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Selected financial optimization models
Bujnovský, Daniel ; Bednář, Josef (referee) ; Popela, Pavel (advisor)
This work is focused on models of optimal asset and liability management. The practical section illustrates various ways of modelling strategies depending on the problem formulation, chosen set of assets and the type of the used optimization technique. The main examples are portfolio immunization and the Yasuda-Kasai model together with the extended version of Markowitz model. The author provides across the work an overview of different financial risks and various tools for their measurement together with possible formulations of expected returns relevant to the studied models. The individual models are compared and often extended by other constraints in order to improve their practical applicability. From the point of view of the mathematical optimization several ways of input data generation are described for example by using the extended Brownian motion. All practical parts go hand in hand with illustrative pictures and codes. The necessary financial and mathematical theory is included as well.
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Contemporary measures of financial risk
Leder, Ondřej ; Hurt, Jan (advisor) ; Zichová, Jitka (referee)
The main goal of this work is to talk about some financial risks and to introduce some methods of measuring them. The most important part of this work is the value at risk, its extension in form of conditional value at risk and introduction of some of its possible alternatives, which are expectile and spectral risk measures. For this it is needed to give a theoretical framework from the theory of probability. Its goal is to show the similarity of expectile and quantile, because value at risk is practicaly a quantile. Another goal of this fork is to show some weak properties of VaR and to practically illustrate the possibility of using expectile as an alternative to VaR. Powered by TCPDF (www.tcpdf.org)
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Quantitative methods in finance
Zboňáková, Lenka ; Hurt, Jan (advisor) ; Zichová, Jitka (referee)
In the present thesis we deal with the quantitative risk measures estimating the influence of market risk on the investments to the financial instruments. The most commonly used measure is Value at Risk which we introduce with its characteristics and modifications. Applying the methods to real data we deal with the problem of approximation of its distribution, especially in the multidimensional cases when the risk factors are dependent on each other. This leads us to explore copula functions that are in the thesis used to include the dependence structures of the risk factors to calculation of the risk measures. Chosen methods of approximation and evaluation of the risk measures are applied to real data and stated with outputs and their comparison.
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Diversification in Data Envelopment Analysis in finance
Macková, Simona ; Branda, Martin (advisor) ; Hurt, Jan (referee)
Title: Diversification in Data Envelopment Analysis in Finance Author: Simona Macková Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Martin Branda, Ph.D., Department of Probability and Ma- thematical Statistics Abstract: This thesis deals with an extension of data envelopment analysis and its application in finance. This method enables to evaluate the efficiency of cho- sen production units based on several inputs and outputs. Administrative fees or risk measures can be used as inputs and expected incomes of observed assets as outputs in financial application. We show basic traditional models in a form of a primary problem of linear programming and a dual problem as well and later compare with diversification models. It is suitable to deal with diversification which enables to consider dependencies between assets in case of finance and in- vestments. Than we get to nonlinear programming problem hence we introduce appropriate risk and return measures to make the problem solvable. Especially, we focus on the conditional value at risk. Next we introduce the model which deals with diversification. We use this on real data of chosen mutual funds. Keywords: Data envelopment analysis, Efficiency, Diversification, Conditional value at risk
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Optimal investment problems solvable using linear programming
Jančařík, Joel ; Branda, Martin (advisor) ; Kopa, Miloš (referee)
Portfolio optimization problem is a classical optimization problem, where the expected return of the portfolio is maximized and the risk is minimized. In this bachelor thesis some LP solvable portfolio optimization models are studied. Application on real life financial data is also included. Model with Conditional Value at Risk, MAD-model and Minimax model are described. In numerical analysis data from Frankfurt Stock Exchange are used and optimization has been made by Wolfram Mathematica 9.0 function LinearProgramming. As a result we got optimal portfolios for eleven different models for each of six minimal expected return constraints. The portfolios have been then evaluated according to the data from next year period. Powered by TCPDF (www.tcpdf.org)
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Cyber risk modelling using copulas
Spišiak, Michal ; Teplý, Petr (advisor) ; Baruník, Jozef (referee)
Cyber risk or data breach risk can be estimated similarly as other types of operational risk. First we identify problems of cyber risk models in existing literature. A large dataset consisting of 5,713 loss events enables us to apply extreme value theory. We adopt goodness of fit tests adjusted for distribution functions with estimated parameters. These tests are often overlooked in the literature even though they are essential for correct results. We model aggregate losses in three different industries separately and then we combine them using a copula. A t-test reveals that potential one-year global losses due to data breach risk are larger than the GDP of the Czech Republic. Moreover, one-year global cyber risk measured with a 99% CVaR amounts to 2.5% of the global GDP. Unlike others we compare risk measures with other quantities which allows wider audience to understand the magnitude of the cyber risk. An estimate of global data breach risk is a useful indicator not only for insurers, but also for any organization processing sensitive data.
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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 Cornish-Fisher 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
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Optimization of reinsurace parameters in insurance
Dlouhá, Veronika ; Branda, Martin (advisor) ; Cipra, Tomáš (referee)
This thesis is dedicated to searching optimal parameters of reinsurance with a focus of quota-share and stop-loss reinsurance. The optimization is based on minimization of value at risk and conditional value at risk of total costs of the insurer for the recieved risk. It also presents a compound random variable and shows various methods of obtaining its probability distribution, for example ap- proximation by lognormal or gamma mixtures distributions or by Panjer recurive method for continuous severity and numerical method of its solution. At the end of the thesis we can find the calculation of the optimal parameters of reinsurance for a compound random variable based on real data. We use various methods to determine probability distribution and premiums. 1
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Alternative risk measures and their applications
Drobuliak, Matúš ; Hurt, Jan (advisor) ; Večeř, Jan (referee)
Title: Alternative risk measures and their applications Author: Matúš Drobuliak Department: Department of Probability and Mathematical Statistics Supervisor: Doc. RNDr. Jan Hurt, CSc., Department of Probability and Mathe- matical Statistics Abstract: The objective of this thesis is to discuss alternative measures of risk. We focused on the expectile value at risk, which we compared with conventional risk measures - namely value at risk and conditional value at risk. We also discussed its properties from the financial point of view. A numerical illustration is included in the thesis. Keywords: Value at risk, Conditional value at risk, Quantile, Expectile, Expectile value at risk iii
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Basic approaches to robust conditional value at risk
Nožička, Michal ; Branda, Martin (advisor) ; Petrová, Barbora (referee)
The work describes conditional value at risk, its robustification with respect to the probability distribution of yields of assets and its applications to optimal portfolio selection. In chapter one there are definitions of conditional value at risk and its generalization throught robustification and also motivation to these definitions. The basic properties of conditional value at risk, mainly coherence and continuity with respect to the parametr of confidence level, are discussed in chapter two. There is also shown that some of these properties are preserved after robustification. The third chapter is dedicated to the derivation of optimization problems of optimal portfolio selection on the basis of conditional value at risk and its robustification. This thesis describes only special cases so that the final problems are solveble by the means of linear programming. The fourth chapter describes particular utilization of these methods with usage of real data from financial markets. Powered by TCPDF (www.tcpdf.org)
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