Bonus-Malus Systems in Automobile Insurance by Jean Lemaire

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By Jean Lemaire

Most insurers around the globe have brought a few type of merit-rating in vehicle 3rd occasion legal responsibility coverage. Such structures, penalizing at-fault injuries through top class surcharges and lucrative claim-free years by means of discount rates, are known as bonus-malus platforms (BMS) in Europe and Asia. With the present deregulation traits that problem such a lot assurance markets worldwide, many businesses might want to boost their very own BMS. the most target of the booklet is to supply them versions to layout BMS that meet their ambitions.
half I of the ebook includes an total presentation of the professionals and cons of merit-rating, a case learn and a evaluation of the various chance distributions that may be used to version the variety of claims in an vehicle portfolio. partly II, 30 structures from 22 varied nations, are evaluated and ranked in accordance with their `toughness' in the direction of policyholders. 4 instruments are created to judge that longevity and supply a tentative type of all structures. Then, issue research is used to mixture and summarize the knowledge, and supply a last rating of all platforms. half III is an updated evaluate of all of the chance types which were proposed for the layout of an optimum BMS. the applying of those types may permit the reader to plot the procedure that's splendid to the habit of the policyholders of his personal assurance corporation. ultimately, half IV analyses a substitute for BMS; the advent of a coverage with a deductible.

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The portfolio was observed in 1976 (Lemaire, 1977b. 1979a) but figures are quite similar today, as claim frequencies in Belgium have remained very stable over the past nineteen years. 1033. Table 3-1. Observed Distribution of Number of Claims in a Portfolio k nk 0 1 2 3 4 >4 96,978 9,240 704 43 9 0 Total 106,974 Let us fit the distribution shown in table 3-1 by a Poisson. Both the maximum likelihood and the moments methods lead to the selection of the observed mean as the estimator of A.. 1011, and multiplying them by the sample size 106,974, lead to the fitted theoretical frequencies presented in column 3 of table 3-2.

In 1992, over 65 percent of the policyholders enjoyed the maximum discount, and over 80 percent of the policies belonged to one of the three highest discount classes. 42 percent of the drivers actually paid a malus. The average policy was situated between classes 2 and 3. Table 2-3. 48 When the distribution of policies in the classes of a BMS becomes so unbalanced, the main reason for the use of an experience-rated premium system -to have the bad drivers pay substantially more premiums over the long run than the good drivers- is essentially lost.

Na+ I) = a! The distribution {Pk; k=O, 1,2, ... l+Q 31 = (k+a-1)(_. )Q(_l )k k = ( k+a-1) k 1+. 1+. PQ qk setting and p q I-p 1 1+. and defining, as generalized combinatorial coefficient, ( k+a-1) = r(k+a) . k r(k+ l)r(a) We obtain a negative binomial distribution, of mean m=a/'t" and variance 0 2 = ~(1+~). The variance of the negative binomial exceeds its mean, a desirable property. ) The skewness coefficient is 32 MODELS FOR CLAIM NUMBER DISTRIBUTIONS The moment-generating function is t < In(1+-r).

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