Actuarial Modelling of Claim Counts Risk Classification, Credibility ...

Credibility Models for Claim Counts 151
3.4.2 Poisson-Gamma Credibility Model
Let us now apply the result contained in Proposition 3.2 to the Poisson-Gamma credibility
model. To this end, assume that i ∼ ama a. Given N i• , we know from (3.4) that i
follows the ama + N i• a+ i• distribution, so that we know from (1.36) that
(
)
[ ]
a+Ni•
a +
E exp−c iTi +1 i N i1 N iTi =
i•
a + i• + c iTi +1
It follows that
and
[ ]
ln E exp−c iTi +1 i N i1 N iTi
(
=−a + N i• ln 1 + c )
iT i +1
a + i•
[
E ln E [ ] ]
exp−c iTi +1 i N i1 N iTi
(
=−a + i• ln 1 + c )
iT i +1
a + i•
Proposition 3.2 then gives
⋆⋆ k i1 k iTi = iTi +1 + k i• − i•
c
(
ln 1 + c )
iT i +1
(3.17)
a + i•
Considering (3.17), we see that ⋆⋆ k i1 k iTi is equal to the a priori expectation iTi +1 =
EN iTi +1 plus a correction term. This correction is positive, so that
⋆⋆ k i1 k iTi > iTi +1
if k i• > i• , that is, if the policyholder reported more claims than expected. Otherwise, the
correction is negative. As with the quadratic loss function, the penalty is caused by an excess
of observed claims over expected ones.
Let us now compare the credibility formulas obtained with a quadratic and exponential
loss function. Since for any c ≥ 0,
(
ln 1 + c )
iT i +1
≤ c iT i +1
a + i• a + i•
it is easily seen that we have
⋆⋆ k i1 k iTi ≤ ⋆ k i1 k iTi if k i• > i•