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

Actuarial Analysis of the French Bonus-Malus System 341
As f ⋆t is the t-fold convolution of a lattice random vector, we will apply the trivariate
extension of De Pril’s algorithm described in Property 9.2. As this algorithm needs a mass
at the origin, we define an auxilliary density function
g ⋆t x y z = f ⋆t x t − y z
Using similar arguments as before we obtain the following recursion
( )
x∑ z∑ t + 1
g ⋆t x y z = e 1+
− 1 g ⋆t x − u y − 1z− wgu 1w
y
u=0 w=0
□
9.3.4 Numerical Illustrations
To illustrate this special case, we use the same parameters as in Section 9.2.8. We assume
that 20 % of the claims concern partial liability, that is q = 02. We numerically solve the
system of two equations for different values of . Table 9.4 gives the results. As before, t
and t decrease with t and an averaging is needed to get a unique set of parameters.
The total income of the company is not much influenced by the value of , and is quite
close to the values listed in Table 9.1.
To obtain unique values for the CRM coefficients, we choose the first age distribution of
Section 9.2.8. The minimization of
E A =
∑
a t t
then gives the values displayed in Table 9.5. The same comments apply to this case.
Specifically, the t s with large values of t play the prominent role, giving optimal
CRM coefficients close to the values obtained with t>20 in Table 9.1.
The values of the optimal CRM coefficients displayed in Table 9.5 are again much smaller
than those implemented by the French law. As before, this is due to the special bonus rule
of the French system (after two consecutive years without claim, the driver goes back to the
initial level of 100 %).
t=1
Table 9.4 Parameters t and t and financial equilibrium for different values of t and .
t = 15 = 20 = 25
t t Financial
equilibrium
t t Financial
equilibrium
t t Financial
equilibrium
1 0.4336 0.1423 09747 0.3316 0.1423 0.9729 0.2674 0.1423 09714
2 0.3253 0.0954 09870 0.2498 0.0953 0.9850 0.2022 0.0953 09832
3 0.2616 0.0727 09984 0.2014 0.0726 0.9965 0.1633 0.0726 09946
4 0.2194 0.0589 10083 0.1692 0.0589 1.0062 0.1374 0.0588 10047
10 0.1128 0.0279 10419 0.0873 0.0279 1.0402 0.0711 0.0279 10386
20 0.0627 0.0149 10634 0.0486 0.0149 1.0624 0.0397 0.0149 10617
30 0.0435 0.0102 10725 0.0337 0.0102 1.0707 0.0276 0.0102 10712