两种配置的故事：尾部风险资本贡献与风险贡献的比较
A tale of two allocations: Risk capital contributions versus risk contributions in the tail
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中文总结 AI 辅助
本文比较了尾部风险资本配置的两种比例方法（CTE与GTE），揭示了它们的一致性条件、排序及渐近行为，并指出差异源于尾部协方差，极端尾部关系由ES与VaR之比决定。
中文摘要 AI 辅助
我们比较了风险组成部分对一组风险的总尾部风险的两种自然的比例贡献概念：在条件尾部期望（CTE）下分配给该组成部分的总尾部风险资本的比例，以及在几何尾部期望（GTE）下该组成部分对总风险的期望实现份额。由此产生的比例配置通常不同。对于任意的随机向量——允许原子、带符号的风险以及任何尾部域——我们确定了配置何时一致，在它们不一致时确定其排序，并刻画了它们的渐近分离。两种比例配置都是给定总量下组成部分的条件风险份额的加权平均值：比例CTE配置按严重性对尾部情景加权，而比例GTE配置则均匀加权。因此，它们的差异是归一化的尾部协方差。比例CTE和GTE配置在整个尾部一致，当且仅当条件风险份额在该处为常数；在温和的单峰条件下，优势由其单调性刻画。在具有异质自然参数的独立指数分散模型中，精确一致仅可能出现在缩放泊松族中；在共单调性下，它等价于比例分位数函数。在极端尾部，比例CTE和GTE配置之间的极限关系由期望损失与风险价值的比率控制：有界性确保共同极限，收敛到1迫使配置合并，而发散可能导致其极限分离。
英文摘要
We compare two natural proportional notions of a risk component's contribution to the aggregate tail risk of a collection of risks: the fraction of aggregate tail risk capital allocated to the component under Conditional Tail Expectation (CTE), and the component's expected realized share of aggregate risk under Geometric Tail Expectation (GTE). The resulting proportional allocations generally differ. For arbitrary random vectors-allowing atoms, signed risks, and any tail domain--we establish when the allocations agree, determine their ordering when they do not, and characterize their asymptotic separation. Both proportional allocations are weighted averages of the conditional risk share of a component given the aggregate: the proportional CTE allocation weights tail scenarios by severity, whereas the proportional GTE allocation weights them uniformly. Their difference is therefore a normalized tail covariance. The proportional CTE and GTE allocations agree throughout a tail exactly when the conditional risk share is constant there; under a mild unimodality condition, dominance is characterized by its monotonicity. Among independent exponential dispersion models with heterogeneous natural parameters, exact agreement is possible only for the scaled Poisson family; under comonotonicity, it is equivalent to proportional quantile functions. In the extreme tail, the limiting relationship between the proportional CTE and GTE allocations is governed by the ratio of Expected Shortfall to Value-at-Risk: boundedness ensures common limits, convergence to one forces the allocations to merge, and divergence can cause their limits to separate.
发表机构
- Department of Mathematics and Statistics, York University（约克大学数学与统计系）
- RISC Foundation（RISC基金会）
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