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

Bonus-Malus Scales 167
of this topic. All the useful results have been derived in an elementary way (the readers
having acquaintance with the theory of Markov chains will rapidly recognize all the classical
machinery taught in textbooks devoted to stochastic processes).
4.1.5 Financial Equilibrium
Exactly as for credibility mechanisms, it is important that the relativities average to 100 %,
resulting in financial equilibrium. This fundamental property is highly desirable: it guarantees
that the introduction of a bonus-malus system has no impact on the yearly premium collection.
The distribution of the amounts paid by the policyholders is modified according to the
reported claims but on the whole, the company gets the same amount of money.
Throughout this chapter, we work with the long-run equilibrium distribution of
policyholders in the bonus-malus levels. We will see that in the long run, the way the
relativities are computed in this chapter ensures that the bonus-malus system is financially
stable. Things are however more complicated in practice. Specifically, some undesirable
phenomena can arise in a transient regime. These issues will be addressed in Chapters 8–9.
4.1.6 Agenda
In Section 4.2, the trajectory of the policyholder accross the bonus-malus levels is modelled
as a Markov chain. Section 4.3 is devoted to transition probabilities, that is, the probability
that the policyholder moves from one level to another over a given time horizon. The longterm
behaviour of the scale is studied in Section 4.4. It is shown there that the proportions
of policyholders in each level of the scale tend to stabilize over time. Various methods to
compute the stationary probabilities are described.
Section 4.5 explains how to compute the relativities using a quadratic loss function. As
for credibility formulas, relativities that are linear in the bonus-malus level are also derived.
Section 4.5.3 examines the interaction between the bonus-malus scale and a priori risk
classification. It is shown there that creating several scales decreases the rating inadequacies.
In Section 4.6, the quadratic loss function is replaced with an exponential one. A
comparison with quadratic relativities is performed, and the influence of the severity
parameter is carefully assessed.
In Section 4.7, we will consider the so-called special bonus rule. According to this rule,
a policyholder who did not report any claim for a certain number of years, and is still in the
malus zone (i.e. in a level with a relativity above 100 %) is automatically sent to the initial
level (i.e. to the level with relativity equal to 100 %). Many compulsory systems formerly
imposed by governments included such a rule. Because of this special rule, the stochastic
process describing the trajectory of the drivers accross the levels is no longer Markovian.
The memoryless property can nevertheless be re-obtained by adding fictitious levels in the
scale. To fix the ideas, we will study the special bonus rule in the former compulsory Belgian
system.
In a competitive market, it can be expected that some policyholders switch from one
insurance company to the other. In a regulated framework, with a unique compulsory bonusmalus
system imposed on all the insurance companies, the drivers will be subject to the
same a posteriori corrections whatever the insurer. If a driver decides to switch to another