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

Efficiency and Bonus Hunger 223
The modelling of claim costs is much more difficult than claim frequencies. There are
several reasons for this: In liability insurance, claims costs are often a mix of moderate
and large claims. Usually, ‘large claim’ means exceeding some threshold, depending on the
portfolio under study. This threshold can be selected using techniques from Extreme Value
Theory, as described in Cebrian, Denuit & Lambert (2003). Large liability claims need
several years to be settled. Only estimates of the final cost appear in the file until the claim
is closed. Moreover, the statistics available to fit a model for claim severities are much more
limited than for claim frequencies, since only 10 % of the policies in the portfolio produced
claims. Finally, the cost of an accident is for the most part beyond the control of a policyholder
since the payments of the insurance company are determined by third-party characteristics.
The degree of care exercised by a driver mostly influences the number of accidents, but in
a much lesser way the cost of these accidents. The information contained in the available
observed covariates is usually much less relevant for claim sizes than for claim counts.
In liability insurance, the settlement of larger claims often requires several years. Much of
the data available for the recent accident years will therefore be incomplete, in the sense that
the final claim cost will not be known. In this case, loss development factors can be used to
obtain a final cost estimate. The average loss severity is then based on incurred loss data.
In contrast to paid loss data (which are purely objective, representing the actual payments
made by the company), incurred loss data include subjective reserve estimates.
The total claim amount generated by policyholder i covered for motor third party liability
can be represented as
where
Ni
small
C ik
N large
i
N small N i∑
i∑
large
S i = C ik + L ik (5.1)
is the number of standard (or small) claims filed by policyholder i
is the cost of the kth standard claim filed by policyholder i
is the number of large claims filed by policyholder i
L ik is the cost of the kth large claim filed by policyholder i.
k=1
All these random variables are assumed to be mutually independent. The random variables
Ni
small and N large
i are analysed as explained in Chapters 1–2. Here, we explain how to model
the C ik s and the L ik s. The first question to be addressed is to separate standard claims and
large claims.
k=1
5.2.2 Determining the Large Claims with Extreme Value Theory
Extreme Claim Amounts
Gamma, LogNormal and Inverse Gaussian distributions (as well as other parametric models)
have often been used by actuaries to fit claim sizes. However, when the main interest is in the
tail of loss severity distributions, it is essential to have a good model for the largest claims.
Distributions providing a good overall fit can be particularly bad at fitting the tails. Extreme
Value Theory and Generalized Pareto distributions focus on the tails, being supported by
strong theoretical arguments.