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Discrimination measures in credit risk
Polak, Michal ; Pešta, Michal (advisor) ; Zahradník, Petr (referee)
Scoring models represent a fundamental tool for the modern management of credit risk. This is mainly due to a significant development in the field of information technology. Such models are used not only when providing credit, but also in strategies relating to the future management of credit risk, or in strategies connected with enforcing receivables. In my thesis I deal with discrimination measures used in the validation of diversification potential of logistic scoring models. At the beginning, I focus on the term 'risk'. Then, I introduce a basic division of scoring models. Next, I describe the method of scoring logistic regression, I concentrate on estimating parameters, their significance and on testing their relevance. For the measurement and illustration of diversification potential of the model I mention the most commonly used methods such as the Lorenz and ROC curve, the Gini coeficient, the c-statistic as well as the Kolmogorov-Smirnov test. Finally, I apply the theoretical knowledge to real data. I design a scoring model and subsequently compare the discrimination measures which it contains. Powered by TCPDF (www.tcpdf.org)
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Double chain ladder
Perichtová, Margaréta ; Pešta, Michal (advisor) ; Mazurová, Lucie (referee)
This thesis deals with one of the biggest problems in non-life insurance and that is forecasting outstanding claims liabilities. Chain ladder method is probably the most often used method for estimating outstanding liabilities. Firstly, we show classic chain ladder method and its deterministic and stochastic form. Secondly, we introduce relatively new method, double chain ladder method which comes from chain ladder method, but in addition it considers number of reported claims, that allow us to count RBNS reserve more precisely and also to count IBNR and RBNS reserves separately. In the end we apply both methods on the real data. We compute point estimate of the reserve by chain ladder method and by double chain ladder method and compare the results. Powered by TCPDF (www.tcpdf.org)
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Discrimination measures in credit risk
Polak, Michal ; Pešta, Michal (advisor) ; Zahradník, Petr (referee)
Scoring models represent a fundamental tool for the modern management of credit risk. This is mainly due to a significant development in the field of information technology. Such models are used not only when providing credit, but also in strategies relating to the future management of credit risk, or in strategies connected with enforcing receivables. In my thesis I deal with discrimination measures used in the validation of diversification potential of logistic scoring models. At the beginning, I focus on the term 'risk'. Then, I introduce a basic division of scoring models. Next, I describe the method of scoring logistic regression, I concentrate on estimating parameters, their significance and on testing their relevance. For the measurement and illustration of diversification potential of the model I mention the most commonly used methods such as the Lorenz and ROC curve, the Gini coeficient, the c-statistic as well as the Kolmogorov-Smirnov test. Finally, I apply the theoretical knowledge to real data. I design a scoring model and subsequently compare the discrimination measures which it contains. Powered by TCPDF (www.tcpdf.org)
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Expectation-Maximization Algorithm
Vichr, Jaroslav ; Pešta, Michal (advisor) ; Zvára, Karel (referee)
EM (Expectation-Maximization) algorithm is an iterative method for finding maximum likelihood estimates in cases, when either complete data include missing values or assuming the existence of additional unobserved data points can lead to more simple formulation of the model. Each of its iterations consists of two parts. During the E step (expectation) we calculate the expected value of the log-likelihood function of the complete data, with respect to the observed data and the current estimate of the parameter. The M step (maximization) then finds new estimate, which will maximize the function obtained in the previous step and which will be used in the next iteration in step E. EM algorithm has important use in e.g. price and manage risk of the portfolio.
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Modern stochastic claims reserving methods in insurance and their comparison
Vosáhlo, Jaroslav ; Pešta, Michal (advisor) ; Mazurová, Lucie (referee)
This thesis deals with an issue of claims reserving for non-life insurance. The issue is approached in a sense of analytical calculation and stochastic modelling. First, Chain-ladder, Bornhuetter-Ferguson, Benktander-Hovinen and Cape-Cod method are introduced. In following chapters, we try to find related stochastic underlying models including Generalized linear models and Mack's distribution-free approaches, we analyze second moments of claims estimates for each of the methods and examine alternative Merz-Wüthrich approach to reserve risk measurement. At the end, bootstrap algorithm and estimates are suggested and simulation results are compared with analytic ones.
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Parametrizace rozdělení škod v neživotním pojištení
Špaková, Mária ; Pešta, Michal (advisor) ; Cipra, Tomáš (referee)
Title: Parameterization of claims distribution in non-life insurance Author: Bc. Mária Špaková Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Michal Pešta Ph.D., MFF UK Abstract: This paper deals with the parameterization of claim size distributions in non-life insurance. It consists of the theoretical and the practical part. In the first part we discuss the usual distributions of claims and their properties. One section is devoted to extreme values distributions. Consequently, we mention the most known methods for parameter estimation - the maximum likelihood method, the method of moments and the method of weighted moments. The last theoretical chapter is focused on some validation techniques and goodness-of-fit tests. In the practical part we apply some of the discussed approaches on real data. However, we concentrate mainly on the large claims modeling - firstly, we select a reasonable threshold for our data and then we fit the claims by the generalized Pareto distribution together with the introduced parameterization procedures. Based on the results of the applied validation methods we will choose appropriate models for the biggest claims. Keywords: parameterization, non-life insurance, claims distribution.
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Technical reserves of non-life insurance in the internal solvency models
Thomayer, Jiří ; Mertl, Jakub (advisor) ; Pešta, Michal (referee)
Title: Technical reserves of non-life insurance in the internal solvency model Author: Bc. Jiří Thomayer Department: Department of Propability and Mathematical Statistics Supervisor: Mgr. Ing. Jakub Mertl Abstract: In this work we study and describe calculation of solvency capital using the standard formula contained in the Directive of the European Union (Solvency II), which should be put into practice in Europe on 1 January 2013. This calcu- lation is described in quantitative impact study 5. We describe a general approach to risk measurement and we show some particular practical measures used to risk measurement. We explain under what conditions the standard formula or its parts can be replaced by internal model. Next, we show disadvantages of using the stan- dard formula and we propose possible internal model to calculate risk premiums and risk reserves in non-life insurance. Finally we apply the proposed model for calculation risk reverses in non-life insurance in practice. Keywords: Standard formula, Risk measurement, Solvency II, Internal model;
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Logistic regression with applications in financial sector
Bílková, Kristýna ; Branda, Martin (advisor) ; Pešta, Michal (referee)
In this bachelor thesis binary logistic regression model is described. Its parameters are estimated by maximum likelihood method. Newton-Raphson's algorithm is used for enumeration of these estimates. There are defined some statistics for testing the significance of the coefficients. Then stepwise regression is desribed. For assessing the quality of the model Pearson's Chi Square Test and Hosmer-Lemeshow's Test of the goodness of fit are defined. Diversification abilitz of the model is illustrated bz the Loreny curve and is quantificated by Gini coefficient, Kolmogorov-Smirnov statistics and generalized coefficient of determination. The theoretical knowledge is applied to insurance area data.
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Structural Equation Modeling
Kuzminskaya, Kseniya ; Pešta, Michal (advisor) ; Lachout, Petr (referee)
Structural Equation Models (SEM) - also called Simultaneous Equation Models - are used to describe relationships among a set of variables. Similarly as in multivariate regression models, some of the variables are treated as predictors and the others as outcomes. However, unlike in a classical regression model, a variable, which is outcome in one equation, can become a predictor in another equation. SEM are even able to handle variables, which are not measured directly but only through their effects. They are often used in econometrics or socio-economics.
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Advanced Techniques of Risk Aggregation
Dufek, Jaroslav ; Justová, Iva (advisor) ; Pešta, Michal (referee)
In last few years Value-at-Risk (Var) is a very popular and frequently used risk measure. Risk measure VaR is used in most of the financial institutions. VaR is popular thanks to its simple interpretation and simple valuation. Valuation of VaR is a problem if we assume a few dependent risks. So VaR is estimated in a practice. In presented thesis we study theory of stochastic bounding. Using this theory we obtain bounds for VaR of sum a few dependent risks. In next part of presented thesis we show how we can generalize obtained bounds by theory of copulae. Then we show numerical algorithm, which we can use to evaluate bounds, when exact analytical evaluate isn't possible. In a final part of presented thesis we show our results on practical examples.
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