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Value-at-Risk Calculation Using Extreme Value Theory
Lipták, Patrik ; Hendrych, Radek (advisor) ; Mazurová, Lucie (referee)
This diploma thesis studies extreme value theory and its application in finan- cial risk management, when focusing on computation of well-known risk measure - Value at Risk (VaR). The first part of the thesis reviews theoretical background. In particular, it rigorously discusses the extreme value theory when emphasi- zing fundamentals theorems and their consequences followed by the summary of methods based on this theory, specifically, Block Maxima method, Hill met- hod and Peaks over Threshold method. Moreover, specific issues that may arise in such applications and ways how to deal with these problems are described. The second part of the thesis contains extensive empirical study, which together with theoretical foundings applies each of the examined method to real market data of the closing prices of Dow Jones Industrial Average stock index, stocks of JPMorgan and stock index Russell 2000 in order to compare methods based on extreme value theory together with the classic methodology RiskMetrics. 1
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Estimation and Application of the Tail Index
Pokorná, Markéta ; Šopov, Boril (advisor) ; Zelený, Tomáš (referee)
Examining the nature of extreme values plays an important role in financial risk management. This thesis investigates tail behaviour of distribution of re- turns using the framework of univariate Extreme Value Theory. The empirical research was conducted on the S&P 500 index and its seven constituents. The goal of this thesis was to use the Hill method to estimate the tail index of the series which characterizes the tail behaviour, especially the speed of the tail decay. To select the tail threshold several graphical methods were performed as they represent empirical measures of model stability. Classical Hill plots as well as alternative Hill plots and smoothing procedure were presented. The threshold choice based on stable regions in the graphs was found to be highly subjective. Hill method modified by Huisman was used instead and the results confirmed that the classical Hill method yields estimates which overestimate the tail thickness. All the examined series were found to have heavy tails with polynomial tail decay. This thesis stressed the need to model the left and the right tail separately as both extreme losses and profits are important depending on whether an investor takes a long or a short position on portfolio. Finally, the tail index was used to demonstrate the need to compute the...
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Stability of the Financial System: Systemic Dependencies between Bank and Insurance Sectors
Procházková, Jana ; Šopov, Boril (advisor) ; Janda, Karel (referee)
The central issue of this thesis is investigating the eventuality of systemic break- downs in the international financial system through examining systemic depen- dence between bank and insurance sectors. Standard models of systemic risk often use correlation of stock returns to evaluate the magnitude of intercon- nectedness between financial institutions. One of the main drawbacks of this approach is that it is oriented towards observations occurring along the central part of the distribution and it does not capture the dependence structure of outlying observations. To account for that, we use methodology which builds on the Extreme Value Theory and is solely focused on capturing dependence in extremes. The analysis is performed using the data on stock prices of the EU largest banks and insurance companies. We study dependencies in the pre- crisis and post-crisis period. The objective is to discover which sector poses a higher systemic threat to the international financial stability. Also, we try to find empirical evidence about an increase in interconnections in recent post- crisis years. We find that in both examined periods systemic dependence in the banking sector is higher than in the insurance sector. Our results also in- dicate that extremal interconnections in the respective sectors increased,...
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Multivariate extreme value models and their application in hydrology
Drápal, Lukáš ; Jarušková, Daniela (advisor) ; Hušková, Marie (referee)
Present thesis deals with the multivariate extreme value theory. First, concepts of modelling block maxima and threshold excesses in the univariate case are reviewed. In the multivariate setting the point process approach is chosen to model dependence. The dependence structure of multivariate extremes is provided by a spectral measure or an exponent function. Models for asymptotically dependent variables are provided. A construction principle from Ballani and Schlather (2011) is discussed. Based on this discussion the pairwise beta model introduced by Cooley et al. (2010) is modified to provide higher flexibility. Models are applied to data from nine hydrological stations from northern Moravia previously analysed by Jarušková (2009). Usage of the new pairwise beta model is justified as it brought a substantial improvement of log-likelihood. Models are also compared with Bayesian model selection introduced by Sabourin et al. (2013). Powered by TCPDF (www.tcpdf.org)
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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
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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
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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.
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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.
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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...
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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.
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