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

220 Actuarial Modelling of Claim Counts
The first step of any actuarial analysis consists in displaying descriptive statistics in order
to figure out the composition of the portfolio and the marginal impact of the rating factors.
In a second stage, available explanatory variables are incorporated to the policyholders’
expected claim frequencies and severities with the help of generalized regression models.
5.1.3 Large Claims and Extreme Value Theory
Large claims generally affect liability coverages. These major accidents require a separate
analysis. The reason for a separate analysis of small (or moderate) and large losses is that
no standard parametric model seems to emerge as providing an acceptable fit to both small
and large claims. The main goal is then to determine an optimal threshold separating the
two types of losses.
Extreme Value Theory and Generalized Pareto distributions can be used to set the value
of this threshold. Specifically, graphical tools including the Pareto index plot and the
Gertensgarbe plot can be used to estimate the threshold defining the large losses. In the
former case, the maximum likelihood estimator of the Pareto tail parameter is computed
for increasing thresholds until it becomes approximately constant. The Gertensgarbe plot
is based on the assumption that the optimal threshold can be found as a change point in
the ordered series of claim costs and that the change point can be identified by means of
a sequential version of the Mann-Kendall test as the intersection point between normalized
progressive and retrograde rank statistics.
5.1.4 Measuring the Efficiency of the Bonus-Malus Scales
As explained in the preceding chapters, the basis of fair ratemaking in motor insurance is
the fact that each policyholder is charged a premium that is proportional to the risk that
he actually represents. The accident proneness of a policyholder being represented by the
relative risk parameter , we expect that a relative change in will have the same impact
on the premium paid to the insurance company. If this is the case then the system is said to
be fully efficient.
Section 5.3 reviews two concepts of efficiency: Loimaranta efficiency and De Pril
efficiency. Both intend to measure how the bonus-malus system responds to a change in the
riskiness of the driver. Loimaranta efficiency is solely based on the stationary probabilities
whereas De Pril efficiency is a transient concept and uses the time value of money (through
discounting).
5.1.5 Bonus Hunger and Optimal Retention
Since the penalty induced by the bonus-malus system is independent of the claim amount,
a crucial issue for the policyholder is therefore 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, known as the hunger for bonus after Philipson (1960), is studied in Section
5.4.
Section 5.4.1 is devoted to the censorship of claim amounts and claim frequencies arising
from bonus-malus systems. Specifically, a statistical model is specified, that takes into