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

158 Actuarial Modelling of Claim Counts
3.5.6 Increasingness in the Linear Credibility Model
Let ̂N iTi +1 be the predictor (3.16) of N iTi +1. It can be shown that N iTi +1 is indeed increasing
in ̂N iTi +1, in the sense that
N iTi +1̂N iTi +1 = p ≼ ST N iTi +1̂N iTi +1 = p ′ whenever p ≤ p ′
⇔ PrN iTi +1 >k̂N iTi +1 = p ≤ PrN iTi +1 >k̂N iTi +1 = p ′ whatever k, provided p ≤ p ′
This means that the linear credibility premium is indeed a good predictor of the future
claim number in model A1–A2. Basically, we prove that increasing the linear credibility
premium (i.e. degrading the claim record of the policyholder) makes the probability of
observing more claims in the future greater.
3.6 Further Reading and Bibliographic Notes
3.6.1 Credibility Models
Credibility theory began with the papers by Mowbray (1914) and Whitney (1918). These
papers purposed to derive a premium which was a balance between the experience of an
individual risk and of a class of risks. An excellent introduction to credibility theory can be
found, e.g., in Goovaerts & Hoogstad (1987), Herzog (1994), Dannenburg, Kaas &
Goovaerts (1996), Klugman, Panjer & Willmot (2004, Chapter 16) and Bühlmann
& Gisler (2005). See also Norberg (2004) for an overview with useful references and
links to Bayesian statistics and linear estimation. The underlying assumption of credibility
theory which sets it apart from formulas based on the individual risk alone is that the risk
parameter is regarded as a random variable. This naturally leads to a Bayesian approach
to credibility theory. The book by Klugman (1992) provides an in-depth treatment of the
question. See also the review papers by Makov ET AL. (1996) and Makov (2002). The
connection between credibility formulas and Mellin transforms in the Poisson case has been
established by Albrecht (1984).
In a couple of seminal papers, Dionne & Vanasse (1989, 1992) proposed a credibility
model which integrates a priori and a posteriori information on an individual basis. The
unexplained heterogeneity was then modelled by the introduction of a latent variable
representing the influence of hidden policy characteristics. Taking this random effect to be
Gamma distributed yields the Negative Binomial model for the claim number. An excellent
summary of the statistical models that may lead to experience rating in insurance can be
found in Pinquet (2000). The nature of serial correlation (endogeneous or exogeneous) is
discussed there.
There are many applications of credibility techniques to various branches of insurance.
Let us mention a nonstandard one, by Rejesus ET AL. (2006). These authors examined
the feasibility of implementing an experience-based premium rate discount in crop
insurance.