By Michel Denuit, Xavier Marechal, Sandra Pitrebois, Jean-Francois Walhin
There are quite a lot of variables for actuaries to contemplate whilst calculating a motorist’s coverage top class, resembling age, gender and sort of car. additional to those elements, motorists’ premiums are topic to event score platforms, together with credibility mechanisms and Bonus Malus platforms (BMSs).
Actuarial Modelling of declare Counts offers a entire remedy of a few of the adventure ranking platforms and their relationships with hazard type. The authors summarize the newest advancements within the box, offering ratemaking platforms, when considering exogenous information.
- Offers the 1st self-contained, sensible method of a priori and a posteriori ratemaking in motor insurance.
- Discusses the problems of declare frequency and declare severity, multi-event structures, and the combos of deductibles and BMSs.
- Introduces contemporary advancements in actuarial technology and exploits the generalised linear version and generalised linear combined version to accomplish possibility classification.
- Presents credibility mechanisms as refinements of business BMSs.
- Provides functional functions with genuine facts units processed with SAS software.
Actuarial Modelling of declare Counts is key interpreting for college kids in actuarial technological know-how, in addition to working towards and educational actuaries. it's also excellent for pros interested by the assurance 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
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. 35) yields the Negative Binomial probability mass function Pr N = k = = a+k−1 ···a k! a+k a k! a a+ d a a+ d a a d a+ d d a+ d k k k=0 1 2 Mixed Poisson Models for Claim Numbers 29 where is the annual expected claim number and d is the length of the observation period (the exposure-to-risk).
Ii) the process has independent increments, that is, for any integer k > 0 and instants 0 ≤ t0 < t1 < t2 < · · · < tk , the random variables N t1 − N t0 , N t2 − N t1 , , N tk − N tk−1 are mutually independent. (iii) and ⎧ 1 − h + o h if k = 0 ⎪ ⎨ h + o h if k = 1 Pr N h = k = ⎪ ⎩ o h if k ≥ 2 where o h is a function of h that tends to 0 faster than the identity, that is, limh 0 o h /h = 0. Intuitively speaking, o h is negligible when h becomes sufficiently small. Assumption (i) implies that the probability of causing an accident is assumed to be the same for every day during any given period (we thus neglect the fact that meteorological conditions, safety conditions and other factors could vary over time).
3, we recall the main probabilistic tools to work with discrete distributions: probability mass function, distribution function, probability generating function, etc. Then, we review some basic counting distributions, including the Binomial and Poisson laws. 4 is devoted to mixture models to account for unobserved heterogeneity. Mixed Poisson distributions are discussed, including Negative Binomial (or Poisson-Gamma), Poisson-Inverse Gaussian and Poisson-LogNormal models. 5 presents the maximum likelihood estimation method.
Actuarial Modelling of Claim Counts: Risk Classification, Credibility and Bonus-Malus Systems by Michel Denuit, Xavier Marechal, Sandra Pitrebois, Jean-Francois Walhin