By Michel Denuit, Xavier Marechal, Sandra Pitrebois, Jean-Francois Walhin
There are a variety of variables for actuaries to think about whilst calculating a motorist’s assurance top rate, equivalent to age, gender and sort of auto. additional to those components, motorists’ premiums are topic to adventure ranking structures, together with credibility mechanisms and Bonus Malus platforms (BMSs).
Actuarial Modelling of declare Counts offers a complete therapy of a number of the event ranking structures and their relationships with chance class. The authors summarize the latest advancements within the box, proposing ratemaking structures, when bearing in mind 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 fresh advancements in actuarial technology and exploits the generalised linear version and generalised linear combined version to accomplish hazard classification.
- Presents credibility mechanisms as refinements of business BMSs.
- Provides functional purposes with genuine info units processed with SAS software.
Actuarial Modelling of declare Counts is key analyzing for college students in actuarial technology, in addition to training and educational actuaries. it's also ultimate 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
When the number of observations n is large, and the success probability q is small, the corresponding Binomial distribution is well approximated by the Poisson distribution with mean = nq. The Poisson distribution is thus sometimes called the law of small numbers because it is the probability distribution of the number of occurrences of an event that happens rarely but has very many opportunities to happen. The parallel with traffic accidents is obvious. The Poisson distribution was discovered by Siméon-Denis Poisson (1781–1840) and published in 1838 in his work entitled Recherches sur la Probabilité des Jugements en Matières Criminelles et Matière Civile (which could be translated as ‘Research on the Probability of Judgments in Criminal and Civil Matters’).
In an actuarial context, this amounts to recognizing that several vehicles can be involved in the same accident, each of the insured drivers filing a claim. Therefore, a single accident may generate several claims. If the number of claims per accident follows a Logarithmic distribution, and the number of accidents over the time interval of interest follows a Poisson distribution, then the total number of claims for the time interval can be modelled with the Negative Binomial distribution. Let us formally establish this result.
K k! k=0 1 k = k0 + 1 k ≥ k1 + 1 k0 k1 Mixed Poisson Models for Claim Numbers 27 Shaked’s Two Crossings Theorem tells us (i) that the mixed Poisson distribution has an excess of zeros compared to the Poisson distribution with the same mean and (ii) that the mixed Poisson distribution has a thicker right tail than the Poisson distribution with the same mean. Probability Generating Function The probability generating function of Poisson mixtures is closely related to the moment generating function of the underlying random effect.
Actuarial Modelling of Claim Counts: Risk Classification, Credibility and Bonus-Malus Systems by Michel Denuit, Xavier Marechal, Sandra Pitrebois, Jean-Francois Walhin