Building on the one-to-one relationship between generalized FGM copulas and multivariate Bernoulli distributions, we prove that the class of multivariate distributions with generalized FGM copulas is a convex polytope. Therefore, we find sharp bounds in this class for many aggregate risk measures, such as value-at-risk, expected shortfall, and entropic risk measure, by enumerating their values on the extremal points of the convex polytope. This is infeasible in high dimensions. We overcome this limitation by considering the aggregation of identically distributed risks with generalized FGM copula specified by a common parameter $p$. In this case, the analogy with the geometrical structure of the class of Bernoulli distribution allows us to provide sharp analytical bounds for convex risk measures.

翻译：基于广义FGM copula与多元伯努利分布之间的一一对应关系，我们证明了具有广义FGM copula的多元分布类构成一个凸多面体。因此，通过枚举该凸多面体极值点上的取值，我们在此类分布中为许多聚合风险度量（如风险价值、期望短缺和熵风险度量）找到了尖锐界。在高维情形下，该方法不可行。为克服这一限制，我们考虑具有由共同参数$p$指定的广义FGM copula的同分布风险聚合问题。在此情形下，通过与伯努利分布类几何结构的类比，我们为凸风险度量提供了尖锐的解析界。