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

Credibility Models for Claim Counts 135
Now, considering the penalty in case one claim is reported, we see that the good driver
who reports one claim during the first year will have to pay 178.4 % of the base premium
to be covered by the insurance company during the second year. The average driver in the
same situation pays 171.2 % of the base premium, and the bad driver 153.1 % of the base
premium. The penalties in the case where an accident is reported to the company are thus
decreasing with the a priori annual expected claim frequencies.
The system appears rather severe. Reporting a claim entails a penalty of between 50 and
75 %, which seems difficult to implement in practice. This is a consequence of the financial
balance property. The weighted averages of all figures (in each infinite row) is equal to
100 %. The modest discounts awarded to the majority of claim-free policyholders have
then to be exactly compensated by the penalties supported by the minority of policyholders
reporting claims to the company. This causes large penalties. The severity of the credibility
corrections also appears in the number of claim free years needed to erase the penalty induced
by an accident: 10 years for the good driver, 7 for the average one and 3 for the bad one.
These periods make sense compared to the average claim number per policy.
3.3.7 Discrete Poisson Mixture Credibility Model
The good driver / bad driver model presented in Section 3.2.1 assumes that the portfolio is
composed of two classes of insured drivers. This can easily be extended to several categories
of drivers (for instance, bad, below average, average, above average, excellent). Specifically,
let us assume that each risk class of the portfolio is made of q categories of insured drivers.
Let us denote as p 1 p 2 p q the proportion of drivers in each of these categories. Then,
⎧
1 with probability p 1
⎪⎨ 2 with probability p 2
i =
(3.5)
⎪⎩
q with probability p q
with 0 < 1 < 2 < ···< q . The number N it of claims caused by policyholder i during year
t is distributed as
PrN it = k =
q∑
exp− it j it j k
p
k! j k= 0 1
j=1
Note that the special case q = 2 gives the good driver / bad driver model.
The expected number of claims reported by policyholder i in year T i + 1 is given by
q∑
q∑
EN iTi +1 = EN iTi +1 = j p j = iTi +1 j p j
j=1
If we know that this policyholder reported k claims during the past T i years, we expect
EN iTi +1N i• = k claims in year T i + 1. The computation of EN iTi +1N i• = k requires the
conditional distribution of N iTi +1 given N i• = k. We get it from
j=1