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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 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.
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Casualty reinsurance exposure rating
Těšínská, Anna ; Zimmermann, Pavel (advisor) ; Hrevuš, Jan (referee)
The main aim of this thesis is a development of ILF curves that can be used in the insurance industry when pricing general third party liability on the Czech market. Based on available data there are first estimated size of loss distribution functions used for following generating process. From generated data the increased limit factors are estimated and with a usage of Riebesell's parameterization ILF curves are derived. A substantial part of the thesis is a compilation of literature and the expansion of the statistical approach for estimating fair ILFs based on these data. Besides, the basis for the curves derivation are chapters describing basic theoretical knowledge in the field of reinsurance - in particular, the description of the basic types of reinsurance contracts, as well as the most common methods of a pricing. There is the whole mechanism of curves derivation described; their own use is then demonstrated with the example based on pseudoreal data.
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Gaussian mixtures in R
Marek, Petr ; Malá, Ivana (advisor) ; Zimmermann, Pavel (referee)
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.
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Modelování přírodních katastrof
Zuzák, Jaroslav ; Zimmermann, Pavel (advisor) ; Masec, Frant (referee)
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.
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Modelling natural catastrophes in insurance
Varvařovský, Václav ; Zimmermann, Pavel (advisor) ; Justová, Iva (referee)
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.
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Modelování rizika rezerv v neživotním pojištění založené na neagregovaných datech
Zimmermann, Pavel ; Kahounová, Jana (advisor) ; Cipra, Tomáš (referee) ; Jedlička, Petr (referee)
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.
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Modeling development of incurred value of claim
Kantorová, Petra ; Zimmermann, Pavel (advisor) ; Hrevuš, Jan (referee)
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.
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Modeling of the Mortality Development in the Czech Republic
Hejdová, Martina ; Zimmermann, Pavel (advisor) ; Čech, Tomáš (referee)
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.
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USE OF STATISTICAL METHODS IN THE BUSINESS VALUATION
Zelenka, Martin ; Zimmermann, Pavel (advisor) ; Sedakov, Vladimir (referee)
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.
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