Risks aggregation in multivariate dependent Pareto distributions

TL;DR

This paper derives explicit formulas for the distribution of aggregated risks modeled by multivariate dependent Pareto distributions, with applications to insurance risk measures and model comparison using real data.

Contribution

It provides the first closed-form expressions for the distribution of aggregated risks under dependent multivariate Pareto type II distributions, including new risk measures and model comparisons.

Findings

Derived closed-form PDFs involving hypergeometric functions.

Calculated risk measures like VaR and TVaR for these models.

Demonstrated improved model fit over classical models using real insurance data.

Abstract

In this paper we obtain closed expressions for the probability distribution function, when we consider aggregated risks with multivariate dependent Pareto distributions. We work with the dependent multivariate Pareto type II proposed by Arnold (1983, 2015), which is widely used in insurance and risk analysis. We begin with the individual risk model, where we obtain the probability density function (PDF), which corresponds to a second kind beta distribution. We obtain several risk measures including the VaR, TVaR and other tail measures. Then, we consider collective risk model based on dependence, where several general properties are studied. We study in detail some relevant collective models with Poisson, negative binomial and logarithmic distributions as primary distributions. In the collective Pareto-Poisson model, the PDF is a function of the Kummer confluent hypergeometric function, and in the Pareto-negative binomial is a function of the Gauss hypergeometric function. Using the data set based on one-year vehicle insurance policies taken out in 2004-2005 (Jong and Heller, 2008), we conclude that our collective dependent models outperform the classical collective models Poisson-exponential and geometric-exponential in terms of the AIC and CAIC statistics.