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

Actuarial Analysis of the French Bonus-Malus System 335
of N • and I • , it suffices to integrate the conditional mass function f ⋆t x y with respect
to the structure function f , that is,
f ⋆t x y =
∫
0
f ⋆t x yf d x > 0 0 ≤ y ≤ t
These quantities can then be used to evaluate Er t t
N • I • t.
9.2.8 Numerical Illustration
We will assume that is Gamma distributed with probability density function given by
(1.35). The parameters a and are estimated on the basis of Portfolio A, that is, ̂ = 01474
and â = 0889.
Table 9.1 displays, for different values of t, the coefficients t and t obtained by solving
the system given in Section 9.2.4. We observe a dramatic decrease of the values of t and
t over time. The last column of the table allows us to verify the financial equilibrium
of the system. The total premium income first decreases to 97.61 % and then increases to
107.29 % after 30 years. The discount per claim-free year decreases from 14.23 % to about
1 %. Similarly, the penalty induced by each reported claim decreases from 61.45 % to 6.09 %.
The a posteriori corrections are therefore considerably softened with time.
The decrease of t and t with time t that is apparent from Table 9.1 can be explained
as follows: The aim is that r t t
be as close as possible to the unknown risk parameter .
Since does not depend on t whereas N • and I • are almost surely nondecreasing with t,
the optimal parameters t and t must decrease to compensate for the increase in N • and I • .
This is why averaging over time is needed.
Table 9.2 gives the CRM coefficient r t t
x y = 1 + t x 1 − t y for different periods
of length t and for different values of the total number of claims x. The index ty means that
we have y claimsfree years during the period 0t. For the sake of comparison, Table 9.3
gives the CRM coefficients obtained from classical Bayesian credibility. In this case, the
a priori annual expected claim frequency is multiplied by a + x/a + t as discussed in
Chapter 3. We observe some large discrepancies between the values listed in Tables 9.2
and 9.3.
We see from Tables 9.2 and 9.3 that the discounts awarded to the policyholders who
did not report any claim (column x = 0 in Tables 9.2 and 9.3) are larger with Bayesian
Table 9.1
values of t.
Parameters t and t and financial equilibrium for different
t t t Financial equilibrium
1 0.6145 0.1423 0.9761
2 0.4595 0.0955 0.9985
3 0.3690 0.0727 1.0001
4 0.3092 0.0589 1.0099
10 0.1585 0.0279 1.0431
20 0.0880 0.0149 1.0638
30 0.0609 0.0102 1.0729