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

Bonus-Malus Systems with Varying Deductibles 287
monetary units. The distribution of S is then determined by Panjer’s algorithm. Claim
amounts are discretized according to Method (7.4).
7.4.3 Annual Deductible
The relativities computed for the scale −1/top are displayed in the second column of Table 7.1
and for the scale −1/ + 2 in Table 7.2. In the pure bonus-malus case, it is thus clear that the
r l s associated with the upper levels are considerable (more than 300 % for level 5 in case of
the scale −1/ + 2). Now, let us replace the r l s in the malus zone (i.e. levels 1 to 5) with an
annual deductible d l . In order to obtain the d l s from equation (7.10), we first discretize the
claim sizes using Method (7.4). Finally, (7.10) is solved numerically using routines available
from the SAS R /IML package.
The third column of each table displays the new relativities. In this case, the maluses
disappear (r l = 100 % for l = 15) and are compensated for by the deductibles listed
in the two last columns. The fourth column shows the deductible to be applied if the loss
amounts are Negative Exponentially distributed and the last column shows the deductible to
be applied if the loss amounts are LogNormally distributed. Since the LogNormal distribution
has a thicker tail than the Negative Exponential one, we expect larger amounts of deductible
for the former. This is indeed the case, as can be seen from Tables 7.1 and 7.2.
The very high r l s in the second column induce high amounts of deductible, even in the
Negative Exponential case. Therefore, this solution seems to be difficult (if not impossible)
to implement in practice.
Table 7.1 Results for an annual deductible varying according to the level occupied in the malus zone
and for the scale −1/top.
Level l r l r l with deductible d l if C 1 ∼ xp d l if C 1 ∼ or
5 1973 % 100 % 18 367 35 253
4 1709 % 100 % 14 028 29 303
3 1507 % 100 % 10 472 23 544
2 1348 % 100 % 7470 17 934
1 1220 % 100 % 4897 12 452
0 547 % 54.7 % 0 0
Table 7.2 Results for an annual deductible varying according to the level occupied in the malus zone
and for the scale −1/ + 2.
Level l r l r l with deductible d l if C 1 ∼ xp d l if C 1 ∼ or
5 3091 % 100 % 34 576 50 794
4 2414 % 100 % 25 076 42 704
3 2077 % 100 % 20 004 37 241
2 1429 % 100 % 9027 20 933
1 1302 % 100 % 6563 16 079
0 624 % 62.4 % 0 0