Dissertations / Theses on the topic 'Claims reserving'

Three methods for claims reserving are compared on two data sets. The first two methods are the commonly used chain ladder method that uses aggregated payments and the relatively new method, double chain ladder, that apart from the payments data also uses the number of reported claims. The third method is more advanced, data on micro-level is needed such as the reporting delay and the number of payment periods for every single claim. The two data sets that are used consist of claims with typically shorter and longer settlement time, respectively. The questions considered are if you can gain anything from using a method that is more advanced than the chain ladder method and if the gain differs from the two data sets. The methods are compared by simulating the reserves distributions as well as comparing the point estimates of the reserve with the real out-of-sample reserve. The results show that there is no gain in using the micro-level method considered. The double chain lad- der method on the other hand performs better than the chain ladder method. The difference between the two data sets is that the reserve in the data set with longer settlement times is harder to estimate, but no difference can be seen when it comes to method choice.
Tre reservsättningsmetoder jämförs på två dataset. De första två metoderna är den välkända chain ladder-metoden som använder sig av aggregerade utbetalningar samt den relativt nya metoden double chain ladder som förutom utbetalningarna använder sig av antalet anmälda skador. Den tredje metoden baseras på mikro-nivå och kräver information om varje enskild skada, såsom anmälningstid och antalet utbetalningsperioder. De två dataseten som används är ett som innehåller skador med typiskt kortare avvecklingstider och ett som innehåller skador med typiskt längre avvecklingstider. Frågorna som behandlas är om man vinner något på att använda en mer avancerad metod än chain ladder och om det skiljer sig åt mellan dataseten. Metoderna jämförs genom simulering av reserven, men också genom att jämföra punktskattningar med den verkliga reserven. Resultaten visar att man I detta fall inte vinner något på att använda mikro-metoden. Double chain ladder å andra sidan presterar bättre än chain ladder. Skillnaden mellan de två dataseten är att det är svårare att estimera reserven när avvecklingstiden är längre, men ingen skillnad ses när det gäller val av metod

One fundamental function of an insurance company revolves around calculating the expected claims costs for which the insurer has to compensate its policyholders for. This is the process of claims reserving which is practised by actuaries using statistical methods. Over the last few decades statistical learning methods have become increasingly popular due to their ability to find complex patterns in any type of data. However, they have not been widely adapted within the insurance sector. In this thesis we evaluate the capability of claims reserving with the method of gradient boosting, a non-parametric statistical learning method that has proven to be successful within multiple other disciplines which has made it very popular. The gradient boosting technique is compared with the generalized linear model(GLM) which is widely used for modelling claims. We compare the models by using a claims data set provided by Länsförsäkringar AB which allows us to train the models and evaluate their performance on data not yet seen by the models. The models were implemented using R. The results show that the GLM has a lower prediction error. Also, the gradient boosting method requires more fine tuning to handle claims data properly while the GLM already possesses certain features that makes it suitable for claims reserving without making as many adjustments in the model implementation. The advantage of capturing complex dependencies in data is not fully utilized in this thesis since we only work with 6 predictor variables. It is more likely that gradient boosting can compete with GLM when predicting more complicated claims.
En av de centrala verksamheterna ett försäkringsbolag arbetar med handlar om att uppskatta skadekostnader för att kunna ersätta försäkringstagarna. Denna procedur kallas reservsättning och utförs av aktuarier med hjälp av statistiska metoder. Under de senaste årtiondena har statistiska inlärningsmetoder blivit mer och mer populära tack vare deras förmåga att hitta komplexa mönster i alla typer av data. Dock har intresset för dessa varit relativt lågt inom försäkringsbranschen till förmån för mer traditionella försäkringsmatematiska metoder. I den här masteruppsatsen undersöker vi förmågan att reservsätta med metoden \textit{gradient boosting}, en icke-parametrisk statistisk inlärningsmetod som har visat sig fungera mycket väl inom en rad andra områden vilket har gjort metoden mycket populär. Vi jämför denna metod med generaliserade linjära modeller(GLM) som är en av de vanliga metoderna vid reservsättning. Vi jämför modellerna med hjälp av ett dataset tillhandahålls av Länsförsäkringar AB. Modellerna implementerades med R. 80\% av detta dataset används för att träna modellerna och resterande 20\% används för att evaluera modellernas prediktionsförmåga på okänd data. Resultaten visar att GLM har ett lägre prediktionsfel. Gradient boosting kräver att ett antal hyperparametrar justeras manuellt för att få en välfungerande modell medan GLM inte kräver lika mycket korrigeringar varför den är mer praktiskt lämpad. Fördelen med att kunna modellerna komplexa förhållanden i data utnyttjas inte till fullo i denna uppsats då vi endast arbetar med sex prediktionsvariabler. Det är sannolikt att gradient boosting skulle ge bättre resultat med mer komplicerade datastrukturer.​

Master of Science
Department of Statistics
Haiyan Wang
Estimating unpaid liabilities for insurance companies is an extremely important aspect of insurance operations. Consistent underestimation can result in companies requiring more reserves which can lead to lower profits, downgraded credit ratings, and in the worst case scenarios, insurance company insolvency. Consistent overestimation can lead to inefficient capital allocation and a higher overall cost of capital. Due to the importance of these estimates and the variability of these unpaid liabilities, a multitude of methods have been developed to estimate these amounts. This paper compares several actuarial and statistical methods to determine which are relatively better at producing accurate estimates of unpaid liabilities. To begin, the Chain Ladder Method is introduced for those unfamiliar with it. Then a presentation of several Generalized Linear Model (GLM) methods, various Generalized Additive Model (GAM) methods, the Bornhuetter-Ferguson Method, and a Bayesian method that link the Chain Ladder and Bornhuetter-Ferguson methods together are introduced, with all of these methods being in some way connected to the Chain Ladder Method. Historical data from multiple lines of business compiled by the National Association of Insurance Commissioners is used to compare the methods across different loss functions to gain insight as to which methods produce estimates with the minimum loss and to gain a better understanding of the relative strengths and weaknesses of the methods. Key

In practice there is a long tradition of actuaries calculating reserve estimates according to deterministic methods without explicit reference to a stochastic model. For instance, the chain-ladder was originally a deterministic reserving method. Moreover, the actuaries often make ad hoc adjustments of the methods, for example, smoothing of the chain-ladder development factors, in order to fit the data set under analysis. However, stochastic models are needed in order to assess the variability of the claims reserve. The standard statistical approach would be to first specify a model, then find an estimate of the outstanding claims under that model, typically by maximum likelihood, and finally the model could be used to find the precision of the estimate. As a compromise between this approach and the actuary's way of working without reference to a model the object of the research area has often been to first construct a model and a method that produces the actuary's estimate and then use this model in order to assess the uncertainty of the estimate. A drawback of this approach is that the suggested models have been constructed to give a measure of the precision of the reserve estimate without the possibility of changing the estimate itself. The starting point of this thesis is the inconsistency between the deterministic approaches used in practice and the stochastic ones suggested in the literature. On one hand, the purpose of Paper I is to develop a bootstrap technique which easily enables the actuary to use other development factor methods than the pure chain-ladder relying on as few model assumptions as possible. This bootstrap technique is then extended and applied to the separation method in Paper II. On the other hand, the purpose of Paper III is to create a stochastic framework which imitates the ad hoc deterministic smoothing of chain-ladder development factors which is frequently used in practice.

This thesis is concerned with the approximations of prediction error and predictive distribution of the best reserve estimate produced by the models which combine information from multiple data sets. Two models are studied . ~; within the GLM framework, i.e. SClmieper's: model proposed by Schnieper (1991) and the MeL method introduced by Quargand Mack (2004). Theoretical and empirical approximation approaches for the MSEP of these two models are discussed and compared. This includes derivations of closed formulae following the approaches introduced by· both Mack (1993) and Murphy (1994) and also the empirical approach, i.e. the bootstrap method. And finally, various models which combine information from multiple data sets are investigated and compared, providing new insights to the claims .reserving area.

Stochastic models for triangular data are derived and applied to claims reserving data. The standard actuarial technique, the so-called "chain-ladder technique" is given a sound statistical foundation and considered as a linear model. This linear model, the '"Chain Ladder Linear Model" is extended to encompass Bayesian, empirical Bayes and dynamic estimation. The empirical Bayes results are given a credibility theory interpretation, and the advantages and disadvantages of the various approaches are highlighted. Finally, the methods are extended to two-dimensional systems and results based on classical time series and Kalman filtering theory are produced. The empirical Bayes estimation results are very useful, practically, and can be compared to the Kalman filter estimates. They have the advantage that no prior information is required: the Kalman filter method requires the state and observation variances to be specified. For illustration purposes the estimates from the empirical Bayes procedure are used. The empirical Bayes results can also be compared with credibility theory estimators, although they retain the general statistical advantages of the linear modelling approach. For the classical theory, unbiased estimates of outstanding claims, reserves and variances are derived, and prediction intervals for total outstanding claims are produced. Maximum likelihood theory is utilised to derive the distributions of quantities relating to the column parameters which have actuarial interpretations. The row totals are also considered. Bayesian estimates of similar quantities are derived for the methods based on Bayes theory.

Mestrado em Ciências Actuariais
Neste relatório será também explicado ao seu leitor o que é que está na base da separação dos sinistros graves dos restantes, bem como dos métodos frequentemente utilizados para o cálculo de reservas para sinistros com danos corporais. Será também exposto o processo de gestão de sinistros graves actualmente utilizado em Portugal e nas restantes sucursais da Liberty. Para chegar à conclusão de qual o melhor método a sugerir serão primeiramente completados os triângulos tanto de pagamentos ocorridos como de frequências esperadas. As reservas agregadas serão depois obtidas utilizando três métodos diferentes, Chain Ladder, Cape Cod e Benktander. Por forma a estimar o desvio padrão associado a cada método utilizado serão simulados diferentes cenários e calculada a diferença entre as reservas agregadas obtidas para 2014 e 2015 (exceptuando o período de 2015 correspondente ao qual em 2014 não foi possível recolher valores de reservas). A esta técnica é dado o nome de OCD, isto é, The One-Year Claim Development. De entre os três métodos utilizados será eleito aquele cuja medida de sensibilidade para um ano (média dos desvios padrão obtidos) for menor, e consequentemente considerar-se-á esse o melhor método para tratar sinistros graves em Portugal. Devo ainda referir que a leitura e compreensão deste relatório pressupõe o conhecimento prévio das bases da actividade seguradora no ramo automóvel, bem como do processo de cálculo de reservas matemáticas.
This report will arrive at a conclusion by explaining to the reader the basic reasoning behind splitting large claims as well as the most common methods for BI reserving. It is assumed that the reader has fundamental understanding of the insurance industry, motor insurance and BI, and the reserving process. It is necessary to explain the practices used by Portugal and other Liberty International countries to form an opinion of those practices by applying them to Portugal's claim information. Furthermore, the question of whether to split large claims or not will be thoroughly evaluated. Finally, there will be an aggregate suggestion as to the best splitting practice and reserving methodology specific for Liberty Seguros Portugal. To do this, several shocked scenarios will be simulated. Additional large claims will be introduced to the total incurred claims triangle and large claim count triangles. The one-year claim development (OCD) will then be compared using different reserving methodologies, the Chain Ladder Method, Cape Cod Method, and Benktander Method. The one-year claim development is measured by the change in the aggregate reserve ultimate between 2014 and 2015 (excluding the 2015 cohort for which no aggregate reserve ultimate was available in 2014). The standard deviation of each method's one-year uncertainty will be calculated by computing the OCD of each method under the three shocked scenarios. The technique that yields the lowest average of standard deviations, called the one-year sensitivity measure by the author, will be selected as the best approach for handling large claims.

Dans la première partie de la thèse, nous nous intéressons à la convergence faible du processus empirique pondéré des copules. Nous fournissons la condition suffisante pour que cette convergence ait lieu vers un processus Gaussien limite. Nos résultats sont obtenus dans un espace de Banach L^p. Nous donnons des applications statistiques de ces résultats aux tests d'adéquation (tests of goodness of fit) pour les copules. Une attention spéciale est portée aux tests basées sur des statistiques de type Cramér-von Mises.Dans un second temps, nous étudions le problème de provisionnement stochastique pour une compagnie d'assurance non-vie. Les méthodes stochastiques sont utilisées afin d'évaluer la variabilité des réserves. Le point de départ pour cette thèse est une incohérence entre les méthodes utilisées en pratique et celles publiées dans la littérature. Pour remédier à cela, nous présentons un outil général de provisionnement stochastique à horizon ultime (Chapitre 3) et à un an (Chapitre 4), basé sur la méthode Chain Ladder
The aim of this thesis is twofold. First, we concentrate on the study of weak convergence of weighted empirical copula processes. We provide sufficient conditions for this convergence to hold to a limiting Gaussian process. Our results are obtained in the framework of convergence in the Banach space $L^{p}$ ($1\leq p <\infty $). Statistical applications to goodness of fit (GOF) tests for copulas are given to illustrate these results. We pay special attention to GOF tests based on Cramér-von Mises type statistics. Second, we discuss the problem of stochastic claims reserving in general non-life insurance. Stochastic models are needed in order to assess the variability of the claims reserve. The starting point of this thesis is an observed inconsistency between the approaches used in practice and that suggested in the literature. To fill this gap, we present a general tool for measuring the uncertainty of reserves in the framework of ultimate (Chapter 3) and one-year time horizon (Chapter 4), based on the Chain-Ladder method

Mestrado em Actuarial Science
O trabalho seguinte foi realizado durante uma colocação de estágio na If Industrial P & C Insurance, em Estocolmo, na Suécia. Este relatório destaca e discute algumas das diferenças entre o seguro industrial e privado e percorre o processo de “Análise do Ano Normal”. A análise avalia os dados das reivindicações com o objetivo de projetar as perdas em um ano no futuro. A Teoria do Risco Colectivo e a Estimação da Máxima Verossimilhança são utilizadas para obter uma estimativa da gravidade das reivindicações. Além disso, as reservas são estimadas usando o método Chain-ladder. A seção final do relatório descreve uma análise de sensibilidade de um modelo para as reservas de ajuste de sinistros. Esta análise mostra o impacto da introdução de dois novos parâmetros, o que explica a parte já desenvolvida das reivindicações abertas.
The following work was carried out during an internship placement at If Industrial P&C Insurance in Stockholm, Sweden. This report highlights and discusses some of the differences between Industrial and Private insurance and walks through the “Normal Year Analysis”-procedure. The analysis assesses the claims data with the goal of projecting the losses one year into the future. Collective Risk Theory and Maximum Likelihood Estimation is used to obtain an estimate of the severity of the claims. In addition, the reserves are estimated, using the Chain-ladder method. The final section of the report describes a sensitivity analysis of a model for the Claims Adjustment Reserves. This analysis shows the impact of introducing two new parameters, which accounts for the already developed part of the open claims.
info:eu-repo/semantics/publishedVersion

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.

Mestrado em Ciências Actuariais
As the final part of the master degree, I started my internship on 25 Febru- ary 2013 at Taylor Fry Consulting Actuaries. During the internship my work mainly involved analyzing the outstanding workers compensation liabilities for self-insured clients. In this report I will describe the process of this anal- ysis. Throughout this report I will explain the procedures I used to value outstanding claims.

This thesis deals with an issue of claims reserving for non-life insurance. The issue is approached in a sense of analytical calculation and stochastic modelling. First, Chain-ladder, Bornhuetter-Ferguson, Benktander-Hovinen and Cape-Cod method are introduced. In following chapters, we try to find related stochastic underlying models including Generalized linear models and Mack's distribution-free approaches, we analyze second moments of claims estimates for each of the methods and examine alternative Merz-Wüthrich approach to reserve risk measurement. At the end, bootstrap algorithm and estimates are suggested and simulation results are compared with analytic ones.

This thesis investigates the usefulness of Bayesian modelling to claims reserving in general insurance. It can be divided into two parts: Bayesian methodology and Bayesian claims reserving methods. In the first part, we review Bayesian inference and computational methods. Several examples are provided to demonstrate key concepts. Deriving the predictive distribution and incorporating prior information are focused on as two important facets of Bayesian modelling for claims reserving. In the second part, we make the following contributions: 1. Propose a compound model as a stochastic version of the payments per claim incurred method. 2. Introduce the Bayesian basis expansion models and Hamiltonian Monte Carlo method to the claims reserving problem. 3. Use copulas to aggregate the doctor benefit and the hospital benefit in the WorkSafe Victoria scheme. All the Bayesian models proposed are first checked by applying them to simulated data. We estimate the liabilities of outstanding claims arising from the weekly benefit, the doctor benefit and the hospital benefit in the WorkSafe Victoria scheme. We compare our results with those from the PwC report. Except for several Markov chain Monte Carlo algorithms written for the purpose in R and WinBUGS, we largely rely on Stan, a specialized software environment which applies Hamiltonian Monte Carlo method and variational Bayes.

碩士
逢甲大學
統計學系
104
The claim reserve is one of the most important liabilities in property-liability insurance companies. And it is closely related to property-liability insurance companies’ solvency and financial stability. Hence, the estimation of claim reserves is the most important issue. Furthermore, insurance law in Taiwan does not impose a unifying standard for claim-reserving methods. In order to achieve the maximum economic efficiency, the claim reserves need to be estimated through actuaries’ past experiences and professional expertise. Because the subjective opinion of actuaries and the adoption of different estimated methods, each insurance company in Taiwan has different methods of setting up its claim reserve. Last but not least, this article compares Mack’s Method and Bootstrap Method on both qualitative and quantitative aspects in dealing with the claim reserves. On the qualitative aspects, the comparison is focused on the underlying structures, assumptions, and estimation mechanics. On the quantitative aspects, both methods are applied to a particular claims data set and the expected value, the standard deviation or standard error of prediction, and the 75th percentile of the Outstanding Claims Liability(OCL) of the data set are computed.

碩士
逢甲大學
統計與精算所
94
Insurance companies should take claims reserving every year, and deposit enough claims reserving to pay off unpaid claims payment in the annual year-end accounting.Claims reserving is the most important liability on balance sheet , and deposit in adequate will affect the financial stability、cash flow, and solvency of an insurance company. This paper introduces how to use past claim amounts and set single cell development factors to fit normal distribution. Based on a Bayesian chain ladder model, we simulate the posterior distribution of claim reserves by MCMC method. We also simulate the future single cell development factors, and then calculate the future claim reserves.

Dissertation presented as the partial requirement for obtaining a Master's degree in Statistics and Information Management, specialization in Risk Analysis and Management
Insurance companies need to have proper technical reserves in order to stay solvent, due to the liabilities that they take responsibility for. The notion of what are proper reserves is a subject studied by insurers and academics. For the non-life insurance the claim reserves, amongst other types of reserves, have great significance. Hence the existence of many kinds of reserving methods, either stochastic or deterministic. This work aims to compare the estimates of claim reserves using the Thomas Mack and the Bootstrap methods for the workers compensation insurance.
Companhias de seguros precisam de ter provisões técnicas apropriadas de forma a se manterem solventes, dados os riscos que estas tomam responsabilidade. A noção de o que é uma provisão adequada é um tema estudado pelas seguradoras. Para o ramo Não-Vida a provisões de sinistros, de entre outros tipos de provisões, têm grande significância. Consequentemente, existem vários métodos para a sua determinação, quer estocásticos, quer determinísticos. O propósito deste trabalho é a comparação das estimativas das provisões de sinistros no ramo de acidentes de trabalho usando os métodos de Thomas Mack e de Bootstrap.

In this thesis stochastic claims reserving under the model of randomly trun- cated data is presented. For modelling the claims, a compound Poisson process is assumed. Introducing a random variable representing the delay between oc- currence and reporting of a claim, a probability model of IBNR claims is built. The fact that some claims are incurred but not reported yet leads to truncated data. Basic results of non-parametric statistical estimation under the model of randomly truncated data are shown, which can be used to obtain an estimate of IBNR claims reserves. Theoretical background is then used for application on real data from Czech Insurers' Bureau. 36

Claims reserving methods usually use data aggregated into development triangles, therefore a lot of information that insurance companies possess remains unused. This thesis shows a triangle-free approach using granular information from a claim by claim database. A statistical model for claims development which can further be used for estimation of reserves is built. The statistical model consists of a counting process that drives claims occurrence, distribution of reporting delay and distribution of claims severity. Several suitable distributions are presented, as well as methods for obtaining their parameters from data. Theoretical apparatus is used for real data. The thesis also pursues comparison of the IBNR reserve estimation using the triangle free approach and distribution free Chain ladder method for real data as well as for simulated data sets. For the data used in this thesis the complexity and data requirements of the triangle free approach are in favor of more preciseness and versatility. Powered by TCPDF (www.tcpdf.org)

This thesis is constituted of three chapters. he first part of my Ph.D. dissertation develops a Bayesian stochastic model for computing the reserves of a non-life insurance company. The first chapter is the product of my research experience as an intern at the Risk Management Department of Fondiaria-Sai S.p.A.. I present a short review of the deterministic and stochastic claims reserving methods currently applied in practice and I develop a (standard) Over-Dispersed Poisson (ODP) Bayesian model for the estimation of the Outstanding Loss Liabilities (OLLs) of a line of business (LoB). I present the model, I illustrate the theoretical foundations of the MCMC (Markov Chain Monte Carlo) method and the Metropolis-Hastings algorithm used in order to generate the non-standard posterior distributions. I apply the model to the Motor Third Party Liabil- ity LoB of Fondiaria-Sai S.p.A.. Moreover, I explore the problem of computing the prudential reserve level of a multi-line non-life insurance company. In the second chapter, then, I present a full Bayesian model for assessing the reserve requirement of multiline Non-Life insurance companies. The model combines the Bayesian approach for the estimation of marginal distribution for the single Lines of Business and a Bayesian copula procedure for their aggregation. First, I consider standard copula aggregation for different copula choices. Second, I present the Bayesian copula technique. Up to my knowledge, this approach is totally new to stochastic claims reserving. The model allows to "mix" own-assessments of dependence between LoBs at a company level and market wide estimates. I present an application to an Italian multi-line insurance company and compare the results obtained aggregating using standard copulas and a Bayesian Gaussian copula. In the second part of my Dissertation I propose a theoretical model that studies optimal capital and organizational structure choices of financial groups which incorporate two or more business units. The group faces a VaR-type regulatory capital requirement. Financial conglomerates incorporate activities in different sectors either into a unique integrated entity, into legally separated divisions or in ownership-linked holding company/subsidiary structures. I model these different arrangements in a structural framework through different coinsurance links between units in the form of conditional guarantees issued by equityholders of a firm towards the debtholders of a unit of the same group. I study the effects of the use of such guarantees on optimal capital structural and organizational form choices. I calibrate model parameters to observed financial institutions' characteristics. I study how the capital is optimally held, the costs and benefits of limiting undercapitalization in some units and I address the issues of diversification at the holding's level and regulatory capital arbitrage. The last part of my Ph.D. Dissertation studies the hedging problem of life insurance policies, when the mortality rate is stochastic. The field developed recently, adapting well-established techniques widely used in finance to describe the evolution of rates of mortality. The chapter is joint work with my supervisor, prof. Elisa Luciano and Elena Vigna. It studies the hedging problem of life insurance policies, when the mortality and interest rates are stochastic. We focus primarily on stochastic mortality. We represent death arrival as the first jump time of a doubly stochastic process, i.e. a jump process with stochastic intensity. We propose a Delta-Gamma Hedging technique for mortality risk in this context. The risk factor against which to hedge is the difference between the actual mortality intensity in the future and its "forecast" today, the instantaneous forward intensity. We specialize the hedging technique first to the case in which survival intensities are affine, then to Ornstein-Uhlenbeck and Feller processes, providing actuarial justifications for this restriction. We show that, without imposing no arbitrage, we can get equivalent probability measures under which the HJM condition for no arbitrage is satisfied. Last, we extend our results to the presence of both interest rate and mortality risk, when the forward interest rate follows a constant-parameter Hull and White process. We provide a UK calibrated example of Delta and Gamma Hedging of both mortality and interest rate risk.

Claims reserving and claims process estimation present classical problems in general insurance. The overall reserves are often determined under the assumption of independence among the lines of business. Though, recently modelling of the dependence among multiple lines of business has become crucial issue of reserving process. In this context, copulae provide a useful tool to construct models which go beyond the classical ones in terms of dependence structure. This thesis deals, in particular, with the copula regression model, its properties and possible applications in general insurance. This approach combines GLM modelling of margins and then expressing the dependence structure using copula. The theoretical methods are illustrated on a real dataset.

This thesis covers stochastic claims reserving in non-life insurance based on individual claims developments. Summarized theoretical methods are applied on data from Czech Insurers' Bureau for educational purposes. The problem of estimation is divided into four parts: oc- curence process generating claims, delay of notification, times between events and payments. Each part is estimated separately based on maximum likelihood theory and final estimates allow us to obtain an estimate of future liabilities distribution. The results are very promis- ing and we believe this method is worth of a further research. Contribution of this work is more rigorous theoretical part and application on data from the Czech market with some new ideas in practical part and simulation. 1

In the presented thesis the issue of dependency between response variables within the subjects in the generalized linear models framework is investigated. Reserving in non-life insurance is a key factor for the financial position of a company. The text introduces the basic actuarial notation, terminology and methods. The main part is focused on panel data framework, especially Generalized Linear Mixed Models (GLMM) as well as Generalized Estimating Equations (GEE), and their application on claims reserving. The aim of this thesis is to show the advantages, disadvantages, limitations and the comparison of these approaches on representative datasets, which were chosen according to results obtained from whole database analysis. Significant focus is on model selection and diagnostics used for this purpose. Finally, the obtained results are summarized in tables, figures and the comparison of the methods is provided. Powered by TCPDF (www.tcpdf.org)

碩士
逢甲大學
統計與精算所
98
The claim reserve is one of the most important liabilities in property-liability insurance companies, and it is closely related to property-liability insurance companies’ solvency and financial stability. Therefore, the estimation of claim reserves is a very important issue. However, insurance law in Taiwan does not impose a unifying standard for claim-reserving methods. In order to achieve the maximum economic efficiency, the claim reserves need to be estimated through actuaries’ past experiences and professional expertise. Since the subjective opinion of actuaries and the adoption of different estimated methods, each insurance company in Taiwan has different methods of setting up its claim reserves. The Chain-Ladder method is the most common and widely applied claim-reserving method. In recent years, considerable attention has been given to the discussion of possible relationships between the Chain Ladder and various stochastic models. In this article, the similarity between the Chain-Ladder method and “the Quasi Overdispersed Poisson model” in GLM was discussed. The Bootstrap technique has proved to be a very useful tool in many fields. In this article, it was used to assess the variability of the claim-reserving predictions, that is, the standard errors of prediction or prediction errors. After obtaining the standard errors of prediction or prediction errors, we constructed upper limits of the claim-reserving predictions for a confidence level of 95 percent. However, the application of the Bootstrap technique to claim reserving is not straightforward and the applications found in the actuarial literature were not the most adequate, because it has larger standard errors of prediction than other estimated methods. This fact has also been shown in this article. In this article, we have introduced the several different methods of estimating claim reserves, such as Chain-Ladder method, Age-to-Age Average method, and under the structure of GLM, simulating many forecasts of the claim reserve through the Bootstrap technique. Using these forecasts and combining initial forecasts estimated by original data, we calculated standard errors of prediction or percentile of prediction errors for each accident year and the aggregate total. We then obtained the upper limits of the claim-reserving predictions.

This thesis covers, in detail, theoretical background of micro-level stochastic model, which includes definition and properties of non-homogeneous Poisson process. This the- ory is then applied to real data generated by MTPL portfolio. Estimates of provisions under micro-level stochastic model are calculated using ordinary Monte Carlo simula- tion method. Results obtained from micro-level stochastic model are compared to Mack Chain-ladder estimates. 1