tUltimate lossClaims paymentsClaims incurredIncurred lossesPrediction uncertainty

1. Paidincurred chain model

1.1. Introduction

The main task of reserving actuaries is to predict ultimate lossratios and outstanding loss liabilities. In general, this predictionis based on past information that comes from different sources ofinformation. In order to get a unified prediction of the outstandingloss liabilities one needs to rank these information channels byassigning credibility weights to the available information. Oftenthis is a difficult task. Therefore, most classical claims reservingmethods are based on one information channel only (for instance,claims payments or incurred losses data).Halliwell (1997, 2009) was probably one of the first who inves-

tigated the problem of combining claims payments and incurredlosses data for claims reserving from a statistical point of view. Theanalysis of Halliwell (1997, 2009) as well as of Venter (2008) isdone in a regression framework.A second approach to unify claims prediction based on claims

payments and incurred losses is the Munich chain ladder (MCL)method. The MCL method was introduced by Quarg and Mack(2004) and their aim is to reduce the gap between the two chainladder (CL) predictions that are based on claims payments and in-curred losses data, respectively. The idea is to adjust the CL fac-tors with incurredpaid ratios to reduce the gap between the two

Corresponding author.E-mail address:mario.wuethrich@math.ethz.ch (M.V. Wthrich).

predictions (see Quarg andMack, 2004; Verdier and Klinger, 2005;Merz andWthrich, 2006 and Liu and Verrall, 2008). The difficultywith the MCL method is that it involves several parameter esti-mations whose precision is difficult to quantify within a stochasticmodel framework.A third approach was presented in Dahms (2008). Dahms

considers the complementary loss ratio method (CLRM) where theunderlying volume measures are the case reserves which is thebasis for the regression and CL analysis. Dahms CLRM can alsobe applied to incomplete data and he derives an estimator for theprediction uncertainty.In this paperwe present a novel claims reservingmethodwhich

is based on the combination of Hertigs log-normal claims re-serving model (Hertig, 1985) for claims payments and of GogolsBayesian claims reserving model (Gogol, 1993) for incurred lossesdata. The idea is to use Hertigs model for the prior ultimateloss distribution needed in Gogols model which leads to apaidincurred chain (PIC) claims reserving method. Using ba-sic properties of multivariate Gaussian distributions we obtain amathematically rigorous and consistentmodel for the combinationof the two information channels claims payments and incurredlosses data. The analysis will attach credibility weights to thesesources of information and it will also involve incurredpaid ra-tios (similar to the MCL method, see Quarg and Mack (2004)). OurPIC model will provide one single estimate for the claims reserves(based on both information channels) and, moreover, it has the ad-vantage that we can quantify the prediction uncertainty and thatit allows for complete model simulations. This means that this PICmodel allows for the derivation of the full predictive distributionInsurance: Mathematics and E

Contents lists availa

Insurance: Mathema

journal homepage: www

Paidincurred chain claims reserving meMichael Merz a, Mario V. Wthrich b,a University of Hamburg, Department of Business Administration, 20146 Hamburg, Germab ETH Zurich, Department of Mathematics, 8092 Zurich, Switzerland

a r t i c l e i n f o

Article history:Received July 2009Received in revised formFebruary 2010Accepted 16 February 2010

Keywords:Claims reservingOutstanding loss liabilities

a b s t r a c t

We present a novel stochasand incurred losses informa(1985)model and Gogols (19Bayesian point of view for thedistribution of the outstandiof the claims reserves and thhand, simulation algorithms0167-6687/$ see front matter 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.insmatheco.2010.02.004conomics 46 (2010) 568579

ble at ScienceDirect

tics and Economics

.elsevier.com/locate/ime

thod

ny

ic model for claims reserving that allows us to combine claims paymentstion. The main idea is to combine two claims reserving models (Hertigs93)model) leading to a log-normal paidincurred chain (PIC)model. Using aparametermodellingwederive in this Bayesian PICmodel the full predictiveng loss liabilities. On the one hand, this allows for an analytical calculatione corresponding conditional mean square error of prediction. On the otherprovide any other statistics and risk measure on these claims reserves.

2010 Elsevier B.V. All rights reserved.

iWe define for j {0, . . . , J} the setsBPj =

{Pi,l : 0 i J, 0 l j

},

B Ij ={Ii,l : 0 i J, 0 l j

},

Bj = BPj B Ij ,the paid, incurred and joint paid and incurred data, respectively,up to development year j.

Model Assumption 1.1 (Log-Normal PIC Model). Conditionally, given 2 = (0, . . . ,J ,0, . . . ,J1, 0, . . . ,J , 0, . . . , J1), we have: the random vector (0,0, . . . , J,J , 0,0, . . . , J,J1) has a mul-tivariate Gaussian distribution with uncorrelated compo-nents given byi,j N

(j,

2j

)for i {0, . . . , J} and j {0, . . . , J},

k,l N(l,

2l

)for k {0, . . . , J} and l {0, . . . , J 1};

Quarg and Mack (2004) found high correlations betweenincurredpaid ratios. In the Example section (see Section 5) wecalculate the implied posterior correlation between thejs andthe ls (see Table 12). Our findings are that these correlationsfor our data set are fairly small (except in regions wherewe have only a few observations). Therefore, we refrain fromintroducing dependence between the components. However,this dependence could be implemented but then the solutionscan only be found numerically and, moreover, the estimation ofthe correlation matrix is not obvious. We choose a log-normal PIC model. This has the advantagethat the conditional distributions of Pi,J , given 2 and Bj,BPj or B

Ij , respectively, can be calculated explicitly. Other

distributional assumptions only allow for numerical solutionsusing simulations with missing data (see, for instance van Dykand Meng, 2001).M. Merz, M.V. Wthrich / Insurance: Mathe

Table 1Left-hand side: cumulative claims payments Pi,j development triangle; Right-hand sPi,J = Ii,J .

of the outstanding loss liabilities. Endowedwith the simulated pre-dictive distribution one is not only able to calculate estimators forthe first two moments but one can also calculate any other riskmeasure, like Value-at-Risk or expected shortfall.Posthuma et al. (2008) were probably the first who studied a

PIC model. Under the assumption of multivariate normality theyformed a claims development chain for the increments of claimspayments and incurred losses. Their model was then treated in thespirit of generalized linear models similar to Venter (2008). Ourmodel will be analyzed in the spirit of the Bayesian chain ladderlink ratio models (see Bhlmann et al., 2009).

1.2. Notation and model assumptions

For the PIC model we consider two channels of information: (i)claims payments, which refer to the payments done for reportedclaims; (ii) incurred losses, which correspond to the reported claimamounts. Often, the difference between incurred losses and claimspayments is called case reserves for reported claims. Ultimately,claims payments and incurred losses must reach the same value(when all the claims are settled).In many cases, statistical analysis of claims payments and

incurred losses data is done by accident years and developmentyears, which leads to the so-called claims development triangles(see Table 1, and Chapter 1 in Wthrich and Merz (2008)). Inthe following, we denote accident years by i {0, . . . , J}and development years by j {0, . . . , J}. We assume that allclaims are settled after the Jth development year. Cumulativeclaims payments in accident year i after j development periodsare denoted by Pi,j and the corresponding incurred losses byIi,j. Moreover, for the ultimate loss we assume Pi,J = Ii,J withprobability 1, which means that ultimately (at time J) they reachthe same value. For an illustration we refer to Table 1. cumulative payments Pi,j are given by the recursionPi,j = Pi,j1 exp

{i,j}, with initial value Pi,0 = exp

{i,0} ;matics and Economics 46 (2010) 568579 569

de: incurred losses Ii,j development triangle; both leading to the same ultimate loss

incurred losses Ii,j are given by the (backwards) recursionIi,j1 = Ii,j exp

{i,j1} , with initial value Ii,J = Pi,J . The components of 2 are independent and j, j > 0 for allj.

Remarks. This PICmodel combines both cumulative paymentsand incurred losses data to get a unified predictor for theultimate loss that is based on both sources of information.Thereby, the model assumption Ii,J = Pi,J guarantees that theultimate loss coincides for claims payments and incurred lossesdata. This means that in this PIC model there is no gap betweenthe two predictors based on cumulative payments and incurredlosses, respectively. This is similar to Section 4 in Posthumaet al.(2008) and to the CLRM (see Dahms, 2008), but this is differentto the MCL method (see Quarg and Mack, 2004). The cumulative payments Pi,j satisfy Hertigs (1985) model,conditional on the parameters2. The model assumption Ii,J =Pi,J also implies that we assume

E[Pi,J2] = E [ Ii,J 2] = exp{ J

m=0m + 2m/2

}, (1.1)

see also (2.2). Henceforth, incurred losses Ii,j satisfy Gogols(1993) model with prior ultimate loss mean E

[Pi,J |2

].

The assumption Ii,J = Pi,J means that all claims are settled afterJ development years and there is no so-called tail developmentfactor. If there is a claims development beyond developmentyear J , then one can extend the PIC model for the estimationof a tail development factor. Because this inclusion of a taildevelopment factor requires rather extended derivations anddiscussions we provide the details in Merz and Wthrich(submitted for publication). We assume conditional independence between all i,js andk,ls. One may question this assumption, especially, becauseOrganisation. In the next section we are going to give the modelproperties and first model interpretations conditional on the

570 M. Merz, M.V. Wthrich / Insurance: Mathe

knowledge of 2. In Section 3 we discuss the estimation of theunderlyingmodel parameters2. In Section 4wediscuss predictionuncertainty and in Section 5 we provide an example. All proofs ofthe statements are given in the Appendix.

2. Simultaneous payments and incurred losses consideration

2.1. Cumulative payments

Our first observation is that, given 2, cumulative paymentsPi,j satisfy the assumptions of Hertigs (1985) log-normal CLmodel (see also Section 5.1 in Wthrich and Merz, 2008). That is,conditional on2, we have for j 0log

Pi,jPi,j1

{BPj1,2

} N (j, 2j ) ,where we have set Pi,1 = 1. This gives the CL property (see alsoLemma 5.2 in Wthrich and Merz, 2008)

E[Pi,j|BPj1,2

] = Pi,j1 exp {j + 2j /2} . (2.1)The tower property for conditional expectations (see, for exam-ple Williams, 1991, 9.7 (i)) then implies for the expected ultimateloss, given {BPj ,2},

E[Pi,J |BPj ,2

] = Pi,j exp{ Jl=j+1

l + 2l /2}. (2.2)

2.2. Incurred losses

The model properties of incurred losses Ii,j are in the spirit ofGogols (1993) model. Namely, given2, the ultimate loss Ii,J = Pi,Jhas a log-normal distribution and, conditional on Ii,J and 2, theincurred losses Ii,j have also a log-normal distribution. This thenallows for Bayesian inference on Ii,J , given B Ij , similar to Lemma4.21 inWthrich andMerz (2008). The key lemma is the followingwell-known property for multivariate Gaussian distributions (seee.g. Appendix A in Posthuma et al., 2008):

Lemma 2.1. Assume (X1, . . . , Xn) is multivariate Gaussian dis-tributed with mean (m1, . . . ,mn) and positive definite covariancematrix . Then we have for the conditional distribution

X1|{X2,...,Xn} N(m1 +1,212,2

(X (2) m(2)) ,

1,1 1,212,22,1),

where X (2) = (X2, . . . , Xn) is multivariate Gaussian with meanm(2) = (m2, . . . ,mn) and positive definite covariance matrix 2,2,1,1 is the variance of X1 and 1,2 = 2,1 is the covariance vectorbetween X1 and X (2).

Lemma2.1 gives the following propositionwhose proof is providedin the Appendix.

Proposition 2.2. Under Model Assumption 1.1 we obtain for 0 j < j+ l Jlog Ii,j+l|{BIj ,2}

N(j+l +

v2j+lv2j

(log Ii,j j

), v2j+l(1 v2j+l/v2j )

),

where the parameters are given by (an empty sum is set equal to 0)

j =Jm=0

m J1n=j

n and v2j =Jm=0

2m +J1n=j

2n .Note that J =Jm=0m and v2J =Jm=0 2m.matics and Economics 46 (2010) 568579

Henceforth, we have the Markov property and we obtain thefollowing corollary:

Corollary 2.3. Under Model Assumption 1.1 we obtain for the ex-pected ultimate loss Ii,J , given {B Ij ,2},

E[Ii,J |B Ij ,2

] = I1ji,j exp{(1 j)

J1l=jl + j

(J + v2J /2

)}

= Ii,j exp{J1l=jl + 2l /2

}

exp{j

(j log Ii,j

J1l=j 2l /2

)},

with credibility weight

j = 1v2J

v2j= 1v2j

J1l=j 2l .

Remark. Compare the statement of Corollary 2.3 with formula(2.2). We see that under Model Assumption 1.1 cumulativepayments Pi,j fulfill the classical CL assumption (2.1) whereasincurred losses Ii,j do in general not satisfy the CL assumption,given2. This is different from theMCLmethodwhere one assumesthat both cumulative payments and incurred losses satisfy the CLassumption (seeQuarg andMack, 2004). At this stage onemayevenraise the question about interesting stochastic models such thatcumulative payments Pi,j and incurred losses Ii,j simultaneouslyfulfill the CL assumption. OurModel 1.1 does not fall into that class.In our model, the classical CL factor gets a correction term

exp

{j

(j log Ii,j

J1l=j 2l /2

)},

which adjusts the CL factor exp{J1

l=j l + 2l /2}to the actual

claims experience with credibility weight j. The smaller thedevelopment year j the bigger is the credibility weight j. Onthe other hand, we could also rewrite the right-hand side ofCorollary 2.3 as

E[Ii,J |B Ij ,2

] = exp{(1 j)(log Ii,j + J1l=jl

)+ j

Jm=0

m

} exp {jv2J /2} ,

the first factor on the right-hand side shows that we considera credibility weighted average between incurred losses log Ii,j +J1l=j l and cumulative payments J =

Jm=0m.

2.3. Cumulative payments and incurred losses

Finally, we would like to predict the ultimate loss Pi,J = Ii,Jwhen we jointly consider payments and incurred losses informa-tion Bj. We therefore apply the full PIC model, given the modelparameters2.

Theorem 2.4. Under Model Assumption 1.1 we obtain for theultimate loss Pi,J = Ii,J , given {Bj,2}, 0 j < J ,log Pi,J |{Bj,2} N

(J + (1 j)(log Pi,j j)+ j(log Ii,j j),

(1 j)(v2J w2j )),

where the parameters are given byj jj =m=0

m and w2j =m=0

2m,

M. Merz, M.V. Wthrich / Insurance: Mathe

and the credibility weight is given by

j =v2J w2jv2j w2j

> 0.

Remarks. The conditional distribution of log Pi,J , given {Bj,2},changes accordingly to the observations log Pi,j and log Ii,j.Thereby is the prior expectation J for the ultimate loss Pi,J =Ii,J updated by a credibility weighted average between the paidresidual log Pi,jj and the incurred residual log Ii,jj, wherethe credibility weight is given by

j =

Jm=j+1

2m

Jm=j+1

2m +J1n=j 2n

.

Analogously, the prior variance v2J w2j is reduced by thecredibility weight 1j, this is typical in credibility theory, seefor example Theorem 4.3 in Bhlmann and Gisler (2005). Theorem 2.4 shows that in our log-normal PIC model wecan calculate analytically the posterior distribution of theultimate loss, given Bj and conditional on 2. Henceforth,we can calculate the conditionally expected ultimate loss, seeCorollary 2.5.

Corollary 2.5 (PIC Ultimate Loss Prediction). Under Model Assump-tion 1.1 we obtain for the expected ultimate loss Ii,J = Pi,J , given{Bj,2},

E[Pi,J |Bj,2

] = Pi,j exp{ Jl=j+1

l + 2l

2

}

exp{j

(logIi,jPi,j (j j)

Jl=j+1

2l

2

)}

= Ii,j exp{J1l=jl

}

exp{(1 j)

(logPi,jIi,j (j j)+

Jl=j+1

2l

2

)}

= exp{(1 j)

(log Pi,j +

Jl=j+1

l

)+ j

(log Ii,j +

J1l=jl

)} exp {(1 j)(v2J w2j )/2} .

Remark. Henceforth, if we consider simultaneously claims pay-ments and incurred losses information, we obtain a correctionterm

exp

{j

(logIi,jPi,j (j j)

Jl=j+1

2l

2

)}(2.3)

to the classical CL predictor E[Pi,J |BPj ,2

]. This adjustment factor

compares incurredpaid ratios and corresponds to the observedresiduals log Ii,jPi,j (j j). For example, a large incurredpaidratio Ii,j/Pi,j gives a large correction term (2.3) to the classical CLpredictor E

[Pi,J |BPj ,2

]. This is a similar mechanism as in the MCL

method that also adjusts the predictors according to incurredpaidratios (see Quarg and Mack, 2004). The last formula in thestatement of Corollary 2.5 shows that we can also understand

the PIC ultimate loss predictor as a credibility weighted averagebetween claims payments and incurred losses information.matics and Economics 46 (2010) 568579 571

3. Parameter estimation

So far, all consideration were done for known parameters 2.However, in general, they are not known and need to be estimatedfrom the observations. Assume that we are at time J and that wehave observations (see also Table 1)DPJ =

{Pi,j : i+ j J

}, D IJ =

{Ii,j : i+ j J

}and

DJ = DPJ D IJ .We estimate the parameters in a Bayesian framework. Thereforewe define the following model:

Model Assumption 3.1 (Bayesian PIC Model). Assume Model As-sumption 1.1 hold true with deterministic 0, . . . , J and 0, . . . ,J1 and

m N(m, s2m

)for {0, . . . , J},

n N(n, t2n

)for n {0, . . . , J 1}.

In a full Bayesian approach one chooses an appropriate prior distri-bution for the whole parameter vector2. We will only use a priordistribution for m and n and assume that m and n are known.This has the advantage that we can analytically calculate the pos-terior distributions that will allow for explicit model calculationsand interpretations.

3.1. Cumulative payments

For claims payments we only need the parameters 8 =(0, . . . ,J). The posterior density of8, givenDPJ , is given by (weset Pi,1 = 1)

u(8|DPJ

) Jj=0

Jji=0exp

{ 12 2j

(j log Pi,jPi,j1

)2}

Jj=0exp

{ 12s2j

(j j

)2}.

This immediately provides the next theorem:

Theorem 3.2. Under Model Assumption 3.1 the posterior distribu-tion of 8, givenDPJ , has independent components with

j|{DPJ } NP,postj = Pj 1](j)

Jji=0log

Pi,jPi,j1

+ (1 Pj )j,

(sP,postj )2 =

(1s2j+ ](j) 2j

)1 ,with ](j) = J j+ 1 and credibility weight Pj =

](j)](j)+ 2j /s2j

.

Henceforth, the posterior mean is a credibility weighted averagebetween the prior mean j and the empirical mean

j =1](j)

Jji=0log

Pi,jPi,j1

,

see also formula (5.2) in Wthrich and Merz (2008). The posteriordistribution of Pi,J , given DPJ , is now completely determined.Moments can be calculated in closed form and Monte Carlosimulation provides the empirical posterior distribution of the

ultimate losses vector (P1,J , . . . , PJ,J)|{DPJ } = (I1,J , . . . , IJ,J)|{DPJ }.In view of (2.2) and Theorem 3.2 the ultimate loss predictor, given

572 M. Merz, M.V. Wthrich / Insurance: Mathe

DPJ , is given by

E[Pi,J |DPJ

]= Pi,Ji

Jl=Ji+1

exp{P,postl + 2l /2+ (sP,postl )2/2

}. (3.1)

3.2. Incurred losses

In this subsection we concentrate on the parameter estimationgiven the data D IJ of incurred losses. We define the underlyingparameter for Ii,J by (see also (1.1))

J = J = Jj=0

j N(J =

Jj=0

j, t2J =Jj=0s2j

),

which is independent from 0, . . . ,J1. For incurred losses wethen only need the parameters 9 = (0, . . . ,J). The posteriordensity of9, givenD IJ , is given by

u(9|D IJ

) Ji=0exp

12v2Ji(

Jn=Ji

n + log Ii,Ji)2

J1j=0

Jj1i=0exp

{ 12 2j

(j + log Ii,jIi,j+1

)2}

Jj=0exp

{ 12t2j

(j j

)2}. (3.2)

This immediately provides the next theorem:

Theorem 3.3. Under Model Assumption 3.1 the posterior distribu-tion of 9, givenD IJ , is a multivariate Gaussian distribution with pos-terior mean post(D IJ ) and posterior covariance matrix (D

IJ ). The

inverse covariance matrix(D IJ )1 = (aIn,m)0n,mJ is given by

aIn,m =(t2n + (J n)2n

)1{n=m} +

nmi=0

v2i for 0 n,m J.

The posterior mean post(D IJ ) = ( I,post0 , . . . , I,postJ ) is obtainedby

post(D IJ ) = (D IJ )(bI0, . . . , bIJ),with vector (bI0, . . . , b

IJ) given by

bIj = t2j j 2jJj1i=0log

Ii,jIi,j+1

ji=0

v2i log IJi,i.

Observe that bIj can be rewritten so that it involves a credibilityweighted average between prior mean j and the incurred lossesobservations, namelybIj =

(t2j + (](j) 1)2j

) [ Ij j + (1 Ij )j

]

ji=0

v2i log IJi,i, (3.3)

with credibility weight

Ij =](j) 1

](j) 1+ 2j /t2j,

and empirical mean

1 Jj1 Ii,j

j =

](j) 1 i=0logIi,j+1

.matics and Economics 46 (2010) 568579

The posterior distribution of Pi,J = Ii,J , givenD IJ , is now completelydetermined. We obtain for the ultimate loss predictor, givenD IJ ,

E[Pi,J |D IJ

] = I1Jii,Ji exp{(1 Ji)

J1l=Ji

I,postl

+ Ji( I,postJ +

v2J

2

)+(sI,posti

)2/2

}, (3.4)

where(sI,posti

)2 = (eIi)(D IJ )eIi ,with eIi = (0, . . . , 0, 1 Ji, . . . , 1 Ji,Ji) RJ+1.3.3. Cumulative payments and incurred losses

Similar to the last section we determine the posterior distribu-tion of2, givenDJ . Observe that

log Ii,j|{BPj ,2} N(j j, v2j w2j

)= N

(J

m=j+1m

J1n=j

n,

Jm=j+1

2m +J1n=j

2n

).

This implies that the joint likelihood function of the dataDJ is givenby (we set Pi,1 = 1)

lDJ (2) =Jj=0

Jji=0

12pijPi,j

exp

{ 12 2j

(j log Pi,jPi,j1

)2}

Ji=1

12pi(v2Ji w2Ji)Ii,Ji

exp{ 12(v2Ji w2Ji)

(Ji Ji log Ii,JiPi,Ji

)2}

J1j=0

Jj1i=0

12pijIi,j

exp

{ 12 2j

(j + log Ii,jIi,j+1

)2}. (3.5)

Under Model Assumption 3.1 the posterior distribution u(2|DJ

)of2, givenDJ , is given by

u(2|DJ

) lDJ (2) Jm=0exp

{ 12s2m

(m m)2}

J1n=0exp

{ 12t2n

(n n)2}. (3.6)

This immediately implies the following theorem:

Theorem 3.4. Under Model Assumption 3.1, the posterior distribu-tion u

(2|DJ

)is a multivariate Gaussian distribution with posterior

mean post(DJ) and posterior covariance matrix (DJ). The inversecovariance matrix(DJ)1 = (an,m)0n,m2J is given by

an,m =(s2n + (J n+ 1)2n

)1{n=m} +

(n1)(m1)i=0

(v2i w2i

)1for 0 n,m J,

aJ+1+n,J+1+m =(t2n + (J n)2n

)1{n=m} +

nmi=0

(v2i w2i

)1for 0 n,m J 1,

an,J+1+m = aJ+1+m,n = (n1)m (

v2i w2i)1i=0for 0 n J, 0 m J 1.

M. Merz, M.V. Wthrich / Insurance: Mathe

The posterior mean post(DJ)=(post0 , . . . ,

postJ ,

post0 , . . . ,

postJ1)

is obtained by

post(DJ) = (DJ)(c0, . . . , cJ , b0, . . . , bJ1),with vector (c0, . . . , cJ , b0, . . . , bJ1) given by

cj = s2j j + 2jJji=0log

Pi,jPi,j1

+J

i=Jj+1

(v2Ji w2Ji

)1logIi,JiPi,Ji

,

bj = t2j j 2jJj1i=0log

Ii,jIi,j+1

Ji=Jj

(v2Ji w2Ji

)1logIi,JiPi,Ji

.

Henceforth, this implies for the expected ultimate loss in theBayesian PIC model, givenDJ , (see also Corollary 2.5)

E[Pi,J |DJ

]= P1Jii,Ji IJii,Ji exp

{(1 Ji)

Jl=Ji+1

postl + Ji

J1l=Ji

postl

}

exp{(1 Ji)

v2J w2Ji2

+ (sposti )2 /2}, (3.7)

where(sposti

)2 = (ei)(DJ)ei,with ei = (0, . . . , 0, 1 Ji, . . . , 1 Ji, 0, . . . , 0, Ji, . . . ,Ji) R2J+1.

4. Prediction uncertainty

The ultimate loss Pi,J = Ii,J is now predicted by its conditionalexpectations

E[Pi,J |DPJ

], E

[Pi,J |D IJ

]or E

[Pi,J |DJ

],

depending on the available information DPJ , DIJ or DJ (see (3.1),

(3.4) and (3.7)). With Theorems 3.23.4 all posterior distributionsin the Bayesian PIC Model 3.1 are given analytically. Thereforeany risk measure for the prediction uncertainty can be calculatedwith a simpleMonte Carlo simulation approach. Here, we considerthe conditional mean square error of prediction (MSEP) as riskmeasure. The conditional MSEP is probably the most popularrisk measure in claims reserving practice and has the advantagethat it is analytically tractable in our context. We derive it forthe posterior distribution, given DJ . The cases DPJ and D

IJ are

completely analogous. The conditional MSEP is given by

msep Ji=1Pi,J |DJ

(E

[Ji=1Pi,J

DJ])

= E( J

i=1Pi,J E

[Ji=1Pi,J

DJ])2DJ

= Var

(Ji=1Pi,J

DJ),see Wthrich and Merz (2008), Section 3.1. For the conditionalMSEP, given the observationsDJ , we obtain:matics and Economics 46 (2010) 568579 573

Theorem 4.1. Under Model Assumption 3.1we have, using informa-tionDJ ,

msep Ji=1Pi,J |DJ

(E

[Ji=1Pi,J

DJ])

=1i,kJ

(e(1Ji)(v

2J w2Ji)1{i=k}+ei(DJ ) ek 1

) E [Pi,J |DJ] E [Pk,J |DJ]

with E[Pi,J |DJ

]given by (3.7).

Similarly we obtain for the conditional MSEP w.r.t. D IJ and DPJ ,

respectively:

Theorem 4.2. Under Model Assumption 3.1we have, using informa-tionD IJ ,

msep Ji=1Pi,J |D IJ

(E

[Ji=1Pi,J

D IJ])

=1i,kJ

(eJiv

2J 1{i=k}+(eIi )(D IJ )eIk 1

) E [Pi,J |D IJ ] E [Pk,J |D IJ ]

with E[Pi,J |D IJ

]given by (3.4). Using informationDPJ we obtain

msep Ji=1Pi,J |DPJ

(E

[Ji=1Pi,J

DPJ])

=1i,kJ

(e(v

2J w2Ji)1{i=k}+(ePi )(DPJ )ePk 1

) E [Pi,J |DPJ ] E [Pk,J |DPJ ]

with E[Pi,J |DPJ

]given by (3.1), ePi = (0, . . . , 0, 1, . . . , 1) RJ+1

and(DPJ ) = diag((sP,postj )2).

5. Example

We revisit the first example given in Dahms (2008) and Dahmset al. (2009) (see Tables 10 and 11). We do a first analysis of thedata under Model Assumption 3.1 where we assume that j and jare deterministic parameters (using plug-in estimates). In a secondanalysis we alsomodel these parameters in a Bayesian framework.

5.1. Data analysis in the Bayesian PIC Model 3.1

Because we do not have prior parameter information andbecause we would like to compare our results to Dahms (2008)results, we choose non-informative priors for m and n. Thismeans that we let s2m and t2n . This then impliesfor the credibility weights Pm = Im = 1 which means that ourclaims payments prediction is based onDPJ only (see Theorem 3.2)and our incurred losses prediction is based onD IJ only (see (3.3)),i.e. no prior knowledge m and n is used. Similarly for the jointPIC prediction no prior knowledge is needed for non-informativepriors, because then the prior values m and n disappear in cmand bn for s2m and t2n , see Theorem 3.4.Henceforth, there remains to provide the values for j and j.

For the choice of these parameters we choose in this subsectionan empirical Bayesian point of view and use the standard plug-in estimators (sample standard deviation estimator, see e.g. (5.3)

in Wthrich and Merz, 2008). Since the last variance parameterscannot be estimated from the data (due to the lack of observations)

43was also already previously observed for other data sets.In Table 5 we compare the corresponding prediction uncertain-

ties measured by the square root of the conditional MSEP. UnderModel Assumption 3.1 these are calculated analytically with Theo-rems 3.23.4, 4.1 and 4.2. First of all, we observe that in our modelthe PIC predictor R(DJ) has a smaller prediction uncertainty com-pared to R(DPJ ) and R(D

IJ ). This is clear because increasing the set

tion approach to the full predictive distribution of the outstandingloss liabilities in the Bayesian PIC Model 3.1. This is done as fol-lows: Firstly, we generatemultivariate Gaussian samples2(t)withmean post(DJ) and covariance matrix (DJ) according to The-orem 3.4. Secondly, we generate samples (I(t)1,J , . . . , I

(t)J,J )|{2(t)} ac-

cording to Theorem2.4. In Table 6weprovide the empirical densityfor the outstanding loss liabilities from 20,000 simulations. More-We observe that in all accident years the PIC reserves R(DJ)based on the whole information DJ are between the estimatesbased on DPJ and D

IJ . The deviations from R(D

IJ ) are comparably

small which comes from the fact that j j.In Table 4 we provide the claims reserves estimates for other

popular methods. We observe that the reserves R(DPJ ) are closeto the ones from CL paid (the differences can partly be explainedby the variance terms 2l /2 in the expected value of log-normaldistributions, see (2.2)). The reserves R(D IJ ) are very close to theones from CL incurred. The PIC reserves R(DJ) from the combinedinformation look similar to the ones from the CLRM and to MCLincurred method. We also mention that for our data set the MCLdoes not really reduce the gap between the two predictions. This

CLRM paid 10,728,771 467,814CLRM incurred 10,728,771 471,873

MCL paid 10,314,181 Not availableMCL incurred 10,761,918 Not available

Furthermore, our prediction uncertainties are comparable tothe ones from the othermodels.Wewould also like tomention thatin the CLRM there are two values for the prediction uncertaintydue to the fact that one can use different parameter estimators inthe CLRM, see Corollary 4.4 in Dahms (2008) (the claims reservescoincide).As mentioned in Section 4 we cannot only calculate the con-

ditional MSEP, but Theorem 3.4 allows for a Monte Carlo simula-6 1,138,623 1,868,664 1,786,947 919,102 1,894,8617 1,638,793 1,997,651 1,942,518 1,498,163 2,020,3108 2,359,939 1,418,779 1,569,657 3,181,319 1,320,4929 1,979,401 2,556,612 2,590,718 1,602,089 2,703,242

Total 10,165,612 10,665,287 10,728,771 10,314,181 10,761,918

we use the extrapolation used in Mack (1993). This gives theparameter choices provided in Table 2.The expected outstanding loss liabilities then provide the PIC

claims reserves:

R(DJ) = E[Pi,J |DJ

] Pi,Ji,if the ultimate loss prediction is based on thewhole informationDJ(and similarly for DPJ and D

IJ , respectively). This gives the claims

reserves provided in Table 3.

Table 5Total claims reserves and prediction uncertainty.

Claims reserves msep1/2

R(DPJ ) 10,511,390 1,559,228R(D IJ ) 10,695,996 421,298R(DJ ) 10,626,108 389,496

CL paid 10,165,612 1,517,480CL incurred 10,665,287 455,794574 M. Merz, M.V. Wthrich / Insurance: Mathe

Table 2Standard deviation parameter choices for j and j .

j 0 1 2 3 4

j 0.1393 0.0650 0.0731 0.0640 0.026j 0.0633 0.0459 0.0415 0.0122 0.008

Table 3Claims reserves in the Bayesian PIC Model 3.1.

R(DPJ ) = E[Pi,J |DPJ

] Pi,Ji1 115,4702 428,2723 642,6644 729,3445 1,284,5456 1,183,7817 1,692,6328 2,407,4389 2,027,245

Total 10,511,390

Table 4Claims reserves from the CL method for claims payments and incurred losses (see Mpayments and incurred losses (see Quarg and Mack, 2004).

CL paidMack (1993)

CL incurredMack (1993)

1 114,086 337,9842 394,121 31,8843 608,749 331,4364 697,742 1,018,3505 1,234,157 1,103,928of information reduces the uncertainty. Therefore, the PIC predic-tor R(DJ) should be preferred within Model Assumption 3.1.matics and Economics 46 (2010) 568579

5 6 7 8 9

0.0271 0.0405 0.0227 0.0494 0.02270.0017 0.0019 0.0011 0.0006

R(D IJ ) = E[Pi,J |D IJ

] Pi,Ji R(DJ ) = E [Pi,J |DJ ] Pi,Ji337,994 337,79931,526 31,686331,526 331,8901,018,924 1,018,3081,102,580 1,104,8161,869,284 1,842,6691,990,260 1,953,7671,465,661 1,602,2292,548,242 2,402,946

10,695,996 10,626,108

ack, 1993), from the CLRM (see Dahms, 2008), and from the MCL method for claims

CLRMDahms (2008)

MCL paidQuarg and Mack (2004)

MCL incurredQuarg and Mack (2004)

314,902 104,606 338,20066,994 457,484 30,850359,384 664,871 330,205981,883 615,436 1,021,3611,115,768 1,271,110 1,102,396over, we compare it to the Gaussian density with the same meanand standard deviation (see also Table 5, line R(DJ)). We observe

that these densities are very similar, the Gaussian density havingslightly more probability mass in the lower tail and slightly lessprobability mass in the upper tail (less heavy tailed). To further in-vestigate this issueweprovide theQQ-Plot in Table 7.Weonly ob-serve differences in the tails (as described above). The lower panelin Table 7 gives the upper tail for values above the 90% quantile.There we see that a Gaussian approximation underestimates thetrue risks. However, the differences are comparably small.In Table 3 we have observed that the resulting PIC reserves

R(DJ) are close to the claims reserves R(D IJ ) from incurred

We now calculate a second example where we double thesestandard deviation parameters, i.e. j = 2j. The results arepresented in Table 8. Firstly, we observe that the conditional MSEPusing information D IJ and DJ strongly increases. This is clear,because doubling the standard deviation parameters increasesthe uncertainty. More interestingly, we observe that the PICreserves for each accident year i = 2, . . . , J are now closerto the claims reserves from cumulative payments (especially foryoung accident years). The reason for this is that increasing thej parameters means that we give less credibility weight to the

I, , , , , , , , , , , , , ,

Table 7QQ-plot from 20,000 simulations of the outstanding loss liabilities with the Gaussian distribution. Upper panel: full QQ-Plot; Lower panel: QQ-Plot of the upper 90%quantile.M. Merz, M.V. Wthrich / Insurance: Mathe

Table 6Empirical density from 20,000 simulations and a comparison to the Gaussian density.losses information only. The reason therefore is that the standarddeviation parameters j for incurred losses are rather small.matics and Economics 46 (2010) 568579 575incurred losses observationsDJ andmore credibility weight to theclaims payments observationsDPJ .

fn=0

Jm=0

j1j exp

{cjj} J1n=0

j1j exp

{cjj} , (5.1)with lDJ (2) given in (3.5). This distribution can no longer behandled analytically because the normalizing constant takes anon-trivial form. But because we can write down its likelihoodfunction up to the normalizing constant, we can still apply theMarkov chain Monte Carlo (MCMC) simulation methodology.MCMC methods are very popular for this kind of problems. Foran introduction and overview to MCMC methods we refer to Gilkset al. (1996), Asmussen and Glynn (2007) and Scollnik (2001).Because MCMC methods are widely used we refrain from describ-ing them in detail. We will use the MetropolisHastings algorithmas described in Section 4.4 inWthrich andMerz (2008). The aim isto construct a Markov chain (2t)t0 whose stationary distribution)

signing appropriate credibility weights to these different channelsof information. The benefits of our method are that

it combines two different channels of information to get aunified ultimate loss prediction; for claims payments observation the CL structure is preservedusing credibility weighted correction terms to the CL factors; for deterministic standard deviation parameters we can calcu-late both the claims reserves and the conditional MSEP analyt-ically; the full predictive distribution of the outstanding loss liabilitiesis obtained from Monte Carlo simulations, this allows one toconsider any risk measure; for stochastic standard deviation parameters all key figures andthe full predictive distribution of the outstanding loss liabilitiesare obtained from the MCMC method. a model extension will allow the inclusion of tail developmentJ1)is given by

u(2|DJ

) lDJ (2) Jm=0exp

{ 12s2m

(m m)2}

J1exp

{ 12t2n

(n n)2}

of variations.

6. Conclusions

We have defined a stochastic PIC model that simultaneouslyconsiders claims payments information and incurred losses infor-mation for the prediction of the outstanding loss liabilities by as-576 M. Merz, M.V. Wthrich / Insurance: Mathe

Table 8Total claims reserves and prediction uncertainty for j and j = 2j .

res. R(DPJ ) res. R(DIJ ) PIC res. R(D

j j j

1 115,470 337,994 337,7992 428,272 31,526 31,6863 642,664 331,526 331,8904 729,344 1,018,924 1,018,3085 1,284,545 1,102,580 1,104,8166 1,183,781 1,869,284 1,842,6697 1,692,632 1,990,260 1,953,7678 2,407,438 1,465,661 1,602,2299 2,027,245 2,548,242 2,402,946

Total 10,511,390 10,695,996 10,626,108

msep1/2 1,559,228 421,298 389,496

Table 9Total claims reserves and prediction uncertainty in the Full Bayesian PIC model for dif

Vco(j) = 1/2j = Vco(j) = 1/2j = 0%Claims reserves R(DJ ) 10,626,108Prediction uncertainty msep1/2 389,496

Finally, in Table 12 we provide the posterior correlation matrixof 2 = (0, . . . ,9,0, . . . ,8), given DJ , which can directlybe calculated from the posterior covariance matrix (DJ) (seeTheorem 3.4). We observe only small posterior correlations whichmay justify the assumption of prior uncorrelatedness.

5.2. Full Bayesian PIC model

A full Bayesian approach suggests that one alsomodels the stan-dard deviation parameters j and j stochastically. We thereforemodify Model Assumption 3.1 as follows: Assume that j and jhave independent gamma distributions with

j (j , cj

)and j

(j , cj

).

Of course, the choice of gamma distributions for the standarddeviation parameters is rather arbitrary and any other positivedistribution would also fit. Then the posterior distribution of theparameters 2 = (0, . . . ,J ,0, . . . ,J1, 0, . . . , J , 0, . . . ,is u (2|DJ . Then, we run this Markov chain for sufficiently long,so that we obtain approximate samples 2t s from that stationarymatics and Economics 46 (2010) 568579

J ) res. R(DPJ ) res. R(DIJ ) PIC res. R(DJ )

j = 2j j = 2j j = 2j115,470 338,025 337,246428,272 31,574 32,212642,664 331,580 333,028729,344 1,019,091 1,016,6371,284,545 1,101,948 1,110,5851,183,781 1,869,904 1,774,0591,692,632 1,981,419 1,882,3412,407,438 1,581,122 1,903,1552,027,245 2,549,115 2,242,048

10,511,390 10,803,778 10,631,310

1,559,228 741,829 614,453

erent coefficients of variations for j and j .

10% 100%

10,589,180 10,701,455392,832 472,449

distribution. This is achieved by defining an acceptance probability

(2t ,2

) = min{1, u (2|DJ) q (2t |2)u(2t |DJ

)q (2|2t)

},

for the next step in the Markov chain, i.e. the move from 2t to2t+1. Thereby, the proposal distribution q (|) is chosen in such away that we obtain an average acceptance rate of roughly 24% be-cause this satisfies certain optimal mixing properties for Markovchains (see Roberts et al., 1997, Corollary 1.2).We apply this algorithm to different coefficients of variations

1/2j and

1/2j of j and j, respectively. Moreover, we keep the

means E[j] = j/cj and E[j] = j/cj fixed and choose themequal to the deterministic values provided in Table 2. The resultsare provided in Table 9. We observe, as expected, an increaseof the prediction uncertainty. The increase from Model 3.1 withdeterministic js and js to a coefficient of variation of 10% ismoderate, but it starts to increase strongly for larger coefficientsfactors, for details see Merz and Wthrich (submitted for pub-lication).

M. Merz, M.V. Wthrich / Insurance: Mathe

Appendix. Proofs

In this appendix we prove all the statements. The proof ofTheorem 3.2 easily follows from its likelihood function (and it isa special version of Theorem 6.4 in Gisler and Wthrich, 2008),therefore we omit its proof.Proof of Proposition 2.2. Note that we only consider conditionaldistributions, given the parameter 2, and for this conditionaldistributions claims in different accident years are independent.Thereforewe can restrict ourselves to one fixed accident year i. Thevector(log Ii,j+l, log Ii,j, log Ii,j1, . . . , log Ii,0

){2}has amultivariate Gaussian distributionwithmean (j+l, j, j1,. . . , 0) and covariance matrix with elements given by: forn m {j+ l, j, j 1, . . . , 0}Cov

(log Ii,n, log Ii,m|2

) = v2n .Henceforth, we can apply Lemma 2.1 to the random variablelog Ii,j+l|{BIj ,2} with parameters m1 = j+l, m(2) = (j, . . . , 0),1,1 = v2j+l, 2,2 is the covariance matrix of X (2) =

(log Ii,j,

. . . , log Ii,0) |{2} and

1,2 =(v2j+l, . . . , v

2j+l) Rj+1.

We obtain from Lemma 2.1 a Gaussian distribution and thereremains the calculation of the explicit parameters of the Gaussiandistribution. Note that the covariancematrix2,2 has the followingform

2,2 =(v2(j+1n)(j+1m)

)1n,mj+1 ,

where (j + 1 n) (j + 1 m) = max{j + 1 n, j + 1 m}.Henceforth, 2,2 has a fairly simple structure which gives a niceform for its inverse

12,2 =(bn,m

)1n,mj+1 ,

with diagonal elements

b1,1 =v2j1

v2j (v2j1 v2j )

,

bn,n =v2jn v2j+2n

(v2j+1n v2j+2n)(v2jn v2j+1n)for n {2, . . . , j},

bj+1,j+1 = 1v20 v21

,

and off-diagonal elements 0 except for the side diagonals

bn,n+1 = 1v2jn v2j+1n

for n {1, . . . , j},

and its symmetric counterpart bn,n1 for n {2, . . . , j + 1}. Thismatrix has the following property

1,212,2 =

(v2j+l/v

2j , 0, . . . , 0

) Rj+1,from which the claim follows. Proof of Corollary 2.3. Proposition 2.2 implies for the conditionalexpectation

E[Ii,J |B Ij ,2

] = exp{J + v2Jv2j

(log Ii,j j

)+ v2J2

(1 v

2J

v2j

)}= exp {J + (1 j) (log Ii,j j)+ jv2J /2}{

J1 } { ( ) }= Ii,j expl=jl exp j log Ii,j j + jv2J /2 .matics and Economics 46 (2010) 568579 577

Finally, observe that

jv2J

J1l=j 2l =

J1l=j 2l

(v2J

v2j 1

)= j

J1l=j 2l .

This completes the proof. Proof of Theorem 2.4. The proof is similar to the one of Proposi-tion 2.2 and uses Lemma 2.1. Again we only consider conditionaldistributions, given the parameter2, thereforewe can restrict our-selves to one fixed accident year i. Using the Markov property ofcumulative payments, we see that it suffices to consider the vector(log Ii,J , log Pi,j, log Ii,j, . . . , log Ii,0

){2} ,which has a multivariate Gaussian distribution with mean(J , j, j, . . . , 0) and covariance matrix similar to Proposi-tion 2.2 but with an additional column and row for

Cov(log Pi,j, log Ii,l|2

) = w2j for l {J, j, . . . , 0}.Henceforth, we can apply Lemma 2.1 to the random variablelog Ii,J |{Bj,2} with parameters m1 = J , m(2) = (j, j, . . . , 0),1,1 = v2J , 2,2 is the covariance matrix of X (2) =

(log Pi,j,

log Ii,j, . . . , log Ii,0){2} and

1,2 =(w2j , v

2J , . . . , v

2J

) Rj+2.Thus, there remains the calculation of the explicit parameters ofthe Gaussian distribution. Note that the covariancematrix2,2 hasnow the following form

w2j for elements in the first column or first row,

v2(j+2n)(j+2m) for elements in the remaining right lower square2 n,m j+ 2.Therefore, 2,2 has again a simple form whose inverse can easilybe calculated and has a similar structure to the one given inProposition 2.2.

1,212,2 =

(v2j v2Jv2j w2j

,v2J w2jv2j w2j

, 0, . . . , 0

)= (1 j, j, 0, . . . , 0) Rj+2.

This implies (note that v2j > v2J > w

2j )

m1 +1,212,2(X (2) m(2)) = J + (1 j) (log Pi,j j)

+j(log Ii,j j

).

Moreover, we have

1,1 1,212,22,1 = v2J (1 j)w2j jv2J= (1 j)(v2J w2j ),

from which the claim follows. Proof of Corollary 2.5. Theorem 2.4 implies

E[Ii,J |Bj,2

]= exp {J + (1 j)(log Pi,j j)+ j(log Ii,j j)+ (1 j)(v2J w2j )/2

}= Pi,j exp

{J

l=j+1l + 2l /2

}exp

{j(j log Pi,j

+ log Ii,j j (v2J w2j )/2)}. Proof of Theorem 3.3. From the likelihood (3.2) it immediatelyfollows that the posterior distribution 9, given D IJ , is Gaussian.

421929i=Jj+1Ji Ji Pi,Ji

Jm=0

maj,m J1m=0

maj,J+1+m!= 0,

and

log u (2|DJ)

j=[t2j j

Jj1i=0

2j logIi,jIi,j+1

J

i=Jj

(v2Ji w2Ji

)1logIi,JiPi,Ji

]J J1

i=1=

1i,kJ

E[E[Pi,J |DJ ,2

]E[Pk,J |DJ ,2

] |DJ]

1i,kJ

E[Pi,J |DJ

]E[Pk,J |DJ

].

Henceforth, this provides

Var

(Ji=1Pi,J

DJ)=

Ji=1e(1Ji)(v

2J w2Ji)E

[E[Pi,J |DJ ,2

]2 |DJ]+ 2

1i

320197Gilks, W.R., Richardson, S., Spiegelhalter, D.J., 1996. Markov Chain Monte Carlo inPractice. Chapman & Hall.

Gisler, A., Wthrich, M.V., 2008. Credibility for the chain ladder reserving method.Astin Bulletin 38 (2), 565600.

Gogol, D., 1993. Using expected loss ratios in reserving. Insurance:Mathematics andEconomics 12 (3), 297299.

Halliwell, L.J., 1997. Cojoint prediction of paid and incurred losses. CAS ForumSummer. pp. 241379.

Halliwell, L.J., 2009. Modeling paid and incurred losses together. CAS E-ForumSpring.

van Dyk, D.A., Meng, X.-L., 2001. The art of data augmentation. Journal ofComputational and Graphical Statistics 10 (1), 150.

Verdier, B., Klinger, A., 2005. JAB chain: a model-based calculation of paid andincurred loss development factors. In: Conference Paper, 36th Astin Colloquium2005. Zrich, Switzerland.

Williams, D., 1991. Probability with Martingales. In: Cambridge MathematicalTextbooks.

Wthrich, M.V., Merz, M., 2008. Stochastic Claims Reserving Methods in Insurance.Wiley.6 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 100 0 07 0 0 0 0 0 0 0 0 2 1 0 0 0 0 0 0 0 100 08 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 0 100

Using Theorem 3.4 provides the claim:

E[E[Pi,J |DJ ,2

]E[Pk,J |DJ ,2

] |DJ]= E [Pi,J |DJ] E [Pk,J |DJ] exp {ei(DJ)ek} .

The proof of Theorem 4.2 is similar to the proof of Theorem 4.1.

References

Asmussen, S., Glynn, P.W., 2007. Stochastic Simulation. Springer.Bhlmann, H., De Felice, M., Gisler, A., Moriconi, F., Wthrich, M.V., 2009. Recursivecredibility formula for chain ladder factors and the claims development result.Astin Bulletin 39 (1), 275306.

Bhlmann, H., Gisler, A., 2005. A Course in Credibility Theory and its Applications.In: Springer Universitext.

Dahms, R., 2008. A loss reserving method for incomplete claim data. Bulletin SwissAssociation of Actuaries 127148.

Dahms, R., Merz,M.,Wthrich,M.V., 2009. Claims development result for combinedclaims incurred and claims paid data. Bulletin Francais dActuariat 9 (18), 539.

Hertig, J., 1985. A statistical approach to the IBNR-reserves in marine insurance.Astin Bulletin 15 (2), 171183.

Liu, H., Verrall, R.J., 2008. Bootstrap estimation of the predictive distributions ofreserves using paid and incurred claims. In: Conference Paper, 38th AstinColloquium 2008. Manchester, UK.

Mack, T., 1993. Distribution-free calculation of the standard error of chain ladderreserve estimates. Astin Bulletin 23 (2), 213225.

Merz, M., Wthrich, M.V., 2006. A credibility approach to the Munich chain-laddermethod. Bltter DGVFM Band XXVII, 619628.

Merz, M.,Wthrich, M.V., 2010. Estimation of tail factors in the paidincurred chainreserving method, Preprint (submitted for publication).

Posthuma, B., Cator, E.A., Veerkamp, W., Zwet, van E.W., 2008. Combined analysisof paid and incurred losses. CAS E-Forum Fall. pp. 272293.

Quarg, G., Mack, T., 2004.Munich chain ladder. Bltter DGVFMBand XXVI, 597630.Roberts, G.O., Gelman, A., Gilks, W.R., 1997. Weak convergence and optimalscaling of random walks Metropolis algorithm. Annals of Applied Probability7, 110120.

Scollnik, D.P.M., 2001. Actuarial modeling with MCMC and BUGS. North AmericanActuarial Journal 5 (2), 96125.

Venter, G.G., 2008. Distribution and value of reserves using paid and incurredtriangles. CAS E-Forum Fall. pp. 348375.M. Merz, M.V. Wthrich / Insurance: Mathe

Table 11Observed incurred losses Ii,j .

i/j 0 1 2 3 4

0 3,362,115 5,217,243 4,754,900 4,381,677 4,136,881 2,640,443 4,643,860 3,869,954 3,248,558 3,102,002 2,879,697 4,785,531 4,045,448 3,467,822 3,377,543 2,933,345 5,299,146 4,451,963 3,700,809 3,553,394 2,768,181 4,658,933 3,936,455 3,512,735 3,385,125 3,228,439 5,271,304 4,484,946 3,798,384 3,702,426 2,927,033 5,067,768 4,066,526 3,704,1137 3,083,429 4,790,944 4,408,0978 2,761,163 4,132,7579 3,045,376

Table 12Posterior correlation matrix corresponding to(DJ ).

0 (%) 1 (%) 2 (%) 3 (%) 4 (%) 5 (%) 6 (%) 7 (%) 8 (%)

0 100 0 0 0 0 0 0 0 01 0 100 2 1 1 0 1 0 12 0 2 100 4 2 1 2 1 23 0 1 4 100 3 3 4 2 44 0 1 2 3 100 2 3 2 45 0 0 1 3 2 100 6 3 76 0 1 2 4 3 6 100 8 187 0 0 1 2 2 3 8 100 188 0 1 2 4 4 7 18 18 1009 0 0 1 1 1 2 6 6 340 0 1 2 1 0 0 1 0 11 0 1 3 3 1 1 1 1 12 0 1 3 5 2 2 2 1 33 0 0 1 1 1 1 2 1 24 0 0 0 1 1 1 2 1 25 0 0 0 0 0 0 1 0 1matics and Economics 46 (2010) 568579 579

5 6 7 8 9

4,094,140 4,018,736 3,971,591 3,941,391 3,921,2583,019,980 2,976,064 2,946,941 2,919,9553,341,934 3,283,928 3,257,8273,469,505 3,413,9213,298,998

9 (%) 0 (%) 1 (%) 2 (%) 3 (%) 4 (%) 5 (%) 6 (%) 7 (%) 8 (%)

0 0 0 0 0 0 0 0 0 00 1 1 1 0 0 0 0 0 01 2 3 3 1 0 0 0 0 01 1 3 5 1 1 0 0 0 01 0 1 2 1 1 0 0 0 02 0 1 2 1 1 0 0 0 06 1 1 2 2 2 1 1 0 06 0 1 1 1 1 0 1 0 034 1 1 3 2 2 1 1 2 1100 0 0 1 1 1 0 1 1 2

0 100 1 1 0 0 0 0 0 00 1 100 2 0 0 0 0 0 01 1 2 100 1 1 0 0 0 01 0 0 1 100 0 0 0 0 01 0 0 1 0 100 0 0 0 00 0 0 0 0 0 100 0 0 0

Paid--incurred chain claims reserving methodPaid--incurred chain modelIntroductionNotation and model assumptions

Simultaneous payments and incurred losses considerationCumulative paymentsIncurred lossesCumulative payments and incurred losses

Parameter estimationCumulative paymentsIncurred lossesCumulative payments and incurred losses

Prediction uncertaintyExampleData analysis in the Bayesian PIC Model 3.1Full Bayesian PIC model

ConclusionsProofsReferences