Analysis Of Credibility Models Based On Integer-valued Autoregressive Time Series

Actuarial science is a subject rooted in practical activities,based on the relevant theories and methods of mathematics and statistics and combined with the general principles of economics,it analyzes and solves the the quantitative problems people may face in the fields of insurance,finance,investment and finance.Pricing for insurance products and ruin theory for surplus process,which are core tasks of actuaries and main research areas of actuarial science,are very important for insurance operators and policymakers to develop their businesses and manage their risks.In this thesis,we also focus on this two interesting topics,our results generalize the existing literature,and could provide more scientific and reasonable theoretical guidance for the insurance companies.Next,we introduce our work in this thesis.I Pricing for insurance productsIn non-life actuarial science,ratemaking process for automobile insurance is done in two steps.In the first step,actuaries design a tariff structure to partition poli-cies when issued according to their risk characteristics,which is often called a priori classification.The policyholders in the same class would pay the same base premium.However,the tariff cells are still quite heterogeneous because there are many important factors that are ruled out of this stage,such as respectfulness towards the law,swiftness of reflexes,aggressiveness behind the wheel or knowledge of the highway code.But it is reasonable to believe that these characteristics are revealed by the number and sizes of claims reported by the policyholders over the successive insurance periods.Thus,to compensate to the inadequacies of the a priori rating system,insurers adjust the premium based on the individual claims experience.Such systems,the second step of ratemaking process,penalizing insureds responsible for one or more accidents by premium surcharges(or maluses),and rewarding claim-free policyholders by awarding them discounts(or bonuses),are called experience rating.Credibility theory is the most important method to obtain the experience rating.In this thesis,we consider more practical credibility models based on two kinds of integer-valued autoregressive time series,and apply them to Bayesian credibility premium updating in the automobile insurance to investigate how the claim history affects the future premium1.Credibility model based on mixed INAR(1)processLet us consider one policyholder from a insurance portfolio and denote by N1,...,NT the number of claims per year submitted by this individual.Our goal is to draw pre-diction on the premium for the subsequent year T +1 of this policyholder.To this end,we assume thatB1.The count variables N1,...,NT,NT+1follow the INAR(1)process:Nt =φoNt-1+ ∈t,t≥2,in which φ∈∈[0,1),and the so-called thinning operator ’o’ is defined by:where {Bt-1,i,t=2,3,...,i= 1,2,...,Nt-1}is a array of i.i.d.Bernoulli random variables with mean φ,and independent of the sequence {N1,N2,...}.Besides,for any ixed t {Bt-1,i,i=1.2,...} and Nt-1 are independent.B2.Given the unobservable heterogeneity(?)=θ,{∈t,t≥2} is a sequence of i.i.d.non-negative integer-valued random variables confirming to the Poisson distribution with mean λθ i.e.Furthermore,{∈t,t≥2} is independent of N1 conditional on(?).B3.The unobservable heterogeneity(?)follows a finite mixture of Erlang distri-butions whose density function is given by parameters of the Erlang distributions,π =(π1,...,πg)with πr>0 and ∑r=1gπr = 1 are the weights used in the mixture,and η is the common scale parameter.B4.The financial balance constraint holds,i.e.(?)The following theorem gives the analytical expression of the conditional distribu-tion of the heterogeneity.Theorem 1 If N1|(?)=θ～Poi(λθ/1-φ),then:(1)For T = 0,the conditional density function of(?)is f(?)(θ;π,m,η)which is defined by(1).(2)Fot T = 1,the conditional density function of(?)given N1=n1 is in which(3)For any T≥2,the conditional density function of(?)given N1=n1,...,NT=nT is f(?)(θ|N1= n1,...,NT=nT)in which mzT,...,z2(nT...,n1)=mr +n1 +…+nT-z2-…-zT,(4)η=η+(T-1)λ+ λ/(1-φ),(5)and γ{θ;mr,η)is defined by(2).From the above theorem,we can derive the analytical formulas of the pure premi-um for the year T + 1 directly.Theorem 2 If N1|(?)=θ～Poi(λθ/1-φ),then:(1)For T = 0,no claim history is available at the beginning of the contract,and P1=E(N1)=λ/1-φ.(2)For T = 1,P2 =φn1+λ(?)1,where the predicted heterogeneity(?)1 is given by in which πr(n1)is defined by(3).(3)For any T ≥ 2,PT+1 =φnT+λ(?)T,where the predicted heterogeneity(?)T is given by in which mzT,...,z2{nT...,n1)are defined by(4),(5)and(6),respectively.2.Credibility model based on SETINAR(2,1)processDenote the number of claims per year submitted by one policyholder from a in-surance portfolio by N1,...,NT,in order to draw prediction on the premium for the subsequent year T + 1 of this policyholder,we assume that:D1.The count variables N1,...,Nt,NT+1 follow the INAR(1)process:Nt I1,t·(φoNt-1)+I2,t ·(φ2oNt-1)+∈t,t ≥ 2,in which I1,t = I{Xt-1≤r},I2,t = 1-I1,t =I{Xt-1>r},where r is a known positive integer,called threshold variable.The thinning operatorφi∈[0,1),i = 1,2,’o’ is defined by where {Bt-1,k(i),t=2,3,...,k=1,2,...}is two arrays of i.i.d.Bernoulli random vari-ables with mean φ,respectively.{Bt-1(1),k,t = 2,3,...,k = 1,2,...},{Bt-1(2),k,t =2,3,…k=1,2,...} and {∈2,∈3,...} are independent.Besides,for any fixed t,{Bt-1k(i),k=1,2,...} and Nt-1 are independent.D2.Given the unobservable heterogeneity(?)=θ,{∈t,t≥2} is a sequence of i.i.d.non-negative integer-valued random variables confirming to the Poisson distribution with mean λθ,i.e.P(∈t = n|(?)= θ)=(λθ)n/n!e-λθ,t≥ 2.Furthermore,{∈t,t≥2} is independent of N1 conditional on(?).D3.The unobservable heterogeneity(?)follows a Ga(α,α)distributions whose density function is given by in which the scale parameter equals to the shape parameter,then we have E((?))= 1,such that the financial balance constraint holds.First,we obtain the analytical expression of the conditional distribution of the heterogeneity given the claims history.Theorem 3 If N1|(?)= θ～Poi(βλθ),with a known positive real number β,then we have:(1)For T = 0,the conditional density function of(?)is f(?)θ defined by(7).(2)For T = 1,the conditional density function of(?)given N1 =n1 is Ga(α+n1,α+βλ),whose density function is(3)For any T≥ 2,the conditional density function of(?)given N1=n1,…,NT=nT is f(?)(θ|N1=n1,...,NT=nT)in whichα1,zT,...,z2(nT...,n1)=α+n1+…+nT-Z2-…-zT,(8)α2=α+(T-1+β)λ,(9)and γ(θ;α1,α2)is the density function of Ga(α1,α2).Furthermore,the analytical formulas of the pure premium for year T + 1 is given as follow.Theorem 4 If N1|(?)=θ～Poi(βλθ),with a k fnown positive real number β,then we have:(1)For T = 0,no claim history is available at the beginning of the contract,and P1 = E(N1)= βλ.(2)For T = 1,P2 =φ1n1I{n1≤r} +φ2n1I{n1>r}+ λ(?)1,where the predicted heterogeneity(?)1 is given by(3)For any T ≥ 2,PT+1=φnTI{nT≤r}+φ2nTI{nT>r}+λ(?)T,where the predicted heterogeneity(?)T is given by in which αi,zT,…,z2(nT…,n1),α2 and πzT,…,z2(nT…,n1)are defined by(8),(9)and(10),respectively.Ⅱ Ruin theoryRuin theory is one of the most active research areas of actuarial science.Ueder some practical conditions,it models the surplus process of the insurance company,analyzes and measures the risk quantitatively,to provide more effective theoretic guide for insurers to give early warning for the bankrupt.In this thesis,we consider a kind of non-standard renewal risk model,in which{Xk,k≥1} is the sequence of claim sizes,and {θk,k ≥ 1} is the sequence of inter-arrival times.Assume that {(Xk,θk),k≥1} forms a sequence of independent and identically distributed(i.i.d.)copies of a generic random pair(X,θ)with marginal distribution functions F(x)on[0,∞)and G(θ)on[0,∞),respectively.Furthermore,the claim arrival times are τk=∑i=1kθi,k∈N,with τ0 = 0.We define the number of claims by then Nt*is a stopping time,and it is easy to know that(1)if τk = t,Nt*= Nt(?)supk{k∈N:τk≤t};(2)if τk≠t,Nt*=Nt + 1.In this way,the aggregate amount of claims over the[0,t]is of the following form:We get the large deviations of St*as follow.Theorem 5 Let the individual claim sizes {Xk,k≥1} be a sequence of i.i.d.random variables with common distribution function F ∈ C,assume that E(X)= μ∈(0,∞)and E(θ)= 1/λ∈(0,∞),Then,for arbitrarily given γ>0,we have uniformly for all x>γλt P(St*-μλt>x)～λtF(x),t→∞,that is to sayThe above result shows that the dependence between X and θ can not change the asymptotic property of the aggregate amount of claims St*.