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

Efficiency and Bonus Hunger 253
First Iteration
Part A Starting from rl 0 l = 0 for l = 0s, the strategy consisting of reporting
all the accidents to the insurer, (5.8) becomes
∑
V l = b l + v exp− k
k! V T k l
which gives the cost V 0 corresponding to the initial strategy.
Part B
k=0
An improved strategy can then be obtained from (5.9) that reduces to
rl 1 l = v 1−t
l = 0 1s.
∑
k=0
exp−1 − t
1 −
(
)
tk
V
k!
Tk+m+1 l − V Tk+m l
Second Iteration
Part A Inserting the rl 1 l s in (5.8) gives the cost associated with this strategy. This
cost will be smaller than the one associated with the initial strategy.
Part B Inserting the new cost in the system (5.9), we find an improved strategy rl 2 l ,
l = 0s.
Subsequent Iterations
The successive insertion of updated retentions and costs in the systems (5.8)–(5.9) produces
a sequence of strategies, with reduced costs.
In all the cases considered in Lemaire (1995), the sequence of the rl k
l
s converges to
the optimal solution with miminum cost. The optimal retention limit is thus a function of
the level l occupied in the scale at the beginning of the insurance year, of the discount
factor v, of the annual expected claim frequency , of the time t of occurrence of
the accident, and of the number m of claims previously reported to the company from
the beginning of the insurance period. The optimal strategy is an increasing function
of t: the optimal retention increases as one approaches the end of the year (and the
premium discount if no accidents are reported). The influence of t on the optimal
retention limit is much weaker than the level l, the discount factor v or . Putting
t = 0 (and so m = 0) greatly simplifies the computation but leaves the retentions almost
unchanged.
The optimal retentions coming from the Lemaire algorithm should not be considered as
being the real threshold above which policyholders report the accident to the insurance
company. Indeed, this algorithm postulates a high degree of rationality behind individual
behaviours. It is enough to have a look at insurance statistics to see that some claims
concern accidents bearing a cost much lower than the optimal retention, which contradicts
the assumptions behind the Lemaire algorithm. The output of the Lemaire algorithm should
be better understood as a measure of toughness for a particular bonus-malus system. Note
that Walhin & Paris (2000,2001) softened this requirement by assuming that there was a
proportion of the policyholders complying with Lemaire claiming rule, and the remainder
reporting all the accidents, whatever their cost.