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

208 Actuarial Modelling of Claim Counts
so that
r l = r l1 = r l2 =···=r lnl
l= 0 1s
n
∑ l
The solution to this problem is the same as above. We merely have to replace l by li .
The results are displayed in Table 4.23. The constant step between two levels of the linear
scale is equal to 99 % and the value of the mean square error is Q = 041779. The major
advantage of the linear scale is that it gives a system that is commercially more acceptable.
i=1
4.8 Change of Scale
4.8.1 Migration from One Scale to Another
Since the 90s insurance markets in the EU have been deregulated. More competition is
allowed. Two related problems arise with the deregulation of the bonus-malus systems. The
first one consists of transferring the policyholders to the new scales. The second one is more
difficult: it consists of transferring a new policyholder to the scale of the company knowing
his level in the scale of his previous insurer.
The aim of the present section is to show how to develop rules allowing the transfer
of a policyholder to a bonus-malus scale knowing his level in his previous bonus-malus
scale. The a posteriori probability density function of given the level L occupied in the
bonus-malus scale is given by
f L=l = PrL = l = f
PrL = l
∑
k
=
k l k dF
∫ ∑k k 0 l k dF
Because we want to move a policyholder from one scale to the other, we should try to put
the policyholder at a level which is as close as possible to his level in his original bonusmalus
scale. By ‘close’, we mean here having the a posteriori random effect as close as
possible.
4.8.2 Kolmogorov Distance
In addition to the total variation distance d TV used previously, we also need the Kolmogorov
distance. The Kolmogorov (or uniform) metric based on the well-known Kolmogorov-
Smirnov statistic (associated with the goodness-of-fit test with that name), is defined
as follows: The Kolmogorov distance d K between the random variables X and Y is
given by
d K X Y = sup F X t − F Y t (4.24)
t∈
Given two random variables X and Y , we have that d K X Y ≤ d TV X Y. This result is
an immediate consequence of (4.11) since F X t = PrX ∈ t +.