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

244 Actuarial Modelling of Claim Counts
De Pril efficiency Eff DeP is thus defined analogously to Eff Loi , substituting V l for r.
Note that Eff DeP l now depends on the starting class l. The initial class can then be
selected so as to maximize Eff DeP l .
Computation
To compute Eff DeP l , we need the derivatives dV l /d of the V l satisfying (5.7).
These derivatives can be obtained by solving the system
dV l
d
= v ∑ exp− k
k!
k=0
This system admits a unique solution.
(( ) k
− 1 V Tk l + dV )
T k l
l= 0s
d
Global Efficiency
At the portfolio level, the efficiency is then obtained by averaging over all the possible
values for , that is,
Eff DeP l = EEff DeP l
De Pril Efficiency in Portfolio A
Table 5.10 displays the De Pril efficiencies associated with the highest level 5 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 %. The discount factor is taken to be v = 1/104. De Pril efficiency behaves roughly
as the Loimaranta efficiency displayed in Table 5.9.
Figure 5.3 displays the De Pril efficiency as a function of the annual expected claim
frequencies for the three bonus-malus systems. We see that the efficiency first increases and
then decreases, reaching its maximum value at about the same frequencies as the Loimaranta
efficiency. The shape of both efficiencies is pretty much the same.
Let us now use the De Pril efficiency to select the optimal starting level. We have computed
in Table 5.11 the values of Eff DeP l and Eff DeP l according to the initial level. We see
that the optimal starting level is 0 for the three −1/top, −1/ + 2 and −1/ + 3 bonus-malus
scales.
Table 5.10 De Pril 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 (starting level: level 5).
Frequency Scale −1/top Scale −1/ + 2 Scale −1/ + 3
0.0928 02192 01871 02252
0.1409 02482 02880 03039
0.2840 02414 04633 03741
Portfolio A 01817 02186 02150