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Modeling the frequency of unreported claims at the policy level
Králová, Eva ; Mazurová, Lucie (advisor) ; Hendrych, Radek (referee)
In this thesis we study policy-level models for unreported claim counts. We assume that the total number of claims on a policy follows a Poisson or negative binomial dis- tribution. The parameters of these distributions depend on the risk exposure introduced in the thesis, we also describe possible methods of calculating the risk exposure. We derive distributions for the number of reported and unreported claims, both of which are dependent on the report lag time of a claim. To estimate the parameters of these distributions, we use the maximum likelihood method. We demonstrate the performance of the models via a simulation study. 1
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Multiline aggregate XL-reinsurance
Šuchová, Martina ; Mazurová, Lucie (advisor) ; Cipra, Tomáš (referee)
This paper focuses on simulation modeling of the total aggregate reinsurer claim S when considering XL.reinsurance for multiple insurance lines. In the first part, this re- insurance structure is defined. In the second part of the paper, the collective model is approached, as well as the definition of copulas (comonotony copula, independence co- pula, Clayton's or Gumbelt's copula), and Sklar's theorem. The last part discusses a simulation study that shows the simulation of aggregate claims when considering inde- pendence as well as dependence of insurance industries. The simulation study outlines the theoretical approach in the introduction, explaining the heuristic algorithm it uses in simulating dependent industries. The conclusion of the study depicts one of the practical applications, and the outputs of the simulations. 1
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Forecasting age distribution of death counts with applications in life insurance pricing
Škopek, Pavel ; Branda, Martin (advisor) ; Mazurová, Lucie (referee)
This thesis deals with the topic of mortality modelling and life insurance pricing. First, basic concepts from the demographic model and life tables are introduced. Following is a description of the Lee-Carter model including three methods for estimating of parameters and predicting of future values. The the- sis also analyses the Renshaw-Haberman model and the method, which uses compositional data analysis including the non-parametric bootstrap for interval estimations. Besides the theoretical part the thesis also contains a practical one, where Czech mortality data are modeled separately for men and women. Based on the data from 1970-2021 we select the best model, predict future values for 30 years ahead and price the life insurance in 2021 and in the following years. 1
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Approximations of the Aggregate Loss Distribution
Antalicová, Viktória ; Mazurová, Lucie (advisor) ; Kříž, Pavel (referee)
This thesis is focused on the approximation of the distribution of aggregate losses. We first present a method for modelling aggregate losses, which involves selecting an appropriate frequency and severity distributions. Next, the computation of aggregate losses as the sum of the respective number of individual losses is explained. In the sec- ond section, we discuss the approximation of the distribution of the simulated aggregate losses. We present the distributions chosen for the approximation, the method for esti- mating the parameters of these distributions, and the subsequent testing of fit of these distributions with the actual distribution of the simulated aggregate losses. In the third chapter we show the results of this approximation and indicate the suitability of using each of the considered distributions for modelling aggregate losses. In the last section, we introduce the Edgeworth approximation as a method for approximating the distribution of aggregate losses. 1
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Distributions of (a,b,0) type in non-life insurance
Zejda, Albert ; Kříž, Pavel (advisor) ; Mazurová, Lucie (referee)
First, a definition of the distribution type (a, b, 0) is introduced. Next, it is shown which known distributions satisfy this definition, the parameters a and b that correspond to them, and specific sets of parameters for each of the distributions are determined. Then, it is proven that no other distributions can satisfy this definition. A maximum likelihood estimation method for estimating the parameters a and b directly from the data is presented. Finally, a simulation study is conducted, in which the probabilities from the estimated distribution type (a, b, 0) from specific data using the maximum likelihood method are compared with the empirical relative frequencies calculated from the data. 1
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Risk reserving based on ODP model
Procházka, Viktor ; Maciak, Matúš (advisor) ; Mazurová, Lucie (referee)
This thesis deals with estimating the outstanding claims reserve, one of important problems of insurance mathematics. It introduces the chain-ladder method as the ba- sic method for estimating the outstanding claims. Besides this method, it also presents models using the Poisson and ODP distributions to describe the increments of run-off triangles, which lead to identical point estimate of the outstanding claims reserve as the chain-ladder method. Additionally, this thesis deals with a simulation study concerned with the properties of these methods and the upper estimate of the outstanding claims in the Poisson and ODP models, where bootstrap estimate is also examined.
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Reverse mortgage
Korotkov, Daniil ; Mazurová, Lucie (advisor) ; Zichová, Jitka (referee)
At this moment, reverse mortgages are relatively new products on the Czech market and this thesis deals with their problematics. In this thesis, we describe the main risks related to reverse mortgages, namely, longevity risk and adverse evolution of property prices. Analyzing these risks we are modelling the underlying property prices, their future behavior along with studying the risk mo- dels such as vector autoregression. In practical part, we focus on estimating the parameters of Lee-Carter model to estimate the distribution of life expectancy. In conclusion we apply the results of the estimation to calculate various charac- teristics of reverse mortgages using the simplied version of the model. 1
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Bonus hunger in motor insurance
Povolná, Eliška ; Mazurová, Lucie (advisor) ; Vejmělka, Petr (referee)
The thesis deals with the analysis of bonus-malus systems used in motor insurance to adjust the amount of the premium depending on the number of claims reported by the driver. It focuses on the mathematical description of a phenomenon called bonus hunger, where a driver prefers not to claim a claim in order not to be placed in a bonus class with a higher premium for the following period. The thesis describes the procedure for choosing the optimal retention using Lemaire's algorithm on the chosen model. In the practical part, the algorithm is implemented in software and values are calculated for a system based on the conditions of one Czech insurance company. 1
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Transient behavior of bonus-malus systems
Tichá, Tereza ; Mazurová, Lucie (advisor) ; Cipra, Tomáš (referee)
This thesis deals with bonus-malus systems in car insurance. First of all, the basic no- tation is introduced and the principles are described on the basis of which these systems can be modeled using homogeneous Markov chains. The development of bonus-malus systems is usually evaluated using various characte- ristics such as relativity or elasticity, which are calculated on the basis of a stationary distribution. However, these calculations only make sense if the stationarity is reached in a reasonable time. However, for real systems, this time is much longer than the time the driver spends in the portfolio. Therefore, an alternative possibility of evaluation using the age correction of the stationary distribution is propo- sed. Finally, the use of stationary and age-corrected distributions is compared for specific examples in the practical part. 1
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Forecasting mortality: Selected actuarial applications
Hric, Patrik ; Hendrych, Radek (advisor) ; Mazurová, Lucie (referee)
This thesis deals with calculation of solvency capital requirement for life longevity risk. We start with defining selected demographic terms. Afterwards we introduce some stochastic mortality models, namely Lee-Carter and Cairns- Blake-Dowd model, which will be applied to real data. Subsequently we review mentioned models, regarding parameters, estimates, forecast and also diagno- sis. The theoretical part is closed by a brief description of Solvency II directive, scheme of solvency capital requirement and also method of life longevity risk calculation. In application part we demonstrate particular calculations related to stochastic mortality models resulting in determining solvency capital requ- irement for life longevity risk based on the data from Czech Statistical Office. Applied methods are mutually compared. 1
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