1. Introduction
When predicting the ultimate reserve amounts, general insurance actuaries use a mix of chain ladder, BornhuetterFerguson (Bornhuetter and Ferguson 1972), and Cape Cod (Bühlmann and Straub 1983) methods. Most commercial reserving software systems propose these three methods. These systems will likely ask the application of chain ladder first to derive a payment or incurred pattern, which will then be used for the application of BornhuetterFerguson or Cape Cod methods. In all the cases, the chain ladder method will be applied on a cumulative triangle. Below is a summary of the stochastic models underlying each of the three methods—chain ladder, BornhuetterFerguson, and Cape Cod.
1.1. Chain ladder
The chain ladder method is applied on cumulative triangles.
Let Table 1.
denote the cumulative claims amount (either paid or incurred) of accident year i after k years of development, where n denotes the most recent accident year. Then denotes the currently known claims amount of accident year i, shown inThe basic chain ladder assumption is that there exist development factors
such thatE(Ci,k+1∣Ci,1,…,Ci,k)=fkCi,k,1≤i≤I,1≤k≤I−1
where the link ratios (agetoage factors) can be estimated as
ˆfk=∑I−kj=1Cj,k+1∑I−kj=1Cj,k,1≤k≤I−1,
under the assumption that
are independent.In this paper,
will denote the estimator of the random variable Mack (1993) shows that the link ratios are unbiased and uncorrelated.Variance of
In the framework of the distributionfree calculation of the standard error of the reserve estimates, several variance models exist. For the purpose of this discussion, we will focus on the Mack standard error.
As for the variance of
Mack (1993) induced that (where denotes the conditional variance of A knowing B) should be proportional to i.e.:Var(Ci,k+1∣Ci,1,…,Ci,k)=Ci,kσ2k,1≤i≤I,1≤k≤I−1
where
ˆσ2k=1I−k−1I−k∑i=1Ci,k(Ci,k+1Ci,k−ˆfk)2for1≤k≤I−2
It can be shown that the estimator (Mack 1993).
is unbiased1.2. BornhuetterFerguson
As mentioned earlier, the BornhuetterFerguson (hereinafter “BF”) is usually applied on cumulative triangles using a pattern derived from the chain ladder method. In this section, we will review the stochastic model underlying the BF method introduced in Mack (2008). In this stochastic model, the BF method should be applied on incremental triangles.
As for the chain ladder method, let
denote the cumulative claims amount (either paid or incurred) of accident year i after k years of development, and be the premium volume of accident year i where n denotes the most recent accident year. Then denotes the currently known claims amount of accident year i. Let further denote the incremental claims amount (with and U_{i} the (unknown) ultimate claims amount of accident year i. Then is the (unknown true) claims reserve for accident year i. Finally, let be the incremental claims amount after development year n (tail development).Bornhuetter and Ferguson (1972) introduced their method to estimate R_{i} as follows:
ˆRBFi=ˆUi(1−ˆzn+1−i)
where
with a prior estimate for the ultimate claims ratio of accident year i, is the estimated percentage of the ultimate claims amount which is expected to be known after development year k.The BF stochastic model developed in Mack (2008) relies on the following assumptions related to the increments :

BF1: All increments
are independent 
BF2: There are unknown parameters x_{i}, y_{k} such that:


BF3: There are unknown proportionality constants
with
On the basis of these three assumptions, the prediction error of BornhuetterFerguson can be estimated (Mack 2008). The prediction error, usually denoted as MSEP (mean squared error of prediction) consists of two components, the process error and the estimation error. Whereas the estimation error basically always can be calculated via the laws of error propagation, for the process error a stochastic model of the claims process was developed by Mack (2008).
Following Mack (2008), we have the following with x_{1},… x_{n} known:
ˆyk=∑n+1−ki=1Si,k∑n+1−ki=1xi
is a best linear unbiased estimate of y_{k} and
ˆs2k=1n−kn+1−k∑i=1(Si,k − xiˆyk)2xi
is an unbiased estimate of s^{2}_{k}.
1.3. Cape Cod
As for the chain ladder and BF methods, we denote the cumulative claims (cumulative payments or incurred losses) in accident year
at the end of development year by and we assume J ≤ I. Let denote the incremental claims, where we set The summation over an index starting from 0 is denoted with a square bracket, for example:C⌊k⌋,j=k∑i=0Ci,j, 0≤k≤I, 0≤j≤J.
We assume that all claims are settled after development year J and therefore the total ultimate claim of accident year i is given by
At time I, we have information in the upper left trapezoid/triangle:DI={Ci,j:i+j≤I, j≤J}
and our goal is to predict the lower right triangle:
DcI={Ci,j:i+j>I, i≤I,j≤J}.
The chain ladder prediction of the ultimate claim
of accident year i > I − J is given byˆCCLi,J=Ci,ι(i)J−1∏j=ι(i)ˆfj
where
ˆfj=C[I−j−1],j+1C[I−j−1],j and ι(i)=min(J,I−i).
The chain ladder development pattern is defined as
ˆβCLj=J−1∏k=jˆf−1k,0≤j≤J−1, ˆβCLJ=1
The Cape Cod predictor (Bühlmann and Straub 1983) for the ultimate claim is given by
ˆCCCi,J=Ci,ι(i)+υiˆq(1−ˆβι(i))
where
The earned premium for accident year i is denoted by
andˆq=∑Ii=0Ci,ι(i)∑Ii=0υiˆβι(i).
is an estimate of and describes the percentage of claims emerging up to development year The incremental development pattern is estimated by
ˆγ0=ˆβ0
ˆγj+1=ˆβj+1−ˆβj, 0≤j≤J−1.
In the original article of Bühlmann and Straub (1983), it is mentioned that the estimation of the development pattern is an unsolved problem. In practice, the development pattern is often estimated by the chain ladder (CL) development pattern given in (7).
Finally, we define the outstanding loss liabilities for accident year i at time I as:
RCCi=Ci,J−Ci,I−i
Model assumptions
Incremental claims
are independent and there exist positive parameters and a development pattern with such thatE[Si,j]=υiqγj
Var[Si,j]=(υiq) σ2j
where
denotes the variance of the random variableFor the estimation of the variance, we need estimates for
Note that^q σ2j=1I−jI−j∑i=01νi(Si,j−υi^γj)2,0≤j≤J, j≠I
is an unbiased estimator for
Note also that the above model assumptions assume that the expected loss ratio q is the same for all accident years.
2. A review of the chain ladder method
In this section, we are going to review the equation (4) of the chain ladder method:
ˆσ2k=1I−k−1I−k∑i=1Ci,k(Ci,k+1Ci,k−ˆfk)2 for 1≤k≤I−2.
Using the definition
and the chain ladder incremental pattern we can reformulate equation (4) asˆσ2k=1I−k−1I−k∑i=1Ci,k(Si,k+1Ci,k−(ˆfk−1))2.
We can see that
ˆwk=ˆβCLk−1−ˆβCLk=1∏J−1l=k−1ˆfl−1∏J−1l=kˆfl=1∏J−1l=kˆfl(ˆfk−1).
Hence,
ˆσ2k=1I−k−1I−k∑i=1Ci,k(Si,k+1Ci,k−ˆwkJ−1∏l=kˆfl)2.
As a result,
ˆσ2k=1I−k−1I−k∑i=1ˆCCLi,JJ−1∏l=kˆfl(Si,k+1ˆCCLi,J−ˆwk)2.
The shape of this equation is very similar to the volatility factor of the BornhuetterFerguson and Cape Cod methods. Despite its unusual shape, it is the same as the usual known equation (4) used to estimate the
but based on incremental triangle.As a conclusion of this section, we can mention that, in practice, actuaries usually estimate their loss ultimates on the basis of cumulative triangles. However, as we have seen, the stochastic underlying models are based on incremental claims. Incremental claims have the advantage of having independence between each triangle cell, which is not the case for cumulative claims. This is why the stochastic underlying models are based on increments. It would therefore be advisable to change the commercial reserving software and provide actuaries with analysis based on incremental claims: this would better reflect the dynamics of the claims movements.
3. Stochastic reserving methods—A unification
Equations (6), (8) and (9) are rewritten below.
BornhuetterFerguson
ˆs2k=1n−kn+1−k∑i=1xi( Si,kxi− ˆyk)2
where x_{i} represents the apriori ultimate of the BF method.
The stochastic model defines:
Cape Cod
^q σ2j=1I−jI−j∑i=0υi(Si,jνi−^γj)2
where
represents the ultimate premium of the CapeCod method.The stochastic model defines:
Chain ladder
ˆσ2k=1I−k−1I−k∑i=1ˆCCLi,JJ−1∏l=kˆfl(Si,k+1ˆCCLi,J−ˆwk)2
where
represents the prediction of the ultimate claimThe stochastic model defines
Var(Si,k+1Ci,1,…,Ci,k)=Ci,kσ2k
The variance of the increment for each model is defined according to these volatility factors. These three equations have the same shape:

The volatility factor always depends on the difference between the increments divided by the ultimate and the estimated pattern.

The volatility factor is always a weighted average of these differences where the weights are either the ultimates (BF, Cape Cod) or derived from the ultimates (chain ladder).
In relation to the first point, in practice, the feeling for the volatility of a line of business always depends on the possibility for the actuary to have confidence in the incurred/payment pattern. When an actuary feels unsure about the incurred/payment pattern, he will say that the line of business is very volatile. On the opposite when the incurred/payment pattern is stable across the accident or underwriting years, he will say that the line of business is not volatile. The three equations reflect therefore the practice. In addition, their similarity shows that the applied method is not a determinant of the volatility of the resulting ultimates.
As for the skewness factors, the definitions are provided below.
BornhuetterFerguson (Dal Moro 2021)
On the basis of the three assumptions described in 1.b (All increments (Mack 2008). In order to estimate the skewness of the BF method, we need a fourth assumption:
are independent, there are unknown parameters x_{i}, y_{k,} there are unknown proportionality constants with the prediction error of BornhuetterFerguson can be estimatedBF4: There are unknown proportionality constants
with SKand
ˆt3k=1n−kn+1−k∑i=1(Si,k − xiˆyk)3x32i
It has to be noted that the skewness of BornhuetterFerguson in the proposed model comes in a distributionfree environment. Once the best estimate, the standard deviation and the skewness of the reserves are estimated, the actuary can fit a distribution of his choice to these first three moments.
Cape Cod (Dal Moro 2022)
As for the skewness of Cape Cod in a distribution free environment, following on the work of Saluz (2015), we assume that there exist positive parameters and a development pattern with such that
E[Xi,j]=υiqγj
SK[Xi,j]=(υiq)32 t3j
where
denotes the third moment of the random variable And corresponds to the UWY premium and to the Cape Cod loss ratio.For the estimation of the skewness, we need estimates for
Note that^q32 t3j=1I−jI−j∑i=01υ32i(Xi,j−υi^γj)3, 0≤j≤J, j≠I
is an unbiased estimator for
As for BornhuetterFerguson, it has to be noted that the skewness of Cape Cod in the proposed model comes in a distributionfree environment. Once the best estimate, the standard deviation and the skewness of the reserves are estimated, the actuary can fit a distribution of his choice to these first three moments.
Chain ladder (Dal Moro 2013)
SK(Ci,k+1∣Ci,1,…,Ci,k)=C3/2i,kSk3k,1≤i≤I,1≤k≤I−2
where:
ˆSk3k=1(I−k−(∑I−ki=1C3/2i,k)2(∑I−ki=1Ci,k)3)I−k∑i=1C3/2i,k(Ci,k+1Ci,k−ˆfk)3 for 1≤k≤I−3
It has to be noted that, like for BornhuetterFerguson and Cape Cod, the skewness of chain ladder in the proposed model comes in a distributionfree environment. Once the best estimate, the standard deviation, and the skewness of the reserves are estimated, the actuary can fit a distribution of his choice to these first three moments.
Like for the volatility factor, the chain ladder skewness factor can be changed to reflect the difference between the increments divided by the ultimate and the estimated pattern.
The skewness factors show, as they do for the volatility factors, a unity in terms of the general shape of the formulae.
As a unification of these three methods, for the volatility and skewness estimators denoted respectively
and we could therefore consider the overall equations:v2k=1m(k)∑ig(ˆUl)(Si,kˆUi−ˆyk)2
where
represents either Ik or Ik1, represents the ultimate, and is a function of the ultimate.sk3k=1o(k)∑ih(ˆUl)(Si,kˆUi−ˆyk)3
with the same variables as for the volatility and o(k) is Ik or the factor before the sum in equation (10).
4. Individual claims analysis for a unified model
Based on the unified notation described above (equations 11 and 12), we see that the reserve risk distribution is defined by the relative position of the percentage of incremental claim to ultimate vs. the pattern that is defining the best estimate. As mentioned earlier, in practice, this is the work of the actuary to try and defend his choice of best estimate patterns. Nowadays, such study of the position of incremental claims vs. best estimate pattern should be looked at individual claims level.
For this purpose, let’s denote
the individual incremental claim amount for claim p for accident year i at the end of development year k and the ultimate claim for claim p for the same accident year and development year. We have then:Si,kˆUi=∑pSICp,i,k∑p^UICCp,i,k=∑pSICp,i,k^UICCp,i,k^UICp,i,k∑p^UICp,i,k
Let’s denote
ap,i,k=^UICCp,i,k∑p^UICp,i,k
Then we have the following (as
:si,kˆUi−ˆyk=∑pap,i,k(SICp,i,k^UTCCp,i,k−ˆyk).
With this last equation, we can see that some information related to the unified volatility and skewness estimated in equations (11) and (12) can be derived from individual claims. In equation (13), for i and k given, the following elements are not random:
Therefore, the risk distribution depends on the shape (volatility and skewness) of the individual claims in year i and development year k.In this context, let’s look at an individual claims triangle and see the consequences of looking at the reserving distribution on such a triangle, shown in Table 2.
Any projection method applied to this triangle (e.g., neural networks, chain ladder …) will just estimate the IBNER (incurred but not enough reported) and will not take into account the IBNYR (incurred but not yet reported). Therefore, in the context of a unified reserving model based on an individual claims triangle, the following procedure will have to be followed:
 Estimate the future number of claims and the volatility and skewness of the distribution of the future number of claims. This can be done on using a chain ladder method (Mack 1993; Dal Moro 2013), shown in Table 3.

Estimate for each development year, the mean, volatility and skewness of individual claim incurred or payment. In this regard, we assume that the exposures remain stable across the UWYs allowing the calculation at development year level and not at development and UWY level.

Having the future number of claims and the individual incurred or payment per development year, it is easy to build the IBNYR risk distribution (see following sections for details).

As for the IBNER, we will rely on Schnieper (1991), which gives a complete description of the way in which IBNER and their volatilities can be estimated based on individual claim information. In a following section, we will extend the calculations to the skewness case.
In the next two sections, we are going to detail the four steps just described. It has to be noted that the most significant amount of reserves will come from the IBNYR as, for each new claim, the full mean ultimate payment has to be reserved.
5. IBNYR based on individual claims analysis
Based on the individual claims’ triangle of Table 2, we can estimate the following parameters:

The mean incurred/payment per development year;

The variance of the incurred/payment per development year;

The skewness of the incurred/payment per development year.
Having estimated the above characteristics, we can use the law of total variance to estimate the overall variance for accident year i:
Var(Ni∑p=1I∑k=I−i+1SICp,i,k)=E(Ni)Var(I∑k=I−i+1SICp,i,k)+Var(Ni )E(I∑k=I−i+1SICp,i,k)2
Due to the independence between the
we haveVar(Ni∑p=1I∑k=I−i+1SICp,i,k)=E(Ni)(I∑k=I−i+1Var(SICp,i,k))+Var(Ni)(I∑k=I−i+1E(SICp,i,k))2
where N_{i} denotes the future number of claims, which can be estimated with a standard chain ladder method from accident year i (see next section for details).
As we have
Var(SICp,i,k)=Var(SICk)for all p and i
E(SICp,i,k)=E(SICk)for all p and i
the overall variance for accident year i is
Var(Ni∑p=1I∑k=I−i+1SICp,i,k)=E(Ni)(I∑k=I−i+1Var(SICk))+Var(Ni)(I∑k=I−i+1E(SICk))2
The same can be done for skewness with the law of total skewness and we get
SK(Ni∑p=1I∑k=I−i+1SICp,i,k)=E(Ni)(I∑k=I−i+1SK(SICk))+SK(Ni)(I∑k=I−i+1E(SICk))3+3 Var(Ni)(I∑k=I−i+1E(SICk))×(I∑k=I−i+1Var(SICk))
6. IBNER based on individual claims analysis
According to Schnieper (1991), the IBNER for accident year i is equal to:
IBNERi=Ci,n+1−i{[n∏j=n+2−i(1−δj)]−1}
where:
ˆδj=∑n+1−ji=1Di,j∑n+1−ji=1Ci,j−1
and D_{i,j} is the decrease in total claims amount between development year j1 and development year j with respect to claims already known in development year j1.
In the case of the triangle shown in Table 2, we would have the following resulting triangle of D_{i,j} with the corresponding shown in Table 5.
In Schnieper (1991), we also have that:
Var(Di,j)=Ci,j−1τ2j
where
ˆτ2j=1n−jn+1−j∑i=1(Di,j−ˆδjCi,j−1)2Ci,j−1
By natural extension, the skewness estimate can be derived as follows:
SK(Di,j)=C32i,j−1ζ3j
where
ˆζ3j=1n−jn+1−j∑i=1(Di,j−ˆδjCi,j−1)3C32i,j−1
As in Schnieper (1991), let’s denote Developing in a Taylor series, we obtain
IBNERi(ˆθi)=IBNERi(θi)+n∑j=n+2−iδ IBNERi(δj)δ δj
Due to the independence of the D_{i,j}, we can calculate the mean standard error (hereinafter “mse”) as follows (Schnieper 1991):
mse(IBNERi(ˆθi))=E(IBNERi(ˆθi)−IBNERi(θi))2=n∑j=n+2−i(δ IBNERi(δj)δ δjθi=ˆθi)2Var(^δj)
where
Var(ˆδj)=τ2j∑n+1−ji=1Ci,j−1
Following equation (16), we have:
δ IBNERi(δj)δ δjθi=ˆθi=−Ci,n+1−in∏k=n+2−ik≠j(1−δk)
which leads to:
mse(IBNERi(ˆθi))=C2i,n+1−in∑j=n+2−i(n∏k=n+2−i(1−δk))2Var(ˆδJ)(1−δj)2
The same can be done for the skewness estimation which leads to the following formula:
SK(IBNERi(ˆθi))=−C3i,n+1−in∑j=n+2−i(n∏k=n+2−i(1−δk))3SK(ˆδj)(1−δj)3
where
SK(ˆδj)=ζ3j∑n+1−ji=1C32i,j−1
7. Estimation of overall skewness and standard deviation
After estimating IBNYR and IBNER by UWY, based on Mack (1993), the overall standard deviation can easily be calculated as per the formula below. Let’s denote s.d.(R_{i}), the standard deviation of the reserve of UWY i (see 1.b for definition of (R_{i}) and s.d.(R) the standard deviation overall all UWYs (R=R_{2} + … + R_{n}). Then:
s.d.(ˆR)2=I∑i=2{s.d.(ˆRi)2+ˆCCLiI(I∑j=i+1ˆCCLjI)I−1∑k=I+1−i2 ˆσ2kˆf2k∑I−kn=1Cn,k}
Let’s denote the following correlations:
ri,j=ˆCCLiIˆCCLjI∑I−1k=I+1−iˆσ2kˆf2k∑I−kn=1Cn,k√Var(ˆCCLiI)Var(ˆCCLjI)
And as
we can write equation (19) in the following form:s.d. (ˆR)2=(s⋅d⋅(^R1)… s.d. (^R1))(1r12…r12…………1)( s.d. (^R1)… s.d. (^R1))
In order to aggregate the standard deviation of the proposed method based on IBNYR and IBNER by UWY, we can use the same correlation matrix on the standard deviations of each UWY to get the overall standard deviation.
As for the overall skewness, before giving a general formula, we are going to limit the estimation to the case of three accident years X1, X2, X3 as shown in the framework of lemma 1 (see appendix B). The question is the estimation of
It develops as follows:SK(X1+X2+X3)=E[((X1+X2+X3)−E(X1+X2+X3))3]=E[(X1+X2+X3)3]−3E[(X1+X2+X3)2]E[(X1+X2+X3)]+2E[(X1+X2+X3)]3
SK(X1+X2+X3)=SK[X1]+SK[X2]+SK[X3]+3E[X1X2(X1+X2)]−6E[X1X2](E[X1]+E[X2])+3E[X1X3(X1+X3)]−6E[X1X3](E[X1]+E[X3])+3E[X2X3(X2+X3)]−6E[X2X3](E[X2]+E[X3])+3E[X1][E(X2)2−Var(X2)]+3E[X1][E(X3)2−Var(X3)]+3E[X2][E(X1)2−Var(X1)]+3E[X2][E(X3)2−Var(X3)]+3E[X3][E(X1)2−Var(X1)]+3E[X3][E(X2)2−Var(X2)]+6E[X1X2X3]+12E[X1]E[X2]E[X3]−6E[X1X2]E[X3]−6E[X1X3]E[X2]−6E[X2X3]E[X1]
Following lemma 1 (see appendix B), we have
E[X21X2]=Cov(X21,X2)+E(X21)E(X2)=E(X1)(1+Var(X1)E(X1)2)Cov(X1,X2)(2+Cov(X1,X2)E(X1)E(X2))+(Var(X1)+E(X1)2)E(X2)
For the following elements of the above equation, we therefore get
E[X1X2(X1+X2)]−2E[X1X2](E[X1]+E[X2])+E[X1][E(X2)2−Var(X2)]+E[X2][E(X1)2−Var(X1)]=E(X1)(1+Var(X1)E(X1)2)Cov(X1,X2)(2+Cov(X1,X2)E(X1)E(X2))+(Var(X1)+E(X1)2)E(X2)+E(X2)(1+Var(X2)E(X2)2)Cov(X1,X2)(2+Cov(X1,X2)E(X1)E(X2))+(Var(X2)+E(X2)2)E(X1)−2(E[X1]+E[X2])(Cov(X1,X2)+E(X1)E(X2))+E[X1][E(X2)2−Var(X2)]+E[X2][E(X1)2−Var(X1)]=Cov(X1,X2)(2+Cov(X1,X2)E(X1)E(X2))×[E(X1)(1+Var(X1)E(X1)2)+E(X2)(1+Var(X2)E(X2)2)]−2(E[X1]+E[X2])Cov(X1,X2)=Cov(X1,X2)(2+Cov(X1,X2)E(X1)E(X2))[Var(X1)E(X1)+Var(X2)E(X2)]+Cov(X1,X2)2[E[X1]+E[X2]E[X1]E[X2]]
In addition, using lemma 2 (see appendix B), we get
E[X1X2X3]+2E[X1]E[X2]E[X3]−E[X1X2]E[X3]−E[X1X3]E[X2]−E[X2X3]E[X1]=E[X1]E[X2]E[X3](2+(1+Cov(X1,X2)E[X1]E[X2])×(1+Cov(X1,X3)E[X1]E[X3])×(1+Cov(X2,X3)E[X2]E[X3]))−E[X3]Cov(X1,X2)−E[X1]E[X2]E[X3]−E[X2]Cov(X1,X3)−E[X1]E[X2]E[X3]−E[X1]Cov(X2,X3)−E[X1]E[X2]E[X3]=r12r13r23√Var(X1)Var(X2)Var(X3)×(√Var(X1)Var(X2)Var(X3)E[X1]E[X2]E[X3]+√Var(X1)r23E[X1]+√Var(X2)r13E[X2]+√Var(X3)r12E[X3])
In the case of three accident years and under the restrictions indicated in lemmas 1 and 2, we therefore find the following aggregate skewness:
SK(X1+X2+X3)=SK[X1]+SK[X2]+SK[X3]+3r12√Var(X1)Var(X2)×[Var(X1)E(X1)+Var(X2)E(X2)][2+r12√Var(X1)Var(X2)E[X1]E[X2]]+3r212Var(X1)Var(X2)[E[X1]+E[X2]E[X1]E[X2]]+3r13√Var(X1)Var(X3)×[Var(X1)E(X1)+Var(X3)E(X3)][2+r13√Var(X1)Var(X3)E[X1]E[X3]]+3r213Var(X1)Var(X3)[E[X1]+E[X3]E[X1]E[X3]]+3r23√Var(X2)Var(X3)×[Var(X2)E(X2)+Var(X3)E(X3)][2+r23√Var(X2)Var(X3)E[X2]E[X3]]