Credibility for the Chain Ladder Reserving Method by Alois Gisler and Mario V. Wüthrich Winterthur Insurance Company and ETH Zürich Abstract: We consider the chain ladder reserving method in a Bayesian set up, which allows for combining individual claims development data with portfolio information as for instance development patterns from industry-wide data. We derive the Bayes estimators and the credibility estimators within this Bayesian framework. We show that the credibility estimators are exact Bayesian in the case of the exponential family with its natural conjugate priors. Finally, we make the link to classical chain ladder and show that by using a non-informative prior we arrive at the classical chain ladder forecasts. However, the estimates for the mean square error of prediction differ from the ones found in the literature. Hence the paper also throws a new light upon the estimator of the mean square error of prediction of the classical chain ladder forecasts and suggests a new estimator. Key-Words: Chain-Ladder, Bayes Statistics, Credibility Theory, Exponential Dispersion Family, Mean Square Error of Prediction.

1

Introduction

Claims reserving is still one of the basic actuarial tasks in the insurance industry. Based on observed claims development figures (complete or incomplete development triangles or trapezoids) actuaries have to predict the ultimate claim amount for different lines of business as well as for the whole insurance portfolio. They are often confronted with the problem that the observed development figures within a given loss development triangle heavily fluctuate due to random fluctuations and a scarce data base. This makes it difficult to make a reliable forecast for the ultimate claim. In such situations, actuaries often rely on industry-wide development patterns rather than on the observed individual data. The question then arises, when and to what extent should one rely on the industry-wide data. A similar question arises when considering different lines of business. If the data of a line of business is too scarce, one typically considers the development pattern of other similar lines of business. Here again the question occurs how much one should rely on the claims experience of other similar lines and how much on the observations of the individual line in question. The mathematical tool to answer such kind of questions is credibility theory. The point is that besides the individual claims triangle there are also other sources of information available (so called collective information such as the development pattern of industrywide data or the development figures of other ”similar” lines of business), which may tell 1

something about the future development of the claims of the individual claims triangle in question. Credibility theory allows for modelling such situations and gives an answer to the question of how to combine individual and portfolio claims information to get a best possible forecast of the individual ultimate claim amount. In claims reserving, various models and methods are found in the literature. In the following, we concentrate on the chain ladder reserving method, which is still one of the best known and most used method in the insurance practice. However, the basic idea of this paper, namely to consider a Bayesian set-up and to use credibility techniques for estimating the ultimate claim amount, can also be transformed to other claims reserving methods.

2

Classical Chain Ladder

Assume that Ci,j denotes the total claim of accident year i ∈ {0, . . . , I} at the end of development period j ∈ {0, . . . , J = I}. Without loss of generality, we assume that the development period is one year. Usually, Ci,j denotes either cumulative claims payments or claims incurred, but it could also be another quantity like for instance the number of reported claims. We assume that the claims development ends at development year J and that therefore Ci,J is the ultimate total claim amount of accident year i. Throughout this paper, Ci,j is referred to as claim of accident year i at the end of development year j and Ci,J as the ultimate claim. The exact definition of ”claim” depends on the situation, respectively, on the claims data considered (cumulative payments or claims incurred). At time I, we have observations (upper left trapezoid) DI = {Ci,j : 0 ≤ i ≤ I, 0 ≤ j ≤ J, i + j ≤ I} ,

(2.1)

and the random variables Ci,j need to be predicted for i + j > I. In particular, we want to estimate for each accident year i the ultimate claim Ci,J and the outstanding liabilities Ri = Ci,J − Ci,I−i .

(2.2)

Cj = (C0,j , C1,j , . . . , CI−j,j )0 ,

(2.3)

We define for j = 0, 1, . . . , J

the column vectors of trapezoid DI , and for k ≤ I − j [k] Sj

=

k X

Ci,j .

(2.4)

i=0

The chain ladder method was originally developed as a deterministic algorithm without having an underlying stochastic model. The basic assumption behind this method is that the column vectors {Cj : j = 0, 1, . . . , J} are, up to random fluctuations, proportional to each other, i.e. Ci,j+1 ' fj Ci,j (2.5)

2

for appropriate constants fj > 0. These factors fj are called chain ladder factors, development factors or age-to-age factors. The chain ladder algorithm is such that at time I the random variables Ci,k for k > I − i are predicted by the chain ladder forecasts CL = Ci,I−i Ci,k

k−1 Y

j=I−i

where

fbj ,

(2.6)

[I−j−1]

fbj =

Sj+1

[I−j−1]

Sj

.

(2.7)

CL and the chain ladder reserve of accident Thus the ultimate claim Ci,J is predicted by Ci,J year i at time I is CL RiCL = Ci,J − Ci,I−i . (2.8)

It is the merit of Mack [9] to have formulated the stochastic model underlying the chain ladder method. Mack’s model relies on the following model assumptions: Model Assumptions 2.1 (Mack) M1 Random variables Ci,j belonging to different accident years i ∈ {0, 1, . . . , I} are independent. M2 There exist constants fj > 0 and σ 2j > 0, such that for all i ∈ {0, 1, . . . , I} and for all j = 0, 1, . . . , J − 1 E [Ci,j+1 | Ci,j ] = fj Ci,j , Var [Ci,j+1 | Ci,j ] = σ 2j Ci,j .

(2.9) (2.10)

Note that Mack’s model is a distribution-free model making only assumptions on the conditional first and second moments. The advantage of an underlying stochastic model is that it does not only yield a point estimate for the ultimate claim but that it also allows for estimating the standard error of the chain ladder prediction. Mack derived such a formula in [9]. This formula was the subject of discussions and of further investigations for example in [1], [2], [12], [5] and [17]. In the literature, there are also found other stochastic models leading to the chain ladder forecasts given by (2.6) and (2.7), in particular the Poisson or overdispersed Poisson model. In this model one considers the non-cumulative claim figures Di,j = Ci,j −Ci,j−1 for j ≥ 1, and Di,j = Ci,j for j = 0. Model Assumptions 2.2 ((overdispersed) Poisson) P1 All random variables Di,j are independent. P2 The random variables Di,j are (overdispersed) Poisson-distributed. P3 There exist constants αi > 0 and β j > 0 such that E [Di,j ] = αi β j . 3

More details on this model and a comparison of other models related to the chain ladder method can be found in [8], [11], [6], and [10]. Even if the (overdispersed) Poisson model (Model Assumptions 2.2) leads to the same chain ladder forecast as in Mack’s model (Model Assumptions 2.1), the two models are by no means identical and have very different properties. Indeed, as pointed out in [11], the (overdispersed) Poisson model deviates in different aspects from the basic idea behind the historical chain ladder method. Therefore it is correct to say that, from the point of view of classical statistics, Mack’s model is the stochastic model underlying the chain ladder method. In this paper we introduce a Bayesian chain ladder model. We see later that under certain conditions and by using non-informative priors, the chain ladder forecasts in the Bayesian model are the same as in the classical chain ladder model. However, the estimators of the mean square error of prediction are different to the ones given in Mack’s model [9]. Moreover, we would like to remark that the Bayesian model considered in this paper is different to the Bayesian models considered for increments Di,j in Verrall [18] and [20]. In the sequel it is useful to define for j = 0, . . . , J Bj = {Ci,k ; i + k ≤ I, k ≤ j} = {C0, C1, . . . , Cj } ⊂ DI ,

(2.11) (2.12)

which is the set of observations up to development period j at time I. It is convenient to switch from the random variables Ci,j to the random variables Yi,j (individual development factors) defined by Ci,j+1 Yi,j = . (2.13) Ci,j Analogously, for j = 0, 1, . . . , J − 1, we denote by Yj = (Y0,j , Y1,j , . . . , YI−j−1,j )0 , the column vectors of the observed Y -trapezoid and by yj = (y0,j , y1,j , . . . , yI−j−1,j )0 a realization of Yj . The chain ladder assumptions (2.9) and (2.10) are then equivalent to E [Yi,j | Ci,j ] = fj , σ 2j Var [Yi,j | Ci,j ] = . Ci,j

(2.14) (2.15)

In the chain ladder methodology and in the underlying stochastic chain ladder model of Mack, only the individual data of a specific development triangle or development trapezoid are considered and modelled. In this paper we also want to make use of a priori information or of portfolio information from other ”similar” risks, from which we can possibly learn something about the unknown development pattern of the considered individual claims data. To do this, we have to consider the chain ladder methodology in a Bayesian set up, which will be the subject of the next section.

4

3

Bayes Chain Ladder

In the Bayesian chain ladder set up, the unknown chain ladder factors fj , j = 0, 1, . . . , J − 1, are assumed to be realizations of independent, real valued random variables Fj . We denote by F = (F0 , F1 , . . . , FJ−1 )0 the random vector of the Fj and by f = (f0 , f1 , . . . , fJ−1 ) a realization of F. In the Bayes chain ladder model it is assumed that conditionally, given F, the chain ladder Model Assumptions 2.1 are fulfilled. Model Assumptions 3.1 (Bayes chain ladder)) B1 Conditionally, given F, the random variables Ci,j belonging to different accident years i ∈ {0, 1, . . . , I} are independent. B2 Conditionally, given F and Bj , the conditional distribution of Yi,j depend only on Ci,j and it holds that E [ Yi,j | F,Bj ] = Fj , σ 2j (Fj ) Var [ Yi,j | F,Bj ] = . Ci,j

(3.1) (3.2)

B3 {F0 , F1 , . . . , FJ−1 } are independent. Remarks: • The conditional expected value of Yi,j , given F and Bj , depends only on the unknown chain ladder factor Fj and not on the chain ladder factors Fk of other development years k 6= j. • In (3.2) Ci,j plays the role of a weight, i.e. the conditional variance of Yi,j , given F and Bj , is inversely proportional to Ci,j . Note that the nominator of (3.2) may depend on Fj . • Of course, the unconditional distribution of F does not depend on DI . The distributions of the Fj ’s are often called structural function. • We define

[I−j−1]

Fbj =

Sj+1

[I−j−1]

Sj

,

which is the estimator of the chain ladder factor fj in the classical chain ladder model (see (2.6)). Then it follows from Model Assumptions (3.1) that i h ¯ ¯ (3.3) E Fbj ¯ F,Bj = Fj , i h ¯ σ 2j (Fj ) ¯ . (3.4) Var Fbj ¯ F,Bj = [I−j−1] Sj 5

• Conditionally, given F, {Ci,j : j = 0, 1, . . . , J} possess the Markov property, i.e. the conditional distribution of Ci,j+1 given {Ci,k : k = 0, 1, . . . , j} depends only on the last observation Ci,j and not on the observations Ci,k for k < j. This is a slightly stronger assumption than assumption M2 of Mack (Model Assumptions 2.1), where only the conditional first and second moments and not the whole conditional distribution depend only on the last observation Ci,j . • Conditionally, given F, {Yi,j : j = 0, 1, . . . , J − 1} are uncorrelated and Yi,j and Yk,l are independent for i 6= k.

(3.5) (3.6)

(3.5) is a well known result of Mack [9]. Note however, that the Yi,j in (3.5) are only uncorrelated but not independent (see Mack et al. [12]). • Our goal is to find best predictors of Ci,j for i + j > I, given the observations DI . Theorem 3.2 Under Model Assumptions 3.1 it holds that a posteriori, given the observations DI , the random variables F0 , F1 , . . . , FJ−1 are independent with posterior distribution given by (3.9) . Proof: To simplify the notation, we use in the remainder of this section the following terminology: for j = 0, 1, . . . , J − 1, the distribution functions of the random variables Fj are denoted by U (fj ) , i.e. we use the same capital U for different distribution functions and the argument fj in U (fj ) says that it is the distribution function of Fj . Analogously, we denote the conditional distributions, given Fj = fj or F = f, by Ffi (.) and Ff (.) , respectively. For instance, Ffj (yi,j | Bj ) is the conditional distribution of Yi,j , given Fj = fj and given Bj . From Model Assumptions 3.1 follows that dFf (y0 , . . . , yJ−1 | B0 ) =

J−1 Y I−j−1 Y j=0

i=0

dFfj (yi,j | Ci,j ),

(3.7)

where Ci,j = yi,j−1 Ci,j−1 for j ≥ 1. For the joint posterior distribution of F given the observations DI we find ( ) J−1 Y I−j−1 Y dFfj (yi,j | Ci,j ) dU (fj ) (3.8) dU (f0 , . . . , fJ−1 | DI ) ∝ j=0

∝

J−1 Y j=0

i=0

dU (fj | DI ) ,

which completes the proof of the theorem.

(3.9) 2

6

Remark: • Note that the conditional distribution of Fj , given DI , depends only on Yj and Cj , where Ci,j , i = 0, . . . , I − j play the role of weights because of the variance condition Var [Yi,j | Ci,j , Fj ] = σ 2j (Fj ) /Ci,j . Indeed, the random variables Yi,j are the only ones in the Y -trapezoid containing information on Fj . Next we derive the Bayes estimator of the ultimate claim. Definition 3.3 An estimator Zb of some random variable Z is said to be better or equal than an estimator Ze if ·³ ·³ ´2 ¸ ´2 ¸ E Zb − Z ≤ E Ze − Z . (3.10) Definition 3.3 means that we use the expected quadratic loss as optimality criterion. The following result is a well known result from Bayesian statistics.

Theorem 3.4 Let Z be an unknown random variable and X a random vector of observations. Then the best estimator of Z is Z Bayes = E [ Z| X] .

(3.11)

Remark: • Z Bayes also minimizes the conditional quadratic loss, i.e. ·³ ´2 ¯¯ ¸ Bayes = arg min E Zb − Z ¯¯ X . Z b

(3.12)

Z

bi,J be a predictor of the ultimate claim Ci,J based on the observations DI . Let C bi,J is defined by Definition 3.5 The conditional mean square error of C · ³ ´ ³ ´2 ¯¯ ¸ bi,J = E C bi,J − Ci,J ¯ DI . mse C ¯ Denote by

bi,J − Ci,I−i bi = C R

the corresponding reserve estimate. Note that ·³ ´ ³ ´ ´2 ¯¯ ¸ ³ bi = E R bi − Ri ¯ DI . bi,J = mse R mse C ¯

(3.13)

(3.14)

(3.15)

From (3.12) follows that

Bayes = E [Ci,J | DI ] Ci,J

(3.16)

is the best estimator minimizing the conditional mean square error (3.13) . Theorem 3.6 Under Model Assumptions 3.1 we have Bayes Ci,J = Ci,I−i

J−1 Y

j=I−i

where FjBayes denotes the Bayes estimator of Fj . 7

FjBayes ,

(3.17)

Remark: • The corresponding reserve estimate is denoted by Bayes − Ci,I−i . RiBayes = Ci,J

Proof: From the posterior independency of the Fj , given DI , (see Theorem 3.2) follows that Yi,j , j = I − i, . . . , J − 1, are also conditionally uncorrelated. Thus we obtain ¯ # " J−1 ¯ Y ¯ Bayes Ci,J = E Ci,I−i Yi,j ¯ DI ¯ j=I−i

= Ci,I−i

J−1 Y

j=I−i

= Ci,I−i

J−1 Y

j=I−i

= Ci,I−i

J−1 Y

j=I−i

= Ci,I−i

J−1 Y

E [ Yi,j | DI ]

E {E [Yi,j | F,DI ]| DI } E [ Fj | DI ]

FjBayes .

j=I−i

2 Next we want to find a formula for the mean square error of RiBayes , which is the same Bayes as the mean square error of Ci,J . Because of the general property

we have

£ ¤ E (X − a)2 = Var [X] + (E [X] − a)2 ·³ ´ ´2 ¯¯ ¸ ³ Bayes Bayes = E Ci,J − Ci,J ¯¯ DI mse Ci,J

= E [Var [Ci,J | F,DI ]| DI ] ·³ ´2 ¯¯ ¸ Bayes +E Ci,J − E [Ci,J | F,DI ] ¯¯ DI .

(3.18)

In the classical approach of Mack [9] the first term corresponds to the process error and the second to the estimation error. Here, this is not so clear any more. Since F is a random vector, the first term is some kind of an ”average” process error (averaged over the set of possible values of F) and the second term is some kind of an ”average” estimation error. Var [Ci,J | F,DI ] = E [Var (Ci,J | F,DI , Ci,J−1 )| F,DI ] + Var (E [Ci,J | F,DI , Ci,J−1 ]| F,DI ) J−2 Y 2 2 = Ci,I−i σ (FJ−1 ) Fj + FJ−1 Var [Ci,J−1 | F,DI ] . (3.19) j=I−i

8

By iterating (3.19) we obtain Var (Ci,J | F,DI ) = Ci,I−i

J−1 X

k=I−i

2 2 FI−i · . . . · Fk−1 · σ 2 (Fk ) · Fk+1 · . . . · FJ−1 .

(3.20)

Formula (3.20) is the same as the formula found by Mack [9], which is not surprising, since conditionally on F, the chain ladder model assumptions of Mack are fulfilled. From (3.20) and since the Fj are conditionally independent, given DI , we obtain for the ”average” process error ( k−1 ) J−1 X Y Y ¯ ¤ J−1 £ 2 £ 2¯ ¤ Bayes E [Var (Ci,J | F,DI )| DI ] = Ci,I−i Fm · E σ (Fk )¯ DI · E Fn ¯ DI . k=I−i

m=I−i

n=k+1

(3.21)

The ”average” estimation error of

Bayes Ci,J

is given by (

)2 ¯¯  ¯ E [Fj | DI ] − Fj ¯¯ DI  ¯ j=I−i j=I−i à J−1 ¯ ! ¯ Y ¯ 2 (3.22) Var Fj ¯ DI . = Ci,I−i ¯

·³ ´2 ¯¯ ¸ Bayes 2 ·E E Ci,J − E [Ci,J | F,DI ] ¯¯ DI = Ci,I−i

J−1 Y

J−1 Y

j=I−i

From (3.18) , (3.21) and (3.22) follows immediately the following result:

Theorem 3.7 The conditional mean square error of the Bayes reserve of accident year i is given by ·³ ´ ´2 ¯¯ ¸ ³ Bayes Bayes = E Ci,J − Ci,J ¯¯ DI mse Ri 2 = Ci,I−i ΓI−i + Ci,I−i ∆B I−i,

(3.23)

where ΓI−i =

J−1 X

k=I−i

∆B I−i = Var

4

(

Ã

k−1 Y

m=I−i

J−1 Y

j=I−i

) J−1 Y ¯ ¯ £ ¤ ¤ FmBayes · E σ 2 (Fk )¯ DI · E Fn2 ¯ DI ,

¯ ! ¯ ¯ Fj ¯ DI . ¯

£

(3.24)

n=k+1

(3.25)

Credibility for Chain Ladder

In the Bayesian set-up, the best predictor of the ultimate claim Ci,J is Bayes Ci,J

= Ci,I−i

J−1 Y

j=I−i

9

FjBayes .

(4.1)

However, to calculate FjBayes one needs to know the distributions of the Fj as well as the conditional distributions of the Ci,j , given F. These distributions are usually unknown in the insurance practice. The advantage of credibility theory is that one needs to know only the first and second moments. It is assumed that these first and second moments exist and are finite for all considered random variables. Given a portfolio of similar risks, these moments can be estimated from the portfolio data. For the results of credibility theory used in this paper we refer the reader to the literature, e.g. to the book by Bühlmann and Gisler [3]. By chain ladder credibility we mean that we replace the FjBayes in (4.1) by credibility estimators FjCred . Definition 4.1 The credibility based predictor of the ultimate claim Ci,J given DI is defined by J−1 Y (Cred) = Ci,I−i FjCred . (4.2) Ci,J j=I−i

Remarks: (Cred)

• Note that we have put the superscript Cred into brackets and that we call Ci a credibility based estimator and not a credibility estimator. By definition a credibility estimator would be a linear function of the observations. However, given the multiplicative structure in the chain ladder methodology, it would not make sense to restrict to linear estimators of Ci,J . • The corresponding reserve estimate is defined by (Cred)

Ri

(Cred)

= Ci,J

− Ci,I−i .

Credibility estimators based on some statistic X are best estimators which are a linear function of the entries of X. For estimating Fj we base our estimator on the observations Yi,j , i = 0, . . . , I −j −1, since these are the only observations of the Y -trapezoid containing information on Fj . Definition 4.2 FjCred

=n

arg min

(j) P (j) Fbj :Fbj =a0 + I−j−1 ai Yi,j i=0

·³ ´2 ¯¯ ¸ b E Fj − Fj ¯¯ Bj . o

(4.3)

n PI−j−1 (j) o (j) b b In other words, is defined as the best estimator out of Fj : Fj = a0 + i=0 ai Yi,j ·³ ´2 ¯¯ ¸ b minimizing E Fj − Fj ¯¯ Bj . FjCred

Remark:

·³ ·³ ´2 ¯¯ ¸ ´2 ¯¯ ¸ • Note that E Fbj − Fj ¯¯ Bj = E Fbj − Fj ¯¯ Cj . 10

Theorem 4.3 (Credibility estimator) i) The credibility estimators of the unknown chain ladder factors Fj are FjCred = αj Fbj + (1 − αj ) fj ,

where

(4.4)

[I−j−1]

Fbj = αj =

Sj+1

[I−j−1]

Sj

(4.5)

,

[I−j−1]

Sj

[I−j−1]

Sj

+

σ2j τ 2j

(4.6)

,

fj = E [Fj ] , £ ¤ σ 2j = E σ 2j (Fj ) , where σ 2j (Fj ) is defined in (3.2) , τ 2j = Var [Fj ] .

(4.7) (4.8) (4.9)

ii) The mean square error of FjCred is E

Remark:

h¡ ¢2 ¯¯ i FjCred − Fj ¯ Bj = αj

σ 2j [I−j−1] Sj

= (1 − αj )τ 2j .

(4.10)

• Note that Fbj is the estimate of the development factor fj in the ”classical” chain ladder model. That is, (4.4) is a credibility weighted average between the classical chain ladder estimator Fbj and the a priori expected value fj .

Proof of the theorem: Conditionally on Bj , the random variables Yi,j , i = 0, 1, . . . , I − j − 1, fulfill the assumptions of the Bühlmann and Straub model (see Section 4.2 in [3]). (4.4) is the well known credibility estimator of Bühlmann and Straub, and the formula of its mean square error is also well known in the credibility literature (see e.g. [3], Chapter 4). 2 The credibility estimator (4.4) depends on the structural parameters fj , σ 2j and τ 2j . These structural parameters can be estimated from portfolio data by using standard estimators (see for instance [3], Section 4.8). For the mean square error we obtain similar to (3.18) ³ ´ ³ ´ (Cred) (Cred) mse Ri = mse Ci,J ·³ ´2 ¯¯ ¸ (Cred) = E Ci,J − Ci,J ¯¯ DI ·³ ´2 ¯¯ ¸ (Cred) − E [Ci,J | F,DI ] ¯¯ DI . = E [ Var [Ci,J | F,DI ]| DI ] + E Ci,J 11

The first summand, the ”average” process error, remains unchanged and is the same as in Theorem 3.7. For the ”average” estimation error we obtain ·³ ´2 ¯¯ ¸ (Cred) − E [Ci,J | F,DI ] ¯¯ DI E Ci,J à !2 ¯¯  J−1 J−1 Y Y ¯ 2 E FjCred − Fj ¯¯ DI  . = Ci,I−i (4.11) ¯ j=I−i j=I−i Hence

´ ³ (Cred) 2 = Ci,I−i ΓI−i + Ci,I−i ∆C mse Ri I−i,

where ΓI−i =

J−1 X

k=I−i

∆C I−i

(

k−1 Y

m=I−i

) J−1 Y ¯ ¯ £ ¤ ¤ FmBayes · E σ 2 (Fk )¯ DI · E Fn2 ¯ DI , £

!2 ¯¯  J−1 J−1 Y Y ¯ = E FjCred − Fj ¯¯ DI  . ¯ j=I−i j=I−i Ã

(4.12)

n=k+1

(4.13)

To find an estimator for the mean square error we make the following approximations in (4.12) and (4.13): FjBayes ' FjCred , ·³ h¡ ´2 ¯¯ ¸ ¢¯ i Bayes Cred 2 ¯ ¯ E Fj − Fj ¯ BI = αj ¯ DI ' E Fj − Fj

(4.14) σ 2j , [I−j−1] Sj

(4.15)

where ' means that the equation is not exactly but only approximately fulfilled. Then we get ·³ ´2 ¯¯ ¸2 ³ ´2 £ 2¯ ¤ Bayes ¯ DI + F Bayes E Fj ¯ DI = E Fj − Fj j ¯ ' αj

σ 2j

[I−j−1]

Sj

¡ ¢2 + FjCred .

à !2 ¯¯  !2 ¯¯  J−1 J−1 Y Bayes Y ¯ ¯ FjCred − Fj ¯¯ DI  ' E  Fj − Fj ¯¯ DI  E ¯ ¯ j=I−i j=I−i j=I−i j=I−i à J−1 ¯ ! ¯ Y ¯ Fj ¯ DI = Var ¯ Ã

J−1 Y

J−1 Y

j=I−i

=

J−1 Y

j=I−i

'

J−1 Y

j=I−i

Ã

αj

J−1 Y £ ¯ ¤ E Fj2 ¯ DI − (E [Fj | DI ])2

σ 2j

[I−j−1]

Sj

j=I−i

¡ ¢2 + FjCred

!

−

J−1 Y

j=I−i

¡ Cred ¢2 . Fj

(Cred)

Thus we have found the following approximation for the mean square error of Ri 12

:

Theorem 4.4

³ ´ (Cred) ∼ 2 mse Ri ∆∗I−i, = Ci,I−i Γ∗I−i + Ci,I−i

where Γ∗I−i =

J−1 X

k=I−i

∆∗I−i =

J−1 Y

j=I−i

(

k−1 Y

m=I−i

FmCred · σ 2k

à ¡ Cred ¢2 Fj + αj

J−1 Y

n=k+1

σ 2j

[I−j−1]

Sj

µ ¡ Cred ¢2 Fn + αn !

−

J−1 Y

j=I−i

(4.16)

σ 2n [I−n−1]

Sn

¶)

,

¡ Cred ¢2 . Fj

(4.17)

(4.18)

Remark: • By replacing in (4.17) and (4.18) σ 2j and the variance components σ 2j and τ 2j in αj by appropriate estimates, we obtain from Theorem 4.4 an estimator for the mean (Cred) square error of Ri . Often the mean square error or the standard error (=square root of the mean square error) of the total reserve is of interest too. Denote by X (Cred) Ri (4.19) R(Cred) = i

(Cred)

= 0 for those i for which I − i ≥ J. the total credibility reserve, where Ri à !2 ¯¯  X (Cred) X ¯ mse(R(Cred) ) = E  Ci,J − Ci,J ¯¯ DI  ¯ i i ¯ " " #¯ # ¯ ¯ X ¯ ¯ = E Var Ci,J ¯ F,DI ¯ DI ¯ ¯ à i !2 ¯¯  X (Cred) X ¯ Ci,J − E [Ci,J | F,DI ] ¯¯ DI  . +E  ¯ i i

(4.20)

Because of the conditional independence of the accident years we get "

E Var

"

X i

¯ #¯ # ¯ ¯ X ¯ ¯ Ci,J ¯ F,DI ¯ DI = E [Var [Ci,J | F,DI ]| DI ] ¯ ¯ i X = Ci,I−i ΓI−i ,

(4.21)

i

where ΓI−i is given by (3.24) . Note that the average process error of R(Cred) is the sum (Cred) of the process errors of Ri . 13

For the second summand, the average estimation error, we obtain à !2 ¯¯  X (Cred) X ¯ E Ci,J − E [Ci,J | F,DI ] ¯¯ DI  ¯ i i ¯ ·³ I ´2 ¯ ¸ X (Cred) E Ci,J − E [Ci,J | F,DI ] ¯¯ DI =

(4.22)

i=0

+2

I I X X

E

i=0 k=i+1

h³ ´³ ´¯ i ¯ (Cred) (Cred) Ci,J − E [Ci,J | F,DI ] Ck,J − E [Ck,J | F,DI ] ¯ DI .

With the approximations (4.14) and (4.15) we get for the terms in the second summand h³ ´³ ´¯ i ¯ (Cred) (Cred) E Ci,J − E [Ci,J | F,DI ] Ck,J − E [Ck,J | F,DI ] ¯ DI ! à J−1 !¯ # "à J−1 J−1 J−1 ¯ Y Y Y Y ¯ FjCred − Fj FlCred − Fl ¯ DI = Ci,I−i Ck,I−k E ¯ j=I−i j=I−i l=I−k l=I−k ! à J−1 !¯ # "à J−1 J−1 J−1 ¯ Y Bayes Y Y Bayes Y ¯ Fj − Fj Fl − Fl ¯ DI ' Ci,I−i Ck,I−k E ¯ j=I−i j=I−i l=I−k l=I−k ¯ ! à J−1 J−1 Y Y ¯¯ Fj , Fl ¯ DI = Ci,I−i Ck,I−k Cov ¯ j=I−i l=I−k ¯ ! à J−1 I−i−1 ¯ Y Bayes Y ¯ Fj Var Fj , ¯ DI = Ci,I−i Ck,I−k ¯ j=I−k

j=I−i

(Cred)

' Ci,I−i Ck,I−k ∆∗I−i ,

(4.23)

where ∆∗I−i is given by (4.12) . In the second last equation we have used ¯ ! à J−1 J−1 Y Y ¯¯ Fj , Fl ¯ DI Cov ¯ j=I−i l=I−k ¯ !¯ # " à J−1 J−1 ¯ Y Y ¯¯ ¯ Fj , Fl ¯ DI , FI−k , . . . , FI−i−1 ¯ DI = E Cov ¯ ¯ j=I−i l=I−k ¯ # " J−1 #¯ ! à " J−1 ¯ ¯ ¯ ¯ Y Y ¯ ¯ ¯ Fj ¯ DI , FI−k , . . . , FI−i−1 , E Fj ¯ DI , FI−k , . . . , FI−i−1 ¯ DI + Cov E ¯ ¯ ¯ j=I−i j=I−k "I−i−1 à J−1 ¯ !¯ # ¯ ¯ Y Y ¯ ¯ =E Fl Var Fj ¯ DI , ¯ DI + 0. ¯ ¯ l=I−k

j=I−i

Thus we have found the following result:

14

Corollary 4.5 mse(R

(Cred)

)'

X i

I I ´ ³ X X (Cred) (Cred) +2 mse Ri Ci,I−i Ck,I−i ∆∗I−i , i=0 k=i+1

³ ´ (Cred) where mse Ri and ∆∗I−i are as in Theorem 4.4.

5

Exact Credibility for Chain Ladder

The case where the Bayes estimator is of a credibility type is referred to as exact credibility in the literature. In this section we consider a class of models for the chain ladder method where this is the case, i.e. where FjBayes = FjCred . The basic assumption of the class of models considered in this section is that, conditionally on F and Bj , the random variables Y0,j , . . . , YI,j are independent with a distribution belonging to the one-parameter exponential dispersion family and that the a priori distribution of Fj belongs to the family of the natural conjugate priors. The exponential dispersion family is usually parameterized by the so called canonical parameter. Hence, instead of F with realizations f we consider in this section the vector Θ of the canonical parameters Θj with realizations ϑ. Of course, as we will see, the two parameters are linked each to the other. Definition 5.1 A distribution is said to be of the exponential dispersion type, if it can be expressed as ¸ · xϑ − b(ϑ) + c(x, ϕ/w) dν(x), x ∈ A ⊂ R. (5.1) dF (x) = exp ϕ/w where ν (.) is either the Lebesgue measure on R or the counting measure, ϕ ∈ R+ is the dispersion parameter, w ∈ R+ is a suitable weight, b(ϑ) is a twice differentiable function with a unique inverse for the first derivative b0 (ϑ). If X has a distribution function of the exponential dispersion type (5.1), then it is well known from standard theory of generalized linear models (GLM) (see e.g. [13] or [4]) that µX = E [X] = b0 (ϑ), ϕ 00 σ 2X = Var (X) = b (ϑ). w By taking the inverse in (5.2) we obtain −1

ϑ = (b0 )

(µX ) = h(µX ),

where h(.) is called the canonical link function. The variance can also be expressed as a function of the mean by ϕ 00 ϕ Var (X) = b (h(µX )) = V (µX ) , w w where V (.) is the so-called variance function. 15

(5.2) (5.3)

(5.4)

(5.5)

Definition 5.2 The class of distributions as defined in (5.1) is referred to as the one (real-valued) parameter exponential class Fexp = {Fϑ : ϑ ∈ M} ,

(5.6)

where M is the canonical parameter space (set of possible values of ϑ). It contains the families b,c ∈ Fexp Fexp

(5.7)

specified by the specific form of b(.) and c(.,.). In the sequel we assume that, conditionally on F and Bj , the random variables Y0,j , . . . , YI,j b,c are independent with a distribution belonging to a one-parameter exponential family Fexp . The following results hold true under this general condition. However, not all exponential families are suited for our problem. In particular, the random variables Yi,j are nonnegative. Hence, only distributions having support on R+ are suitable for the chain-ladder situation. A subclass of the exponential class having this property are the Tweedie models with p ≥ 1. The Tweedie models are the subclass of the one parameter exponential class with variance function V (µ) = µp . (5.8) They are defined only for p outside the interval 0 < p < 1. For p ≤ 0 they have positive probability mass on the whole real line. The family of the Tweedie models include in particular the following distributions: • p = 0 : Normal-distribution • p = 1 : (Overdispersed) Poisson distribution • 1 < p < 2 : Compound Poisson distribution with Gamma distributed claim amounts, where the shape parameter of the Gamma distribution is γ = (2 − p)/(p − 1). • p = 2 : Gamma distribution The Tweedie models were also considered in Ohlsson and Johansson [15] in connection with calculating risk premiums in a multiplicative tariff structure. In that paper, there is a good summary on the properties of this family. For the chain ladder, the case 1 ≤ p ≤ 2 is of particular interest. Model Assumptions 5.3 (Exponential Familiy and conjugate priors) E1 Conditional on Θ=ϑ and Bj , the random variables Y0,j , . . . , YI,j are independent with distribution of the exponential type given by (5.1) with specific functions b(.) and c(., .), dispersion parameter ϕj and weights wi,j = Ci,j .

16

E2 Θ0, . . . , ΘJ−1 are independent with densities (with respect to the Lebesgue measure) " (0) # fj ϑ − b(ϑ) (0) + d(fj , η 2j ) , (5.9) uj (ϑ) = exp η 2j (0) fj

where factor.

and

η 2j

h i (0) 2 are hyperparameters and where exp d(fj , η j ) is a normalizing

Remarks: • From Assumption E1 follows that fϑj (yi,j | Bj ) = exp

½

yi,j ϑj − b(ϑj ) ϕj /Ci,j

¾

¡ ¢ a yi,j , ϕj /Ci,j .

(5.10)

• From Assumption E1 also follows that E [Yi,j | Θj ] = b0 (Θj ) = Fj , ϕj ϕj = V (Fj ) . Var (Yi,j | Θj , Ci,J ) = b00 (Θj ) Ci,j Ci,j

(5.11) (5.12)

Hence conditionally, given Fj , the chain ladder assumption M2 of Mack (Model Assumptions 2.1) are fulfilled. The difference to the Mack assumptions is, that specific assumptions on the whole conditional distribution and not only on the first and second conditional moments are made. For instance, in the Tweedie model with p = 1, Yi,j are assumed to be conditionally over-dispersed Poisson distributed. Note however, that this conditional assumptions are not comparable with the Model Assumptions 2.2 and much closer to the idea of the historical chain ladder methodology. b • The distributions of Θ0, . . . , ΘJ−1 with densities (5.9) belong to the family Uexp of b,c the natural conjugate prior distributions to Fexp , which is given by

© ª uγ (ϑ) : γ = (x0 , η 2 ) ∈ R × R+ , · ¸ x0 ϑ − b(ϑ) 2 + d(x0 , η ) , ϑ ∈ M. uγ (ϑ) = exp η2 b Uexp =

• The Gaussian model and the Gamma model studied in Gisler [5] are special cases of the exponential dispersion model defined above. • The model is formulated in terms of individual development factors Yi,j . We could also formulate it in terms of Ci,j . Then, for given ϑ, we would obtain the time series model studied in Murphy [14], Barnett-Zehnwirth [1] or Buchwalder et al. [2].

17

Theorem 5.4 Under Model Assumptions 5.3 and if the region M is such that uj (ϑ) disappears on the boundary of M then it holds that i) (0)

E [Fj ] = fj . ii) where

(0) FjBayes = FjCred = αj Fbj + (1 − αj ) fj ,

(5.13)

[I−j−1]

Fbj = αj =

Sj+1

[I−j−1]

Sj

(5.14)

,

[I−j−1]

Sj

[I−j−1]

Sj

+

σ2j τ 2j

(5.15)

,

σ 2j = ϕj E [b00 (Θj )] = ϕj E [V (Fj )] ,

(5.16)

τ 2j

(5.17)

0

= Var [b (Θj )] = Var (Fj ) .

Remarks: • Note that FjBayes in Theorem 5.4 coincides with the credibility estimator of Theorem 4.3. • For the Tweedie models with p ≥ 0 the conditions and the result of the theorem are fulfilled for p = 0 (Normal-distribution), for 1 < p < 2 (compound Poisson) and for p = 2 (Gamma), but not for p > 2 (see Ohlsson and Johansson[15]). Proof of the theorem: The result of the theorem follows directly from well known results in the actuarial literature. It was first proved by Jewell [7] in the case without weights. A proof for the case with weights can for instance be found in [3]. 2 0

Theorem 5.5 If, in addition to the conditions of Theorem 3.4, uj (ϑ) disappears on the boundary of M then i)

·³ ´2 ¯¯ ¸ Bayes E Fj − Fj ¯¯ Bj = αj

σ 2j [I−j−1] Sj

= (1 − αj )τ 2j .

(5.18)

ii) The conditional mean square error of the reserve RiBayes of accident year i is given by ·³ ´2 ¯¯ ¸ (Cred) Bayes ) = E Ci,J − Ci,J ¯¯ DI mse(Ri ¸ ·³ ´2 ¯¯ (Cred) ¯ − Ci,J ¯ BI−i ' E Ci,J 2 ∆∗I−i, = Ci,I−i Γ∗I−i + Ci,I−i

18

(5.19)

where Γ∗I−i =

J−1 X

k=I−i

(

2 ∆∗I−i = Ci,I−i

k−1 Y

m=I−i

"

J−1 Y

j=I−i

FmCred · σ 2k

J−1 Y

n=k+1

à ¡ Cred ¢2 Fj + αj

µ ¡ Cred ¢2 Fn + αn σ 2j

[I−j−1]

Sj

iii) The mean square error of the total reserve RBayes = mse(R

Bayes

)'

X

mse(RiBayes )

+2

i

I I X X

!

P

i

−

σ 2n [I−n−1]

Sn

J−1 Y

j=I−i

¶)

,

# ¡ Cred ¢2 . Fj

RiBayes is (Bayes)

Ci,I−i Ck,I−i ∆I−i .

(5.20)

i=0 k=i+1

Remarks: • Note, that Γ∗I−i and ∆∗I−i are the same as in Theorem 4.4. • For the Tweedie models for p ≥ 0 the condition and the result of Theorem 5.5 are fulfilled for p = 0 (Normal-distribution) and for 1 < p < 2 (compound Poisson). For p = 2 (Gamma-distribution) it is only fulfilled for η 2 < 1. In this case the natural conjugate prior uj (ϑ) is again a Gamma-distribution, and η 2 < 1 means, that the shape parameter γ of this Gamma-distribution is smaller than one. Proof of the theorem: Since FjBayes is a credibility estimator it follows from Theorem 4.3, that (5.18) holds true, if all random variables are square integrable. Thus, it remains to prove that Fj is square integrable. From (5.9) we get 1 η2 1 u00j (ϑ) = 4 η u0j (ϑ) =

³ ´ (0) fj − b0 (ϑ) uj (ϑ) , ³ ´2 1 (0) fj − b0 (ϑ) uj (ϑ) + 2 b00 (ϑ) uj (ϑ) . η

Since u0j (ϑ) disappears on the boundaries of M, we have Z ³ Z ´2 1 1 (0) 0 0 = 4 fj − b (ϑ) uj (ϑ) dϑ + 2 b00 (ϑ) uj (ϑ) dϑ η M η M 1 1 = 4 Var (Fj ) + 2 ϕj E [V (Fj )] . η η Hence, Fj is square integrable. The proof of (5.19) is the same as in Theorem 4.4. The proof of (5.20) is the same as the derivation of Corollary 4.5. 2

19

6

Link to Classical Chain Ladder

In this section we consider the same ”exponential family” Bayes models as in Section 5. But now we look at the situation of non-informative priors, i.e. we consider the limiting case for τ 2j → ∞. Theorem 6.1 Under Model Assumptions 5.3 and if uj (ϑ) disappears on the boundaries of M, then for τ 2j → ∞ (non-informative prior) [I−j−1]

FjBayes

= c Fj =

RiBayes = RiCL .

Sj+1

[I−j−1]

Sj

= fbj ,

(6.1) (6.2)

Remarks: • Note, that fbj are the estimates of the chain ladder factors in the classical chain ladder model. Hence in this case the Bayes chain ladder forecasts are the same as the classical chain ladder forecasts and the resulting Bayes reserves are the same as the classical chain ladder reserves. • For the Tweedie models with p ≥ 0 the result holds true for p = 0 (Normal) and 1 < p ≤ 2. The cases p = 0 and p = 2 have already been considered in Gisler [5]. Proof: The credibility weights αj → 1 for τ 2j → ∞. The result then follows immediately from Theorem 5.4. 2 The next result shows how we can estimate the mean square error of prediction in the limiting case of non-informative priors, which is in this case the mean square error of the classical chain ladder forecast. Thus the following result gives another view on the estimation of the mean square error of the classical chain ladder method and suggests a different estimator to the ones found so far in the literature. The estimation of the mean square error has been the topic of several papers in the ASTIN Bulletin 36/2 (see [2], [12], [5], [17]). As discussed in Gisler [5], we believe that this Bayesian approach is the appropriate way to look at the conditional situation and to estimate the conditional mean square error. From Theorem 5.5 and Theorem 6.1 we obtain immediately the following result: Theorem 6.2 Under Model Assumptions 5.3 and if uj (ϑ) and u0j (ϑ) disappear on the boundaries of M, then it holds, for τ 2j → ∞ (non-informative prior), that the conditional mean square error of the chain ladder reserves RiCL can be estimated by ¢ ¡ 2 bI−i + Ci,I−i b I−i , (6.3) mse d RiCL = Ci,I−i Γ ∆ 20

where bI−i = Γ

b I−i = ∆ Remarks:

J−1 X

k=I−i J−1 Y

j=I−i

(

Ã

k−1 Y

m=I−i 2

fbj +

fc b2k m·σ

J−1 Y

n=k+1

σ b2j [I−j−1] Sj

!

−

µ 2 fbn + J−1 Y

j=I−i

[I−n−1]

Sn

2 fbj ,

σ b2n

¶)

,

(6.4)

(6.5)

where σ b2j are appropriate estimators for σ 2j .

• For the Tweedie models, the result holds true for p = 0 (Normal) and for 1 < p < 2, but not for p = 2, since η 2 has to be smaller than one for p = 2. In particular the case 1 < p < 2 (compound Poisson with Gamma ”claim sizes”) seems to us a fairly adequate and good model conditional for a chain-ladder claim development process. • These estimators should be compared to the ones suggested by Mack [9] and BuchbI−i of the ”average” process error is slightly bigger walder et al. [2]. The estimate Γ [I−n−1] 2 b I−i of the ”average” estimation because of the terms σ bn /Sn . The estimate ∆ error is the same as the one called conditional resampling by Buchwalder et al. [2], but different to the one of Mack [9]. Finally, from Theorem 6.1, Theorem 6.2 and Corollary 4.5 we obtain P Corollary 6.3 The mean square error of the total reserve RCL = i RiCL can be estimated by I I X X ¡ ¢ X ¢ ¡ bk,I−i ∆ b I−i , mse d RiCL + 2 Ci,I−i C (6.6) mse d RCL = i

i=0 k=i+1

where

bk,I−i = Ck,I−k C

I−i−1 Y

j=I−k

fbj

¡ ¢ b I−i are as in Theorem 6.2 and where fbj are the classical chain and where mse d RiCL and ∆ ladder factors given by (6.1). Remark:

• This result should again be compared with the estimator in Mack [9] and Buchwalder et al. [2].

21

7

Numerical Example

For pricing and profit-analysis of different business units it is necessary to set up the reserves for each of these business units (BU). In Appendix A trapezoids of cumulative payments of the business line contractors all risks insurance for different BU of Winterthur Insurance Company are given (for confidentiality purposes the figures are multiplied with a constant). The aim is to determine for this line of business the reserves for each business unit. Thus we have a portfolio of similar loss development figures, which is suited to apply the theory presented in this paper. The following table shows the estimated values of σ 2j and of the development factors Fj estimated by classical chain ladder and by the credibility estimators of Section 4. The structural parameters fj , σ 2j and τ 2j have been estimated by the ”standard estimators”, which can be found e.g. in Section 4.8 of [3]. table of results j= 2 σj

0 336.53

1 34.739

2 7.828

3 5.934

4 0.426

5 4.342

6 4.248

7 0.239

8 0.097

9 product 0.154

FjCL

2.111

1.119

1.031

1.013

1.004

1.001

0.993

0.998

1.000

0.999

2.452

2.270

1.233

0.982

1.024

1.012

0.981

0.962

1.003

0.996

1.000

2.686

2.111

1.189

0.996

1.015

1.004

1.001

0.984

0.998

1.000

0.999

2.499

2.133

1.094

1.032

1.002

0.998

1.000

1.014

0.999

1.000

0.990

2.416

2.111

1.111

1.033

1.012

1.003

1.001

0.997

0.998

1.000

0.997

2.444

2.189

1.138

1.037

1.042

1.003

1.000

0.999

1.002

1.000

1.000

2.700

2.111

1.134

1.036

1.016

1.004

1.001

0.994

0.998

1.000

0.999

2.509

2.108

1.070

1.054

1.013

1.004

1.015

0.996

0.995

1.000

1.000

2.429

2.111

1.084

1.050

1.013

1.004

1.001

0.994

0.998

1.000

0.999

2.426

1.930

1.114

1.018

0.995

1.002

0.997

0.999

0.997

1.002

1.000

2.172

2.111

1.119

1.021

1.010

1.004

1.001

0.995

0.998

1.000

0.999

2.429

3.008

1.190

1.146

1.006

1.000

0.979

0.996

1.000

1.000

1.004

4.041

2.111

1.139

1.064

1.013

1.004

1.001

0.993

0.998

1.000

0.999

2.578

all BU A B C D E F

Fj Fj

cred

Fj Fj

CL

cred

Fj Fj

CL

cred

Fj Fj

CL

cred

Fj Fj

CL

cred

Fj Fj

CL

CL

cred

From the above table we can see that the chain ladder (CL) and the credibility (Cred) estimates can differ quite substantially (see for instance the estimate of F0 for BU F). The estimate of the variance component τ 2j became negative for j = 0. Therefore F0Cred is identical to F0CL of all BU. The values of the above result table can be visualized by looking at the following graphs showing the resulting loss development pattern of a ”normed reference year” characterized

22

by a payment of one in development year 0. Chain Ladder Development Pattern

1.4 1.35 1.3 1.25

BU A BU B BU C BU D BU E BU F

1.2 1.15 1.1 1.05 1 0

1

2

3

4

5

6

7

8

9

Development Year j

Credibility Development Pattern

1.25 1.2 1.15

BU A BU B BU C BU D BU E BU F

1.1 1.05 1 0

1

2

3

4

5

6

7

8

9

Development Year

From these graphs one can see the smoothing effect on the chain ladder procedure on the estimates. The next two graphs show for the business units A and E the same kind of development patterns. ”CL Portfolio” is the one obtained by the chain ladder factors of the portfolio data (total of all BU), ”CL BU” the one obtained by the chain ladder factors of the BU and ”Cred BU” the one obtained by the credibility estimated chain ladder factors of the BU. One can see out of these figures that Cred BU is somewhere in between ”CL

23

Portfolio” and ”CL BU”. BU A 1.300 1.250 1.200 1.150 CL Portfolio Cred BU CL BU

1.100 1.050 1.000 0

1

2

3

4

5

6

7

8

9

8

9

development year

BU E 1.200 1.180 1.160 1.140 1.120 1.100 1.080

CL Portfolio

1.060

Cred BU CL BU

1.040 1.020 1.000 0

1

2

3

4

5

6

7

develoopment year

Finally the next table shows the estimated reserves and the estimates of the square root of the mean square error (mse). BU A B C D E F sum

estimated reserves CL Cred 486 504 235 244 701 517 1'029 899 495 621 40 25 2'987 2'810

1/2

estimated mse CL 657 135.1% 288 122.7% 411 58.6% 844 82.1% 397 80.2% 140 347.0% 1'254 42.0%

Cred 498 98.9% 402 164.3% 520 100.6% 729 81.1% 596 95.9% 149 595.8% 1'261 44.9%

The mse of the chain ladder estimate was calculated by applying the results of Theorem 6.2. However, they only differ very little from the values obtained with the estimator 24

suggested by Mack [9]. These findings are similar to the findings in Buchwalder et alias [2], Section 4.2 and Table 5. The mse for the sum of the reserves over all business units is obtained by summing up the mse of the business units. For the sum over all business units, the difference between the ”chain ladder” reserves and the ”credibility” reserves is 6%, and for the business units it varies between 4% and 38%. Hence the differences between the chain ladder and the credibility estimated reserves can be quite substantial and may have a considerable impact on the profit and loss of the individual BU. The chain ladder method can also be applied to the portfolio trapezoid (data from the total of all business units). By doing so, the total reserve obtained is 2’746 and the estimated mse1/2 is 48.4%. This shows once more the well known fact that the chain ladder method is not additive. Credibility could only be applied to the portfolio triangle, if the structural variance components were a priori known or fixed in a pure Bayesian way. In our paper, we have followed the empirical credibility approach and estimated the structural parameters from the portfolio data. Therefore we could not do a credibility estimate based on the trapezoid of the portfolio data.

Acknowledgment: The authors thank F. Schnelli for having carried through the calculations of the numerical example in Section 7.

Addresses of authors: Alois Gisler Chief Actuary Non-Life Winterthur Insurance Company P.O. Box 357 CH 8401 Winterthur

Mario Valentin Wüthrich Departement Mathematik HG F42.2 Rämistrasse 101 CH 8092 Zürich

Email: [email protected]

Email: [email protected]

25

Appendix A

Loss Development Data

The following trapezoids of cumulative payements show the loss development of the line building engineering for different business units. For confidentiality reasons the data were multiplied with some constant. BU A acc. year 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006

dev.year 0 118 124 556 1'646 317 242 203 492 321 609 492 397 523 1'786 241 327 275 89 295 151 315

1

2 1'232 863 3'494 2'492 890 1'218 622 1'685 1'728 1'283 1'622 868 1'475 3'370 536 577 529 378 396

3 1'266 890 2'998 2'507 890 1'224 639 1'668 1'863 1'294 1'672 889 1'489 3'029 596 583 529 382

4 1'266 914 2'983 2'612 950 1'229 667 1'753 1'877 1'253 1'672 964 1'489 3'211 652 583 541

5 1'397 916 3'018 2'612 990 1'249 647 1'742 1'877 1'255 1'672 964 1'489 3'289 652 583

6 1'397 941 2'458 2'608 990 1'249 647 1'804 1'877 1'255 1'672 964 1'489 3'325 652

7 1'397 941 2'458 1'755 990 1'249 647 1'804 1'877 1'255 1'672 964 1'489 3'325

8 1'492 941 2'470 1'755 990 1'249 647 1'804 1'877 1'255 1'621 964 1'489

9 1'492 865 2'470 1'755 990 1'249 647 1'804 1'877 1'255 1'621 964

10 1'492 865 2'470 1'755 990 1'249 647 1'804 1'877 1'255 1'621

1

2

3

4

5

6

7

8

9

442 1'077 717 572 834 1'246 1'008 580 815 805 641 847 830 1'134 925 1'924 1'003 680 352 418

541 1'085 834 813 1'048 1'272 1'061 630 871 906 833 854 978 1'064 915 2'034 1'137 682 418

541 1'178 849 875 1'072 1'353 1'061 670 859 969 842 915 1'034 1'202 957 1'897 1'164 686

528 1'212 849 878 1'088 1'285 1'061 672 867 971 842 918 1'048 1'202 953 1'897 1'196

528 1'217 850 910 1'088 1'285 1'071 672 777 971 842 918 1'048 1'210 953 1'897

528 1'217 850 912 1'088 1'285 1'071 672 777 971 842 918 1'048 1'210 953

528 1'217 850 1'096 1'088 1'285 1'071 672 777 971 842 918 1'048 1'210

528 1'217 850 1'089 1'088 1'285 1'071 672 777 971 842 918 1'048

528 1'217 850 1'089 1'088 1'285 1'071 672 777 971 842 918

10 528 1'217 850 986 1'088 1'285 1'071 672 777 971 842

487 657 2'204 2'351 886 919 612 1'405 1'149 1'109 1'627 793 1'098 2'951 465 622 520 327 301 406

BU B acc. year 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006

dev.year 0 92 451 404 203 352 504 509 229 324 508 354 431 205 522 567 1'238 355 312 246 91 130

26

BU C acc. year 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006

dev.year 0 268 268 385 251 456 477 405 443 477 581 401 474 649 911 508 389 373 276 465 343 254

1

2

3

4

5

6

7

8

9

456 520 968 742 905 1'286 999 932 1'046 1'146 997 778 1'420 1'935 1'054 790 998 853 820 622

485 577 1'017 795 1'162 1'376 1'172 952 1'336 1'316 1'229 939 1'707 2'304 1'101 868 1'091 932 859

483 579 1'019 931 1'164 1'376 1'196 965 1'362 1'362 1'248 1'321 1'709 2'307 1'071 909 1'155 948

483 579 1'019 931 1'164 1'373 1'196 984 1'375 1'391 1'281 1'366 1'709 2'309 1'071 1'569 1'201

483 579 1'019 931 1'164 1'373 1'210 992 1'375 1'391 1'284 1'392 1'709 2'309 1'071 1'569

483 579 1'019 931 1'164 1'373 1'210 1'012 1'375 1'391 1'264 1'392 1'709 2'309 1'071

483 579 1'019 931 1'164 1'373 1'210 1'012 1'375 1'391 1'264 1'392 1'638 2'362

483 579 1'019 931 1'191 1'373 1'210 1'012 1'375 1'391 1'264 1'392 1'638

483 579 1'019 931 1'191 1'373 1'210 1'012 1'375 1'391 1'264 1'392

10 483 579 1'019 931 1'191 1'373 1'210 1'012 1'375 1'391 1'264

1 1'022 873 1'137 1'289 2'158 2'164 2'666 2'458 2'014 2'990 2'063 6'630 1'587 7'710 3'647 2'658 3'054 1'212 1'661 1'486

2 1'066 1'057 1'234 1'418 2'910 2'446 2'946 2'892 2'459 3'235 2'378 6'850 1'780 6'596 3'699 3'063 3'335 1'247 1'816

3 1'086 1'076 1'460 1'574 3'071 2'570 3'008 3'502 2'466 3'795 2'368 6'885 1'794 7'201 3'780 3'036 3'438 1'215

4 1'094 1'082 1'475 1'578 3'213 2'578 3'021 3'629 2'554 3'816 2'384 6'923 1'838 7'292 3'773 3'093 3'438

5 1'094 1'082 1'588 1'634 3'199 2'558 3'022 3'664 2'554 3'841 2'368 6'923 1'838 7'292 3'773 3'095

6 1'094 1'082 1'586 2'250 3'052 2'558 3'019 3'887 2'554 3'842 2'373 6'923 1'838 7'292 3'773

7 1'094 1'082 1'586 2'044 3'052 2'558 3'019 3'867 2'540 3'860 2'373 6'923 1'865 7'292

8 1'094 1'082 1'586 2'044 3'052 2'558 3'019 3'697 2'540 3'860 2'373 6'923 1'865

9 1'094 1'082 1'586 2'044 3'052 2'558 3'019 3'697 2'540 3'860 2'373 6'923

10 1'094 1'082 1'586 2'044 3'052 2'558 3'019 3'697 2'540 3'860 2'373

1

2 1'057 1'643 2'204 1'309 1'776 2'628 1'922 2'219 2'553 2'660 2'341 2'260 2'736 2'667 1'575 1'034 1'046 1'689 877

3 1'106 1'717 2'488 1'442 1'823 2'743 1'863 1'921 2'598 2'640 2'420 2'226 2'759 2'655 1'603 1'049 1'123 1'779

4 1'130 1'720 2'507 1'467 1'827 2'294 1'886 1'931 2'598 2'639 2'516 2'226 2'760 2'655 1'654 1'049 1'143

5 1'130 1'724 2'509 1'467 1'832 2'338 1'886 1'944 2'598 2'641 2'516 2'215 2'766 2'650 1'654 1'050

6 1'138 1'724 2'510 1'477 1'833 2'358 1'886 1'947 2'598 2'659 2'431 2'215 2'688 2'650 1'675

7 1'131 1'724 2'510 1'477 1'833 2'358 1'886 1'867 2'598 2'659 2'431 2'059 2'737 2'824

8 1'131 1'724 2'436 1'477 1'833 2'358 1'886 1'867 2'598 2'659 2'431 2'059 2'737

9 1'131 1'724 2'436 1'477 1'833 2'358 1'886 1'867 2'598 2'659 2'468 2'059

10 1'131 1'724 2'436 1'477 1'833 2'358 1'886 1'867 2'598 2'659 2'468

BU D acc. year 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006

dev.year 0 330 327 304 426 750 761 1'119 917 905 1'761 824 4'364 493 4'092 1'733 1'261 1'517 778 727 561 459

BU E acc. year 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006

dev.year 0 486 867 1'285 395 802 966 759 1'136 1'467 1'309 877 1'004 1'351 906 563 417 322 1'047 497 1'021 302

964 1'669 1'925 994 1'468 1'967 1'766 2'139 2'243 2'521 2'170 1'963 2'579 2'341 1'450 1'006 836 1'656 843 1'237

27

BU F acc. year 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006

dev.year 0 18 20 20 88 3 11 17 73 48 98 38 42 64 57 85 212 56 25 19 25 7

1

2 64 73 70 133 180 79 66 216 213 153 529 140 95 144 178 341 152 44 137 45

3 64 103 318 133 214 80 105 218 253 153 557 141 95 169 188 357 187 103 140

4 64 153 328 133 214 82 172 218 386 158 632 141 102 178 186 371 246 178

5 64 155 328 133 215 81 172 218 400 158 639 141 102 178 186 371 246

6 64 155 328 133 215 81 172 218 400 158 639 141 102 178 186 371

7 64 155 328 133 215 81 188 218 317 158 639 141 102 178 186

8 64 155 328 133 215 81 188 218 304 158 639 141 102 178

9 64 155 328 133 215 81 188 218 304 158 639 141 102

10 64 155 328 133 215 81 188 218 304 158 639 141

64 155 328 133 215 81 199 218 304 158 639

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[2] Buchwalder, M., H. Bühlmann, M. Merz and M.V. Wüthrich (2006). The mean square error of prediction in the chain ladder reserving method (Mack and Murphy revisited). ASTIN Bulletin 36/2, 521-542.

[3] Bühlmann, H. and A. Gisler (2005). A Course in Credibility Theory and its Aapplications. Universitext, Springer Verlag.

[4] Dobson, A.J. (1990). An Introduction to Generalized Linear Models. Chapman and Hall, London.

[5] Gisler A. (2006). The estimation error in the chain-ladder reserving method: a Bayesian approach. ASTIN-Bulletin 36/2, 554-565.

[6] Hess, K. and K.D. Schmidt (2000). Acomparison of models for the chain-ladder method. Dresdner Schriften zur Versicherungsmathematik, 3/2000, Technische Universität Dresden.

[7] Jewell W.S. (1974). Credible means are exact Bayesian for exponential families. ASTIN Bulletin 8, 77-90.

[8] Mack, T. (1991). A simple parametric model for rating automobile insurance or estimating IBNR claims reserves. ASTIN Bulletin 21/1, 93-109.

[9] Mack, T. (1993). Distribution-free calculation of the standard error of chain ladder reserve estimates. ASTIN Bulletin 23/2, 213—225.

28

[10] Mack, T. (1994). Which stochastic model is underlying the chain ladder method ? Insurance: Mathematics and Economics 15, 133-138.

[11] Mack, T. and G. Venter (2000). A comparison of stochastic models that reproduce chain ladder reserve estimates. Insurance: Mathematics and Economics 26, 101-107.

[12] Mack, T., G. Quarg and C. Braun (2006). The mean square error of prediction in the chain ladder reserving method - a comment. ASTIN Bulletin 36/2, 543-552.

[13] McCullagh, P. and J.A. Nelder (1989). Generalized Linear Models. Chapman and Hall, Cambridge, 2nd edition.

[14] Murphy, D.M. (1994). Unbiased loss development factors. Proc. CAS, Vol. LXXXI, 154— 222.

[15] Ohlsson, E. and B. Johannson (2006). Exact credibility and Tweedie models. ASTIN Bulletin 36/1, 121-133.

[16] Schmidt, K.D. and A. Schnaus (1996). An extension of Mack’s model for the chain-ladder method. ASTIN Bulletin 26, 247-262.

[17] Venter, G. (2006). Discussion of the mean square error of prediction in the chain ladder reserving method. ASTIN Bulletin 36/2, 566-572.

[18] Verrall, R.J. (2000). An investigation into stochastic claims reserving models and chainladder technique. Insu

[19] rance: Mathematics and Economics 26, 91-99. [20] Verrall, R.J. (2004). A Bayesian generalized linear model for the Bornhuetter-Ferguson method of claims reserving. North American Act. J. 813, 67-89.

29