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Modeling of large claims in a non-life insurance company

Aleksandra Budzowska 1Ryszard Kutner 2

1. UNIQA T.U. S.A., Jutrzenki 139, Warszawa 02-231, Poland
2. Warsaw University, Faculty of Physics, Hoża 69, Warszawa 00-681, Poland


The large claims, which are difficult to forecast, are a significant part of the financial result of an insurance company. Their influence can be limited by choosing the proper reinsurance program, for which the knowledge of the large claims distribution is needed. The information about this distribution comes from its realisation - that is, from the historical data of claims. In order to improve the quality of the prediction, the separation of claims into attritional and large claims is required. The threshold between attritional and large claims becomes a crucial parameter of the estimation.

In this work we used the plot of empirical complementary cumulative distribution function (CCDF) in the range of attritional claims to find the threshold of large claims. In the process of parameters' estimation of attritional claims distribution, the threshold value was the value of claim, which - when included in the sample of attritional claims - was giving considerably worse quality of the fit. The second method of finding the threshold of large claims was fitting the global distribution to the empirical CCDF. The global distribution was a combination of different distributions depending on the range of claims - in this work the lognormal distribution was used for small claims and Pareto distribution for average claims. As a distribution for large claims five distributions were considered: gamma distribution, lognormal distribution, generalized Pareto distribution, Weibull distribution and inverse Gaussian distribution.

Having determined the value of the threshold we used two methods of estimation of the remaining parameters - the method of maximising of the likelihood function and the method of minimising of the least squares of the difference between theoretical and emipirical cumulative distribution function (CDF). The main challenge was to verify the quality of estimation - the large claims empirical sample consisted of 40 claim values and the ordinary statistical methods were inadequate. Instead, we used the average and standard deviation from the distribution of maximum value in the sample. Thoee measures were found using Monte Carlo simulation for each of the five fitted distributions of large claims with the assumption that the random sample in Monte Carlo simulation has the same cardinality as the cardinality of empirical sample. These measures were compared with the actual maximum value of the empirical sample. The results of this comparison were thoroughly discussed.


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Related papers

Presentation: Oral at 6 Ogólnopolskie Sympozjum "Fizyka w Ekonomii i Naukach Społecznych", by Aleksandra Budzowska
See On-line Journal of 6 Ogólnopolskie Sympozjum "Fizyka w Ekonomii i Naukach Społecznych"

Submitted: 2012-01-22 17:02
Revised:   2012-01-22 18:09