| We discussed premium rating of Bonus-malus system in the first part .In most companies,the importance of the auto-insurance prompted their actuaries to look for different tariff systems that distribute the exact weight of each risk among portfolios.An initial approach to solve this problem would be to divide our portfolio into homogeneous sub-portfolios according to some variables chosen as influencing factors.We can then determinate a premium for each sub-portfolio.This approach is called tariff-segmentation.Next,credibility theory and its bonus-malus system appear to solve heterogeneity in our portfolio,in a different way from tariff-segmentation.Let us consider a given car insurance portfo-lio,where the policyholders are usually differentiated by two sorts of " process ": a set of a priori factors(like power of the car,age and sex of the driver,age of driving license) and a posteriori classification scheme according to bonus-malus rules.Obviously,there are some factors that cannot measured or introduced on the rates in order to calculate a priori premiums in a tariff-segmentation.Some of these unmeasured factors are very important in the claims amounts,and for the determination of each premium.The bonus-malus is necessary to get a better approximation,taking into account the experience of each risk.There are three chapters in the first part.In the first chapter,we introduce the NCD systems in short word,and we review the literature on NCD systems.In the second chapter,we use an entropy loss function and subsequently obtain the result for the Beta-Negative Binomial model and Logarithmic Normal-Poisson model.The bonus-malus system has the following assumptions:l.Let Nij be the claim number of risk i{i = 1, ? ? ■ , /) in year j(j = 1, ■ ? ? , t).riij is a realization of the random variable A^.To each risk i belongs a risk parameter Gi.Given 6j the {Nij-,j = 1, ■ ■ ■ ,t} are independent.2.{6i : i = 1, ???,/} are i.i.d.with EinBi = 1we got some conclusions as follows:Theorem 2.1 The solution of the problemunder the constraint I?[Ingt(Nn, ? ? ? , iVit)] = l,is given bytf (nil,? ?" ,n?) = exp{l + ￡ ln￡(e?|nn,■ ■ ■ ,nit) - -^[{aE{e*\Nlu ■ ? ? ,Nit)\}.Remark l.limc^oo<￡(na, ■■ ?,nu) — e,the case corresponds to the " no experience rating " case. If the factor c tends to oo,then all risks have to pay the same premium.There are no bonuses or malues.Remark 2.1imc^oo&*(n?i> ?? ? ,nit) = exp^lne^!, ? ? ? ,Nit)}.Under the constraint E\\ngi(Nn, ■ ■■ ,Nit)) = EQnBi) = l,We want to know if we overrate the risk or underrate it. We obtain the result as follows:(1)0 < c < i,Egr(Nn,-- ■ ,NU) < E(Bi), underrate the risk.(2)c = 1,Eg*(Na,- ■ ■ ,Nit) < ￡(6,), underrate the risk.(3)c -+ oo, limc_cx) In Eg*t (Na, ■ ? ■ , Nit) < E(Qi), underrate the risk.(4)c > 1, we don't judge.Theorem 2.3 If 0;~ Beta(a,f3) and EinGi = 1,^19, are independent with Nij\Qi ~ NB(r,@i), then the optimal estimate for the bonus-malus factor is given by-v n n^1! r(rt + a + nu + 0) 1 _? T(rt + a + N* + 0) ^gtina, ■ ■ ■ ,nit) = exP{l + - in r(e + rt + (> + Wfa + /9) - ^ r(c + rt + a + Nu + 0)?-Theorem 2.5 If?* ~ LN(l,a2),Nij\Qi are independent with Nij\&i ~ Poisson(QiXij), then the optimal estimate for the bonus-malus factor is given byc ￡[exp(-A,,,ex)] cIn chapter 3,we use a Linex loss function and subsequently obtain the result for the Poisson-Inverse Gaussian model.The assumptions are the same to chapter 2,but E[Qi] = 1. We obtain some result as follows:Theorem 3.6 The solution of the problem mxngE{\ (e-c0"(w"--'Ar?>-e?) - 1)] under the constraint E[gi(Nn,- ,Nit)] = l,is given byS*(na, ? ■" ,n*) = 1 - - ■ E\tRE(e^*\Na, ■ ■ ■ ,Nit)} + - ■ ln￡(ec6<|nii, ? ? ? ,n?). c c\(\a, ,it)} c cTheorem 3.8 If e< ~ IG(1,P) and %|9i are independent with Nij\Qi ~ Pozss 0, and IG(-y, 5) denotes the density of the Inverse Gaussian distribution given byz-le-*^*), z>0. It can be verified that the resulting unconditional density for j/j is given byf(yi) = J!?y?-16e+le*'{T2 + 2^)3"f K^^Syf-f + 2tiVi), 6,y > 0, Vi > 0. y 27TI (a) 2In chapter 5,we estimate the parameters of the model by an EM algo-rithm.We consider a sample (yi,Xi),i — 1, ,500,where j/j is the response and Xi a vector of covariates.We obtain the sample by making the random number .In our case,the missing data are simply the realizations of the unobserved mixing parameter 0j for each data point.If one augments the unobserved data 0;to the observed data (j/i,Xj),for i = I,--- ,n,then the complete data log-likelihood takes the formLc = J = (a- -)t=l i=l i=l" 62Let f(y\a, 6) — fnjy!/a *e 6y and let the prior distribution of the parameter 0 be the GIG(X, 7, S) distribution.The posterior distribution $\y is a GIG{\ + a, \/y^ + 2yiti, 6) distribution.So,an EM type algorithm for the Gamma-IG distribution can be described as follows:E-step: Let w = (6,7, (3) be the vector of the parameters to be estimated.Given the values of the parameters after the fcth iteration,say u/fc) ,then- nlnr(a) + nln<5 - nIn v^ir + n6-y - ^tiyiE{di\u{-k\yi,ni=\M-step: Update 0,5,7 respectively usingt=it=in5(k+DUntil |/?(fc+1> - (3^\ < 5 x 10-5, we stop the iteration.< 5 x lO"5, J7(fc+1) - l(k)\ < 10"1,... |
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