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

240 Actuarial Modelling of Claim Counts
The total pure premium of a policyholder belonging to the reference class is then
E31676 + E4306 = E35982
Note that the fitted mean equal to E 368 018.80 is close to the observed mean of the 17 large
losses recorded in Table 5.2.
Recall that the analysis of large claims performed in this chapter is based on an estimation
of their final cost six months after the end of the observation year 1997. We thus work with
incurred losses (payments plus reserves). In practice, the company should maintain a data
base recording the costs of large claims that have occurred in the past, corrected for the
different sources of inflation (reinsurance companies can often provide valuable assistance
to the ceding companies in this respect). The typical price for a reinsurance treaty covering
motor third party liability insurance losses in excess of E 350 000 represents about 5 % of
the total motor premium income of a Belgian insurance company, so that the expected cost
of large claims computed in Portfolio C seems to be of the right order of magnitude.
Remark 5.5 If the sum of the individual pure premiums obtained above exceeds the
observed total loss for the insurance portfolio during the reference period, or if we expect
larger losses in the future (because, e.g., of different sources of inflation), we can then
keep the same relative premium amounts applied to the anticipated future total claim cost.
This allows us to incorporate in the individual premiums observed trends in the total claim
amount.
5.3 Measures of Efficiency for Bonus-Malus Scales
The elasticity of a bonus-malus system measures its response to a change in the expected
claim frequency or expected aggregate claim amount. We expect that the premium paid by
the policyholders subject to bonus-malus scales is increasing in the expected claim frequency
or total claim amount. The rate of increase is related to the concept of efficiency.
5.3.1 Loimaranta Efficiency
Definition
Let us denote as r the average relativity once stationarity has been reached, for a
policyholder with annual expected claim frequency , i.e.
r =
s∑
l r l
l=0
The Loimaranta efficiency Eff Loi is then defined as the elasticity of the relative premium
induced by the bonus-malus system, that is,
Eff Loi =
dr
r
d
=
d ln r
d ln