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

130 Actuarial Modelling of Claim Counts
to the posterior distribution of i . Past claims history N i1 = k 1 N iTi = k Ti
modifies the
distribution of i , and this modified distribution is used as a new mixing law for the number
of claims N iTi +1 for year T i + 1.
3.3.3 Bayesian Credibility Premium
We are looking for the function ⋆ of N i1 N iTi that is the closest to i , i.e. minimizing
[ (i
E − N i1 N iTi ) ] 2
over all the measurable functions T i
→ . Proposition 3.1 gives the solution of this
optimization problem:
⋆ N i1 N iTi = E [ i
∣ ∣ N i1 N iTi
]
In general, the posterior mean of i given N i1 = k 1 N iTi = k Ti
is given by
E [ i
∣ ∣ N i1 = k 1 N iTi = k Ti
]
=
∫ +
=
(
∏Ti
0
∫ +
0
)
t=1 PrN it = k t i = dF
(
∏Ti
)
t=1 PrN it = k t i = dF
∫ +
0
exp− i• k •+1 dF
∫ +
0
exp− i• k • dF (3.2)
This a posteriori expectation thus appears as the ratio of two Mellin transforms Mk =
Eexp− i i k of i. It is interesting to note that the a posteriori expectation depends
only on the total number k • of accidents caused in the past T i years of insurance, and not on
the history of these claims. This was expected from Property 3.1. This is a characteristic of
the credibility models with static random effects.
The Bayesian credibility premium is simply the mean of the predictive distribution. It is
given by
E [ ∣
] ∫
N iTi +1
∣N i1 = k 1 N iTi = k Ti = iTi +1dF k •
0
= iTi +1E [ ∣ ]
i N i1 = k 1 N iTi = k Ti
where the posterior expectation of i is given by (3.2). The posterior expected claim number
EN iTi +1N i1 N iTi is obtained by multiplying iTi +1 by the correction coefficient
E i N i1 N iTi . This approach always yields financial balance, since
[
E E [ ∣ ] ]
i N i1 = k 1 N iTi = k Ti
= E i = 1
so that the corrections average to unity.