|
|
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 Value-at-Risk and Expected Shortfall predictive performance. We conduct several out-of-sample 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, Value-at-Risk, Expected Shortfall, Out-of-Sample Backtesting Author's e-mail ozturkdurmus@windowslive.com Supervisor's e-mail ies.avdulaj@gmail.com
|
|
|
Stability of the Banking Sector: Dependence Beyond Correlation
Fiala, Tomáš ; Šopov, Boril (advisor) ; Zelený, Tomáš (referee)
We analyze systemic risk of banks in countries of the so-called Visegrad Group (V4). Particularly, we focus on the relationship between a foreign mother and its local subsidiary which we compare to the relationship between subsidiaries in a given V4 country. We find that the systemic risk between two subsidiaries is higher than that between a mother and the respective subsidiary. In our analysis, we employ a technique stemming from a nonparametric multivariate Extreme Value Theory which is distribution free. Thus, our results are robust to heavy tails. Keywords Extreme value theory, systemic risk, financial stability, Visegrad Group, mother-subsidiary re- altionship Author's e-mail tomas.fiala@gjn.cz Supervisor's e-mail boril.sopov@gmail.com
|
|
|
Operational risk modelling
Mináriková, Eva ; Mazurová, Lucie (advisor) ; Hlubinka, Daniel (referee)
In the present thesis we will firstly familiarize ourselves with the term of operational risk, it's definition presented in the directives Basel II and Solvency II, and afterwards with the methods of calculation Capital Requirements for Operational Risk, set by these directives. In the second part of the thesis we will concentrate on the methods of modelling operational loss data. We will introduce the Extreme Value Theory which describes possible approaches to modelling data with significant values that occur infrequently; the typical characteristic of operational risk data. We will mainly focus on the model for threshold exceedances which utilizes Generalized Pareto Distribution to model the distribution of those excesses. The teoretical knowledge of this theory and the appropriate modelling will be applied on simulated loss data. Finally we will test the ability of presented methods to model loss data distributions.
|
|
|
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 semi-parametric 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 quantile-based market risk measures, Value at Risk and Expected Shortfall, are investigated. The main conclusion is that heavy-tailed 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 Semi-parametric Estimation, Statistics of Extremes, Extreme Value Theory, Market Risk, Value at Risk, Expected Shortfall.
|
|
|
Extreme value theory: Empirical analysis of tail behaviour of GARCH models
Šiml, Jan ; Šopov, Boril (advisor) ; Kocourek, David (referee)
This thesis investigates the capability of GARCH-family models to capture the tail properties using Monte Carlo simulation in framework of Conditional Extreme Value Theory. Analysis is carried out for three different GARCH-type models: GARCH, EGARCH, GJR-GARCH using Normal and Student's t-distributed innovations on four well-known stock market indices: S&P 500, FTSE 100, DAX and Nikkei 225. After conducting 3000 simulations of every estimated model, the Hill estimate of shape parameter implied by the GARCH-type models will be calculated and the models' performance will be assessed based on histograms, descriptive statistics and Root Mean Squared Error of simulated Hill estimates. Interesting results and im- plications for further research have been identified. Firstly, we highlight the Normal distribution's inappropriate nature in this case and its inability to capture the tail properties. Furthermore, GJR-GARCHT with t-distributed innovations is identified to be the best model, closely followed by other t-distributed GARCH-type models. Finally, a pattern in all Q-Q plots forecasting the simulation study results is appar- ent, with the exception of the DAX. This anomalous behaviour therefore necessitated further analysis and a significant right tail influence was recorded. Even though Hill estimates...
|
|
|
Analysis of extreme values
Vyhlídka, Jan ; Hendrych, Radek (advisor) ; Antoch, Jaromír (referee)
The goal of this thesis is to introduce basic concepts of the extreme value theory. The first chapter describes two fundamentally different approaches - block maxima and peaks over threshold models. Furthermore, it presents generalized extreme value distribution and generalized Pareto distribution. Moreover, relevant theorems and characteristics that are tied to these probabilistic distributions are discussed. The second chapter is a survey of various methods of parameter estimation of discussed distributions. The last chapter shows a simple application of how extreme value theory can be applied in finance on selected shares listed on the Prague Stock Exchange.
|
|
|
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 best-practices 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 well-known caveats Value-at-Risk remains a predominant risk measure used in the context of operational risk management. We describe several serious drawbacks of Value-at-Risk 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...
|
|
|
Unemployment Duration in the Czech Republic Through the Lens of Survival Analysis
Čabla, Adam ; Malá, Ivana (advisor) ; Komárková, Lenka (referee) ; Popelka, Jan (referee)
In the presented thesis the aim is to apply methods of survival analysis to the data from the Labour Force Survey, which are interval-censored. With regard to this type of data, I use specific methods designed to handle them, especially Turnbull estimate, weighted log-rank test and the AFT model. Other objective of the work is the design and application of a methodology for creating a model of unemployment duration, depending on the available factors and its interpretation. Other aim is to evaluate evolution of the probability distribution of unemployment duration and last but not least aim is to create more accurate estimate of the tail using extreme value theory. The main benefits of the thesis can include the creation of a methodology for examining the data from the Labour Force Survey based on standard techniques of survival analysis. Since the data are internationally comparable, the methodology is applicable at the level of European Union countries and several others. Another benefit of this work is estimation of the parameters of the generalized Pareto distribution on interval-censored data and creation and comparison of the models of piecewise connected distribution functions with solution of the connection problem. Work brought empirical results, most important of which is the comparison of results from three different data approaches and specific relationship between selected factors and time to find a job or spell of unemployment.
|
|
|
Oceňování zajištění škodního nadměrku v neživotním pojištění
Hrevuš, Jan ; Marek, Luboš (advisor) ; Cipra, Tomáš (referee) ; Zimmermann, Pavel (referee)
Probably the most frequently used definition of reinsurance is insurance for insurance companies, by reinsurance the cedant (insurance company) cedes part of the risk to the reinsurer. Reinsurance plays nowadays a crucial role in insurance industry as it does not only reduce the reinsured's exposure, but it can also significantly reduce the required solvency capital. In past few decades various approaches to reinsurance actuarial modelling were published and many actuaries are nowadays just reinsurance specialized. The thesis provides an overview of the actuarial aspects of modelling a non-life per risk and for motor third party liability per event excess of loss reinsurance structure, according to the author's knowledge no study of such wide scope exists and various aspects have to be found in various fragmented articles published worldwide. The thesis is based on recent industry literature describing latest trends and methodologies used, the theory is compared with the praxis as the author has working experience from underwriting at CEE reinsurer and actuarial reinsurance modelling at global reinsurance broker. The sequence of topics which are dealt corresponds to sequence of the steps taken by actuary modelling reinsurance and each step is discussed in detail. Starting with data preparation and besides loss inflation, more individual claims development methods are introduced and own probabilistic model is constructed. Further, burning cost analysis and probabilistic rating focused on heavy tailed distributions are discussed. A special attention is given to exposure rating which is not commonly known discipline among actuaries outside of reinsurance industry and different methodologies for property and casualty exposure modelling are introduced including many best practice suggestions. All main approaches to the reinsurance modelling are also illustrated on either real or realistically looking data, similar to those provided by European insurance companies to their reinsurers during renewal periods.
|
|
|
Modelování extrémních hodnot
Shykhmanter, Dmytro ; Malá, Ivana (advisor) ; Luknár, Ivan (referee)
Modeling of extreme events is a challenging statistical task. Firstly, there is always a limit number of observations and secondly therefore no experience to back test the result. One way of estimating higher quantiles is to fit one of theoretical distributions to the data and extrapolate to the tail. The shortcoming of this approach is that the estimate of the tail is based on the observations in the center of distribution. Alternative approach to this problem is based on idea to split the data into two sub-populations and model body of the distribution separately from the tail. This methodology is applied to non-life insurance losses, where extremes are particularly important for risk management. Never the less, even this approach is not a conclusive solution of heavy tail modeling. In either case, estimated 99.5% percentiles have such high standard errors, that the their reliability is very low. On the other hand this approach is theoretically valid and deserves to be considered as one of the possible methods of extreme value analysis.
|
guest ::
