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

328 Actuarial Modelling of Claim Counts
In words, N • is the total number of claims reported by the policyholder during the period
0t and I • is the number of years without any claim reported to the company. Note that
the CRM coefficients depend on t, so that the penalties and discounts may change for every
0t period.
To obtain the parameters t and t , we minimize the expected squared difference between
the ‘true’ relative premium and the relative premium r t t
applicable to the policyholder
according to the French-type bonus-malus system. More specifically, for a policyholder
observed during t years, and having filed N 1 N 2 N t claims, we aim to determine t and
t so as to minimize the objective function
t = − r N • I • t 2
with respect to the arguments and . We therefore have to solve the first order conditions
t = 0
and
t = 0
which rewrites as
⎧
⎨ [ ]
N • 1 + N •−1 1 − I • − 1 +
N •1 −
I • = 0
⎩
[ ]
I • 1 + N • 1 −
I • −1 − 1 + N • 1 −
I • = 0
⎧
⎨ [ [ ]
N • 1 + N •−1 1 − •] I = N• 1 + 2N •−1 1 − 2I •
⇐⇒
⎩
[ I • 1 + ] N • 1 −
I • −1
= [ I • 1 + ] 2N • 1 −
2I • −1
(9.2)
9.2.4 Computation of the CRMs at Time t
Let us define the conditional probability generating function of the random couple N • I •
given = as
1 2 = N •
1 I •
2
=
The conditional independence assumption of N 1 N 2 N t allows us to write
1 2 =
=
t∏
j=1
We can then rewrite the system (9.2) as
⎧
⎨
⎩
[
N j
1 I j
2
]
∣
∣ =
(
e − 2 − 1 + e −1− 1) t
2 10 1 + 1 − = 10
2 1 + 1 −
2 01 1 + 1 − = 01
2 1 + 1 −