Actuarial Modelling of Claim Counts Risk Classification, Credibility ...

126 Actuarial Modelling of Claim Counts
reported during the first year are displayed in Table 3.1. This elementary example contains
all the ingredients of experience rating.
3.2.2 Credibility Models Incorporating a Priori Risk Classification
This chapter aims to design merit rating plans in accordance with the a priori ratemaking
structure of the insurance company. Specifically, let us consider a portfolio with n policies,
each one observed during T i periods. Let N it (with mean EN it = it ) be the number of claims
reported by policyholder i during year t, i.e. during the period t − 1t, i = 1 2n,
t = 1 2T i . By convention, time 0 corresponds to the issuance of the policy. We thus
face a nested structure: each policyholder generates a sequence N i = N i1 N i2 N iTi T of
claim numbers. It is reasonable to assume independence between the series N 1 N 2 N n
(at least in motor third party liability insurance), but we expect some positive dependence
inside the N i s.
The ith policy of the portfolio, i = 1 2n, is represented by a sequence
i N i1 N i2 N i3 . At the portfolio level, the sequences i N i1 N i2 N i3 are
assumed to be independent for i = 1 2n. The risk parameter i represents the
risk proneness of policyholder i, i.e. unknown risk characteristics of the policyholder
having a significant impact on the occurrence of claims; it is regarded as a random
variable. Given i = , the random variables N i1 N i2 N i3 are assumed to be independent.
Unconditionally, these random variables are dependent since their behaviour depends on the
common i .
The very basic tenets of a credibility model for claim counts are as follows:
(i) a conditional distribution for the number of claims, that is, for the N it s given i = ;
(ii) a distribution function F for the risk parameters 1 n to describe how the
conditional distributions vary accross the portfolio;
(iii) a loss function whose expectation has to be minimized in order to find the optimal
experience premium.
Let us briefly comment on these three aspects. In motor third party liability insurance
portfolios, the Poisson distribution often provides a good description of the number of claims
incurred by an individual policyholder during a given reference period (one year, say): the
set of all Poisson assumptions should at least provide (locally in time) a good approximation
to the accident generating mechanism. Given i = , the annual numbers of claims N it for
policyholder i are assumed to be independent and to conform to a Poisson distribution with
mean it . As in Chapter 2, it is a known function of the exposure-to-risk and possibly other
covariates.
Let us now consider the choice of F . Traditionally, actuaries have assumed that the
distribution of values among all drivers is well approximated by a two-parameter Gamma
distribution. The resulting probability distribution for the number of claims is Negative
Binomial. Other classical choices for F include the Inverse-Gaussian and the LogNormal
distributions, as explained in Chapters 1–2.
Regarding (iii), quadratic and exponential loss functions will be considered in this chapter.
This leads to the following model.