
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.


Optimal portfolio selection under Expected Shortfall optimisation with Random Matrix Theory denoising
Šíla, Jan ; Šopov, Boril (advisor) ; Baruník, Jozef (referee)
This thesis challenges several concepts in finance. Firstly, it is the Markowitz's solution to the portfolio problem. It introduces a new method which de noises the covariance matrix  the cornerstone of the portfolio management. Random Matrix Theory originates in particle physics and was recently intro duced to finance as the intersection between economics and natural sciences has widened over the past couple of years. Often discussed Efficient Market Hypothesis is opposed by adopting the assumption, that financial returns are driven by Paretian distributions, in stead of Gaussian ones, as conjured by Mandelbrot some 50 years ago. The portfolio selection is set in a framework, where Expected Shortfall replaces the standard deviation as the risk measure. Therefore, direct optimi sation of the portfolio is implemented to be compared with the performance of the classical solution and its denoised counterpart. The results are evalu ated in a controlled environment of Monte Carlo simulation as well as using empirical data from S&P 500 constituents. 1


Setting of the exposition limit for brokerage companies using risk quantifying methods
Nováková, Kateřina ; Zouhar, Jan (advisor) ; Holý, Vladimír (referee)
This thesis is focused on brokerage companies, the middlemen of investors' demands for buying and selling currency pair, and their problems of determining the proper limit on their currency exposition. The limit value has an influence on company's gains and losses resulting from exchange rate movement. The thesis describes a method how to quantify the risk of loss utilizing common financial indicators like Value at Risk and Expected Shortfall. The aim of the~thesis is to describe how to estimate these indicators. Real data supplied by brokerage company are used for calculation. The replication method is applied for determining the values of indicators, data for calculation are stochastically simulated. All calculations are implemented in R.


Impacts of new regulatory requirements for market risk
Vojkůvka, Adam ; Witzany, Jiří (advisor) ; Brodani, Jana (referee)
The aim of this master thesis is analyze the impact of new regulatory requirements for market risk in terms of internal approach of the selected portfolio. The first part deals with the definition and calculation methods of risk measures Value at Risk and Expected Shortfall. Furthermore, this part is dedicated to model backtesting and determination of the stress period. The second part describes the development of Basel IIII regulatory requirements for market risk with a focus on internal approaches. The third part focuses on the calculation and subsequent analysis of current and new regulatory reguirements for market risk using the historical simulation method, variance and covariance method and Monte Carlo simulation.


Measuring Extremes: Empirical Application on European Markets
Öztürk, Durmuş ; Avdulaj, Krenar (advisor) ; Janda, Karel (referee)
This study employs Extreme Value Theory and several univariate methods to compare their ValueatRisk and Expected Shortfall predictive performance. We conduct several outofsample backtesting procedures, such as uncondi tional coverage, independence and conditional coverage tests. The dataset in cludes five different stock markets, PX50 (Prague, Czech Republic), BIST100 (Istanbul, Turkey), ATHEX (Athens, Greece), PSI20 (Lisbon, Portugal) and IBEX35 (Madrid, Spain). These markets have different financial histories and data span over twenty years. We analyze the global financial crisis period sep arately to inspect the performance of these methods during the high volatility period. Our results support the most common findings that Extreme Value Theory is one of the most appropriate risk measurement tools. In addition, we find that GARCH family of methods, after accounting for asymmetry and fat tail phenomena, can be equally useful and sometimes even better than Extreme Value Theory based method in terms of risk estimation. Keywords Extreme Value Theory, ValueatRisk, Expected Shortfall, OutofSample Backtesting Author's email ozturkdurmus@windowslive.com Supervisor's email ies.avdulaj@gmail.com


Estimations of risk with respect to monthly horizon based on the twoyear time series
Myšičková, Ivana ; Houfková, Lucia (advisor) ; Zichová, Jitka (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.


Heavy Tails and Market Risk Measures: the Case of the Czech Stock Market
Bulva, Radek ; Zápal, Jan (advisor) ; Bubák, Vít (referee)
One of the stylized facts about the behaviour of financial returns is that they tend to exhibit more probability mass in the tails of the distribution than would be suggested by the normal distribution. This phenomenon is called heavy tails. The first part of this thesis focuses on examining the tails of a distribution of returns on Czech stock market index PX. Parametric and semiparametric approaches to estimation of the tail index, a measure of heaviness of tails, are applied and compared. The results indicate that the tails behave in a way one would expect from an emerging market stock index. In the second part of the thesis, implications for two quantilebased market risk measures, Value at Risk and Expected Shortfall, are investigated. The main conclusion is that heavytailed alternatives should be preferred to the normal distribution in order to avoid serious underestimation of risks embedded in the underlying process. JEL classification: C13, C14, C16, G15; Keywords: Heavy Tails, Parametric and Semiparametric Estimation, Statistics of Extremes, Extreme Value Theory, Market Risk, Value at Risk, Expected Shortfall.


The use of coherent risk measures in operational risk modeling
Lebovič, Michal ; Teplý, Petr (advisor) ; Doležel, Pavel (referee)
The debate on quantitative operational risk modeling has only started at the beginning of the last decade and the bestpractices are still far from being established. Estimation of capital requirements for operational risk under Advanced Measurement Approaches of Basel II is critically dependent on the choice of risk measure, which quantifies the risk exposure based on the underlying simulated distribution of losses. Despite its wellknown caveats ValueatRisk remains a predominant risk measure used in the context of operational risk management. We describe several serious drawbacks of ValueatRisk and explain why it can possibly lead to misleading conclusions. As a remedy we suggest the use of coherent risk measures  and namely the statistic known as Expected Shortfall  as a suitable alternative or complement for quantification of operational risk exposure. We demonstrate that application of Expected Shortfall in operational loss modeling is feasible and produces reasonable and consistent results. We also consider a variety of statistical techniques for modeling of underlying loss distribution and evaluate extreme value theory framework as the most suitable for this purpose. Using stress tests we further compare the robustness and consistency of selected models and their implied risk capital estimates...


Portfólio Value at Risk a Expected Shortfall s použitím vysoko frekvenčních dat
Zváč, Marek ; Fičura, Milan (advisor) ; Janda, Karel (referee)
The main objective of this thesis is to investigate whether multivariate models using Highfrequency data provide significantly more accurate forecasts of Value at Risk and Expected Shortfall than multivariate models using only daily data. Our objective is very topical since the Basel Committee announced in 2013 that is going to change the risk measure used for calculation of capital requirement from Value at Risk to Expected Shortfall. The further improvement of accuracy of both risk measures can be also achieved by incorporation of highfrequency data that are rapidly more available due to significant technological progress. Therefore, we employed parsimonious Heterogeneous Autoregression and its asymmetric version that uses highfrequency data for the modeling of realized covariance matrix. The benchmark models are chosen well established DCCGARCH and EWMA. The computation of Value at Risk (VaR) and Expected Shortfall (ES) is done through parametric, semiparametric and Monte Carlo simulations. The loss distributions are represented by multivariate Gaussian, Student t, multivariate distributions simulated by Copula functions and multivariate filtered historical simulations. There are used univariate loss distributions: Generalized Pareto Distribution from EVT, empirical and standard parametric distributions. The main finding is that Heterogeneous Autoregression model using highfrequency data delivered superior or at least the same accuracy of forecasts of VaR to benchmark models based on daily data. Finally, the backtesting of ES remains still very challenging and applied Test I. and II. did not provide credible validation of the forecasts.


Řízení rizik v komerční pojišťovně
Strýček, Tomáš
The diploma thesis deals with current issues of risk management in a selected insurance company. The thesis is conceptually divided into two parts the literature recherche and the empirical part. The first section introduces the individual risks and the basic methods of the quantification of the risks which affect the functioning of commercial insurances. A new system of European insurance regulation, Solvency II, is also described. The empirical part of the diploma thesis deals with the risk quantification of the selected insurance company according to the standard and internal model. The thesis is concluded with the evaluation of the risk management in the selected insurance company and of the company preparedness for the regulatory regime Solvency II. Based on this quantification, the recommendations are put forward to improve the risk management of the selected insurer.
