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

Efficiency and Bonus Hunger 221
account the fact that only ‘expensive’ claims are reported to the insurance company. We
will consider that each policyholder has his own unknown retention limit, depending on the
level occupied inside the bonus-malus scale as well as on observable characteristics (like
age or gender, for instance). The policyholder reports the accident to the company only if its
cost exceeds the retention limit. A regression model accounting for the fact that we observe
the maximum between the accident cost and the retention limit is then fitted to the observed
claim data. We then recover probability models for the accident costs (whereas formerly,
we modelled claim costs). The claim frequencies can also be corrected in order to obtain
accident frequencies (this is done in Section 5.4.2).
Section 5.4.3 examines the optimal claiming strategy, that should be followed by rational
policyholders. A strategy for each policyholder can be defined by a vector rl 0 rl s T
where rl l is the retention limit for the policyholder occupying level l in the bonus-malus
scale. This means that the cost of any accident of amount less than rl l is borne by the
policyholder in level l. The claims causing higher costs are reported to the insurer. The
problem is to determine optimal values for the rl l s. This can be done using the Lemaire
algorithm. The optimal retention limits depend on the level occupied in the scale, on the
annual expected claim frequency as well as on a discount rate. Note that the Lemaire
algorithm gives the optimal retention limit obtained by means of dynamic programming.
The resulting strategy should be adopted by rational policyholders, but may differ from the
one empirically observed in insurance portfolios. The optimal retentions obtained from the
Lemaire algorithm can also be seen as a measure of the toughness of the bonus-malus system.
A system that induces large rl l s is more severe than another one yielding moderate retention
limits. As such, the rl l s can also be used to measure the efficiency of the bonus-malus system.
The claim costs play an important role in this chapter. As explained above, modelling
claim sizes is a difficult issue because of the strong heterogeneity and of the presence of
large claims. Nevertheless, it seems reasonable to agree that large claims will always be
reported to the company, so that only moderate claims are subject to bonus hunger.
In this chapter, we work within a given bonus-malus scale, whose levels are numbered
from 0 to s, with fixed relativities r 0 r s (the r l s have been computed as explained in
Chapter 4, or have been derived from marketing considerations, but are treated as given for
the whole chapter).
5.1.6 Descriptive Statistics for Portfolio C
The numerical illustrations of this chapter are based on the observation of a Belgian motor
third party liability insurance portfolio during the year 1997. This portfolio, henceforth
referred to as Portfolio C, comprised 163 660 policies.
The following variables are available for portfolio C: As far as policyholders’
characteristics are concerned, we know the Gender (male or female), the age (variable Ageph,
four classes: 18–24, 25–30, 31–60 and > 60), the place of residence (variable City, rural or
urban) and the Use of the car (private or professional). Concerning the insured vehicle, we
know its age (variable Agev, four classes: 1–2, 3–5, 5–10 and > 10 years), the type of Fuel
(petrol or gasoil) and its Power (three classes: < 66 kW, 66–110 kW and > 110kW). About
the type of contract, we know whether the premium payment has been split up (variable
Premium split, with payment once a year, or more than once a year) and the type of Coverage
(motor third party liability only, or motor third party liability together with some more