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

56 Actuarial Modelling of Claim Counts
The maximum likelihood estimator of j is then obtained from
̂j =
=
∑
iAgei=j k i
∑
iAgei=j d i
# of claims filed by policyholders in age category j
total exposure-to-risk (in year) for age category j
Of course, the reliability of ̂ j depends on the magnitude of the exposure-to-risk appearing
in the denominator.
We see in Figure 2.2 that the observed annual claim frequency decreases with age. The
young drivers are riskier as their observed annual claim frequency is 21.3 %. Old drivers are
safer with an observed annual claim frequency of 10.8 %. We notice that the policyholders
aged between 25 and 30, with an observed annual claim frequency of 15.5 %, tend to
report more claims than the policyholders aged between 31 and 60 (observed annual claim
frequency of 12.3 %).
The analysis conducted in this paragraph is often referred to as a one-way analysis: the
effect of Age on claim frequencies is studied without taking account of the effect of other
variables. The major flaw with one-way analyses is that they can be distorted by correlations.
For instance, one can imagine that the majority of young policyholders split the payment
of the insurance premiums (for budget reasons). If more claims are filed by young drivers,
a one-way analysis of Split may show higher claim frequencies for drivers having split
their premium payment. However, this may result from the fact that such drivers are in
general the high-risk young policyholders. Premium differentials based on one-way analyses
of Split and Age would double-count the effect of Age. Multivariate methods (such as the
Poisson regression approach discussed below) adjust for correlations between explanatory
variables. The correlations existing between explanatory variables explain why the policies
are not uniformly distributed over risk classes but cluster in some specified highly populated
classes.
Gender
It is common to include the gender of the main driver in the actuarial ratemaking. Note
however that some states have banned the use of this rating factor (as well as of age, for
instance). The reason is that age and gender are out of the policyholders’ control, in contrast
to many other covariates (like the power of the car, or the driving area). The latter may thus
be freely used for ratemaking purposes, but some limitations are needed for the former.
In Portfolio A, there are 9358 male policyholders (representing 64.7 % of the portfolio)
and 5147 female policyholders (representing 35.3 % of the portfolio). Figure 2.3 suggests a
higher annual claim frequency for males (observed annual claim frequency of 15.2 %) than
for females (observed annual claim frequency of 13.6 %).
District
Figure 2.4 gives the distribution of the policyholders according to the district where they
live. We see that 8664 policyholders (representing 59.8 % of the portfolio) live in an urban
area and 5841 policyholders (representing 40.2 % of the portfolio) live in a rural one. The
urban policyholders have a larger observed annual claim frequency (15.7 %) than the rural
ones (13.0 %).