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Optimal investment problems solvable using linear programming
Jančařík, Joel ; Branda, Martin (advisor) ; Kopa, Miloš (referee)
Portfolio optimization problem is a classical optimization problem, where the expected return of the portfolio is maximized and the risk is minimized. In this bachelor thesis some LP solvable portfolio optimization models are studied. Application on real life financial data is also included. Model with Conditional Value at Risk, MAD-model and Minimax model are described. In numerical analysis data from Frankfurt Stock Exchange are used and optimization has been made by Wolfram Mathematica 9.0 function LinearProgramming. As a result we got optimal portfolios for eleven different models for each of six minimal expected return constraints. The portfolios have been then evaluated according to the data from next year period. Powered by TCPDF (www.tcpdf.org)
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Scenario generation by the moment fitting method
Koláčková, Hana ; Dupačová, Jitka (advisor) ; Branda, Martin (referee)
The thesis presents four methods for scenario generating leading to the resulting discrete probability distribution that replicates given values of the moments. The first method uses heuristic algorithm, the second method generates by symmetrically distributing values around the mean value, the third one is based on solving the system of nonlinear equations and finally the last method is based on goal programming. Next section describes the nature of problems solved by the goal programming. It also details possible ways of parameter specification to allow control of the computational complexity. In the last part of the thesis the results of several suitable methods for chosen types of problem are compared. Powered by TCPDF (www.tcpdf.org)
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Zobecněné lineární a aditivní modely v pojišťovnictví
Rusnák, Peter ; Branda, Martin (advisor) ; Pešta, Michal (referee)
In this thesis we describe the theory of generalized linear models and demon- strate its applications in non-life insurance. We also introduce some methods com- monly used to estimation of regression parameters and hypothesis testing . Further- more, we discuss possible extensions of GLM by introducing tools for reparametriza- tion of predictors which leads to new classes of models, concretely to segmented generalized linear models and generalized additive models. Consequently, we derive models appropriate for actuarial praxis using the real insurance data. In practical part of this thesis we illustrate the use of appropriate software for calculating the parameters of GAM and find way how to use open source statistical program .
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Penalizační metody ve stochastické optimalizaci
Kálosi, Szilárd ; Branda, Martin (advisor) ; Kaňková, Vlasta (referee)
The submitted thesis studies penalty function methods for stochastic programming problems. The main objective of the paper is to examine penalty function methods for deterministic nonlinear programming, in particular exact penalty function methods, in order to enhance penalty function methods for stochastic programming. For this purpose, the equivalence of the original de- terministic nonlinear and the corresponding penalty function problem using arbi- trary vector norm as the penalty function is shown for convex and invex functions occurring in the problems, respectively. The obtained theorems are consequently applied to multiple chance constrained problems under finite discrete probability distribution to show the asymptotic equivalence of the probabilistic and the cor- responding penalty function problems. The practical use of the newly obtained methods is demonstrated on a numerical study, in which a comparison with other approaches is provided as well. 1
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Claims count modeling in insurance
Škoda, Štěpán ; Branda, Martin (advisor) ; Pešta, Michal (referee)
1 Abstract: The present work investigates techniques of insurence ratemaking accor- ding to the claims counts of policyholders on the basis of information contained in policies. At the beginning, we provide a closer examination of the theory of genera- lized linear models, which have wide range of applications in the field of actuarial modeling. The second chapter presents the basic Poisson regression model as well as some particular verification methods. Specifically, deviance and Wald test could be found here and furthermore also important results for residuals. The third chapter contains information on alternative approaches to modeling the claim frequencies and at the end the GEE method, that can be applied in case of panel data, is de- scribed. The numerical study based on real insurace data in last part of this diploma thesis illustrate's previously described techniques which were obtained with the help of statistical software SAS.
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Penzijní modely
Kalaš, Martin ; Cipra, Tomáš (advisor) ; Branda, Martin (referee)
The thesis is concerned with the problem of sustainable spending towards the end of the human life cycle, which is a substantial quantitative problem in the pension framework. We gradually build a model, which coherently links the three key factors of retirement planning: uncertain length of human life, uncertain investment returns and spending rates. Within the framework of our intuitive model, we apply the method of moment matching to derive an approximation for the probability of individual's retirement ruin. The accuracy of presented approximation is analyzed via extensive Monte Carlo simulations. A numerical case study using Czech data is provided, including calculated values for the probability of ruin and maximal sustainable spending rate under various combinations of wealth-to-spending ratios and investment portfolio characteristics.
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Stochastic modelling of mortality development
Škerlík, Peter ; Mazurová, Lucie (advisor) ; Branda, Martin (referee)
In the presented diploma thesis we study possibilities of forecasting mortality rates and we explain the most used models to measure it. The longevity and mortality risk are characterized and options for transfer of risks to other subjects are suggested. Further we applied LifeMetrics tool to predict mortality and quantify longevity risk in our data set, also possibilities of its usage are described in more details. The aim of the work is to provide the reader with sufficient amount of theoretical information about the used stochastic models for mortality prediction. Also the work may be helpful to gain deeper knowledge about longevity risk.
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Multi - event Bonus - Malus System
Kaplanová, Martina ; Mazurová, Lucie (advisor) ; Branda, Martin (referee)
This work deals with bonus - malus systems for automobile insurance that distinguishtypes of claim. The first part of this work is definition of bonus - malus systems that do not distinguish types of claim and then their expansion just to multi - event bonus - malus systems. The main focus of the work is computation of stationary distribution for different systems, which means the distribution of classes in which the system stabilizes. Furthermore, there are several simulations of trajectory of insured through the system based on the number and type of accidents that they have caused. Finally, relative frequencies of classes in which insured is at the end of the simulation and the stationary distribution of the system are compared. Powered by TCPDF (www.tcpdf.org)
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Mean absolute deviation risk measure
Janouchová, Petra ; Kozmík, Václav (advisor) ; Branda, Martin (referee)
This bachelor thesis considers the mean absolute deviation as a risk me- asure. It deals with its properties and its application in the case of the asset allocation problem. The Markowitz model is described and we demonstrated the relation between our model with mean absolute deviation and the Mar- kowitz model. We study the influence of changes in the input data for the linear model with mean absolute deviation. The primary data used in this thesis are historical relative rates of profit of shares in the Prague Stock Ex- change. The testing is done on the selected subsets of scenarios from primary data and the stability is discussed in conclusion.
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Vehicle Routing Problem
Kafka, Ondřej ; Branda, Martin (advisor) ; Hanzák, Tomáš (referee)
The thesis deals with optimization problems which arise at distribution planning. These problems can often be easily formulated as integer programming problems, but rarely can be solved using mixed integer programming techniques. Therefore, it is necessary to study the efficiency of heuristic algorithms. The main focus of the thesis is on the vehicle routing problem with time windows. A tabu search algorithm for this problem was developed and implemented. It uses integer programming to solve the set partitioning problem in order to find optimal distribution of all customers into feasible routes found during the search. The results of the classical integer programming approach, basic insertion heuristic and presented tabu search algorithm are compared in a numerical study.
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