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

152 Actuarial Modelling of Claim Counts
and
⋆⋆ k i1 k iTi ≥ ⋆ k i1 k iTi if k i• < i•
Let us define
exp c = 1 (
c ln 1 + c )
iT i +1
a + i•
to be the weight given to k i• in the a posteriori evaluation (3.17) in the Poisson-
Gamma model. We have that lim c→+ exp c = 0. Moreover, routine calculations show
that d/dc exp c < 0, so that the weight given to the observed average claim number
decreases as c increases. This provides an intuitive meaning of the parameter c: if c
increases, then the a posteriori merit-rating scheme becomes less severe, and at the limit
for c →+, the premium no longer depends on the incurred claims. If the asymmetry
factor c tends to + then all the risks within the same tariff class pay the same premium:
there is no longer an experience rating. Conversely, the weight given to past claims
under an exponential loss function tends to the weight under a quadratic loss function
as c → 0.
3.4.3 Linear Credibility
Another possibility is to determine a linear credibility premium based on an exponential
loss function, by considering predictors of the form b 0 + ∑ T i
t=1 b jN it . The b j s minimize the
Lagrangian function
[
b 0 b 1 ··· b t = E exp ( − c iTi +1 i − b 1 N i1 − b 2 N i2 −···−b Ti
N iTi − b 0 )]
[
]
− E b 0 + b 1 N i1 + b 2 N i2 +···+b Ti
N iTi − iTi +1
Setting to 0 the derivatives of with respect to , b 0 b 1 b Ti
in the Poisson-Gamma case, i.e. (3.17).
yields the same result as
3.4.4 Numerical Illustration
Let us now illustrate the use of the exponential loss function in credibility. To this end, let us
consider the Negative Binomial fit to Portfolio A described in Table 2.7. Thus, i is taken
to be ama a distributed, with estimated parameter â = 1065.
Formula (3.17) allows us to compute the a posteriori correction as a function of the
number T i of coverage periods and of the total number of claims k • filed to the company.
The results obtained with c = 1 are displayed in Table 3.13 for a good driver, in Table 3.14
for an average driver, and in Table 3.15 for a bad driver. Tables 3.16–3.18 are the analogues
for c = 5.
Let us first compare the a posteriori corrections listed in Table 3.13 with those of
Table 3.2 corresponding to a quadratic loss function (i.e. to c = 0). We see that the
application of an exponential loss function slightly reduces the penalties in case of claims