By Michel Denuit, Xavier Marechal, Sandra Pitrebois, Jean-Francois Walhin
ISBN-10: 0470026774
ISBN-13: 9780470026779
There are quite a lot of variables for actuaries to contemplate whilst calculating a motorist’s coverage top class, similar to age, gender and sort of car. additional to those elements, motorists’ premiums are topic to adventure ranking platforms, together with credibility mechanisms and Bonus Malus structures (BMSs).
Actuarial Modelling of declare Counts offers a entire therapy of some of the adventure score platforms and their relationships with probability type. The authors summarize the newest advancements within the box, proposing ratemaking structures, while bearing in mind 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 mixtures of deductibles and BMSs.
- Introduces contemporary advancements in actuarial technology and exploits the generalised linear version and generalised linear combined version to accomplish chance classification.
- Presents credibility mechanisms as refinements of business BMSs.
- Provides functional purposes with actual information units processed with SAS software.
Actuarial Modelling of declare Counts is key studying for college kids in actuarial technology, in addition to working towards and educational actuaries. it's also preferrred for execs thinking about the coverage undefined, utilized mathematicians, quantitative economists, monetary engineers and statisticians.
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Additional resources for Actuarial Modelling of Claim Counts: Risk Classification, Credibility and Bonus-Malus Systems
Example text
Success or failure, alive or dead, male or female, 0 or 1. The probability of success is q. The probability of failure is 1 − q. If N is Bernoulli distributed with success probability q, which is denoted as N ∼ er q , we have ⎧ ⎪ ⎨ 1 − q if k = 0 p k q = q if k = 1 ⎪ ⎩ 0 otherwise. There is thus just one parameter: the success probability q. 7) The probability generating function is N It is easily seen that N z = 1 − q × z0 + q × z1 = 1 − q + qz 0 = p 0 q and N 0 q = p 1 q , as it should be. 8) Actuarial Modelling of Claim Counts 14 Binomial Distribution The Binomial distribution describes the outcome of a sequence of n independent Bernoulli trials, each with the same probability q of success.
Specifically, N 0 = Pr N = 0 and dk dtk N z z=0 = k! 5 Convolution Product A key feature of probability generating functions is related to the computation of sums of independent discrete random variables. Considering two independent counting random variables N1 and N2 , their sum is again a counting random variable and thus possesses a probability mass function as well as a probability generating function. 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.
In other words, the measurability for any x ∈ , where X condition X −1 − x ∈ ensures that the actuary can make statements like ‘X is less than or equal to x’ and quantify their likelihood. Random variables are mathematical formalizations of random outcomes given by numerical values. An example of a random variable is the amount of a claim associated with the occurrence of an automobile accident. A random vector X = X1 X2 Xn T is a collection of n univariate random variables, X1 , X2 , , Xn , say, defined on the same probability space Pr .