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

Multi-Event Systems 269
Table 6.2 Evolution of relativities and pure premiums (taking the average cost of a claim with
material damage only (mat) as the monetary unit, and assuming that claims with bodily injuries (bod)
are on average ten times more expensive) if no claim has been reported.
Time Good driver Bad driver
Relativity
mat
Relativity
bod
Pure
premium
Relativity
mat
Relativity
bod
Pure
premium
1 92.9 % 94.2 % 16.8 % 81.5 % 84.6 % 45.1 %
2 86.7 % 89.1 % 15.8 % 68.7 % 73.9 % 38.8 %
3 81.3 % 84.6 % 15.0 % 59.3 % 66.0 % 34.1 %
4 76.6 % 80.6 % 14.2 % 52.1 % 59.9 % 30.6 %
5 72.3 % 77.1 % 13.5 % 46.5 % 55.1 % 27.7 %
6 68.5 % 73.9 % 12.8 % 41.9 % 51.1 % 25.4 %
7 65.0 % 71.0 % 12.3 % 38.2 % 47.8 % 23.5 %
8 61.9 % 68.4 % 11.8 % 35.0 % 45.0 % 21.9 %
9 59.1 % 66.0 % 11.3 % 32.3 % 42.5 % 20.5 %
10 56.5 % 63.8 % 10.9 % 30.0 % 40.4 % 19.3 %
Table 6.2 displays the results for the case where no claim is reported for 10 years. The
first column gives the coefficient to be applied on mat
it
, the second column gives the
coefficient to be applied on bod
it
and the third column gives the premium to be charged
if the average cost of a material damage claim is 1 and the average cost of a bodily
injury claim is 10 (the monetary unit is thus the average cost of a claim with material
damage only, and claims with bodily injuries are assumed to be on average ten times
more expensive than claims with material damage only). The first three columns are for
the good driver and the next three are for the bad driver. We see that the correction
coefficients are always smaller for the claims with material damage only than for the claims
with bodily injuries. This is due to the fact that the former claims occur more frequently
than the latter ones, so that not reporting any claim with material damage only entails
more premium discount. As explained previously, the discounts are always larger for a
bad driver than for a good one. However, the premiums always stay higher for the bad
drivers.
Table 6.3 considers the case where the policyholder reported a single claim with
material damage only, for 10 years. Finally, Table 6.4 considers the case where the
policyholder reported a single claim with bodily injuries, for 10 years. Comparing these
two tables, we see that the premium amount is larger if a claim with bodily injuries has
been reported, compared to the case where a claim with material damage only has been
reported. The cost of the claim is thus taken into account in the premium correction.
The correction coefficients are always larger for the good driver than for the bad one, as
explained previously. It is also interesting to note that reporting a claim of one type always
increases the probability of reporting a claim of the other type. There is thus a double
effect when updating the premium: the frequency of claims of the same type as the one
that has been reported is increased, but the frequency of claims of the other type gets
inflated, too.