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Claim inflation in car insurance
Neumann, Vojtěch ; Kříž, Pavel (advisor) ; Cipra, Tomáš (referee)
This thesis explores the practical use of generalized linear models. The aim of the thesis is to analyze the claims inflation for Motor Third Party Liability Insurance. For this purpose, current data from a Czech insurance company are provided. In the thesis, a generalized linear model is constructed in detail based on specified criteria. From the model, the effect of inflation is identified and its value for the given period is determined. 1
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Parameter estimation for fractional Brownian motion
Hartman, Štěpán ; Kříž, Pavel (advisor) ; Čoupek, Petr (referee)
This bachelor's thesis deals with a mathematical object called fractional Brownian motion, which has substantial applications in a wide variety of disciplines including, next to theoretical and financial mathematics, the fields of biology, geography, or information technology. This concept is a generalization of a standard Brownian motion, in which we do not assume the independence of its increments. In this thesis we define said object and explore its basic properties. Subsequently, we discuss the estimators of its Hurst index. We suggest a correction of one of the methods of constructing the estimator and demonstrate its effectiveness using both simulated and real-life data. 1
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Design and verification of an intervention exercise program aimed at improving the fitness of recreational athletes
KŘÍŽ, Pavel
The aim of this work is to compile an intervention program aimed at improving fitness. The goal was achieved. According to the proposal, the intervention exercise program can be practiced by everyone and anywhere, even if the gyms are closed, for example for pandemic reasons. Eleven people took part in the exercise exercise program, and the control group also included eleven people. The EUROFIT test battery, consisting of nine tests, was chosen within the methodology. From the results, it should be emphasized that the experimental group improved on average by 0,1 try for endurance training standing on one leg, 1.89 seconds for tapping, also improved in the overhang in the forward bend by 3.23 centimeters, 13, 96 in the jump from the place, 3.17 repetitions in the sit-ups exercise, 4.79 seconds in the endurance in the squat, the only deterioration of the experimental group occurred in the shuttle run by 0.2 seconds, in the last dynamometry measurement test there was an improvement of 7.14 Newtons. In the control group, there was a slight deterioration in the first test by 0.1 attempts, we could also observe a deterioration in the tapping test by 1.33 seconds, a slight improvement of 0.98 centimeters occurred in the forward bend, the group only further deteriorated in all tests. Specifically, by 0.54 centimeters in the jump, by 0.84 repetitions in the sit-up, 2.56 seconds in the endurance of the push-up, 0.29 in the shuttle run and by 2.23 Newtons in the dynamometry. All research assumptions have been met.
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Optimal FInancial Payoffs Maximizing Utility Function
Kožnar, František ; Večeř, Jan (advisor) ; Kříž, Pavel (referee)
The goal of this thesis is to characterize payoffs that maximize expected utility function in different market setups. One can solve this problem in its generality in terms of a function of a likelihood ratio between the subjective measure of an agent P and a risk neutral measure Q. Such payoffs should be transformed to the function of the terminal stock price. The question is what measure P should be chosen, the natural candidates would correspond to either the frequentist or the Bayesian choice of the parameters. The thesis should provide a link to the Kelly Criterion in the binomial evolution of the stock price and to the Merton's Portfolio Problem in the geometric Brownian motion exam- ple showing the possible extensions of these well known problems in the novel Bayesian setup. The thesis should discuss pricing and hedging of these contracts together with their asymptotic behavior. 1
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Malliavin operators for real-valued Gaussian random variables and their applications
Kubát, Martin ; Kříž, Pavel (advisor) ; Čoupek, Petr (referee)
In this thesis, we introduce Malliavin Operators. We will focus on derivative, di- vergence and Ornstein-Uhlenbeck operator to study properties of transformed Gaussian random variables. We will explain all concepts in detail and add some typical examples. Then we will use Malliavin operators in the proofs of famous Poincaré inequality and variance expansions. The technique of the last proofs provides a good general approach how to solve similar problems with understanding Malliavin Operators. 1
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Chain-ladder method as maximum likelihood estimator in Poisson model
Wagner, Vojtěch ; Kříž, Pavel (advisor) ; Pešta, Michal (referee)
First, the distribution-free chain-ladder is introduced. Then, the Poisson model is in- troduced. It is proven that the total reserves for one accident year given by the maximum likelihood estimation applied to the Poisson model lead to the identical reserves as the reserves derived from the distribution-free chain-ladder used in the Poisson model. Later, inadequacies of the Poisson model are discussed. Hessian matrices of the log-likelihood evaluated at the Poisson estimators are analyzed. The question whether the inverse of the Fisher information matrix approximates the real covariance matrix of the Poisson esti- mators is explored. Comparing the variance of the total reserves derived from the inverse of the Fisher information and the real covariance matrix leads to negative conclusion, that the former does not approximate the latter well. 1
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Stein's method for normal approximation of random variables
Strnad, Martin ; Kříž, Pavel (advisor) ; Nagy, Stanislav (referee)
The Stein's method is a collection of probabilistic techniques for answering the ques- tion as to how far the probability distributions of random variables are from each other. This thesis only concerns the basics of the approach. We use the Kolmogorov distance and the total variation distance to formalize the concept of the distance between mea- sures. We focus on the normal distribution for which we first find a suitable differential operator, often called Stein operator, that bears much information. Not only does it charactize the Gaussian measure, it also gives us a means to quantify the distance from another random variable's distribution. Finally, we apply the method to prove the clas- sic Berry-Esseen inequality for a sum of independent and identically distributed random variables. 1
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Difference and differential equations in life insurance
Kirešová, Katarína ; Kříž, Pavel (advisor) ; Pešta, Michal (referee)
The diploma thesis deals with the calculation of life insurance reserves, higher mo- ments and the distribution function of future payments of reserves using difference and differential equations. In the beginning, the basic theory of a stochastic process, insu- rance model, cash flow, and reserve is summarized. After that, equations themselves are derived; first in general and then for four specific types of insurance. Subsequently, a cal- culation of premiums is presented for each type of insurance. The next two chapters deal with the calculation of higher moments and the distribution function. After deriving the formulas for four types of insurance, the reserves, standard deviations, and distribution functions are calculated for specific values and then they are compared with the Monte Carlo simulation. The conclusion contains pros and cons of the method compared to the simulation. 1
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Quantifying Mortality and Longevity Risk by Means of Stochastic Models
Plotnikova, Valeriya ; Mazurová, Lucie (advisor) ; Kříž, Pavel (referee)
In this thesis we investigate the structure of the generalized age-period-cohort mortality model and we comment on the key components of its structure. As an example of the generalized age- period-cohort model we take a closer look at the widely used Lee-Carter mortality model. We further construct mortality models for the Czech male and female populations, by using a certain procedure that involves expert judgment. To project mortality rates we choose the most suitable time series processes for the selected parameters in the model. Finally, we describe and implement the value at risk framework for the longevity risk, which is one of the possible applications where the obtained mortality models can be used in practice. In particular, we investigate how much a temporary life annuity liability might change based on new information over the course of one year.
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