We offer a new perspective on risk aggregation with FGM copulas. Along the way, we discover new results and revisit existing ones, providing simpler formulas than one can find in the existing literature. This paper builds on two novel representations of FGM copulas based on symmetric multivariate Bernoulli distributions and order statistics. First, we detail families of multivariate distributions with closed-form solutions for the cumulative distribution function or moments of the aggregate random variables. We order aggregate random variables under the convex order and provide methods to compute the cumulative distribution function of aggregate rvs when the marginals are discrete. Finally, we discuss risk-sharing and capital allocation, providing numerical examples for each.

中文翻译：

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使用 FGM copula 进行风险聚合

我们通过 FGM copulas 提供了风险聚合的新视角。在此过程中，我们发现新结果并重新审视现有结果，提供比现有文献中更简单的公式。本文建立在基于对称多元伯努利分布和阶数统计的 FGM copula 的两种新颖表示上。首先，我们详细介绍了具有闭合形式解的多元分布族，用于累积分布函数或聚合随机变量的矩。我们在凸顺序下对聚合随机变量进行排序，并提供计算边际离散时聚合 rvs 的累积分布函数的方法。最后，我们讨论了风险分担和资本分配，并为每一个提供了数字示例。