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Risk measures - sensitivity and dynamics
Branda, Martin ; Polívka, Jan (referee) ; Lachout, Petr (advisor)
Risk measures are subject to many scientific papers and monographs published on financial portfolio optimization problem within stochastic programming. Currently there are many functionals which measure risk of random future losses according to risk managers preferences. However, their sensitivity is studied less commonly, especially according to possible changes of input data or with respect to the portfolio allocation. This thesis deals with sensitivity of two frequently discussed measures - Value at Risk (VaR) and Conditional Value at Risk (CVaR). Explicit contamination bounds for relative VaR optimization problem are expressed using general results of parametric optimization valid for quadratic programming. A numerical study and a heuristic algorithm for correlation matrices stressing are involved. Sensitivity of VaR and CVaR is studied through their derivatives with respect to the portfolio allocation. Assumptions for the derivatives are formulated, Hessians introduced and convexity is discussed. At last, some dynamic risk measures for multi-period investory models are proposed.
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Seeming regression of economic indices
Komzáková, Magdalena ; Zvára, Karel (referee) ; Lachout, Petr (advisor)
In the time series analysis it often appears that two or more time series influence each other. When the generating stochastic processes of these series do not have stationary structure but they are stochastically non-stationary, i.e. the characteristic polynomial has a unit root, it happens that the regression modelling the dependence of some absolutely independent series gives statistically significant parameter estimations and statistics used to judge the model fitting do not indicate anything about its impropriety. This phenomenon is called seeming regression (spurious regression) and is solved with the theory of cointegration. We can say that when the series are cointegrated, their model shows their real dependence, not only the seeming one. Due to this fact, cointegration tests are also used for testing for the presence of seeming regression. These tests are based on unit root tests in generating process (or on stationarity tests), because time series can be cointegrated only if their linear combination is a stationary series.
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Multicriteria games
Tichá, Michaela ; Dlouhý, Martin (advisor) ; Lachout, Petr (referee) ; Čičková, Zuzana (referee)
Theory of multicriteria games is a special field of game theory, when one or more players have at least two payoff functions and want to maximize simultaneously. The work introduces a number of new findings. It examined the concept of finding equilibria in pure strategies in noncooperative multicriteria game. It is possible to find all the equilibria in pure strategies by full search and solving two linear programs for each point. Furthermore, two linear programs are formulated for verifying that a selected point is the equilibrium of the game or not. In the noncooperative games is also introduced the concept that with knowledge of the equilibrium of bimatrix game determines preferences of the players. Although finding the equilibrium point of the bimatrix game is nonlinear problem, finding the preferences is linear problem. The latest findings in the noncooperative games is a generalization of the concept that solves multicriteria game by assigning weights to each criterion of each player. The work demonstrates that it may not be necessarily linear weights, but it can be more general function that describes the player's preference. The remaining part is devoted to knowledge in cooperative games. There is considered that the players know their preferences and are able to express them by weights. The game with known preferences is defined and solved with the use of bargaining theory. Then it is generalized to a case where players have more payoff functions, from which they can choose. Finally, the multicriteria case of voting game is defined. It is designed completely new concept, which selects the winning coalition in the voting game. This concept is then applied to the real situation after the elections to the Chamber of Deputies in 2013.
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Total cost of a construction - a mathematical model
Bauer, K. ; Lachout, Petr
The paper presents a mathematical model for a total cost of a larger construction, e.g. a factory, a department store, a speed-way, etc. The total cost is divided into several subcosts according to specific features. Each particular subcost is considered to be random by nature, and, consequently, the total cost itself is random in our model. We aim to give a tool convenient for decision making.
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