
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. 32KPMS) 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.


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.


Multivariate claim numbers models
Zušťáková, Lucie ; Mazurová, Lucie (advisor) ; Cipra, Tomáš (referee)
Multidimensional frequency models can be used for modeling number of claims from different branches which are somehow dependent on each other. As in the onedimensional case Poisson distribution and negative binomial distribution are primarily used for modeling multidimensional claim counts data, only they are extended to higher dimensions. The generalization of multi dimensional distributions is often done using socalled shock variables, where one random variable is included in all dimensions of a random vector which models claim counts. The more comprehensive approach to modeling dependence uses copulas. Comparison of these models is done on a simulated data of number of claims from two different car insurance guarantees.


Aggregation of dependent risks
Asipenka, Anna ; Mazurová, Lucie (advisor) ; Omelka, Marek (referee)
In this thesis we are interested in the calculation of economic capital for the to tal loss which is the sum of partial dependent losses, whose dependence structure is described by Archimedean and hierarchical Archimedean copulas. Firstly, the concept of economic capital and the ways of its aggregation are introduced. Then the basic definitions and properties of copulas are listed, as well as the depen dence measures. After that we work with definition and properties of Archimedean copulas and their simulation. We also mention the most popular families of Ar chimedes copulas. Next, hierarchical Archimedean copulas are defined, as well as the algorithm for their sampling. Finally, we present methods for estimating the parameters of copulas and the recursive algorithm for estimating the hierarchical Archimedean copula structure. In the last chapter we perform simulation studies of selected models using hierarchical Archimedes copulas. 1


Modelling mortality by causes of death
Valter, Boris ; Mazurová, Lucie (advisor) ; Hurt, Jan (referee)
The aim of this thesis is to provide an overview of methods used in causeofdeath mortality analysis and to demonstrate the application on real data. In Chapter 1 we present the continuous model based on the force of mortality and review the approach using copula functions. In Chapter 2 we focus on the multinomial logit model formulated for causespecific mortality data, discuss life tables construction and derive life expectancy. In Chapter 3 we apply the multinomial logit model on the data from Czech Statistical Office. We identify the regression model, check its assumptions, present the outputs including the fitted life expectancy, and predicted mortality rates. Later in Chapter 3 we consider several stress scenarios in order to demonstrate the impact of shocked mortality rates on the life expectancy.


Risk aggregation allowing for skewness
Šimonová, Soňa ; Mazurová, Lucie (advisor) ; Zichová, Jitka (referee)
The main objective of this thesis is to examine different methods of calcula tion of economic capital for an insurance company which allow for skewness. For calculating the economic capital we use two alternative risk measures Value at Risk (VaR) and Conditional Value at Risk (CVaR). The first part of the thesis is concerned with deriving exact formulae for VaR and CVaR for normally distribu ted losses and describing the modification of these formulae using CornishFisher approximation. Next, the method using lognormal model with a parameter cap turing skewness is discussed. The parameter is used for deriving a formula for skewness of a sum of losses. The approximation of the sum is thus obtained and is used for deriving formulae for VaR and CVaR for aggregated losses. Finally, the methods are compared numerically using R software. 1


Composite distributions of loss sizes
Karatun, Ksenia ; Mazurová, Lucie (advisor) ; Zichová, Jitka (referee)
In this work, we deal with composite distributions that can be used to model loss sizes in some specific classes of nonlife insurance. The first part contains definition of the general composite model and its special features. The second part describes models that are made up by piecing together Weibull distribution and distributions belonging to a family of transformed beta distributions. The third part describes algorithm that computes the maximum likelihood estimators for parameters of composite distribution and criteria of the relative quality of statistical models. In the last part we apply composite models to two real data sets. 1


Reverse mortgage
Korotkov, Daniil ; Mazurová, Lucie (advisor) ; Večeř, Jan (referee)
ČSOB Pojišťovna, a. s., člen holdingu ČSOB Veřejné 1 / 1 20.7.2018 Abstract: At this moment, reverse mortgages are relatively new products on the Czech market and this thesis deals with their problematics. In this thesis, we describe the main risks related to reverse mortgages, namely, longevity risk and adverse evolution of property prices. Analyzing these risks we are modelling the underlying property prices, their future behavior, discount factors along with studying the risk models such as vector autoregression, hedonic model, repeatsales and WillsSherris model. In practical part, we focus on estimating the parameters of LeeCarter model and autoregression model of zerocoupon government bond as well as applying the results of the estimation to calculate various characteristics of reverse mortgages.


Risk quantification in annuity insurance
Berdák, Vladimír ; Mazurová, Lucie (advisor) ; Branda, Martin (referee)
The thesis examines the impact of individual risks on an annuity product. It focuses on the deffered whole life annuity and on two basic risks, which affect the overall loss the most. These are interest rate risk and longevity risk. We choose standard deviation (σ), value at risk (VaR) and expected shortfall (ES) at different confidence levels for target risk measures. Hoeffding decomposition is used to split the overall loss. Then Euler allocation principle will show the distribution of individual risks for different entry ages.


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.
