in multivariate applications as well as in the evaluation of other commonly used risk measures. Numerical illustrations on synthetic and real-world data showcase the practical utility of our approach.

翻译：条件风险价值（CVaR）是一种广泛用于量化极端损失影响的风险度量。由于缺乏代表性样本，CVaR对底层分布的尾部极为敏感。为应对此敏感性，常采用分布鲁棒优化（DRO）方法，该方法在一组可能的数据分布上评估最坏情况下的CVaR度量。然而，若DRO公式选择不当，可能导致尾部风险的严重低估。本文旨在利用极值理论，构建一种能产生代表性最坏情况CVaR评估的DRO公式：该公式既可避免上述缺陷，又能确保最坏情况评估不会严重高估真实CVaR。我们从理论上证明，即使在分布尾部样本稀缺的情况下，所提公式仍可直接基于数据实现，仅需校准单个标量参数。研究还表明，该公式可轻松扩展至多元应用场景及其他常用风险度量的评估中，以提供尾部风险鲁棒性。基于合成数据与真实数据的数值算例验证了本方法的实用价值。