# The Bühlmann–Straub Estimation of Claim Means in Random B-F Reserve Model

Authors: Zhang Yi; Wen Limin; Li Zhilong
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In the B-F reserve model, it is a very critical step to estimate the claim means of the accident year. However, the traditional method uses the prior estimators of the claim means based on the personal experience of actuaries or historical data. This method inevitably carries the subjectivity of the actuary himself. In this paper, a stochastic B-F model is established, and a prior distribution is constructed for the claim means in the accident year. The idea of the credibility theory is used to derive the linear Bayesian estimators of claim means. Finally, the empirical Bayesian method is used to estimate the first two moments of the prior distribution, and the empirical Bayesian estimators of the claim means and the corresponding reserves are derived. The estimators obtained in this paper do not depend on the specific forms of the sample distribution and the prior distribution and can be used directly in practice. In the numerical simulation, our estimates are compared with the traditional B-F estimates and the chain ladder estimates. It is verified that the estimates given in this paper have small mean square error.


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## Body

## 1. Introduction

The liability reserve of the policy refers to the amount estimated by the actuary to pay the insurance claims due in the future. The insurance company must ensure that it has sufficient capital reserves so that it has funds to pay future claims. In the non-life insurance business, the reserve is the largest debt account on the balance sheet, which is directly related to the level of the insurance company’s profit index and the solvency ability. Due to the influence of many complex factors, the assessment of non-life insurance business reserves has been a hot and difficult issue in actuarial theory and applications.Among the many liability reserve models, the distribution-free chain ladder (CL) and Bornhuetter–Ferguson (B-F) method are still the most frequently used claims reserving methods in practice. Refer to Mack [1] and Bornhuetter and Ferguson [2]. In order to estimate the amount of reserves, the actuary classified the claim by the year in which the accident occurred. If the vertical axis is considered as the year of occurrence of the accident, and the horizontal axis is taken as the development year after the accident, the claim data will form an upper triangle. In the chain ladder method, it is assumed that each progress year has a stable development factor. Based on this upper triangular data, the claims in each accident year are predicted and the reserve estimates are obtained. This method is called “chain ladder method”. Because the chain ladder method is simple to calculate and the calculation result is approximately accurate, it is widely used in the reserve assessment in the non-life insurance company.Actually, the early chain ladder method has nonrandom form which is called as “nonrandom chain ladder method”. However, some researchers began to question this method since it has no relevant theoretical support. Furthermore, it cannot give a good explanation. This question was well resolved in Mack [1]. He gives a stochastic model of the chain method, which not only obtains the same claim reserve estimate as the “nonrandom chain ladder method,” but also obtains the error of the estimate. So far, the chain ladder method is still a mainstream approach to reserve estimates. Related studies include Quarg and Mack [3], Hürlimann [4], Pešta and Hudecová [5], Riegel [6], Leonardo et al. [7], Boratyńska [8], Wahl et al. [9], and references therein.Note that, in the chain ladder method, it is assumed that there is a development factorfj in the development year j, such that(1)ECi,j+1Ci0,…,Cij=fjCij,where Ci,j represents the cumulative claims in accident year i∈0,1,…,I at the end of development year j∈0,1,…,J, and we assume that J≤I. For the B-F method, it is assumed that there are given estimates μ^i of the expected ultimate claim of each accident year i and the development pattern γj in progress year j such that(2)ECi,j+kCi0,…,Cij=Cij+βj+k−βjμiforj=1,2,…andj+k≤J,with βj=∑k=0jγk.Different from the chain ladder method, the B-F method attempts to obtain a reserve estimate by combining the prior information of the random parameters and the sample information of loss. Due to the complexity of the risk, claims arising from different policies take different values which form a certain probability distribution. Thus, the estimation and statistical inferences of reserve fall into the Bayesian framework. Note that if the prior estimates of claim means are taken asμ^i=μi for i∈0,1,…,I, the B-F estimator of the reserve is exactly the same as that of the chain ladder estimator. Therefore, in a sense, the B-F model is a generalization of the ladder method under the Bayesian framework. The reserve model for the B-F method has attracted many researchers in recent years. Related discussion can refer to Verrall [10], Alai et al. [11], Saluza et al. [12], Jochen and Anne [13], and references therein. Obviously, the estimates of claim means μi are very crucial in the reserve assessment of the B-F model. A common method is to assume that a prior estimate already exists for μi,i=0,1,…,I, for example, Gogol [14] and Verrall [10]. It is noted that the prior estimates μ^i for i∈0,1,…,I are generally determined by the actuary based on the expert opinions or some empirical information which are determined with greater subjectivity. If the prior estimates are appropriately chosen, an accurate estimate of the reserve can be obtained. On the contrary, a greater error is brought into the estimates of reserve if the prior estimates are wrongly chosen.Therefore, it is very necessary to establish the Bayesian model forμi and get appropriate estimates for μi by the use of Bayesian statistical inference procedures. In the Bayesian statistical model, it is still very difficult how to choose the appropriate prior distribution for μi. In this paper, however, we will introduce the credibility theory to estimate the claim means and derive the linear Bayesian estimators of μi. The estimates obtained depend only on the first two moments of the prior distribution and do not depend on the specific prior distribution. The credibility theory is a premium pricing model by using both sample information and prior information, which is widely used in non-life insurance actuarial science. One can refer to Bühlmann [15], Bühlmann and Gisler [16], Wen et al. [17], Wen et al. [18], and Tsai and Wu [19] for more detailed introduction of credibility theory. About the research studies of liability reserve by using credibility theory, one can refer to Benktander [20], Hovinen [21], Hesselager and Witting [22], Vylder [23], Gisler and Wüthrich [24], Taylor [25], and so on.The rest of the paper is arranged as follows. Section2 gives the assumptions and notations of the model. Section 3 gives the nonhomogeneous and homogeneous credibility estimates of the claim means. In addition, the estimates of structural parameters in credibility factors are derived, and the empirical Bayesian estimates of reserves are discussed. In the fourth section, we give a numerical example to illustrate the calculations of the empirical Bayesian estimators and to compare the credibility estimators obtained in this paper with the traditional chain ladder method and B-F method. Finally, a real example is given to show the results derived in this paper.


## 2. Bühlmann–Straub Model for Claim Mean

In the loss triangle data of the liability reserve, it is assumed that the claims for the different accident years are independent of each other and the cumulative claim for each yearCij satisfies the conditions of the B-F model:(3)ECi,0μ=μiβ0,i=0,1,2,…,I,ECi,j+kCi,0,…,Ci,j,μ=Ci,j+βj+k−βjμi,for j+k≤J, where μ=μ0,…,μIT are claim means for every accident year, and βj=∑k=0jγj with γj>0, where the superscript “T ” indicates the transpose of the vector. Without loss of generality, one can assume that βJ=∑k=0Jγk=1.According to the double expectation formula, we get(4)ECi,jμ=EECi,jCi,0,…,Ci,j−1,μμ=μiβj.Let(5)Xi,j=Ci,j−Ci,j−1denote the incremental claims. So the incremental claim Xi,j satisfies(6)EXi,jμ=μiγj,where γj=βj−βj−1.We assume that the claim means are independent random variables with the same prior distribution. Therefore, we propose the following random B-F model.Assumption 1.
There are functionσ2μi and fixed numbers wij for i∈0,1,2,…,I, j∈0,1,2,…,J such that(7)EXi,jμ=γjμi,VarXi,jμ=σ2μiwij,(8)CovXi,j,Xi,k=0forj≠k,k=0,1,…,J.Assumption 2.
Theμ0,μ1,…,μI are assumed to be independent variables with the same prior distribution πμ.Assumption 3.
Assume that claims arising from different accident years are independent of each other.Remark 1.
According to Assumptions(1)–(3), this model is similar to the Bühlmann–Straub assumptions in the credibility theory. See, for example, Bühlmann and Gisler [16]. So this model is also called the Bühlmann–Straub model for claim means.
At timeI, we have the sample information:(9)DI=Cij,i+j≤I,j≤J=Xij,i+j≤I,j≤J,and prior information πμ. In this paper, our purpose is to estimate μi and predict(10)DIc=Cij,i+j>I,i≤I,j≤J.
The outstanding loss liabilities for accident yeari at time I are given by(11)Ri=CiJ−Ci,I−i,I−J+1≤i≤I,and the total outstanding loss liabilities of all accident years are given by(12)R=∑i=I−J+1IRi.
Obviously, if the square loss is used, by minimizing the conditional expectation(13)ECi,J−gDI2DI,we get the optimal predictors of CiJ as(14)C^iJ∗=Ci,I−i+1−β^I−iEμiDI,for i∈0,1,…,I, where β^I−i is the chain ladder estimator of βI−i, i.e.,(15)β^j=∏k=jJ−1f^k−1,j=0,1,2,…,J.
Here, the estimators of chain ladder factorfj are given by(16)f^j=∑i=0I−j−1Ci,j+1∑i=0I−j−1Ci,j,j=1,2,…,J−1.
For a detailed introduction to the estimation of development factors, refer to Gisler and Wüthrich [24].


## 3. Credibility Estimation of Reserves

### 3.1. Nonhomogeneous Credibility Estimation of Claim Mean

According to (14), in order to predict the ultimate loss Ci,J, the μ^i=EμiDI must be solved. In this paper, we introduce the following notations:(17)Eμi=ν0,Varμi=τ02,Eσ2μi=σ02,and constrain the estimate of μi to the linear function of the sample DI. For convenience, it is assumed that I=J in the following. With slight modification, the models can be extended to the case of I≥J.Under the square loss function, we solve the optimization problem,(18)minα,αst∈REμi−α−∑s=0I∑t=0I−sαstXst2and get the following theorem.Theorem 1.
Under Assumptions (1)–(3), by minimizing (18), the nonhomogeneous credibility estimators of μi are given by(19)μi^BS=ZiwX¯iw+1−Ziwν0,for i∈0,1,2,…,I, where(20)X¯iw=1wi⋅∑j=0I−iwijγjXij,Ziw=τ02wi⋅σ02+τ02wi⋅,and wi⋅=∑j=0I−iwijγj2.Remark 2.
In Theorem1, if the weights are taken as wij=1/γj2 for any i and j, we have(21)EXi,jγjμ=μi,VarXi,jγjμ=σ2μiwijγj2.
In addition, we denoteYij=Xij/γj and get(22)EYi,jμ=μi,VarYi,jμ=σ2μiwijγj2,which are exactly the Bühlmann–Straub models for conditional means in credibility theory. In addition, if we take wij=1/γj2, the assumptions are simplified to Bühlmann’s model for claims Yi,j,i=0,1,…,I,j=0,1,…,J. See Bühlmann and Gisler [16] for more detail.Proposition 1.
Theμ^iBS is an unbiased estimator of μi, i.e., Eμ^iBS=Eμi=ν0, and the mean square error is given by(23)MSEμ^iBS=τ02σ02σ02+τ02∑j=0I−iwijγj2.By insertingμ^iBS into (14), the estimators of ultimate loss Ci,J for i∈0,1,…,I can be derived, and the estimator of reserve can also be derived, which is stated as the following proposition.Proposition 2.
Under Assumptions(1)–(3), if the estimator of μi is taken as μ^iBS, the estimator of ultimate loss CiJ for i∈0,1,…,I is given by(24)C^iJBS=Ci,I−i+1−β^I−iZiwX¯iw+1−Ziwν0.
In addition, the estimators of reserveRi and total reserve are given by(25)R^iBS=1−β^I−iZiwX¯iw+1−Ziwν0,(26)R^BS=∑i=1I1−β^I−iZiwX¯iw+1−Ziwν0,respectively.


### 3.2. Homogeneous Credibility Estimation

In practice, we note thatμ0 in formula (24) is unknown and needs to be estimated by the samples. Similar to the nonhomogeneous credibility estimation method, we define the homogeneous linear function class of DI as(27)LeX=μ^i=∑s=0I∑t=0I−sαstXstwith∑s=0I∑t=0I−sγtαst=1.and solve the optimal estimator of μi in LeX. We get the following theorem.Theorem 2.
By solving the optimization problem,(28)minα,αst∈REμi−∑s=0I∑t=0I−sαstXst2,∑s=0I∑t=0I−sαstγt=1,we get the optimal homogeneous estimators of μi as(29)μ^iwhom=ZiwX¯iw+1−Ziwν0^,i=0,1,…,I,where(30)ν0^=∑k=0IZkwX¯kw∑k=0IZkw.Remark 3.
According to the optimal homogeneous credibility estimators ofμi for i∈0,1,…,I, the homogeneous credibility of the corresponding estimators of ultimate loss CiJ is given by(31)C^iJwhom=Ci,I−i+1−β^I−iZiwX¯iw+1−Ziwν0^.and the homogeneous credibility estimators of reserve Ri are(32)R^iwhom=1−β^I−iZiwX¯iw+1−Ziwν0^,for i∈0,1,2,…,I. As a result, the estimator of the total reserve is given by(33)R^whom=∑i=0I1−β^I−iZiwX¯iw+1−Ziwν0^.


### 3.3. The Estimation of Variance Parameters

The credibility estimator of the total reserve is still unable to use in practice because they still contain unknown structural parameters such asτ02 and σ02. In this part, we will give the unbiased estimators of these structural parameters.Proposition 3.
An estimator ofσ02 is given by(34)σ02^=1I∑i=0I−11I−i∑j=0I−iwijXij−γjX¯iw2.
In addition, theσ02^ is an unbiased estimator of σ02 when γj,j=0,1,…,J−1 are given.Proposition 4.
An estimator ofτ02 is given by(35)τ02^=w⋅⋅w⋅⋅2−∑i=0I−1wi⋅2∑i=0I−1wi⋅X¯iw−X¯¯w2−Iσ02^,where(36)X¯¯w=1w⋅⋅∑i=0I−1wi⋅Xi¯w,w⋅⋅=∑i=0I−1wi⋅,wi⋅=∑j=0I−iwijγj2.
Furthermore, theτ02^ is an unbiased estimator of τ02 if γj,j=0,1,…,J−1 are given constants.Remark 4.
If we denotewij′=1/γj2 and take wij=wij′, some symbols degenerate to wi′⋅=∑j=0I−iwij′γj2=I−i+1, w⋅⋅′=∑i=0I−1wi⋅=1/2II+1,(37)X¯iw′=1I−i+1∑j=0I−iYij,X¯¯w′=2II+1∑i=0I−1I−i+1Y¯i.
DenoteY¯i=1/I−i+1∑j=0I−iYij and Y¯¯=2/II+1∑i=0I−1I−i+1Y¯i, and thus, the estimators σ02^ and τ02^ can be simplified as follows:(38)σ02^=1I∑i=0I−11I−i∑j=0I−i1γjYij−Y¯i2,τ02^=63I+2I−1∑i=0I−1I−i+1Y¯i−Y¯¯2−Iσ02^,respectively.
In practice, the estimatorsγ^j for j∈0,1,…,J are given in terms of the chain ladder method. By inserting γ^j into the estimators of σ02 and τ02, then we get the estimators σ02˜ and τ02˜. However, the estimators σ02^^ and τ02^^ are no longer unbiased. In the actual data, it is possible that τ02˜<0, we usually take τ02˜+=maxτ02˜,0 as the ultimate estimator of τ02.
Finally, by insertingγ^j,j=0,1,…,J−1 and σ02˜ and τ02˜+ into the estimator of ν0, thus the ultimate estimator of ν0 is given by(39)ν0˜=∑k=0IZ^kwX¯kw∑k=0IZ^kw,where(40)Z^iw=τ02˜+∑j=0I−iwijγ^j2σ02˜+τ02˜+∑j=0I−iwijγ^j2.
Thus, the estimators of ultimate lossCiJ and reserves Ri are given by(41)C^iJexp=Ci,I−i+1−β^I−iZ^iwX¯iw+1−Z^iwν0˜,(42)R^iexp=1−β^I−iZ^iwX¯iw+1−Z^iwν0˜,for i∈0,1,2,…,I. Correspondingly, the credibility estimator of the total reserve is(43)R^exp=∑i=0I1−β^I−iZ^iwX¯iw+1−Z^iwν0˜.
The estimatorR^exp is called the empirical Bayesian estimator of total reserve. Obviously, the empirical Bayesian estimator does not depend on any unknown parameters and can be used directly in practice.


## 3.1. Nonhomogeneous Credibility Estimation of Claim Mean

According to (14), in order to predict the ultimate loss Ci,J, the μ^i=EμiDI must be solved. In this paper, we introduce the following notations:(17)Eμi=ν0,Varμi=τ02,Eσ2μi=σ02,and constrain the estimate of μi to the linear function of the sample DI. For convenience, it is assumed that I=J in the following. With slight modification, the models can be extended to the case of I≥J.Under the square loss function, we solve the optimization problem,(18)minα,αst∈REμi−α−∑s=0I∑t=0I−sαstXst2and get the following theorem.Theorem 1.
Under Assumptions (1)–(3), by minimizing (18), the nonhomogeneous credibility estimators of μi are given by(19)μi^BS=ZiwX¯iw+1−Ziwν0,for i∈0,1,2,…,I, where(20)X¯iw=1wi⋅∑j=0I−iwijγjXij,Ziw=τ02wi⋅σ02+τ02wi⋅,and wi⋅=∑j=0I−iwijγj2.Remark 2.
In Theorem1, if the weights are taken as wij=1/γj2 for any i and j, we have(21)EXi,jγjμ=μi,VarXi,jγjμ=σ2μiwijγj2.
In addition, we denoteYij=Xij/γj and get(22)EYi,jμ=μi,VarYi,jμ=σ2μiwijγj2,which are exactly the Bühlmann–Straub models for conditional means in credibility theory. In addition, if we take wij=1/γj2, the assumptions are simplified to Bühlmann’s model for claims Yi,j,i=0,1,…,I,j=0,1,…,J. See Bühlmann and Gisler [16] for more detail.Proposition 1.
Theμ^iBS is an unbiased estimator of μi, i.e., Eμ^iBS=Eμi=ν0, and the mean square error is given by(23)MSEμ^iBS=τ02σ02σ02+τ02∑j=0I−iwijγj2.By insertingμ^iBS into (14), the estimators of ultimate loss Ci,J for i∈0,1,…,I can be derived, and the estimator of reserve can also be derived, which is stated as the following proposition.Proposition 2.
Under Assumptions(1)–(3), if the estimator of μi is taken as μ^iBS, the estimator of ultimate loss CiJ for i∈0,1,…,I is given by(24)C^iJBS=Ci,I−i+1−β^I−iZiwX¯iw+1−Ziwν0.
In addition, the estimators of reserveRi and total reserve are given by(25)R^iBS=1−β^I−iZiwX¯iw+1−Ziwν0,(26)R^BS=∑i=1I1−β^I−iZiwX¯iw+1−Ziwν0,respectively.


## 3.2. Homogeneous Credibility Estimation

In practice, we note thatμ0 in formula (24) is unknown and needs to be estimated by the samples. Similar to the nonhomogeneous credibility estimation method, we define the homogeneous linear function class of DI as(27)LeX=μ^i=∑s=0I∑t=0I−sαstXstwith∑s=0I∑t=0I−sγtαst=1.and solve the optimal estimator of μi in LeX. We get the following theorem.Theorem 2.
By solving the optimization problem,(28)minα,αst∈REμi−∑s=0I∑t=0I−sαstXst2,∑s=0I∑t=0I−sαstγt=1,we get the optimal homogeneous estimators of μi as(29)μ^iwhom=ZiwX¯iw+1−Ziwν0^,i=0,1,…,I,where(30)ν0^=∑k=0IZkwX¯kw∑k=0IZkw.Remark 3.
According to the optimal homogeneous credibility estimators ofμi for i∈0,1,…,I, the homogeneous credibility of the corresponding estimators of ultimate loss CiJ is given by(31)C^iJwhom=Ci,I−i+1−β^I−iZiwX¯iw+1−Ziwν0^.and the homogeneous credibility estimators of reserve Ri are(32)R^iwhom=1−β^I−iZiwX¯iw+1−Ziwν0^,for i∈0,1,2,…,I. As a result, the estimator of the total reserve is given by(33)R^whom=∑i=0I1−β^I−iZiwX¯iw+1−Ziwν0^.


## 3.3. The Estimation of Variance Parameters

The credibility estimator of the total reserve is still unable to use in practice because they still contain unknown structural parameters such asτ02 and σ02. In this part, we will give the unbiased estimators of these structural parameters.Proposition 3.
An estimator ofσ02 is given by(34)σ02^=1I∑i=0I−11I−i∑j=0I−iwijXij−γjX¯iw2.
In addition, theσ02^ is an unbiased estimator of σ02 when γj,j=0,1,…,J−1 are given.Proposition 4.
An estimator ofτ02 is given by(35)τ02^=w⋅⋅w⋅⋅2−∑i=0I−1wi⋅2∑i=0I−1wi⋅X¯iw−X¯¯w2−Iσ02^,where(36)X¯¯w=1w⋅⋅∑i=0I−1wi⋅Xi¯w,w⋅⋅=∑i=0I−1wi⋅,wi⋅=∑j=0I−iwijγj2.
Furthermore, theτ02^ is an unbiased estimator of τ02 if γj,j=0,1,…,J−1 are given constants.Remark 4.
If we denotewij′=1/γj2 and take wij=wij′, some symbols degenerate to wi′⋅=∑j=0I−iwij′γj2=I−i+1, w⋅⋅′=∑i=0I−1wi⋅=1/2II+1,(37)X¯iw′=1I−i+1∑j=0I−iYij,X¯¯w′=2II+1∑i=0I−1I−i+1Y¯i.
DenoteY¯i=1/I−i+1∑j=0I−iYij and Y¯¯=2/II+1∑i=0I−1I−i+1Y¯i, and thus, the estimators σ02^ and τ02^ can be simplified as follows:(38)σ02^=1I∑i=0I−11I−i∑j=0I−i1γjYij−Y¯i2,τ02^=63I+2I−1∑i=0I−1I−i+1Y¯i−Y¯¯2−Iσ02^,respectively.
In practice, the estimatorsγ^j for j∈0,1,…,J are given in terms of the chain ladder method. By inserting γ^j into the estimators of σ02 and τ02, then we get the estimators σ02˜ and τ02˜. However, the estimators σ02^^ and τ02^^ are no longer unbiased. In the actual data, it is possible that τ02˜<0, we usually take τ02˜+=maxτ02˜,0 as the ultimate estimator of τ02.
Finally, by insertingγ^j,j=0,1,…,J−1 and σ02˜ and τ02˜+ into the estimator of ν0, thus the ultimate estimator of ν0 is given by(39)ν0˜=∑k=0IZ^kwX¯kw∑k=0IZ^kw,where(40)Z^iw=τ02˜+∑j=0I−iwijγ^j2σ02˜+τ02˜+∑j=0I−iwijγ^j2.
Thus, the estimators of ultimate lossCiJ and reserves Ri are given by(41)C^iJexp=Ci,I−i+1−β^I−iZ^iwX¯iw+1−Z^iwν0˜,(42)R^iexp=1−β^I−iZ^iwX¯iw+1−Z^iwν0˜,for i∈0,1,2,…,I. Correspondingly, the credibility estimator of the total reserve is(43)R^exp=∑i=0I1−β^I−iZ^iwX¯iw+1−Z^iwν0˜.
The estimatorR^exp is called the empirical Bayesian estimator of total reserve. Obviously, the empirical Bayesian estimator does not depend on any unknown parameters and can be used directly in practice.


## 4. Numerical Simulation

In this paper, we give the Bühlmann–Straub credibility estimators of claim means, as well as the credibility estimator of the total reserve. In addition, the estimators of development factors and the structural parameters are discussed. Finally, the empirical Bayesian estimator of the reserve is obtained. In the following, we will use numerical simulation to verify two problems. First, we will compare and check the differences between the credibility estimates and Bayesian estimates of claim means; second, we will examine the difference between the empirical Bayesian estimates and the credibility estimates of reserves.In the simulations, we takewij=1/γj2 and denote Yij=Xij/γj. In addition, we assume that(44)θi=1μi,i=0,1,2,…,I,are i.i.d. random variable with Gaα,β distribution, and the density function of θi is given by(45)πθ=βαΓαθα−1exp−βθ,θ>0.Givenμi, the Yij,j=0,1,2,… are independent and distributed as follows:(46)fyθi=θiexp−θiy,y>0.So we can easily get the following:(47)ν0=βα−1,τ02=β2α−12α−2,σ02=β2α−1α−2,Zi=I−i+1I−i+1+α−1.In addition, we takeI=J=6, α=2, β=1, γj=0.1 for j=0,1,…,J. Therefore, the posterior distribution of θi is given by(48)θiDI∼Gaα+∑j=0I−iYij,β+I−i+1.In the simulation progress, we first generate a claim sequenceYi,j,i,j=0,1,…,I based on the gamma distribution Gaα,β. In the second, the Xi,j,Ci,j can be calculated for i,j∈0,1,…,I. Thus, we get the following estimates:(1)
Chain ladder estimates (CL):Ci,JCL=Ci,I−ifI−i…fJ−1.(2)
B-F estimates (B-F):Ci,JB−F=Ci,I−i+1−βI−iμ^i, where μ^i are the prior estimates of μi for i∈0,1,…,I. Since there is no prior information, we directly take μ^i=ν0 for i∈0,1,…,I.(3)
Bayesian estimates (BE):(49)C^i,J∗=Ci,I−i+1−βI−iEμiDI=Ci,I−i+1−βI−iβ+I−i+1α+∑j=0I−iYij−1.(4)
Credibility estimates (CE):(50)CiJcred=Ci,I−i+1−βI−iZiY¯i+1−Ziν0.Under 10000 simulations, the mean predictor error of CL, B-F, BE, and CE estimates is listed in the following tables.The MSE in Table1 is calculated under given parameters γj,βj,fj and ν0,τ02,σ02, while in Table 2 the MSE of empirical Bayesian version is calculated by inserting the estimator of the structure parameters with simulated data, and the prior estimators of B-F method are given by μ^i=ν0 for i∈0,1,2,…,I.Table 1
The MSE of CL, B-F, BE, and CE under Bayesian setup.
i0123456CL09.37333.46521.86220.35491.31311.1630B-F00.14410.25782.07590.87481.50951.4992BE00.20770.49542.55471.58582.37682.2290CE00.09930.15550.52300.52791.16231.1992Table 2
The MSE of CL, B-F, BE, and CE under empirical Bayesian setup.
i0123456CL025.639415.26812.92900.94220.64481.9643B-F00.05240.14360.37540.78140.75401.1165Cred00.07740.20120.42900.73000.81691.2150Obviously, from Table1, the mean square error of the credibility estimates is smaller than that of CL estimates and B-F estimates. Among these estimates, the CL estimates have the largest mean square error. The main reason is that CL estimates ignore the use of prior information. In conclusion, the simulation results show that the credibility estimation is a good choice within all these estimation methods in the random B-F model.In Table2, we estimate the parameters γj,βj,fj and ν0,τ02,σ02 by using simulated data DI. In addition, the CL estimates, B-F estimates, and credibility estimates are derived correspondingly. At this time, the mean square error is increased due to the estimates of the structural parameters used in the empirical Bayesian setup. While the known prior estimates μ^i=ν0 are still be used, the resulting credibility estimates have a slightly larger mean square error than the B-F estimator. However, the MSE of empirical Bayesian credibility estimator is still smaller than that of the CL estimates.In the following, we apply this method to calculate the reserves in a practical example of an insurance company. The data come from Table 2.2 on page 20 of Alai et al.’s study. [11]. Based on the insurance claim data, we calculated the reserve estimates for the three methods including B-F method, chain ladder method, and credibility estimation. The results are listed as follows, where the prior estimates of the B-F method are taken as the same of Alai et al. [11], and the weights in credibility models are taken as wij=1/γj2 for i∈0,1,2,…,I and j∈0,1,2,…,J (Table 3).Table 3
Comparison of reserve estimates for the three methods.
PoliciesiPrior estimatesEstimates of ultimate lossEstimates of loss reservesμ^iC^iJBFC^iJCLC^iJCreR^iBFR^iCLR^iCre0116531011114812411148124111481240001113673061066431610663318106614181612413226.2715126.292109629651066274910662008106615012699825749.5426257.453106167629761643975860697593663757535298.4234538.474110448819882350987221898864859543499568.7385301.62511480700101137771009224710086401178024150648.41156494.25611413572962332895681439575260341305293238.44286092.48711126527883030187053788733654574089477442.86449163.5281098654889673758691971879107913186461142349.781043238.98911618437104439539626383980551947683844129950.963950811.42Total73565796367473.416047024.48From the table above, the total reserve obtained by the B-F method is larger than that of credibility estimates and CL estimates, mainly because the actuary obtains a larger value of the prior estimatesμi for i∈0,1,2,…,I. The reserve obtained by the credibility method is larger than that of CL estimates but is smaller than that of B-F estimates. Obviously, the reserve estimate obtained by the B-F method depends on the selection of the prior distribution, which is subjectively determined by the actuary, while the chain ladder method only uses sample information and ignores the use of prior information. The credibility estimates use both the sample information and prior information. However, the estimates derived by the credibility method do not depend on the specific distribution form of the prior distribution, which avoids the subjectivity of the selection of the prior distribution. Therefore, the credibility estimators are recommended to be used in the assessment of the liability reserve in insurance companies.



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*Source: 6062906-2020-05-07.xml*