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The impact of stochastic dominance generator on a portfolio efficiency
Dungl, Martin ; Dupačová, Jitka (referee) ; Kopa, Miloš (advisor)
This contribution focuses on the sets of efficient portfolios and their properties, given the class of utility functions. Firstly the basic concepts of stochastic dominance are recalled, then we limit our attention to the scenario approach and the concept of portfolio eeffciency is introduced. Di erent de nitions of efficient portfolios are taken into account - we call them admissibility, strict admissibility and optimality. We summarize the most important results concerning the shapes of sets composed by portfolios efficient according to the above mentioned approaches. The problem of path connectedness of the sets of optimal portfolios with limited short sales is analyzed. We prove that the set of optimal portfolios is path connected under the assumption that the generator of the stochastic dominance is a path connected set of strictly concave utility functions. Then the convexity of the set of optimal portfolios with respect to the set of exponential utility functions, in case of allowed short sales, is analyzed. We conclude that the sets of optimal portfolios generally need not be convex and we prove the necessary and sufficient condition of convexity.
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The transportation problem, its generalizations and applications in probability and statistics
Doležel, Pavel ; Kopa, Miloš (referee) ; Dupačová, Jitka (advisor)
Author describes a specific optimization problem-the transportation problem and analyzes relevant solution methods. Several methods of solving the transportation problem are listed, applied or introduced and applications of the transportation poblem in the theory of probability and mathematical statistics are presented, namely the statistical sorting in L1 norm and re-construction of contingency tables. Special interest is devoted to several modifications of ordinary transportation problem, mainly the multiindex transportation problem. The crucial part of the work are selected applications of the transportation problem to particular problems and showing some algorithms used for finding solutions.
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Portfolio efficiency with continuous probability distribution of returns
Kozmík, Václav ; Dupačová, Jitka (referee) ; Kopa, Miloš (advisor)
Present work deals with the portfolio selection problem using mean-risk models. The main goal of this work is to investigate the convergence of approxi mate solutions using generated scenarios to the analytic solution and its sensitivity to chosen risk measure and probability distribution. The considered risk measures are: variance, VaR, cVaR, absolute deviation and semivariance. We present analytical solutions for all risk measures under the assumption of normal or Student distribution. For log-normal distribution, we use the approximate assumption that the sum of log-normal random variables has log-normal distribution. Optimization models for discrete scenarios are derived for all risk measures and compared with analytical solution. In case of approximate solution with scenarios, we repeat the procedure multiple times and present our own approach to nding the optimal solution using the cluster analysis. All optimization models are written in GAMS language. Testing and estimating are realized using an application developed in C++ language.
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Applications of evolution and differential games
Chvoj, Martin ; Kopa, Miloš (referee) ; Lachout, Petr (advisor)
This work deals with advance parts of game theory and their applications in financial area and follows a bachelor work Applications of game theory in economics written by Martin Chvoj at 2008. At the beginning there is a brief introduction into classical game theory, which uses a concept of Nash equilibrium. The work requires knowledge of this theory. In following chapters, modern ways of game theory are described in detail. These chapters deal with basic concepts of evolutionary and differential games which are supported by a number of examples and application possibilities. At the end of the work, there is an original example which models an income tax system as a differential game. This model uses a Czech tax legal framework for year 2009 and statistical data, which can be found at Czech bureau of statistics.
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