Credibility for the Chain Ladder Reserving Method

by Alois Gisler and Mario V. WüthrichWinterthur Insurance Company and ETH Zürich

Abstract:We consider the chain ladder reserving method in a Bayesian set up, which allows forcombining individual claims development data with portfolio information as for instancedevelopment patterns from industry-wide data. We derive the Bayes estimators and thecredibility estimators within this Bayesian framework. We show that the credibility esti-mators are exact Bayesian in the case of the exponential family with its natural conjugatepriors. Finally, we make the link to classical chain ladder and show that by using anon-informative prior we arrive at the classical chain ladder forecasts. However, the esti-mates 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 ofprediction of the classical chain ladder forecasts and suggests a new estimator.

Key-Words: Chain-Ladder, Bayes Statistics, Credibility Theory, Exponential DispersionFamily, Mean Square Error of Prediction.

1 Introduction

Claims reserving is still one of the basic actuarial tasks in the insurance industry. Basedon observed claims development figures (complete or incomplete development trianglesor trapezoids) actuaries have to predict the ultimate claim amount for different lines ofbusiness as well as for the whole insurance portfolio. They are often confronted with theproblem that the observed development figures within a given loss development triangleheavily fluctuate due to random fluctuations and a scarce data base. This makes it difficultto make a reliable forecast for the ultimate claim. In such situations, actuaries often relyon industry-wide development patterns rather than on the observed individual data. Thequestion 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 lineof business is too scarce, one typically considers the development pattern of other similarlines of business. Here again the question occurs how much one should rely on the claimsexperience of other similar lines and how much on the observations of the individual linein 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 informationavailable (so called collective information such as the development pattern of industry-wide 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 inquestion. Credibility theory allows for modelling such situations and gives an answer tothe question of how to combine individual and portfolio claims information to get a bestpossible 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 thebest known and most used method in the insurance practice. However, the basic ideaof this paper, namely to consider a Bayesian set-up and to use credibility techniques forestimating the ultimate claim amount, can also be transformed to other claims reservingmethods.

2 Classical Chain Ladder

Assume that Ci,j denotes the total claim of accident year i ∈ {0, . . . , I} at the end ofdevelopment period j ∈ {0, . . . , J = I}. Without loss of generality, we assume that thedevelopment period is one year. Usually, Ci,j denotes either cumulative claims paymentsor claims incurred, but it could also be another quantity like for instance the number ofreported claims. We assume that the claims development ends at development year Jand that therefore Ci,J is the ultimate total claim amount of accident year i. Throughoutthis paper, Ci,j is referred to as claim of accident year i at the end of development year jand 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). Attime 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 wantto estimate for each accident year i the ultimate claim Ci,J and the outstanding liabilities

Ri = Ci,J − Ci,I−i. (2.2)

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

Cj = (C0,j, C1,j, . . . , CI−j,j)0, (2.3)

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

S[k]j =

kXi=0

Ci,j. (2.4)

The chain ladder method was originally developed as a deterministic algorithm withouthaving an underlying stochastic model. The basic assumption behind this method is thatthe column vectors {Cj : j = 0, 1, . . . , J} are, up to random fluctuations, proportional toeach 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, devel-opment factors or age-to-age factors. The chain ladder algorithm is such that at time Ithe random variables Ci,k for k > I − i are predicted by the chain ladder forecasts

CCLi,k = Ci,I−ik−1Yj=I−i

bfj, (2.6)

where bfj = S[I−j−1]j+1

S[I−j−1]j

. (2.7)

Thus the ultimate claim Ci,J is predicted by CCLi,J and the chain ladder reserve of accidentyear i at time I is

RCLi = CCLi,J − Ci,I−i. (2.8)

It is the merit of Mack [9] to have formulated the stochastic model underlying the chainladder 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} areindependent.

M2 There exist constants fj > 0 and σ2j > 0, such that for all i ∈ {0, 1, . . . , I} and forall j = 0, 1, . . . , J − 1

E [Ci,j+1|Ci,j] = fj Ci,j, (2.9)

Var [Ci,j+1|Ci,j] = σ2j Ci,j. (2.10)

Note that Mack’s model is a distribution-free model making only assumptions on theconditional 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 errorof the chain ladder prediction. Mack derived such a formula in [9]. This formula was thesubject 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 βj > 0 such that E [Di,j] = αiβj.

3

More details on this model and a comparison of other models related to the chain laddermethod 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 byno 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 thehistorical chain ladder method. Therefore it is correct to say that, from the point of viewof classical statistics, Mack’s model is the stochastic model underlying the chain laddermethod.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 theBayesian model are the same as in the classical chain ladder model. However, the estima-tors 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 paperis 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} (2.11)

= {C0,C1, . . . ,Cj} ⊂ DI , (2.12)

which is the set of observations up to development period j at time I. It is convenient toswitch from the random variables Ci,j to the random variables Yi,j (individual developmentfactors) defined by

Yi,j =Ci,j+1Ci,j

. (2.13)

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, (2.14)

Var [Yi,j|Ci,j] =σ2jCi,j

. (2.15)

In the chain ladder methodology and in the underlying stochastic chain ladder model ofMack, only the individual data of a specific development triangle or development trapezoidare considered andmodelled. In this paper we also want to make use of a priori informationor of portfolio information from other ”similar” risks, from which we can possibly learnsomething about the unknown development pattern of the considered individual claimsdata. 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. Wedenote 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, givenF, 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 accidentyears i ∈ {0, 1, . . . , I} are independent.

B2 Conditionally, given F and Bj, the conditional distribution of Yi,j depend only onCi,j and it holds that

E [Yi,j|F,Bj] = Fj, (3.1)

Var [Yi,j|F,Bj] =σ2j (Fj)

Ci,j. (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 unknownchain ladder factor Fj and not on the chain ladder factors Fk of other developmentyears k 6= j.

• In (3.2)Ci,j plays the role of a weight, i.e. the conditional variance of Yi,j, givenF and Bj, is inversely proportional to Ci,j. Note that the nominator of (3.2) maydepend on Fj.

• Of course, the unconditional distribution of F does not depend on DI . The distrib-utions of the Fj’s are often called structural function.

• We define bFj = S[I−j−1]j+1

S[I−j−1]j

,

which is the estimator of the chain ladder factor fj in the classical chain laddermodel (see (2.6)). Then it follows from Model Assumptions (3.1) that

Eh bFj ¯̄̄F,Bji = Fj, (3.3)

Varh bFj ¯̄̄F,Bji =

σ2j (Fj)

S[I−j−1]j

. (3.4)

5

• Conditionally, given F, {Ci,j : j = 0, 1, . . . , J} possess the Markov property, i.e. theconditional distribution of Ci,j+1 given {Ci,k : k = 0, 1, . . . , j} depends only on thelast observation Ci,j and not on the observations Ci,k for k < j. This is a slightlystronger assumption than assumptionM2 of Mack (Model Assumptions 2.1), whereonly the conditional first and second moments and not the whole conditional distri-bution depend only on the last observation Ci,j.

• Conditionally, given F,

{Yi,j : j = 0, 1, . . . , J − 1} are uncorrelated and (3.5)

Yi,j and Yk,l are independent for i 6= k. (3.6)

(3.5) is a well known result of Mack [9]. Note however, that the Yi,j in (3.5) are onlyuncorrelated 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 observa-tions DI , the random variables F0, F1, . . . , FJ−1 are independent with posterior distributiongiven by (3.9) .

Proof:To simplify the notation, we use in the remainder of this section the following termi-

nology: for j = 0, 1, . . . , J − 1, the distribution functions of the random variables Fj aredenoted by U (fj) , i.e. we use the same capital U for different distribution functions andthe argument fj in U (fj) says that it is the distribution function of Fj. Analogously, wedenote the conditional distributions, given Fj = fj or F = f , by Ffi (.) and Ff (.) , respec-tively. For instance, Ffj (yi,j| Bj) is the conditional distribution of Yi,j, given Fj = fj andgiven Bj.From Model Assumptions 3.1 follows that

dFf (y0, . . . ,yJ−1| B0) =J−1Yj=0

I−j−1Yi=0

dFfj(yi,j|Ci,j), (3.7)

where Ci,j = yi,j−1Ci,j−1 for j ≥ 1. For the joint posterior distribution of F given theobservations DI we find

dU (f0, . . . , fJ−1| DI) ∝J−1Yj=0

(I−j−1Yi=0

dFfj(yi,j|Ci,j) dU (fj))

(3.8)

∝J−1Yj=0

dU (fj| DI) , (3.9)

which completes the proof of the theorem. 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 conditionVar [Yi,j|Ci,j, Fj] = σ2j (Fj) /Ci,j. Indeed, the random variables Yi,j are the only onesin the Y -trapezoid containing information on Fj.

Next we derive the Bayes estimator of the ultimate claim.

Definition 3.3 An estimator bZ of some random variable Z is said to be better or equalthan an estimator eZ if

E

·³ bZ − Z´2¸ ≤ E ·³ eZ − Z´2¸ . (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 obser-vations. Then the best estimator of Z is

ZBayes = E [Z|X] . (3.11)

Remark:

• ZBayes also minimizes the conditional quadratic loss, i.e.

ZBayes = argminbZ E

·³ bZ − Z´2 ¯̄̄̄X¸ . (3.12)

Let bCi,J be a predictor of the ultimate claim Ci,J based on the observations DI .Definition 3.5 The conditional mean square error of bCi,J is defined by

mse³ bCi,J´ = E ·³ bCi,J − Ci,J´2 ¯̄̄̄DI¸ . (3.13)

Denote by bRi = bCi,J − Ci,I−i (3.14)

the corresponding reserve estimate. Note that

mse³ bCi,J´ = mse³bRi´ = E ·³ bRi −Ri´2 ¯̄̄̄DI¸ . (3.15)

From (3.12) follows thatCBayesi,J = E [Ci,J | DI ] (3.16)

is the best estimator minimizing the conditional mean square error (3.13) .

Theorem 3.6 Under Model Assumptions 3.1 we have

CBayesi,J = Ci,I−iJ−1Yj=I−i

FBayesj , (3.17)

where FBayesj denotes the Bayes estimator of Fj.

7

Remark:

• The corresponding reserve estimate is denoted byRBayesi = CBayesi,J − Ci,I−i.

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

CBayesi,J = E

"Ci,I−i

J−1Yj=I−i

Yi,j

¯̄̄̄¯DI

#

= Ci,I−iJ−1Yj=I−i

E [Yi,j| DI ]

= Ci,I−iJ−1Yj=I−i

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

= Ci,I−iJ−1Yj=I−i

E [Fj| DI ]

= Ci,I−iJ−1Yj=I−i

FBayesj .

2

Next we want to find a formula for the mean square error of RBayesi , which is the sameas the mean square error of CBayesi,J . Because of the general property

E£(X − a)2¤ = Var [X] + (E [X]− a)2

we have

mse³CBayesi,J

´= E

·³CBayesi,J − Ci,J

´2 ¯̄̄̄DI¸

= E [Var [Ci,J |F,DI ]| DI ]+E

·³CBayesi,J − E [Ci,J |F,DI ]

´2 ¯̄̄̄DI¸. (3.18)

In the classical approach of Mack [9] the first term corresponds to the process error andthe second to the estimation error. Here, this is not so clear any more. Since F is a randomvector, the first term is some kind of an ”average” process error (averaged over the set ofpossible 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)

= Ci,I−i σ2 (FJ−1)J−2Yj=I−i

Fj + F2J−1Var [Ci,J−1|F,DI ] . (3.19)

8

By iterating (3.19) we obtain

Var (Ci,J |F,DI) = Ci,I−iJ−1Xk=I−i

FI−i · . . . · Fk−1 · σ2 (Fk) · F 2k+1 · . . . · F 2J−1. (3.20)

Formula (3.20) is the same as the formula found by Mack [9], which is not surprising, sinceconditionally 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

E [Var (Ci,J |F,DI)| DI ] = Ci,I−iJ−1Xk=I−i

(k−1Y

m=I−iFBayesm · E £σ2 (Fk)¯̄DI¤ · J−1Y

n=k+1

E£F 2n¯̄DI¤) .(3.21)

The ”average” estimation error of CBayesi,J is given by

E

·³CBayesi,J −E [Ci,J |F,DI ]

´2 ¯̄̄̄DI¸= C2i,I−i · E

( J−1Yj=I−i

E [Fj| DI ]−J−1Yj=I−i

Fj

)2 ¯̄̄̄¯̄DI

= C2i,I−iVar

ÃJ−1Yj=I−i

Fj

¯̄̄̄¯DI

!. (3.22)

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 iis given by

mse³RBayesi

´= E

·³CBayesi,J − Ci,J

´2 ¯̄̄̄DI¸

= Ci,I−iΓI−i + C2i,I−i∆BI−i, (3.23)

where

ΓI−i =J−1Xk=I−i

(k−1Y

m=I−iFBayesm · E £σ2 (Fk)¯̄DI¤ · J−1Y

n=k+1

E£F 2n¯̄DI¤) , (3.24)

∆BI−i = Var

ÃJ−1Yj=I−i

Fj

¯̄̄̄¯DI

!. (3.25)

4 Credibility for Chain Ladder

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

CBayesi,J = Ci,I−iJ−1Yj=I−i

FBayesj . (4.1)

9

However, to calculate FBayesj one needs to know the distributions of the Fj as well as theconditional distributions of the Ci,j, given F. These distributions are usually unknown inthe insurance practice. The advantage of credibility theory is that one needs to know onlythe first and second moments. It is assumed that these first and second moments existand are finite for all considered random variables. Given a portfolio of similar risks, thesemoments can be estimated from the portfolio data. For the results of credibility theoryused in this paper we refer the reader to the literature, e.g. to the book by Bühlmannand Gisler [3].By chain ladder credibility we mean that we replace the FBayesj in (4.1) by credibility

estimators FCredj .

Definition 4.1 The credibility based predictor of the ultimate claim Ci,J given DI isdefined by

C(Cred)i,J = Ci,I−i

J−1Yj=I−i

FCredj . (4.2)

Remarks:

• Note that we have put the superscript Cred into brackets and that we call C(Cred)i

a credibility based estimator and not a credibility estimator. By definition a credi-bility estimator would be a linear function of the observations. However, given themultiplicative structure in the chain ladder methodology, it would not make senseto restrict to linear estimators of Ci,J .

• The corresponding reserve estimate is defined byR(Cred)i = C

(Cred)i,J − Ci,I−i.

Credibility estimators based on some statistic X are best estimators which are a linearfunction of the entries of X. For estimating Fj we base our estimator on the observationsYi,j, i = 0, . . . , I−j−1, since these are the only observations of the Y -trapezoid containinginformation on Fj.

Definition 4.2

FCredj = argminn bFj : bFj=a(j)0 +PI−j−1i=0 a

(j)i Yi,j

oE·³ bFj − Fj´2 ¯̄̄̄Bj¸ . (4.3)

In other words, FCredj is defined as the best estimator out ofnbFj : bFj = a(j)0 +

PI−j−1i=0 a

(j)i Yi,j

ominimizing E

·³ bFj − Fj´2 ¯̄̄̄Bj¸ .Remark:

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

10

Theorem 4.3 (Credibility estimator)

i) The credibility estimators of the unknown chain ladder factors Fj are

FCredj = αj bFj + (1− αj) fj, (4.4)

where

bFj =S[I−j−1]j+1

S[I−j−1]j

, (4.5)

αj =S[I−j−1]j

S[I−j−1]j +

σ2jτ2j

, (4.6)

fj = E [Fj] , (4.7)

σ2j = E£σ2j (Fj)

¤, where σ2j (Fj) is defined in (3.2) , (4.8)

τ 2j = Var [Fj] . (4.9)

ii) The mean square error of FCredj is

Eh¡FCredj − Fj

¢2 ¯̄̄Bji = αjσ2j

S[I−j−1]j

= (1− αj)τ2j . (4.10)

Remark:

• Note that bFj is the estimate of the development factor fj in the ”classical” chainladder model. That is, (4.4) is a credibility weighted average between the classicalchain ladder estimator bFj 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 assump-

tions of the Bühlmann and Straub model (see Section 4.2 in [3]). (4.4) is the well knowncredibility estimator of Bühlmann and Straub, and the formula of its mean square erroris 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, σ2j and τ 2j .These structural parameters can be estimated from portfolio data by using standardestimators (see for instance [3], Section 4.8).For the mean square error we obtain similar to (3.18)

mse³R(Cred)i

´= mse

³C(Cred)i,J

´= E

·³C(Cred)i,J − Ci,J

´2 ¯̄̄̄DI¸

= E [Var [Ci,J |F,DI ]| DI ] +E·³C(Cred)i,J − E [Ci,J |F,DI ]

´2 ¯̄̄̄DI¸.

11

The first summand, the ”average” process error, remains unchanged and is the same asin Theorem 3.7. For the ”average” estimation error we obtain

E

·³C(Cred)i,J −E [Ci,J |F,DI ]

´2 ¯̄̄̄DI¸

= C2i,I−iE

à J−1Yj=I−i

FCredj −J−1Yj=I−i

Fj

!2 ¯̄̄̄¯̄DI

. (4.11)

Hencemse

³R(Cred)i

´= Ci,I−iΓI−i + C2i,I−i∆

CI−i,

where

ΓI−i =J−1Xk=I−i

(k−1Y

m=I−iFBayesm · E £σ2 (Fk)¯̄DI¤ · J−1Y

n=k+1

E£F 2n¯̄DI¤) , (4.12)

∆CI−i = E

à J−1Yj=I−i

FCredj −J−1Yj=I−i

Fj

!2 ¯̄̄̄¯̄DI

. (4.13)

To find an estimator for the mean square error we make the following approximationsin (4.12) and (4.13):

FBayesj ' FCredj , (4.14)

E

·³Fj − FBayesj

´2 ¯̄̄̄DI¸' E

h¡Fj − FCredj

¢2 ¯̄̄BIi = αjσ2j

S[I−j−1]j

, (4.15)

where ' means that the equation is not exactly but only approximately fulfilled.Then we get

E£F 2j¯̄DI¤ = E

·³Fj − FBayesj

´2 ¯̄̄̄DI¸2+³FBayesj

´2' αj

σ2j

S[I−j−1]j

+¡FCredj

¢2.

E

à J−1Yj=I−i

FCredj −J−1Yj=I−i

Fj

!2 ¯̄̄̄¯̄DI

' E

à J−1Yj=I−i

FBayesj −J−1Yj=I−i

Fj

!2 ¯̄̄̄¯̄DI

= Var

ÃJ−1Yj=I−i

Fj

¯̄̄̄¯DI

!

=J−1Yj=I−i

E£F 2j¯̄DI¤− J−1Y

j=I−i(E [Fj| DI ])2

'J−1Yj=I−i

Ãαj

σ2j

S[I−j−1]j

+¡FCredj

¢2!− J−1Yj=I−i

¡FCredj

¢2.

Thus we have found the following approximation for the mean square error of R(Cred)i :

12

Theorem 4.4mse

³R(Cred)i

´ ∼= Ci,I−iΓ∗I−i + C2i,I−i∆∗I−i, (4.16)

where

Γ∗I−i =J−1Xk=I−i

(k−1Y

m=I−iFCredm · σ2k

J−1Yn=k+1

µ¡FCredn

¢2+ αn

σ2n

S[I−n−1]n

¶), (4.17)

∆∗I−i =J−1Yj=I−i

áFCredj

¢2+ αj

σ2j

S[I−j−1]j

!−

J−1Yj=I−i

¡FCredj

¢2. (4.18)

Remark:

• By replacing in (4.17) and (4.18) σ2j and the variance components σ2j and τ 2j in αjby appropriate estimates, we obtain from Theorem 4.4 an estimator for the meansquare error of R(Cred)i .

Often the mean square error or the standard error (=square root of the mean squareerror) of the total reserve is of interest too. Denote by

R(Cred) =Xi

R(Cred)i (4.19)

the total credibility reserve, where R(Cred)i = 0 for those i for which I − i ≥ J.

mse(R(Cred)) = E

ÃXi

C(Cred)i,J −

Xi

Ci,J

!2 ¯̄̄̄¯̄DI

= E

"Var

"Xi

Ci,J

¯̄̄̄¯F,DI

#¯̄̄̄¯DI

#

+E

ÃXi

C(Cred)i,J −

Xi

E [Ci,J |F,DI ]!2 ¯̄̄̄¯̄DI

. (4.20)

Because of the conditional independence of the accident years we get

E

"Var

"Xi

Ci,J

¯̄̄̄¯F,DI

#¯̄̄̄¯DI

#=

Xi

E [Var [Ci,J |F,DI ]| DI ]

=Xi

Ci,I−iΓI−i, (4.21)

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

13

For the second summand, the average estimation error, we obtain

E

ÃXi

C(Cred)i,J −

Xi

E [Ci,J |F,DI ]!2 ¯̄̄̄¯̄DI

=

IXi=0

E

·³C(Cred)i,J −E [Ci,J |F,DI ]

´2 ¯̄̄̄DI¸

(4.22)

+ 2IXi=0

IXk=i+1

Eh³C(Cred)i,J − E [Ci,J |F,DI ]

´³C(Cred)k,J −E [Ck,J |F,DI ]

´¯̄̄DIi.

With the approximations (4.14) and (4.15) we get for the terms in the second summand

Eh³C(Cred)i,J − E [Ci,J |F,DI ]

´³C(Cred)k,J − E [Ck,J |F,DI ]

´¯̄̄DIi

= Ci,I−iCk,I−kE

"ÃJ−1Yj=I−i

FCredj −J−1Yj=I−i

Fj

!ÃJ−1Yl=I−k

FCredl −J−1Yl=I−k

Fl

!¯̄̄̄¯DI

#

' Ci,I−iCk,I−kE"Ã

J−1Yj=I−i

FBayesj −J−1Yj=I−i

Fj

!ÃJ−1Yl=I−k

FBayesl −J−1Yl=I−k

Fl

!¯̄̄̄¯DI

#

= Ci,I−iCk,I−kCov

ÃJ−1Yj=I−i

Fj,J−1Yl=I−k

Fl

¯̄̄̄¯DI

!

= Ci,I−iCk,I−kI−i−1Yj=I−k

FBayesj Var

ÃJ−1Yj=I−i

Fj,

¯̄̄̄¯DI

!' Ci,I−iC(Cred)k,I−k ∆

∗I−i, (4.23)

where ∆∗I−i is given by (4.12) . In the second last equation we have used

Cov

ÃJ−1Yj=I−i

Fj,J−1Yl=I−k

Fl

¯̄̄̄¯DI

!

= E

"Cov

ÃJ−1Yj=I−i

Fj,J−1Yl=I−k

Fl

¯̄̄̄¯DI , FI−k, . . . , FI−i−1

!¯̄̄̄¯DI

#

+Cov

ÃE

"J−1Yj=I−i

Fj

¯̄̄̄¯DI , FI−k, . . . , FI−i−1

#, E

"J−1Yj=I−k

Fj

¯̄̄̄¯DI , FI−k, . . . , FI−i−1

#¯̄̄̄¯DI

!

= E

"I−i−1Yl=I−k

Fl Var

ÃJ−1Yj=I−i

Fj

¯̄̄̄¯DI ,

!¯̄̄̄¯DI

#+ 0.

Thus we have found the following result:

14

Corollary 4.5

mse(R(Cred)) 'Xi

mse³R(Cred)i

´+ 2

IXi=0

IXk=i+1

Ci,I−iC(Cred)k,I−i ∆

∗I−i,

where mse³R(Cred)i

´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 credibilityin the literature. In this section we consider a class of models for the chain ladder methodwhere this is the case, i.e. where FBayesj = FCredj .The basic assumption of the class of models considered in this section is that, condition-

ally on F and Bj, the random variables Y0,j, . . . , YI,j are independent with a distributionbelonging to the one-parameter exponential dispersion family and that the a priori dis-tribution 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 vectorΘ of the canonical parameters Θj with realizations ϑ. Of course, as we will see, the twoparameters are linked each to the other.

Definition 5.1 A distribution is said to be of the exponential dispersion type, if it can beexpressed as

dF (x) = exp

·xϑ− b(ϑ)

ϕ/w+ c(x,ϕ/w)

¸dν(x), x ∈ A ⊂ R. (5.1)

whereν (.) 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 wellknown from standard theory of generalized linear models (GLM) (see e.g. [13] or [4]) that

µX = E [X] = b0(ϑ), (5.2)

σ2X = Var (X) =ϕ

wb00(ϑ). (5.3)

By taking the inverse in (5.2) we obtain

ϑ = (b0)−1 (µX) = h(µX), (5.4)

where h(.) is called the canonical link function.The variance can also be expressed as a function of the mean by

Var (X) =ϕ

wb00(h(µX)) =

ϕ

wV (µX) , (5.5)

where V (.) is the so-called variance function.

15

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 familiesF b,cexp ∈ Fexp (5.7)

specified by the specific form of b(.) and c(.,.).

In the sequel we assume that, conditionally onF and Bj, the random variables Y0,j, . . . , YI,jare independent with a distribution belonging to a one-parameter exponential family F b,cexp.The following results hold true under this general condition. However, not all exponen-tial families are suited for our problem. In particular, the random variables Yi,j are non-negative. Hence, only distributions having support on R+ are suitable for the chain-laddersituation.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 withvariance function

V (µ) = µp. (5.8)

They are defined only for p outside the interval 0 < p < 1. For p ≤ 0 they have positiveprobability mass on the whole real line. The family of the Tweedie models include inparticular 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 connectionwith calculating risk premiums in a multiplicative tariff structure. In that paper, there isa good summary on the properties of this family. For the chain ladder, the case 1 ≤ p ≤ 2is 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 independentwith 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)

uj(ϑ) = exp

"f(0)j ϑ− b(ϑ)

η2j+ d(f

(0)j , η

2j)

#, (5.9)

where f (0)j and η2j are hyperparameters and where exphd(f

(0)j , η

2j)iis a normalizing

factor.

Remarks:

• From Assumption E1 follows that

fϑj (yi,j| Bj) = exp½yi,j ϑj − b(ϑj)

ϕj/Ci,j

¾a¡yi,j,ϕj/Ci,j

¢. (5.10)

• From Assumption E1 also follows that

E [Yi,j|Θj] = b0 (Θj) = Fj, (5.11)

Var (Yi,j|Θj, Ci,J) = b00 (Θj)ϕjCi,j

= V (Fj)ϕjCi,j

. (5.12)

Hence conditionally, given Fj, the chain ladder assumption M2 of Mack (ModelAssumptions 2.1) are fulfilled. The difference to the Mack assumptions is, thatspecific assumptions on the whole conditional distribution and not only on the firstand second conditional moments are made. For instance, in the Tweedie modelwith p = 1, Yi,j are assumed to be conditionally over-dispersed Poisson distrib-uted. Note however, that this conditional assumptions are not comparable with theModel Assumptions 2.2 and much closer to the idea of the historical chain laddermethodology.

• The distributions of Θ0, . . . ,ΘJ−1 with densities (5.9) belong to the family Ubexp ofthe natural conjugate prior distributions to F b,cexp, which is given by

U bexp =©uγ (ϑ) : γ = (x0, η

2) ∈ R×R+ª ,uγ (ϑ) = exp

·x0ϑ− b(ϑ)

η2+ d(x0, η

2)

¸, ϑ ∈M.

• The Gaussian model and the Gamma model studied in Gisler [5] are special casesof the exponential dispersion model defined above.

• The model is formulated in terms of individual development factors Yi,j. We couldalso formulate it in terms of Ci,j. Then, for given ϑ, we would obtain the time seriesmodel 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)E [Fj] = f

(0)j .

ii)FBayesj = FCredj = αj bFj + (1− αj) f

(0)j , (5.13)

where

bFj =S[I−j−1]j+1

S[I−j−1]j

, (5.14)

αj =S[I−j−1]j

S[I−j−1]j +

σ2jτ2j

, (5.15)

σ2j = ϕj E [b00(Θj)] = ϕj E [V (Fj)] , (5.16)

τ 2j = Var [b0 (Θj)] = Var (Fj) . (5.17)

Remarks:

• Note that FBayesj in Theorem 5.4 coincides with the credibility estimator of Theorem4.3.

• For the Tweedie models with p ≥ 0 the conditions and the result of the theorem arefulfilled for p = 0 (Normal-distribution), for 1 < p < 2 (compound Poisson) and forp = 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 thecase with weights can for instance be found in [3]. 2

Theorem 5.5 If, in addition to the conditions of Theorem 3.4, u0j(ϑ) disappears on the

boundary of M then

i)

E

·³FBayesj − Fj

´2 ¯̄̄̄Bj¸= αj

σ2j

S[I−j−1]j

= (1− αj)τ2j . (5.18)

ii) The conditional mean square error of the reserve RBayesi of accident year i is givenby

mse(RBayesi ) = E

·³C(Cred)i,J − Ci,J

´2 ¯̄̄̄DI¸

' E

·³C(Cred)i,J − Ci,J

´2 ¯̄̄̄BI−i

¸= Ci,I−iΓ∗I−i + C

2i,I−i∆

∗I−i, (5.19)

18

where

Γ∗I−i =J−1Xk=I−i

(k−1Y

m=I−iFCredm · σ2k

J−1Yn=k+1

µ¡FCredn

¢2+ αn

σ2n

S[I−n−1]n

¶),

∆∗I−i = C2i,I−i

"J−1Yj=I−i

áFCredj

¢2+ αj

σ2j

S[I−j−1]j

!−

J−1Yj=I−i

¡FCredj

¢2#.

iii) The mean square error of the total reserve RBayes =P

iRBayesi is

mse(RBayes) 'Xi

mse(RBayesi ) + 2IXi=0

IXk=i+1

Ci,I−iC(Bayes)k,I−i ∆I−i. (5.20)

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 arefulfilled for p = 0 (Normal-distribution) and for 1 < p < 2 (compound Poisson). Forp = 2 (Gamma-distribution) it is only fulfilled for η2 < 1. In this case the naturalconjugate prior uj (ϑ) is again a Gamma-distribution, and η2 < 1 means, that theshape parameter γ of this Gamma-distribution is smaller than one.

Proof of the theorem:Since FBayesj 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 issquare integrable. From (5.9) we get

u0j (ϑ) =1

η2

³f(0)j − b0 (ϑ)

´uj (ϑ) ,

u00j (ϑ) =1

η4

³f(0)j − b0 (ϑ)

´2uj (ϑ) +

1

η2b00 (ϑ)uj (ϑ) .

Since u0j (ϑ) disappears on the boundaries of M, we have

0 =1

η4

ZM

³f(0)j − b0 (ϑ)

´2uj (ϑ) dϑ+

1

η2

ZM

b00 (ϑ)uj (ϑ) dϑ

=1

η4Var (Fj) +

1

η2ϕjE [V (Fj)] .

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 limitingcase for τ 2j →∞.

Theorem 6.1 Under Model Assumptions 5.3 and if uj(ϑ) disappears on the boundariesof M, then for τ 2j →∞ (non-informative prior)

FBayesj = cFj = S[I−j−1]j+1

S[I−j−1]j

= bfj, (6.1)

RBayesi = RCLi . (6.2)

Remarks:

• Note, that bfj are the estimates of the chain ladder factors in the classical chainladder model. Hence in this case the Bayes chain ladder forecasts are the same asthe classical chain ladder forecasts and the resulting Bayes reserves are the same asthe classical chain ladder reserves.

• For the Tweedie models with p ≥ 0 the result holds true for p = 0 (Normal) and1 < 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 thelimiting case of non-informative priors, which is in this case the mean square error ofthe classical chain ladder forecast. Thus the following result gives another view on theestimation of the mean square error of the classical chain ladder method and suggestsa different estimator to the ones found so far in the literature. The estimation of themean 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 approachis the appropriate way to look at the conditional situation and to estimate the conditionalmean 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 theboundaries of M, then it holds, for τ 2j →∞ (non-informative prior), that the conditionalmean square error of the chain ladder reserves RCLi can be estimated by

dmse ¡RCLi ¢= Ci,I−ibΓI−i + C2i,I−i b∆I−i, (6.3)

20

where

bΓI−i =J−1Xk=I−i

(k−1Y

m=I−icfm · bσ2k J−1Y

n=k+1

µ bfn2 + bσ2nS[I−n−1]n

¶), (6.4)

b∆I−i =J−1Yj=I−i

Ãbfj2 + bσ2jS[I−j−1]j

!−

J−1Yj=I−i

bfj2, (6.5)

where bσ2j are appropriate estimators for σ2j .Remarks:

• For the Tweedie models, the result holds true for p = 0 (Normal) and for 1 < p < 2,but not for p = 2, since η2 has to be smaller than one for p = 2. In particular thecase 1 < p < 2 (compound Poisson with Gamma ”claim sizes”) seems to us a fairlyadequate and good model conditional for a chain-ladder claim development process.

• These estimators should be compared to the ones suggested by Mack [9] and Buch-walder et al. [2]. The estimate bΓI−i of the ”average” process error is slightly biggerbecause of the terms bσ2n/S[I−n−1]n . The estimate b∆I−i of the ”average” estimationerror 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

Corollary 6.3 The mean square error of the total reserve RCL =P

iRCLi can be esti-

mated by

dmse ¡RCL¢ =Xi

dmse ¡RCLi ¢+ 2

IXi=0

IXk=i+1

Ci,I−i bCk,I−i b∆I−i, (6.6)

where bCk,I−i = Ck,I−k I−i−1Yj=I−k

bfjand where dmse ¡RCLi ¢

and b∆I−i are as in Theorem 6.2 and where bfj are the classical chainladder factors given by (6.1).

Remark:

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

21

7 Numerical Example

For pricing and profit-analysis of different business units it is necessary to set up thereserves for each of these business units (BU). In Appendix A trapezoids of cumulativepayments of the business line contractors all risks insurance for different BU of WinterthurInsurance Company are given (for confidentiality purposes the figures are multiplied witha constant). The aim is to determine for this line of business the reserves for each businessunit. Thus we have a portfolio of similar loss development figures, which is suited to applythe theory presented in this paper.The following table shows the estimated values of σ2j and of the development factors

Fj estimated by classical chain ladder and by the credibility estimators of Section 4. Thestructural parameters fj,σ2j and τ 2j have been estimated by the ”standard estimators”,which can be found e.g. in Section 4.8 of [3].

table of resultsj= 0 1 2 3 4 5 6 7 8 9 productσj

2336.53 34.739 7.828 5.934 0.426 4.342 4.248 0.239 0.097 0.154

all BU FjCL 2.111 1.119 1.031 1.013 1.004 1.001 0.993 0.998 1.000 0.999 2.452

FjCL

2.270 1.233 0.982 1.024 1.012 0.981 0.962 1.003 0.996 1.000 2.686

Fjcred

2.111 1.189 0.996 1.015 1.004 1.001 0.984 0.998 1.000 0.999 2.499

FjCL

2.133 1.094 1.032 1.002 0.998 1.000 1.014 0.999 1.000 0.990 2.416

Fjcred

2.111 1.111 1.033 1.012 1.003 1.001 0.997 0.998 1.000 0.997 2.444

FjCL

2.189 1.138 1.037 1.042 1.003 1.000 0.999 1.002 1.000 1.000 2.700

Fjcred

2.111 1.134 1.036 1.016 1.004 1.001 0.994 0.998 1.000 0.999 2.509

FjCL

2.108 1.070 1.054 1.013 1.004 1.015 0.996 0.995 1.000 1.000 2.429

Fjcred

2.111 1.084 1.050 1.013 1.004 1.001 0.994 0.998 1.000 0.999 2.426

FjCL

1.930 1.114 1.018 0.995 1.002 0.997 0.999 0.997 1.002 1.000 2.172

Fjcred

2.111 1.119 1.021 1.010 1.004 1.001 0.995 0.998 1.000 0.999 2.429

FjCL

3.008 1.190 1.146 1.006 1.000 0.979 0.996 1.000 1.000 1.004 4.041

Fjcred

2.111 1.139 1.064 1.013 1.004 1.001 0.993 0.998 1.000 0.999 2.578

E

F

A

B

C

D

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 F

Cred0 is

identical to FCL0 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

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

0 1 2 3 4 5 6 7 8 9

Development Year j

BU ABU BBU CBU DBU EBU F

Credibility Development Pattern

1

1.05

1.1

1.15

1.2

1.25

0 1 2 3 4 5 6 7 8 9

Development Year

BU ABU BBU CBU DBU EBU F

From these graphs one can see the smoothing effect on the chain ladder procedure onthe estimates.

The next two graphs show for the business units A and E the same kind of develop-ment patterns. ”CL Portfolio” is the one obtained by the chain ladder factors of theportfolio data (total of all BU), ”CL BU” the one obtained by the chain ladder factors ofthe BU and ”Cred BU” the one obtained by the credibility estimated chain ladder factorsof the BU. One can see out of these figures that Cred BU is somewhere in between ”CL

23

Portfolio” and ”CL BU”.

BU A

1.000

1.050

1.100

1.150

1.200

1.250

1.300

0 1 2 3 4 5 6 7 8 9

development year

CL PortfolioCred BUCL BU

BU E

1.0001.0201.0401.0601.0801.1001.1201.1401.1601.1801.200

0 1 2 3 4 5 6 7 8 9

develoopment year

CL PortfolioCred BUCL BU

Finally the next table shows the estimated reserves and the estimates of the squareroot of the mean square error (mse).

BUCL Cred

A 486 504 657 135.1% 498 98.9%B 235 244 288 122.7% 402 164.3%C 701 517 411 58.6% 520 100.6%D 1'029 899 844 82.1% 729 81.1%E 495 621 397 80.2% 596 95.9%F 40 25 140 347.0% 149 595.8%sum 2'987 2'810 1'254 42.0% 1'261 44.9%

estimated mse1/2

CL Credestimated reserves

The mse of the chain ladder estimate was calculated by applying the results of Theorem6.2. 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 unitsis 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% and38%. Hence the differences between the chain ladder and the credibility estimated reservescan be quite substantial and may have a considerable impact on the profit and loss of theindividual 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 theestimated mse1/2 is 48.4%. This shows once more the well known fact that the chainladder 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 Bayesianway. In our paper, we have followed the empirical credibility approach and estimated thestructural parameters from the portfolio data. Therefore we could not do a credibilityestimate based on the trapezoid of the portfolio data.

Acknowledgment:

The authors thank F. Schnelli for having carried through the calculations of the numericalexample in Section 7.

Addresses of authors:

Alois Gisler Mario Valentin WüthrichChief Actuary Non-Life Departement MathematikWinterthur Insurance Company HG F42.2P.O. Box 357 Rämistrasse 101CH 8401 Winterthur CH 8092 Zürich

Email: alois.gisler@winterthur.ch Email: mario.wuethrich@math.ethz.ch

25

Appendix

A Loss Development Data

The following trapezoids of cumulative payements show the loss development of the linebuilding engineering for different business units. For confidentiality reasons the data weremultiplied with some constant.

BU Adev.year

acc. year 0 1 2 3 4 5 6 7 8 9 101986 118 487 1'232 1'266 1'266 1'397 1'397 1'397 1'492 1'492 1'4921987 124 657 863 890 914 916 941 941 941 865 8651988 556 2'204 3'494 2'998 2'983 3'018 2'458 2'458 2'470 2'470 2'4701989 1'646 2'351 2'492 2'507 2'612 2'612 2'608 1'755 1'755 1'755 1'7551990 317 886 890 890 950 990 990 990 990 990 9901991 242 919 1'218 1'224 1'229 1'249 1'249 1'249 1'249 1'249 1'2491992 203 612 622 639 667 647 647 647 647 647 6471993 492 1'405 1'685 1'668 1'753 1'742 1'804 1'804 1'804 1'804 1'8041994 321 1'149 1'728 1'863 1'877 1'877 1'877 1'877 1'877 1'877 1'8771995 609 1'109 1'283 1'294 1'253 1'255 1'255 1'255 1'255 1'255 1'2551996 492 1'627 1'622 1'672 1'672 1'672 1'672 1'672 1'621 1'621 1'6211997 397 793 868 889 964 964 964 964 964 9641998 523 1'098 1'475 1'489 1'489 1'489 1'489 1'489 1'4891999 1'786 2'951 3'370 3'029 3'211 3'289 3'325 3'3252000 241 465 536 596 652 652 6522001 327 622 577 583 583 5832002 275 520 529 529 5412003 89 327 378 3822004 295 301 3962005 151 4062006 315

BU Bdev.year

acc. year 0 1 2 3 4 5 6 7 8 9 101986 92 442 541 541 528 528 528 528 528 528 5281987 451 1'077 1'085 1'178 1'212 1'217 1'217 1'217 1'217 1'217 1'2171988 404 717 834 849 849 850 850 850 850 850 8501989 203 572 813 875 878 910 912 1'096 1'089 1'089 9861990 352 834 1'048 1'072 1'088 1'088 1'088 1'088 1'088 1'088 1'0881991 504 1'246 1'272 1'353 1'285 1'285 1'285 1'285 1'285 1'285 1'2851992 509 1'008 1'061 1'061 1'061 1'071 1'071 1'071 1'071 1'071 1'0711993 229 580 630 670 672 672 672 672 672 672 6721994 324 815 871 859 867 777 777 777 777 777 7771995 508 805 906 969 971 971 971 971 971 971 9711996 354 641 833 842 842 842 842 842 842 842 8421997 431 847 854 915 918 918 918 918 918 9181998 205 830 978 1'034 1'048 1'048 1'048 1'048 1'0481999 522 1'134 1'064 1'202 1'202 1'210 1'210 1'2102000 567 925 915 957 953 953 9532001 1'238 1'924 2'034 1'897 1'897 1'8972002 355 1'003 1'137 1'164 1'1962003 312 680 682 6862004 246 352 4182005 91 4182006 130

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BU Cdev.year

acc. year 0 1 2 3 4 5 6 7 8 9 101986 268 456 485 483 483 483 483 483 483 483 4831987 268 520 577 579 579 579 579 579 579 579 5791988 385 968 1'017 1'019 1'019 1'019 1'019 1'019 1'019 1'019 1'0191989 251 742 795 931 931 931 931 931 931 931 9311990 456 905 1'162 1'164 1'164 1'164 1'164 1'164 1'191 1'191 1'1911991 477 1'286 1'376 1'376 1'373 1'373 1'373 1'373 1'373 1'373 1'3731992 405 999 1'172 1'196 1'196 1'210 1'210 1'210 1'210 1'210 1'2101993 443 932 952 965 984 992 1'012 1'012 1'012 1'012 1'0121994 477 1'046 1'336 1'362 1'375 1'375 1'375 1'375 1'375 1'375 1'3751995 581 1'146 1'316 1'362 1'391 1'391 1'391 1'391 1'391 1'391 1'3911996 401 997 1'229 1'248 1'281 1'284 1'264 1'264 1'264 1'264 1'2641997 474 778 939 1'321 1'366 1'392 1'392 1'392 1'392 1'3921998 649 1'420 1'707 1'709 1'709 1'709 1'709 1'638 1'6381999 911 1'935 2'304 2'307 2'309 2'309 2'309 2'3622000 508 1'054 1'101 1'071 1'071 1'071 1'0712001 389 790 868 909 1'569 1'5692002 373 998 1'091 1'155 1'2012003 276 853 932 9482004 465 820 8592005 343 6222006 254

BU Ddev.year

acc. year 0 1 2 3 4 5 6 7 8 9 101986 330 1'022 1'066 1'086 1'094 1'094 1'094 1'094 1'094 1'094 1'0941987 327 873 1'057 1'076 1'082 1'082 1'082 1'082 1'082 1'082 1'0821988 304 1'137 1'234 1'460 1'475 1'588 1'586 1'586 1'586 1'586 1'5861989 426 1'289 1'418 1'574 1'578 1'634 2'250 2'044 2'044 2'044 2'0441990 750 2'158 2'910 3'071 3'213 3'199 3'052 3'052 3'052 3'052 3'0521991 761 2'164 2'446 2'570 2'578 2'558 2'558 2'558 2'558 2'558 2'5581992 1'119 2'666 2'946 3'008 3'021 3'022 3'019 3'019 3'019 3'019 3'0191993 917 2'458 2'892 3'502 3'629 3'664 3'887 3'867 3'697 3'697 3'6971994 905 2'014 2'459 2'466 2'554 2'554 2'554 2'540 2'540 2'540 2'5401995 1'761 2'990 3'235 3'795 3'816 3'841 3'842 3'860 3'860 3'860 3'8601996 824 2'063 2'378 2'368 2'384 2'368 2'373 2'373 2'373 2'373 2'3731997 4'364 6'630 6'850 6'885 6'923 6'923 6'923 6'923 6'923 6'9231998 493 1'587 1'780 1'794 1'838 1'838 1'838 1'865 1'8651999 4'092 7'710 6'596 7'201 7'292 7'292 7'292 7'2922000 1'733 3'647 3'699 3'780 3'773 3'773 3'7732001 1'261 2'658 3'063 3'036 3'093 3'0952002 1'517 3'054 3'335 3'438 3'4382003 778 1'212 1'247 1'2152004 727 1'661 1'8162005 561 1'4862006 459

BU Edev.year

acc. year 0 1 2 3 4 5 6 7 8 9 101986 486 964 1'057 1'106 1'130 1'130 1'138 1'131 1'131 1'131 1'1311987 867 1'669 1'643 1'717 1'720 1'724 1'724 1'724 1'724 1'724 1'7241988 1'285 1'925 2'204 2'488 2'507 2'509 2'510 2'510 2'436 2'436 2'4361989 395 994 1'309 1'442 1'467 1'467 1'477 1'477 1'477 1'477 1'4771990 802 1'468 1'776 1'823 1'827 1'832 1'833 1'833 1'833 1'833 1'8331991 966 1'967 2'628 2'743 2'294 2'338 2'358 2'358 2'358 2'358 2'3581992 759 1'766 1'922 1'863 1'886 1'886 1'886 1'886 1'886 1'886 1'8861993 1'136 2'139 2'219 1'921 1'931 1'944 1'947 1'867 1'867 1'867 1'8671994 1'467 2'243 2'553 2'598 2'598 2'598 2'598 2'598 2'598 2'598 2'5981995 1'309 2'521 2'660 2'640 2'639 2'641 2'659 2'659 2'659 2'659 2'6591996 877 2'170 2'341 2'420 2'516 2'516 2'431 2'431 2'431 2'468 2'4681997 1'004 1'963 2'260 2'226 2'226 2'215 2'215 2'059 2'059 2'0591998 1'351 2'579 2'736 2'759 2'760 2'766 2'688 2'737 2'7371999 906 2'341 2'667 2'655 2'655 2'650 2'650 2'8242000 563 1'450 1'575 1'603 1'654 1'654 1'6752001 417 1'006 1'034 1'049 1'049 1'0502002 322 836 1'046 1'123 1'1432003 1'047 1'656 1'689 1'7792004 497 843 8772005 1'021 1'2372006 302

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BU Fdev.year

acc. year 0 1 2 3 4 5 6 7 8 9 101986 18 64 64 64 64 64 64 64 64 64 641987 20 73 103 153 155 155 155 155 155 155 1551988 20 70 318 328 328 328 328 328 328 328 3281989 88 133 133 133 133 133 133 133 133 133 1331990 3 180 214 214 215 215 215 215 215 215 2151991 11 79 80 82 81 81 81 81 81 81 811992 17 66 105 172 172 172 188 188 188 188 1991993 73 216 218 218 218 218 218 218 218 218 2181994 48 213 253 386 400 400 317 304 304 304 3041995 98 153 153 158 158 158 158 158 158 158 1581996 38 529 557 632 639 639 639 639 639 639 6391997 42 140 141 141 141 141 141 141 141 1411998 64 95 95 102 102 102 102 102 1021999 57 144 169 178 178 178 178 1782000 85 178 188 186 186 186 1862001 212 341 357 371 371 3712002 56 152 187 246 2462003 25 44 103 1782004 19 137 1402005 25 452006 7

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