this methode for reserving in actuarial

1. University of Waterloo
Waterloo, Ontario N2L 3G1
Email: iipr@uwaterloo.ca
INSTITUTE OF INSURANINSTITUTE OF INSURANCE ANDCE AND
PENSION RESEARCHPENSION RESEARCH
The Munich Chain-Ladder Method:
A Bayesian Approach
Enrique de Alba
08-07

2. 2007 IIPR Reports
2007-01 “On the Gerber-Shiu Discounted Penalty Function in the Sparre Andersen Model with and
Arbitrary Interclaim Time Distribution”, David Landriault and Gordon Willmot
2007-02 “On the analysis of a multi-threshold Markovian risk model”, Andrei Badescu, Steve Drekic, and
David Landriault
2007-03 “Valuing CDO’s of Bespoke Portfolios with Implied Multi-Factor Models”, Dan Rosen and David
Saunders
2007-04 “Analytical Methods for Hedging Systematic Credit Risk with Linear Factor Portfolios”, Dan
Rosen and David Saunders
2007-05 “On the Time Value of Absolute Ruin with Debit Interest”, Jun Cai
2007-06 “The Compound Poisson Surplus Model with Interest and Liquid Reserves: Analysis of the
Gerber-Shiu Discounted Penalty Function”, Jun Cai, Runhuan Feng, and Gordon E. Willmot
2007-07 “Optimal Reinsurance under VaR and CTE Risk Measures”, Jun Cai, Ken Seng Tan, Chengguo
Weng, and Yi Zhang
2007-08 “Designing a Social Security Pension System”, Robert L. Brown
2007-09 “Negative Effects of the Canadian GIS Clawback and Possible Mitigating Alternatives”, Diana
Chisholm and Rob Brown
2007-10 “Issues in the Issuance of Enhanced Annuities”, Robert L. Brown and Patricia Scahill
2007-11 “Estimating the Variance of Bootstrapped Risk Measures”, Joseph H.T. Kim and Mary R. Hardy
2007-12 “A Capital Allocation Based on a Solvency Exchange Option”, Joseph H.T. Kim and Mary R.
Hardy
2008 IIPR Reports
2008-01 “On the regulator-insurer-interaction in a structural model”, Carole Bernard and An Chen
2008-02 “Evidence of Cohort Effect on ESRD Patients Based on Taiwan’s NHIRD”, Juang Shing-Her,
Chiao Chih-Hua, Chen Pin-Hsun, and Lin Yin-Ju
2008-03 “Measurement and Transfer of Catastrophic Risks. A Simulation Analysis”, Enrique de Alba,
Jesús Zúñiga, and Marco A. Ramírez Corzo
2008-04 “An Accelerating Quasi-Monte Carlo Method for Option Pricing Under the Generalized
Hyperbolic Lévy Process”, Junichi Imai and Ken Seng Tan
2008-05 “Conditional tail moments of the Exponential Family and its transformed distributions”, Joseph
H.T. Kim
2008-06 “Fast Simulation of Equity-Linked Life Insurance Contracts with a Surrender Option”, Carole
Bernard and Christiane Lemieux
2008-07 “The Munich Chain-Ladder Method: A Bayesian Approach, Enrique de Alb

3. The Munich Chain-Ladder Method: a Bayesian
Approach
Enrique de Alba
Actuarial Science Department, ITAM
Department of Statistics and Actuarial Science , U. of Waterloo
e-mail: dealba@itam.mx
Abstract
In non-life insurance the traditional chain-ladder method for claims reserving is
widely used and its results frequently serve as benchmark. From the actuarial point
of view, reserving is a problem of estimation, or more precisely, forecasting. As in
many fields the estimation or prediction methods can range from very simple
deterministic techniques to some very sophisticated ones, based on stochastic
models. This has also been the case with the chain-ladder method. In its initial
form it is a simple deterministic procedure, but it has given rise to numerous
developments that intend to provide stochastic formulations that in some way
reproduce those of the deterministic scheme while satisfying stochastic
assumptions that allow the user to evaluate his results. One of the most recent
formulations of the chain-ladder method is the Munich Chain Ladder (MCL) which
is aimed at optimizing the simultaneous use of paid and incurred claims data.
While both of these sources of claims data should lead to the same ultimate claims
they typically produce different results. To date several modifications to this
method have been proposed. In this paper we present a Bayesian approach to the
MCL and compare the results to other formulations. Computations are carried out
via MCMC using WinBUGS.

4. 1
The Munich Chain-Ladder Method: a Bayesian
Approach
Enrique de Alba
Actuarial Science Department, ITAM
Department of Statistics and Actuarial Science , U. of Waterloo
e-mail: dealba@itam.mx
Introduction
In non-life insurance the traditional chain-ladder method for claims reserving is
widely used and its results frequently serve as benchmark when proposing new
procedures. From the actuarial point of view, reserving is a problem of estimation,
or more precisely, forecasting. Hence, as in many fields, the estimation or
prediction methods used can range from very simple deterministic techniques to
some very sophisticated ones, based on stochastic models. This has also been the
case with the chain-ladder method. In its original form it is a simple deterministic
procedure, but it has given rise to numerous developments that intend to provide
stochastic formulations that in some way reproduce those of the deterministic
scheme, while satisfying stochastic assumptions that allow the user to evaluate his
results, England and Verrall (2002), Hess and Schmidt (2002), Kaas et al. (2001),
Mack (1993, 1994), Renshaw and Verrall (1998) and Verrall (1990, 2000), among
others.
One of the most recent formulations of the chain-ladder method is the Munich
Chain Ladder (MCL) which is aimed at optimizing the simultaneous use of paid and
incurred claims data, Quarg and Mack (2004). While both of these sources of
claims data should lead to the same ultimate claims they typically produce different
results. To date, several alternative formulations to this method have been
proposed, Jedlicka (2007), Merz and Wuthrich (2006) and Verdier and Klinger
(2005).
In this paper we present a Bayesian approach to the MCL and compare the results
to other formulations. The Bayesian approach to inference has now been used
extensively and successfully in actuarial science, Klugman (1992), Makov (2001),
Scollnik, (2001); and more specifically in claims reserving, de Alba (2002, 2006),
de Alba and Nieto-Barajas (2008), Ntzoufras and Dellaportas (2002) and Verrall
(2004).

5. 2
The Munich Chain Ladder Model
We use standard notation in claims reserving, but differentiating between paid and
incurred claims. So that P
itZ = incremental amount of paid claims in the t-th
development year corresponding to accident year i. Thus },...,1,,...,1;{ stniZP
it ==
where s = maximum number of development years it takes to completely pay out
the total amount of claims corresponding to a given exposure year. In this paper
we do not need to assume 0>P
itZ for all i = 1,…,n and t = 1,…,s. Most claims
reserving methods usually assume that n=s and that we know the values P
itZ
for 1+≤+ nti . The known values for paid claims are presented in the form of a
run-off triangle, Table 1. In addition to incremental paid claims we usually define a
corresponding set of cumulative paid claims ∑=
=
j
t
P
itij ZP
1
, njni ,...,1,,...,1 == . An
important quantity is "ultimate paid claims", which for accident year i is defined as
∑=
==
n
t
P
itini ZPUP
1
i.e. "ultimate paid claims" is the total amount of claims that will
eventually be paid out, over all n development years for those claims
corresponding to accident year i.
Table 1
year of development year
origin 1 2 ...... t ... n-1 n
1 P
Z11
P
Z12
... P
tZ1
P
nZ 1,1 −
P
nZ1
2 P
Z21
P
Z22
... P
tZ2
P
nZ 1,2 −
-
3 P
Z31
P
Z32
... P
tZ3
-
: - -
n-1 P
nZ 1,1−
P
nZ 2,1−
- -
n P
nZ 1
- -
Analogous definitions are given for incurred claims in the MCL, that is let that I
itZ =
incremental amount of incurred claims in the t-th development year corresponding
to accident year i for all i = 1,…,n, t = 1,…,n. and we know the values I
itZ
for 1+≤+ nti . The known values for incurred claims can also be presented in the
form of a run-off triangle. In addition to incremental incurred claims we define the
corresponding set of cumulative incurred claims ∑=
=
j
t
I
itij ZI
1
, njni ,...,1,,...,1 == , as
well as is "ultimate incurred claims", which for accident year i is defined as
∑=
==
n
t
I
itini ZIUI
1
.

6. 3
Given the two triangles, we want to estimate (forecast) the unknown values of P
itZ
and I
itZ , i = 1,…,n, t = 1,…,n for 1+>+ nti . Usually, the CL method is applied to
both, cumulative payments and claims separately and since in both cases we are
estimating the same total ultimate outstanding claim amount, we would expect that
the two estimates be very similar for both cumulative paid and incurred claims
data. Nevertheless, this is often not the case so that the estimates based on
different data (paid and incurred claims) may differ substantially.
Efforts have been made to solve this problem. Halliwell (1997) used a constrained
regression approach. His regression model uses all data simultaneously, paid and
incurred claims, but he imposes a double constraint. One first constraint is on the
estimates of the regression parameters to assure that the predicted outstanding
total loss values are the same for both paid and incurred losses. That is if P
itZˆ and
I
itZˆ are the estimates of the incremental claims then they must satisfy
∑∑ ∑∑= I
it
P
it ZZ ˆˆ , where both summations are for i = 1,…,n, t = 1,…,n
with 1+>+ nti . The other constraint is on the predictions of ultimate claims. It is
required that the estimates satisfy ii IUPU ˆˆ = , for i=2,…,n. Needles to say he ends
up with a complicated model that may become even more so if any of the common
problems in regression are present, such as autocorrelation. Consequently, the
distribution of total outstanding claims is not obtained.
The original Munich chain-ladder (MCL) model was proposed by Quarg and Mack
(2004). It is also aimed at reducing the gap between estimated claims reserves
based on cumulative payments data and estimated claims reserves based on
claims incurred data in a chain-ladder framework. The basic structure of the MCL
model is the same as Mack’s distribution-free chain ladder model, Mack (1993). So
the chain ladder development factors in the MCL model are obtained by Mack’s
distribution-free approach. However, the MCL model adjusts the chain ladder
development factors using the correlations between the observed paid and
incurred claims. Thus it combines information on both paid and incurred cumulative
claims. This is done using paid/incurred ratios and incurred/paid ratios:
ij
ij
ij
ij
ij
ij
P
I
Qand
I
P
Q == −1
.
These ratios are used to smooth the predictions: if a paid/incurred ratio ijQ is
below-average (above-average) then it leads to an above-average (below-average)
development factor for paid claims and/or a below-average (above-average)
development factor for incurred claims, Quarg and Mack (2004). For example, if
the incurred amount is high relative to the paid amount, it means that relatively high
paid losses will be expected for the following years. This information can be

7. 4
incorporated by comparing the paid to incurred ratios for each business year to the
average ratio over business years. An analogous argument applies to incurred/paid
claims ratios 1−
ijQ . This way the adjusted development factors for both paid and
incurred claims are different not only by development years, as is the case in the
traditional chain-ladder, but also for different accident years. Formally, inclusion of
these two ratios implies additional model assumptions relative to those of the
original CL model as made by Mack (1993). The assumptions of the MCL are given
in the Appendix. Assumptions (A1)-(A3) are the classical CL assumptions for the
paid and incurred claims processes as stated in Mack's model, Mack (1993).
Assumptions (A4)-(A5) are additional assumptions made for the MCL model, in
Quarg and Mack (2004). In particular (A4) includes the following equation
( )
( ))]([
)(
)(
)( 11
2/11
2/1
sQEQ
sQVar
s
P
P
Var
fs
P
P
E iisis
iis
i
is
ij
PP
jsi
is
ij
P
P
P
B −−
−→ −⋅






⋅+=





λ , (1)
where P
jsf → stands for the true value of the development factor between
development periods s and j, with s < j, as defined in Mack (1993). The parameter
P
λ is the slope of the regression line in the regression of the ratios isij PP / on 1−
ijQ
(both standardized). Note that it does not depend on the development year. Here
Pi(s) represents all the information on paid claims available up to period s, which is
used in estimating paid claims for period j; and Bi(s) represents all the information
on both paid and incurred claims available up to that same period. An analogous
expression is assumed to hold for incurred claims. Both equations are used
iteratively to estimate outstanding claims, paid and incurred, and hence compute
the required reserves. The original paper does not show how to compute prediction
error, or the mean square error of prediction (MSEP).
Several modifications and/or extensions to the MCL have been proposed. Merz
and Wüthrich (2006) use a credibility approach to estimate claims reserves, and
show that the model assumptions underlying the MCL method can be reduced to
the usual model assumptions of the classical chain-ladder model of Mack (1993) in
a linear Bayesian framework. They further show that MCL estimators are locally
optimal linear predictors in this context, i.e. they minimize the mean square error of
prediction, at each iteration. Their best linear-affine predictor for ijP is the
following.
2/1
11,
1,,2/1
111,1,1
}|{
]|[
}|{}|,{ˆ
P
jji
P
jijiijiP
jij
P
jjiijji
P
j
P
ij
IVar
IEI
PVarIPCorrPfZ
−−
−−−
−−−−−
−
⋅⋅+=
B
B
BB (2)
where P
jf stands for the true value of the j-th development factor as defined in
Mack (1993), and P
jB represents all the information on paid claims available up to

8. 5
period j. A similar expression is obtained for incurred claims. Notice that if in
equation (1) we let s = j-1 and multiply through by 1, −jiP we essentially get equation
(2). No applications are given in Merz and Wüthrich (2006). In fact Wutrhich and
Merz (2008) indicate that this is rather difficult since the dependence structure
between the paid and incurred triangles is non-trivial and because the method uses
some ad-hoc estimators for the parameters.
Jedlicka (2007) discusses several extensions to the Munich Chain Ladder Method.
He suggests the use of robust estimation methods for the parameters in the
regressions used in the original MCL. He also proposes an expression to compute
the mean square error that will enable one to specify a safety margin of the
reserve, in a manner similar to Mack (1993). In addition he presents what is called
a multivariate generalization of MCL method based on previous extensions of the
traditional chain ladder method, Kremer (2005). It is stated that in all cases they
can be applied to both paid and incurred claims, but no examples are provided.
In another extension of the MCL Liu and Verrall (2008) show how the predictive
distribution may be estimated using bootstrapping for both incurred and paid
losses. In their paper they describe an adapted bootstrap approach, combined with
simulation for two dependent data sets. This paper presents a new bootstrap
approach to the estimation of the prediction distributions of reserves produced by
the MCL model. In order to produce a bootstrap distribution, they apply
bootstrapping methods to dependent data, where correlation between data sets is
considered. The key problem that they have to deal with is how to meet the
requirement of not breaking the observed dependence between paid and incurred
claims. They solve the problem by grouping four sets of residuals calculated in the
MCL model, Quarg and Mack (2004).
Verdier and Klinger (2005) proposed an alternative formulation of the MCL, the
JAB Chain. They indicate that while the MCL as proposed Quarg and Mack (2004)
uses an iterative procedure on the normalized residuals, they base their proposal
for the JAB Chain on a linear mixed model that allows the actuary to get the full
development in a single step and to use the related theoretical results for standard
error estimation. The original MCL requires past figures for both paid and incurred,
but also uses the paid to incurred ratios. The key idea to use reserving information
on incurred amounts to improve reserve estimates for paid losses. In particular
considering that for recent accident years, the largest part of the information is in
the incurred claims. They argue that if incurred claims are relatively high, then
relatively high paid losses are to be expected in the following years, and this
information can be taken into account by comparing the paid to incurred ratios for
each business year to the overall average ratio.
Some characteristics of the JAB Chain that differentiate it from the original MCL
are:

9. 6
• time varying slopes: in the original MCL presentation, the slopes used in the
regression model are assumed to be constant over development years. The
JAB Chain allows for varying slopes.
• integration into one model: the MCL uses reserving information for incurred
losses to model the corresponding paid amounts. Paid claims reserves are
similarly used to estimate incurred losses. The JAB Chain considers that
while the incurred process may be informative for paid losses, the converse
does not necessarily hold. JAB Chain considers that the incurred process
should be modeled separately. It is assumed that incurred claims contain all
the historical information.
• joint estimation of all components: while the MCL is carried out in several
steps the JAB Chain uses a single model in which all the parameters are
estimated simultaneously.
Based on the previous arguments Verdier and Klinger (2005) make the following
assumptions.
a) The cumulative paid amount jiP, depends on the previous paid amount 1, −jiP
and on the previous P/I ratio 1, −jiQ .
b) The incurred amount Ii;j depends only on the previous incurred amount Ii;j-1.
where i=2,...,n and j=2,...,n.
Their model is formulated in linear state space form. The observation equation is
jijjijjjiji qQPP ,,,1, ))(( εβα +−⋅+⋅=+ (3a)
or
[ ]


⋅
−=
+−⋅+=
−=
+
ji
P
jnjjiji
ji
jjijj
ji
ji
P
nji
P
qQ
P
P
,1...1,,
,
,
,
1,
2
,0~}{
1...1,
where)(
σε
ε
βα . (3b)
In contrast to the MCL, this model uses only the P/I ratios, which can easily be
interpreted as the share of the global claims (paid and reserves) which has already
been paid, and it is generally between 0 and 1, Quarg and Mack (2004).
Equation (3) can be derived from the usual CL model equation, by adding the
second term )( , jjij qQ −⋅β that corresponds to a correction that is made due to the
deviation of the P/I ratio from its average. The parameter αj is equivalent to the
development factor P
jf in the CL model.
The transition equations of the state space model are defined as follows

10. 7
j
j
jj
jj
β
α
εββ
εαα
+=
+=
+
+
1
1
(4)
where
.
In addition the restriction βn-1 = 0 is necessary because αn-1 and βn-1 are not
identifiable. The parameters are estimated by means of Penalized Least Squares,
i.e. minimizing
where
1
∑=
=
n
i
iMM
2
,,,1,
1
1
, ))(( jijjijjijji
n
j
jii PqQPPM −−−= +
−
=
∑ βαω








+−+−+ ∑∑
−
=
−+
−
=
+
3
1
2
2
2
1
2
1
2
2
12
)(
1
)(
1 n
j
njj
n
j
jj βββ
σ
αα
σ βα
. (5)
In the linear state space model formulation σα and σβ can be chosen so that the
parameters αj and βj vary smoothly. This is more evident in equation (5) where a
trade-off between goodness of fit (the first, least-squares term) and smoothness
(two last terms), where high variation in the αj and βj is penalized. The weights are
computed as .)( 12 −
= P
jijij P σω They suggest two procedures for determining the
values for σα and σβ .
The different versions and modifications of the MCL try to improve on the original
formulation. The main advantage in many of them is that a measure of the
uncertainty can be obtained, be it the mean square error or the complete
distribution. With regards to the use of credibility theory in general, and in particular
to justify results of the MCL, it is interesting to quote from Venter as given in
Halliwell (1997): “An apparent advantage of credibility over Bayesian analysis is
that distributional assumptions are not needed for credibility. That is, credibility
gives the best linear least squares answer for any distribution, whereas a Bayesian
analysis will be different for different distributions. There are two problems with this
conclusion, however. First, . . when the Bayesian estimates are not linear, as in the
1...1
),0(~
),0(~
−=∀ n
N
N
j
j
j
ββ
αα
σε
σε

11. 8
case of most highly skewed distributions, credibility errors can be substantially
greater than the Bayes’ errors. Second, when Bayes’ estimates are linear functions
of the data, … credibility analysis gives the same answer as assuming normal (or
gamma) distributions and doing a Bayesian analysis, and it gives a useful answer
only in those cases where normal or gamma distributions would be reasonable”.
And he adds “An advantage of Bayesian analysis is that it gives a distribution
around the estimate, so that the degree of likely deviation from the estimate can be
quantified. “
Model Assumptions
The Bayesian model we present here is structured along the lines of some of those
described in the previous section. It is a parametric Bayesian model in which we
assume that the link ratios for paid claim amounts follow a lognormal distribution,
Let ijL be the random link ratio between age j and 1+j for the i-th accident year.
That is
,1,...,2,1,1,...,2,1,
1,
−=−==
+
njni
P
P
L
ij
ji
ij
i.e. ),(~ 2
jjij LNL σµ and set )( jijjjj qQ −+= βαµ so that we have a structure
similar to Verdier and Klinger (2005) with the addition of the lognormal
distributional assumption. Assuming the ijL are independent it can be seen that
when ,,...,, 1210 −== njjβ the model is similar to the one proposed in Han and
Gau (2008). Hence
))],([(~ 2
jjijjjij qQnLNL σβα −+l (6)
with i=1,...,n-1, j=1,...,n-1 and 1+≤+ nji .
Thus given the data in B the likelihood function is
∏∏
+−
=
−+−−
+
−
=
=
1
1
)}](log{[log
2
1
1,
1
1
2
2
2
1
)|,,(
in
j
qQL
jji
n
i
jijjjij
j
e
L
L
βα
σ
σπ
Bσβα (7)
where ),...,(' 11 −= nααα , ),...,(' 11 −= nβββ and ),...,( 2
1
2
1 −= nσσσ are vectors of
unknown parameters, and B represents the information available on the ijL , or

12. 9
equivalently on ijP . Notice that the variance is constant by accident years but
changes over development years.
Assuming lognormality as we do has the advantage that the probability of negative
cumulative paid claims is ruled out by the model. Whereas under the assumption
of normality negative values could in principle occur. In addition this assumption
has interesting implications for the distribution of the cumulative paid claims, ijP . It
can be shown that their implied distribution is also lognormal.
Bayesian Formulation of the Model
In order to carry out the Bayesian analysis we must specify prior distributions for
the parameters, Bernardo and Smith (1994). We assume that these vectors are
mutually independent a–priori and thus specify
)()()(),,( σβασβα ffff ⋅⋅= ,
with the following definitions of these priors
∏
−
=
−=
1
2
11
n
i
iifff )|()()( αααα
taking ),(~ 2
1 αα σµα N and ),(~| 2
1 ασααα iii N+
∏
−
=
−−−− +
=
1
2
2
1
2
1 2
12
2
12
2
1
2
1 n
i
ii
eef
)()(
)(
αα
σ
α
µα
σ
α
α
α
α
σπσπ
α
A similar prior structure is imposed on )(βf , i.e. ),(~ 2
1 ββ σµβ N and
),(~| 2
1 βσβββ iii N+ . These prior distribution assumptions are equivalent to the
transition equations given in (4) for the JAB Chain, Verdier and Klinger (2005).
However, because of the priors )( 1αf and )( 1βf it is not necessary to impose the
zero constraint 01 =−nβ that they include. The prior specification for σ
is ∏
−
=
=
1
1
2
n
i
iff )()( σσ with each ),(~2
baGaInviσ an inverse gamma with parameters
(a,b) . The (hyper-) parameters of the prior distributions are assumed known, or if a
hierarchical model is used then they are assigned their own distribution.
Hence with these prior distributions and the likelihood equation given in equation
(7) the posterior distribution for the parameters is given by

13. 10
)(
1
)|,,(
1
1
1
1
)}](log{[log
2
1 2
2
σσβα fef
n
i
in
j
qQL
j
jijjjij
j
⋅








∝ ∏ ∏
−
=
+−
=
−+−− βα
σ
σ
B
∏∏
−
=
−−−−−
=
−−−− ++
×
1
1
)(
2
1
)(
2
1
1
1
)(
2
1
)(
2
1 2
12
2
12
2
12
2
12
n
i
n
i
iiii
eeee
ββ
σ
µβ
σ
αα
σ
µα
σ β
β
βα
α
α
. (8)
This posterior distribution is used to obtain the predictive distribution for the lower
right triangle of unknown paid claims, de Alba (2004). To estimate outstanding
claims for the lower right hand triangle we use the predictive posterior distribution.
To this purpose the past (known) data in the upper triangle, B, where
111 +≤+== njindnjni a,,...,,,..., , are used to predict the observations in the
lower triangle ijP , by means of the posterior predictive distribution. To do this we
use assumption of independence between accident years and so analyze for each
one separately.
Let )...(' ,2,1, niiniinii LLL +−+−=L be the link ratios of the i-th accident year that are in the
lower right hand triangle, i.e. they are unobserved. Also, since we do not have the
ijQ for the lower right hand triangle we set jij qQ = , for
ninjni ,...,and,,..., 12 +−== . Hence, if we let ),,(' σβαθ = be a vector that
contains all the parameters in the model, then we can write
∏
−
+−=
−−
+− =
1
1
}]log{[log
2
1
,
1
2
2
2
1
),|(
n
inj
L
jji
P
ini
jij
j
e
L
f
α
σ
σπ
BθL .
Furthermore, analogous to the definition of i'L above, let )...(' ,3,2, niiniinii PPP +−+−=P
be the paid claims in the lower triangle corresponding to accident year i. A
straightforward, although laborious, change of variable yields the predictive
distribution
∏
−
+−=
−−−
+
+−
+
=
1
1
)]log(log[log
2
1
1,
1
2
12
2
1
),|(
n
inj
PP
jji
P
ini
jijij
j
e
P
f
α
σ
σπ
BθP , (9)
for .,...,ni 2= The posterior predictive distribution for the outstanding claims of the
i-th accident year is then obtained as
∫ +−=
Θ
θθθPP dfff P
inii )|(),|()|( 1 BBB , .,...,2 ni = (10)

14. 11
Once we have the posterior predictive distribution as in (10), the distribution of the
total outstanding claims in the i-th accident year, )|( Bif M , can be obtained from
(6) by taking ∑+−=
==
n
inj
ijiii P
2
' PM l , where il is an (i-1)x1 vector of ones; and finally
the distribution of total outstanding claims, )|( BMf , can be obtained using
∑=
=
n
i
i
2
MM . In principle, the derivation of these distributions should be
straightforward. However they can not be obtained analytically. We use Markov
chain Monte Carlo (MCMC) with WinBUGS to carry out the analyses, Scollnik
(2001) and Spiegelhalter et al. (2003).
Examples
In this section we apply our Bayesian version of the Munich chain-ladder model to
three sets of data. The first one is the data used in the original MCL of Quarg and
Mack (2004). The second data set is presented and analyzed in Liu and Verrall
(2008). It is indicated that it uses market data from Lloyd’s which have been
scaled to assure confidentiality. The data are aggregated for paid and incurred
claims, for two Lloyd’s syndicates, categorized at risk level. They present it to
illustrate that the MCL model does not necessarily produce better results in all
situations. It is provided as an example where the data have more variability; they
are ‘jumpy’, in their words.
The first data set is given in Appendix 2. We provide both paid and incurred claims,
as well as the P/I ratios. These are plotted below in Figure 1. The thicker dotted
line is the average for each development year. Clearly, they begin between 0.5 and
0.65 in development year 1, and increase to above 0.90 by development year 4,
and continue growing. This is what is to be expected, according to Quarg and
Mack (2004).

15. 12
Figure 1
Paid/Incurred Ratios
0.5
0.6
0.7
0.8
0.9
1
1 2 3 4 5 6 7
Development Year
Ratio
Application of the Bayesian model using non informative priors we get a posterior
distribution which combined with the predictive distribution gives the reserve
estimates. This is done directly in WinBUGS generating 5000 replications after a
burn-in set has been eliminated. The results are given in the second to last column
in Table 2.
Table 2. Bootstrap Reserves, MCL Reserves and Bayesian MCL Reserves
Bootstrap Munich CL Traditional CL Bayesian MCL (paid)
Paid Incurred Paid Incurred Paid Incurred Mean Median Std.
Dev.
Year 1 0 43 0 43 - - - - -
Year 2 35 95 35 96 32 97 80 49 468
Year 3 106 128 103 135 158 2,061 303 143 1,274
Year 4 275 317 269 326 332 1,590 598 247 2,076
Year 5 294 287 289 302 408 1,229 691 365 1,965
Year 6 672 649 646 655 924 1,426 1,175 757 2,163
Year 7 5,512 5,655 5505 5,606 4,084 8,192 4,436 3,815 3,005
Total 6,893 7,175 6846 7,163 5,938 14,595 7,283 6,763 5,678
The table shows estimates obtained for both the mean and the standard deviation
of the predictive distribution in each accident year. The means are larger than the
estimates obtained by the direct MCL, and the bootstrap estimates, Liu and Verrall
(2008), in all accident years except the last. However, the estimate for the total is
fairly close. The mean is larger due to the resulting skewed predictive distributions.

16. 13
Figure 2 shows the predictive distribution of total outstanding claims. Even so, the
value is close to the one obtained for incurred claims in both the original MCL and
the bootstrapped model. Estimated total reserves are 5,938 and 14,595 for paid
and incurred claims when using the traditional CL method. Notice that with the
exception of Accident years two and three, the Bayesian estimates are between
these two values.
Figure 2
Paid Claims
-20.0 0.0 20.0
0.0
0.05
0.1
As indicated above, the data of Example 2 is taken from Liu and Verrall (2008). It is
also given in Appendix 2. Figure 3 shows the P/I ratios for these data. Again, the
dark dotted line is the average per development year. Here also they begin small in
development year 1, but with a broader range between them. They only increase
to around 0.80 by development year 6, and continue growing. The average shows
this general trend, but the individual lines are much more dispersed than in the
previous case. This will result in more uncertainty in the estimates.

17. 14
Figure 3
Paid/Incurred Ratios
0.00
0.20
0.40
0.60
0.80
1.00
1 2 3 4 5 6 7 8 9 10
Development Year
Ratio
Table 3 has the same structure of Table 2. It shows estimates obtained for both the
mean and the standard deviation of the predictive distribution in each accident
year. There is no clear pattern in the means of the Bayesian estimates with respect
to the other methods. For some accident years the estimate from one method is
higher than from another method and vice versa in other accident years.
However, the estimate for the total mean is fairly close to the total incurred claims
reserves for the Bootstrap. On the other hand, the median of the predictive
distribution for the total claims is close to the estimate of total incurred claims,
under the MCL. All of these estimates are larger that those obtained by the
traditional CL, where estimated total reserves are 65,986 and 89,463 for paid and
incurred claims, respectively. All these differences are probably due to the
variability of the data. This was to be expected given the initial dispersion of the
ratios. The predictive distribution for total paid claims is given in Figure 4.

18. 15
Table 3. Bootstrap Reserves, MCL Reserves and Bayesian MCL Reserves
Bootstrap Munich CL Traditional CL Bayesian MCL (paid)
Paid Incurred Paid Incurred Paid Incurred Mean Median Std.
Dev.
Year 1 - 212 - 212 - - - -
Year 2 48 255 46 258 64 256 661 -9 5,613
Year 3 4,177 3,945 3,197 3,974 4,221 3,961 4,430 2,295 13,380
Year 4 7,319 4,548 6,692 4,306 2,804 4,314 5,754 4,107 9,488
Year 5 18,366 9,007 17,223 8,200 2,195 8,273 8,362 6,666 8,330
Year 6 10,708 8,492 10,456 8,314 6,854 8,325 9,812 7,988 8,744
Year 7 14,291 11,553 14,430 11,219 7,849 11,262 12,620 10,770 9,239
Year 8 9,670 8,845 9,004 9,051 12,313 8,966 17,410 14,480 12,470
Year 9 23,980 19,987 23,584 19,185 9,137 19,409 11,700 9,772 8,090
Year 10 27,901 24,542 28,190 24,633 20,551 24,697 16,560 13,030 13,890
Total 116,459 91,386 112,822 89,351 65,986 89,463 87,300 77,270 63,740
Figure 4. Predictive Density of Paid Claim Reserves (Lloyd’s data)
Paid Claims
-100.0 0.0 100.0 200.0
0.0
0.005
0.01
In Example 3 we use data provided by Wütrhich and Merz (2008) and it is also
included in Appendix 2. The plot of the P/I ratios, along with their average, is given
in Figure 5. In this case all the lines are very close to one another and they grow to
0.90 by development year 2. The behavior is much more stable than the previous
example. It is actually more stable than even the first example.

19. 16
Figure 5.
Paid/Incurred Ratios
0.5
0.6
0.7
0.8
0.9
1
1.1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Development Year
Ratio
Table 4 has the same structure as the two previous tables except for the fact that
we do not have results for a bootstrapped CL. It shows estimates obtained for both
the mean and the standard deviation of the predictive distribution in each accident
year. None of the estimates is consistently larger or smaller than the others. The
original MCL produces the expected effect, that paid claims reserves become
larger and incurred claims reserves smaller, bringing them closer. The Bayesian
MCL increases the total reserves for paid claims but it is in the general range. The
differences between methods are small, total reserves for all methods are fairly
close to each other. This is probably due to the reduced variability of the data, but
mostly to the fact that the P/I ratios get near one (90%) by development year 2.
The Bayesian results based on the median of the predictive distribution are very
close to those of the CL method. The predictive distribution for total paid claims is
given in Figure 6.

20. 17
Table 4. MCL Reserves and Bayesian Reserves
Munich CL Traditional CL Bayesian MCL (paid)
Paid Incurred Paid Incurred Mean Median Std. Dev.
Year 2 5 22 6 22 93 -6 7,027
Year 3 9 -57 15 -58 -47 1 3,248
Year 4 18 233 20 232 221 13 7,863
Year 5 36 309 36 309 144 26 6,794
Year 6 220 534 213 551 239 209 4,206
Year 7 99 -223 227 -520 246 225 5,127
Year 8 179 237 226 128 319 207 6,197
Year 9 281 405 322 330 372 301 6,121
Year 10 603 736 677 637 1,209 633 38,180
Year 11 724 937 712 908 870 722 13,170
Year 12 808 899 1,003 698 1,220 975 12,470
Year 13 858 864 1,403 322 1,858 1,377 33,360
Year 14 2,335 2,539 2,153 2,712 2,395 2,217 9,018
Year 15 2,458 2,534 3,100 1,909 2,819 2,810 4,866
Year 16 5,717 5,842 6,139 5,509 6,158 5,961 11,640
Year 17 37,813 37,944 38,306 37,594 38,100 37,910 9,943
Total 52,162 53,755 54,559 51,282 56,210 53,740 94,950
Figure 6. Predictive Density of Paid Claim Reserves (Wüthrich and Merz data)
Paid Claims
0.0 50.0 100.0 150.0
0.0
0.01
0.02
0.03
0.04

21. 18
Final Comments
We have presented a Bayesian version of the MCL that is based on the proposal of
Verdier and Klinger (2005), the JAB Chain. The model assumes a lognormal
distribution for both the link ratios of paid claims and for paid claims amounts. It
includes a correction to the regular development factors that depends on the
deviations of the Paid/Incurred ratios from their average per development year.
This is essentially in accordance with the ideas put forward by Quarg and Mack
(2004) in the original MCL. In addition the formulation includes all the elements of
the JAB Chain: it allows different slopes of the regression line in development
years, the parameters are analyzed simultaneously in the Bayesian inference
process and there is only one model to estimate. The Bayesian model also allows
for varying degrees of smoothness in the factors slope parameters: the s'α and
s'β of the model. The smoothness is implicitly regulated through the variance
parameters in this prior distribution. We allow the model to estimate them by
deriving their posterior distribution rather than setting them equal to a given
arbitrarily fixed value.
Extensions of the proposed model are possible. Simultaneous estimation of a
model for incurred claims can be done in principle. But more importantly, other
structures of the parameter jα ’s and , sj 'β such as trends, can also be included in
the model.

22. 19
References
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New York.
de Alba, E. (2002), Bayesian Estimation of Outstanding Claim Reserves, North
American Actuarial Journal 6(4), 1-20.
de Alba, E. (2004), Bayesian Claims Reserving, in Encyclopedia of Actuarial
Science, Volume 1, pp. 146–153
de Alba, E. (2006), Claims Reserving When There Are Negative Values in the
Runoff Triangle: Bayesian analysis using the three-parameter log-normal
distribution, North American Actuarial Journal 10 (3), 1–15.
de Alba, E. and Nieto-Barajas, L.E. (2008), Claims reserving: A correlated
Bayesian model, Insurance: Mathematics and Economics,
doi:10.1016/j.insmatheco.2008.05.007
England P. D. and Verrall R. J. (2002). Stochastic Claims Reserving in General
Insurance (with discussion). British Actuarial Journal, 8, 443-544
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errors in claims reserving, Insurance: Mathematics and Economics 25, 281-293.
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Han, J. and Gau, G. (2008), Estimation of loss reserves with lognormal
development factors, Insurance: Mathematics and Economics 42, Issue 1, 389-
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Hess, K.T. and Schmidt, K.D. (2002), A comparison of models for the chain-ladder
method, Insurance: Mathematics and Economics 31, 351-364.
Jedlicka, P. (2007). Various extensions based on Munich Chain Ladder Method,
37th
ASTIN Colloquium, Orlando, Florida
Kaas, R., Goovaerts, M., Dhaene, J. and Denuit, M. (2001), Modern Actuarial Risk
Theory, Kluwer Academic Publishers.
Klugman, S.A. (1992), Bayesian Statistics in Actuarial Science, Kluwer: Boston.
Kremer, E. (2005), The correlated chain ladder method for reserving in case of
correlated claims development, Blatter der DGVFM, Vol. 27, No. 2, 315-322.

23. 20
Liu, H. and Verrall, R. (2008). Bootstrap Estimation of the Predictive Distributions
of Reserves Using Paid and Incurred Claims, 38th
ASTIN Colloquium.
Mack, T. (1993), Distribution-Free calculation of the standard error of chain ladder
reserve structure, ASTIN Bulletin 23, 213-225.
Mack, T. (1994), Which Stochastic Model is Underlying the Chain Ladder Method?,
Insurance: Mathematics and Economics 15, 133-138.
Makov, U.E. (2001), “Principal applications of Bayesian Methods in Actuarial
Science: A Perspective”, North American Actuarial Journal 5(4), 53-73.
Merz, M. and Wuthrich, M.V. (2006). Various extensions based on Munich Chain
Ladder, Method, Blätter der DGVFM, Band XXVII, Vol. 4, 619-628.
Ntzoufras, I. and Dellaportas, P. (2002), Bayesian Modeling of Outstanding
Liabilities Incorporating Claim Count Uncertainty, North American Actuarial Journal
6(1), 113-136.
Quarg, G. and Mack, T. (2004). Munich Chain Ladder, Blätter der DGVFM, Band
XXVI, Vol. 4, 597-630.
Renshaw, A.E. and R. J. Verrall (1998), A stochastic model underlying the chain-
ladder technique, British Actuarial Journal 4,(IV) 905-923.
Scollnik, D.P.M. (2001), Actuarial Modeling With MCMC and BUGS, North
American Actuarial Journal 5(2), 96-124.
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2003). BUGS: Bayesian inference using Gibbs sampling. MRC Biostatistics Unit,
Cambridge, England. www.mrc-bsu.cam.ac.uk/bugs/
Spiegelhalter, D.J., Thomas, A., Best, N.G. and Gilks, W.R. (2001) WinBUGS 1.4:
Bayesian Inference using Gibbs Sampling, Imperial College and MRC Biostatistics
Unit, Cambridge, UK. http://www.mrc-bsu.cam.ac.uk/bugs.
Verdier, B. and Klinger, A. (2005). JAB Chain: A model based calculation of paid
and incurred development factors, 36th ASTIN Colloquium.
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the chain-ladder technique, Insurance: Mathematics and Economics, Vol. 26, 91-
99.

24. 21
Verrall, R.J. (2004), A Bayesian Generalized Linear Model for the Bornhuetter-
Ferguson Method of Claims Reserving, North American Actuarial Journal 8(3), 67-
89.
Wüthrich, M. V. and Merz, M. (2008), Stochastic Claims Reserving Methods in
Insurance, Wiley

25. 22
APPENDIX 1
In this Appendix we state the assumptions of the original Munich Chain Ladder
model of Quarg and Mack (2004). However we state only those that are relevant
for the formulation and understanding of the Bayesian model described in this
paper. Essentially this means we state the assumptions made on paid claims.
There is another parallel set of assumptions on incurred claims that are not stated
here.
Assumption A1. (Expectation) For Tts ∈, , with 1+= st there exists a constant
0>→
P
tSf such that for ni ,...,1=
P
⋅→=







tsi
si
ti
fs
P
P
)(|
,
,
PEEEE
This corresponds to assumption PE in Quarg and Mack (2004). There is an
analogous assumption (IE) for incurred claims.
Assumption A2. (Variance) For Tts ∈, , with 1+= st there exists a constant
0≥→
P
tsσ such that for ni ,...,1=
si
P
ts
i
si
ti
P
s
P
P
,
2
,
, )(
)(| →
=






 σ
PVarVarVarVar .
This corresponds to assumption PV in Quarg and Mack (2004). There is an
analogous assumption (IV) for incurred claims.
Assumption A3. (Independence) The random variables corresponding to paid
claims of different accident years are stochastically independent, i.e.
{ }njP j ,...,1;1 = , { }njP j ,...,1;2 = and { }njPnj ,...,1; = are mutually stochastically
independent.
This corresponds to assumption PU in Quarg and Mack (2004). There is an
analogous assumption (IU) for incurred claims.
Assumption A4. There exists a constant P
λ , such that for Tts ∈, , with 1+= st
anda ll ni ,...,1=
( )))(|(
)(|
)(|
)(| 1
,
1
,1
,
,
,
,
,
sQQ
sQ
s
P
P
fs
P
P
isisi
isi
i
si
ti
PP
tsi
si
ti
P
P
P
B −−
−→ −⋅








⋅+=







EEEEEEEE
σ
λ
where isisis IPQ /= .

26. 23
This corresponds to proposition PQ in Quarg and Mack (2004). There is an
analogous proposition (IQ) for incurred claims.
Assumption A5. Both cumulative payments and claims incurred are mutually
independent across different accident years, i.e. the sets { }njIP jj ,...,;, 111 = ,
{ }njIP jj ,...,;, 122 = ,…,{ }njIP njnj ,...,;, 1= are independent.
This corresponds to proposition PIU in Quarg and Mack (2004).

27. 24
APPENDIX 2
1. Data set from Quarg and Mack (2004). Cumulative data.
Table 1. Paid Claim Data
1 2 3 4 5 6 7
1 576 1804 1970 2024 2074 2102 2131
2 866 1948 2162 2232 2284 2348
3 1412 3758 4252 4416 4494
4 2286 5292 5724 5850
5 1868 3778 4648
6 1442 4010
7 2044
Table 2. Incurred Claim Data
1 2 3 4 5 6 7
1 978 2104 2134 2144 2174 2182 2174
2 1844 2552 2466 2480 2508 2454
3 2904 4354 4698 4600 4644
4 3502 5958 6070 6142
5 2812 4882 4852
6 2642 4406
7 5022
Table 3. Paid/Incurred Claims Ratios.
1 2 3 4 5 6 7
1 0.5890 0.8574 0.9231 0.9440 0.9540 0.9633 0.9802
2 0.4696 0.7633 0.8767 0.9000 0.9107 0.9568
3 0.4862 0.8631 0.9051 0.9600 0.9677
4 0.6528 0.8882 0.9430 0.9525
5 0.6643 0.7739 0.9580
6 0.5458 0.9101
7 0.4070

28. 25
2. Lloyd’s Data from Liu and Verrall (2008). Cumulative data.
Table 4. Scaled Aggregate Paid Claims at Risk Level
1 2 3 4 5 6 7 8 9 10
1 184 1845 3748 5400 6231 9006 9699 10008 10035 10068
2 155 1483 3768 7899 8858 13795 15360 15895 19333
3 676 2287 10635 16102 22177 28825 29828 30700
4 67 367 2038 2879 6329 14366 16201
5 922 1693 3523 4641 6431 8325
6 22 488 3424 5649 7813
7 76 435 1980 5062
8 24 1782 3881
9 39 745
10 306
Table 5. Scaled Aggregate Incurred Claims at Risk Level
1 2 3 4 5 6 7 8 9 10
1 1530 8238 10564 12332 12173 10576 10630 10316 10325 10280
2 1505 6247 8728 10500 15241 16720 16845 16829 19675
3 2505 6150 17937 22143 29511 33336 32162 31500
4 204 2748 9984 13167 16523 17807 18959
5 2285 4361 6432 8834 12092 15309
6 269 5549 7214 12422 13581
7 1271 2657 6187 11004
8 298 3533 6423
9 2023 5415
10 1779
Table 6. Paid/Incurred Claims Ratios
1 2 3 4 5 6 7 8 9 10
1 0.1203 0.2240 0.3548 0.4379 0.5119 0.8516 0.9124 0.9701 0.9719 0.9794
2 0.1030 0.2374 0.4317 0.7523 0.5812 0.8251 0.9118 0.9445 0.9826
3 0.2699 0.3719 0.5929 0.7272 0.7515 0.8647 0.9274 0.9746
4 0.3284 0.1336 0.2041 0.2187 0.3830 0.8068 0.8545
5 0.4035 0.3882 0.5477 0.5254 0.5318 0.5438
6 0.0818 0.0879 0.4746 0.4548 0.5753
7 0.0598 0.1637 0.3200 0.4600
8 0.0805 0.5044 0.6042
9 0.0193 0.1376
10 0.1720

29. 3. Data set from Wuthrich and Merz (2008)
Table 7. Cumulative Paid Claims
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
1 46.726 64.768 65.412 65.663 66.008 66.462 67.208 67.252 67.258 67.279 67.281 67.282 67.292 67.307 67.313 67.331 67.336
2 48.658 71.816 73.514 73.946 74.137 74.254 74.765 74.793 74.820 74.821 74.830 74.901 74.901 74.901 74.901 74.898
3 53.455 79.454 81.188 82.236 83.203 83.901 84.924 85.387 85.522 86.105 86.106 86.107 86.286 86.286 86.286
4 61.851 91.040 92.205 93.297 93.375 93.544 93.597 93.647 95.128 95.139 95.201 95.201 95.631 95.675
5 65.971 89.125 92.019 92.563 92.688 93.713 93.721 93.731 93.690 93.662 93.644 93.644 93.811
6 64.913 89.369 91.819 92.350 93.306 93.811 93.831 93.841 93.818 93.849 93.844 93.848
7 64.019 87.951 89.930 92.170 93.606 93.726 93.725 93.658 93.655 93.698 93.702
8 60.412 86.978 88.988 89.685 89.917 89.896 89.877 89.843 89.832 89.832
9 60.994 87.799 89.708 90.470 91.300 92.482 92.605 92.829 92.830
10 82.391 118.384 120.920 121.490 122.399 123.373 123.432 123.446
11 75.977 109.038 111.630 112.733 112.848 112.968 112.980
12 74.212 110.220 112.615 113.427 113.777 113.805
13 65.557 96.596 99.317 102.561 103.019
14 66.116 100.415 109.417 110.900
15 66.782 97.728 100.447
16 71.205 103.052
17 72.624

30. 2
Table 8. Cumulative Incurred Claims
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
1 71.229 71.262 68.803 68.511 67.572 68.448 67.682 67.680 67.670 67.642 67.564 67.526 68.183 68.158 68.162 68.144 68.074
2 86.774 82.525 79.191 77.111 77.776 76.851 76.061 75.507 75.436 75.422 75.385 75.086 75.064 75.046 75.006 74.997
3 86.064 90.815 88.342 89.265 87.915 87.314 87.903 87.063 86.988 86.651 86.557 86.524 86.420 86.405 86.334
4 98.419 101.600 99.502 96.215 96.996 96.625 98.179 98.794 98.087 97.998 97.936 97.872 96.041 96.068
5 106.154 102.136 99.693 98.418 96.443 95.520 95.381 95.349 95.032 95.029 94.656 94.387 94.288
6 102.815 100.942 100.835 97.769 96.331 95.520 95.268 95.556 95.196 95.148 94.894 94.882
7 104.442 100.028 98.028 97.167 94.689 94.391 94.091 93.928 93.917 93.777 93.789
8 102.366 98.160 93.327 92.097 91.215 91.162 91.169 91.123 90.742 90.678
9 100.672 95.277 96.141 95.002 94.102 94.235 94.208 94.219 94.000
10 134.428 131.328 127.963 126.999 127.355 126.368 126.068 125.538
11 125.793 124.528 122.220 121.751 119.238 115.689 115.380
12 121.351 124.244 120.442 118.423 118.232 116.084
13 110.666 111.664 111.508 109.246 105.643
14 119.939 117.876 118.237 117.462
15 112.185 112.416 107.240
16 114.117 116.216
17 119.140

31. 3
Table 9. Paid/Incurred ratios:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
1 0.656 0.909 0.951 0.958 0.977 0.971 0.993 0.994 0.994 0.995 0.996 0.996 0.987 0.988 0.988 0.988 0.989
2 0.561 0.870 0.928 0.959 0.953 0.966 0.983 0.991 0.992 0.992 0.993 0.998 0.998 0.998 0.999 0.999
3 0.621 0.875 0.919 0.921 0.946 0.961 0.966 0.981 0.983 0.994 0.995 0.995 0.998 0.999 0.999
4 0.628 0.896 0.927 0.970 0.963 0.968 0.953 0.948 0.970 0.971 0.972 0.973 0.996 0.996
5 0.621 0.873 0.923 0.941 0.961 0.981 0.983 0.983 0.986 0.986 0.989 0.992 0.995
6 0.631 0.885 0.911 0.945 0.969 0.982 0.985 0.982 0.986 0.986 0.989 0.989
7 0.613 0.879 0.917 0.949 0.989 0.993 0.996 0.997 0.997 0.999 0.999
8 0.590 0.886 0.954 0.974 0.986 0.986 0.986 0.986 0.990 0.991
9 0.606 0.922 0.933 0.952 0.970 0.981 0.983 0.985 0.988
10 0.613 0.901 0.945 0.957 0.961 0.976 0.979 0.983
11 0.604 0.876 0.913 0.926 0.946 0.976 0.979
12 0.612 0.887 0.935 0.958 0.962 0.980
13 0.592 0.865 0.891 0.939 0.975
14 0.551 0.852 0.925 0.944
15 0.595 0.869 0.937
16 0.624 0.887
17 0.610

32. INSTITUTE OF INSURANCE AND PENSION RESEARCH
CORPORATE SPONSORS
AEGON Canada, Inc.
CANNEX Financial Exchanges Ltd.
Co-operators
Dominion of Canada General
Eckler Partners Ltd.
Economical Mutual Insurance Co.
Empire Life
Equitable Life Insurance Co.
Gilliland Gold Young Consulting Inc.
Great-West Life Assurance Co.
Hewitt Associates
ING Canada Inc
Mercer Human Resources Consulting
Munich Reinsurance Co.
Oliver Wyman
PPI Financial Group
RBC Life Insurance Co of Canada
RGA Life Reinsurance Co. of Canada
Sun Life Assurance Co. of Canada
Swiss Re Life and Health Canada
Towers Perrin