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

Bonus-Malus Scales 187
Table 4.3 Numerical characteristics for the system −1/top and for Portfolio A.
Level l PrL = l r l = EL = l
without a priori
ratemaking
r l = EL = l
with a priori
ratemaking
EL = l
5 128 % 1973 % 1812% 163%
4 97 % 1709 % 1599% 158%
3 77 % 1507 % 1439% 155%
2 62 % 1348 % 1313% 152%
1 52 % 1220 % 1209% 150%
0 585% 547% 612% 141%
a priori risk classification. Similarly, the relativity attached to the highest level of 197.3 %
gets reduced to 181.2 %. The severity of the a posteriori corrections is thus weaker once the
a priori ratemaking is taken into account in the determination of the r l s. The last column
of Table 4.3 indicates the extent to which a priori and a posteriori ratemakings interact.
The numbers in this column are computed as (4.15). The average a priori expected claim
frequency clearly increases with the level l occupied by the policyholder.
Example 4.12 (−1/Top Scale, Portfolio B) The results for the bonus-malus scale −1/top
are displayed in Table 4.4. We only give the relativities computed by taking into account
the a priori risk classification that is taken from Table 2.16 with ̂ = 0677. We see that
in Portfolio B, only 46.6 % of the policyholders occupy level 0. The relativities are now
less dispersed, ranging from 70.6 % to 146.9 % (instead of 61.2 % to 181.2 %). Again, the
last column indicates that a priori risk classification and a posteriori premium corrections
interact.
Example 4.13 (−1/+2 Scale, Portfolio A) Results are displayed in Table 4.5 which is
the analogue of Table 4.3 for the bonus-malus scale −1/ + 2. The bonus-malus system is
perhaps too soft since the vast majority of the portfolio (about 71 %) clusters in the super
bonus level 0. The higher levels are occupied by a very small minority of drivers. Such a
system does not really discriminate between good and bad drivers. Consequently, only those
Table 4.4 Numerical characteristics for the system −1/top and for portfolio B.
Level l PrL = l r l = EL = l EL = l
with a priori
ratemaking
5 163 % 1469% 195%
4 125 % 1293% 193%
3 99 % 1172% 191%
2 80 % 1080% 190%
1 66 % 1007% 189%
0 466% 706% 184%