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

Efficiency and Bonus Hunger 241
Computation
The computation of Eff Loi requires the determination of the derivative of r with
respect to the annual expected claim frequency . This derivative is given by
dr
d
= s∑
l=0
d l
d
r l
so that its computation requires the derivative of the stationary probabilities l with
respect to the annual expected claim frequency . To get d l /d, it suffices to
differentiate (4.8): we thus have to solve the linear system
⎧
⎪⎨
d T
= dT
d d
P + T dP
d
⎪⎩
∑ s d l
l=0
d = 0
with respect to the d l /ds.
Global Efficiency
So far, we have defined the Loimaranta efficiency for a given value of the expected annual
claim frequency. To get a value for the portfolio, we have to account for its composition
with respect to rating factors as well as its residual heterogeneity. Hence, the Loimaranta
efficiency for the portfolio is obtained by averaging over all the possible values for as
[ ( ) ]
Eff Loi = E Eff Loi
Loimaranta Efficiency in Portfolio A
Table 5.9 displays the Loimaranta efficiencies for a good driver, with annual expected claim
frequency 9.28 %, for an average driver with annual expected claim frequency 14.09 %, and
for a bad driver with annual expected claim frequency 28.40 %. For the −1/top bonus-malus
scale, the efficiency is larger for the average driver than for the good and bad ones. On
the contrary, for the −1/+ 2 and −1/+ 3 bonus-malus scales, the efficiencies appear to
increase from the good to the average driver, and from the average driver to the bad one.
The global efficiencies listed in Table 5.9 are rather poor, ranging from 23.23 % for the
−1/top bonus-malus scale to 28.39 % in the −1/+ 2 bonus-malus scale. This means that
these bonus-malus systems weakly respond to a change in the underlying claim frequency.
Table 5.9 Loimaranta efficiency for three types of insured drivers (a good driver, with annual
expected claim frequency 9.28 %, an average driver with annual expected claim frequency 14.09 %,
and a bad driver with annual expected claim frequency 28.40 %) and global efficiency for Portfolio
A, for the −1/top, −1/+ 2 and −1/+ 3 bonus-malus scales.
Frequency Scale −1/top Scale −1/ + 2 Scale −1/ + 3
0.0928 02865 02380 02987
0.1409 03144 03793 04008
0.2840 02901 06204 04733
Portfolio A 02323 02839 02775