Claims Reserving Using Tweedie'S Compound Poisson Model: BY Ario Üthrich
Claims Reserving Using Tweedie'S Compound Poisson Model: BY Ario Üthrich
Claims Reserving Using Tweedie'S Compound Poisson Model: BY Ario Üthrich
p
i i
( )
.
r
w y
p p
g
m m
1
1 2
,
,
i
i j
i i
p p
i j
1 2
=
+
-
-
-
-
- -
j j
j
j
!
!
J
L
K
K
K
N
P
O
O
O
(4.5)
336 MARIO V. WUTHRICH
From this we obtain the profile likelihood for p and g, resp.,
i
> r 0
j
ij
!
a k as
L
m
(p) =L(m, p,
p
) = ( ) log r
w
g 1 1
,
i
i
i j
+ -
p
z
j
! f p
(4.6)
. log log r
p p
y
r g G
2
1
1
, ,
i
i
i
i j i j
g
+
- -
-
j
j
j
! !
J
L
K
K
f `
N
P
O
O
p j
Given m, the parameter p is estimated maximizing (4.6).
c) Finally we combine a) and b). The main advantage of our parametrization
is (as already mentioned above) the orthogonality of m and (, p). m can be
estimated as if (, p) were known and vice versa. Alternating the updating
procedures for m and (, p) leads to an efficent algorithm: Set initial value
p
(0)
and estimate m
(1)
via a). Then estimate p
(1)
from m
(1)
via (4.6), and iterate this
procedure. We have seen that typically one obtains very fast convergence of
(m
(k)
, p
(k)
) to some limit (for our examples below we needed only 4 iterations).
4.2. Dispersion modelling
So far we have always assumed that is constant over all cells (i, j). If we con-
sider the definitions (3.3) and (3.4) we see that every factor which increases l
increases the mean m and decreases the dispersion because p (1,2). Increas-
ing the average payment size t increases both the mean and the dispersion.
Changing l and t such that l
1p
t
2p
remains constant has only an effect on the
mean m. Hence it is necessary to model both the mean and the dispersion in
order to get a fine structure, i.e. model m
ij
and
ij
for each cell (i,j) individually
and estimate p. Such a model has been studied in the context of tarification
by Smyth-Jrgensen [8].
We do not further follow these ideas here since we have seen that in our sit-
uation such models are over-parametrized. Modelling the dispersion parame-
ters while also trying to optimize the power of the variance function allows
too many degrees of freedom: e.g. if we apply the dispersion modelling model
to the data given in Example 6.1 one sees that p is blown up when allowing
the dispersion parameters to be modelled too. It is even possible that there is no
unique solution when modelling
ij
and p at the same time (in all our examples
we have observed rather slow convergence even when choosing meaningful
initial values which indicates this problematic).
5. MEAN SQUARE ERROR OF PREDICTION
To estimate the mean square error of prediction (MSEP) we proceed as in
England-Verrall [1]. Assume that the incremental payments C
ij
are independent,
CLAIMS RESERVING USING TWEEDIES COMPOUND POISSON MODEL 337
and C
ij
%
are unbiased estimators depending only on the past (and hence are
independent from C
ij
). Assume j
ij
is the GLM estimate for j
ij
=log m
ij
, then (see
e.g. [1], (7.6)-(7.7))
. j z
C E C C C C
w w
MSEP Var Var
Var m m
C i i i i i
i
i
p
i i i
2
2
i
$ .
= - = +
+
j
j
j j j j j
j j
% % %
a a ` a
` `
k k j k
j j
< F
(5.1)
The last term is usually available from standard statistical software packages,
all the other parameters have been estimated before. The first term in (5.1) denotes
the process error, the last term the estimation error.
The estimation of the MSEP for several cells (i, j) is more complicated since
we obtain correlations from the estimates. We define D to be the unknown tri-
angle in our run-off pattern. Define the total outstanding payments
. C C C and C
( , ) ( , )
i
i j
i
i j D D
= =
! !
j j
! !
%
(5.2)
Then
2
( )
, .
j
j j
z E C w w
w w
MSEP Var
Cov
C C m m
m m
!
( , ) ( , )
( , ) ( , )
( , ),( , ) ,
C i i
i
p
i j
i i i
i j
i
i j i j
i j i j
i j i i j i j i j
D D
D
2
1
1 1 2 2
1 1 2 2
1 1 2 2 2 1 1 2 2
$ . = - +
+
! !
!
j
j
j j
! !
!
_ ` `
`
i j j
j
9 C
The evaluation of the last term needs some care: Usually one obtains a covari-
ance matrix for the estimated GLM parameters log(i) and log f ( j). This
covariance matrix needs to be transformed into a covariance matrix for j with
the help of the design matrices.
6. EXAMPLE
Example 6.1.
We consider Swiss Motor Insurance datas. We consider 9 accident years over
a time horizon of 11 years. Since we want to analyze the different methods
rather mechanically, this small part of the truth is already sufficient for drawing
conclusions.
338 MARIO V. WUTHRICH
Remark: As weights w
i
we take the number of reported claims (the number of
IBNyR claims with reporting delay of more than two years is almost zero for
this kind of business).
a) Tweedies compound Poisson model with constant dispersion.
We assume that Y
i j
are independent withY
i j
ED
(p)
(m
i j
, /w
i
) (see (4.1)). Define
the total outstanding payments C as in (5.2). If we start with initial value
p
(0)
=1.5 (1,2) and then proceed the estimation iteration as in Subsection 4.1,
we observe that already after 4 iterations we have sufficiently converged to
equilibrium (for the choice of p one should also have a look at Figure 1):
CLAIMS RESERVING USING TWEEDIES COMPOUND POISSON MODEL 339
TABLE 6.2
OBSERVATIONS FOR THE NORMALIZED INCREMENTAL PAYMENTS Y
ij
=C
ij
/ w
i
.
y
ij
Development period j
AY i 0 1 2 3 4 5 6 7 8 9 10
0 157.95 65.89 7.93 3.61 1.83 0.55 0.14 0.22 0.01 0.14 0.00
1 176.86 60.31 8.53 1.41 0.63 0.34 0.49 1.01 0.38 0.23
2 189.67 60.03 10.44 2.65 1.54 0.66 0.54 0.09 0.19
3 189.15 57.71 7.77 3.03 1.43 0.95 0.27 0.61
4 184.53 58.44 6.96 2.91 3.46 1.12 1.17
5 185.62 56.59 5.73 2.45 1.05 0.93
6 181.03 62.35 5.54 2.43 3.66
7 179.96 55.36 5.99 2.74
8 188.01 55.86 5.46
TABLE 6.3
NUMBER OF PAYMENTS R
ij
AND VOLUME w
i
.
r
i j
Development period j
AY i 0 1 2 3 4 5 6 7 8 9 10 w
i
0 6229 3500 425 134 51 24 13 12 6 4 1 112953
1 6395 3342 402 108 31 14 12 5 6 5 110364
2 6406 2940 401 98 42 18 5 3 3 105400
3 6148 2898 301 92 41 23 12 10 102067
4 5952 2699 304 94 49 22 7 99124
5 5924 2692 300 91 32 23 101460
6 5545 2754 292 77 35 94753
7 5520 2459 267 81 92326
8 5390 2224 223 89545
Figure 1: Profile likelihood function L
m
(p) (see (4.6)).
For p =1.1741 the GLM output is as follows: Dispersion