92 R.J. Verrall Insurance: Mathematics and Economics 26 2000 91–99
being used and the data for which it is suitable. In order to go beyond the simple reserve estimates of the chain-ladder technique, it is necessary to specify a stochastic model. Mack 1993 and Renshaw and Verrall 1998 are examples
of papers which do this, and this paper will give a number of other specifications of models which reproduce the chain-ladder estimates. The various models make different assumptions and it is hoped that this paper will clarify
some of the issues confronting a practitioner when they have to decide which model to use to estimate the variability of the chain-ladder reserve estimates.
In contrast with the chain-ladder technique, the separation technique was defined first as a stochastic model by Verbeek 1972 and only after this was an easy algorithm produced by Taylor 1977, which is akin to the chain-ladder
technique. The chain-ladder technique and the separation technique are very closely related see Verrall, 1996 and the split between a stochastic model and an algorithm for producing point estimates applies to both cases. In the
case of the separation technique, a stochastic model was defined first and a simple algorithm derived later. This is a logical way to proceed and it is possible to return to the stochastic model and adjust the assumptions in the light of
the data observed. In the days of fast computers, simple algorithms have less importance. This is the philosophy of our approach: the most important thing is the stochastic model, and simple algorithms are useful only if computing
facilities are not available. This does not imply that the connection between the stochastic model and the reserve estimates produced by the simple algorithm should be ignored.
We begin with a model which assumes that the data have a Poisson distribution. While this is unrealistic since it implies that the data must be positive integers, it is straightforward to relax this, and it makes the exposition easier
to understand. Without loss of generality, we assume that the data consist of a triangle of incremental claims: C
ij
: j = 1, . . . , n − i + 1; i = 1, . . . n .
The cumulative claims are defined by: D
ij
=
j
X
k=1
C
ik
, and the development factors of the chain-ladder technique are denoted by
{λ
j
: j = 2,. . . ,n }.
2. The Poisson model
Consider the following model, which assumes a multiplicative structure for expected incremental claims. C
ij
∼ independent Poisson with E[C
ij
] = x
i
y
j
and
n
X
k=1
y
k
= 1. 2.1
Clearly x
i
= E[D
in
], which is expected ultimate cumulative claims up to the latest development year so far observed. The parameters
n y
j
: j = 1, . . . , n; P
n j =1
y
j
= 1 o
will be referred to as the “column parameters”. The column parameters can be interpreted as the proportions of ultimate claims which emerge in each development
year. This model can be reparameterised as follows: C
ij
∼ Poisson with E[C
ij
] = z
i
y
j
P
n−i+1 k=1
y
k
and
n
X
k=1
y
k
= 1. 2.2
In this case z
i
= E[D
i ,n − i + 1
], which is the expected value of cumulative claims up to the latest development year observed in accident year i. Using this parameterisation, the likelihood is:
L =
n
Y
i=1 n−i+1
Y
j =1
z
i
y
j
S
n−i+1 C
ij
e
−zi yj Sn−i+1
C
ij
, where S
m
=
m
X
k=1
y
k
. 2.3
R.J. Verrall Insurance: Mathematics and Economics 26 2000 91–99 93
Thus, L =
n
Y
i=1
z
P
n−i+1 j =1
C
ij
i
exp −
z
i
P
n−i+1 j =1
y
j
S
n−i+1 n−i+1
Y
j =1
y
j
S
n−i+1 C
ij
C
ij
=
n
Y
i=1
z
D
i,n−i+1
i
e
−z
i
n−i+1
Y
j =1
1 C
ij
y
j
S
n−i+1 C
ij
=
n
Y
i=1
z
D
i,n−i+1
i
e
−z
i
D
i,n−i+1
D
i,n−i+1
Q
n−i+1 i=1
C
ij n−i+1
Y
j =1
y
j
S
n−i+1 C
ij
.
2.4 L can be written as L = L
D
L
C
, where L
D
=
n
Y
i=1
z
D
i,n−i+1
i
e
−z
i
D
i,n−i+1
, 2.5
and L
C
=
n
Y
i=1
D
i,n−i+1
Q
n−i+1 i=1
C
ij n−i+1
Y
j =1
y
j
S
n−i+1 C
ij
. 2.6
Thus, L is the product of the likelihood of z
i
and the likelihood for n
y
j
: j = 1, . . . , n; P
n j =1
y
j
= 1 o
. The first term is the probability for D
i ,n − i + 1
and the second term is the probability for {C
ij
: j = 1,. . . ,n − i + 1 }, conditional
on D
i ,n − i + 1
. From this, it can be seen that the maximum likelihood estimate of z
i
is D
i ,n − i + 1
. It is also clear that the same estimates of
n y
j
: j = 1, . . . , n; P
n j =1
y
j
= 1 o
are obtained, whether L or L
C
is maximised. The connection between these estimates and the estimates of the development factors in the chain-ladder technique
is as follows. It can be shown that the column parameters of this model, n
y
j
: j = 1, . . . , n; P
n j =1
y
j
= 1 o
, are related to the development factors of the chain-ladder technique via λ
j
= S
j
S
j − 1
see Verrall, 1991. Thus, the maximum likelihood estimates of the development factors, ˆλ
j
, can be obtained from the maximum likelihood estimates of the column parameters in the multiplicative model, ˆ
y
j
: ˆλ
j
= P
j k=1
ˆ y
k
P
j −1 k=1
ˆ y
k
. It can be shown that this gives the same development factor estimates as the chain-ladder technique,
ˆλ
j
= P
n−j +1 i=1
D
ij
P
n−j +1 i=1
D
i,j −1
. 2.7
Hence, in this context, the chain-ladder technique can be seen as an algorithm for producing the maximum likelihood estimates of the column parameters for either the unconditional likelihood of
{C
ij
: j = 1,. . . ,n − i + 1; i = 1,. . . ,n
}, L, or for the conditional likelihood, conditioning on {D
i ,n − i + 1
: i = 1,. . . ,n }, L
C
. A number of points should be noted:
1. It should be emphasised here that this does not imply that it is necessary to view the chain-ladder estimates as arising from this model: there certainly exist other formulations which will give rise to the same estimates of
the development factors.
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
