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

Efficiency and Bonus Hunger 251
Moreover, the random variable N large
i is independent from Mi
small Ni small . Both M i and
N i are mixed Poisson distributed. Specifically, given i = , M i ∼ oi˜ i + large
i and
N i ∼ oi i + large
i . Note that the variable that has been analysed in the preceding chapters
is N i , the number of accidents reported to the insurer, and not M i .
5.4.3 Lemaire Algorithm for the Determination of Optimal Retention
Limits
In the preceding section, we explained how to correct the cost of claims to obtain the accident
costs. This also allowed us to switch from claim frequencies to accident frequencies. To this
end, we estimated the retention limits that were used by the policyholders on the basis of the
costs of the claims they filed to the insurance company. The aim of this section is somewhat
different. Having the distribution of the accident costs and of the accident frequencies, we
would like to determine the optimal claiming strategy (which may differ from the observed
claiming strategy inferred in the previous section).
For each level of the scale, a critical claim size is determined: if the cost of the claim falls
below this critical threshold then the rational policyholder should not report the accident to
the company. Conversely if the cost exceeds this threshold, the rational policyholder should
report the claim to the company. Note the close similarity with deductibles: under coherent
behaviour, the bonus-malus scale is equivalent to a set of deductibles depending on the level
occupied in the scale.
Cost of Non-Reported Accidents
Let rll be the optimal retention for a policyholder with expected annual accident
frequency occupying level l in the bonus-malus scale. Here, rll is not a random
variable, but an unknown constant to be determined.
Assume that this policyholder has caused an accident with cost x at time t, 0≤ t