By Michel Denuit, Xavier Marechal, Sandra Pitrebois, Jean-Francois Walhin
There are quite a lot of variables for actuaries to contemplate while calculating a motorist’s assurance top class, akin to age, gender and kind of auto. extra to those components, motorists’ premiums are topic to event ranking structures, together with credibility mechanisms and Bonus Malus structures (BMSs).
Actuarial Modelling of declare Counts offers a finished remedy of some of the adventure ranking platforms and their relationships with danger type. The authors summarize the latest advancements within the box, proposing ratemaking platforms, when considering exogenous information.
- Offers the 1st self-contained, useful method of a priori and a posteriori ratemaking in motor insurance.
- Discusses the problems of declare frequency and declare severity, multi-event platforms, and the mixtures of deductibles and BMSs.
- Introduces fresh advancements in actuarial technological know-how and exploits the generalised linear version and generalised linear combined version to accomplish threat classification.
- Presents credibility mechanisms as refinements of industrial BMSs.
- Provides useful purposes with actual info units processed with SAS software.
Actuarial Modelling of declare Counts is vital interpreting for college kids in actuarial technological know-how, in addition to working towards and educational actuaries. it's also superb for execs interested by the assurance undefined, utilized mathematicians, quantitative economists, monetary engineers and statisticians.
Read or Download Actuarial Modelling of Claim Counts: Risk Classification, Credibility and Bonus-Malus Systems PDF
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Additional info for Actuarial Modelling of Claim Counts: Risk Classification, Credibility and Bonus-Malus Systems
The probability mass function of N1 + N2 is obtained as follows: We obviously have that k Pr N1 + N2 = k = Pr N1 = j N2 = k − j j=0 for any integer k. Since N1 and N2 are independent, their joint probability mass function factors to the product of the univariate probability mass functions. This simply comes from Pr N1 = j N2 = k − j = Pr N1 ≤ j N2 ≤ k − j − Pr N1 ≤ j N2 ≤ k − j − 1 − Pr N1 ≤ j − 1 N2 ≤ k − j + Pr N1 ≤ j − 1 N2 ≤ k − j − 1 = Pr N1 ≤ j Pr N2 ≤ k − j − Pr N2 ≤ k − j − 1 − Pr N1 ≤ j − 1 Pr N2 ≤ k − j − Pr N2 ≤ k − j − 1 = Pr N2 = k − j Pr N1 ≤ j − Pr N1 ≤ j − 1 = Pr N1 = j Pr N2 = k − j The probability mass function of N1 + N2 can thus be obtained from the discrete convolution formula k Pr N1 + N2 = k = Pr N1 = j Pr N2 = k − j k=0 1 j=0 For large values of k, a direct application of the discrete convolution formula can be rather time-consuming.
There are several models that lead to the Negative Binomial distribution. A classic example arises from the theory of accident proneness which was developed after Greenwood & Yule (1920). This theory assumes that the number of accidents suffered by an individual is Poisson distributed, but that the Poisson mean (interpreted as the individual’s accident proneness) varies between individuals in the population under study. If the Poisson mean is assumed to be Gamma distributed, then the Negative Binomial is the resultant overall distribution of accidents per individual.
4) where p · is a known function depending on a set of parameters . 4) is usually called the probability mass function. Different functional forms lead to different discrete distributions. This is a parametric model. → 0 1 of N gives for any real threshold x, the probability The distribution function FN for N to be smaller than or equal to x. 4) and where x denotes the largest integer n such that n ≤ x (it is thus the integer part of x). 4), FN also depends on . 3 Moments There are various useful and important quantities associated with a probability distribution.