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

222 Actuarial Modelling of Claim Counts
Table 5.1 Descriptive statistics of the claim costs (only strictly
positive values) for portfolio C.
Statistic
Value
# observations 18 176
Minimum 27.02
Maximum 1 989 567.9
Mean 1810.63
Standard deviation 17 577.83
25th percentile 145.02
Median 598.17
75th percentile 1464.75
90th percentile 3021.87
95th percentile 4268.06
99th percentile 19 893.68
Skewness 85.08
optional coverages). In addition to these covariates, we know the number of claims filed by
each policyholder during 1997, the exposure-to-risk from which these claims originated, as
well as the resulting total claim amount. The information recorded in the data base dates
from the end of June 1998 (6 months after the end of the observation period). Hence, most
of the ‘small’ claims are settled and their final cost is known. However, for the large claims,
we work here with incurred losses (payments made plus reserve).
Descriptive statistics for claim costs are displayed in Table 5.1; we have at our disposal
18 176 observed individual claim costs, ranging from E27.02 to almost E2 000 000, with a
mean of E1810.63. We see in Table 5.1 that 25 % of the recorded claim costs are below
E145.02, that half of them are smaller than E598.17, and that 90 % of them are less than
E3021.87. The interquartile range is E1319.73. The observed claim cost distribution is highly
asymmetric, with a skewness coefficient of about 85.
5.2 Modelling Claim Severities
5.2.1 Claim Severities in Motor Third Party Liability Insurance
In nonlife business, the pure premium is the expected cost of all the claims that policyholders
will file during the coverage period (under the assumption of the Law of Large Numbers).
Let S i , i = 1n, be the total claim amount relating to policy number i. The S i s are
assumed to be independent and identically distributed with common mean . The Law of
Large Numbers ensures that
[
Pr S n = 1 n
n∑
i=1
]
S i → as n →+ = 1
Under the conditions of the Law of Large Numbers, the pure premium is thus the expected
claim amount.