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Risk aversion in portfolio efficiency
Puček, Samuel ; Branda, Martin (advisor) ; Kopa, Miloš (referee)
This thesis deals with selecting the optimal portfolio for a risk averse investor. Firstly, we present the risk measures, specifically spectral risk me- asures which consider an individual risk aversion of the investor. Then we propose a diversification-consistent data envelopment analysis model. The model is searching for an efficient portfolio with respect to second-order sto- chastic dominance. The crux of the thesis is a model based on the theory of multi-criteria optimization and spectral risk measures. The presented mo- del is searching for an optimal portfolio suitable for the investor with a given risk aversion. In addition, the optimal portfolio is also consistent with second- order stochastic dominance efficiency. The topic of the practical part is a nu- merical study in which both models are implemented in MATLAB. Models are applied to a dataset from real financial markets. Personal contribution lies in comparing the diversification-consistent data envelopment analysis model and model based on multi-criteria optimization, both with respect to second order stochastic dominance efficiency.
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Minimax in scheduling problems under uncertainty
Jeliga, Jan ; Branda, Martin (advisor) ; Lachout, Petr (referee)
In this work, we deal with fixed interval scheduling problems with the possibility of random delay of the end of the tasks (FIS). First, we pre- sent the basic deterministic FIS problems and ways to solve them. Next, we introduce the concept of minimax and present two well-known and one new FIS problem under uncertainty, when random task delays are conside- red to belong to a certain uncertainty set. Next, we deal with the solution of previously presented FIS problems for five chosen uncertainty sets. We present both previously achieved and original results. The work concludes with a summary of a numerical study of two problems. First, we explore the possibility of Lagrange relaxation application to the first presented problem. Next we explore the quality of approximation allowing to solve the later of problems as LP. 1
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Optimization problems with decision-dependent uncertainty
Šípka, Stanislav ; Branda, Martin (advisor) ; Lachout, Petr (referee)
In practical optimization problems, uncertainty in parameter values is often present. This uncertainty needs to be taken in account when taking real-life de- cisions. Such issues, where the parameters of the problem lie in the sets with a given shape, can be solved by a type of linear optimization called robust linear optimization. Special cases of these robust optimization are problems, where the sets depend on decisions. In this thesis we will focus on these special problems. The main aim of this thesis is to reformulate the classical form of the problems, leading to formulations which can be solved by standard computational software. We will then use these formulations in numerical study, focusing on behavior of robust shortest path in graphs. 1
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Exact penalization in optimization
Šešulka, Marek ; Branda, Martin (advisor) ; Kopa, Miloš (referee)
This thesis deals with one of the possible different approaches to solving nonlinear optimization problems by convertion to finding non-bounded extrema of function, where constrains are transfered to objective function via penalty function. We will introduce exterior penalty function method and appropriate algorithm for solving this type for problems. The thesis also deals with exact penalty functions, which do not requires limit approximation of the penalty pa- rameter to infinity. Then we deal with integer binary nonlinear progamming, where several suitable penalty functions are presented to solve this type of pro- blem. In the numerical part, the thesis deals with the minimization of risk at the specifed minimum expected return on the sparse portfolio. We observe the effect of changing the penalty parameter on the results of ten different minimization problems calculating risk of sparsity portfolios. 1
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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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Market model with random inputs
Krch, Ivan ; Lachout, Petr (advisor) ; Branda, Martin (referee)
The thesis deals with market models with random inputs represented by the newsvendor problem for which the randomness is given through a random number of customers. Presented work is divided into three chapters. In the first chapter we present the elementar newsvendor problem as stochastic programming problem with a fixed recourse. In the second chapter we present the multiplayer game theory adapted to the newsvendors problem. Moreover, in the second chapter we extend the problem by the second newsvendor on the market and in the third chapter we generalize the problem for n newsvendors on the market. We deal with the situations that arise in the chapters two and three from the game theory point of view and we study characteristics of a Nash equilibrium. Presented theory is demonstrated on illustrative examples in the ends of the two last chapters. 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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Solving methods for bilevel optimization problems
Lžičař, Jiří ; Kopa, Miloš (advisor) ; Branda, Martin (referee)
The presented thesis discusses bilevel programming problems with the focus on solution algorithms. Bilevel programming problem is a hierarchical programming problem, where constraints contain another programming problem. We formulate basic bilevel optimization theory and describe three types of so- lution algorithms for bilevel programming problems: Algorithms based on KKT reformulation where the lower level is replaced by its KKT conditions, algorithms based on optimal value function where the bilevel programming problem is re- duced to a single level problem using the optimal value function of the lower level problem, and algorithms solving linear bilevel programming problems. Using real data for portfolio optimization bilevel programming problems, we compare ability to solve the problems and computing time of some of the pre- sented algorithms. 1
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A verification of an approximation of the continuous double auction by a sequence of call auctions.
Kubík, Petr ; Šmíd, Martin (advisor) ; Branda, Martin (referee)
The thesis deals with two kinds of double auction - with the continuous auction and a sequence of call auctions. We explain their rules and we define their models. We present results of simulations of the both kinds of double auction - the aim is to look for the call auction with such parameters that the prices and the traded volume of the continuous auction are approximated best. Finally, in a theoretical part, we characterize the dis- tribution of the order book in the continuous auction and then we specify the joint distribution of the price and the traded volume in the call auction (the distribution of bid, ask and the traded volume given by the continuous auction may be immediately devised from the distribution of the order book).
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Multivariate stochastic dominance and its application in portfolio optimization problems
Petrová, Barbora ; Kopa, Miloš (advisor) ; Ortobelli, Sergio (referee) ; Branda, Martin (referee)
Title: Multivariate stochastic dominance and its application in portfolio optimization Problems Author: Barbora Petrová Department: Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Ing. Miloš Kopa, Ph.D., Department of Probability and Mathematical Statistics Abstract: This thesis discusses the concept of multivariate stochastic dominance, which serves as a tool for ordering random vectors, and its possible usage in dynamic portfolio optimization problems. We strictly focus on different types of the first-order multivariate stochastic dominance for which we describe their generators in the sense of von Neumann-Morgenstern utility functions. The first one, called strong multivariate stochastic dominance, is generated by all nondecreasing multivariate utility functions. The second one, called weak multivariate stochastic dominance, is defined by relation between survival functions, and the last one, called the first-order linear multivariate stochastic dominance, applies the first-order univariate stochastic dominance notion to linear combinations of marginals. We focus on the main characteristics of these types of stochastic dominance, their relationships as well as their relation to the cumulative and marginal distribution functions of considered random vectors. Formulated...
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