非寿险精算Aggregate loss models.pptx

1、Ch4. Aggregate loss models,Introduction (9.1 and 9.2) General property of collective risk model (9.2) Computation the collective risk model Direct calculate (9.4) Approximation (9.5) Recursive method (9.6) Inversion method (9.8) Calculation with approximation distribution (9.9) Individual risk model

2、 (9.11),Ch4.1 Aggregate loss models,The aim of insurance is to combine individual risk into an aggregate risk to further diverse loss in the risk pool. Its important to model the aggregate loss for insurance business. An example from individual risk to aggregate risk see page199, example 9.1.,Ch4.1

3、Aggregate loss models,Collective risk model: For a random number , individual payments are i.i.d. sequence The random sum is the aggregate loss Individual risk model: loss amount for contracts are independent but not identically distributed. The partial sum represents the aggregate loss.,Ch4.2 Colle

4、ctive risk model,General property of collective risk model Pf Pgf Mgf Mean Variance Book P203-P205 Section 9.3,Ch4.3.1 Direct calculate,P217 Example 9.7 exponential severity + Negative binomial frequency P218 Example 9.8 exponential severity + any frequency P219 Example 9.9 Mixed Erlang severity + a

5、ny frequency P220 Example 9.10 Severity closed under convolution,Ch4.3.2 Approximation,Normal distribution approximation Lognormal distribution approximation,Ch4.3.3 Recursive method,Severity distribution is defined on integer and with (a,b,1) class frequency distribution(Th9.8) with (a,b,0) class f

6、requency distribution(Co9.9) with compound frequency models with premier and secondary distribution from (a,b,1) and (a,b,0) class, the aggregate dis. can be derived P228 Example 9.12,Ch4.3.3 Recursive method,For compound frequency models with pgf and severity with pgf Then the pgf of aggregate clai

7、ms S is And its probability function the can be derived as Step1 calculate by Step2 use recursive formula to derive Step3 calculate by Step4 use recursive formula to derive from,Ch4.3.3 Recursive method,Severity distribution is continuous for (a,b,1) class frequency, the pf of aggregate satisfies Vo

8、lterra integral equation for (a,b,0) class frequency, the same Volterra integral equation is also derived. Note: numerical solution for Volterra integral equation is available but complicated.,Ch4.3.3 Recursive method,Other solution for continuity severity Method of rounding (mass dispersal) Method

9、of local moment matching locate point masses at points so that the first p(1 or 2) moment are preserved. Then,Ch4.3.4 Inversion methods,Fast Fourier Transform (P242) Direct numerical inversion (p244),Ch4.3.5 calculate approximation dis.,from discrete dis. to continuous dis. Arithmetic distribution (P246) let probability on point jh uniformly spread over the interval (jh