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

Actuarial Analysis of the French Bonus-Malus System 339
giving the reduced penalty in case of a partial liability claim. Now the CRM coefficient for
the time period 0t is
with
r t t t
N 1• N 2• I 12• t= 1 + t N 1•
1 + t N 2•
1 − t I 12•
N 1• =
N 2• =
I 12• =
t∑
N 1j
j=1
t∑
N 2j
j=1
t∑
j=1
{ 1ifN1j = N
I j with I j =
2j = 0
0 otherwise
We will assume that t = t with fixed by the actuary. The value of describes the way
a claim with full liability is penalized, compared to a claim with partial liability. Then the
CRM coefficient becomes
r t t
N 1• N 2• I 12• t= 1 + t N 1•
1 + t N 2•
1 − t I 12•
In order to obtain t and t we have now to minimize the objective function
t = [( − r N 1• N 2• I 12• t ) 2]
with respect to the parameters and . The first order conditions are
⎧
[ (
1 − I 12• N1• 1 + N1•−1 1 + N 2• + N2• 1 + N2•−1 1 + 1•)]
N
⎪⎨
= [ (
1 − 2I 12• N1• 1 + 2N1•−1 1 + 2N 2• + N2• 1 + 2N2•−1 1 + 1•)]
2N
[ 1 + N 1• 1 +
N 2•I12• 1 − I 12•−1 ]
⎪⎩
= [ 1 + 2N 1• 1 +
2N 2•I12• 1 − ] 2I 12•−1
Let us define
[
1 2 3 = N 1•
1 N 2•
2 I 12•
3
]
∣
∣ =
to be the conditional probability generating function of the random vector N 1• N 2• I 12•
given = . We clearly have that
( ))
1 2 3 =
(e − 3 − 1 + e 1−q 1 +q 2 −1 t