Risk aggregation with FGM copulas

TL;DR

This paper introduces new methods for risk aggregation using FGM copulas, offering simplified formulas, novel representations, and practical tools for risk management and capital allocation.

Contribution

It presents new representations of FGM copulas based on symmetric Bernoulli distributions and order statistics, with closed-form solutions for aggregate distributions.

Findings

Derived simpler formulas for risk aggregation with FGM copulas

Established methods for ordering aggregate risks under convex order

Provided numerical examples for risk-sharing and capital allocation

Abstract

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.