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

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318 Actuarial Modelling of Claim Counts Table 8.16 Relativities computed on the basis of the transient maximum accuracy criterion (influence of the maturity of the portfolio). Level l Mature Young Old Steady state portfolio ¯r l portfolio ¯r l portfolio ¯r l distribution r l 5 2326 % 2186 % 2459 % 2714% 4 1643 % 1440 % 1846 % 2185% 3 1318 % 1164 % 1521 % 1925% 2 1145 % 1066 % 1227 % 1388% 1 994% 938 % 1070 % 1286% 0 710% 738% 693% 685% The influence of the maturity of the portfolio is illustrated in Table 8.16. We notice that the relativities are more severe when the age of the portfolio is greater. Choice of the Initial Level We have computed the efficiency with different starting levels. Table 8.17 indicates that level 2isthe ¯Q-optimal level ( ¯Q-efficiency of 1.2330). The evolution of the expected financial income when using an initial level l = 2 is as follows: I 0 = 1406% I 1 = 1413% I 2 = 1019% I 3 = 1009% I 4 = 1044% I 5 = 1000% I 6 = 1000% I 7 = 1009% I 8 = 999% Table 8.17 Choice of the initial class for the −1/ + 2 scale. Starting level Level of the resulting scale Efficiency ē 5 4 3 2 1 0 5 2071 % 1231 % 1074% 952% 839% 634% 11532 4 2167 % 1774 % 1118 % 1002% 878% 651% 11942 3 2313 % 1851 % 1582 % 1059% 941% 675% 12187 2 2582 % 1969 % 1675 % 1349 % 1009% 711% 12330 1 2675 % 2102 % 1777 % 1404 % 1276% 749% 12321 0 2749 % 2140 % 1847 % 1447 % 1317% 759% 12251
Transient Maximum Accuracy Criterion 319 8.6 Super Bonus Level 8.6.1 Mechanism As explained above, the companies operating in Belgium continue to use the former compulsory bonus-malus scale, despite the deregulation of the a posteriori corrections. Slight modifications have nevertheless been brought to the system. For marketing purposes, and to keep the best drivers in the portfolio, many insurance companies operating in Belgium have allowed the insured drivers reaching level 0 to ‘claim for free’: whatever the number of accidents reported to the company, they stay in level 0. This makes the stationary distribution degenerate: all its probability mass is concentrated at 0. Therefore, computations must be based on the transient distribution. The transition probabilities for the Belgian bonus-malus scale with a super bonus level 0 are the same as before, except for the line corresponding to level 0, which is replaced with a line of 0s and a single 1 on the diagonal. 8.6.2 Initial Distributions We use the following initial distributions: the uniform distribution (1/23 of the portfolio in each level), the top distribution (78 % of the policyholders are concentrated in level 22 and the remaining 22 % are spread over levels 0 to 21) and the steady-state distribution displayed in Table 8.1. Starting with the stationary distribution gives an idea of the influence of the introduction of a super bonus level in an existing bonus-malus scale. 8.6.3 Transient Relativities The results for the uniform initial distribution are given in Tables 8.18 (transient distributions) and 8.19 (corresponding relativities). We note that after 50 years, the majority of the policyholders (76.0 %) are in the super bonus level. Most of the other policyholders are in the upper levels. They correspond to the very bad drivers. Concerning the transient relativities, we see that the super bonus level has a greater relativity than level 1 during the first 20 years. This side-effect of the introduction of a super bonus level is undesirable. Table 8.20 presents the transient relativities for the top initial distribution. The relativities also show particular patterns as policyholders principally move by clusters in the scale. For example, the relativities of levels 0 to 2 are not ordered. The transient relativities for the steady-state initial distribution are given in Table 8.21. Once again, the transient relativities are not always in ascending order. Table 8.22 shows the ¯r l s for the three initial distributions. The results for the steady-state initial distribution are not in ascending order. Indeed, the relativity associated with the super bonus level is larger than the relativities associated with levels 1 to 5. Therefore, we see that the introduction of a super bonus level in an actual scale leads to some undesirable side-effects: concentration of the majority of the policyholders in the super bonus level after a few years and optimal relativities no longer ordered for the lowest levels (these side-effects could be overcome with the introduction of a linear scale).
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Contents Foreword Preface Notation
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Contents vii 2.4 Overdispersion 79
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Contents ix 4.2.3 Trajectory 170 4.
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Contents xi 7.4 Numerical Illustrat
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Part I Modelling Claim Counts Actua
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Page 375 and 376: Bibliography Albrecht, P. (1983a).
Page 377 and 378: Bibliography 347 Daengdej, J., Luko
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Page 381 and 382: Bibliography 351 Norberg, R. (1976)
Page 383: Bibliography 353 Walhin, J.-F., & P
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