 

Oceňování zajištění škodního nadměrku v neživotním pojištění
Hrevuš, Jan ; Marek, Luboš (advisor) ; Cipra, Tomáš (referee) ; Zimmermann, Pavel (referee)
Probably the most frequently used definition of reinsurance is insurance for insurance companies, by reinsurance the cedant (insurance company) cedes part of the risk to the reinsurer. Reinsurance plays nowadays a crucial role in insurance industry as it does not only reduce the reinsured's exposure, but it can also significantly reduce the required solvency capital. In past few decades various approaches to reinsurance actuarial modelling were published and many actuaries are nowadays just reinsurance specialized. The thesis provides an overview of the actuarial aspects of modelling a nonlife per risk and for motor third party liability per event excess of loss reinsurance structure, according to the author's knowledge no study of such wide scope exists and various aspects have to be found in various fragmented articles published worldwide. The thesis is based on recent industry literature describing latest trends and methodologies used, the theory is compared with the praxis as the author has working experience from underwriting at CEE reinsurer and actuarial reinsurance modelling at global reinsurance broker. The sequence of topics which are dealt corresponds to sequence of the steps taken by actuary modelling reinsurance and each step is discussed in detail. Starting with data preparation and besides loss inflation, more individual claims development methods are introduced and own probabilistic model is constructed. Further, burning cost analysis and probabilistic rating focused on heavy tailed distributions are discussed. A special attention is given to exposure rating which is not commonly known discipline among actuaries outside of reinsurance industry and different methodologies for property and casualty exposure modelling are introduced including many best practice suggestions. All main approaches to the reinsurance modelling are also illustrated on either real or realistically looking data, similar to those provided by European insurance companies to their reinsurers during renewal periods.


Analýza rizik a oceňování energetických retailových kontraktů
Hron, Jiří ; Marek, Luboš (advisor) ; Málek, Jiří (referee) ; Krtička, Jiří (referee)
The presented dissertation is focused on the applications of statistical methods and approaches applied in the energy business. The need for the modeling of energy risks arose only recently when the energy business was opened to competition. Therefore, the primary aim of the dissertation is to clarify the main principles of the energy business which are necessary for understanding both risk principles and motivation of the proposed models. I am largely focused on retail risks, i.e., the risks associated with delivery to endconsumers. In particular, I deal with energy contracts providing volume flexibility, recalled as swing options in the literature. Therefore, the second issue on which I am focusing is a group of demanddriven swing options whose more systematic analysis in the portfolio context has not been published so far. Examining the risk, I apply the deductive (probabilistic) analysis which reveals interesting relations between correlations. The practical applications also require inductive considerations resulting in the construction of statistical estimators relying on historical data. I propose an estimator of the volumetric correlation based on a classical theory whose bias is investigated via MC simulation. To analyze a particular volumeprice correlation, I introduced the notion of robust dependency. Applying bootstrap procedures, robust dependency can be used both for testing purposes and for sensitivity analysis of the sample correlation. There are many works available devoted to energy price models which are different from the price models applied on financial markets. Therefore, the third target of the dissertation is an empirical statistical analysis of both power and natural gas Czech spot prices which can serve as a basis for the development of price models adapted to the Czech market environment. Finally, the fourth aim is the evaluation of power contracts which is very specific. The outputs of the model are both a synthetic market price and a hedging strategy. The model is designed to provide flexibility in practical applications.


DRG system
Vraná, Lenka ; Marek, Luboš (advisor) ; Mašek, Petr (referee)
This thesis is focused on description of the DRG (Diagnosis Related Groups) classification system and its application as the inpatient care financing tool. The objectives of this thesis are to sum up the possible applications of DRG system, to describe the calculation of indicators needed for health care payment in 2012 and to suggest some improvements to the algorithm to enhance the quality of data processing in the years ahead. The first part of this thesis is theoretical and it includes especially the history of the DRG classification system and an explanation of the basic concepts and the calculation methods. In the second part there is the description of the data file, which was used for the calculation of the inpatient care payments for 2012, its processing (specifically the determination of the relative weights of DRG) and the possibility to automate the whole solution.


Approaches to Functional Data Clustering
Pešout, Pavel ; Marek, Luboš (advisor) ; Trešl, Jiří (referee) ; Palát, Milan (referee)
Classification is a very common task in information processing and important problem in many sectors of science and industry. In the case of data measured as a function of a dependent variable such as time, the most used algorithms may not pattern each of the individual shapes properly, because they are interested only in the choiced measurements. For the reason, the presented paper focuses on the specific techniques that directly address the curve clustering problem and classifying new individuals. The main goal of this work is to develop alternative methodologies through the extension to various statistical approaches, consolidate already established algorithms, expose their modified forms fitted to demands of clustering issue and compare some efficient curve clustering methods thanks to reported extensive simulated data experiments. Last but not least is made, for the sake of executed experiments, comprehensive confrontation of effectual utility. Proposed clustering algorithms are based on two principles. Firstly, it is presumed that the set of trajectories may be probabilistic modelled as sequences of points generated from a finite mixture model consisting of regression components and hence the densitybased clustering methods using the Maximum Likehood Estimation are investigated to recognize the most homogenous partitioning. Attention is paid to both the Maximum Likehood Approach, which assumes the cluster memberships to be some of the model parameters, and the probabilistic model with the iterative ExpectationMaximization algorithm, that assumes them to be random variables. To deal with the hidden data problem both Gaussian and less conventional gamma mixtures are comprehended with arranging for use in two dimensions. To cope with data with high variability within each subpopulation it is introduced twolevel random effects regression mixture with the ability to let an individual vary from the template for its group. Secondly, it is taken advantage of well known KMeans algorithm applied to the estimated regression coefficients, though. The task of the optimal data fitting is devoted, because KMeans is not invariant to linear transformations. In order to overcome this problem it is suggested integrating clustering issue with the Markov Chain Monte Carlo approaches. What is more, this paper is concerned in functional discriminant analysis including linear and quadratic scores and their modified probabilistic forms by using random mixtures. Alike in KMeans it is shown how to apply Fisher's method of canonical scores to the regression coefficients. Experiments of simulated datasets are made that demonstrate the performance of all mentioned methods and enable to choose those with the most result and time efficiency. Considerable boon is the facture of new advisable application advances. Implementation is processed in Mathematica 4.0. Finally, the possibilities offered by the development of curve clustering algorithms in vast research areas of modern science are examined, like neurology, genome studies, speech and image recognition systems, and future investigation with incorporation with ubiquitous computing is not forbidden. Utility in economy is illustrated with executed application in claims analysis of some life insurance products. The goals of the thesis have been achieved.


Statisitcal models of the renewable energy market
Kozma, Petr ; Marek, Luboš (advisor) ; Coufal, Jan (referee) ; Hronová, Stanislava (referee) ; Janáček, Kamil (referee)
An efficient application and development of renewable energy sources is one of the most important contribution to the energetic balance of the human society. Anyhow, statistical model of the renewable energy market, which would fundamentally explain relevant economical rules related to these perspective energetic resources, is not clearly known up to now. Nevertheless, the relevant statistical data concerning application of solar energy (photovoltaic and thermosolar heating) are available for the last twenty years. Based on the economic models, statistical data concerning sales of photovoltaic models and thermosolar collectors sales have been analysed in this work. It has been shown that the model of constant elasticity predicts an exponential increase which will slow down when a certain level of annual cumulative sales was reached. The model of constant elasticity was found to be successful to interpret past sales data. In the approach of variable elasticity model the parameter of the elasticity has been modified as a function of variables such as market volume, price and time through the statistical evaluation. It enabled to calculate initial, saturation and competitive market conditions, as well. Whereas the constant elasticity demand model describes exponential growth of sales and installations, which was characteristic for the beginning of the application of these renewable resources of energy, the variable elasticity demand model describes a more realistic situation, where cumulative sales either increase or decrease and prices vary subsequently. Simple growth model of unlimited demand based on the growing sales is not realistic and could not be feasible in the long term. The market elasticity could be understood as a real economical parameter representing percentual market increase or decrease at a given time; in the variable demand elasticity model, the constant elasticity is replaced by a function of a market volume, price and time. In this case, we can estimate model parameters for the different market conditions: growth, saturation and decrease. The function representing the capital adequacy in the generalized market model has also been deliberated. Statistical models have been used to determine cumulative sales and market prices of photovoltaic modules and thermosolar collectors. Moreover, model parameters have been used for the calculation of the realized photovoltaic and thermo solar projects' capital adequacy on the renewable energy market. By using model parameters, renewable energy market forecast up to 2020 has been estimated. We have used generalized market model to credibly estimate future renewable energy market until 2020; as well as extend model parameterization on other resources of renewable energy (water and wind, geothermal sources, biomass) and set prices of energy produced from these renewable sources. Potential energetic savings have been estimated for households (apartments and private houses), who can be relevant consumers of energy from renewable sources. We have performed statistical findings on randomly selected files, where we have reached a real energy consumption, to prove this. This research allowed us to perform a real estimate of a renewable energy contribution to the total energy balance. We have successfully proved that linearly growing capital adequacy function, with an annual growth between 2.5% and 3.0%, is reflecting the renewable energy market sufficiently and is fully in line with an average growth of the total energy consumption. Renewable energy share on the total energy balance will grow substantially to reach a level of 15% in 2015 on the world market and a level of 8% in the Czech Republic for the same period with a perspective to reach a level of 11% in 2020 respectively. Assuming this level of renewable energy on the total production will lead to a decrease of CO2 emissions by three million of tones in 2015 and by four million of tones in 2020. Final reach of this status quo is fully predicted by our statistical model for renewable energy market.


Transformations of random variables
Šára, Michal ; Marek, Luboš (advisor) ; Malá, Ivana (referee)
This bachelor thesis deals with the transformation of random variables,which plays a significant partv in the theory of probability. The main aim of this paper is to show few methods and techniques which are used when transforming random variables. At the very beginning of this paper one can find a definition and practical examples of the LebesgueStieltjes integral and probability measure, which are nowdays present in every book dealing with modern explanation of theory of probability.


Statistics and sportsbook
Schatral, Jan ; Marek, Luboš (advisor) ; Mazouch, Petr (referee)
This bachelor thesis describes the history and fundamental terminology of sportsbook. In theoretical part it describes how to become successful bookmaker and explains specific procedures. In practical part are these procedures applied to real outputs and there's given a solution how to process analysis of sportsbook on the rebound. There are further analysed the lottery and betting games as a complex unit, in which it's finding specific trend and it also deals with the return rate in single games. The last part of the thesis describes frequency testing of relative hypotheses, which are based on favorite betting analysis. During this testing there is compared this frequency in individual seasons and also in a single season.


Managing financial risks in an insurence company
Čech, Tomáš ; Marek, Luboš (advisor) ; Branda, Martin (referee)
The graduation thesis addresses the problems of managing and measuring of financial risks in activities of insurance companies. The first chapter handles the definitions of the financial risk and it classification. The second chapter defines a random variable returns of measure of financial assets. Sets up formulas of the return measure and also focuses on problem of time aggregation. The third chapter theoretically describes methodology of value at risk as the most widely used method for measuring and managing risk by insurance companies and regulatory authority. The fourth chapter contains an empirical study from practice which compares the two basic method of computing value at risk. The fifth chapter is the main part of the graduation thesis and focuses on verifying of the model and his imperfections. It verifies also achievements of initial assumptions. The sixth chapter targets on possibilities of extension value at risk method by liquidity risk incorporation.


Covariance extension of Chainladder method
Žváčková, Lenka ; Marek, Luboš (advisor) ; Hasil, Jakub (referee)
This diploma thesis deals with technical reserves in nonlife insurance, in particular with provisions for future claim payments for damages that have occurred, but has not yet been reported to the insurance company. This type of provision is known by the acronym IBNR. After the introductory section containing a general introduction to the issue of claims reserving in nonlife insurance different approaches to modeling of IBNR reserves are briefly presented. Subsequently, full attention is given to Chainladder method, which is most frequently used in the actuarial practise for the purpose of claims reserving. This method is then presented progressively from its simplest form of a simple computing algorithm followed by Mack's stochastic model to the last theoretical part of this part describing extended form of Chainladder method with relations between different groups of insurance portfolio included. In the very last section, all the lessons are demonstrated on real data to give readers an idea of how the process of claims reserving works is in the common actuarial practice.
