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Recently the eld of actuarial mathematics has experienced a large development due to a signi cant increase of demands for insurance and nancial risk quanti cation due to the fact that the implementation of a complex of rules of international reporting standards (IFRS) and solvency reporting (Solvency II) has started. It appears that the key question for solvency measuring is determination of probability distribution of future cash ows of an insurance company. Solvency is then reported through an appropriate risk measure based e.g. on a percentile of this distribution. While as present popular models are based solely on aggregated data (such as total loss development from a certain time period), the main objective of this work is to scrutinize possibilities of modelling of the reserve risk (i.e. roughly said, the distribution of the ultimate incurred value of claims that have already happened in the past) based directly on individual claims. These models have not yet become popular and to the author's knowledge an overview of such models has not been published previously. The assumptions and speci cation of the already published models were compared to the practical experience and some inadequacies were pointed out. Further more a new reserve risk model was constructed which is believed to have practically more suitable assumptions and properties than the existing models. Theoretical aspects of the new model were studied and distribution of the ultimate incurred value (the modelled variable) was derived. An emphasis was put also on practical aspects of the developed model and its applicability in the case of industrial use. Therefore some restrictive assumptions which might be considered realistic in variety of practical cases and which lead to a signi cant simpli cation of the model were identi ed throughout the work. Furthermore, algorithms to reduce the number of the necessary calculations were developed. In the last chapters of the work, an e ort was devoted to the methods of the estimation of the considered parameters respecting practical limitations (such as missing observations at the time of modelling). For this purpose, survival analysis was (amongst other methods) applied.

This thesis is focused on practical application of two methods used in non-life insurance, Nested Monte Carlo and Least squares Monte Carlo. Best estimate and 99.5% quantile was calculated using both methods and results was compared. Both methods are similar in estimates and therefore can be used for computation of capital requirement. Least squares Monte Carlo seem more favourable, because it significantly reduces computation time.

This thesis deals with reserving issues in non-life insurance. The main aim is to compare methods used to estimate outstanding claims reserves and delineate their advantages, disadvantages, extensions and limitations. Initially, in the theoretical section, there are mathematically formulated chosen models and their assumptions, which are then applied to real data in the practical section. Finally, the findings acquired from both parts are utilized for the comparison of applicability of methods and the identification of their further distinctions and commonalities.

In this thesis, we consider several statistical techniques applicable to claim frequency models of an MTPL portfolio with a focus on overdispersion. The practical part of the work is focused on the application and comparison of the models on real data represented by an MTPL portfolio. The comparison is presented by the results of goodness-of-fit measures. Furthermore, the predictive power of selected models is tested for the given dataset, using the simulation method. Hence, this thesis provides a combination of the analysis of goodness-of-fit results and the predictive power of the models.

This bachelor thesis deals with the exploring the business of crowdfunding, a relatively new way of financing using modern technology. The motivation of potentional investors or people needing investment to employ this method of funding is also discussed, comparing the course of specific similar campaigns in the countries named above, and the influence of the general campaign success. The theoretical part of the thesis defines crowdfunding its types, properties and the technological environment in which it currently operates. Selected crowdfunding platforms are also described, along with, the specifications that significantly impact, the structure of customers and investors. The practical component of the thesis deals with the comparison of the reward-based crowdfunding in the Czech Republic, the Slovak Republic and the USA. The survey was conducted on the basis of the statistical analysis of the collected data, along with a questionnaire aimed at deducing the opinions of the public in relation to crowdfunding. The conclusion of this work is devoted to the formulation of knowledge gained from the empirical part of the work applicable to the local environment.

Using Gaussian mixtures is a popular and very flexible approach to statistical modelling. The standard approach of maximum likelihood estimation cannot be used for some of these models. The estimates are, however, obtainable by iterative solutions, such as the EM (Expectation-Maximization) algorithm. The aim of this thesis is to present Gaussian mixture models and their implementation in R. The non-trivial case of having to use the EM algorithm is assumed. Existing methods and packages are presented, investigated and compared. Some of them are extended by custom R code. Several exhaustive simulations are run and some of the interesting results are presented. For these simulations, a notion of usual fit is presented.

The traditional reserve estimation by an insurance company is based on the aggregated data. However, new trend is to utilize all the information available and analyse each claim separately. This way the application of claims specific features, such as non-proportional reinsurance or policy limits, is possible. The aim of this thesis is to construct the reserving model based on the individual claims. Following the recent legislative changes, the reserve risk has been redefined from ultimate claim horizon to a one-year risk horizon. Hence, the next task is to setup simulation model to calculate one year horizon reserve risk by updating the estimates based on new observations collected over one year. This is a typical task for Bayesian approach, therefore the model components are estimated using the tools of Bayesian statistics.

Title: Some possibilities of heteroskedasticity modeling with applications to non-life insurance Author:Petra Pavlačková Department: Department of Probability and Mathematical Statistics Supervisor: Ing. Zimmermann Pavel, Ph.d. Abstract: This thesis deals with the possibilities of modeling heteroskedasticity using generalized linear models. It summarizes the assumption for these models and their application in practice. It shows the practical need for these models. Furthermore, the thesis deals with the modeling of variance using other methods than generalized lienar models - such as generalized additive models or local regression. Comparison of methods is graphically demonstrated. Keywords: Dispersion parameter, variance function, Joint modelling of mean and dispersion

The thesis deals with natural disasters from the statistical point of view and treats them as extremal observations. Basics of classical extreme value theory will be summarized and new approach based on maximum entropy principle will be proposed. Both methods will be used in order to analyze real discharge data observed at the river Vltava.

The thesis deals with a mortality modelling in the Czech Republic. The main aim of this thesis is to describe historical development of mortality in the Czech Republic on the basis of multinomial regression. In a theoretical part the basic knowledge demanded for model construction divided into two chapters can be found. In the first chapter the basic difference between linear and generalized linear model is described. Second chapter is devoted to a logistic regression. Here we proceed from simple, binomial variable and then we generalize it for multinomial variable. Practical part deals with a construction of a described model. In the third chapter data (type and source) are described and in the fourth chapter characterizes all three models itself (two of them are descriptive and the third one is used for simple prediction). The prediction is added rather for completeness than for the importance of the work itself because it was not the main goal.

This thesis introduces various approaches to natural catastrophe risk assessment in (re)insurance environment. Most emphasis and further elaboration is put on probabilistic models in comparison to the standard model as proposed by Solvency II. The outcomes of natural catastrophe modeling play an important role in the design of proper actuarial models related to catastrophe risk. More specifically it is shown that they can be entirely understood in a wider actuarial context, namely risk theory. Within the Solvency II framework, probabilistic model outcomes are translated by means of the proposed decomposition methodology putting them into a similar language of the standard formula in order to create the ability to compare different results implied by either probabilistic model or standard formula. This enables both comparison of the implied dependence structure of probabilistic model to standardized correlations assumed in Solvency II, and scenario year loss factors of Solvency II to implied damage factors of probabilistic models in defined cresta zones. The introduced decomposition methodology is illustrated by flood and windstorm model outcomes calculated on exposure data of Czech insurance companies and compared to the respective standard formula parameters and outcomes. Finally, other applications of the proposed decomposition methodology are introduced, such as measurement of diversification effect or blending of different results calculated by different models or even approaches to natural catastrophe risk assessment.

The aim of this paper is to outline the possibility of application of statistical methods for business valuation. This paper provides a basic overview of the subject in particular mathematical statistical point of view. The first chapter contains an introduction to the field of business valuation are presented valuation areas where it is possible to use different statistical methods. In the following parts of my work it is possible to find a description of methods and a brief description of the problem. The work is mainly focused on the analysis of time series. At the end of the theoretical part of the time series analysis problems of application of regression models are mentioned as well as the difficulties of its application practice. Potential solutions of these problems are mentioned. The final chapter is devoted to practical demonstration of application of the proposed methods on real data for a selected company. The work presents unique suitability of statistical methods in business valuation and demonstrates their practical application.

This diploma project is focused on the estimation of incurred value of claim and probability of the claim remaining opened (not settled) in the specific stage of the insurance settlement process. The change of incurred value of claim means the change of settlement process stage. Generalized linear model is used for modelling these changes. Classical linear regression model also belongs into this theory, which is its special case, just with stricter premises. Generalized linear model among others allows solving the problem of heteroscedasticity in the unusual way using joint model. This model is applied in the practical part of this piece of work. Logistic regression is the part of the generalized linear model theory, which helps to model the probability of the claim remaining opened in this piece of work. The model outcome is presented in graphic way, especially the graphs containing probability that levels of given claim will occur in certain range.

In the present work we study classes of homogenous policy contracts by CEIOPS definition and its specified risk characteristics. First part of the thesis study the risk measures and the methods used to measure these risks. As the main risk categories we study underwriting and reserve risk. In the second part of the thesis we analyse these classes by its risk characteristics and cluster them in homogenous groups. At the end we outline the characteristic features of each group for better understanding the result of presented cluster analysis.

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 non-life 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.

My thesis theme from the branch of insurance industry has been chosen for a reason of my long-term experience. The particular theme of "accident insurance" has been then chosen and based on the fact that in general it belongs to one of the most important types of insurance and everybody should arrange for it. In the part of theory I would like to clarify basic terms of the field of accident insurance as well as those of insurance industry as a whole. At the same time I would like to illustrate how and where an insurance contract can be concluded and what one should be aware of before and in the course of concluding such an insurance contract. An integral part of the theory will also be legislation dealing with the insurance industry and its development. The practical part will be focused on looking for different insurance amounts people should be insured for. In addition there will be given a particular example of effecting an accident insurance of more people as well as of a person and various insurance companies' offers in my place of living will be analysed. In conclusion let me make basic calculations concerning the insurance clear and analyse the data that will be at my disposal.

Quantification of risks is one of the pillars of the contemporary insurance industry. Natural catastrophes and their modelling represents one of the most important areas of non-life insurance in the Czech Republic. One of the key inputs of catastrophe models is a spatial dependence structure in the portfolio of an insurance company. Copulas represents a more general view on dependence structures and broaden the classical approach, which is implicitly using the dependence structure of a multivariate normal distribution. The goal of this work, with respect to absence of comprehensive monographs in the Czech Republic, is to provide a theoretical basis for use of copulas. It focuses on general properties of copulas and specifics of two most commonly used families of copulas -- Archimedean and elliptical. The other goal is to quantify difference between the given copula and the classical approach, which uses dependency structure of a multivariate normal distribution, in modelled flood losses in the Czech Republic. Results are largely dependent on scale of losses in individual areas. If the areas have approximately a "tower" structure (i.e., one area significantly outweighs others), the effect of a change in the dependency structure compared to the classical approach is between 5-10% (up and down depending on a copula) at 99.5 percentile of original losses (a return period of once in 200 years). In case that all areas are approximately similarly distributed the difference, owing to the dependency structure, can be up to 30%, which means rather an important difference when buying the most common form of reinsurance -- an excess of loss treaty. The classical approach has an indisputable advantage in its simplicity with which data can be generated. In spite of having a simple form, it is not so simple to generate Archimedean copulas for a growing number of dimensions. For a higher number of dimensions the complexity of data generation greatly increases. For above mentioned reasons it is worth considering whether conditions of 2 similarly distributed variables and not too high dimensionality are fulfilled, before general forms of dependence are applied.