Aplikace teorie extrémních hodnot a kopul pro modelování operačního rizika pojišťovny

The aim of the dissertation is to calculate the solvency capital requirement for operational risks in the insurance company using modern statistical methods such as extreme value theory and copula theory. The dissertation focuses on modelling a unique real data set of insurance claims arising from operational risk of an anonymous insurance company operating in the CEE region. The unique database consists of 4245 insurance claims recorded during 2010 to 2018. Using extreme value theory, a comprehensive model is developed to estimate losses from operational risk events using available historical data. Claims with low frequency but extremely high severity were modelled using a generalised Pareto distribution and compared with representatives of the family of generalised extreme distributions - the Frechet, Weibull and Gumbel distributions. The dissertation compares extended graphical and numerical approaches to estimating the threshold value, which is the key parameter for the method of excesses over the threshold. This comparison is complemented by the proposal of own innovative approach that combines the graphical approach with the numerical one - maximizing the log-likelihood in the area defined by the L-moment graph. In addition, a collective risk model is used for claims occurring with high frequency but low claim severity. Using the bootstrapping principle, 100,000 one-year predictions of the claims portfolio are generated to calculate the value-at-risk. Due to the correlation structure of the operational risk categories, it is not possible to determine the total solvency capital requirement by simply summing the values-at-risk across the categories. The final part of the paper is therefore devoted to the problem of modelling the dependence between the components of the random vector of each type of operational risk. We then aggregate the marginal distributions of the individual variables and information about their correlation structure using n-dimensional copula theory. By combining insights from extreme value theory and copula theory on a real data set, a comprehensive approach to estimating the solvency capital requirement for the operational risk area of the insurance company has been built. The results of the work suggest that the application of extreme value theory and copula theory to the insurer's internal real database yields a saving in the capital requirement compared to the regulator's recommended calculation via a standard formula. In fact, after applying the diversification effect via the Frank copula, there is a significant decrease in the capital requirement. The standard formula therefore does not always provide an appropriate method of calculation.