Credibility Premium Under Asymmetric Loss Function

Credibility Premium Under Asymmetric Loss FunctionThe calculation of insurance premium is an important work in insurance. We have many methods to calculate insurance premium. The credibility premium theoryis one of the most widly using comprehensive theories in the world. Credibility theory thinks that risk classification in insurance involves unobserved risk characteristics,these characteristics are usually modelled by the introduction of a random effect in the classification process. Consequently, a posteriori analysis following claims experience is an interesting task and at each insured period, the random effects can be updated for past claim experience, revealing some individual information.Credibility theory is a common approach to calculate insurance premium based on the policyholder's past, experience and the experience of the entire group of policyholders.It is a weighted average of the two classes datas. The weighted factor is called credibility factor, and the insurance premium is called credibility premium. The popular formulas in credibility theory take premium as a weighted sum of the average experience of the policyholder and the average of the entire collection of policyholders.These formulas are easy to understand and simple to apply due to their linear properties. This method is widely used in commercial property or liability insurance and group health or life insurance.Let X denote total claim of a. policyholder. The distribution of X depends on the parameterθ, whereθvaries across policyholders and maybe vector valued. Let X_i, i = 1,2,…, denote total claims of a policyholder in the i^th policy period. If & is given, X_i's are independent and identically distributed. Therefore, the value ofθ completely detemines the claim distribution of the policyholder. LetX_n= (X₁,X₂,…, X_n)∈(?)It is the policyholder's claim experience in the first n periods. Generally, credibility estimators Y is a real valued function of the given information, i.e. Y(X_n). In BUUUUUUhlmann's classical credibility theory (1967, 1970), the loss function U is taken to be the traditional squared error loss function.U[Y(X_n),μ(θ)] = [Y(X_n)-μ(θ)]².The resulting credibility premiumY(X_n) = E[μ(θ)|X_n], (1)which is the posterior expected value ofμ(θ).If we constrain Y to be a liner function of the prior claim data, we can use L to present Y. Restricting the credibility premium to be a liner combination of the prior claims, L(X_n) is given asL(X_n) = z(?) + (1-z)μ, z =n/(n+k), (2)where (?) = 1/n (?)X_i is the sample mean,μ(θ) = E[X|θ],μ= E[μ(θ)] = E[X],v(θ) = V ar(X|θ),v = E[v(θ)],a = V ar[μ(θ)] = E[(μ(θ) -μ)²],k = v/a. Therefore, k = v/a = (E[V ar(X|θ)]/(V ar[E(X|θ)],k is the expected value of conditional variance to the variance of conditional means. Generally, z is called as the credibility factor.In classical credibility theory, the loss function is taken to be the traditional symmetric loss functions, eg. squared error loss function. But in some estimation a.nd prediction problems, use of symmetric loss functions may be inappropriate, as has been recognized in the literature-see, for example, Ferguson (1967), Zellner and Geisel (1968), Aitchison and Dunsmore (1975), Varian (1975), and Berger (1980). Regarding to this case, we propose three asymmetic loss functions: balance loss function, entropy loss funtion and Linex loss function, to calculate the credibility premium in this paper.In chapter 2, we will introduce a balance loss function which is supposed by Zellner.where 0≤ω≤1 is given,δ=δ(X₁,X₂,…,X_n) is a estimation ofθandΔis a decision-space. In this section, we focus on the balance loss function, and study the optimal credibility premium and the optimal linear premium under the balance loss function.Theorem 2.1 (Credibility Premium under Balance Loss Function) Let X_i , i=1 , 2 ,…, denote total claims of a policyholder in the i^th policy period. The distribution of X_i depends on the parameterδ, whereδvarise across policyholders and maybe vector valued. Denote X_n=(X_i, X₂,…, X_n )∈(?) as the policyholder's claim experience in the first n periods. For the loss function U₁(θ,δ), the optimal premium Y₁(X_n) is given byY₁(X_n) =ω(?) + (1 -ω)E[μ(θ)|X_n], (4)where (?) = 1/2 (?)X_n is the sample mean.Theorem 2.2(The Optimal Linear Premium under Balance Loss Function) For the loss function U₁(θ,δ), the optimal linear premium L₁(X_n) is given byL₁(X_n) = z(?) + (1- z)μ, z = (n+ωk)/(n+k). (5)where (?) = 1/n (?) X_i is the sample mean,μ(θ) = E[X|θ],μ= E[μ(θ)] = E[X],v(θ) =V ar(X|θ),v=E[v(θ)],a = V ar[μ(θ)] = E[(μ(θ) -μ)²], k = v/a.In chapter 3, we will introduce an entropy loss function which is supposed by (cf. Dey,1999).U₄(θ,δ) =δlogδ/θ+ (1 -δ) log (1-δ)/(1-θ),(?)δ∈Δ. (6) U₅(θ,δ) =θlogθ/δ+ (1 -θ)log (1-θ)/(1-δ),(?)δ∈Δ. (7)whereδ=δ(X₁, X₂,…,X_n) is a estimation ofθandΔis a decision-space. In this section, we focus on the entropy loss function, and stud}' the optimal credibility premium and the optimal linear premium under the entropy loss function.Theorem 3.1 (Credibility Premium under Entropy1 Loss Function) Let X_i, i=1 , 2 ,…, denote total claims of a policyholder in the i^th policy period. The distribution of X_i depends on the parameter 6, where 6 varise across policy holders and maybe vector valued. Denote X_n= (X₁, X₂,…,X_n )∈(?) as the policyholder'sclaim experience in the first n periods. For the loss function U₄(θ,δ). the optimal premium Y₄ (X_n) is given byTheorem 3.2(The Optimal Linear Premium under Entropy¹ Loss Function) For the loss function U₄(θ,δ), the optimal linear premium L₄(X_n) is given byL₄(X_n) = z(?) + (1 - z)μ, z = n/(n+k), (9)where (?) = 1/n (?) X_i is the sample mean,μ(θ) = E[X|θ],μ= E[μ(θ)) = E[X],v(θ) =V ar(X|θ), v =E[v(θ)], a = V ar[μ(θ)] = E[(μ(θ) -μ)²], k = u/a.Theorem 3.3 (Credibility Premium under Entropy² Loss Function) Let X_i , i=1, 2,…, denote total claims of a policyholder in the i^th policy period. The distribution of X_i depends on the parameterδ, whereδvarise across policyholders and maybe vector valued. Denote X_n=( X₁, X₂,…,X_n )∈(?) as the policyholder's claim experience in the first n periods. For the loss function U₅(θ,δ), the optimal premium Y₅(X_n) is given byY₅(X_n) = E[μ(θ)|X_n]. (10) Theorem 3.4(The Optimal Linear Premium under Entropy² Loss Function) For the loss function U₅(θ,δ), the optimal linear premium L₅(X_n) is given byL₅(X_n) = z(?) + (1- z)μ, z =n/(n+k), (11)where (?) = 1/n (?) X_i is the sample mean,μ(θ) = E[X|θ],μ= E[μ(θ)] = E[X], v(θ) =V ar(X|θ), v = E[v(θ)], a = V ar[μ,(θ)] = E[(μ(θ) -μ)²], k = v/a.In some estimation and prediction problems, use of symmetric loss functions may be inappropriate, as has been recognized in the literature. That is, a given positive error may be more serious than a given negative error of the same magnitude or vice versa. Varian introduced a very useful asymmetric LINEX loss function that rises approximately exponentially on one side of zero and approximately linearly on the other side in his applied study of real estate assessment.In chapter 4, we will introduce a Linex loss function which is supposed by Varian (1975).U₆(θ,δ) = (δ/θ)^α* -α^* logδ/θ- 1,(?)δ∈Δ. (12)In this section, we focus on the Linex loss function, and study the optimal credibility premium and the optimal linear premium under the Linex loss function.Theorem 4.1 ( Credibility Premium under Linex Loss Function) Let X_i, i=1 , 2 ,…, denote total claims of a policyholder in the i^th policy period. The distribution of X_i depends on the parameterδ, whereδvarise across policyholders and maybe vector valued. Denote X_n=( X₁, X₂,…,X_n )∈(?) as the policyholders claim experience in the first n periods. For the loss function U₆(θ,δ), the optimal premium Y₆(X_n) is given byY₆(X_n) = (E[μ^-α*(θ)|X_n])⁽-1/(α^*). (13)Particularly, whenα^* is equal to 1, the credibility premium is given byY₇(x_n) = 1/(E[μ^-1(θ)|X_n]). (14) Theorem 4.2(The Optimal Linear Premium under Linex Loss Function) For the loss function U₆(θ,δ). the optimal linear premium L₆(X_n) is given byL₆(X_n) = z(?) + (1-z)μ. (15)The credibility factor z is given byParticularly, whenα^* is equal to one, the credibility factor is given bywhere (?) = 1/n (?) X_i is the sample mean,μ(θ) = E[X|θ],μ= E[μ(θ)] = E[X],v(θ) = V ar(X|θ) v =E[v(θ)],a = V ar[μ(θ)] = E[(μ(θ) -μ)²], k =v/a.In chapter 5, if we have r groups of data, each group has m_i datas:x₁₁,x₁₂,…,x_1m1,x₂₁,x₂₂,…,x_2m2,…x_r1, x_r2,…, x_rmr.Using the moment method of estimation, the estimators of credibility formulas under the asymmetric loss functions.