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# Econometric models for insurance applications: essays on Bayesian mortality models, heavy tails and extreme value statistics with censored data.

Life insurers, pension funds, health care providers and social security institutions face increasing expenses due to continuing improvements of mortality rates. The actuarial and demographic literature has introduced a myriad of (deterministic and stochastic) models to forecast mortality rates of single and multiple populations.

Chapter 2 in this PhD thesis presents aBayesian analysis of two related multi-population mortality models of log-bilinear type, designed for two or more populations. Using a larger set of data, multi-population mortality models allow joint modelling and projection of mortality rates by identifying characteristics shared by all sub-populations as well as subpopulation specific effects on mortality. This is important when modeling and forecasting mortality of malesand females, regions within a country and when dealing with index-based longevity hedges. Our first model is inspired by the two factor Lee & Carter model of Renshaw and Haberman (2003) and the common factor model of Carter and Lee (1992). The second model is the augmented common factor model of Li and Lee (2005). This chapter approaches both models in a statistical way, using a Poisson distribution for the number of deaths at a certain age and in a certain time period. Moreover, we use Bayesian statistics to calibrate the models and to produce mortality forecasts. Appendix A contains the technicalities necessary for Markov Chain Monte Carlo ([MCMC]) simulations. We also provide software implementation (in R) for the models discussed. Key benefits of this approachare multiple. We jointly calibrate the Poisson likelihood for the number of deaths and the times series models imposed on the time dependent parameters, we enable full allowance for parameter uncertainty and we are able to handle missing data as well as small sample populations. We compare and contrast results from both models to the results obtained with a frequentist single population approach and a least squares estimation of the augmented common factor model.

Chapter 3 continues the study of multi-population stochastic mortality models as a strategy for achieving robust and coherent projections of mortality rates. This chapter presents a Bayesian analysis of two coherent multi-population models of log-bilinear type, designed for two or more populations, while allowing for dependence between these populations. The first model is inspired by Cairns et al. (2011b) and Enchev et al.(2016), and the second is the well known Li & Lee model, proposed by Li and Lee (2005). For both models we identify the parameters through appropriate constraints and we avoid the multi-step calibration strategy that is currently used in the literature. We assume a Poisson distribution for the number of deaths at a certain age and in a certain period and include full dependency between the period effects. As such, we extend earlier work (including Chapter 2 in this thesis) where period effects are considered independent. Moreover, we utilize the Kannisto parametric mortality law to close the generated mortality scenarios for higher ages and provide projections of important demographic markers, such as period and cohort life expectancy. Appendix B contains the technicalities necessary for Markov Chain Monte Carlo ([MCMC]) simulations. We also provide software implementation (in R) for the models discussed. We finally present a case study using five European countries which are geographically close andshare similar socio-economic characteristics. In the field of risk theoryand risk management an important concept is the calculation of the so called ruin probability or probability of default. The default of an insurance company is more likely to occur in the presence of large claims. In insurance mathematics we use the theory of heavy-tailed distributions to describe the possibility of large claims. Even though the presence of a significant number of large claims becomes more realistic in the heavy-tailed framework, the calculation of the ruin probability is more difficult in the presence of distributions with heavy tails. Therefore solid mathematical properties of heavy tails are necessary. To enable the derivation of these properties we split the class of heavy tails into smaller sub-classes. Popular sub-classes are the regular varying subclass and the subexponential subclass.

In Chapter 4 we use the properties of the Matuszewska indices to show asymptotic results of tails of distributions and their corresponding hazard rate functions. We discuss the relation between membership of the classes of dominatedly or extended rapidly varying tail distributions and corresponding hazard rate conditions. We establish the max-sum equivalence property of a subclass of subexponential distributions. For the class of distributions with extended rapidly varying tails we demonstrate the convolution closure property. In certain insurance business lines, such as motor third party liability insurance, large claims may emerge. Since large losses may jeopardize the solvability of an insurance or reinsurance company, tools from extreme value statistics are studied within actuarial risk measurement. However, when claims need a (very) long time to close, the ultimate claim amount is not known at the time of the evaluation of the portfolio but is (right) censored. This complicates the statistical analysis of claim amount data. Motivated by Beirlant et al. (2007) and Einmahl et al. (2008), Chapter 5 considers bias reduced estimators for the extreme value index and tail probability estimators when the censored as well as the censoring distribution belong to the Fréchet domain. Our solution is based on second order refined peaks-over-threshold modelling as developed by Beirlant et al. (2009). Finite sample simulations show the improvement over the existing estimators, and applications to tail probability estimation are considered. The asymptotic normality of the proposed bias reduced estimator of the extreme valueindex is established.