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

196 Actuarial Modelling of Claim Counts
Remark 4.4 If no a priori ratemaking is in force, the expressions (4.20) and (4.22) are
equal to those derived in Denuit & Dhaene (2001), that is
Eexp−cL = l =
∫ +
0
exp−c l dF
PrL = l
and
[
]
E ln Eexp−cL =
=
s∑
PrL = l ln Eexp−cL = l
l=0
s∑
PrL = l ln
l=0
(∫ +
0
exp−c l dF
PrL = l
)
4.6.2 Fixing the Value of the Severity Parameter
Let us briefly explain a possible criterion to fix the value of the parameter c. First, note that
lim r exp
l c→0
= EL = l = r quad
l
where r quad
l
is the relativity obtained with a quadratic loss function. Letting c tend to 0 thus
yields Norberg’s approach. In other words, the bonus-malus scale becomes more severe as
c decreases. Now, the ratio of the variances of the premiums obtained with an exponential
and a quadratic loss is given by
Vr exp
L
Vr quad
L = 1 V [ ln E exp−cL ]
c 2 V [ ] = % ≤ 100 %
E L
The fact that the ratio of the variances is less than unity comes from the Jensen inequality.
The idea is then to select the variance of the premium in the new system as a fraction of the
corresponding variance under a quadratic loss (for instance = 25, 50 or 75 %). Of course,
other procedures can be applied. For instance, the actuary could select the value of r 0 ,orof
r s , and then compute c in order to match this value.
4.6.3 Linear Relativities
In practice, a linear scale of the form rl
lin = + l, l = 0 1s, could be desirable. Let
us now indicate how Gilde & Sundt’s (1989) approach can be extended using exponential
loss functions. The aim is now to minimize the objective function
[
= E exp ( − c − − L )]