Multivariate matrix-exponential affine mixtures and their applications in risk theory

TL;DR

This paper introduces a new class of multivariate matrix-exponential affine mixtures with properties useful for actuarial calculations, applicable to various risk management problems.

Contribution

It proposes a novel class of multivariate matrix-exponential affine mixtures with explicit properties for actuarial applications, including risk measures and capital allocation.

Findings

Closed under size-biased Esscher transform and other distributions

Explicit formulas for actuarial quantities

Application to risk measures and capital allocation

Abstract

In this paper, a class of multivariate matrix-exponential affine mixtures with matrix-exponential marginals is proposed. The class is shown to possess various attractive properties such as closure under size-biased Esscher transform, order statistics, residual lifetime and higher order equilibrium distributions. This allows for explicit calculations of various actuarial quantities of interest. The results are applied in a wide range of actuarial problems including multivariate risk measures, aggregate loss, large claims reinsurance, weighted premium calculations and risk capital allocation. Furthermore, a multiplicative background risk model with dependent risks is considered and its capital allocation rules are provided as well. We finalize by discussing a calibration scheme based on complete data and potential avenues of research.