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Analysis of several acceleration techniques for life insurance liability value determination
Drahokoupil, Matěj ; Pešta, Michal (advisor) ; Branda, Martin (referee)
The aim of the diploma thesis is to apprise the reader with a basic life insur- ance projection method which is used for the valuation of insurance company's liabilities. The basic projection method can be extremely time consuming in practise so another two variance reduction methods and their combination are presented to obtain either more precise liabilities estimation, or to reduce the time required for the projection. The presented methods are antithetic variate method, control-variate method and their combination later called integrated control-variate method. The final outcome of the thesis is simulation experi- ment which evaluates the liabilities of the group of policies and comparison of the presented variance reduction methods. 1
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Multivariate generalized autoregressive conditional heteroscedasticity models
Nováková, Martina ; Pešta, Michal (advisor) ; Maciak, Matúš (referee)
This master thesis deals with extension of the univariate GARCH model to multivari- ate models. We present individual models and deal with methods of their estimation. Then we describe some statistical tests for diagnosting the models. We have programmed in the statistical software R one of them - the Ling-Li test. Afterwards we apply selected models to real data of stock market index S&P 500, stock market index Russell 2000 and stocks of crude oil. For the GO-GARCH model, we compare all available estimation methods and show their differences. Then we compare the results of all models with each other and also with univariate models in terms of estimates of conditional variances, estimates of conditional correlations and also in terms of computational complexity. 1
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Gradual change model
Míchal, Petr ; Hlávka, Zdeněk (advisor) ; Pešta, Michal (referee)
The thesis aims at change-point estimation in gradual change models. Methods avail- able in literature are reviewed and modified for point-of-stabilisation (PoSt) context, present e.g. in drug continuous manufacturing. We describe in detail the estimation in the linear PoSt model and we extend the methods to quadratic and Emax model. We describe construction of confidence intervals for the change-point, discuss their interpre- tation and show how they can be used in practice. We also address the situation when the assumption of homoscedasticity is not fulfilled. Next, we run simulations to calculate the coverage of confidence intervals for the change-point in discussed models using asymp- totic results and bootstrap with different parameter combinations. We also inspect the simulated distribution of derived estimators with finite sample. In the last chapter, we discuss the situation when the model for the data is incorrectly specified and we calculate the coverage of confidence intervals using simulations. 1
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Testing equality of means by confidence intervals
Jandl, Vojtěch ; Kulich, Michal (advisor) ; Pešta, Michal (referee)
We deal with testing the equality of means using confidence intervals. Firstly, we introduce the methods of testing that have already been published. The advantage of these methods is that one can present the underlying confidence intervals alongside the result of the test without doing further calculations. In the second part we discuss the necessary assumptions and by that we extend the Noguchi's method to discrete distribu- tions. Also, we derive a generalization of the Noguchi's method for testing the equality of other parameters than means, based on the assumption of asymptotic normality of their consistent estimates. Lastly, we conduct a simulation study in order to compare the methods we discussed. We found out that the Noguchi's method is a worthy alternative to the often-used Welch test bearing the advantage of being able to present extra visual output in the form of the underlying confidence intervals. In comparison to other methods the Noguchi's method yields better results in the case of unequal or small sample sizes. Unlike other methods it can also be used for testing in the paired sample case. 1
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Various change point estimation methods
Šimonová, Soňa ; Pešta, Michal (advisor) ; Hušková, Marie (referee)
This thesis aims to give a comprehensive account of some of the most recent methods of a change point estimation. The literature on the change point estimation shows a variety of approaches to deal with this subject. Among them, tests based on the popular CUSUM process, likelihood ratio tests, wild binary segmentation and some of the most recent techniques on the change point estimation in panel data are all covered by this paper. The case of dependent panels is discussed as well. The practical part of the study is focused on application of the wild binary segmentation method on weekly log-returns of the Dow Jones stock index. Firstly, we fit a GARCH model to the analysed time series. We next use the wild binary segmenatation method to detect structural changes in the mean of the original time series. Next, we apply the same method to the residuals from the GARCH fit. We analyse several penalization criteria proposed by previous studies and evaluate their effects on the estimated number and locations of the change points in the given data set. 1
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Total Least Squares and Their Asymptotic Properties
Chuchel, Karel ; Pešta, Michal (advisor) ; Antoch, Jaromír (referee)
Tato práce se zabývá metodou úplně nejmenších čtverc·, která slouží pro odhad parametr· v lineárních modelech. V práci je uveden základní popis metody a její asymptotické vlastnosti. Je vysvětleno, jakým zp·sobem lze v konceptu metody využít neparametrický bootstrap pro hledání odhadu. Vlastnosti bootstrap od- had· jsou pak simulovány na pseudo náhodně vygenerovaných datech. Simulace jsou prováděny pro dvourozměrný parametr v r·zných nastaveních základního modelu. Jednotlivé bootstrap odhady jsou v rovině řazeny pomocí Mahalanobis a Tukey statistical depth function. Simulace potvrzují, že bootstrap odhad dává dostatečně dobré výsledky, aby se dal využít pro reálné situace.
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Stochastic claims reserving with double chain ladder
Javůrková, Tereza ; Pešta, Michal (advisor) ; Zichová, Jitka (referee)
This thesis deals with an important problem of insurance which is forecasting outstanding claims liabilities. It describes the Chain-Ladder method, the basic method for forecasting outstanding claims, and then it's extention to Double Chain-Ladder method. It also uses the number of reported claims for a beter estimate. The final forecast is calculated from the IBNR and RBNS reserves which are estimated separetly. Finly we aplly those methods to a real life dataset. The results shows differences betwen those two methods and different ways of programming. 1
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Multivariate Pareto distribution
Novytskyi, Oleksandr ; Mazurová, Lucie (advisor) ; Pešta, Michal (referee)
Title: Multivariate Pareto distribution Author: Oleksandr Novytskyi Department: Department of Probability and Mathematical Statistics (305. 32- KPMS) Supervisor: RNDr. Lucie Mazurová, Ph.D., Department of Probability and Mathematical Statistics (305. 32-KPMS) Abstract: This bachelor thesis focuses on three methods of constructing multiva- riate Pareto distribution, i.e. multivariate distribution, where marginal distributi- ons are univariate Pareto distributions. We provide survival and density functions for these models, which are used for the numerical studies and valuation of insu- rance product, specifically a yearly life annuity paid to each insured in the group, whose remaining life time is given by the multivariate Pareto distribution. Keywords: multivariate distribution, Pareto distribution, survival function, density, life annuity.
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Stochastic approaches to distributions of aggregated claims
Kirešová, Katarína ; Pešta, Michal (advisor) ; Mazurová, Lucie (referee)
Bachelor thesis deals with the calculation of the distribution of an aggregated claim: at first generally and afterward, focusing on life portfolio in an indivi- dual model. Three methods are compared: De Pril recursion, Kornya's method, and Panjer algorithm. We assess assumptions and derive formulae for particular methods. Methods are compared in terms of time complexity and precision of computations. We also deal with the calculation of the expected value and va- riance. Eventually, examples and simulations, which we used to determine the best method of calculation of the distribution of aggregated claim in an arbitrary portfolio, are preceded.
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Micro-level stochastic claims reserving
Rathouský, Marek ; Pešta, Michal (advisor) ; Vitali, Sebastiano (referee)
This thesis covers, in detail, theoretical background of micro-level stochastic model, which includes definition and properties of non-homogeneous Poisson process. This the- ory is then applied to real data generated by MTPL portfolio. Estimates of provisions under micro-level stochastic model are calculated using ordinary Monte Carlo simula- tion method. Results obtained from micro-level stochastic model are compared to Mack Chain-ladder estimates. 1
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