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Importance Sampling methods in solving optimization problems
Zavřel, Lukáš ; Kozmík, Václav (advisor) ; Kopa, Miloš (referee)
Present work deals with the portfolio selection problem using mean-risk models where analysed risk measures include variance, VaR and CVaR. The main goal is to approximate solution of optimization problems using simulation techniques like Monte Carlo and Importance Sampling. For both simulation techniques we present a numerical study of their variance and efficiency with respect to optimal solution. For normal distribution with particular expected value and variance the values of parameters for sampling using Importance Sampling method are empirically deduced and they are consequently used for solving a practical problem of choice of optimal portfolio from ten stocks, when their weekly historical prices are available. All optimization problems are solved in Wolfram Mathematica program. Powered by TCPDF (www.tcpdf.org)
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Portfolio efficiency with continuous probability distribution of returns
Kozmík, Václav
Present work deals with the portfolio selection problem using mean-risk models. The main goal of this work is to investigate the convergence of approximate 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 finding 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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Choice of the risk-aversion coefficient in optimization
Janásková, Eliška ; Kopa, Miloš (advisor) ; Lachout, Petr (referee)
Cílem této práce je studovat chování portfolia slo¾eného z daných akcií pro rùzné parametry averze k riziku. Nejprve popí¹eme, jaké vlastnosti by mìla splòovat vhodná míra rizika a poté uká¾eme, které z nich tyto vlastnosti opravdu splòují. Pøedstavíme Markowitzùv model a Mean-CVaR model, které slou¾í k optimalizaci portfolia. Z historických dat poté pomocí Mean-CVaR modelu urèíme pro dané akcie jejich zastoupení v optimálním portfoliu v závislosti na parametru averze k riziku a podíváme se, jak by si toto portfolio vedlo v následujících obdob ích. Na základì tìchto výpoètù budeme diskutovat výbìr vhodného parametru. Powered by TCPDF (www.tcpdf.org)
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Importance Sampling methods in solving optimization problems
Zavřel, Lukáš ; Kozmík, Václav (advisor) ; Kopa, Miloš (referee)
Present work deals with the portfolio selection problem using mean-risk models where analysed risk measures include variance, VaR and CVaR. The main goal is to approximate solution of optimization problems using simulation techniques like Monte Carlo and Importance Sampling. For both simulation techniques we present a numerical study of their variance and efficiency with respect to optimal solution. For normal distribution with particular expected value and variance the values of parameters for sampling using Importance Sampling method are empirically deduced and they are consequently used for solving a practical problem of choice of optimal portfolio from ten stocks, when their weekly historical prices are available. All optimization problems are solved in Wolfram Mathematica program. Powered by TCPDF (www.tcpdf.org)
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Portfolio efficiency with continuous probability distribution of returns
Kozmík, Václav
Present work deals with the portfolio selection problem using mean-risk models. The main goal of this work is to investigate the convergence of approximate 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 finding 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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