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

Transient Maximum Accuracy Criterion 301
Table 8.4 Evolution of the transient relativities for the bottom initial distribution for the former
compulsory Belgian bonus-malus scale.
Level l
r 0
l
r 10
l
r 20
l
r 30
l
r 40
l
r 50
l
r l
22 00 % 3049 % 2902 % 2818 % 2774 % 2750 % 2715%
21 00 % 2681 % 2629 % 2566 % 2529 % 2508 % 2474%
20 00 % 2429 % 2402 % 2361 % 2334 % 2318 % 2291%
19 00 % 2075 % 2240 % 2215 % 2189 % 2172 % 2141%
18 00 % 1901 % 2071 % 2066 % 2050 % 2038 % 2014%
17 00 % 1776 % 1929 % 1933 % 1926 % 1919 % 1903%
16 00 % 1813 % 1841 % 1827 % 1819 % 1814 % 1803%
15 00 % 1866 % 1788 % 1745 % 1729 % 1722 % 1714%
14 00 % 1408 % 1607 % 1633 % 1637 % 1636 % 1631%
13 00 % 1482 % 1566 % 1562 % 1557 % 1554 % 1549%
12 00 % 1341 % 1513 % 1499 % 1486 % 1480 % 1472%
11 00 % 1426 % 1493 % 1455 % 1430 % 1417 % 1402%
10 00 % 1486 % 1474 % 1419 % 1384 % 1364 % 1340%
9 00 % 1153 % 1293 % 1285 % 1272 % 1264 % 1255%
8 00 % 1223 % 1263 % 1231 % 1206 % 1192 % 1173%
7 00 % 1255 % 1203 % 1185 % 1158 % 1140 % 1114%
6 00 % 1268 % 1181 % 1152 % 1120 % 1099 % 1067%
5 00 % 1270 % 1159 % 1122 % 1087 % 1064 % 1028%
4 00% 994% 920% 884% 861% 847% 826%
3 00% 981% 901% 863% 839% 824% 802%
2 00% 966% 848% 838% 816% 801% 778%
1 00% 950% 827% 816% 794% 779% 755%
0 00% 566% 498% 475% 465% 460% 451%
that even if the ultimate distribution of the policyholders in the scale is the same whatever
the initial distributions, the transient relativities are affected by these distributions. Starting
with a uniform distribution of the policyholders in the scale (Table 8.2), the r n
l
s increase
to the r l s for levels 1 to 22, and they decrease to the limit for l = 0. The same phenomenon
arises when starting with the top distribution (Table 8.3), but the difference between the
r 10
l
s and the asymptotic r l s is now larger. On the contrary, if the bottom distribution is used
(Table 8.4) then the r n
l
s decrease to their limit r l .
The relativities ¯r l computed using (8.6) are given in Table 8.5 according to the initial
distribution of the policyholders in the scale and for a uniform distribution of age of policy
over 50 years, that is, a n = 1/50 for n = 150. For the sake of completeness, the steady
state relativities are displayed in the last column. When a uniform initial distribution of the
policyholders is used, the relativities based on the transient maximum accuracy criterion are
smaller than the steady state relativities, except for level 0. This is the case for all levels if
the top distribution is assumed. On the contrary, if the bottom distribution is used then the
¯r l s are larger than the corresponding r l s.
In order to figure out the impact of the maturity of the portfolio on the relativities,
we considered two alternative age structures to the uniform age of policy distribution
used so far. The three distributions (henceforth referred to as mature, young and old