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

122 Actuarial Modelling of Claim Counts
contracts that belong to a (more or less) heterogeneous portfolio, where there is limited or
irregular claim experience for each contract but ample claim experience for the portfolio.
Credibility theory can be seen as a set of quantitative tools that allows the insurers to perform
experience rating, that is, to adjust future premiums based on past experience. In many
cases, a compromise estimator is derived from a convex combination of a prior mean and
the mean of the current observations. The weight given to the observed mean is called the
credibility factor (since it fixes the extent to which the actuary may be confident in the
data).
3.1.3 Limited Fluctuation Theory
There are different types of credibility mechanisms: limited fluctuations credibility and
greatest accuracy credibility. Limited fluctuation credibility theory was developed in the early
part of the 20th century in connection with workers compensation insurance by Mowbray
(1914). It provides a mechanism for assigning full or partial credibility to a policyholder’s
experience. In the former case, the policy is rated on the basis of its own claims history,
whereas in the latter case, a weighted average of past experience and grand mean is used
by the insurer. Although the limited fluctuation approach provides simple solutions to the
problem, it suffers from a lack of theoretical justification. We will not consider this approach
in this book. Instead, we will consider the greatest accuracy credibility theory formalized by
Bühlmann (1967,1970).
3.1.4 Greatest Accuracy Credibility
The idea behind greatest accuracy credibility theory can be summarized as follows: Tariff
cells include policyholders with similar underwriting characteristics; each of them is viewed
as homogeneous with respect to the underwriting characteristics used by the insurance
company. Of course, the risks in the cell are not truly homogeneous: there still remains some
heterogeneity in each of the tariff cells, as explained in the preceding chapters. To reflect this
heterogeneity, the relative risk level of each policyholder in the rating cell is characterized by
a risk parameter , but the value of varies by policyholder. If = 50 % then the expected
number of claims reported by this policyholder is half of the claim frequency corresponding
to the rating cell, whereas if = 300 % then the expected number of claims for this individual
is three times the claim frequency of the rating cell. Of course, even if assuming the existence
of such a is reasonable, it is not observable and the actuary can never know its true value
for a given policyholder.
Because varies by policyholder, there is a distribution function F giving the proportion
of policyholders in the portfolio with relative risk level less than or equal to a certain
threshold. Stated another way, F represents the probability that a policyholder picked at
random from the portfolio has a risk parameter that is less than or equal to . The connection
with the random effect introduced in the statistical models of Chapter 2 to account for the
residual heterogeneity is now clear: this random effect becomes the random risk parameter
for a policyholder picked at random from the portfolio (the distribution function of is
F ). Even if the risk parameter remains unknown, the distribution function F can be
estimated from data, as explained in Chapter 2. Once estimated, the heterogeneity model can
be used to perform prediction on longitudinal data and allows for experience rating in motor