3.1 INTRODUCTION

In this chapter, we introduce collective risk models. Just as in Chapter 2, we calculate the distribution of the total claim amount in a certain time period, but now we regard the portfolio as a collective that produces a claim at random points in time. We write

(3.1)

where N denotes the number of claims and X _i is the ith claim, and by convention, we take S = 0 if N = 0. So, the terms of S in (3.1) correspond to actual claims; in (2.25), there are many terms equal to zero, corresponding to the policies which do not produce a claim. The number of claims N is a random variable, and we assume that the individual claims X _i are independent and identically distributed. We also assume that N and X _i are independent. In the special case that N is Poisson distributed, S has a compound Poisson distribution. If N is (negative) binomial distributed, then S has a compound ( negative) binomial distribution.

In collective models, some policy information is ignored. If a portfolio contains only one policy that could generate a high claim, this term will appear at most once in the individual model (2.25). In the collective model (3.1), however, it could occur several times. Moreover, in collective models we require the claim number N and the claim amounts X _i to be...