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

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Actuarial Modelling of Claim Counts
bodily injury and other severe claims. Not incorporating claim sizes in bonus-malus systems
and a priori risk classification requires an (implicit) assumption of independence between
the random variables ‘number of claims’ and ‘cost of a claim’, as well as the belief that the
latter does not depend on the driver’s characteristics. This means that the actuarial practice
considers that the cost of an accident is, for the most part, beyond the control of a driver: a
cautious driver reduces the number of accidents, but for the most part cannot control the cost
of these accidents (which is largely independent of the mistake that caused it). This belief
will be challenged in Chapter 5.
The penalty induced by the majority of bonus-malus systems being independent of the
claim amount, policyholders have to decide whether it is profitable or not to report small
claims (in order to avoid an increase in premium). Cheap claims are likely to be defrayed
by the policyholders themselves, and not to be reported to the company. This phenomenon
is known as the hunger for bonus and censors claim amounts and claim frequencies. In
Chapter 5, a statistical model is specified, that takes into account the fact that only ‘expensive’
claims are reported to the insurance company. Retention limits for the policyholders are
determined using the Lemaire algorithm.
In a few bonus-malus systems, however, reporting a ‘severe’ claim (typically, a claim
with bodily injuries) entails a more severe penalty than reporting a ‘minor’ claim (typically,
a claim with material damage only). In the system in force in Japan before 1993, claims
involving bodily injuries were penalized by four levels, while claims with property damage
only were penalized by only two levels. Bonus-malus systems using different types of events
to update premium amount will be examined in Chapter 6.
In Chapter 7, we examine an innovative system using variable deductibles rather than
premium relativities. It differs from the systems studied in preceding chapters in that
it mixes elements of both a conventional bonus-malus system and a set of deductibles
depending on the level occupied in the bonus-malus scale. The first system is a conventional
discount system with loss of discount in the case where a claim at fault is reported.
The second system also has a variable discount scale, which can increase with claim-free
experience. However, there is no stepback of the discount on claim, only a stepback of the
deductible.
Aims of This Book
About ten years after the seminal book ‘Bonus-Malus Systems in Automobile Insurance’
by Professor Jean Lemaire, we aim to offer a comprehensive treatment of the various
experience rating systems applicable to automobile insurance and their relationships with
risk classification.
We hope that the present book will be useful for students in actuarial science, actuaries
(both practitioners and in academia) and more generally for all the persons involved in
technical problems inside insurance companies or consulting firms. For the first time, systems
taking into account the exogeneous information are presented in an actuarial textbook.
Many numerical illustrations carried out with advanced statistical softwares allow for a deep
understanding of the concepts.
The present book is the result of a close and fruitful collaboration between the Institute of
Actuarial Science of the Université Catholique de Louvain, Louvain-la-Neuve, Belgium, its
spin-off consulting firm Reacfin SA and the reinsurance company Secura, based in Brussels.