Large deviations for a class of multivariate heavy-tailed risk processes used in insurance and finance

TL;DR

This paper develops a large deviations framework for multivariate heavy-tailed risk vectors, providing asymptotic tail event approximations and an optimization method for risk sharing schemes in finance and insurance.

Contribution

It introduces a non-parametric approach to model dependence in heavy-tailed multivariate risks and derives a large deviations principle under minimal assumptions.

Findings

Asymptotic tail event approximations for multivariate heavy-tailed vectors.

A large deviations principle for sums of such vectors.

An optimization method for risk sharing schemes.

Abstract

Modern risk modelling approaches deal with vectors of multiple components. The components could be, for example, returns of financial instruments or losses within an insurance portfolio concerning different lines of business. One of the main problems is to decide if there is any type of dependence between the components of the vector and, if so, what type of dependence structure should be used for accurate modelling. We study a class of heavy-tailed multivariate random vectors under a non-parametric shape constraint on the tail decay rate. This class contains, for instance, elliptical distributions whose tail is in the intermediate heavy-tailed regime, which includes Weibull and lognormal type tails. The study derives asymptotic approximations for tail events of random walks. Consequently, a full large deviations principle is obtained under, essentially, minimal assumptions. As an application, an optimisation method for a large class of Quota Share (QS) risk sharing schemes used in insurance and finance is obtained.