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

252 Actuarial Modelling of Claim Counts
The expected cost of a non-reported accident for a policyholder occupying level l is
l = 1
p l
∫ rll
y=0
yfydy
This policyholder will pay on average l × − l per period, because of the accidents
not reported to the company. Let us assume that the accident occurrences are uniformly
distributed over the year (so that on average they occur in the middle of the year). The
average annual total cost borne by a policyholder in level l is then
CTrll = b l + v 1 2 l − l
where b l is the premium paid at the beginning of the year, subject to the bonus-malus scale.
Let V l be the present value of all the payments made by a policyholder with annual
expected claim frequency occupying level l. The V l s are obtained from
∑
V l = CTrll + v q l kV Tk l l = 0 1s (5.8)
k=0
If the policyholder reports all the accidents to the company, the system (5.8) coincides with
(5.7). The system (5.8) admits a unique solution. For a given set of optimal retentions, the
V l s give the cost of the strategy, according to the level occupied in the scale.
Lemaire Algorithm
Let us consider a policyholder in level l who just caused an accident with cost x at time t,
0 ≤ t ≤ 1. There are two possibilities:
(1) Either he does not claim for the accident and the expected present cost is
v −t CTrll + x + v 1−t
∑
k=0
q l
(
k1 − t
)
VTk+m
l
where m is the number of claims that the policyholder has already filed during the year.
(2) Or he reports the accident to the company and the expected present cost is
v −t CTrll + v 1−t
∑
k=0
q l
(
k1 − t
)
VTk+m+1
l
The retention limit rll is the claim amount x for which the policyholder is indifferent
between the two possibilities: the optimal retentions thus solve
rll = v 1−t
∑
k=0
( ) ( )
q l k1 − t V Tk+m+1 l − V Tk+m l (5.9)
for l = 0 1s. Note that (5.9) does not provide an explicit expression for the optimal
retention since rll also appears in the q l k1 − ts.
The optimal strategy is obtained using the following algorithm: