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

214 Actuarial Modelling of Claim Counts
processes, but that can be made Markovian by increasing the number of states as we did in
Section 4.7.
The notion of distance found a second life in probability in the form of metrics in
spaces of random variables and their probability distributions. The study of limit theorems
(among other questions) made it necessary to introduce functionals evaluating the nearness
of probability distributions in some probabilistic sense. In this chapter, the total variance
and Kolmogorov distances have been used in connection with bonus-malus systems. We
refer the interested reader to Chapter 9 of Denuit ET AL. (2005) for a detailed account of
probability metrics and their applications in risk theory.
The premium relativities are traditionally computed with the help of a quadratic loss
function, in the vein of Norberg (1976). Other loss functions nevertheless also deserve
consideration, such as the exponential loss function applied by Denuit & Dhaene (2001)
to the computation of the relativities. For the sake of completeness, let us mention that
the absolute value loss function has been successfully applied to the determination of the
relativities by Heras, Vilar & Gil (2002) and Heras, Gil, Garcia-Pineda & Vilar (2004).
If in a given market, companies start to compete on the basis of bonus-malus systems,
many policyholders could leave the portfolio after the occurrence of an accident, in order to
avoid the resulting penalties. Those attritions can be incorporated in the model by adding a
supplementary level to the Markov chain (in the spirit of Centeno & Andrade e Silva
(2001)). Transitions from a level of the bonus-malus scale to this state represent a policyholder
leaving the portfolio whereas transitions from this state to any level of the bonus-malus scale
mean that a new policy enters the portfolio.
It has been assumed throughout this chapter that the unknown expected claim frequencies
were constant and that the random effects representing hidden characteristics were timeinvariant.
Dropping these assumptions makes the determination of the relativities much
harder. We refer the interested reader to Brouhns, Guillén, Denuit & Pinquet (2003)
for a thorough study of this general situation. A fundamental difference with the traditional
approaches is that we lose the homogeneity of the chain in a dynamic segmented environment.
Indeed, if the observable characteristics of the policyholders are allowed to vary in time, the
claim frequencies are no longer constant and the trajectory of the policyholder in the bonusmalus
scale is no longer described by a homogeneous Markov chain, but well described
by a non-homogeneous one. Consequently, the classical techniques based on stationary
distributions cannot be applied to the problem of determining the relativities. Brouhns,
Guillén, Denuit & Pinquet (2003) propose a computer-intensive method to calibrate
bonus-malus scales. Their paper clearly illustrates the strong complementarity of a priori
and a posteriori ratemakings. The main originality of their approach is to compare on the
basis of real data four different credibility models: static versus dynamic heterogeneity, with
and without recognizing a priori risk classification. The impact of the different assumptions
becomes clear in the numerical illustrations.
Andrade e Silva & Centeno (2005) suggested the use of geometric relativities instead
of linear ones. Specifically, under a quadratic loss function, the relativity associated with
level l becomes r geo
l
= l , with and positive. As pointed out in Remark 4.3 for linear
relativities, finding the geometric relativities amounts to finding the best approximation l
to the Bayesian relativities, that is, and solve
min E[ (EL
−
L ) 2 ]