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

7
Bonus-Malus Systems with
Varying Deductibles
7.1 Introduction
In this chapter, we compare bonus-malus systems to deductibles. Specifically, we design a
system in which the policyholders in the malus zone are allowed to choose at each renewal
between a premium surcharge (induced by relativities associated with the bonus-malus scale)
or a deductible in case of a claim during the forthcoming year. If the deductible is selected,
this induces a strong incentive to careful driving. According to signal theory, drivers opting
for the deductible are expected to be better drivers (on average) than those paying the
premium surcharge induced by the upward mode in the bonus-malus scale.
Bonus-malus systems do not take claim amounts into account so that a posteriori
corrections only rely on the number of claims. Holtan (1994) suggested the use of very high
deductibles that may be borrowed by the policyholder from the insurance company. Although
technically acceptable, this approach obviously causes considerable practical problems. While
Holtan (1994) assumes a high deductible which is constant for all policyholders, and thus
independent of the level they occupy in the bonus-malus scale at the claim occurrence time,
the present chapter lets the deductible vary between the levels of the bonus-malus scales, and
also considers a mixed case setup of both premium and deductible surcharge after a claim.
Specifically, the a posteriori premium correction induced by the bonus-malus scale is
replaced with a deductible (in whole or in part). To each level of the bonus-malus scale is
attached an amount of deductible, applied to the claims filed during the coverage period.
Combining bonus-malus scales with varying deductibles presents a number of advantages:
(1) according to signal theory, policyholders choosing varying deductible should be good
drivers;
Actuarial Modelling of Claim Counts: Risk Classification, Credibility and Bonus-Malus Systems
S. Pitrebois and J.-F. Walhin © 2007 John Wiley & Sons, Ltd
M. Denuit, X. Maréchal,