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

170 Actuarial Modelling of Claim Counts
future (the level for year t + 1) depends on the present (the level for year t and the number
of accidents reported during year t) and not on the past (the complete claim history and
the levels occupied during years 1 2t− 1). Sometimes, fictitious levels have to be
introduced in order to meet this memoryless property. Indeed, in some bonus-malus systems,
policyholders occupying high levels are sent to the starting class after a few years without
claims. This issue will be addressed in Section 4.7.
4.2.3 Trajectory
New drivers start in level l 0 of the scale. Note that experienced drivers arriving in the
portfolio are not necessarily placed in level l 0 , but in a level corresponding to their claim
history or to the level occupied in the bonus-malus scale used by a competitor. This problem
will be dealt with in Section 4.8.
The trajectory of the policyholder in the bonus-malus scale is modelled by a sequence
L 1 L 2 of random variables valued in 0 1s, such that L k is the level occupied
during the k + 1th year, i.e. during the time interval k k + 1. Since movements in the
scale occur once a year (at policy anniversary), the policyholder occupies level L k from
time k until time k + 1. Once the number N k of claims reported by the policyholder during
k − 1k is known, this information is used to reevaluate the position of the driver in the
scale. We supplement the sequence of the L k s with L 0 = l 0 .
The L k s obviously depend on the past numbers of claims N 1 N 2 N k reported by
the policyholder. If we denote as ‘pen’ the penalty induced by each claim (expressed as a
number of levels), then the L k s obey the recursion
{ maxLk−1 − 1 0 if N
L k =
k = 0
minL k−1 + N k × pens if N k ≥ 1
{
= max min { L k−1 − 1 − I k + N k × pens } }
0
where
{ 1ifNk ≥ 1
I k =
0 otherwise
indicates whether at least one claim has been reported in year k. This is an example of a
stochastic recursive equation. This representation of the L k s clearly shows that the future
trajectory of the policyholder in the scale is independent of the levels occupied in the past,
provided that the present level is given. This conditional independence property is at the
heart of Markov models.
The stochastic recursive equations given above assume that the bonus is lost in case at
least one claim is filed with the company. In some cases (like the former compulsory Belgian
bonus-malus scale), the bonus is granted in any case. The L k s then obey the recursion
{
L k = max min { L k−1 − 1 + N k × pens } }
0
This means that the first claim is penalized by pen−1 levels, and the subsequent ones by
pen levels.