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

Efficiency and Bonus Hunger 227
test is equal to 261 %, so that the result is significant at the 5 % level. Considering the two
analyses, the threshold for being qualified as a large claim is set to E100 000.
5.2.3 Generalized Pareto Fit to the Costs of Large Claims
Maximum Likelihood
Now that the threshold defining the large losses has been determined as E100 000, we model
the excesses over E100 000 with the help of a Generalized Pareto distribution. Descriptive
statistics for the cost of large claims of Portfolio C are displayed in Table 5.2. The mean is
equal to E364 6062. The limited number of large losses (17 for Portfolio C) does not allow
for incorporating exogeneous information in these amounts. Therefore, the same parameters
and are used for all the large losses. These parameters are estimated by maximum
likelihood.
Let us now fit the Generalized Pareto model to the excesses over E100 000. To this end,
we use maximum likelihood theory, and we maximize the likelihood function given by
=
∏
ix i >100 000
( (
1
1 + −
1
x −1
i − 100 000) )
The log-likelihood to be maximized is
(
L =−ln #x i x i > 100 000 − 1 + 1 ) ( ∑
ln 1 + )
x i − 100 000
ix i >100 000
where #x i x i > 100 000 = 17 in Portfolio C. This optimization problem requires numerical
algorithms. There are different approaches to getting starting values for the parameters
and . A natural approach consists of using moment conditions (that is, we equate sample
mean and sample variance to their theoretical expressions involving and ). The values
Table 5.2 Descriptive statistics of the cost of large claims
(Portfolio C).
Statistic
Value
Length 17
Minimum
104 3867
Maximum
1 989 5679
Mean
364 6062
Standard deviation
439 8827
25th percentile
140 0324
Median
252 2317
75th percentile
407 4776
90th percentile
499 7274
95th percentile
797 8475
99th percentile
1 751 2238
Skewness
29