A Deeper Look Into an Insurance Risk Model with Two Types of Claims and FGM Copula Dependency

Abstract

This paper investigates a continuous-time, two types of claims risk model where the dependence between claim sizes and inter-claim times is structured using a Farlie–Gumbel–Morgenstern (FGM) copula. The methodology begins with the construction of a Lundberg’s equation and the determination of its non-negative roots. Subsequently, the integro-differential equation for the ruin probability is derived, from which the Laplace transform of the ruin probability is obtained. For the specific case of exponentially distributed claim sizes, an explicit analytical expression for the ruin probability is derived to examine the effects of dependence parameters and distributional characteristics. A series of numerical experiments with varying FGM copula parameters demonstrate that the ruin probability decreases as the initial surplus increases and is significantly influenced by the strength of the dependence structure. From a practical perspective, distinguishing between claim types allows insurers to identify which category poses the greatest threat to solvency, thereby supporting more targeted underwriting and accurate capital allocation strategies.