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

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246 Actuarial Modelling of Claim Counts Table 5.11 De Pril efficiency according to the starting level, for the −1/top, −1/ + 2 and −1/ + 3 bonus-malus scales. Initial level −1/top scale Eff DeP l 00928 Eff DeP l 01409 Eff DeP l 02804 Eff DeP l 0 02711379 03018164 0287755 02230948 1 02644811 02952273 02826536 02179881 2 02558848 02863492 02746372 02109572 3 02454854 02755109 02648401 02025042 4 02332928 02628293 02537603 01927769 5 02191762 0248209 02414355 01817001 Initial level −1/ + 2 scale Eff DeP l 00928 Eff DeP l 01409 Eff DeP l 02804 Eff DeP l 0 02168178 03422846 05689437 0262244 1 02159419 03399076 05617233 02597218 2 02156974 03369153 05488944 02558833 3 02120306 03274338 05247412 02474823 4 0202593 0311189 04963046 02351387 5 01870807 02878845 04633192 02186229 Initial level −1/ + 3 scale Eff DeP l 00928 Eff DeP l 01409 Eff DeP l 02804 Eff DeP l 0 02733728 03689638 04526559 02605588 1 02691461 0362999 04449703 02563432 2 02648276 03556598 04328504 02508412 3 02572639 03446637 0417896 02429854 4 02429893 03261114 03971923 02302266 5 02252077 03038548 03740766 02149911 5.4 Bonus Hunger and Optimal Retention 5.4.1 Correcting the Estimations for Censoring As explained in the introduction, the policyholders subject to a bonus-malus mechanism tend to self-defray minor accidents to avoid premium surcharges. This means that the number of accidents is a censored variable: the insurer only knows the number of claims filed by the insured drivers, and not the number of accidents they caused. We develop here a simple statistical model allowing for censorship in the observed claim costs (and thus also in the observed numbers of claims reported to the insurer). The claiming threshold is considered here as a random variable, specific to each policyholder and with a distribution depending on the level occupied in the bonus-malus scale at the beginning of the observation period as well as on observable characteristics. Specifically, let us consider the LogNormal model for moderate claim sizes: the claim costs are then seen as independent and identically distributed realizations of LogNormal random variables in each risk class. Now, each policyholder in this class reports an accident
Efficiency and Bonus Hunger 247 to the insurer if its cost exceeds a random threshold, assumed to be LogNormally distributed with parameters specific to the level occupied in the scale. Note that we deal here with moderate claim sizes only. The reason is that large claims are not subject to bonus hunger and are systematically reported to the insurer. The frequency of large claims will have to be added to the corrected frequency of moderate claims to get the actual number of accidents caused each year. Let l i be the level occupied by policyholder i in the bonus-malus scale at the beginning of the period, and let RL i l i ∼ N or i 2 be the random optimal retention, with a linear predictor of the form i = 0 + p∑ j x ij + fl i j=1 specific to policyholder i, where the function f· expresses the effect of occupying level l i in the bonus-malus scale. Considering the values obtained for the optimal retention in the literature (reaching a maximum somewhere in the middle of the scale, and decreasing when approaching uppermost and lowermost levels), we will use here a quadratic effect f of l i . Note that other approaches are possible (see the references in the closing section for more details). This means that policyholder i will report all the accidents with a cost larger than RL i l i , and defray himself all those with a cost less than RL i l i . At the portfolio level, the RL i l i s are assumed to be independent. Now, let CA ik be the cost of the kth accident caused by policyholder i. We assume that for each i the random variables CA i1 CA i2 are ∑ independent and identically distributed, with CA ik ∼ N or i 2 , where i = 0 + p j=1 jx ij . Moreover, the CA ik s and RL i l i are mutually independent. We consider here the explanatory variables selected in the LogNormal analysis of the censored claim costs, presented in Table 5.5. Now, denoting as c i1 c ini the costs of the n i moderate claims filed by policyholder i, the likelihood is 2 = ∏ n ∏ i f i c ik f i c ik = in i >0 k=1 where f i · denotes the probability density function of CA ik given CA ik >RL i l i . Each factor involved in the likelihood can be written is 1 √ exp (− ln c ( ) ik − i 2 ln cik − i 2cik 2 2 ) ( 1 − − ) i − √ i 2 + 2 The estimators of the parameters , , 2 , and are determined by maximizing the likelihood 2 . This basic model could be refined in different respects. Firstly, the retention limit could depend on the number of claims previously filed by the policyholder during the same year. The retention for the second claim depends on the level to which the policyholder is
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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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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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