-
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:[email protected]
(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,
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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
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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.
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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