
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.


Microlevel stochastic claims reserving
Rathouský, Marek ; Pešta, Michal (advisor) ; Vitali, Sebastiano (referee)
This thesis covers, in detail, theoretical background of microlevel stochastic model, which includes definition and properties of nonhomogeneous Poisson process. This the ory is then applied to real data generated by MTPL portfolio. Estimates of provisions under microlevel stochastic model are calculated using ordinary Monte Carlo simula tion method. Results obtained from microlevel stochastic model are compared to Mack Chainladder estimates. 1


Actuarial and Exposurebased Models for Hail Peril
Drobuliak, Matúš ; Pešta, Michal (advisor) ; Hlubinka, Daniel (referee)
Title: Actuarial and Exposurebased Models for Hail Peril Author: Bc. Matúš Drobuliak Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Michal Pešta, Ph.D., Department of Probability and Mathe matical Statistics Abstract: This thesis covers an introduction to catastrophe modelling and focuses on statistical methods for extreme events. This includes methods of estimating parameters of claim distribution with a focus on probability weighted moments estimation technique. Furthermore, times series modelling, skew tdistribution, and two model clustering techniques are examined as well. This is later utilised in the practical application part of this thesis, which uses real data provided by an insurance company operating in the Czech Republic. Probability distribution fitting of large claims caused by hailstorms and Monte Carlo simulation of future losses are demonstrated later. Keywords: Catastrophe modelling, Hail peril, Probability weighted moments, Extreme events, ARMAGARCH, Monte Carlo simulation iii


Counterparty risk in reinsurance
Kohout, Marek ; Cipra, Tomáš (advisor) ; Pešta, Michal (referee)
The main goal of this Bachelor thesis is to present a survey of methods for cal culating the required capital to cover the default risk of reinsurers in the frame work of the regulatory system Solvency II in EU. The methods are based on socalled common shock principle which is preferred in the case of portfolios with a smaller number of heterogeneous counterparties (e. g. reinsurers). In difference from (Hendrych and Cipra, 2018) the case with flexible weights of particular reinsurers given by their LGD (loss given default) is considered. One discusses the results of extensive numerical study comparing particular methods. 1


Claims reserve volatility and bootstrap with aplication on historical data with trend in claims development
Malíková, Kateřina ; Pešta, Michal (advisor) ; Zichová, Jitka (referee)
This thesis deals with the application of stochastic claims reserving methods to given data with some trends in claims development. It describes the chain ladder method and the generalized linear models as its stochastic framework. Some simple functions are suggested for smoothing the origin and development period coefficients from the estimated model. The extrapolation is also considered for estimation of the unobserved tail values. The residual bootstrap is used for the reparameterized model in order to get the predictive distribution of the estimated reserve together with its standard deviation as a measure of volatility. Solvency capital requirement in one year time horizon is also calculated. 1


Estimations of risk with respect to monthly horizon based on the twoyear time series
Myšičková, Ivana ; Houfková, Lucia (advisor) ; Pešta, Michal (referee)
The thesis describes commonly used measures of risk, such as volatility, Value at Risk (VaR) and Expected Shortfall (ES), and is tasked with creating models for measuring market risk. It is concerned with the risk over daily and over monthly horizons and shows the shortcomings of a squarerootoftime approach for converting VaR and ES between horizons. Parametric models, geometric Brownian motion (GBM) and GARCH process, and nonparametric models, historical simulation (HS) and some its possible improvements, are presented. The application of these mentioned models is demonstrated using real data. The accuracy of VaR models is proved through backtesting and the results are discussed. Part of this thesis is also a simulation study, which reveals the precision of VaR and ES estimates.


Tests for Combination of Correlation Coefficients
Kulíšková, Michaela ; Hudecová, Šárka (advisor) ; Pešta, Michal (referee)
This bachelor thesis is focused on tests for correlation coefficients. Fundamental knowledge about correlation coefficient are reminded as well as tests for correlation coefficient based on estimated correlation coefficient. Then three main methods for combining more correlation coefficients  Fisher`s method, Linear combination test with Ztransformation and Hotelling transformation  are described, simulated and compared. These tests have several assumptions such as that k correlation coefficients are known as well as the range of random samples from which they were calculated plus it is assumed that these correlation coefficients are equal.


ksample problem with ordered alternative
Nováková, Martina ; Hlávka, Zdeněk (advisor) ; Pešta, Michal (referee)
In this thesis we deal with ksample problem with ordered alternative. At the beginning of the thesis isotonic regression is introduced. We use isotonic regression for maximum likelihood estimation of ordered parameters. In the second chapter, we describe the χ2 and E 2 tests that use the knowledge of isotonic regression and are based on the likelihood ratio. The exact null hypothesis distributions of their test statistics are derived in detail. The onesided studentized range test is also further described. At the end of the thesis, we show the use of the E 2 test on the real data. 1


Loss reserving for individual claimbyclaim data
Bednárik, Vojtěch ; Pešta, Michal (advisor)
This thesis covers stochastic claims reserving in nonlife insurance based on individual claims developments. Summarized theoretical methods are applied on data from Czech Insurers' Bureau created for educational purposes. The problem of estimation is divided into four parts: occurence process generating claims, delay of notification, times between events and payments. Each part is estimated separately based on maximum likelihood theory and final estimates allow us to obtain an estimate of future liabilities distribution. The results are very promising and we believe this method is worth of a further research. Contribution of this work is more rigorous theoretical part and application on data from the Czech market with some new ideas in practical part and simulation. 1
