Insurance Applications of Some New Dependence Models derived from Multivariate Collective Models

TL;DR

This paper develops new dependence models for joint claim sizes in insurance portfolios, providing tractable parametric families and applications to real insurance data, along with theoretical properties analysis.

Contribution

It introduces a novel class of dependence models based on collective claims, allowing for parameter-dependent distributions and practical applications in insurance risk modeling.

Findings

Three applications to insurance data demonstrate model flexibility.

Distributional and asymptotic properties of the models are established.

Models effectively capture dependence in joint claim sizes.

Abstract

Consider two different portfolios which have claims triggered by the same events. Their corresponding collective model over a fixed time period is given in terms of individual claim sizes $(X_i,Y_i), i\ge 1$ and a claim counting random variable $N$. In this paper we are concerned with the joint distribution function $F$ of the \ece{largest claim sizes} $(X_{N:N}, Y_{N:N})$. By allowing $N$ to depend on some parameter, say $\theta$, then $F=F(\theta)$ is for various choices of $N$ a tractable parametric family of bivariate distribution functions. We present three applications of the implied parametric models to some data from the literature and a new data set from a Swiss insurance company. Furthermore, we investigate both distributional and asymptotic properties of $(X_{N:N}, Y_{N:N})$.