Large Deviations for a Class of Multivariate Heavy-Tailed Risk Processes Used in Insurance and Finance

Miriam Hägele,Jaakko Lehtomaa

doi:10.3390/jrfm14050202

Miriam Hägele, Jaakko Lehtomaa

Open Access

https://doi.org/10.3390/jrfm14050202

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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.

Highlights

We examine the probability of the asymptotic event that the random walk exceeds a threshold in a selected norm in order to prove a large deviations theorem
The auxiliary results study the projection of the random walk to a one-dimensional setting and its asymptotics
We look at the quota share reinsurance from two different perspectives

Summary

Introduction

Applications in finance and insurance require multivariate models with heavy-tailed distributions to accurately describe multivariate risks This includes understanding the possible dependence types of large observations. We concentrate on the less studied situation where the large observations can be found from any direction and where the tails are not as heavy as regularly varying tails Such situations appear naturally in the case of financial returns of portfolios since the tails are often observed to have a lognormal type distribution Hardy (2001); Jensen and Maheu (2018); Tegnér and Poulsen (2018) and the observations can be present in all orthants Lehtomaa and Resnick (2020). D since typically the insurance company keeps some share for every line of business Under this assumption, Matrix Q is invertible

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