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

186 Actuarial Modelling of Claim Counts
=
∑k w k
∫ +
0
l k dF
∑k w k
∫ +
0
l k dF (4.13)
It is easily seen that
[ ]
Er L = E EL = E = 1
resulting in financial equilibrium once steady state is reached.
Remark 4.2 If the insurance company does not enforce any a priori ratemaking system,
all the k s are equal to E = and (4.13) reduces to the formula
r l =
∫ +
0
l dF
∫ +
0
l dF
(4.14)
that has been derived in Norberg (1976).
The way a priori and a posteriori ratemakings interact is described by
EL = l = ∑ k
k Pr = k L = l
= ∑ k
=
k
PrL = l = k w k
PrL = l
∑k k w k
∫ +
0
l k dF
∑k w k
∫ +
0
l k dF (4.15)
If EL = l is indeed increasing in the level l, those policyholders who have been granted
premium discounts at policy issuance (on the basis of their observable characteristics) will
also be rewarded a posteriori (because they occupy the lowest levels of the bonus-malus
scale). Conversely, the policyholders who have been penalized at policy issuance (because
of their observable characteristics) will cluster in the highest bonus-malus levels and will
consequently be penalized again.
Example 4.11 (−1/Top Scale, Portfolio A) The results for the bonus-malus scale −1/top
are displayed in Table 4.3. Specifically, the values in the third column are computed with
the help of (4.14) with â = 0889 and ̂ = 01474. Those values were obtained in Section 1.6
by fitting a Negative Binomial distribution to the portfolio observed claim frequencies given
in Table 1.1. Integrations have been performed numerically with the QUAD procedure of
SAS R /IML. The fourth column is based on (4.13) with â = 1065 and the ̂ k s listed in
Table 2.7.
Once the steady state has been reached, the majority of the policies (58.5 % of the
portfolio) occupy level 0 and enjoy the maximum discount. The remaining 41.5 % of the
portfolio are distributed over levels 1–5, with about 13 % in level 5 (those policyholders who
just claimed). Concerning the relativities, the minimum percentage of 54.7 % when the a
priori ratemaking is not recognized becomes 61.2 % when the relativities are adapted to the