By Michel Denuit, Xavier Marechal, Sandra Pitrebois, Jean-Francois Walhin
There are quite a lot of variables for actuaries to think about whilst calculating a motorist’s coverage top rate, similar to age, gender and kind of car. additional to those components, motorists’ premiums are topic to event score platforms, together with credibility mechanisms and Bonus Malus platforms (BMSs).
Actuarial Modelling of declare Counts provides a finished remedy of some of the event score structures and their relationships with threat type. 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 platforms, and the mixtures of deductibles and BMSs.
- Introduces fresh advancements in actuarial technology and exploits the generalised linear version and generalised linear combined version to accomplish probability classification.
- Presents credibility mechanisms as refinements of industrial BMSs.
- Provides useful functions with actual information units processed with SAS software.
Actuarial Modelling of declare Counts is vital examining for college students in actuarial technological know-how, in addition to training and educational actuaries. it's also perfect 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
10) k=1 where M ∼ in n − 1 q . Furthermore, with M as defined before, n E N2 = k=1 n! kq k 1 − q k−1 ! n−k ! e. its variance is smaller than its mean : V N = nq 1 − q ≤ E N = nq. 8), raised to the nth power. This was expected since the Binomial random variable N can be seen as the Mixed Poisson Models for Claim Numbers 15 sum of n independent Bernoulli random variables with equal success probability q. 7) (in the latter case, since the variance is additive for independent random variables). 12), we also see that having independent random variables N1 ∼ in n1 q and N2 ∼ in n2 q , the sum N1 + N2 is still Binomially distributed.
4), N · depends on . If any function that is known to be a probability generating function is expanded as a power series in z, then the coefficient of zk must be pk for the corresponding distribution. An alternative way of obtaining the probabilities is by repeated differentiation of N with respect to z. 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.
Specifically, let us assume that given = , N t t ≥ 0 is a homogeneous Poisson process with rate . Then N t t ≥ 0 is a mixed Poisson process, and for any s t ≥ 0, the probability that k events occur during the time interval s t is Pr N t + s − N s = k = = 0 0 Pr N t + s − N s = k exp − t = dF tk dF k! that is, N t + s − N s ∼ oi t . Note that, in contrast to the Poisson process, mixed Poisson processes have dependent increments. Hence, past number of claims reveal future number of claims in this setting (in contrast to the Poisson case).
Actuarial Modelling of Claim Counts: Risk Classification, Credibility and Bonus-Malus Systems by Michel Denuit, Xavier Marechal, Sandra Pitrebois, Jean-Francois Walhin