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

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188 Actuarial Modelling of Claim Counts Table 4.5 Numerical characteristics for the system −1/ + 2 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 44 % 3091 % 2714% 185% 4 47 % 2414 % 2185% 171% 3 44 % 2077 % 1925% 164% 2 87 % 1429 % 1388% 153% 1 71 % 1302 % 1286% 151% 0 706% 624% 685% 142% policyholders in level 0 get some discount whereas occupancy of any level 1–5 implies some penalty. Again, the a posteriori corrections are softened when a priori risk classification is taken into account in the determination of the r l s. The comments made for the scale −1/top still apply to this bonus-malus scale. Example 4.14 (−1/+2 Scale, Portfolio B) Results are displayed in Table 4.6 which is the analogue of Table 4.4 for the bonus-malus scale −1/ + 2. The comparison with Portfolio A yields the same comments as before. Example 4.15 (−1/+3 Scale, Portfolio A) Let us now make the −1/ + 2 bonus-malus scale more severe: to this end, each claim is now penalized by 3 levels (instead of 2 in the −1/ + 2 system). The numerical results are displayed in Table 4.7. We see that less policyholders occupy level 0 (64.5 % compared with 70.6 % with the −1/+2 system), and that the upper levels are now more populated. Drivers in level 0 deserve more bonus compared to the −1/+2 system (they pay 57.8 % of the base premium compared to 62.4% in the non-segmented case, and 64.2 % compared to 68.5 % in the segmented case). Also, the maximal penalties get reduced when claims are more severely penalized. Example 4.16 (−1/+3 Scale, Portfolio B) Let us now consider Portfolio B where each claim is penalized by 3 levels (instead of 2 in the −1/ + 2 system). The numerical results Table 4.6 Numerical characteristics for the system −1/ + 2 and for Portfolio B. Level l PrL = l r l = EL = l with a priori ratemaking EL = l 5 54 % 2232% 204% 4 60 % 1713% 199% 3 58 % 1489% 196% 2 115 % 1121% 191% 1 93 % 1050% 190% 0 620% 747% 185%
Bonus-Malus Scales 189 Table 4.7 Numerical characteristics for the system −1/ + 3 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 73 % 2571 % 2308 % 17.4 % 4 59 % 2194 % 2009 % 16.7 % 3 90 % 1517 % 1452 % 15.5 % 2 73 % 1365 % 1331 % 15.2 % 1 60 % 1240 % 1230 % 15.0 % 0 645% 578% 642 % 14.1 % Table 4.8 Numerical characteristics for the system −1/ + 3 and for Portfolio B. Level l PrL = l r l = EL = l with a priori ratemaking EL = l 5 92 % 1840% 200% 4 78 % 1566% 197% 3 117 % 1174% 192% 2 95 % 1086% 190% 1 78 % 1016% 189% 0 540% 720% 184% are displayed in Table 4.8. The comparison with Portfolio A shows that the relativities are much less dispersed here. 4.5.3 Interaction between Bonus-Malus Systems and a Priori Ratemaking Since the relativities attached to the different levels are the same whatever the risk class to which the policyholders belong, those scales overpenalize a priori bad risks. Let us explain this phenomenon, put in evidence by Taylor (1997). Over time, policyholders will be distributed over the levels of the bonus-malus scale. Since their trajectory is a function of past claims history, policyholders with low a priori expected claim frequencies will tend to gravitate to the lowest levels of the scale. Conversely for individuals with high a priori expected claim frequencies. Consider for instance a policyholder with a high a priori expected claim frequency, a young male driver living in a urban area, say. This driver is expected to report many claims (this is precisely why he has been penalized a priori) and so to be transferred to the highest levels of the bonus-malus scale. On the contrary, a policyholder with a low a priori expected claim frequency, a middle-aged lady living in a rural area, say, is expected to report few claims and so to gravitate to the lowest levels of the scale. The level occupied by the policyholders in the bonus-malus scale can thus be partly explained by their observable characteristics included in the price list. It is thus fair to isolate that part of the information contained in the level occupied by the policyholder that
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Actuarial Modelling of Claim Counts
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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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3 Credibility Models for Claim Coun
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Credibility Models for Claim Counts
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8 Transient Maximum Accuracy Criter
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Transient Maximum Accuracy Criterio
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Bibliography Albrecht, P. (1983a).
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Bibliography 347 Daengdej, J., Luko
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Bibliography 349 Greenwood, M., & Y
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Bibliography 351 Norberg, R. (1976)
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Bibliography 353 Walhin, J.-F., & P
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