Collective risk models with FGM dependence

TL;DR

This paper investigates collective risk models with FGM copula dependence, providing new representations for moments and transforms, and analyzing how dependence affects aggregate risk measures.

Contribution

It introduces a novel approach linking FGM copulas to Bernoulli distributions, enabling easier analysis of moments and dependence effects in risk models.

Findings

Dependence impacts aggregate risk measures significantly.

FGM copula induces moderate dependence but can cause substantial risk measure variations.

New representations simplify analysis of collective risk models with FGM dependence.

Abstract

We study copula-based collective risk models when the dependence structure is defined by a Farlie-Gumbel-Morgenstern (FGM) copula. By leveraging a one-to-one correspondence between the class of FGM copulas and multivariate symmetric Bernoulli distributions, we find convenient representations for the moments and Laplace-Stieltjes transform for the aggregate random variable defined from collective risk models with FGM dependence. We examine different components of this collective risk model, aiming to better understand the impact of the assumed dependence between a claim's frequency and severity. Relying on stochastic ordering, we analyze the impact of dependence on the aggregate claim amount random variable. Even if the FGM copula may only induce moderate dependence, we illustrate through numerical examples that the cumulative effect of FGM dependence can lead to substantial variations in key risk measures on aggregate random variables defined from collective risk models.