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Stochastic DEA and dominance
Majerová, Michaela ; Kopa, Miloš (advisor) ; Dupačová, Jitka (referee)
At the beginning of this thesis we discuss DEA methods, which measure efficiency of Decision Making Units by comparing weighted inputs and outputs. First we describe basic DEA models without random inputs and outputs then stochastic DEA models which are derived from the deterministic ones. We describe more approaches to stochastic DEA models, for example using scenario approach or chance constrained programming problems. Another approach for measuring efficiency employs stochastic dominance. Stochastic dominance is a relation that allows to compare two random variables. We describe the first and second order stochastic dominance. First we consider pairwise stochastic efficiency, then we discuss the first and second order stochastic dominance portfolio efficiency. We describe different tests to measure this type of efficiency. At the end of this thesis we study efficiency of US stock portfolios using real historical data and we compare results obtained when using stochastic DEA models and stochastic dominance. Powered by TCPDF (www.tcpdf.org)
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Semi - infinite programming: theory and portfolio efficiency application
Klouda, Lukáš ; Kopa, Miloš (advisor) ; Lachout, Petr (referee)
Title: Semi-infinite programming: theory and portfolio efficiency application Author: Bc. Lukáš Klouda Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Ing. Miloš Kopa, PhD. Supervisor's e-mail address: kopa@karlin.mff.cuni.cz Abstract: The thesis deals with application of semi-infinite programming to a portfolio efficiency testing. The summary of semi-infinite programming, first and second order optimality conditions and duality in linear semi-infinite programming is presented. The optimization problem for a portfolio efficiency testing with respect to the second order stochastic dominance under assumption of discrete, normal, Students and general elliptical distribution is formulated. Conditional value at risk(CVaR) is used as the risk measure, because of its consistency with the second order stochastic dominance relation. Efficiency of index PX with respect to the second order stochastic dominance is tested. The tests are performed using the program GAMS.
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High-order stochastic dominance
Mikulka, Jakub ; Kopa, Miloš (advisor) ; Branda, Martin (referee)
The thesis deals with high-order stochastic dominance of random variables and portfolios. The summary of findings about high-order stochastic dominance and portfolio efficiency is presented. As a main part of the thesis it is proven that under assumption of both normal and gamma distribution the infinite-order stochastic dominance is equivalent to the second-order stochastic dominance. The necessary and sufficient condition for the infinite-order stochastic dominance portfolio efficiency is derived under the assumption of normality. The condition is used in the empirical part of the thesis where parametrical approach to the portfolio efficiency is compared to the nonparametric scenario approach. The derived necessary and sufficient condition is based on the assumption of normality; therefore we use two sets of data, one with fulfilled assumption of normality and the other for which the assumption of normality was unambigously rejected. Consequently, the influence of fulfillment of the normality assumption on the results of the necessary and sufficient condition for portfolio efficiency is estimated.
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Efficiency of representative portfolios using data envelopment analysis
Junová, Jana ; Kopa, Miloš (advisor) ; Branda, Martin (referee)
In this work, several data envelopment analysis (DEA) models are used to assess efficiency of US representative portfolios. We consider a portfolio to be efficient if no other surpasses it in minimizing risk or maximizing return. This property is precisely defined in the work and it can be well detected by DEA models. DEA models assuming constant return-to-scale (CRS) as well as variable return-to- scale (VRS) are described here. A model with directional measure is also presented. Four of the VRS models are transformed into diversification consistent (DC) models. In the empirical part, CVaRs on multiple levels are used as risk measures and expected return as a return measure typically. Results acquired using different DEA models to assess efficiency of portfolios are compared. DC models are stronger than their classical VRS counterparts. The DC models identified as efficient only the portfolio with the highest expected return. On the contrary, VRS models classified as efficient more portfolios which differ in riskiness. Their results could be interesting if an investor wanted to choose only one portfolio based on its riskiness.
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Stochastic DEA and dominance
Majerová, Michaela ; Kopa, Miloš (advisor) ; Dupačová, Jitka (referee)
At the beginning of this thesis we discuss DEA methods, which measure efficiency of Decision Making Units by comparing weighted inputs and outputs. First we describe basic DEA models without random inputs and outputs then stochastic DEA models which are derived from the deterministic ones. We describe more approaches to stochastic DEA models, for example using scenario approach or chance constrained programming problems. Another approach for measuring efficiency employs stochastic dominance. Stochastic dominance is a relation that allows to compare two random variables. We describe the first and second order stochastic dominance. First we consider pairwise stochastic efficiency, then we discuss the first and second order stochastic dominance portfolio efficiency. We describe different tests to measure this type of efficiency. At the end of this thesis we study efficiency of US stock portfolios using real historical data and we compare results obtained when using stochastic DEA models and stochastic dominance. Powered by TCPDF (www.tcpdf.org)
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High-order stochastic dominance
Mikulka, Jakub ; Kopa, Miloš (advisor) ; Branda, Martin (referee)
The thesis deals with high-order stochastic dominance of random variables and portfolios. The summary of findings about high-order stochastic dominance and portfolio efficiency is presented. As a main part of the thesis it is proven that under assumption of both normal and gamma distribution the infinite-order stochastic dominance is equivalent to the second-order stochastic dominance. The necessary and sufficient condition for the infinite-order stochastic dominance portfolio efficiency is derived under the assumption of normality. The condition is used in the empirical part of the thesis where parametrical approach to the portfolio efficiency is compared to the nonparametric scenario approach. The derived necessary and sufficient condition is based on the assumption of normality; therefore we use two sets of data, one with fulfilled assumption of normality and the other for which the assumption of normality was unambigously rejected. Consequently, the influence of fulfillment of the normality assumption on the results of the necessary and sufficient condition for portfolio efficiency is estimated.
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Semi - infinite programming: theory and portfolio efficiency application
Klouda, Lukáš ; Kopa, Miloš (advisor) ; Lachout, Petr (referee)
Title: Semi-infinite programming: theory and portfolio efficiency application Author: Bc. Lukáš Klouda Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Ing. Miloš Kopa, PhD. Supervisor's e-mail address: kopa@karlin.mff.cuni.cz Abstract: The thesis deals with application of semi-infinite programming to a portfolio efficiency testing. The summary of semi-infinite programming, first and second order optimality conditions and duality in linear semi-infinite programming is presented. The optimization problem for a portfolio efficiency testing with respect to the second order stochastic dominance under assumption of discrete, normal, Students and general elliptical distribution is formulated. Conditional value at risk(CVaR) is used as the risk measure, because of its consistency with the second order stochastic dominance relation. Efficiency of index PX with respect to the second order stochastic dominance is tested. The tests are performed using the program GAMS.
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Value at Risk application to FSD portfolio efficiency testing
Kopa, Miloš
The paper deals with efficiency testing of a given portfolio with respect to all other portfolios that can be created from the considered set of assets. The efficiency is based on the first order stochastic dominance (FSD) relation. A necessary and sufficient condition for the first order stochastic dominance criterion is expressed in terms of Value at Risks (VaRs). Consequently a FSD portfolio efficiency test based on VaRs is formulated. Contrary to the usual case, a general discrete distribution of portfolio returns is assumed what makes the test computationally more demanding comparing to the equiprobable scenarios case. Therefore we present a tractable reformulation of this test that turns constraints on VaRs into classical mixed-integer nonlinear programming problem.
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