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

258 Actuarial Modelling of Claim Counts
De Pril (1979) defined L n l k t as the amount that the actual accident must exceed in
order to justify the filing of a claim, if the policyholder is at time t of period n in level l and
has already filed k claims. The optimal value of L n l k t is determined to minimize the
discounted expectation of the total future costs (premiums and self-defrayed accidents) for
the policyholder. After De Pril (1979), De Pril & Goovaerts (1983) determined bounds
for the optimal critical claim size when only incomplete information about the claim amount
distribution is available. They considered −1/top bonus-malus scales.
Dellaert ET AL. (1990) proved that under mild conditions the optimal decision rule is
to claim for damages with amount above a certain limit. In some instances, policyholders
are allowed to decide at the end of an insurance year which damages occurred during the
year should be claimed; see, e.g., Martin-Lof (1973). This means that the policyholder
has perfect information about the number of accidents and the corresponding damages at
the moment he/she decides which damages to claim. This situation has been investigated
by Dellaert ET AL. (1991). Let us also mention that Dellaert ET AL. (1993) considered
damage insurance (where, in addition to the bonus hunger phenomenon, the optimal stopping
rule to terminate the insurance has to be determined).
Holtan (2001) envisaged the loss of bonus after a claim as a rate of interest paid from
the customer to the insurer, and studied the hunger for bonus from this viewpoint.
Optimal claiming rules have also been considered in Operational Research, using Markov
decision processes. When a driver is involved in a motor accident decisions have to be made
as to whether or not a claim should be made. Hastings (1976) considered this problem as
a Markov decision process with the expected cost over a finite horizon (where the relevant
costs are repair costs and premium costs) as objective function. See also Haehling von
Lanzenauer (1974) and Hey (1985). Norman & Shearn (1980) proposed the following
simple rule of thumb that is shown to work well in their case: irrespective of when the
accident occurs, claim only if the amount of the claim exceeds the difference over the next
4 years between the total premiums payable if a claim is made and those payable if it is not,
assuming that no further claims will be made. Chappell & Norman (1989) demonstrated
that this simple rule was less efficient in the case of protected bonus. Kolderman &
Volgenant (1985) examined the same problem under the assumption that only one claim
is needed to change the insurance premium category, and any extra claims in the year have
no effect on the current repair estimate for an accident.
Walhin & Paris (2000) derived the actual claim amount and frequency distributions
within a bonus-malus system. As explained above, policyholders should defray the small
claims to avoid the penalties induced by the bonus-malus system. Consequently, there are
more accidents than the number of claims filed by the insurer: the insurance data are censored.
The kind of censorship is nevertheless very particular, and much more complicated than the
phenomena encountered in classical nonlife problems (where losses are censored because
they exceed some policy limit, or fall below a given deductible). The procedure described
in Section 5.4.1 is taken from Denuit, Maréchal, Pitrebois & Walhin (2007a), where
alternative approaches to obtain uncensored accident distributions can be found.
Even if the bonus-hunger phenomenon has been extensively studied in connection with
bonus-malus scales, the same idea applies to credibility systems. See, e.g., Norberg (1975)
and Sundt (1988) for an illustration.