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

Efficiency and Bonus Hunger 229
Application to the Claim Costs Recorded in Portfolio C
The initial values obtained with the the moment method applied to large claims of Portfolio
C are ̂ 0 = 0319 and ̂ 0 = 180 1767. The maximum likelihood estimates of the Generalized
Pareto parameters are ̂ = 04152 and ̂ = 156 7374. Different starting values have been
used and the convergence always occurred.
Note that the limited sample size for the large losses does not allow us to draw reliable
conclusions about major claims (in particular, large sample properties of the maximum
likelihood estimators cannot be invoked with a sample size as small as 17). In practice, the
insurance company must gather the large losses together in a data base to perform detailed
analysis. The amounts of these large losses have to be corrected for different sources of
inflation. The assistance of a reinsurance company is useful in this respect, especially for
insurers with small to moderate portfolios.
5.2.4 Modelling the Number of Large Claims
The number of large claims N large
i for policyholder i is modelled using the oi large
i
distribution. This is in line with the fact that large claims occur purely at random. Poisson
regression is then used to incorporate the information available about policyholder i,
summarized in a vector xi
T = x i1 x ip consisting of explanatory variables (assumed
here to be categorical and coded by means of binary variables), in the expected frequency
of large claims through a linear predictor by means of an exponential link function.
The regression coefficients are estimated with Poisson regression. All the explanatory
variables introduced in Section 5.1.6 have been excluded from the model. Only the intercept
remained in the Poisson regression model. This is not surprising with such a small number of
policies producing a large claim. We obtained ̂ 0 =−90555, with a standard deviation equal
to 0.2425. The resulting frequency of large claims is
̂
large
i
= 00117 % for all policyholders.
Remark 5.1 (Logistic regression) In most cases, policyholders report 0 or just 1 large
claim. Therefore, the number of large claims could also be modelled with a binary variable
instead of a Poisson count. The probability that policyholder i reports (at least) a large claim
can also be modelled with the help of logistic regression. Specifically, let us define the
binary random variable J i
{ 0 if policyholder i does not report any large claim during the observation period
J i =
1 if policyholder i reports at least one large claim during the observation period
The aim is to model the probability PrJ i = 0 = q i x i that policyholder i does not report
any large claim during the coverage period.
Since q i x i ∈ 0 1, we resort to some distribution function F to link q i x i to the linear
predictor 0 + ∑ p
j=1 jx ij , that is
(
)
p∑
p∑
q i x i = F 0 + j x ij ⇔ 0 + j x ij = F −1 q i x i
j=1
j=1