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

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.

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.

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

[pt] Esta dissertação tem como objetivo principal a apresentação, discussão e implementação de um modelo de espaço de estado, derivado do modelo desenvolvido por De Jong & Zenhwirth, no cenário de estimação de reservas IBNR. O modelo visa obter uma distribuição para as reservas e seu desvio padrão, que permite obter um intervalo de confiança para a estimativa. Também são propostas extensões para o modelo. / [en] The main purpose of this master thesis is the presentation, discussion and implementation of a state space model, derived from the De Jong & Zehnwirth model, on the IBNR Reserve estimation scenario. The model tries to obtain a distribution for the reserves and its standard deviation as well, allowing the cofidence interval estimation. Extensions for the model are also discussed.

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.

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)

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

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.