Random Shifting and Scaling of Insurance Risks

Enkelejd Hashorva,Lanpeng Ji

doi:10.3390/risks2030277

Abstract

Random shifting typically appears in credibility models whereas random scaling is often encountered in stochastic models for claim sizes reflecting the time-value property of money. In this article we discuss some aspects of random shifting and random scaling in insurance focusing in particular on credibility models, dependence structure of claim sizes in collective risk models, and extreme value models for the joint dependence of large losses. We show that specifying certain actuarial models using random shifting or scaling has some advantages for both theoretical treatments and practical applications.

Highlights

Random shifting and random scaling in insurance applications are natural phenomena for latent unknown risk factors, time-value of money, or the need of allowing financial risks to be dependent.In this contribution, we are concerned with three principal stochastic models related to credibility theory, ruin theory, and extreme value modeling of large losses.In credibility theory (e.g., [1]) often stochastic models are defined via a conditional argument.As an illustration, consider the classical Gaussian model assuming that the conditional random variableX|Θ = θ has the normal distribution N (θ, σ 2 )
We show that modeling claim sizes by a class of Dirichlet random sequences can be done in the framework of a tractable random scale model
With motivation from credibility theory, we propose to consider a new class of multivariate distributions called LP Dirichlet distributions, which is naturally introduced by letting the parameter p in our definition above to be random

Summary

Introduction

Random shifting and random scaling in insurance applications are natural phenomena for latent unknown risk factors, time-value of money, or the need of allowing financial risks to be dependent. If the claim sizes are assumed to be dependent, particular dependence structures for finite n (like the one in Equation (5)) lead to randomly scaled independence structures; this is further illustrated below in our Theorem 1. Like m dependence or common shock models, can be introduced by this simple transformation of independent risks These are only a few possibilities that lead to tractable dependence structures with certain appeal to actuarial applications; see [1,8,9,10,11,12,13,14,15] and the references therein.

Credibility Premium in Random Shift Models

Discussions and Extensions

Conclusions

Background