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

Transient Maximum Accuracy Criterion 305
EL A =
VL A =
∑
a n
n=1 l=0 k
∑
8.3.3 Financial Balance
a n
n=1 l=0 k
s∑ ∑ ∫ +
w k
0
lp n
l
kdF
s∑ ∑ ∫ +
w k l − EL A 2 p n kdF
0
We know from Chapter 4 that when the relativities are derived from the asymptotic criterion,
the bonus-malus system is financially balanced when the steady state has been reached. The
only way of keeping this financial balance during the transient behaviour of the bonusmalus
scale is to change the relativities applied to the policyholders so that they are equal
to the r n
l
s in each period n. For commercial reasons, it seems difficult to adopt such a
strategy.
But, when the number of levels and the transistion rules of the bonus-malus system have
been fixed, it is possible to check how the expected financial income evolves with respect to
the financial balance for the different periods n = 1 2 until the steady state is reached.
If the relativities in force are the r l s computed on the basis of the asymptotic maximum
accuracy criterion, then the expected financial income for policies with age n is
s∑
I n = r l PrL n = l
=
l=0
l=0
k
0
l
s∑ ∑ ∫ +
w k r l p n
l kdF (8.7)
where the p n
l
·s are computed using (4.6).
Alternatively, if the relativities are the ¯r l s computed on the basis of the transient maximum
accuracy criterion, then the expected financial income for policies with age n is
Ī n =
=
s∑
¯r l PrL n = l
l=0
s∑ ∑ ∫ +
w k
l=0
k
0
¯r l p n
l kdF (8.8)
The evolution of the expected income, until steady state has been reached, is probably
one of the most important parameters to take into account.
Example 8.3 (Former Belgian Compulsory Bonus-Malus Scale) Table 8.8 displays the
evolution of the expected financial income I n (computed with the steady state relativities)
according to the initial distribution of the policyholders. It slowly converges to 100 %. The
choice of the uniform initial distribution or the top initial distribution leads to an expected
financial income greater than 100 %. Indeed, too many policyholders (with respect to the
steady state situation) are in the malus levels, thus providing a greater income to the company.
Conversely, with a bottom initial distribution, the expected financial income is smaller than
100 % as too many policyholders are in the bonus levels.