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

2
Risk Classification
2.1 Introduction
2.1.1 Risk Classification, Regression Models and Random Effects
Motor ratemaking is essentially about classifying policies according to their risk
characteristics. The classification variables are called a priori variables (as their values can
be determined before the policyholder starts to drive). In motor insurance, they include the
age, gender and occupation of the policyholders, the type and use of their car, the place
where they reside and sometimes even the number of cars in the household, marital status,
or the colour of the vehicle.
These observable risk characteristics are typically seen as nonrandom covariates. Other risk
characteristics are unobservable and must be seen as unknown parameters or, in the vein of
credibility theory, latent variables with a common distribution. The literature about premium
rating in motor insurance comprises two mainstream approaches: (i) the first one disregards
observable covariates altogether and lumps all the individual characteristics into random
latent variables and (ii) the second one disregards random individual risk characteristics and
tries instead to catch all relevant individual variations by covariates. Chapter 1 adopted the
first approach. The present chapter combines both views, employing contemporary, advanced
data analysis.
If the data are subdivided into risk classes determined by a priori variables, actuaries work
with figures which are small in exposure and claim numbers. Therefore, simple averages will
be suspect and regression models are needed. Regression analyses the relationship between
one variable and another set of variables. This relationship is expressed as an equation that
predicts a response variable (the expected number of claims filed by a given policyholder)
from a function of explanatory variables and parameters (involving a linear combination
of these explanatory variables and parameters, called a linear predictor). The parameters
are estimated so that a measure of the goodness-of-fit is optimized (the log-likelihood, in
Actuarial Modelling of Claim Counts: Risk Classification, Credibility and Bonus-Malus Systems
S. Pitrebois and J.-F. Walhin © 2007 John Wiley & Sons, Ltd
M. Denuit, X. Maréchal,