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

52 Actuarial Modelling of Claim Counts
are kept unchanged. Analysing insurance data, the actuary is able to draw conclusions
about the number of claims filed by policyholders subject to a specific a posteriori
ratemaking mechanism. The actuary is not able to draw any conclusions about the number
of accidents caused by these insured drivers. We will come back to this important issue in
Chapter 5.
2.1.4 Agenda
This chapter is devoted to a priori ratemaking, and focusses on claim frequencies. To make
the discussion more concrete, we analyse the statistics from a couple of motor insurance
portfolios. We first work with cross-sectional data (i.e. data gathered during one year of
observation) from a Belgian motor third party liability insurance portofolio observed during
the year 1997 (called Portfolio A, already used in Chapter 1). We also show how it is
possible to build a ratemaking on the basis of panel data. To this end, we use another Belgian
portfolio (called Portfolio B) for which the data have been collected during 3 years (from
1997 to 1999).
To fix the ideas, in Section 2.2 we present the data observed during the year 1997 and the
different explanatory variables available for Portfolio A. We give a detailed description of
all the variables and we have a first look at their influence on the risk borne by the insurer.
Then, in Section 2.3, we show how it is possible to build an a priori ratemaking thanks to
a Poisson regression. We illustrate the technique on the data from Portfolio A. Section 2.4
addresses the problem of overdispersion. A random effect is added to the covariates to
account for residual heterogeneity (vector Z i in the preceding discussion). We examine
three classical models: the Poisson-Gamma (or Negative Binomial) model, the Poisson-
Inverse Gaussian model and the Poisson-LogNormal model. These are extensions of the
corresponding models presented in Chapter 1, to incorporate exogenous information about
policyholders.
In Section 2.9, we develop ratemaking techniques using panel data. Portfolio B that has
been observed during three consecutive years is used for the numerical illustrations. GEE and
maximum likelihood are used to estimate the parameters involved in models for longitudinal
data. The final Section 2.10 offers an extensive discussion of topics not covered in this
chapter, together with appropriate references.
2.2 Descriptive Statistics for Portfolio A
2.2.1 Global Figures
The data relate to a Belgian motor third party liability insurance portfolio observed during
the year 1997. The data set (henceforth referred to as Portfolio A, for brevity) comprises
14 505 policies. The observed claim number distribution in the portfolio has been described
in Table 1.1. The observed mean claim frequency for Portfolio A is 14.6 %.
2.2.2 Available Information
The following information is available on an individual basis: in addition to the number of
claims filed by each policyholder (variable Nclaim) and the exposure-to-risk from which