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

194 Actuarial Modelling of Claim Counts
Table 4.12 Linear relativities with risk classification for the system −1/ + 2
and for Portfolio B.
Level l Unconstrained relativities r l Linear relativities rl
lin
with a priori ratemaking
with a priori ratemaking
5 2232 % 2047%
4 1713 % 1785%
3 1489 % 1522%
2 1121 % 1260%
1 1050% 998%
0 747% 736%
The discrete approximations listed in Tables 3.5–3.6 can then be used in this formula.
In the case where has a unimodal probability density function, the mixed uniform
approximations of Tables 3.8–3.9 can also be used.
4.6 Relativities with an Exponential Loss Function
4.6.1 Bayesian Relativities
This section proposes an asymmetric loss function with one parameter that reflects the
severity of the bonus-malus system. In order to reduce the maluses obtained with a quadratic
loss, keeping a financially balanced system, we resort on an exponential loss function. Such
loss functions have been applied in Section 3.4 in the classical credibility setting. Our purpose
here is to apply exponential loss functions to determine the optimal relativities.
When using the exponential loss function, the goal is now to minimize
[
]
Q exp = E exp−c − r L
(4.18)
under the financial balance constraint Er L = 1. The parameter c>0 determines the
‘severity’ of the bonus-malus scale. The loss (4.18) puts more weight on the errors
resulting in an overestimation of the premium (i.e. r L >), than on those coming from
an underestimation. Consequently, the maluses are reduced, as well as the bonuses since
financial stability has been imposed.
Let us derive the general solution of (4.18).
Proposition 4.1
The solution of the constrained optimization problem (4.18) is
r exp
L
= 1 + 1 c
( [
]
)
E ln Eexp−cL − ln Eexp−cL (4.19)
Proof
First, note that
( [
])
exp ( ) expc exp E ln E exp−cL
cr exp
L =
E [ exp−cL ]