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

Multi-Event Systems 275
P 2 = exp−q 2 q 1 exp−q 1
P 3 = q 2 2
exp−q
2
2 exp−q 1
and i represents the sum of all the elements in row i. Specifically,
1 = P 0 + P 1 + P 2 + P 3
2 = P 0 + P 1 + P 2
3 = P 0 + P 1
4 = P 0
Computation of the Relativities
Portfolio A Table 6.5 gives for each of the six levels of the bonus-malus scale the proportion
of the portfolio in that level (column 2) and the relativity attached to that level (column 3)
for the system −1/ + 2/ + 3. About 70 % of the portfolio is in level 0 and enjoys a discount
of about 30 %. The rest of the portfolio is spread out among levels 1–5. The r l s range from
68.0 % to 262.6 %.
In order to compare the results with those of traditional bonus-malus scales, we have
also recalled the results given by the three other scales already considered in the previous
chapters. For all of them, each claim-free year is rewarded by one level down in the scale.
The first bonus-malus system penalizes each claim (with or without bodily injuries) by
two levels up in the scale (−1/ + 2 bonus-malus scale), the second one by three levels up
(−1/ + 3 bonus-malus scale) and the third one sends the policyholder to the maximal level
after one claim (−1/top bonus-malus scale).
Clearly, the more the claims are penalized, the more the policyholders occupying the
lowest levels are awarded discounts, and the less the policyholders in the upper part of the
scale are penalized. The scale −1/ + 2/ + 3 is closer to the scale −1/ + 2. This is because
the majority of the claims only induce material damage. Nevertheless, r 5 is reduced from
271.4 % to 262.6 % when the claims with bodily injuries are more severely penalized.
Table 6.5 Results for the bonus-malus systems −1/ + 2/ + 3, −1/ + 2, −1/ + 3 and −1/top for
Portfolio A.
Level l −1/ + 2/ + 3 −1/ + 2 −1/ + 3 −1/top
l r l l r l l r l l r l
5 47 % 2626% 44 % 2714% 73 % 2308% 128 % 1812%
4 47 % 2158% 47 % 2185% 59 % 2009% 97 % 1599%
3 49 % 1821% 44 % 1925% 90 % 1452% 77 % 1439%
2 86 % 1380% 87 % 1388% 73 % 1331% 62 % 1313%
1 71 % 1278% 71 % 1286% 60 % 1230% 52 % 1209%
0 700% 680% 706% 685% 645% 642% 585% 612%