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

92 Actuarial Modelling of Claim Counts
starts as soon as some policy characteristics are modified (think for instance of a policyholder
house moving for a company using postcode as rating factor, a policyholder’s wedding for
a company using marital status, or simply the policyholder buying a new car). Moreover,
in the year the policy is issued and in the one it is possibly cancelled the length of the
observation period is generally less than unity.
We 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 , but this assumption is very questionable inside the N i s. Regarding a
priori ratemaking, the dependence between the components of each N i is a nuisance (in the
statistical sense). This means that, at this stage, we are not interested in accurately modelling
this dependence, but we must take it into account when estimating the regression coefficients.
The idea now is to incorporate in the N it s exogenous information (like age, gender, power
of the car, and so on) summarized in the vectors x it ; to this end, we resort to a regression
model for longitudinal data.
The distributional assumption for the random component of the regression model has to
account for the non-negativity of the data, as well as their integer values. We begin with
Poisson regression and assume that the N it s conform to the Poisson distribution with a
mean that can be written as an exponential function of a linear combination 0 + ∑ p
j=1 jx itj
of the explanatory variables x it , with unknown regression coefficients to be estimated
from the data. Despite its prevalence as a starting point in the analysis of count data, the
Poisson specification is often inappropriate because of unobserved heterogeneity and failure
of the independence assumption if the data consist in repeated observations on the same
policyholders. A convenient way to take this phenomenon into account is to introduce a
random effect into the model.
Remark 2.5 Before embarking on a panel analysis pooling together the observations
relating to several years, it is interesting to first work year by year to assess the stability of
the effect of each rating variable on the annual expected claim frequency. Specifically, the
vector of the regression coefficients is estimated on the basis of each calendar year and the
components are checked for their stability over time. Only stable coefficients are interesting
for the purpose of ratemaking. Rating factors with unstable regression coefficients should be
excluded from the risk classification scheme. In some cases, a time trend is visible for some
estimated regression coefficients (this is typically true for the intercept 0 ). A time effect
can then be incorporated into the model to account for coefficients with trends.
2.9.2 Descriptive Statistics for Portfolio B
The analysis in this section is based on an insurance portfolio containing 20 354 policies and
observed during 3 years (from 1997 to 1999). We have 45 350 observations available as not
all the policies have been in force for 3 years. For each policy and for each year, we know
the exposure-to-risk, the number of claims filed and some other explanatory variables:
Gender: Policyholder’s gender (male–female)
Age: Policyholder’s age (18–22, 23–30 and over 30)
Power: The power of the vehicle (less than 66 kW, 66–110 kW and more than 110 kW)
Size: The size of the city where the policyholder lives (large, middle or small), and
Colour: The colour of the vehicle (red or other).