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Problem of the nearest correlation matrix
Sotáková, Martina ; Pešta, Michal (advisor) ; Maciak, Matúš (referee)
This work deals with the problem of finding the correlation matrix closest to the given symetric matrix, the distance of which is measured considering the Frobenius norm. The theoretical part of the thesis describes a method used for finding the solution to this problem based on the dual approach and application of Newton method. The method is further modified for other cases. In the practical part we apply the theory to simple math problems.
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Bonus - Malus System with Deductibles
Kubát, Petr ; Mazurová, Lucie (advisor) ; Pešta, Michal (referee)
This thesis deals with the option of substitution of malus surcharge on pre- mium in a classical bonus - malus system with deductible. Firstly, we clarify the basic principles of bonus - malus systems, then we show how to model the expec- ted claim amount of the insureds based on their characteristics and we explain how to correctly select values of premium discounts and surcharges in the classes of bonus - malus systems. Next we clarify the concept of deductible and introduce the technique of its application on these systems. Finally we show the practical application of deductible on two models of bonus - malus systems and we evaluate and compare the results. 1
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Mixed Poisson models for claim counts
Hauptfleisch, Filip ; Pešta, Michal (advisor) ; Hendrych, Radek (referee)
The thesis summarizes the theory of mixed Poisson models. Poisson distri- bution is one of the popular distributions in modelling count data, but its use is limited because it requires equidispersion. Because of this we introduce both con- tinuous and finite mixtures. From continuous mixtures the main representative is the negative binomial model, which arises as Poisson Gamma mixture, while from discrete models we deal mainly with zero-inflated models and hurdle models. For these models we use the maximum likelihood estimates of their parameters. In the end we apply these models to fit automobile insurance data from Australia, where we use MLE to fit Poisson regression, negative binomial regression and Poisson hurdle regression.
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Portfolio Management with Multiple Benchmarks
Navrátil, Robert ; Večeř, Jan (advisor) ; Pešta, Michal (referee)
Portfolio Management with Multiple Benchmarks Bc. Robert Navrátil Abstract: In this thesis, we study a maximal volatility portfolio that treats all assets in a symmetric way and related option contract. To preserve symmetry we need numeraire that treats all assets symmetrically. We choose market index with equal weights. In case of two assets we focus on a variation of a passport option on the portfolio. The optimal strategy for the investor is the mentioned maximal volatility portfolio. We extend the known optimal strategy for the option to a richer class of convex payoff functions. We also show a modification of the optimal strategy for maximizing the probability of ending above or at a desired level. We later extend the symmetric market model to case of three assets, which can be even further extended to an arbitrary number of assets. The three asset model requires more parameters than are observable from the data, however we show indistinguishably of the model on the choice of parameters under very natural conditions. Both numerical simulations and an application on real data is provided. 1
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Statistical inference in varying coefficient models
Splítek, Martin ; Maciak, Matúš (advisor) ; Pešta, Michal (referee)
Tato práce se zabývá modely s promìnlivými koe cienty se za- mìøením na statistickou inferenci. Hlavní my¹lenkou tìchto modelù je pou¾ití regresních koe cientù, mìnících se v závislosti na nìjakém modi kátoru vlivu, namísto konstantních koe cientù klasické lineární regrese. Nejprve si de nujeme tyto modely a jejich odhadové procedury, kterých bylo doposud publikováno nì- kolik variant. K odhadu se pou¾ívá lokální regrese nebo rùzné druhy splajnù { vyhlazovací, polynomiální èi penalizované. Od metody odhadu se následnì od- víjí i daná statistická inference, ke které uvedeme odvozené vychýlení, rozptyl, asymptotickou normalitu, kon denèní pásma a testování hypotéz. Hlavním cílem na¹í práce je kompaktnì shrnout vybrané metody a jejich inferenci. Na závìr je navr¾ena proceduru pro výbìr promìnných.
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Tweedie models for pricing and reserving
Smolárová, Tereza ; Pešta, Michal (advisor) ; Cipra, Tomáš (referee)
This presented thesis deals with applications of Tweedie compound Poisson model in non-life insurance pricing and claims reserving. Tweedie models are exponen- tial dispersion models with power mean-variance relationships and compound Poisson distribution is a particular Tweedie model. The interest in Tweedie com- pound Poisson model is motivated by its applications to generalized linear models (GLMs) and generalized estimation equations (GEE). The purpose of this thesis is to construct pricing and claims reserving models in which the response variables follow Tweedie compound Poisson model. Theoretical approaches are applied on the real datasets. 1
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Archimedean copulas
Vedyushenko, Anna ; Pešta, Michal (advisor) ; Omelka, Marek (referee)
The thesis deals with Archimedean copulas which are very popular nowadays due to easy construction and their appealing properties. At first it introduces the general definition of a copula and also shows its fundamental properties. After that the definition and the basic properties of an Archimedean copula are discussed. The paper also describes some of the commonly used families of Archi- medean copulas. Then several methods of parameter estimation for Archimedean copulas are shown. Finally, we make a study of two real datasets where the distri- bution of the data is estimated based on the procedures described in the thesis. 1
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Statistical tests for VaR and CVaR
Mirtes, Lukáš ; Pešta, Michal (advisor) ; Večeř, Jan (referee)
The thesis presents test statistics of Value-at-Risk and Conditional Value-at-Risk. The reader is familiar with basic nonparametric estimators and their asymptotic distributions. Tests of accuracy of Value-at- Risk are explained and asymptotic test of Conditional Value-at-Risk is derived. The thesis is concluded by process of backtesting of Value-at-Risk model using real data and computing statistical power and probability of Type I error for selected tests. Powered by TCPDF (www.tcpdf.org)
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Projection of mortality tables and their influence on insurance embedded value
Filka, Jakub ; Pešta, Michal (advisor) ; Cipra, Tomáš (referee)
We study development of mortality tables from 1950 to present in Czech Republic. Our aim is to look at the 6 basic models, which can be potentially used to describe behavior of dying for people over 60 years. Models that are being investigated vary from generally accepted Gompertz-Makeham model to logistic models of Thatcher and Kannisto. We also introduce Coale-Kisker and Heligman- Pollard model. Our analysis is concentrated mostly on projecting abilities of given models to the highest ages. Especially for women, where data do not show such dispersion as in the case of men, there is a visible trend that can be described better by using logistic models instead of Gompertz-Makeham model, which has a tendency to overestimate the probabilities of dying in higher ages. Keywords: projection of mortality tables, Gompertz-Makeham, logistic models 1
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Claims reserving with copulae for multiple lines of business
Valentovičová, Katarína ; Pešta, Michal (advisor) ; Mazurová, Lucie (referee)
Claims reserving and claims process estimation present classical problems in general insurance. The overall reserves are often determined under the assumption of independence among the lines of business. Though, recently modelling of the dependence among multiple lines of business has become crucial issue of reserving process. In this context, copulae provide a useful tool to construct models which go beyond the classical ones in terms of dependence structure. This thesis deals, in particular, with the copula regression model, its properties and possible applications in general insurance. This approach combines GLM modelling of margins and then expressing the dependence structure using copula. The theoretical methods are illustrated on a real dataset.
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