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Mortality in high ages
Malá, Kateřina ; Mazurová, Lucie (advisor) ; Cipra, Tomáš (referee)
In this thesis, we study modelling of mortality in high ages by several approaches. Some of mentioned models take into account the phenomenon of mortality deceleration. Further, we present several ways of estimating of exposed to risk in (almost) extinct cohorts. We focus especially on the survivor ratio method but we also mention the MD method and the DG method. Finally, we perform a numerical study.
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Seasonal exponential smoothing
Rábek, Július ; Cipra, Tomáš (advisor) ; Zichová, Jitka (referee)
This thesis deals with the issues of time series modeling, where seasonal component is present. Principles of basic seasonal exponential smoothing methods: simple and double exponential smoothing, Holt's method, which are applicable on time series without seasonality, are described in the beginning. For seasonal time series, Holt-Winters exponential smoothing is the most suitable method. This method is introduced in both of its versions and the usage of either version depends on the characteristics of the seasonal component. Furthermore, state space modeling is presented as a statistical framework for exponential smoothing methods, joined with a discussion of some selected problems related with practical implementation of these techniques together with suggestions of their solution. Finally, Holt-Winters method on two real data time series with seasonality is presented.
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Ensemble learning methods for scoring models development
Nožička, Michal ; Witzany, Jiří (advisor) ; Cipra, Tomáš (referee)
Credit scoring is very important process in banking industry during which each potential or current client is assigned credit score that in certain way expresses client's probability of default, i.e. failing to meet his or her obligations on time or in full amount. This is a cornerstone of credit risk management in banking industry. Traditionally, statistical models (such as logistic regression model) are used for credit scoring in practice. Despite many advantages of such approach, recent research shows many alternatives that are in some ways superior to those traditional models. This master thesis is focused on introducing ensemble learning models (in particular constructed by using bagging, boosting and stacking algorithms) with various base models (in particular logistic regression, random forest, support vector machines and artificial neural network) as possible alternatives and challengers to traditional statistical models used for credit scoring and compares their advantages and disadvantages. Accuracy and predictive power of those scoring models is examined using standard measures of accuracy and predictive power in credit scoring field (in particular GINI coefficient and LIFT coefficient) on a real world dataset and obtained results are presented. The main result of this comparative study is that...
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Reinsurance optimization using stochastic programming and risk measures
Došel, Jan ; Branda, Martin (advisor) ; Cipra, Tomáš (referee)
Title: Reinsurance optimization using stochastic programming and risk measures Author: Jan Došel Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Martin Branda, Ph.D., Department of Probability and Mathe- matical Statistics Abstract: The diploma thesis deals with an application of a stochastic progra- mming in a reinsurance optimization problem in terms of a present regulatory framework of the insurance companies within the European Union, i.e. Solvency II. In this context, the reinsurance does not only transfer a portion of the risk to the reinsurer but also reduces an amout of required capital. The thesis utilizes certain risk measures and their properties, premium principles and non-linear in- teger programming. In the theoretical part, there are basic terms from Solvency II, reinsurance, risk measures and the comonotonicity of random variables descri- bed and the optimization problem itself is derived. The approach is then applied in the practical part on data of Czech Insurers' Bureau using the GAMS software. Finally, a stability of the solution is tested depending on several parameters. Keywords: reinsurance optimization, stochastic programming, Solvency II, risk measures 1
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Median in some statistical methods
Bejda, Přemysl ; Cipra, Tomáš (advisor) ; Hlávka, Zdeněk (referee) ; Víšek, Jan Ámos (referee)
Median in some statistical methods Abstract: This work is focused on utilization of robust properties of median. We propose variety of algorithms with respect to their breakdown point. In addition, other properties are studied such as consistency (strong or weak), equivariance and computational complexity. From practical point of view we are looking for methods balancing good robust properties and computational complexity, be- cause these two properties do not usually correspond to each other. The disser- tation is divided to two parts. In the first part, robust methods similar to the exponential smoothing are suggested. Firstly, the previous results for the exponential smoothing with ab- solute norm are generalized using the regression quantiles. Further, the method based on the classical sign test is introduced, which deals not only with outliers but also detects change points. In the second part we propose new estimators of location. These estimators select a robust set around the geometric median, enlarge it and compute the (iterative) weighted mean from it. In this way we obtain a robust estimator in the sense of the breakdown point which exploits more information from observations than standard estimators. We apply our approach on the concepts of boxplot and bagplot. We work in a general normed vector...
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Median in some statistical methods
Bejda, Přemysl ; Cipra, Tomáš (advisor)
Median in some statistical methods Abstract: This work is focused on utilization of robust properties of median. We propose variety of algorithms with respect to their breakdown point. In addition, other properties are studied such as consistency (strong or weak), equivariance and computational complexity. From practical point of view we are looking for methods balancing good robust properties and computational complexity, be- cause these two properties do not usually correspond to each other. The disser- tation is divided to two parts. In the first part, robust methods similar to the exponential smoothing are suggested. Firstly, the previous results for the exponential smoothing with ab- solute norm are generalized using the regression quantiles. Further, the method based on the classical sign test is introduced, which deals not only with outliers but also detects change points. In the second part we propose new estimators of location. These estimators select a robust set around the geometric median, enlarge it and compute the (iterative) weighted mean from it. In this way we obtain a robust estimator in the sense of the breakdown point which exploits more information from observations than standard estimators. We apply our approach on the concepts of boxplot and bagplot. We work in a general normed vector...
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Smooth Transition Autoregressive Models
Khýr, Miroslav ; Zichová, Jitka (advisor) ; Cipra, Tomáš (referee)
The aim of this work is describing theory of smooth transition autoregressive models, namely LSTAR and ESTAR models. The essential part of the work is devoted to the derivation of tests for linearity against the alternative of the re- levant nonlinear model. There is also shown how to estimate the parameters of these models along with the selection procedure between the LSTAR and the ESTAR model. A simulation study was carried out, which deals with the power of linearity tests. At the end of the thesis, we applied the theory to some real data and we estimated the appropriate model for their representation. 1
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Modelling dependent lives
Pavčová, Eva ; Mazurová, Lucie (advisor) ; Cipra, Tomáš (referee)
Title: Modelling Dependent Lives Author: Eva Pavčová Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Lucie Mazurová, Ph.D., Department of Probability and Mathematical Statistics Abstract: In this thesis, we model the dependence between the remaining lifetimes of a husband and wife using a specific Markov model. We examined the impact of the dependence on the net single premium using the specific Markov model that captures the long-term dependence between lifetimes of the two considered lives. Using this model we have calculated 10-year joint-life annuity due and 10-year last-survivor annuity due considering the age rage (37, 80) in case of dependence and also independence of the two considered lives. The calculations were based on the dataset related to the Czech population in 2015. The impact of the dependence between the remaining lifetimes of the husband and wife was found to be not significant. Keywords: positive quadrant depedence, multiple life insurance premiums, depen- dent lifetimes, joint-life annuity, last-survivor annuity, joint-life and last-survivor models
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Optimization of reinsurace parameters in insurance
Dlouhá, Veronika ; Branda, Martin (advisor) ; Cipra, Tomáš (referee)
This thesis is dedicated to searching optimal parameters of reinsurance with a focus of quota-share and stop-loss reinsurance. The optimization is based on minimization of value at risk and conditional value at risk of total costs of the insurer for the recieved risk. It also presents a compound random variable and shows various methods of obtaining its probability distribution, for example ap- proximation by lognormal or gamma mixtures distributions or by Panjer recurive method for continuous severity and numerical method of its solution. At the end of the thesis we can find the calculation of the optimal parameters of reinsurance for a compound random variable based on real data. We use various methods to determine probability distribution and premiums. 1
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Cointegration and EC model
Asipenka, Hanna ; Cipra, Tomáš (advisor) ; Zichová, Jitka (referee)
The thesis deals with the concept of cointegration of time series and related error correction model. First, we introduce the basic definitions and theorems that are necessary for understanding the subject of other chapters. Then we focus on the definition of cointegration and the issue of tests for cointegration. Next, we define the error correction model in general in the vector autoregression as well. We will show and prove Granger's representation theorem, which will allow the construction of the EC model in the next section of the chapter. Finally, we apply the written theory to real time series. We perform cointegration tests and construct the relevant EC model. 1
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