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

Bonus-Malus Systems with Varying Deductibles 285
Equation (7.10) is to be solved for all levels l such that r l > 100 % (that is, for all levels
in the malus zone). So, the premium surcharge r l − 1EC 1 is replaced with an annual
deductible d l . The relation (7.10) ensures that the substitution is actuarially fair.
In practice, equation (7.10) does not possess an explicit solution so that numerical
techniques have to be used. Panjer’s algorithm is employed to derive the distribution of
S. The claim amounts are discretized according to Method (7.4) and the probability mass
is concentrated on 500 points. Then, the equation can be solved using appropriate routines
available in the IML package of SAS R .
7.3.2 Per Claim Deductible
Of course, the deductible could also be applied to each claim filed by the policyholder.
The indifference principle invoked above will again be used to determine the amount of
the deductible. Considering a policyholder in level l, he will have to pay r l EC 1 if he is
subject to the a posteriori corrections induced by the bonus-malus scale. If, on the contrary,
a fixed deductible d l is applied per claim, he will have to pay EC 1 as well as minC k d l
for each of the claims C k reported to the company. Note that the expected number of claims
is now r l because past claims history is used to update the claim frequency distribution.
According to the indifference principle, the amount of deductible d l for a policyholder in
level l is the solution to the equation
)
r l EC 1 = EC 1 + r l
(EC 1 C 1