The Aggregate Risk and Completely Mixable Distributions

总风险是精算学和定量风险管理中的普遍兴趣。一个重要且具有挑战性的场景是，个体风险具有已知的边际分布，但依赖结构未知。这种情况源于这样一个事实：在实践中，边缘分布可以相对容易地用统计和金融工具建模，但通过可用数据准确捕捉依赖结构要困难得多。由于依赖结构未知，银行和保险公司仍然难以根据行业规则抵消累积风险。由于这种关注，研究人员正试图量化风险度量的最优范围。在过去数十年中，这些问题在各种假设下得到了广泛的考虑，并且证明它们非常具有挑战性。最近，一个新的工具，称为完全混合分布（CMD），被引入来处理这些问题。Copula函数是一种多元函数，用于对相关结构进行建模，在量化总体风险方面也发挥着重要作用。该方法考虑的风险度量包括风险价值、条件尾期望、总风险的方差、期望效用和止损溢价等重要的风险量。

The aggregate risk is of general interest in actuarial science and quantitative risk management. An important and challenging scenario is when the individual risks have known marginal distributions but an unknown dependence structure. This scenario arises from the fact that marginal distributions can be relatively easily modeled with statistical and financial tools in practice, but the dependence structure is much more difficult to capture accurately through available data. With the dependence structure unknown, it remains difficult for the banks and insurance companies to offset the aggregated risks according to industrial regulations. Due to such a concern, researchers are trying to quantify optimal bounds on risk measures of aggregate risk. Such problems were considered extensively with various assumptions in the past few decades and they were proven to be very challenging. Recently, a new tool, called the Completely Mixable Distributions (CMD), was introduced to deal with those problems. Copulas are multivariate functions that are used to model dependence structure, which also play an important role in quantifying aggregate risk. The measures considered in this proposal include important risk quantities such as the Value-at-Risk, Conditional Tail Expectation, the variance, the expected ultility and the stop-loss premium of the aggregate risk.