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

Risk Classification 53
these claims originate (i.e. the number of days the policy has been in force during 1997,
variable Expo), we know
Age : Policyholder’s age (four categories: 1 = ‘between 18 and 24’, 2 = ‘between 25 and
30’, 3 = ‘between 31 and 60’, 4 = ‘more than 60’)
Gender : Policyholder’s gender (two categories: 1 = ‘woman’, 2 = ‘man’)
District : kind of district where the policyholder lives (two categories: 1 = ‘urban’, 2 =
‘rural’)
Use : Use of the car (two categories: 1 = ‘private use, i.e. leisure and commuting’, and 2 =
‘professional use’)
Split : premium split (two categories: 1 = ‘premium paid once a year’ and 2 = ‘premium
split up’).
In practice, insurers have at their disposal much more information about their
policyholders. Here, we focus on these few explanatory variables for pedagogical purposes,
to ease the exposition of ideas.
We see that all the explanatory variables listed above are categorical, i.e. they can be
used to partition the portfolio into homogeneous classes with respect to the variables. Such
explanatory variables are called factors, each factor having a number of levels. In practice,
there are also continuous covariates. We explain in the last section of this chapter how to
deal with such explanatory variables.
2.2.3 Exposure-to-Risk
The majority of policies are in force during the whole year. However, in some cases, the
observation period does not last for the entire year. This is the case, for instance, for new
policyholders entering the portfolio during the observation period, and in case of policy
cancellations. It is also common in practice to start a new period if some changes occur
in the observable characteristics of the policies (for instance, the policyholder moves from
a rural to an urban area and the company uses the rating variable District). The policy
is then represented as two different lines in the data base, and observations are recorded
separately for the two periods (the policy number allows the actuary to track these changes).
Note that independence is lost in this case, and allowance for panel data is preferable (as
in Section 2.9). This variety of situations is taken into account in the Poisson process, by
multiplying the annual expected claim frequency by the length of the observation period, as
explained in Chapter 1.
In Portfolio A, the average coverage period is 298.98 days. Figure 2.1 gives an idea of the
distribution of the exposure-to-risk in the portfolio. About 65 % of the policies have been
observed during the whole year 1997. Considering the distribution of the exposure-to-risk,
we see that policy issuances and lapses are randomly spread over the year. The distribution
of the policies in force during less than one year is roughly uniform over [0,365].
It is worth mentioning that the policies that are just issued often differ from those in the
portfolio. This is why it may be preferable to conduct a separate analysis for this type of
policy.