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

288 Actuarial Modelling of Claim Counts
7.4.4 Per Claim Deductible
In this case, there is an analytical solution when the claims are Negative Exponentially
distributed: the deductible d l involved in (7.11) is simply given by ln r l times the
expected claim cost. No explicit solution is available when claim amounts are LogNormally
distributed, and numerical procedures have to be used in this case to find the
deductibles d l .
Table 7.3 displays the results obtained for a deductible per claim for the scale −1/top
and Table 7.4 for the scale −1/ + 2, when the premium paid by the policyholder is
held constant whatever the claim history. As was the case for the annual deductible,
the second column gives the relativities associated with each level of the scale in the
case of a classical bonus-malus system. The third column gives the relative premium
in the case of a scale with the deductible system. The fourth column shows the
deductible to be applied if the loss amounts are Negative Exponentially distributed and
the last column shows the deductible to be applied if the loss amounts are LogNormally
distributed.
Again, the amounts of deductible displayed in the last two columns are very high compared
to the annual premium, especially for the scale −1/ + 2. This results from the severe r l s
listed in column 2. In order to get acceptable amounts of deductible, keeping the financial
stability of the system, in the next section we will combine softened penalties in the malus
zone with moderate deductibles.
Table 7.3 Results for a deductible per claim varying according to the level occupied in the malus
zone for the scale −1/top.
Level l r l r l with deductible d l if C 1 ∼ xp d l if C 1 ∼ or
5 1973 % 100 % 14 041 16 254
4 1709 % 100 % 11 073 12 191
3 1507 % 100 % 8474 8941
2 1348 % 100 % 6170 6288
1 1220 % 100 % 4108 4080
0 547 % 54.7 % 0 0
Table 7.4 Results for a deductible per claim varying according to the level occupied in the malus
zone for the scale −1/ + 2.
Level l r l r l with deductible d l if C 1 ∼ xp d l if C 1 ∼ or
5 3091 % 100 % 23 317 31 592
4 2414 % 100 % 18 209 22 633
3 2077 % 100 % 15 101 17 803
2 1429 % 100 % 7376 7651
1 1302 % 100 % 5451 5502
0 624 % 62.4 % 0 0