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

302 Actuarial Modelling of Claim Counts
Table 8.5 Relativities computed on the basis of the transient maximum accuracy criterion for the
former compulsory Belgian bonus-malus scale, and for different initial distributions.
Level l Uniform Top Bottom Steady state
distribution ¯r l distribution ¯r l distribution¯r l distribution r l
22 2661 % 2411 % 2835 % 2715%
21 2344 % 1854 % 2552 % 2474%
20 2086 % 1640 % 2326 % 2291%
19 1874 % 1475 % 2137 % 2141%
18 1701 % 1345 % 1983 % 2014%
17 1561 % 1242 % 1856 % 1903%
16 1448 % 1157 % 1750 % 1803%
15 1356 % 1088 % 1660 % 1714%
14 1281 % 1029 % 1589 % 1631%
13 1219% 980 % 1537 % 1549%
12 1166% 937 % 1483 % 1472%
11 1118% 898 % 1426 % 1402%
10 1075% 864 % 1371 % 1340%
9 1038% 835 % 1337 % 1255%
8 1001% 808 % 1287 % 1173%
7 963% 781 % 1228 % 1114%
6 927% 753 % 1172 % 1067%
5 892% 727 % 1120 % 1028%
4 818% 689% 997% 826%
3 780% 658% 941% 802%
2 745% 630% 893% 778%
1 713% 602% 850% 755%
0 458% 401% 525% 451%
portfolios, respectively) are summarized in Table 8.6. Table 8.7 displays the bonus-malus
relativities obtained with these different age structures and a uniform initial distribution of
the policyholders in the bonus-malus scale. We see that the older the portfolio, the closer
the ¯r l s to the corresponding r l s.
8.3.2 Linear Scales
Sometimes, it is desirable to have the same relative penalty associated with each level. To this
end, the actuary can linearize the ¯r l s, as suggested by Gilde & Sundt (1989). The optimal
linear relativity ¯r
l
lin = +l, l = 0 1s, in the transient case is thus the solution of the
minimization of
E [ − r A
L A
2] = E [ − − L A 2]
It is easy to check that the solution of this optimization problem is
= CL A
VL A
and = E − CL A
EL
VL A A