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Mathematical models for transportation problems
Brzobohatý, Jan ; Hrabec, Dušan (referee) ; Popela, Pavel (advisor)
This bachelor's thesis deals with stochastic dominance. The goal is to lay the foundations for defining stochastic dominance, to describe its properties and to explain this term on simple examples. Another goal is to apply this term to network problems with random transport price. Examples in this thesis also contain solutions and code to find these solutions written in GAMS language.
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Mean-Risk Optimization Problem via Scalarization, Stochastic Dominance, Empirical Estimates
Kaňková, Vlasta
Many economic and financial situations depend simultaneously on a random element and on a decision parameter. Mostly it is possible to influence the above mentioned situation by an optimization model depending on a probability measure. We focus on a special case of one-stage two objective stochastic “Mean-Risk problem”. Of course to determine optimal solution simultaneously with respect to the both criteria is mostly impossible. Consequently, it is necessary to employ some approaches. A few of them are known (from the literature), however two of them are very important: first of them is based on a scalarizing technique and the second one is based on the stochastic dominance. First approach has been suggested (in special case) by Markowitz, the second approach is based on the second order stochastic dominance. The last approach corresponds (under some assumptions) to partial order in the set of the utility functions.\nThe aim of the contribution is to deal with the both main above mentioned approaches. First, we repeat their properties and further we try to suggest possibility to improve the both values simultaneously with respect to the both criteria. However, we focus mainly on the case when probability characteristics has to be estimated on the data base.
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Robust approaches in portfolio optimization with stochastic dominance
Kozmík, Karel ; Kopa, Miloš (advisor) ; Lachout, Petr (referee)
We use modern approach of stochastic dominance in portfolio optimization, where we want the portfolio to dominate a benchmark. Since the distribution of returns is often just estimated from data, we look for the worst distribution that differs from empirical distribution at maximum by a predefined value. First, we define in what sense the distribution is the worst for the first and second order stochastic dominance. For the second order stochastic dominance, we use two different formulations for the worst case. We derive the robust stochastic dominance test for all the mentioned approaches and find the worst case distribution as the optimal solution of a non-linear maximization problem. Then we derive programs to maximize an objective function over the weights of the portfolio with robust stochastic dominance in constraints. We consider robustness either in returns or in probabilities for both the first and the second order stochastic dominance. To the best of our knowledge nobody was able to derive such program before. We apply all the derived optimization programs to real life data, specifically to returns of assets captured by Dow Jones Industrial Average, and we analyze the problems in detail using optimal solutions of the optimization programs with multiple setups. The portfolios calculated using...
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Stochastic Programming Problems in Asset-Liability Management
Rusý, Tomáš ; Kopa, Miloš (advisor)
The main objective of this thesis is to build a multi-stage stochastic pro- gram within an asset-liability management problem of a leasing company. At the beginning, the business model of such a company is introduced and the stochastic programming formulation is derived. Thereafter, three various risk constraints, namely the chance constraint, the Value-at-Risk constraint and the conditional Value-at-Risk constraint along with the second-order stochastic dominance constraint are applied to the model to control for riski- ness of the optimal strategy. Their properties and their effects on the optimal decisions are thoroughly investigated, while various risk limits are considered. In order to obtain solutions of the problems, random elements in the model formulation had to be approximated by scenarios. The Hull - White model calibrated by a newly proposed method based on maximum likelihood esti- mation has been used to generate scenarios of future interest rates. In the end, the performances of the optimal solutions of the problems for unconsid- ered and unfavourable crisis scenarios were inspected. The used methodology of such a stress test has not yet been implemented in stochastic programming problems within an asset-liability management. 1
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Stochastic dominance in portfolio optimization
Paulik, Marek ; Kopa, Miloš (advisor) ; Branda, Martin (referee)
The main topic of this thesis is the application of stochastic dominance constrains to portfolio optimization problems. First, we recall Markowitz model. Then we present portfolio selection problems with stochastic dominance constraints. Finally, we compare performance of these two approaches in an empirical study presented in the last chapter.
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Analysis of portfolio efficiency sets
Fehérová, Veronika ; Kopa, Miloš (advisor) ; Lachout, Petr (referee)
Pøedlo¾ená práce se zabývá dvìma pøístupy øe¹ení problému volby portfolia. Prvním jsou čmean-riskÿ modely, které minimalizují riziko pro pøedem zvolený výnos nebo maximalizují výnos pro pevnì stanovené riziko. Druhým je princip stochastické dominance, úzce související s teorií u¾itku. Cílem této diplomové práce je zkoumat vztah mezi mno¾inami e cientních portfolií, které jsou øe¹e- ním v obou pøístupech. Pro kvanti kaci rizika se kromì základních mìr jako jsou rozptyl, V aR nebo CV aR v práci uva¾ují i spektrální míry, zohledòující sub- jektivní postoj investora k riziku. Uká¾eme, za platnosti jakých podmínek jsou modely minimalizující spektrální míry konzistentní se stochastickou dominancí druhého øádu (SSD). Aplikujeme Kopa-Postùv test, který je jedním z více testù na SSD e cienci portfolia, na reálná data z americké burzy cenných papírù a SSD e cientní portfolia porovnáme s e cientními portfoliami získanými minimalizací CV aR-u uva¾ovaného na rùznych hladinách spolehlivosti. 1
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New Trends in Stochastic Programming
Szabados, Viktor ; Kaňková, Vlasta (advisor) ; Lachout, Petr (referee)
Stochastic methods are present in our daily lives, especially when we need to make a decision based on uncertain events. In this thesis, we present basic approaches used in stochastic tasks. In the first chapter, we define the stochastic problem and introduce basic methods and tasks which are present in the literature. In the second chapter, we present various problems which are non-linearly dependent on the probability measure. Moreover, we introduce deterministic and non-deterministic multicriteria tasks. In the third chapter, we give an insight on the concept of stochastic dominance and we describe the methods that are used in tasks with multidimensional stochastic dominance. In the fourth chapter, we capitalize on the knowledge from chapters two and three and we try to solve the role of portfolio optimization on real data using different approaches. 1
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Stochastic Programming Problems in Asset-Liability Management
Rusý, Tomáš ; Kopa, Miloš (advisor) ; Lachout, Petr (referee)
The main objective of this thesis is to build a multi-stage stochastic pro- gram within an asset-liability management problem of a leasing company. At the beginning, the business model of such a company is introduced and the stochastic programming formulation is derived. Thereafter, three various risk constraints, namely the chance constraint, the Value-at-Risk constraint and the conditional Value-at-Risk constraint along with the second-order stochastic dominance constraint are applied to the model to control for riski- ness of the optimal strategy. Their properties and their effects on the optimal decisions are thoroughly investigated, while various risk limits are considered. In order to obtain solutions of the problems, random elements in the model formulation had to be approximated by scenarios. The Hull - White model calibrated by a newly proposed method based on maximum likelihood esti- mation has been used to generate scenarios of future interest rates. In the end, the performances of the optimal solutions of the problems for unconsid- ered and unfavourable crisis scenarios were inspected. The used methodology of such a stress test has not yet been implemented in stochastic programming problems within an asset-liability management. 1
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Stochastic dominance generated by a decreasing absolute risk aversion
Mrázková, Adéla ; Kopa, Miloš (advisor) ; Lachout, Petr (referee)
The aim of the thesis is to describe first order stochastic dominance, second order stochastic dominance and then to motivate and describe stochastic dominance generated by utility functions with a decreasing absolute risk aversion. A numerical application of described methods follows. Efficiency in the meaning of stochastic dominance generated by utility functions with a decreasing absolute risk aversion and second order stochastic dominance is tested. Connection between the results is clarified and used methods are compared in the meaning of computational demands. Powered by TCPDF (www.tcpdf.org)
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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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