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

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288 Actuarial Modelling of Claim Counts 7.4.4 Per Claim Deductible In this case, there is an analytical solution when the claims are Negative Exponentially distributed: the deductible d l involved in (7.11) is simply given by ln r l times the expected claim cost. No explicit solution is available when claim amounts are LogNormally distributed, and numerical procedures have to be used in this case to find the deductibles d l . Table 7.3 displays the results obtained for a deductible per claim for the scale −1/top and Table 7.4 for the scale −1/ + 2, when the premium paid by the policyholder is held constant whatever the claim history. As was the case for the annual deductible, the second column gives the relativities associated with each level of the scale in the case of a classical bonus-malus system. The third column gives the relative premium in the case of a scale with the deductible system. The fourth column shows the deductible to be applied if the loss amounts are Negative Exponentially distributed and the last column shows the deductible to be applied if the loss amounts are LogNormally distributed. Again, the amounts of deductible displayed in the last two columns are very high compared to the annual premium, especially for the scale −1/ + 2. This results from the severe r l s listed in column 2. In order to get acceptable amounts of deductible, keeping the financial stability of the system, in the next section we will combine softened penalties in the malus zone with moderate deductibles. Table 7.3 Results for a deductible per claim varying according to the level occupied in the malus zone for the scale −1/top. Level l r l r l with deductible d l if C 1 ∼ xp d l if C 1 ∼ or 5 1973 % 100 % 14 041 16 254 4 1709 % 100 % 11 073 12 191 3 1507 % 100 % 8474 8941 2 1348 % 100 % 6170 6288 1 1220 % 100 % 4108 4080 0 547 % 54.7 % 0 0 Table 7.4 Results for a deductible per claim varying according to the level occupied in the malus zone for the scale −1/ + 2. Level l r l r l with deductible d l if C 1 ∼ xp d l if C 1 ∼ or 5 3091 % 100 % 23 317 31 592 4 2414 % 100 % 18 209 22 633 3 2077 % 100 % 15 101 17 803 2 1429 % 100 % 7376 7651 1 1302 % 100 % 5451 5502 0 624 % 62.4 % 0 0
Bonus-Malus Systems with Varying Deductibles 289 7.4.5 Annual Deductible in the Mixed Case Another solution would be to keep the system of maluses but to reduce the penalties r l applied to the policyholders in the malus zone, for example by choosing = 20 %. In exchange for these reduced r l s, the policyholders are subject to an annual deductible on the claims they will eventually file. Equation (7.12) used to compute the annual deductible then becomes r l EC 1 = 80 %r l EC 1 + ESS 100 %. Tables 7.5 and 7.6 display the numerical results. The r l s displayed in column 3 are equal to 80 % of those in column 2, except for the bonus level 0. Note that the reduced level 1 for the scale −1/top enters the bonus zone. The annual deductibles in the Negative Exponential case (column 4) are now reasonable (about twice the annual premium for most levels) but those in the LogNormal case remain considerable (up to five times the pure premium). 7.4.6 Per Claim Deductible in the Mixed Case If we combine softened penalties for the bonus-malus system with deductibles per claim, Equation (7.13) becomes 20 %EC 1 = EC 1 C 1
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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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Part III Advances in Experience Rat
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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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