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

166 Actuarial Modelling of Claim Counts
4.1.2 The Nature of Bonus-Malus Scales
When a merit rating plan is in force, the amount of premium paid by the policyholder
depends on the rating factors of the current period but also on claim history. In practice, a
bonus-malus scale consists of a finite number of levels, each with its own relative premium.
New policyholders have access to a specified level. After each year, the policy moves up
or down according to transition rules and to the number of claims at fault. The premium
charged to a policyholder is obtained by applying the relative premium associated to his
current level in the scale to a base premium depending on his observable characteristics
incorporated into the price list.
4.1.3 Relativities
The problem addressed in this chapter is the determination of the relative premiums attached
to each of the levels of the scale when a priori classification is used by the company. The
relativity associated to level l is denoted as r l . The meaning is that a policyholder occupying
level l in the bonus-malus scale has to pay r l times the base premium to be covered by the
insurance company.
The severity of the a posteriori corrections must depend on the extent to which amounts
of premiums vary according to observable characteristics of policyholders. The key idea
is that both a priori classification and a posteriori corrections aim to create tariff cells
as homogeneous as possible. The residual heterogeneity inside each of these cells being
smaller for insurers incorporating more variables in their a priori ratemaking, the a posteriori
corrections must be softer for those insurers.
The framework of credibility theory, with its fundamental notion of randomly distributed
risk parameters, was employed in analysis of bonus-malus systems by Pesonen as early as
1963. In this chapter, we will keep the framework of Definition 3.1. According to Norberg
(1976), once the number of classes, the starting level and the transition rules have been fixed,
the optimal relativity associated with level l is determined by maximizing the asymptotic
predictive accuracy. Formally, the relativities minimize the mean squared deviation between
a policy’s expected claim frequency and its premium in the year t as t →+. The optimal
relativity for level l is thus equal to the conditional expected risk parameter for an infinitely
old policy, given that the policy is in level l.
4.1.4 Bonus-Malus Scales and Markov Chains
In most of the commercial bonus-malus systems, the knowledge of the current level and the
number of claims during the current period suffice to determine the next level in the scale.
So the future (the level for year t + 1) depends only on the present (the level for year t and
the number of accidents reported during year t) and not on the past. This is closely related to
the memoryless property of the Markov chains. If the claim numbers in different years are
(conditionally) independent then the trajectory of a given policyholder in the bonus-malus
scale will be a (conditional) Markov chain. Sometimes, fictitious levels have to be introduced
to recover the memoryless property.
The treatment of bonus-malus scales is best performed in the framework of Markov
chains. This chapter is nevertheless self-contained, and does not require any prior knowledge