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

260 Actuarial Modelling of Claim Counts
by rewarding drivers who do not cause an accident, or incur a traffic law violation. It is
based on several types of events (major and minor at-fault accidents and traffic violations).
Specifically, each policyholder is assigned a level between 9 and 35, based on his driving
record during the previous six years. A new driver begins at level 15 (relativity of 100 %).
Occupying any level below 15 entails a premium discount, while above level 15 the driver
pays a surcharge. For each incident-free year of driving, the policyholder goes down one
level. The driver will move up a certain number of levels based on the type of incident:
two levels for a minor traffic violation, three levels for a minor at-fault accident, four levels
for a major at-fault accident and five levels for a major traffic violation. The Massachusetts
system ‘forgets’ all incidents after six years.
This chapter addresses the actuarial modelling of such systems, with penalties depending
on different types of events. As before, the modelling uses the concept of Markov Chains.
We will see that under mild assumptions, the trajectory of each policyholder in the scale
can be modelled with the aid of discrete-time Markov processes. The relativities associated
with each level will then be computed using the maximum accuracy principle discussed in
Chapter 4.
6.2 Multi-Event Credibility Models
6.2.1 Dichotomy
Let Nit
mat be the number of claims with material damage only, reported by policyholder i
during period t. Similarly, let Nit
bod be the number of claims with bodily injuries, and let
N tot
it
= N mat
it
+ N bod
it
be the total number of claims. Policyholder i, i = 1n, is assumed to have been observed
during T i periods. In the previous chapters, Nit
tot was the variable of interest. Here, a dichotomy
is operated, and we study Nit
mat and Nit
bod separately.
6.2.2 Multivariate Claim Count Model
Overdispersion and possible dependence between N mat
it
correlated random effects i
mat
i
mat = i
mat , we assume that
where mat
it
and bod
i
N mat
it
such that E mat
i
∼ oi ( mat
it
= d it expscore mat
it
, and given i
bod
N bod
it
∼ oi ( bod
it
mat
i
and Nit
bod
= E bod
)
= i
bod , we assume that
)
bod
i
are introduced via possibly
= 1. Specifically, given
where bod
it
= d it expscore bod
it
. Both scores are linear combinations of explanatory variables
specific to policyholder i and year t (summarized in a vector x it ). Specifically,
score mat
it
= mat
p∑
0
+
j=1
mat
j
x itj
i