CREDIBILITY MODELS WITH DEPENDENCE

JOURNAL OF INDUSTRIAL AND MANAGEMENT OPTIMIZATION Volume 11, Number 2, April 2015

doi:10.3934/jimo.2015.11.365 pp. 365–380

CREDIBILITY MODELS WITH DEPENDENCE STRUCTURE OVER RISKS AND TIME HORIZON

Weizhong Huang Department of Mathematics, Shanghai Maritime University Shanghai, China

Xianyi Wu1 Center of International Finance and Risk Management Department of Statistics and Actuarial Science East China Normal University, Shanghai, China

(Communicated by Christian-Oliver Ewald) Abstract. In this paper, the B¨ uhlmann and B¨ uhlmann-Straub’s credibility models with a type of dependence structures over risks and over time are discussed. The inhomogeneous and homogeneous credibility estimators of risk premium were derived. The inhomogeneous credibility estimators for the existing credibility models with common effects are extended to slightly more general versions. The results obtained shake the classical meaning of the term “credibility premiums”.

1. Introduction. Credibility is an experience ratemaking technique developed in insurance industries to determine a future premium of a risk based on experience of both the policyholder’s own and all the individuals in the portfolio, to which the particular policy belongs. While the whole experience is used to settle an overall premium level for this portfolio, the experience from every particular individuals serves to distinguish individual premiums so as to reflect the variations of risk features over individuals. Modern credibility theory is believed to be attributed to the reputable contributions by B¨ uhlmann (1967) and B¨ uhlmann and Straub (1970) who based traditional credibility notion on modern theory of Bayesian statistics and thus interpreted the classical credibility factor by means of the ratio of prior and sample variances. Remarkable subsequent developments in this line include Hachemeister (1975)’s regression models that take into account certain explanatory variables to reflect more specifically the differences among risks, Jewell (1975)’s hierarchical models which split large portfolio that are obviously inhomogeneous into sub-portfolio (sectors) of approximately homogeneous risks, Danneburg (1995)’s cross classification credibility models that are determined by qualitative risk factors, Frees et al. (1999, 2001)’s explanation of credibility techniques by means of contemporary statistical 2010 Mathematics Subject Classification. Primary: 91B30, 62P05; Secondary: 97M30. Key words and phrases. Credibility model, orthogonal projection, credibility factor, dependence. The authors are supported by NSFC grant 71371074 and Shanghai Philosophy and Social Science Foundations grant 2020BJB004. 1 Corresponding to: [email protected].

365

366

WEIZHONG HUANG AND XIANYI WU

theory of longitudinal data analysis, Pan et al. (2008) and Wen, Wu and Zhao (2009)’s treatment on the credibility estimates on Esscher premums, and so on; see, e.g., B¨ uhlmann and Gisler (2005) for a comprehensive treatment of credibility theory. In the prevailing credibility models, a fundamental assumption is that the claims are independent over individuals (referred to as independence over risks) and for each particular individual, the claims are also conditionally independent given the risk feature of the individual (conditional independence over time). In mathematical words, let X1 , X2 , . . . , XK denote a portfolio of K risks under observation. Under the classical credibility setting, each individual contributes a sequence of claims experience Xi = (Xi1 , Xi2 , . . . Xini ) over ni time periods, the distribution of which is characterized by its risk parameter Θi such that the vectors (Xi , Θi ), i = 1, 2, . . . , K, are mutually independent and, for each i, Xi1 , Xi2 , . . . Xini are conditionally independent given Θi . While such an independence structure may be appropriate in numerous practical situations, it is far from being universal in this complex world and in fact it has been recognized that there exist many important insurance scenarios where these dominant assumptions are certainly violated. On one hand, many researchers have explored certain types of conditional dependence over time, including Bolanc´e et al. (2003) who estimated and tested autoregressive specifications for dynamic random effects in a frequency risk model, Purcaru and Denuit (2002, 2003) who studied a type of dependence induced by the introduction of common latent variables in the annual numbers of claims reported by policyholders and revealed how the dependence structure affects the credibility estimate in the Poisson claim frequency models, Frees et al. (1999, 2001) showed how to produce credibility predictors for linear longitudinal and panel data models and Frees and Wang (2005) considered a generalized linear model framework for modeling marginal claims distributions, which allows dependence characterized by the Student-t copula to model the dependence over time for a class of risks, among others. On the other hand, in the communities of insurance economics and actuarial science, it has been commonly recognized that the dependence across risks are frequently encountered in real life practice. Examples include house insurance for which geographic proximity of the insureds may result in exposures to common catastrophes, and motor vehicle insurance where one accident may involve several insureds. There have been many remarkable efforts in the existing actuarial literature to study the impacts of dependent risks in various aspects; see e.g., Dhaene and Goovaerts (1996), Dhaene et al (2002a,b), Lu and Zhang (2004), M¨ uller (1997), Wang et al. (1997), Wu and Zhou (2006), and the references therein. It is believed that Yeo and Valdez (2006) are the first who proposed a credibility model with claim dependence across risks characterized by common effects (stochastic latent risk parameters) which affect all individuals and investigated the corresponding credibility premiums under normally distributed claim amounts. Wen, Wu and Zhou (2009) extended Yeo and Valdez (2006) model by dropping the normality assumption so as to allow arbitrarily distributed risk claims. However, for the experience ratemaking by means of credibility techniques, it appears that the existing results have so far been reported only on two separated directions: one makes focus on dependence over risks but adopts the prevailing conditional independence over time horizon, whereas the other is dedicated only to the conditional dependence but with cross-sectional dependence neglected, and, as a matter of fact, there have not yet been any research efforts that accommodate

CREDIBILITY MODELS WITH DEPENDENCE STRUCTURE

367

simultaneously the dependence cross risks and the conditional dependence over the time horizons. Hence, the existing theory cannot apply in such insurance situations as here discussed: In automobile third-party liability insurance, the whole portfolios are usually divided into certain sub-groups by a number of classification features, such as age, gender, occupation, marital status and smoking behaviors of the policyholders, the types, use and even colors of their cars, so as to make the policies in the same group are as homogeneous as possible. In such countries as Mainland China with the presence of high unbalance in economical development, geographical proximity may be an important factor that makes the risks in a same group of certain dependence, similar to the house insurance case discussed by Yeo and Valdez (2006). Meanwhile, for a fixed policy, due to the use of loss preventing mechanism in premium schemes such as no-claim-discount, there may exist certain short-term dependence relationships between the losses over insurance periods: increased claims in an insured period will cause an increase in the premium of the next period and thus drive the policyholder to make his/her efforts to prevent accidents or decrease the loss in accidents as possible as he/she can. In order to provide certain actuarial solutions for the situations similar to what we just discussed, in this paper, we investigate a more general dependence structure to accommodate the dependence over time and cross risks. It turns out that our model covers those of Wen, Wu and Zhou (2009) and Yeo and Valdez (2006) as special cases and can also be related to the extensive literature, e.g., Purcaru and Denuit (2002, 2003), Frees et al. (1999, 2001), Frees and Wang (2005), regarding risks with conditional dependence over time periods. To be specific, we examine the B¨ uhlmann and B¨ uhlmann-Straub’s settings with dependence structure over risks and time horizon to establish the credibility formulae for those two classical models under this new general dependence structure. The differences between this general, Wen, Wu and Zhou (2009)’s and the classical B¨ uhlmann (1967)’s models are highlighted in Remark 1. The rest of the paper is arranged as follows. In Section 2, some preliminaries are prepared. Section 3 derives the credibility formulae for the classical B¨ uhlmann’s models with dependence structure over risks and time. The credibility models with common effects are also discussed there. The results are extended to the B¨ uhlmannStraub model in Section 4.

2. Preliminaries. Consider a portfolio of K insured individuals, of which each individual i is associated with a sequence of claim experience Xij over ni time periods j = 1, 2, . . . , ni . Write Xi = (Xi1 , Xi2 , . . . , Xini )0 for every i = 1, 2, . . . , K and X = (X1 , X2 , . . . , XK )0 . The distribution of each Xi is characterized by its risk parameter, say Θi . Write Θ = (Θ1 , · · · , ΘK ). It means the distribution of Xi given Θ = (Θ1 , · · · , ΘK ) depends only on Θi . We are concerned with the prediction of a future claim Xi,ni +1 for every individual i ∈ {1, 2, . . . , K}, taking into account all observed claim experiences X = (X1 , X2 , . . . , XK )0 . To be specific, we will discuss both inhomogeneous and homogeneous credibility premiums of an arbitrary future claim Xi,ni +1 , which are denoted bi,n +1 and X b hom and defined as the solutions of by X i i,ni +1 min E[(Xi,ni +1 − a0 −

a0 ,aj

K X j=1

a0j Xj )2 ],

(1)

368

WEIZHONG HUANG AND XIANYI WU

and min E[(Xi,ni +1 − aj

K X

k X a0j Xj )2 ] subject to E( a0j Xj ) = E(Xi,ni +1 ),

j=1

j=1

respectively, where a0 ∈ R, aj ∈ Rnj . Write K X  L(X, 1) = a0 + a0i Xi , a0 ∈ R, aj ∈ Rnj , i = 1, 2, . . . , K

(2)

(3)

i=1

to indicate the linear space generated by the elements of X and the constant 1, and K k X X Le (X) = a0j Xj : aj ∈ Rnj , i = 1, 2, . . . , K, E a0j Xj = EXi,ni +1 (4) j=1

j=1

Pk the linear space spanned by the elements of X satisfying the constraint E( j=1 a0j Xj ) = E(Xi,ni +1 ). Then the standard theory (cf., e.g., B¨ uhlmann and Gisler (2005) and Wen, Wu and Zhou (2009)) states that the inhomogeneous and homogeneous credibility premiums of Xi,ni +1 are the orthogonal projection of Xi,ni +1 on L(X, 1) and Le(X) respectively, i.e.,

hom bi,n +1 = Pro Xi,n +1 |L(X, 1) and X bi,n X = Pro Xi,ni +1 |Le (X) . (5) i i i +1 To the end of computing these projections, we recall the following facts which can be found also in, e.g., B¨ uhlmann and Gisler (2005) and Wen, Wu and Zhou (2009). For the two closed subspace M and M 0 such that M ⊂ M 0 ⊂ L2 ,
Pro(Y |M ) = Pro Pro(Y |M 0 )|M .

Lemma 2.1.

For random vectors Y and X, the orthogonal projection Y on L(X, 1) is
−1 Pro Y |L(X, 1) = E(Y ) + Cov(Y, X) [Cov(X)] (X − EX), and on Le (X) is
−1 Pro Y |Le (X) = Cov(Y, X) [Cov(X)] X −1

+

E(Y ) − Cov(Y, X) [Cov(X)]


EX (EX)0

−1

(EX)0 [Cov(X)] EX Especially for a degenerated random variable a, we have

−1

[Cov(X)]

X.

−1


Pro a|Le (X) =

a(EX)0 [Cov(X)] −1

(EX)0 [Cov(X)]

EX

X.

3. B¨ uhlmann’s model with dependence. We first formulate the model of the joint distribution of Θ = (Θ1 , · · · , ΘK ) and X that features the dependence structure both in time horizon and among risks by the following Assumption 1. Assumption 1. 1. Given Θ = (Θ1 , · · · , ΘK ), the random variables Xi , i = 1, 2, . . . , K, have conditional moments: E(Xi |Θi ) = µ(Θi )1ni , Cov(Xi |Θi ) = σ 2 (Θi )Ini + ρ(Θi )1ni 10ni , where Ini is the ni × ni identity matrix and 1ni the ni -dimensional column vector for which every component is 1.

CREDIBILITY MODELS WITH DEPENDENCE STRUCTURE

369

2. The risk parameters Θ = (Θ1 , · · · , ΘK ) are random variables satisfying   E [µ(Θi )] = µi , E σ 2 (Θi ) = σi2 , E [ρ(Θi )] = ρi , Cov [µ(Θi ), µ(Θj )] = τij . We below need the notations → − µ = (µ , · · · , µ )0 , R = (τ ) 1

K

ij K×K


and Gi = σi2 − ρi Ini + ρi 1ni 10ni .

(6)

Here we would like to note that the classical B¨ uhlmann’s credibility model can be deduced from Assumption 1 by setting ρ(Θi ) = 0, µ1 = µ2 = · · · = µK , σ12 = σ22 = 2 · · · = σK and τij = 0 for all i, j = 1, 2, . . . , K, i 6= j. Moreover, by taking ρ(Θi ) = 0 and τij = η 2 if i = j and γ if i 6= j for certain constant η 2 and γ (satisfying γ ≤ η 2 ), one obtains the so-called credibility model with common effects discussed by, e.g., Wen, Wu and Zhou (2009). Hence, the model formulated here extends the model by Wen, Wu and Zhou (2009). See Assumption 2 and the following Theorem 3.4 for more specific discussion. We are concerned with the prediction of future claims F = (X1,n1 +1 , . . . , XK,nK +1 )0 , taking use of all information from historical claims X1 , X2 , . . . , XK . To be more specific, our objective is to solve the problem of minimizing mean squared errors under distributional Assumption 1. Theorem 3.2 below gives the inhomogeneous credibility premium for the model formulated in Assumption 1. The following lemma can help to smooth the proof of that theorem. Lemma 3.1. Under Assumption 1: 0 0 (1) The covariance matrix of the random vector X = (X10 , · · · , XK ) is ΣXX =

diag(1n1 , · · · , 1nK )R diag(10n1 , · · · , 10nK ) + diag(G1 , · · · , GK ).

where diag indicates block diagonal matrix. Consequently, the inverse of ΣXX can be represented as  −1 0 −1 Σ−1 − Ψ R−1 + Σ−1 Ψ, (7) XX = G where G = diag(G1 , · · · , GK ), R and Gi , i = 1, 2, . . . , k are defined in (6), 1 σ 2 +(n −1)ρK σ 2 +(n −1)ρ1 , · · · , K nKK ) and Ψ = diag( σ2 +(nn11−1)ρ1 , · · · , Σ = diag( 1 n11 1

1

nK 2 +(n −1)ρ σK K K

). (2) The covariance matrix of X and F is ΣF X = (R + Ω)diag(10n1 , · · · , 10nK ), where Ω = diag(ρ1 , · · · , ρK ), the diagonal matrix of ρ1 , ρ2 , . . . , ρK . Proof. (1). Denote
G(Θi ) = σ 2 (Θi ) − ρ(Θi ) Ini + ρ(Θi )1ni 10ni . Note that


0 E(X |Θ) = diag(1n1 , · · · , 1nK ) µ(Θ1 ), · · · , µ(ΘK )

and Cov(X |Θ) = diag(G(Θ1 ), · · · , G(ΘK )). It is easy to check that

ΣXX =Cov E(X |Θ) + E Cov(X |Θ) =diag(1n1 , · · · , 1nK ) R diag(10n1 , · · · , 10nK ) + diag(G1 , · · · , GK ).

(8)

370

WEIZHONG HUANG AND XIANYI WU

(2). Consequently, by the well-known matrix inverse formula (A + BCD)−1 = A−1 − A−1 B(C −1 + DA−1 B)−1 DA−1 ,

(9)

it follows that Σ−1 XX −1  = diag(1n1 , · · · , 1nK )R diag(10n1 , · · · , 10nK ) + diag(G1 , · · · , GK ) −1 −1 −1 =diag(G−1 1 , · · · , GK ) − diag(G1 1n1 , · · · , GK 1nK )·  −1 −1 −1 0 R + diag(10n1 G−1 · 1 1n1 , · · · , 1nK GK 1nK ) −1 0 diag(10n1 G−1 1 , · · · , 1nK GK ),

(10)

Because, in view of (8), Gi = (σi2 − ρi )Ini + ρi 1ni 10ni , applying equality (9) again leads to ni 1ni 10ni 1 . (11) I ni − 2 G−1 = 2 i σi − ρi (σi − ρi ) [σi2 + (ni − 1)ρi ] Consequently, 10ni G−1 = i

1 ni 10 and 10ni G−1 . i 1ni = 2 σi2 + (ni − 1)ρi ni σi + (ni − 1)ρi

(12)

Inserting (11) and (12) into (10) yields the desired (7). (3). The assertion can be verified as what follows. First note that Cov [E(F |Θ), E(X|Θ)]     µ (Θ1 ) µ (Θ1 )     = Cov  ...  , diag(1n1 , · · · , 1nK )  ...  µ (ΘK ) =R

diag(10n1 , · · ·

µ (ΘK ) , 10nK ).

Furthermore, as
Cov(F, X|Θ) = diag ρ(Θ1 ), · · · , ρ(ΘK ) diag(10n1 , · · · , 10nK ), it is easy to see that ΣF X = Cov [E(F |Θ), E(X|Θ)] + E [Cov(F, X|Θ)] = (R + Ω) diag(10n1 , · · · , 10nK ). This completes the proof. With the assistance of this lemma, we below present the inhomogeneous credibility estimator of the future claims. Theorem 3.2. Under Assumption 1, the inhomogeneous credibility estimator of future claims F = (X1,n1 +1 , · · · , XK,nK +1 )0 is − Fb = ZX + (IK − Z)→ µ,

(13)

where Z = (R + Ω)(R + Σ)−1 is the credibility factorPmatrix, IK is the identity ni matrix of size K and X = (X1 , · · · , Xk )0 with Xi = n1i j=1 Xij .

CREDIBILITY MODELS WITH DEPENDENCE STRUCTURE

371

Proof. By equality (5) and Lemmas 2.1 and 3.1, the inhomogeneous credibility estimator of future claims F can be computed by
Fb =proj F | L(X, 1)
=E(F ) + Cov(F, X)Cov(X, X)−1 X − E(X)
− =→ µ + (R + Ω)diag(10n1 , · · · , 10nK ) G−1 − Ψ(R−1 + Σ−1 )−1 Ψ0 ·   X1 − µ1 1n1   .. (14) .  . XK − µK 1nK Because 10ni G−1 = i

10n i σi2 +(ni −1)ρi

(the first equality in Equation (12)) and  n1 n2 diag(10n1 , · · · , 10nK )Ψ =diag , ,··· , σ12 + (n1 − 1)ρ1 σ22 + (n2 − 1)ρ2  nK 2 + (n − 1)ρ σK K K =Σ−1 ,

we have



 X1 − µ1 1n1   .. → − −1 diag(10n1 , · · · , 10nK )G−1   = Σ (X − µ ) . XK − µK 1nK

(15)

and 

 X1 − µ1 1n1   .. diag(10n1 , · · · , 10nK )Ψ(R−1 + Σ−1 )−1 Ψ0   . XK − µK 1nK → − −1 −1 −1 −1 −1 = Σ (R + Σ ) Σ (X − µ ).

(16)

Inserting (15) and (16) into (14), we obtain − − Fb =→ µ + (R + Ω)(IK + Σ−1 R)−1 Σ−1 (X − → µ)  − −1 =(R + Ω)(R + Σ) X + IK − (R + Ω) (R + Σ)−1 → µ → − =ZX + (I − Z) µ . K

This completes the proof. Remark 1. Because Ω and Σ are both diagonal matrices, it is obvious that Z = (R + Ω)(R + Σ)−1 is diagonal if and only if so is R. This indicates that the presence of the conditional dependence over time horizon make the credibility factor highly different from that of Wen, Wu and Zhou (2009). Moreover, the inhomogeneous credibility premium of individual i depends only on her own claims experience if and only if µ(Θi ), i = 1, 2, . . . , K are mutually uncorrelated. In this situation,
ni (τii + ρi ) Z = diag , i = 1, 2, . . . , K , (17) 2 ni τii + σi + (ni − 1)ρi Especially as the claims are also uncorrelated over time horizon, the credibility factors Z can be reduced to
ni τii Z = diag , i = 1, 2, . . . , K , (18) 2 ni τii + σi

372

WEIZHONG HUANG AND XIANYI WU

which coincides with the commonly known results for classical B¨ uhlmann’s model. − The inhomogeneous credibility premiums need the value of the total mean → µ computed over the prior distribution. When it is unknown, B¨ uhlmann (1967) suggested to use homogeneous credibility premiums to instead. Next, we discuss the − homogeneous credibility premiums under condition → µ = µ0 α ~ for certain known α ~. − This is a slightly broader case than B¨ uhlmann has ever discussed: → µ = µ0 1K . − − Theorem 3.3. Under Assumption 1, if → µ = µ0 → α , the homogeneous credibility estimator of future claims F = (X1,n1 +1 , · · · , X1,nK +1 )0 is µ∗ , Fbhom = ZX + (IK − Z)b

(19)

where the credibility factor matrix Z are defined as the same as in theorem 3.2 but µ b∗ is defined as → − α 0 (R + Σ)−1 X → − µ b∗ = → α. − − α 0 (R + Σ)−1 → α Proof. By (5) and lemma 2.1, we have 
Fbhom = proj proj F | L(X, 1) |Le (X) . Thus by (13) in Theorem 3.2, we get
− µ |Le (X) . Fbhom = ZX + (IK − Z)proj → By Lemma 2.1, we know −
→ µ (EX)0 Σ−1 − XX X proj → µ |Le (X) = . −1 0 (EX) ΣXX EX Because − → − −1 0 (EX)0 G−1 EX = → µ 0 diag(10n1 G−1 1 1n1 , · · · , 1nK GK 1nK ) µ = (EX)0 G−1 EX − − =→ µ 0 Σ−1 → µ and − − (EX)0 Ψ(R−1 + Σ−1 )−1 Ψ0 EX = → µ 0 Σ−1 (R−1 + Σ−1 )−1 Σ−1 → µ, It follows that
0 −1 (EX)0 Σ−1 − Ψ(R−1 + Σ−1 )−1 Ψ0 EX XX EX = (EX) G = (EX)0 G−1 EX − (EX)0 Ψ(R−1 + Σ−1 )−1 Ψ0 EX − − − − =→ µ 0 Σ−1 → µ −→ µ 0 Σ−1 (R−1 + Σ−1 )−1 Σ−1 → µ − − =→ µ 0 (R + Σ)−1 → µ. We next compute the term (EX)0 Σ−1 XX X. First note the decomposition 0 −1 (EX)0 Σ−1 X − (EX)0 Ψ(R−1 + Σ−1 )−1 Ψ0 X. XX X = (EX) G

(20)

CREDIBILITY MODELS WITH DEPENDENCE STRUCTURE

373

Because, by again the first equality in (12), 

10n1 G−1 1 X1





10n 1 σ12 +(n1 −1)ρ1

X1



    → .. .. −0  = µ  . .   10n 10nK G−1 X K K K X 2 +(n −1)ρ K σK K K   n1 X1 σ 2 +(n −1)ρ  1 1. 1  → → − 0 =− .. = µ  µ 0 Σ−1 X  nK XK σ 2 +(nK −1)ρK

 − (EX)0 G−1 X = → µ0

K

and − (EX)0 Ψ(R−1 + Σ−1 )−1 Ψ0 X = → µ 0 Σ−1 (R−1 + Σ−1 )−1 Σ−1 X, (EX)0 Σ−1 XX X can be represented as − → − 0 −1 X − → µ 0 Σ−1 (R−1 + Σ−1 )−1 Σ−1 X (EX)0 Σ−1 XX X = µ Σ − =→ µ 0 (R + Σ)−1 X. Therefore, → − → −
µ (EX)0 Σ−1 µ 0 (R + Σ)−1 X → − − XX X proj → µ |Le (X) = = µ → − − −1 0 µ 0 (R + Σ)−1 → µ (EX) ΣXX (EX) − − Especially, if → µ = µ0 → α , we have → − α 0 (R + Σ)−1 X → − α. µ b∗ = → − − α 0 (R + Σ)−1 → α This ends the proof. − Here we would like to note that if → α = 1K , ρi = 0 and τij = 0 for i 6= j, then B¨ uhlmann’s classical results can be derived. As an extension to Yeo and Valdez (2006), Wen, Wu and Zhou (2009) discussed the credibility model with common effects. It is a special case of the models with dependence across risks discussed above. Here we address a specialized version of Assumption 1, which extends also Wen, Wu and Zhou (2009) by allowing certain correlation among claims over time horizon. The assumptions are given as the following. Assumption 2. (1) The common effect is represented by a random variable Λ. Given Λ, the random vectors (Xi , Θi ) are mutually independent. (2) For fixed individual risk i, given the common effect Λ and the structure parameter Θi , the claims Xi = (Xi1 , Xi2 , . . . , Xini )0 are such that E(Xi |Λ, Θi ) = µ(Λ, Θi )1ni , Cov(Xi |Λ, Θi ) = (σ 2 (Λ, Θi ) − ρ(Λ, Θi ))Ini + ρ(Λ, Θi )1ni 10ni . The following notations are also necessary for the theorem below: µi (Λ) = E[µ(Θi , Λ)|Λ], µi = E[µi (Λ)], ai = Var[µi (Λ)] a = Cov[µi (Λ), µj (Λ)], τi2 (Λ) = Var[µ(Θi , Λ)|Λ], τi2 = E[τi2 (Λ)], σi2 (Λ) = E[σ 2 (Θi , Λ)|Λ], σi2 = E[σi 2 (Λ)], E [ρ(Λ, Θi )] = ρi ,

(21)

374

WEIZHONG HUANG AND XIANYI WU

Theorem 3.4. Under assumption 2, the inhomogeneous credibility estimator of future claim Xi,ni +1 is f

bi,n +1 = Zi1 X i + Zi2 X + (1 − Zi1 )µi − Zi2 µf . X i

(22)

where : (1). Zi1 = ci fi−1 , Zi2 = (a − sci fi−1 − asf )f , : (2). ci = τi2 + ai − a + ρi , fi = τi2 + ai − a +
−1 s = a1 + f , and f

: (3). X =

ΣK i=1 fi Xi , f

f

µ =

σi2 +(ni −1)ρi , ni

−1 f = ΣK i=1 fi ,

ΣK i=1 fi µi . f

Especially, if µ1 = · · · = µk = µ, we obtain f

bi,n +1 = Zi1 X i + Zi2 X + (1 − Zi1 − Zi2 )µ. X i

(23)

Proof. Note that the credibility factor matrix Z in Theorem 3.2 is (R+Ω)(R+Σ)−1 . Under Assumption 2, 2 R = diag(τ12 + a1 − a, · · · , τK + aK − a) + a1K 10K ,

Ω and Σ are as same as in Lemma 3.1, so we get easily R + Ω = diag(c1 , · · · , cK ) + a1K 10K , and R + Σ = diag(f1 , · · · , fK ) + a1K 10K . By the formula (A + BCD)−1 = A−1 − A−1 B(C −1 + DA−1 B)−1 DA−1 , we get −1 −1 0 −1 −1 (R + Σ)−1 = diag(f1−1 , · · · , fK ) − s(f1−1 , · · · , fK ) (f1 , · · · , fK ),

so the credibility factor matrix is −1  c−1 1 f1   −1 −1 Z = diag(f1−1 c1 , · · · , fK cK ) − s  ...  (f1−1 , · · · , fK )



−1 c−1 K fK

−1 −1 + a1K (f1−1 , · · · , fK ) − asf 1K (f1−1 , · · · , fK )

(24)

− bi,n +1 is the i-th component of ZX + (IK − Z)→ As X µ , the i-th component of ZX i is f

f

ci fi−1 X i + (a − sci fi−1 − asf )f X = Zi1 X i + Zi2 X , f − and the i-th component of (IK − Z)→ µ are µi and Zi1 µi + Zi2 µ . So we get f

bi,n +1 = Zi1 X i + Zi2 X + (1 − Zi1 )µi − Zi2 µf . X i The proof is completed. Remark 2. If we let µi = µ, ρi = 0, a = ai , we can readily get the result of Wen, Wu and Zhou (2009) from Theorem 3.4. Under this common effects model and its general form defined by Assumption 1, unlike in the classical B¨ uhlmann’s model, the overall mean X plays no longer any role. Instead, every individual means Xi have important contribution to accounting for dependence effect in the credibility estimator.

CREDIBILITY MODELS WITH DEPENDENCE STRUCTURE

375

4. An extension to B¨ uhlmann-Straub’s model. The credibility model with natural weights was developed by B¨ uhlmann and Straub (1970) and hence is known as the B¨ uhlmann-Straub model. There have been broad applications of this model in insurance practice and thus it has been one of the building blocks of credibility theory. In this section, we give the credibility estimators of B¨ uhlmann-Straub model with dependence structure over both risks and time horizon. We need new assumption and notation that are different from the ones in Section 3. The details are the following. Assumption 3. 1. Given Θ = (Θ1 , · · · , ΘK ), the random variables Xi , i = 1, 2, . . . , K, have conditional moments E(Xi |Θi ) = µ(Θi )1ni , Cov(Xi |Θi ) = σ 2 (Θi )/wij Ini + ρ(Θi )1ni 10ni , where Ini is the ni × ni identity matrix and 1ni the ni -dimensional column vector for which every component is 1. 2. The risk parameters Θ = (Θ1 , · · · , ΘK ) are random variables satisfying   E [µ(Θi )] = µi , E σ 2 (Θi ) = σi2 , E [ρ(Θi )] = ρi , Cov [µ(Θi ), µ(Θj )] = τij . We will use the following notations regarding the weights: ∗ wij wij ∗ ∗ ∗ ∗∗ , W = diag(w , · · · , w ), w = , i i1 in ij i (σi2 − wij ρi ) (1 + ρi wi·∗ ) ni ni X X ∗∗ ∗ ∗∗ ∗∗ ∗∗ wij , wij , Wi∗∗ = diag(wi1 , · · · , win ), w = wi·∗ = i· i ∗ wij =

j=1

j=1 w

Pni

∗∗ ∗∗ Σ−1 = diag(w1· , · · · , wK· ), X i =

X

w

w

∗∗ j=1 wij Xij wi·∗∗

,

w

= (X 1 , · · · , X K )0 .

In this section, we will establish the theory for B¨ uhlmann-Straub’s model, which is parallel to that for the B¨ uhlmann model as discussed in section 3. Firstly, the following lemma is a counterpart of Lemma 3.1. Lemma 4.1. Under Assumption 3: 0 0 (1) The covariance matrix of random vector X = (X10 , · · · , XK ) is w 0 0 ΣXX = diag(Gw 1 , · · · , GK ) + diag(1n1 , · · · , 1nK ) R diag(1n1 , · · · , 1nK )

(25)

where R is defined as same as in Lemma 3.1, and Gw i = diag


σ2 σ2 σi2 − ρi , i − ρi , · · · , i − ρi + ρi 1ni 10ni . wi1 wi2 wini

(2) The inverse matrix of matrix ΣXX is w −1 Σ−1 − Φ(R−1 + L−1 )−1 Φ0 , XX = (G )

where ∗∗ w Φ = diag(W1∗∗ 1n1 , · · · , WK 1nK ), Gw = diag(Gw 1 , · · · , GK ), ∗∗ ∗∗ L−1 = diag(w1· , · · · , wK· ).

(26)

(3) The covariance matrix between X and F is ΣF X = (R + Ω)diag(10n1 , · · · , 10nK ), where Ω is defined as same as in Lemma 3.1.

(27)

376

WEIZHONG HUANG AND XIANYI WU

Proof. (1). Let
σ 2 (Θi ) σ 2 (Θi ) σ 2 (Θi ) − ρ(Θi ), − ρ(Θi ), · · · , − ρ(Θi ) ωi1 ωi2 ωini 0 + ρ(Θi )1ni 1ni

Gw (Θi ) =diag

and w Gw i = E[G (Θi )] = diag


σi2 σ2 σ2 − ρi , i − ρi , · · · , i − ρi + ρi 1ni 10ni . ωi1 ωi2 ωini

We can write ΣXX with no difficult as

ΣXX =Cov E(X |Θ) + E Cov(X |Θ) w =diag(1n1 , · · · , 1nk ) R diag(10n1 , · · · , 10nK ) + diag(Gw 1 , · · · , GK ).

(2). By again the formula (A + BCD)−1 = A−1 − A−1 B(C −1 + DA−1 B)−1 DA−1 , it follows that
−1 0 1 −1 (Gw + wi·∗ = Wi∗ − Wi∗ 1ni 1ni Wi∗ , i ) ρi Simple algebra gives −1 (Gw 1ni = i )

1 W ∗ 1n = Wi∗∗ 1ni 1 + ρi wi·∗ i i

and −1 10ni (Gw 1ni = i )

ni X

∗∗ wij = wi·∗∗ .

j=1

Hence,
w −1 −1 −1 Σ−1 − diag (Gw 1n1 , · · · , (Gw 1nK · 1) K) XX =(G )
−1 −1 −1 R−1 + diag(10n1 (Gw 1n1 , · · · , 10nK (Gw 1 nK ) · 1) K)
0 w −1 0 w −1 diag 1n1 (G1 ) , · · · , 1nK (GK )
−1 ∗∗ =(Gw )−1 − diag(W1∗∗ 1n1 , · · · , WK 1nK ) R−1 + L−1 · ∗∗ diag(10n1 W1∗∗ , · · · , 10nK WK )

=(Gw )−1 − Φ(R−1 + L−1 )−1 Φ0 .

Theorem 4.2. Under Assumption 3, the inhomogeneous credibility estimator of future claims F = (X1,n1 +1 , · · · , X1,nK +1 )0 are w − Fcw = Z w X + (IK − Z w )→ µ, −1

w

where Z = (R + Ω) (R + L)

(28)

and L is defined as in display (26).

Proof. Similar with the (14) in the proof of the theorem 3.2, we have − Fcw =→ µ + (R + Ω)diag(10n1 , · · · , 10nK )((Gw )−1 − Φ(R−1 + L−1 )−1 Φ0 )·   X1 − µ1 1n1   ..  . . XK − µK 1nK

(29)

CREDIBILITY MODELS WITH DEPENDENCE STRUCTURE

377

Because  X1 − µ1 1n1   .. diag(10n1 , · · · , 10nK )(Gw )−1   . XK − µK 1nK   0 1n1 W1∗∗ (X1 − µ1 1n1 )   .. = , . 

∗∗ (XK 10nK WK

(30)

− µK 1nK )

and  X1 − µ1 1n1   .. diag(10n1 , · · · , 10nK )Φ(R−1 + L−1 )−1 Φ0   . XK − µK 1nK   0 ∗∗ 1n1 W1 (X1 − µ1 1n1 )  −1   .. = L−1 R−1 + L−1 ,  . 

∗∗ (XK 10nK WK

we have

(31)

− µK 1nK )

  w  10n1 W1∗∗ (X1 − µ1 1n1 ) X 1 − µ1     .. ..  = . . . w 0 ∗∗ 1nK WK (XK − µK 1nK ) X K − µK 

(32)

Inserting (30), (31) and (32) into (29), we obtain w − − Fcw =→ µ + (R + Ω)(E + L−1 R)−1 L−1 (X − → µ)
− w =(R + Ω) (R + L)−1 X + IK − (R + Ω) (R + L)−1 → µ − w → w w =Z X + (I − Z ) µ . K

The proof is thus complete. − − Theorem 4.3. Under Assumption 3, if → µ = µ0 → α , the homogeneous credibility estimator of future claims F = (X1,n1 +1 , · · · , X1,nK +1 )0 are w Fbhom = Z w X + (IK − Z w )b µ∗ ,

(33)

w

where the credibility factor matrix Z are defined the same as in Theorem 4.2 and → − α 0 (R + L)−1 X → − α. µ b∗ = → − − α 0 (R + L)−1 → α Proof. Because Assumption 3 is similar with Assumption 1, here we give only a sketch of the proof. By lemma 2.1, and (13), we have
− Fbhom = Z w X + (I − Z w )proj → µ |L (X) K

and

e

−
→ µ (EX)0 Σ−1 − XX X proj → µ |Le (X) = . −1 0 (EX) ΣXX EX

Further by lemma 4.1, w −1 Σ−1 − Φ(R−1 + L−1 )−1 Φ0 . XX = (G )

Thus

− − − ∗∗ ∗∗ → (EX)0 (Gw )−1 EX = → µ 0 diag(w1· , · · · , wK· )− µ =→ µ 0 L−1 → µ.

(34)

378

WEIZHONG HUANG AND XIANYI WU

Moreover, because (EX)0 Φ(R−1 + L−1 )−1 Φ0 EX − ∗∗ ∗∗ ∗∗ ∗∗ → =→ µ 0 diag(w1· , · · · , wK· )(R−1 + Σ−1 )−1 diag(w1· , · · · , wK· )− µ − → − 0 −1 −1 −1 −1 −1 → = µ L (R + L ) L µ , w X1 w − − ∗∗ ∗∗  ..  µ 0 L−1 X , (EX)0 (Gw )−1 X = → µ 0 diag(w1· , · · · , wK· ) .  = →



w

XK and (EX)0 Φ(R−1 + L−1 )−1 Φ0 X w X1 − ∗∗ ∗∗ ∗∗ ∗∗  ..  =→ µ 0 diag(w1· , · · · , wK· )(R−1 + Σ−1 )−1 diag(w1· , · · · , wK· ) . 



w

XK − − =→ µ 0 L−1 (R−1 + L−1 )−1 L−1 → µ, we obtain the projection as w → − → −
µ (EX)0 Σ−1 µ 0 (R + L)−1 X → → − − XX X = → proj µ |Le (X) = µ. − − 0 (R + L)−1 → µ µ (EX)0 Σ−1 (EX) XX

We are next going to consider a special risk structure, under which the risks are correlated, in the following theorem, similar to Theorem 3.4 Theorem 4.4. In the assumption 3, if Cov [µ(Θi ), µ(Θj )] = ρ, for all i 6= j, then the inhomogeneous credibility estimator of a future claim Xi,ni +1 is w w w w bi,n X = Zi1 X i + Zi2 X i +1

where

w Zi1

ρ + wi·∗∗ ,

=

w −1 cw , i (fi )

fw

fw

w w µ + (1 − Zi1 )µi − Zi2

.

(35)

2 w 2 = (a − sci (fiw )−1 − asf )f w , cw i = τii − ρ + ρi , fi = τii − w w w f K w K w
−1 f Σ f X Σ f µ w −1 ΣK , sw = ρ1 + f w , X = i=1f wi i , µ = i=1f wi i . i=1 (fi ) w Zi2

fw = Especially, if µ1 = · · · = µk = µ, we get

w w w bi,n X = Zi1 X i + Zi2 X i +1

fw

w w + (1 − Zi1 − Zi2 )µ.

(36)

Proof. The credibility factor Z w is (R + Ω)(R + L)−1 , 2 2 R = diag(τ11 − ρ, · · · , τKK − ρ)IK + ρ1K 10K ,

and Ω is same as in Theorem 3.4, the structure of L is the same as Σ in Theorem 3.4. Imitating the proof of Theorem 3.4, we can obtain Theorem 4.4. 5. Conclusions. In this paper, we actually extended the credibility models to accommodate dependence structure across risks as well as over time horizon. By the orthogonal projection technique, we derive the inhomogeneous and homogeneous credibility estimators of risk premiums for B¨ uhlmann’s model with this dependence structure. Then the inhomogeneous credibility estimators for common effect models are addressed. Parallel results are obtained with the inhomogeneous and homogeneous credibility estimators for B¨ uhlmann-Straub’s models.

CREDIBILITY MODELS WITH DEPENDENCE STRUCTURE

379

Though the inhomogeneous credibility estimators allow the form of weighted sums in a matrix version (see Theorem 3.2), our results reveal an important fact that the weighted sums are no longer universal form for credibility premiums under the models investigated in this paper. Instead, with dependence structures, the credibility premiums can only be linear functions of the claims experience. This observation may have a possibility to shake the classical meaning of “credibility”, under which the premiums are always expressed as (1 − z) × overall mean + z × individual mean, where the weight z (commonly known as credibility factor) is considered to reflect the credibility of the actuaries on the claims experience. For more complex dependence structure, it would be possible that one can not have a “credibility form” for the credibility premiums any longer. REFERENCES [1] C. Bolanc´ e, M. Guill´ en, M. Denuit and J. Pinquet, Bonus-malus scales in segmented tariffs with stochastic migration between segments, Insurance: Mathematics and Economics, 33 (2003), 273–282. [2] H. B¨ uhlmann, Experience rating and credibility, Astin Bulletin, 4 (1967), 199–207. [3] H. B¨ uhlmann and E. Straub, Glaubw¨ udigkeit f¨ ur Schadens¨ aze, Bulletin of the Swiss Association of Actuaries, 70 (1970), 111–133. [4] H. B¨ uhlmann and A. Gisler, A Course in Credibility Theory and its Applications, Springer, Netherlands, 2005. [5] D. R. Dannenburg, Crossed classification credibility models, Transactions of the 25th International Congress of Actuaries, 4 (1995), 1–35. [6] J. Dhaene, M. Denuit, M. J. Goovaerts, R. Kaas and D. Vyncke, The concept of comonotonicity in actuarial science and finance: Theory, Insurance: Mathematics and Economics, 31 (2002), 3–33. [7] J. Dhaene, M. Denuit, M. J. Goovaerts, R. Kaas and D. Vyncke, The concept of comonotonicity in actuarial science and finance: Applications, Insurance: Mathematics and Economics, 31 (2002), 133–161. [8] J. Dhaene and M. J. Goovaerts, Dependency of risks and stop-loss order, Astin Bulletin, 26 (1996), 201–212. [9] E. W. Frees, V. R. Young and Y. Luo, A Longitudinal Date Analysis Interpretation of Credibility models, Insurance: Mathematics and Economics, 24 (1999), 229–247. [10] E. W. Frees, V. R. Young and Y. Luo, Case studies using panel data models, North American Actuarial Journal, 5 (2001), 24–42. [11] C. A. Hachemeister, Credibility for regression models with application to trend, In Credibility, theory and application. Proceedings of the Berkeley Actuarial Research Conference on credibility, Academic Press, New York, (1975), 129–169. [12] W. S. Jewell, The use of collateral data in credibility theory: A hierarchical model, Giorndle dell’lstituto Italianodegdi Attuari, 38 (1975), 1–16. [13] T. Y. Lu and Y. Zhang, Generalized correlation order and stop-loss order, Insurance: mathematics and economics, 35 (2004), 69–76. [14] A. M¨ uller, Stop-loss order for portfolios of dependent risks, Insurance: Mathematics and Economics, 21 (1997), 219–223. [15] M. Pan, R. Wang and X. Wu, On the consistency of credibility premiums regarding Esscher principle, Insurance: Mathematics and Economics, 42 (2008), 119–126. [16] O. Purcaru and M. Denuit, On the dependence induced by frequency credibility models, Belgian Actuarial Bulletin, 2 (2002), 73–79. [17] O. Purcaru and M. Denuit, Dependence in dynamic claim frequency credibility models, Astin Bulletin, 33 (2003), 23–40. [18] S. S. Wang, V. R. Young and H. H. Panjer, Axiomatic characterization of insurance prices, Insurance: Mathematics and Economics, 21 (1997), 173–183. [19] L. Wen, X. Wu and X. Zhao, The credibility premiums under generalized weighted loss functions, Journal of Industrial and Management Optimization, 5 (2009), 893–910.

380

WEIZHONG HUANG AND XIANYI WU

[20] L. Wen, X. Wu and X. Zhou, The credibility premiums for models with dependence induced by common effects, Insurance: Mathematics and Economics, 44 (2009), 19–25. [21] X. Wu and X. Zhou, A new characterization of distortion premiums via countable additivity for comonotonic risks, Insurance: Mathematics and Economics, 38 (2006), 324–334. [22] K. L. Yeo and E. A. Valdez, Claim dependence with common effects in credibility models, Insurance: Mathematics and Economics, 38 (2006), 609–629.

Received November 2012; 1st revision October 2013; 2nd revision March 2014. E-mail address: [email protected] E-mail address: [email protected]