Actuarial Modelling of Claim Counts: Risk Classification, by Michel Denuit, Xavier Marechal, Sandra Pitrebois,

By Michel Denuit, Xavier Marechal, Sandra Pitrebois, Jean-Francois Walhin

There are a variety of variables for actuaries to think about while calculating a motorist’s coverage top rate, akin to age, gender and kind of car. extra to those elements, motorists’ charges are topic to event score structures, together with credibility mechanisms and Bonus Malus platforms (BMSs).

Actuarial Modelling of declare Counts provides a complete remedy of many of the event score platforms and their relationships with possibility category. The authors summarize the latest advancements within the box, proposing ratemaking structures, when taking into consideration exogenous information.

The text:

  • 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 combos of deductibles and BMSs.
  • Introduces contemporary advancements in actuarial technological know-how and exploits the generalised linear version and generalised linear combined version to accomplish possibility classification.
  • Presents credibility mechanisms as refinements of industrial BMSs.
  • Provides sensible purposes with actual information units processed with SAS software.

Actuarial Modelling of declare Counts is vital interpreting for college students in actuarial technological know-how, in addition to working towards and educational actuaries. it's also excellent for pros occupied with the coverage undefined, utilized mathematicians, quantitative economists, monetary engineers and statisticians.

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Extra resources for Actuarial Modelling of Claim Counts: Risk Classification, Credibility and Bonus-Malus Systems

Sample text

17). This ends the proof. When the hypotheses behind a Poisson process are verified, the number N 1 of claims hitting a policy during a period of length 1 is Poisson distributed with parameter . e. Pr N t + s − N s = k = exp − t tk k=0 1 2 k! Exposure-to-Risk The Poisson process setting is useful when one wants to analyse policyholders that have been observed during periods of unequal lengths. Assume that the claims occur according to a Poisson process with rate . If the policyholder is covered by the company for a period of length d then the number N of claims reported to the company has probability mass function Pr N = k = exp − d d k!

If X is the total amount of claims generated by some policyholder, FX x is the probability that this policyholder produces a total claim amount of at most E x. The distribution function FX corresponds to an estimated physical probability distribution or a well-chosen subjective probability distribution. e. e. limh 0 F x + h = F x , and (v) Pr a < X ≤ b = F b − F a , for any a < b. Henceforth, we denote as F ·− the left limit of F , that is, F x− = lim F x = Pr X < x Suppose that X1 X2 Xn are n random variables defined on the same probability space Pr .

3 Maximum Likelihood Estimation All the models implemented in this book are parametric, in the sense that the probabilities are known functions depending on a finite number of (real-valued) parameters. The Binomial, Poisson and Normal models are examples of parametric distributions. The first step in the analysis is to select a reasonable parametric model for the observations, and then to estimate the underlying parameters. The maximum likelihood estimator is the value of the parameter (or parameter vector) that makes the observed data most likely to have occurred given the data generating process assumed to have produced the observations.

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