The discrete-time risk model with correlated classes of business

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Abstract

The discrete-time risk model with correlated classes of business is examined. Two different relations of dependence are considered. The impact of the dependence relation on the finite-time ruin probabilities and on the adjustment coefficient is also studied. Numerical examples are presented.

Introduction

In most actuarial literature related to risk theory, the assumption of independence between classes of business in an insurance book of business is made. In practice, however, there are situations in which this assumption is not verified. In the case of a catastrophe such as an earthquake for example, the damages covered by homeowners and private passenger automobile insurance cannot be considered independent. Papers that treat of a relation of dependence between classes of business include Ambagaspitiya, 1998, Cummins and Wiltbank, 1983, Wang, 1998.

In the present paper, we study the probability of ruin in the discrete-time risk model proposed by Bühlmann (1970) and also presented in Bowers et al., 1997, Klugman et al., 1998, Rolski et al., 1999. We first give a brief description of the discrete-time model and we define the probability of ruin over finite and infinite-time within this model. Then, we use a Poisson common shock model and a negative binomial component model, proposed by Wang (1998), to introduce a relation of dependence between classes of business. We present numerical examples to illustrate the impact of the introduction of a relation of dependence on the probability of ruin. We also examine its influence on the adjustment coefficient.

Section snippets

The discrete time model

Assume the discrete-time process {Un,n=0,1,2,} where Un is the surplus for a book of business of an insurer at time n(n=0,1,2,) which is defined as Un=u+cn-Sn,where u is the initial surplus, c the premium income received during each period and Sn the total claim amounts over the first n periods. It is also assumed that Sn=W1+W2++Wn,where Wi represents the total claim amounts for the book of business in the period i and {Wi,i=1,2,} is a sequence of independent and

Numerical examples

We study the impact on the probability of ruin of a relation of dependence between two classes of business of an insurance book of business. In a first example, we consider the aggregation of the classes of business via a common shock model and in a second one, the aggregation is made via the Negative Binomial model with common component. We compare the ruin probability ψ(u,1,20) obtained for different relations of dependence between the classes of business. The differences result from the

Dependence and the adjustment coefficient

The purpose of this section is to measure the impact of the dependence on the infinite-time ruin probability through the adjustment coefficient.

Acknowledgements

This research was funded by individual operating grants from the Natural Sciences and Engineering Research Council of Canada and by a joint grant from the Chaire en Assurance L’Industrielle-Alliance (Université Laval).

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