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

150 Actuarial Modelling of Claim Counts
⋆⋆ X 1 X 2 X T = T+1 + 1 c
(E [ ln E [ exp−c T+1 X 1 X 2 X T
]]
− ln E [ exp−c T+1 X 1 X 2 X T
] )
Proof
Starting from
[ (
E exp − c ( T+1 − X 1 X 2 X T ))]
[
= E exp cX 1 X 2 X T E [ ] ]
exp−c T+1 X 1 X 2 X T
[ (
)]
= E exp cX 1 X 2 X T − ⋆⋆ X 1 X 2 X T
( [
× expc T+1 exp E ln E [ ] ])
exp−c T+1 X 1 X 2 X T
Now, let us apply Jensen’s inequality to get
[ (
E exp − c ( T+1 − X 1 X 2 X T ))]
( [
])
≥ exp cE X 1 X 2 X T − ⋆⋆ X 1 X 2 X T
( [
expc T+1 exp E ln E [ ] ])
exp−c T+1 X 1 X 2 X T
Because of the constraint on the expectation of the s, the first exponential is 1, yielding
[ (
E exp − c ( T+1 − X 1 X 2 X T ))]
( [
≥ expc T+1 exp E ln E [ ] ])
exp−c T+1 X 1 X 2 X T
[ (
= E exp − c ( T+1 − ⋆⋆ X 1 X 2 X T ))]
which is the expected result.
□
Remark that in Proposition 3.2 the constraint is made in order to guarantee the financial
equilibrium.
Let us now apply the result contained in Proposition 3.2 to the credibility problem. In this
case,
X t = N it and t i = it i
so that the optimal predictor of N iTi +1 for the exponential loss function is of the form
⋆⋆ N i1 N iTi = iTi +1 + 1 (E [ ln E [ ]]
exp−c
c
iTi +1 i N i1 N iTi
− ln E [ ] )
exp−c iTi +1 i N i1 N iTi