94 R.J. Verrall Insurance: Mathematics and Economics 26 2000 91–99
2. It is straightforward to extend the model so that it applies to positive data, which does not necessarily consist solely of positive integers. This can be done by using the quasi-log-likelihood, and details are contained in
Renshaw and Verrall 1998. 3. This model is not necessarily the one which should be used for all data sets. Again, Renshaw and Verrall 1998
contains some discussion of how to identify a suitable model to use in practice. 4. At this point, it is not clear how negative incremental claims should be dealt with. It is possible to estimate the
column parameters, and the development factors when some of the incremental claims are negative, although certain software packages may have difficulty with this. We believe that the treatment of negative incremental
claims is a very important subject and we defer further discussion until Section 5. 5. It is crucial to note that the same estimates of the development factors are obtained whether the unconditional
likelihood, L, or the conditional likelihood, L
C
is used. It is not possible to refer back to the original formulation of the chain-ladder technique to decide which likelihood to use, and hence it is necessary for a decision to be
made by the practitioner. This final point, number 5, is discussed in more detail in Section 4. First, we show how to write this model in
recursive form.
3. A recursive model
The chain-ladder technique obtains the estimate of D
i ,j
j n − i + 1, from the observed value of D
i ,n − i + 1
in a recursive way:
ˆ D
i,n−i+2
= ˆλ
n−j +2
D
i,n−i+1
ˆ D
i,j
= ˆλ
j
ˆ D
i,j −1
j = n − i + 3, . . . , n. For comparison purposes, and also to inform the choice of model when some of the incremental claim amounts
are negative, it is useful to consider the model in Section 2 in recursive form. In order to do this, we use a Bayesian formulation and concentrate on the estimation of z
i
. For the purposes of simplicity of exposition, we consider just one row of data:
C
i1
, C
i2
, . . . , C
i,n−i+1
. We drop the i suffix, and write the model for C
j
given zj as: C
j
|zj ∼ Poisson with mean zj y
j
S
j
, 3.1
where zj = E[D
j
] is the expected value of aggregate claims up to development year j. Note that in the recursive formulation of the model, it is necessary to attach the label j to z, since the definition of z changes as each datum
is received. In this notation, the row parameter in Section 2 would be written z
i
n − i + 1. zj can be related to zj − 1 as follows:
zj = E D
j
= E D
j −1
+ E C
j
= zj − 1 + zj y
j
S
j
. Hence
zj = zj − 1
1 − y
j
S
j
= zj − 1S
j
S
j −1
. 3.2
Thus, the conditional distribution of C
j
given zj − 1 is C
j
|zj − 1 ∼ Poisson with mean zj − 1y
j
S
j −1
. 3.3
R.J. Verrall Insurance: Mathematics and Economics 26 2000 91–99 95
Now, we use the conjugate prior gamma distribution for zj − 1, and suppose we have the distribution of zj − 1, conditional on the information received up to development year j − 1:
zj − 1|C
1
, C
2
, . . . , C
j −1
∼ Γ α, β, 3.4
for some parameters α and β which will be determined below. Using standard Bayesian analysis, the posterior distribution of zj − 1 is
zj − 1|C
1
, C
2
, . . . , C
j −1
, C
j
∼ Γ α + C
j
, β + y
j
S
j −1
. 3.5
Since we have a relationship between zj and zj − 1, given in Eq. 3.2, we can obtain the distribution of zj, conditional on the information received up to development year j by a straightforward transformation as follows. If
zj − 1 ∼ Γ a, b then
zj ∼ Γ a,
bS
j −1
S
j
. 3.6
From 3.5 and 3.6, zj |C
1
, C
2
, . . . , C
j
∼ Γ α + C
j
, β +
y
j
S
j −1
S
j −1
S
j
. 3.7
This completes a recursive estimation procedure, and we may now derive the distribution of zj | C
1
,C
2
,. . . ,C
j
for all j by considering first the case j = 1. In order to do this, we require a suitable prior distribution for z1. A non-informative improper prior distribution is f z1 ∝ z1
−1
, and C
1
| z1 ∼ Poisson with mean z1, since y
1
= S
1
. Again, a standard Bayesian analysis yields the posterior distribution of z1, conditional on z1:
z1|C
1
∼ Γ C
1
, 1. 3.8
This starts the recursion, and it straightforward to prove by induction, from 3.4 and 3.7 that zj |C
1
, C
2
, . . . , C
j
∼ Γ D
j
, 1. 3.9
This is true for j = 1, as can be seen from 3.8, suppose it is true for j − 1. Then from 3.7 zj |C
1
, C
2
, . . . , C
j
∼ Γ D
j −1
+ C
j
, 1 +
y
j
S
j −1
S
j −1
S
j
, and hence zj | C
1
,C
2
,. . . ,C
j
∼ Γ D
j
,1, as required. Now consider the distribution of C
j
, conditional on the information received up to development year j − 1. For this, it is necessary to integrate out zj − 1as follows:
f C
j
|C
1
, C
2
, . . . , C
j −1
= Z
f C
j
|zj − 1f zj − 1|C
1
, C
2
, . . . , C
j −1
dzj − 1. 3.10
There is a subtle difference in the treatment here from Section 2. Since zj is a random variable, C
j
are only independent conditional on zj. It is well known that 3.10 gives a negative binomial distribution, in this case with
parameters p = S
j −1
S
j
and k = D
j − 1
: f C
j
|C
1
, C
2
, . . . , C
j −1
= Γ D
j
C
j
Γ D
j −1
S
j −1
S
j D
j −1
y
j
S
j C
j
. 3.11
96 R.J. Verrall Insurance: Mathematics and Economics 26 2000 91–99
Thus, in recursive form, the mean and variance of C
j
, conditional on the information received up to development year j − 1 are
D
j −1
y
j
S
j −1
and D
j −1
y
j
S
j
S
j −1 2
, respectively.
3.12 Again, this can be generalised by appealing to the quasi-log-likelihood, and it represents a further formulation
of a stochastic model which will give the same estimates of outstanding claims as the chain-ladder technique. Finally, we write the recursive model in terms of the development factors, λ
j
= S
j
S
j − 1
. C
j
| C
1
,C
2
,. . . ,C
j − 1
has mean and variance λ
j
− 1D
j −1
and λ
j
λ
j
− 1D
j −1
, respectively.
3.13 Hence, noting that D
j
= D
j − 1
+ C
j
, the mean and variance of D
j
| C
1
,C
2
,. . . ,C
j − 1
are λ
j
D
j −1
and λ
j
λ
j
− 1D
j −1
, respectively.
3.14 These moments define a recursive model which will reproduce the reserves given by the chain-ladder technique.
Certain positivity constraints exist, and this is discussed further in Section 5. The fact that the model can be reparameterised as in 3.14, with the mean in a particularly simple form, is one of the key reasons why the
chain-ladder technique is easy to apply. It happens because the column parameters represent separate factors, and can be replaced by another set of factors, the development factors. This point is discussed further in Section 6. We
first consider the conditional and unconditional likelihoods in more detail.
4. The conditional and unconditional likelihoods
