# Aggregation of Dependent Risks in Mixtures of Exponential Distributions and Extensions

## TL;DR

This paper derives analytical formulas for the distribution of sums of dependent risks modeled by mixtures of exponential distributions, with applications to risk measures and extensions to gamma mixtures.

## Contribution

It provides new analytical expressions for the pdf and cdf of aggregated dependent risks in mixtures of exponential distributions, including specific families and model extensions.

## Key findings

- Analytic formulas for pdf and cdf of aggregated risks
- Application to Pareto, Gamma, Weibull, and inverse Gaussian mixtures
- Discussion on risk measures and ruin probabilities

## Abstract

The distribution of the sum of dependent risks is a crucial aspect in actuarial sciences, risk management and in many branches of applied probability. In this paper, we obtain analytic expressions for the probability density function (pdf) and the cumulative distribution function (cdf) of aggregated risks, modeled according to a mixture of exponential distributions. We first review the properties of the multivariate mixture of exponential distributions, to then obtain the analytical formulation for the pdf and the cdf for the aggregated distribution. We study in detail some specific families with Pareto (Sarabia et al, 2016), Gamma, Weibull and inverse Gaussian mixture of exponentials (Whitmore and Lee, 1991) claims. We also discuss briefly the computation of risk measures, formulas for the ruin probability (Albrecher et al., 2011) and the collective risk model. An extension of the basic model based on mixtures of gamma distributions is proposed, which is one of the suggested directions for future research.