LOSS RESERVING METHODS
PHILIPP ARBENZ AND ROBERT SALZMANN
ABSTRACT
We introduce the hybrid chain ladder method, a distribution-free stochastic loss reserving method
that allows for a weighted combination of two approaches. The first approach is data driven and
resembles the Chain-Ladder method. The second approach uses expert estimates of ultimate losses
similar to the Bornhuetter-Ferguson method. Reserving actuaries often have concerns about data
quality or know of particular events that cause unusual effects. Such external knowledge has to be
considered in setting the weights between the two approaches. The weights can be set differently for
each accident and development year. We give predictors for the ultimate claims and estimators for
the prediction error and the uncertainty in the claims development result. An implementation of the
method in an Excel spreadsheet is available atwww.RiskLab.ch/hclmethod.
KEYWORDS
Stochastic claims reserving, distribution-free method, Chain-Ladder, Bornhuetter-Ferguson, mean
square error of prediction, claims development result.
1. INTRODUCTION
The Chain-Ladder (CL) and Bornhuetter-Ferguson (BF) methods are standard methods in claims
reserving. When only the CL and BF methods are viable choices in reserving practice, choosing only
one of them may not be adequate for a given data set. For instance, claims reserving triangles in
which CL is appropriate for some accident years, whereas BF is suitable for the others are frequently
encountered in practice.
There are many different approaches to embed CL and BF into stochastic frameworks, which allow
to deduce statistically consistent claims predictors and assess the uncertainty of the claims
develop-ment. However, CL and BF generally impose incompatible assumptions on the claims developdevelop-ment.
For instance, claim increments are independent in BF, whereas they are correlated in CL. Therefore,
switching between CL and BF on a accident year-wise basis leads to inconsistencies.
In this paper we introduce the hybrid chain ladder (HCL) method, a class of distribution-free
sto-chastic loss reserving models that allows for weighted combinations of two methodologies. The first
methodology is multiplicative in structure, resembling CL, and relies purely on the claims data. The
weights between the two methodologies can be set differently for each accident and development
year in accordance with the actuary’s assessment.
We have implemented HCL in a ready-to-use Excel spreadsheet which can be downloaded at
http://www.RiskLab.ch/hclmethod
This spreadsheet allows to test all features of the method and provides several case studies including
the one discussed in this paper.
Organisation of the paper.Section 2 motivates our modelling approach. Section 3 introduces the
model and derives the structure of the mean and the variance of the claims process. Section 4
de-fines estimators of the unknown parameters and predictors for future claims. Section 5 and Section 6
give estimators of the uncertainty in ultimate claims predictions and the claims development result,
respectively. We extend the model in Section 7 to incorporate uncertainty in the choice of prior
ulti-mates. Section 8 provides guidance how to choose the remaining model parameters. A case study is
presented in Section 9. We outline limitations and possible extensions in Section 10 and conclude in
Section 11.
2. MOTIVATION
We first introduce some notation. LetCi,jdenote the cumulative claims of accident yeari=1, . . . ,I
and development yearj=0, . . . ,J withJ≤I−1. We assume that the ultimate claim for accident year
iis given byCi,J, i.e. there is no further development after development yearJ. Moreover, denote by
DI=©Ci,j : 1≤i≤I, 0≤j≤J,i+j≤Iª,
the trapezoid of observations up to theI-th accounting year, illustrated in Figure 1.
Ci
,
J
u
lt
im
ate
claims
1 j J
I i
DI
ac
cident
y
ear
i
1
2
to be predicted 2
0
observations ofCi,j development yearj
. . . . . .
3 .. .
.. .
FIGURE1. The trapezoidDI of cumulative claims known in accounting yearI. The
lower right triangle is yet unknown and has to be predicted.
Stochastic reserving methods aim to predict the yet unknown lower triangleDIc, i.e.Ci,j∉DI, along
Ci,Jfor each accident yeari. For CL and BF, this is done as follows. In brief, CL predictsCi,J by
b
CiC L,J =Ci,I−ifIb−i+1· · ·fJb,
where the fbj are estimators of the age-to-age factors fj. These factors indicate the average relative
increase of the cumulative claims in one accident year from one development year to the next
devel-opment year. On the other hand, BF predictsCi,Jby
b
CiB F,J =Ci,I−i+µi(γbI−i+1+ · · · +γbJ),
where theµiis a prior estimate ofE[Ci,J] and theγbjare estimates ofγj, which denote the fraction of
claims expected in development yearj.
The CL predictor has a multiplicative structure, whereas the BF predictor is additive. Moreover
the two approaches represent two extreme positions of data reliance and expert opinion. These
dif-ferences impose restrictions on the applicability of the two approaches. In practice, one often
en-counters claims reserving triangles in which CL is appropriate for some accident years, whereas BF
is suitable for the others. As stated in Neuhaus (1992), this can be caused by data sparsity, e.g.,
tri-angles which have missing entries or zero cumulatives for recent accident years. Another reason is
illustrated in the following example. Suppose we are given a short-tailed insurance runoff triangle
DI. Assume the triangle behaves nicely to apply CL, except that in accident yeark, the diagonal value Ck,I−k is twice as large as fI−kCk,I−k−1. The latter value represents the expectation ofCk,I−k in the
CL model, given the information in the previous accounting year. In the following, we illustrate
dif-ferent possible underlying reasons for such an observationCk,I−k, which imply different appropriate
reserves.
• There was a legal change, allowing each insured of accident yearka doubled indemnification.
Then the predicted ultimateCkb ,J should also be doubled. In this case, CL is appropriate, as
b
CkC L,J scales proportionally toCk,I−k.
• The increaseCk,I−k−fI−kCk,I−k−1 is caused by a single event ("outlier"), which is not sys-tematic and is not expected to happen again. In this case, the predicted ultimate should be
increased by this difference, which is realised by BF.
• The increase is due to an exceptional early commutation, which causes a claim to be paid
earlier than usual. In this case, the predicted ultimate from the previous accounting year
should not be changed at all.
The reserving actuary often knows these underlying reasons for seemingly unusual behaviour in a
triangle. However, such information cannot be extracted neither fromDInor from prior estimates of
the ultimate claims. Therefore, actuarial judgement is necessary and appropriate when choosing the
reserving method.
Due to these reasons, it is common practice for reserving actuaries to switch accident year-wise
according toCbiC L,J orCbiB F,J. Such approaches are easily applied, as both CL and BF can be quickly
implemented in a spreadsheet. Commercial reserving software such as EMB ResQ™and Milliman
ReservePro®allow to set reserves as a weighted average between different reserving methodologies,
with weights different for each accident year.
The crucial assumptions behind CL as introduced in Mack (1993) are thatE[Ci,j|Ci,j−1]=fjCi,j−1 and var(Ci,j|Ci,j−1)=σ2jCi,j−1, where theσ
2
j are variance parameters. BF as introduced in Mack
(2008a) assumes increments to be independent, such thatE[Ci,j|Ci,j−1]=Ci,j−1+µiγjand var(Ci,j− Ci,j−1)=σ2jµi. As a consequence, we can calculate the conditional covariance of consecutive
incre-ments:
cov¡
Ci,j+2−Ci,j+1,Ci,j+1−Ci,j
¯ ¯Ci,j
¢
=
¡
fj+2−1¢σ2j+1Ci,j, for CL,
0, for BF.
This shows that the assumptions behind CL and BF are incompatible. However, these assumptions
are applied to the whole triangle when estimating parameters. Thus, when switching accident
year-wise between CL and BF, one cannot avoid applying two conflicting sets of assumptions on the
trian-gleDI. Due to the different nature of the two methodologies, there is no straight forward stochastic
representation of an accident year-wise weighting of CL and BF within a distribution-free model.
Reserving models that allow for a combination of CL-style and BF-style reserves have already been
studied in the literature. A first credibility approach is given by the Benktander-Hovinen method,
see Benktander (1976). In Neuhaus (1992) and Mack (2000) this credibility mixture of CL and BF is
further studied but they do not specify a concrete parametric model and do not deduce a parameter
estimation error. Alai (2010) provides a solution within a generalised linear model (GLM) framework.
Bayesian approaches are proposed in Verrall (2004) and Section 4.3.2 of Wüthrich and Merz (2008),
respectively.
The HCL method as proposed here allows to freely choose a weighting between a multiplicative
and an additive behaviour circumventing the problem of conflicting assumptions within a
distribution-free framework. The following three aspects highlight the main differences between HCL and the
models referenced above.
• In the existing literature the weight between CL and BF is generally interpreted as a
credibil-ity weight, which is chosen to minimise the prediction uncertainty. This approach neglects
special situations often encountered in practice as discussed above. On the contrary, the HCL
method sees this weight as a model selection parameter through which the reserving actuary
can determine whether a multiplicative structure as in CL or an additive structure as in BF is
more appropriate to predict the ultimate claim.
• The HCL method provides a stochastic model that allows parameter estimation, claims
calculation of uncertainty in the claims development result (CDR). Furthermore, an
easy-to-use implementation is available.
• HCL is based on a distribution-free framework. This enables us to avoid assumptions needed
in distribution based methods, such as positivity of incremental claims (in contrast to
over-dispersed Poisson and GLM) or distributional assumption (in contrast to Bayesian methods).
3. THEHCLMODEL
In this section we introduce the distribution-free Markov chain model defining the HCL model.
Let (γ0,γ1, . . . ,γJ) denote the incremental claims pattern and define the cumulative pattern asβj=
Pj
k=1γj, j =0, . . . ,J. We can then interpret the assumptions on the conditional expectation of an increment asE[Ci,j|Ci,j−1]=βj/βj−1Ci,j−1=(1+γj/βj−1)Ci,j−1for CL and asE[Ci,j|Ci,j−1]=Ci,j−1+
γjµifor BF. The main idea of HCL is to take a weighted average of the above expressions
E[Ci,j|Ci,j−1]=αi,j
µ
1+ γj
βj−1
¶
Ci,j−1+(1−αi,j)
¡
Ci,j−1+γjµi
¢
=Ci,j−1+γj
µ αi,j
Ci,j−1
βj−1 +
(1−αi,j)µi
¶
,
whereαi,jdenotes the weight for accident yeariand development yearj.
Model 3.1(HCL Model). There exist parameters
• γjandσ2j>0for j=0, . . . ,J , • βj>0for j=0, . . . ,J−1,
• αi,j∈[0, 1]for i=1, . . . ,I and j=1, . . . ,J , • µi>0for i=1, . . . ,I ,
such that(C1,j)j=0,...,J, . . . , (CI,j)j=0,...,Jare independent Markov processes with
• E[Ci,0]=γ0µiandE[Ci,j|Ci,j−1]=Ci,j−1+γj
³ αi,j
Ci,j−1
βj−1 +(1−αi,j)µi
´
for j=1, . . . ,J and
• var(Ci,0)=σ20µiandvar(Ci,j|Ci,j−1)=σ2jµifor j=1, . . . ,J .
¦
Note that although the parametersγjandβjare closely related, we assume them to be distinct and
independent. The reason for this assumption is to allow a derivation of claims predictors and MSEP
analogous to the deductions in Mack (1993) for CL. The precise relationship and the way to link the
estimates of theγjandβjwill be explained in Section 8.1.
The expectationE[Ci,j−Ci,j−1|Ci,j−1] of the incremental claimCi,j−Ci,j−1conditional on the pre-vious yearCi,j−1is equal to the product ofγjandmi,j, where
• γjis the fraction of the ultimate claim expected in development year j, • mi,j is a volume measure of accident yeari, defined bymi,0=µiand
mi,j=αi,j Ci,j−1
βj−1 +
¡
1−αi,j
¢
Themi,jrepresent a weighted average of
• µi, which is an a-priori estimate of the expected ultimate claimE[Ci,J] that is independent of
the claims process, and
• Ci,j−1/βj−1, which scales linearly in the previous observationCi,j−1. The proportionality
fac-torβj−1is an estimate of the ratio of cumulative claims up to yearj−1 to ultimate claims.
We have that, forαi,j close to 0,E[Ci,j|Ci,j−1] inherits similar features as in the BF model, whereas forαi,jclose to 1,E[Ci,j|Ci,j−1] is similar to the CL model. Indeed, we show below that in the extreme
casesαi,j=1 orαi,j=0 for allj=I−i+1, . . . ,Jwe have
E[Ci,J|DI]=E[Ci,J|Ci,I−i]=
Ci,I−i Q I−i<j≤J
βj−1+γj
βj−1 , ifαi,j=1,
Ci,I−i+µi P I−i<j≤Jγj
, ifαi,j=0.
In particular, forαi,j=1 we get a multiplicative structure as in the CL model whereas forαi,j=0 we
get an additive structure in correspondence to BF type models.
In Section 8.2 we give a thorough discussion about the choice of theαi,j’s which corresponds to the
choice of the model within the class of HCL models. Compared to the situation where only CL and
BF are admissible models, in the HCL model, the choice of theαi,j’s replaces the selection process
deciding between CL and BF.
The following theorems deduce the structure of mean and variance of theCi,j. We identify empty
sums with 0 and empty products with 1. To simplify notation, let
ξi,j=1+αi,j
γj
βj−1
for i=1, . . . ,I, j=1, . . . ,J.
Theξi,jtake a similar role as the age-to-age factors fj in the CL model in representing the influence
of the realisation ofCi,j−1on the expectation ofCi,j.
Theorem 3.2. For0≤k≤j≤J we have
E[Ci,j|Ci,k]=Ci,k
Y
k<m≤j
ξi,m+µi
X
k<n≤j
Ã
(1−αi,n)γn
Y
n<m≤j
ξi,m
!
. (3.1)
Proof. The casek= j is trivial. We assume by induction that (3.1) holds for some fixed j andk.
Then
E[Ci,j|Ci,k−1]=E
£
E[Ci,j|Ci,k]
¯ ¯Ci,k−1
¤
=E
"
Ci,k
Y
k<m≤j
ξi,m+µi
X
k<n≤j
Ã
(1−αi,n)γn
Y
n<m≤j
ξi,m
!¯ ¯ ¯ ¯ ¯
Ci,k−1
#
=¡
γk(1−αi,k)µi+Ci,k−1ξi,k¢
Y
k<m≤j
ξi,m+µi
X
k<n≤j
Ã
(1−αi,n)γn
Y
n<m≤j
ξi,m
!
=Ci,k−1
Y
k−1<m≤j
ξi,m+µi
X
k−1<n≤j
Ã
(1−αi,n)γn
Y
n<m≤j
ξi,m
!
Other possibilities to express the conditional expectationE[Ci,j|Ci,k] are
E[Ci,j|Ci,k]=Ci,k+
X
k<n≤j
γn
µ
(1−αi,n)µi+αi,nE
[Ci,n−1|Ci,k]
βn−1
¶
,
E[Ci,j|Ci,k]=Ci,k+
X
k<n≤j
γn
µ αi,n
Ci,k
βn−1+
(1−αi,n)µi
¶ Y
n<m≤j
ξi,m.
Theorem 3.3. For0≤k≤j≤J we have
var(Ci,j|Ci,k)=µi
X
k<n≤j
σ2
n
Y
n<m≤j
ξ2
i,m. (3.2)
Proof. The casek=j is trivial. We assume by induction that (3.2) holds for some j andk. By the
law of total variance we have
var(Ci,j|Ci,k−1)=E
£
var(Ci,j|Ci,k)
¯ ¯Ci,k−1
¤
+var¡
E[Ci,j|Ci,k]
¯ ¯Ci,k−1
¢
=E
" µi
X
k<n≤j
σ2
n
Y
n<m≤j
ξ2
i,m
¯ ¯ ¯ ¯ ¯
Ci,k−1
#
+var
Ã
Ci,k
Y
k<m≤j
ξi,m
¯ ¯ ¯ ¯ ¯
Ci,k−1
!
=µi
X
k<n≤j
σ2
n
Y
n<m≤j
ξ2
i,m+µiσ
2
k
Ã
Y
k<m≤j
ξi,j
!2
=µi
X
k−1<n≤j
σ2
n
Y
n<m≤k
ξ2
i,m.
We see that mean and variance are similar in structure to the CL model, in the sense that they can be
expressed in terms of some volume terms inflated by factorsξi,j(resembling age-to-age factors).
Sim-ilar results are also obtained in the model given in Schnieper (1991), see also Section 10.2 in Wüthrich
and Merz (2008).
Note that for fixed i, the variances var(Ci,j|Ci,j−1) all have the same proportionality factorµi.
Hence theσ2j can be directly compared, unlike in the CL model, where the proportionality factors are different.
4. PARAMETER ESTIMATION AND CLAIMS PREDICTION
In this section we define estimators forγj andσ2jbased on the information given inDI. These will
imply a predictor ofCi,J. For the deduction of estimators of the model parameters and other
quanti-ties, we will first assume that the parametersβj,µiandαi,jare known constants and hencemi,jcan
be calculated, conditionally givenCi,j−1. This assumption on theµi will be weakened in Section 7
and the estimation of theβjand an approach how to set theαi,jwill be illustrated in Section 8.
First, we defineΓi,0=Ci,0/µiand
Γi,j=
Ci,j−Ci,j−1
mi,j
for j>0. (4.1)
Moreover, we define the weightsωi,jwhereωi,0=µiand
ωi,j=
σ2
j
var(Γi,j|Ci,j−1)= mi2,j
µi
In case there arei and j such thatCi,j−1<0, we assume thatαi,j is small enough (e.g.αi,j =0) to
ensuremi,j>0.
In the following, conditioning onCi,−1denotes conditioning on the empty set, i.e.E[Γi,0|Ci,−1]=
E[Γi,0]. Note thatE[Γi,j|Ci,j−1]=γjand that theωi,j are independent ofσ2j. Using theΓi,j, we define
the following linear estimators ofγjandσ2jbased onDI. Forj=0, . . . ,J, let
b γI
j=
1
ΩI
j I−j
X
i=1
ωi,jΓi,j, (4.2)
b σ2
j=
1
I−j−1
ÃI−j X
i=1
ωi,j(Γi,j−γb
I j)
2
!
,
whereΩIj =PI−j
i=1ωi,j. The superscriptI indicates the fact thatγb
I
j is based on the data inDI. Later,
we will define an estimatorγb
I+1
j based on the data available in accounting yearI+1. IfJ =I−1,
there is only one observation in the last development year and the denominator in σb
2
J becomes
zero. In that case, we propose to estimateσ2J with the extrapolation used by Mack (1993), i.e.σb2J=
min{σb
2
J−2,σb
2
J−1,σb
4
J−1/σb
2
J−2}.
The following theorem shows several nice properties of the estimatorsγb
I jandσb
2
j, which are
analo-gous to the characteristics of the estimatorsfbjof the age-to-age factorsfjin the CL model.
Theorem 4.1. We have that
(1) E[γb
I j]=γj.
(2) cov(γb
I j,γb
I
k)=0for j6=k.
(3) Among all linear estimators,γbIjhas minimal variance andvar(γbIj|Bj−1)=σ2j/ΩIj. (4) E[σb
2
j]=σ2j.
Proof. (1) For j =0, the claim immediately follows from the definition of Model 3.1. For j ≥1
unbiasedness ofγb
I
jdirectly follows fromE[Γi,j]=E[E[Γi,j|Ci,j−1]]=γjand 1/Ω I j
PI−j
i=1ωi,j=1.
(2) LetBk={Ci,j:Ci,j∈DI,j≤k} and note thatB−1= ;. Suppose j<kwithout loss of generality. Asγb
I
k isBk-measurable, we get
cov(γbIj,γbIk)=E[γbIjγbkI]−E[γbIj]E[γbkI]=EhE[γbIjγbIk|Bk−1]
i
−γjγk
=Ehγb
I jE[γb
I k|Bk−1]
i
−γjγk=E[γb
I
j]γk−γjγk=0.
(3) This is a direct consequence of Lemma 3.4 in Wüthrich and Merz (2008).
(4) Note thatB−1= ;. ForE[σb
2
j|Bj−1], we obtain
Ehσb
2
j
¯ ¯ ¯Bj−1
i
= 1 I−j−1
I−j
X
i=1
ωi,jE
· ³
Γi,j−γb
I j
´2¯¯ ¯ ¯Bj−1
¸
For the conditional expectations in the sum above, we have
E
·³
Γi,j−γb
I j
´2¯¯ ¯ ¯Bj−1
¸
=var¡
Γi,j
¯ ¯Bj−1
¢
−2 cov³Γi,j,γb
I j
¯ ¯ ¯Bj−1
´
+var³γb
I j
¯ ¯ ¯Bj−1
´
= σ
2
j
ωi,j−
σ2
j
ΩI
j
.
Summing up yieldsE[σb2j|Bj−1]=σ2j. From the tower property for conditional expectations, we get
E[σb
2
j]=E[E[σb
2
j|Bj−1]]=σ2j.
In order to predict the ultimate claimCi,J, we replace all theγjinE[Ci,J|DI]=E[Ci,J|Ci,I−i] (see (3.1))
with the correspondingγb
I j:
b
CiI,J=Ci,I−i
Y
I−i<m≤J
b ξI
i,m+µi
X
I−i<n≤J
Ã
(1−αi,n)γb
I n
Y
n<m≤J
b ξI
i,m
!
, (4.3)
whereξbIi,j=1+αi,jγb
I j/βj−1.
In the extreme case whereαi,j=0, forj=I−i+1, . . . ,J, we have
b
CiI,J=Ci,I−i+µi
X
I−i<j≤J
b γI
j. (4.4)
This resembles the expression in the pure BF case. If furthermoreαi,j=0 for alliandj, the estimates
b γI
jare equal to the raw (unsmoothed) estimates deduced in Mack (2008a), hence also the predictions
b
CiI,Jcoincide.
If converselyαi,j=1 forj=I−i+1, . . . ,J, we have
b
CiI,J=Ci,I−i
Y
I−i<j≤J
βj−1+γb
I j
βj−1
. (4.5)
The estimatesγb
I
jare not equal to the CL estimatesγb
C L
j ofγj, even ifαi,j=1 for alliand j. However,
numerical examples show that for triangles with few outliers, they are very close and hence also the
predictionsCbiI,Jare close to the CL predictionsCbC Li,J.
To simplify notation we rewrite (3.1) and (4.3) as
E[Ci,J|DI]=
X
I−i≤n≤J
κI i,n
Y
n<m≤J
ξi,m and Cb
I i,J=
X
I−i≤n≤J
b κI
i,n
Y
n<m≤J
b ξI
i,m,
where
κI i,n=
Ci,I−i, forn=I−i,
µi(1−αi,n)γn, forn>I−i,
b κI
i,n=
Ci,I−i, forn=I−i,
µi(1−αi,n)γb
I
n, forn>I−i.
Of course, the properties ofγb
I
jshown in Theorem 4.1 directly translate to analogous properties ofξbiI,j
5. ASSESSING THE PREDICTION ERROR
In this section, we provide a measure of uncertainty of the predicted aggregate ultimate claims
PI
i=1CbiI,J. This is important, as we do not only want to determine the reserves but also measure their
precision in predicting the outstanding claims. A popular meassure of uncertianty among actuaries
is the so called conditional mean square error of prediction (MSEP), defined as
msep
à I X
i=1
b
CiI,J
!
=E
" Ã I X
i=1
³
b
CiI,J−Ci,J
´ !2¯
¯ ¯ ¯ ¯
DI
#
,
which can be decomposed into process variance and parameter estimation error (PEE)
msep
à I X
i=1
b
CiI,J
!
=var
à I
X
i=1
Ci,J
¯ ¯ ¯ ¯ ¯
DI
!
+
à I X
i=1
³
b
CiI,J−E[Ci,J|DI]
´ !2
.
Note that the above expression corresponds to the MSEP of the aggregate reserves as, conditionally
onDI, the reserves are a deterministic shift ofPIi=1CbiI,J. An analogous decomposition holds for the
MSEP of one accident year, i.e., msep(CbiI,J). Fori≤I−J, we haveCi,J∈DI. Hence, we can omit the i≤I−J terms in the sum above.
The process variance can be decoupled into var(PI
i=1Ci,J|DI)=
PI
i=1var(Ci,J|DI) as accident years are assumed to be independent. For estimation, we replace the unknown quantities in (3.2) by their
respectiveDIestimates and get
c
var¡Ci,J|DI
¢
=µi
X
I−i<n≤J
b σ2
n
Y
n<m≤J
³
b ξI
i,m
´2
. (5.1)
In order to estimate the PEE, we use a similar approach as used in Theorem 3 in Mack (1993). That
is, we linearise the PEE and decompose it into a sum of expectations of which most drop out due to
pairwise independence under a suitable change of conditioning, see Appendix A. To that end, fori≥
I−J+1 andn=I−i, . . . ,J, we defineΨi,n=κIi,nQn<m≤Jξi,m, itsDI estimatorΨbiI,n=κbIi,n Q
n<m≤JξbIi,m
and
ψi,n,k=
(bκIi,n−κiI,n)Q
n<m≤Jξi,m, fork=n,
b κI
i,n
³ Q
n<m<kξbIi,m ´ ³
b ξI
i,k−ξi,k
´ ¡Q
k<m≤Jξi,m¢, forn<k≤J.
Note that using the above defined variables, we can decomposeE[Ci,J|DI] andCbiI,J, asE[Ci,J|DI]=
PJ
n=I−iΨi,n,CbiI,J= PJ
n=I−iΨbiI,n andΨbIi,n−Ψi,n=PJk=nψi,n,k. We can now rewrite the aggregate PEE
as
peeΣ=
à I X
i=1
³
b
CiI,J−E[Ci,J|DI]
´ !2
= X
I−J+1≤i1≤I I−i1≤n1≤J
X
I−J+1≤i2≤I I−i2≤n2≤J
³
b
ΨI
i1,n1−Ψi1,n1
´ ³
b
ΨI
i2,n2−Ψi2,n2
´
= X
I−J+1≤i1≤I I−i1≤n1≤J
n1≤k1≤J
X
I−J+1≤i2≤I I−i2≤n2≤J
n2≤k2≤J
ψi1,n1,k1ψi2,n2,k2.
In Appendix A we show that by a suitable change of conditioning and replacing all unknown
quan-tities with their estimates we get the estimators
á ψi1,n1,k1ψi2,n2,k2=
0, ifk16=k2,
b
ΨI
i1,n1Ψb I
i2,n2bi1,n1,kbi2,n2,k
b
σ2 k ΩI k
, ifk1=k2=k,
where
bi,n,k=
αi,k βk−1ξbI
i,k
, ifk>n,
1
b
γI k
, ifk=n,n>I−i,
0, ifk=n=I−i.
This leads to the following estimates of the one-year and aggregate PEE
d
peei= X I−i≤n1≤J
n1≤k1≤J
X
I−i≤n2≤J n2≤k2≤J
á ψi,n1,k1ψi,n2,k2,
d
peeΣ= X I−J+1≤i1≤I
I−i1≤n1≤J n1≤k1≤J
X
I−J+1≤i2≤I I−i2≤n2≤J
n2≤k2≤J
á ψi1,n1,k1ψi2,n2,k2.
We use the direct and well known approach of Mack (1993) in order to get an estimation of the PEE.
Note that there are alternative procedures to estimate the PEE such as e.g. the conditional approach
given in Buchwalder et al. (2006), which however does not lead to significantly different estimates.
The MSEP is finally estimated bymsep( CbiI,J)=var(c Ci,J|DI)+peediand
msep
à I X
i=1
b
CiI,J
!
= I
X
i=1
c
var(Ci,J|DI)+peedΣ.
6. CLAIMS DEVELOPMENT RESULT
After each year, i.e. at the end of an accounting year, we receive additional information on the
claims runoff. This means that the available dataDI is augmented with an additional diagonal to
DI+1={Ci,j: 1≤i≤I, 0≤j≤J,i+j≤I+1}. According to this new information, we can update our
prediction of the ultimate claimCi,J. New solvency regimes (e.g., Solvency II and the Swiss Solvency
Test) require protection for each year (one-year perspective) against the risk of possible changes in
subsequent ultimate claim predictions. In this section, we provide estimates of the uncertainty in
those changes.
Similar to (4.3) we defineCbiI,+J1 fori≥I−J+1 which is an estimate ofE[Ci,J|DI+1] at timeI+1,
based onDI+1:
b
CiI,+J1=Ci,I−i+1
Y
I−i+1<m≤J
b ξI+1
i,m+µi
X
I−i+1<n≤J
Ã
(1−αi,n)γb
I+1
n
Y
n<m≤J
b ξI+1
i,m
!
Theγb
I+1
j are estimates ofγjbased on the information inDI+1, asγb
I+1 0 =γb
I
0and forj≥1
b γI+1
j =
1 ΩI+1
j
I−j+1
X
i=1
ωi,jΓi,j=
ΩI
j ΩI+1
j
b γI
j+
ωI−j+1,j ΩI+1
j
ΓI−j+1,j,
whereΩI+j 1=PI−j+1
i=1 ωi,j=Ω
I
j+ωI−j+1,jandξbiI,+j1=1+αi,jγb
I+1
j /βj−1. Note thatΩI+
1
j isDI-measurable.
The CDR viewed from timeI denotes the difference in estimated ultimate claims for subsequent
accounting years
C DRΣ= I
X
i=1
³
b
CiI,J−CbiI+,J1 ´
,
and for a single accident yeari we haveC DRi=CbiI,J−CbiI,+J1. Note that for givenDI andi≤I−J, we
haveC DRi=0, whereas fori≥I−J+1,C DRiis stochastic viewed from timeI.
Other publications such as Wüthrich et al. (2009) define theC DRasPI
i=1(E[Ci,J|DI]−E[Ci,J|DI+1]), our definition is an estimate thereof. For a detailed interpretation of the CDR in a Bayesian setting we
refer to Bühlmann et al. (2009).
As we consider bothCbiI,JandCbiI,+J1to be conditional best estimates, we estimateC DRi=0. We want
to assess the uncertainty in the CDR, measured by its second moment,C DRUΣ=E[(C DRΣ)2|DI] as
well as the accident year-wise uncertaintyC DRUi=E[(C DRi)2|DI]. To do so, we define
b κI+1
i,n =
Ci,I−i+1, forn=I−i+1,
µi(1−αi,n)γb
I+1
n , forn>I−i+1,
b
ΨI+1
i,n =bκ
I+1
i,n
Y
n<m≤J
b ξI+1
i,m.
With the above formulas and (6.1) we getCbiI+,J1= P
I−i+1≤n≤JΨbI+i,n1. For the CDR we then have
C DRi= X I−i≤n≤J
b
ΨI
i,n−
X
I−i+1≤n≤J
b
ΨI+1
i,n =ΨbIi,I−i+ X
I−i+1≤n≤J
³
b
ΨI
i,n−ΨbiI+,n1 ´
.
We now rewriteC DRi=PI−i+1≤n≤JΘi,nandΘi,n=PkJ=nθi,n,k, where
Θi,n=
b
ΨI
i,n−1+ΨbIi,n−ΨbiI,+n1, forn=I−i+1,
b
ΨI
i,n−ΨbiI+,n1, forn>I−i+1,
θi,n,k=
ei,n,kQn<m≤JξbIi,m, fork=n,
b κI+1
i,n
Q
n<m<kξbiI,+m1ei,n,kQk<m≤JξbiI,m, fork>n,
with
ei,n,k=
αi,k/βk−1(γb
I k−γb
I+1
k ), fork>n,
µi(1−αi,n)γb
I
n−µi(1−αi,n)γb
I+1
n , fork=n,n>I−i+1, Ci,I−iξbIi,n
−1+µi(1−αi,n)γb
I
TheC DRUΣexpressed in terms ofθi,n,k is given by
C DRUΣ=E
" Ã
X
I−J+1≤i≤I
X
I−i+1≤n≤J
X
n≤k≤J
θi,n,k
!2¯ ¯ ¯ ¯ ¯
DI
#
= X
I−J+1≤i1≤I I−i1+1≤n1≤J
n1≤k1≤J
X
I−J+1≤i2≤I I−i2+1≤n2≤J
n2≤k2≤J
E£
θi1,n1,k1θi2,n2,k2
¯ ¯DI
¤
,
where thei≤I−Jterms have been omitted sinceC DRi=0 in that case.
In Appendix B, we prove by the help of approximations and a suitable change of conditioning, that
we can estimateE[θi1,n1,k1θi2,n2,k2|DI] with
b
E£
θi1,n1,k1θi2,n2,k2|DI
¤
=
0, fork16=k2,
e
ΨI+1
i1,n1gi1,n1,k1Ψe I+1
i2,n2gi2,n2,k2µI−k1+1σb
2
k1, fork1=k2,
where
e
ΨI+1
i,n =
b
ΨI
i,n, forn>I−i+1,
b
ΨI
i,n−1+ΨbIi,n, forn=I−i+1,
gi,n,k=
1
b
ξI i,k
αi,k βk−1
ωI−k+1,k ΩI+1
k
1
mI−k+1,k, fork>n,
1
b
γI n
ωI−k+1,k ΩI+1
k
1
mI−k+1,k, fork=n,n>I−i+1,
1
Ci,I−iξbIi,I−i+1+µi(1−αi,I−i+1)bγII−i+1
, fork=n=I−i+1.
The terms withk=n=I−i+1 cover the variability of the next diagonal {Ci,I−I+1:i=I−J+1, . . . ,I}, i.e., can be seen as the process variance component of the CDR. The other terms cover the variability
of theγjestimates.
With those approximations we define estimates ofC DRUΣandC DRUiby
à
C DRUΣ= X I−J+1≤i1≤I I−i1+1≤n1≤J
n1≤k1≤J
X
I−J+1≤i2≤I I−i2+1≤n2≤J
n2≤k2≤J
b
E£
θi1,n1,k1θi2,n2,k2|DI
¤
,
à
C DRUi=
X
I−i+1≤n1≤J n1≤k1≤J
X
I−i+1≤n2≤J n2≤k2≤J
b
E£
θi,n1,k1θi,n2,k2|DI
¤
.
Whenαi,j =0 for alli and j, the HCLγj estimates as defined in (4.2) are equivalent to the raw
estimatesγbj as defined in Mack (2008a) and Saluz et al. (2012). Hence, our formula can be used to
estimate theC DRUof the pure BF method. To our awareness, the only other work deducing such an
7. ESTIMATING THEµi AND INCORPORATING THEIR UNCERTAINTY
The HCL method requires prior estimatesµiof the expected ultimate claimsE[Ci,J]. In most cases,
the product of (planned) premium volume and expected loss ratio is considered as a viable estimate.
Since the prior estimatesµifor the HCL method are of the same nature as the prior estimates for the
BF method, the same policy guidelines as in the BF method should apply. Namely, theµi should be
independent ofDI.
The source of information that determines theµi often comprises inaccuracy. Therefore, the PEE
should take this additional uncertainty into account. In the BF model, Mack (2008a) embeds theµi
in the distribution-free framework, by assuming some variance var(µi). A similar approach is used by
Saluz et al. (2012) in a distributional framework. In order to estimate the PEE, Mack (2008a) assumes
b
γj and µi to be independent. As the µi appear in the quotient ofΓi,j, we avoid this assumption
and present a simple scenario based approach instead, which naturally captures the dependence
between theγbj andµi. Of course, the scenarios ofµi can be fitted to any assumption on mean and
variance. We consider (µ1, . . . ,µI) to be a realisation of a random vectorµtaking values in a finite set
of scenarios {µ1, . . . ,µS} and reformulate Model 3.1 as being conditional on (µ1, . . . ,µI)=µs for some s∈{1, . . . ,S}. The equations deduced in the previous sections continue to hold under the following
model (but must be thought of as being conditional on theµi).
Note that we could also assumeµto have a continuous distribution. Then the calculation would
require simulation.
Model 7.1(HCL Model with uncertainµi). There exist parameters
• γjandσ2j>0for j=0, . . . ,J , • βj>0, for j=0, . . . ,J−1,
• αi,j∈[0, 1]for i=1, . . . ,I and j=1, . . . ,J ,
• and a discrete random vectorµ∈(0,∞)I satisfyingP[µ=µs]=ps>0, s∈{1, 2, . . . ,S}forµs∈
(0,∞)IandPS
s=1ps=1.
Conditionally onµ=(µ1, . . . ,µI), the (C1,j)j=0,...,J, . . . , (CI,j)j=0,...,J are independent Markov processes with
• E[Ci,0|µ]=γ0µiandE[Ci,j|Ci,j−1,µ]=Ci,j−1+γj
³ αi,j
Ci,j−1
βj−1 +(1−αi,j)µi
´
for j=1, . . . ,J ;
• var(Ci,0|µ)=σ20µiandvar(Ci,j|Ci,j−1,µ)=σ2jµifor j=1, . . . ,J .
¦
We can now run the estimation procedure for each possible realisation of the random vectorµ.
For fixedµi, we are in the framework of Model 3.1, which allows us to calculate estimates of
ulti-mate claims etc. For the caseµ=µs, these estimates are denoted with a superscripts, such asCb
I,s i,J,
c
vars(Ci,J|DI),peed
s
ΣandC DRUà
We calculate unconditional estimates by averaging out over the different possible outcomes ofµ,
using weights given by the probabilitiesps.
• BecauseE[Ci,J]=E[E[Ci,J|µ]], we estimate
b
CiI,J= S
X
s=1
b
CiI,,Jµsps.
• By the law of total variance, var(Ci,J|DI)=E[var(Ci,J|DI,µ)]+var(E[Ci,J|DI,µ]), thus
c
var(Ci,J|DI)= S
X
s=1
c
vars(Ci,J|DI)ps+ S
X
s=1
³
b
CiI,,Js−Cb
I i,J
´2
ps.
• For the PEE we have thatCb
I,s
i,J is close toE[Ci,J|DI,µs], which allows to approximatepeedi= PS
s=1peed
s
ipsandpeedΣ= PS
s=1peed
s
Σps, respectively.
• AsE[C DRi|µ]≈0, the uncertainty in the CDR can be estimated byC DRUià = PS
s=1C DRUà
s ips
andC DRUàΣ= PS
s=1C DRUà
s
Σps, respectively.
Note that the distribution of theµi can be fitted to any assumption on mean and variance. To
determine an adequate amount of uncertainty in theµi, we suggest the following possible sources of
information:
• Experts can determine scenariosµsbased on socio-economic scenarios. Using expert judge-ment unavoidably involves psychological effects, due to the expert’s subjectivity. These effects
should be taken care of, see for instance Meyer and Booker (2001).
• Literature suggests to use a coefficient of variation (CoV(µi)=pvar(µi)/E[µi]) between 5%
and 10%, see Section 6.6.2 in Wüthrich and Merz (2008). This is also based on regulatory
guidelines, see FINMA (2006).
• Saluz et al. (2012) suggest a data based approach in order to determine the uncertainty in the
µi. The two case studies considered in Saluz et al. (2012) induce a CoV(µi) between 5% and
10%.
• Regulators may enforce socio-economic scenarios, see FINMA (2006).
• Industry data, such as provided by theSchedule Pdatabase in the US, can be used to
deter-mine a prudent amount of uncertainty in theµi, see NAIC (2010).
Note that we assume that the scenarios implied byµdo not change over time.
8. COMMENTS FOR PRACTITIONERS
In Model 3.1 the parametersβj andαi,jare assumed to be fixed and given constants. In this
sec-tion, we discuss practical approaches to set theβjandαi,j.
8.1. Setting theγj’s andβj’s. It might be surprising that in Model 3.1,γjandβj are introduced as
distinct parameters although their role is very similar in describing the runoff pattern of the claims.
(1993). In the following we consider the relation between these two parameters. Although theβjare
regarded in Model 3.1 as exogenous estimates, we propose an approach for estimation based onDI.
For given prior estimatesµiand a known runoff patternγj, we would expect a realistic stochastic
claims reserving method to satisfy
E[Ci,j]=¡γ0+ · · · +γj¢µi, for all 0≤j≤J, (8.1)
E[Ci,J]=µi. (8.2)
I.e. theγjandµiprovide best estimates for the runoff process.
The following lemma yields a relation between theβjandγjsuch that (8.1) and (8.2) hold.
Lemma 8.1. Supposeγ0+ · · · +γJ=1andγ0+ · · · +γj=βjfor all j<J . Then(8.1)and(8.2)hold.
Proof.Fix somei. Forj=0,E[Ci,0]=γ0µiholds by definition. We proceed by induction onj.
E£
Ci,j¤=E£E£Ci,j
¯ ¯Ci,j−1
¤¤
=E£Ci,j−1ξi,j+γj¡1−αi,j¢µi¤
=¡
γ0+ · · · +γj−1¢µi
µ
1+αi,j
γj
βj−1
¶
+γj¡1−αi,j¢µi
=¡
γ0+ · · · +γj¢µi+µiγjαi,j
µγ
0+ · · · +γj−1
βj−1 −
1
¶
.
The second summand above is equal to zero ifγ0+ · · · +γj−1=βj−1.
Due to Lemma 8.1, we propose the following adaptions to the HCL method to be used in practice.
• For initial estimates of theγbIj, use (4.2). Then, smooth and/or rescale theγbIjsuch thatPJ
j=0γb
I j=
1 holds. Smoothing techniques are described for instance in Mack (2008a) and Section 11.2 in
Wüthrich and Merz (2008). This adaption may also comprise the inclusion of tail factors, see
for instance Mack (1999).
• Fix theβjsuch thatβj=Pk=j 0γb
I
kholds. This can be achieved by running the whole estimation
procedure several times and iteratively pluggingPj
k=0γb
I
k intoβj. In our numerical examples,
this took no more than 5 iterations to converge to a precision of 10−10.
This approach attaches additional data dependence and uncertainty to theβj. However, we
as-sume that these additional sources of errors are negligible compared to the error terms covered in
Section 5, 6 and 7. One rationale to do so is that the estimatedβjis most stable for late development
years. For smallj, when the relative error of theβjis potentially high, but in the same time theαi,jare
in general rather small when chosen as proposed in Section 8.2, which diminishes the final impact.
Indeed, numerical tests confirm this statement.
8.2. Setting theαi,j’s. The choice of theαi,jreflect the actuary’s assessment of the behaviour of the
claims process. In the same time, theαi,jcontrol the reliance of the HCL method on the dataDI. We
• Fori+j≤I, theαi,j control the data reliance in theparameter estimation. Eachαi,j gives a
measure of the predictive power ofCi,j−1/βj−1as an estimator of the ultimate claimCi,J. • Fori+j>I, theαi,j control the predictive power of the diagonal valueCi,I−i of the ultimate
claimCi,J. In other words, how sensitive thepredictionCbiI,J is to deviations ofCi,I−i.
Recognising the two distinct manners of influence, we make the following simple proposals to set
theαi,j.
• Fori+j ≤I, we suppose the runoff to become more stable for higher development years,
i.e.αi,j increases to 1 for increasing j. A simple and practical approach that satisfies this
requirement, is to set
αi,j=βj−1 for all i+j≤I.
We highlight that theαi,j as given above are development year-wise constant. Note that this
approach is tail-adaptive, in the sense that for triangles with a longer runoff, the weight of the
µiin the estimation of theγb
I
jfor initial years will be higher than in triangles with a shorter tail. • On the other hand, fori+j>I, we propose to set theαi,jaccident year-wise constant as
αi,j=αei for all i+j>I,
whereαei∈[0, 1] forI−J<i≤I have to be determined by the actuary. In that way, the
data-reliance of the reserves estimates can be set separately for each accident year. In order to
recover a behaviour analogous to Benktander (1976), one has to setαei=βI−i.
This approach is illustrated in Figure 2. Note that it reduces the number ofα-parameters that have to
be determined by the actuary fromI×J toJ. The actuary can choose theαeiaccording to his belief,
whether a multiplicative or an additive structure is more appropriate for a particular accident year.
Possible reasons to come to such a choice are given in the example in Section 2. However, the actuary
is free to choose otherαi,jappropriate for the claims triangle at hand.
One drawback affecting the robustness of the CL method is that theCi,jappear in the denominator
of the parameter estimators. These estimators can strongly deviate if someCi,j are small, zero, or
even negative for early development years in long tailed claims triangles, see Busse et al. (2010). In
the HCL model, theCi,j−1also appear in the denominator of theΓi,j, but for early development years
their weight (namelyαi,j) is generally low for long tailed triangles, if theαi,jare set as outlined above.
Therefore, the problem with unstable parameter estimators does not occur.
Considering the case where only CL and BF models are admissible choices, a "super-expert" chooses
one and declares it as the true model. In the HCL model, the selection of theαi,jcorresponds to the
model choice. In the deduction of the estimates of msep andC DRU, theαi,jare assumed to be
con-stant and known. If model risk is to be taken into account, the sensitivity of the results with respect
dev
elo
p
ment
y
ear
-w
ise
c
o
n
st
ant
J
I 1
0
αi,jused for prediction
αi,
j
u
sed
for
estimat
ion
j . . .
. . .
i
.. . .. .
accident year-wise constant
FIGURE2. A practical approach to set theαi,j: Fori+j≤I development year-wise
constant and increasing, fori+j >I accident year-wise constant according to the
actuary’s chosenαei.
the choice of theαi,jshould be motivated by a minimizing argument for the reserves or a measure of
uncertainty, as it is done in credibility approaches, see Neuhaus (1992).
For sufficiently smallxj, we haveQ
(1+xj)−1≈Pxj. Applying these approximations to (4.3), we
get for the reserves of a single accident yearithat
b
CiI,J−Ci,I−i≈αei Ã
Ci,I−i
Y
I−i<j≤J
βj−1+γb
I j
βj−1 −
Ci,I−i
!
+(1−αei) Ã
µi
X
I−i<j≤J
b γI
j
!
. (8.3)
Recalling (4.4) and (4.5), we see that the reservesCbiI,J−Ci,I−i are approximately the weighted average
between the completely-data-reliant approach and the completely-expert-reliant approach.
Equation 8.3 also shows that the HCL claims estimate is very similar to the Benktander-Hovinen
estimate, except for a different interpretation of the weights, see Mack (2000) or Section 4.1.1 in
Wüthrich and Merz (2008).
9. CASE STUDY
In this section, we present a case study illustrating the use of the HCL method applied to the
tri-angle given in Table 1. We analyse the same loss tritri-angle andµias in Mack (2006) and Mack (2008a).
However, our results cannot be directly compared to the results therein, as Mack (2008a) manually
increases theγj for the last development years and also adds a tail factor. The triangle represents
claims from a general liability excess line of business, which is very long tailed with more than 80%
of the claims payments being expected in the develpment yearsj>2. In general CL type methods do
not work robustly for data coming from business subject to strong volatility and long tailed
develop-ment. Therefore, we assume that the classical CL method is not an adequate model for the triangle
considered in this case study. The results of the CL method applied to this dataset is given in the Excel
i \ j 0 1 2 3 4 5 6 7 8 9 1 0 11 12 µi e αi 1 23 4 4’8 77 11 ’12 6 14’ 65 6 2 1’ 195 2 3’9 32 26 ’4 78 2 8’ 293 28 ’62 8 2 8’7 38 28’ 75 6 28 ’7 82 2 8’ 781 32 ’2 99. 9 2 1’ 99 4 6 ’9 30 1 1’ 755 1 7’9 35 25 ’5 94 27 ’54 5 32 ’65 5 3 3’2 66 34’ 04 2 34 ’45 1 3 4’4 99 35 ’82 6 40 ’2 79. 1 1 3 -7 5 3’1 33 10’ 98 6 1 8’ 113 2 3’4 73 2 7’3 49 30 ’77 5 3 2’2 15 33 ’49 8 3 3’5 65 35’ 18 1 40 ’6 34. 6 1 4 23 6 2’4 38 6 ’56 3 11’ 56 6 1 5’ 755 2 4’8 19 27 ’0 21 2 9’ 085 32 ’32 9 3 3’5 08 39 ’6 04. 3 1 5 97 6 5’6 95 15 ’09 2 28’ 34 5 3 4’ 451 3 9’4 26 42 ’4 75 4 7’ 194 49 ’90 9 58 ’4 40. 6 1 6 -7 30 2 ’62 3 15 ’5 27 26 ’16 9 42 ’66 0 51’ 54 6 5 8’ 774 67 ’28 6 81 ’3 46. 9 1 7 53 9 5’7 77 20 ’67 8 45’ 54 3 6 5’ 817 8 3’5 86 116 ’5 20 16 3’ 258 .7 1 8 72 5 1 5’6 25 50 ’30 1 93’ 89 6 14 6’ 517 17 3’9 97 26 8’ 150 .6 0 9 31 2 6’7 54 50 ’35 0 1 39’ 05 2 17 7’ 864 33 1’ 893 .1 0 10 2’ 98 8 12 ’9 09 3 3’ 266 6 7’8 51 19 3’ 519 .8 0 11 26 0 7’4 41 29 ’64 3 16 9’ 559 .7 0 12 99 4 4’0 43 15 7’ 381 .6 0 13 2’ 41 1 15 6’ 150 .7 0 bγ j ≈ 0. 7% 4 .8% 1 3.9 % 2 0.8 % 16 .6 % 11 .8% 13 .9% 7.6 % 4. 6% 1 .4% 1.7 % 2 .2% 0 .0 % T A B L E 1 . A tr ian gle o f cumu lat iv e p ai d losses Ci , j an d pr ior u ltimat es µi fr om a gen er al liab ility ex c ess lin e of b u sin e ss . This tr iang le is the same as u sed in M ack (2 006 ) and M ac k (20 08a). O n th e rig ht ,w e g iv e th e eαi u se d in model HCL 1 an d HC L 2 . As an app ro x ima te fo r the bγj of a ll models , w e g iv e the bγj of H CL 1. HCL 1 ( S = 1, e αi ) H CL 2 ( S = 3, e αi ) HCL 3 ( S = 1,
αi,
j = 0) HC L 4 ( S = 1,
αi,
j = 1) M ac k BF i reser v es p msep p C D R U reser v es p msep p C D R U reser v es p msep p C D R U reser v es p msep p C D R U reser v es p msep 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 -1 1’2 94 8 64 -1 1’2 97 866 -1 1’2 73 8 49 -2 1’3 92 930 -1 1’2 73 3 79 9 1’ 70 8 89 0 79 9 1 ’71 1 89 1 84 2 1’ 68 4 87 5 95 6 1 ’82 2 93 4 84 2 1’ 68 4 4 1’ 38 5 1’9 84 922 1’ 38 4 1’9 87 92 2 1’ 47 6 1’9 47 886 1’ 66 0 2’ 097 94 7 1’ 47 6 1’9 47 5 2’ 82 0 2’7 70 652 2’ 81 9 2’7 76 65 2 2’ 93 0 2’6 86 618 3’ 38 8 2’ 935 68 3 2’ 93 0 2’6 86 6 7’ 44 0 4’1 78 1’ 786 7’ 43 6 4’1 94 1’ 79 0 7’ 66 1 3’9 34 1’ 593 8’ 99 0 4’ 503 1 ’97 0 7’ 66 1 3’9 34 7 24 ’8 06 8 ’29 1 3 ’6 47 24 ’7 92 8 ’35 6 3 ’6 61 27 ’2 82 7 ’89 0 3 ’14 6 30 ’2 97 9 ’27 1 4 ’2 75 27 ’2 82 7 ’89 0 8 84 ’3 55 18 ’64 6 10 ’1 38 84 ’4 14 20 ’05 2 10 ’1 67 81 ’8 21 16 ’39 0 8 ’95 5 98 ’7 94 24 ’30 8 14 ’8 15 81 ’8 21 13 ’63 8 9 14 3’ 623 2 3’8 93 7’ 368 14 3’ 686 2 6’6 54 7’ 419 14 0’ 449 2 0’9 05 6’ 484 17 1’ 007 3 4’7 93 1 5’ 524 14 0’ 449 1 6’4 89 10 11 5’ 799 1 7’6 50 7’ 086 11 5’ 823 1 9’7 46 7’ 165 11 4’ 154 1 5’8 44 6’ 855 13 1’ 612 3 2’4 04 2 0’ 859 11 4’ 154 1 3’3 29 11 13 6’ 677 1 8’5 98 8’ 704 13 6’ 685 2 0’9 15 8’ 800 13 5’ 915 1 7’0 81 8’ 484 16 6’ 073 5 5’1 13 4 3’ 260 13 5’ 915 1 4’6 93 12 14 8’ 719 1 8’1 73 3’ 819 14 8’ 720 2 0’6 73 3’ 911 14 8’ 522 1 6’8 73 4’ 163 84 ’9 30 89 ’38 4 73 ’5 85 14 8’ 522 1 4’6 80 13 15 5’ 088 1 8’5 40 3’ 905 15 5’ 089 2 1’1 06 3’ 916 15 5’ 060 1 7’2 99 3’ 970 27 0’ 331 17 3’3 32 13 0’ 123 15 5’ 060 1 5’1 00 P 82 1’ 509 8 9’2 53 1 8’ 226 82 1’ 644 10 6’5 48 1 8’ 365 81 6’ 112 7 9’1 46 1 7’ 011 96 8’ 036 23 6’1 97 15 8’ 553 81 6’ 112 6 1’9 22 T A B L E 2 . R eser v es est im ates ,squ ar e root o f M S EP an d squar e ro ot of th e u n cer taint y in th e CDR for tr iang le giv en in T ab le 1. W e u se th e models HCL 1 (one sc e n ar io ,
αi,
j thr ough e αi ), HC L 2 (th ree sc enar ios ,
αi,
j th rough eαi ), HC L 3 (one sc enar io ,
αi,
j = 0), HCL 4 (on e scen ar io ,
αi,
