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

Credibility Models for Claim Counts 125
A priori (i.e. at time t = 0 for a policyholder without any claim record), the actuary is not
able to distinguish between good and bad drivers. The expected number of claims is then
given by
PrGood G + PrBad B = 009
Considering a policyholder who reported k claims during the first year, it is nevertheless
possible to compute the probability that he is a good driver: calling upon Bayes’ Theorem
yields
PrGoodk claims =
=
Prk claimsGood PrGood
Prk claimsGood PrGood + Prk claimsBad PrBad
exp− G k G PrGood
exp− G k G PrGood + exp− B k B PrBad
We get the values listed in Table 3.1 for increasing ks. Clearly, these probabilities are
decreasing with the number of claims reported. If the policyholder does not report any claim
during the first year, then the probability that he is a good driver is increased from 60 %
a priori to 62.37 % a posteriori. As soon as one claim is reported during the first year,
this probability decreases from 60 % a priori to 35.59 % a posteriori. The more claims are
reported, the less likely that the policyholder is a good driver. The a posteriori probability of
being a good driver remains nevertheless positive whatever the number of claims reported
to the insurance company.
A posteriori (i.e. at time t = 1), the actuary knows the number k of claims reported by the
policyholder during the year and should incorporate this additional information in the price
list. Specifically, if k claims have been reported, the expected number of claims for year
two is
PrGoodk claims G + PrBadk claims B
The claim record of the policyholder thus modifies the weights assigned to G and B . The
values of the expected number of claims for year two according to the number k of claims
Table 3.1 Expected numbers of claims for year two in the good driver/bad driver
model given the number k of claims reported during the first year.
# of claims k
reported during
year 1
PrGoodk
claims (%)
PrBadk
claims (%)
Expected
number of
claims for year 2
0 6237 3763 00876
1 3559 6441 01144
2 1555 8445 01344
3 578 9422 01442
4 200 9800 01480
5 068 9932 01493