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

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238 Actuarial Modelling of Claim Counts As explained in Chapter 2, overdispersion can be detected by using the statistics T 1 , T 2 or T 3 presented in Section 2.4.6. The values obtained with Portfolio C are T 1 = 2004, T 2 = 1456 and T 3 = 1289. All the associated p-values are less than 10 −4 leading to the rejection of the null hypothesis in favour of a mixed Poisson model. In order to take the residual heterogeneity into account, we fitted a Negative Binomial regression model to the Portfolio C. The results are given in Table 5.7. All the variables and levels are still relevant. The log-likelihood is now equal to −61 393.3 and is better than with Poisson regression. Resulting Price List The resulting price list is obtained thanks to the Negative Binomial model for the frequency (presented in Table 5.7) and to the LogNormal model for the average cost of the standard claims (presented in Table 5.5). Neglecting the large claims, the pure premium of the reference class is obtained by ( ) exp freq 0 + cost 0 + 2 = E31676 (5.6) 2 This amount corresponds to the pure premium of a male policyholder living in an urban area, aged between 31 and 60, driving a car older than 10 years, using gasoil and with a power greater than 110 kW, paying his premium in several installments and having opted for a more extensive coverage than only the compulsory motor third party liability insurance. To obtain the pure premium (neglecting the large claims) for a policyholder belonging to another risk class, the percentages of Table 5.8 must be applied to the base pure premium Table 5.7 Results of the Negative Binomial regression of the claim counts recorded in Portfolio C. Variable Level Coeff Std error Wald 95 % conf limit Chi-sq Pr>Chi-sq Intercept −14626 00720 −16037 −13216 41306
Efficiency and Bonus Hunger 239 Table 5.8 Price list for the standard claims in Portfolio C. Variable Level Influence of frequency Influence of cost Total influence Ageph 18–24 19033 % 11908 % 22664 % Ageph 25–30 14339 % 10000 % 14339 % Ageph > 60 7997 % 11949 % 9556 % Ageph 31–60 10000 % 10000 % 10000 % Gender Female 10622 % 10000 % 10622 % Gender Male 10000 % 10000 % 10000 % Agev 0–2 10000 % 8844 % 8844 % Agev 3–5 9397 % 8093 % 7605 % Agev 6–10 10000 % 8907 % 8907 % Agev > 10 10000 % 10000 % 10000 % Fuel Petrol 8256 % 10000 % 8256 % Fuel Gasoil 10000 % 10000 % 10000 % Premium split No 7594 % 10000 % 7594 % Premium split Yes 10000 % 10000 % 10000 % Coverage MTPL only 11266 % 10000 % 11266 % Coverage More 10000 % 10000 % 10000 % City Rural 7969 % 10683 % 8514 % City Urban 10000 % 10000 % 10000 % Power < 66 kW 7445 % 10000 % 7445 % Power 66–110 kW 8260 % 10000 % 8260 % Power > 110 kW 10000 % 10000 % 10000 % (5.6) as a function of the characteristics of the policyholder. For example, the pure premium of a woman aged between 25 and 30, living in a rural region, driving a car older than 10 years, using petrol and with a power less than 66 kW, paying her premium once a year and being covered only for the MTPL will be equal to E31676 × 10622 % (correction for being a female policyholder) × 14339 % (correction for being aged between 25 and 30) × 8514 % (correction for living in a rural area) × 8256 % (correction for using petrol) × 7445 % (correction for driving a low-power car) × 7594 % (correction for paying the premium once a year) × 11266 % (correction for buying MTPL coverage only) = E21601 Finally, to obtain the total pure premium of a policyholder, the expected cost of large claims (which is independent of the explanatory variables) must be added. The value is equal to 00117 % × E368 01880 = E4306
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Actuarial Modelling of Claim Counts
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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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4 Actuarial Modelling of Claim Coun
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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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