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

Credibility Models for Claim Counts 149
and put
w • =
n∑
w t and ˜X n = 1 n∑
w
w j X j
•
Clearly, ˜X n is the weighted average of the X j s. The minimum of
[ (
E − a − b˜X n) ] 2
t=1
t=1
on all the couples a b is obtained for
a =
2
2 + w • M 2 and b = w •M 2
2 + w • M 2
To apply this general result to the Poisson case, we define X j = N ij / ij which gives i =
i , = 1, w j = ij , 2 = 1, M 2 = V i . The best linear approximation to i is then given
by ̂N iTi +1/ iTi +1.
3.4 Credibility Formulas with an Exponential Loss Function
3.4.1 Optimal Predictor
This section purposes to describe an alternative approach based on an exponential loss
function. The exponential loss function is asymmetric and possesses one parameter that
reflects the severity of the credibility correction. This allows us to soften the a posteriori
corrections in case of claims, keeping the financial balance.
When the new premium amount is fixed by the insurer, two kinds of errors may arise: either
the policyholder is undercharged and the insurance company loses its money or the insured is
overcharged and the insurer is at risk of losing the policy. In order to penalize large mistakes to
a greater extent, it is usually assumed that the loss function is a nonnegative convex function
of the error. The loss is zero when no error is made and strictly positive otherwise. The loss
function is generally taken to be quadratic as in the preceding section. Among other choices
we find also the absolute loss and the 4-degree loss; see, e.g., Lemaire & Vandermeulen
(1983). The problem with these two last losses is that the resulting bonus-malus systems are
unbalanced.
We give here a technical result involving an exponential loss function. It is the analogue
of Proposition 3.1.
Proposition 3.2
Under the conditions of Proposition 3.1, the minimum of
[ (
E exp − c ( T+1 − X 1 X 2 X T ))]
on all the measurable functions T → satisfying the constraint
EX 1 X 2 X T = T+1 is obtained for