Title: Risk capital calculation in invesment life insurance Author: Bc. Tomáš Coufal Department/Institute: Department of Probability and Mathematical Statis- tics Supervisor of the master thesis: Mgr. Josef Lukášek Supervisor's e-mail address: Josef.Lukasek@allianz.cz Abstract: Unit linked insurance is a modern and flexible life insurance product. The last decade was marked by the raising popularity of unit linked insurance. The discussions conserning the impact of the new directive Sol- vency II on the life insurance business focus mainly on the traditional life insurance. This paper examines the issue of the calculation of the risk capital for unit linked insurance. Analysis of the impact of different death guaran- tees, forms of premium payment, time to maturity and dynamic policyholder bahaviour on the risk capital is presented. Keywords: Unit linked insurance, Solvency II, Risk capital, Solvency capital requirement

In insurance risk theory, the surplus of an insurance company is modelled to monitor and quantify its risks. With the outgo of claims and inflow of premiums, the insurer needs to determine what financial portfolio ensures the soundness of the company’s future while satisfying the shareholders’ interests. It is usually assumed that the net profit condition (i.e. the expectation of the process is positive) is satisfied, which then implies that this process would drift towards infinity. To correct this unrealistic behaviour, the surplus process was modified to include the payout of dividends until the time of ruin. Under this more realistic surplus process, a topic of growing interest is determining which dividend strategy is optimal, where optimality is in the sense of maximizing the expected present value of dividend payments. This problem dates back to the work of Bruno De Finetti (1957) where it was shown that if the surplus process is modelled as a random walk with ± 1 step sizes, the optimal dividend payment strategy is a barrier strategy. Such a strategy pays as dividends any excess of the surplus above some threshold. Since then, other examples where a barrier strategy is optimal include the Brownian motion model (Gerber and Shiu (2004)) and the compound Poisson process model with exponential claims (Gerber and Shiu (2006)). In this thesis, we focus on the optimality of a barrier strategy in the more general Lévy risk models. The risk process will be formulated as a spectrally negative Lévy process, a continuous-time stochastic process with stationary increments which provides an extension of the classical Cramér-Lundberg model. This includes the Brownian and the compound Poisson risk processes as special cases. In this setting, results are expressed in terms of “scale functions”, a family of functions known only through their Laplace transform. In Loeffen (2008), we can find a sufficient condition on the jump distribution of the process for a barrier strategy to be optimal. This condition was then improved upon by Loeffen and Renaud (2010) while considering a more general control problem. The first chapter provides a brief review of theory of spectrally negative Lévy processes and scale functions. In chapter 2, we define the optimal dividends problem and provide existing results in the literature. When the surplus process is given by the Cramér-Lundberg process with a Brownian motion component, we provide a sufficient condition on the parameters of this process for the optimality of a dividend barrier strategy. Chapter 3 focuses on the case when the claims distribution is given by a univariate mixture of Erlang distributions with a common scale parameter. Analytical results for the Value-at-Risk and Tail-Value-at-Risk, and the Euler risk contribution to the Conditional Tail Expectation are provided. Additionally, we give some results for the scale function and the optimal dividends problem. In the final chapter, we propose an expectation maximization (EM) algorithm similar to that in Lee and Lin (2009) for fitting the univariate distribution to data. This algorithm is implemented and numerical results on the goodness of fit to sample data and on the optimal dividends problem are presented.

In insurance risk theory, the surplus of an insurance company is modelled to monitor and quantify its risks. With the outgo of claims and inflow of premiums, the insurer needs to determine what financial portfolio ensures the soundness of the company’s future while satisfying the shareholders’ interests. It is usually assumed that the net profit condition (i.e. the expectation of the process is positive) is satisfied, which then implies that this process would drift towards infinity. To correct this unrealistic behaviour, the surplus process was modified to include the payout of dividends until the time of ruin. Under this more realistic surplus process, a topic of growing interest is determining which dividend strategy is optimal, where optimality is in the sense of maximizing the expected present value of dividend payments. This problem dates back to the work of Bruno De Finetti (1957) where it was shown that if the surplus process is modelled as a random walk with ± 1 step sizes, the optimal dividend payment strategy is a barrier strategy. Such a strategy pays as dividends any excess of the surplus above some threshold. Since then, other examples where a barrier strategy is optimal include the Brownian motion model (Gerber and Shiu (2004)) and the compound Poisson process model with exponential claims (Gerber and Shiu (2006)). In this thesis, we focus on the optimality of a barrier strategy in the more general Lévy risk models. The risk process will be formulated as a spectrally negative Lévy process, a continuous-time stochastic process with stationary increments which provides an extension of the classical Cramér-Lundberg model. This includes the Brownian and the compound Poisson risk processes as special cases. In this setting, results are expressed in terms of “scale functions”, a family of functions known only through their Laplace transform. In Loeffen (2008), we can find a sufficient condition on the jump distribution of the process for a barrier strategy to be optimal. This condition was then improved upon by Loeffen and Renaud (2010) while considering a more general control problem. The first chapter provides a brief review of theory of spectrally negative Lévy processes and scale functions. In chapter 2, we define the optimal dividends problem and provide existing results in the literature. When the surplus process is given by the Cramér-Lundberg process with a Brownian motion component, we provide a sufficient condition on the parameters of this process for the optimality of a dividend barrier strategy. Chapter 3 focuses on the case when the claims distribution is given by a univariate mixture of Erlang distributions with a common scale parameter. Analytical results for the Value-at-Risk and Tail-Value-at-Risk, and the Euler risk contribution to the Conditional Tail Expectation are provided. Additionally, we give some results for the scale function and the optimal dividends problem. In the final chapter, we propose an expectation maximization (EM) algorithm similar to that in Lee and Lin (2009) for fitting the univariate distribution to data. This algorithm is implemented and numerical results on the goodness of fit to sample data and on the optimal dividends problem are presented.

The thesis gives the analysis of the life insurance market in Ukraine. The first part is dedicated to the main characteristics of the Ukrainian insurance market with detailed specification of its participants, regulatory and theoretical basement of life insurance. The second part introduces deep analysis of the main insurance indexes of the market and also the analysis of particular insurance companies with reference to their market shares and product supplies. The last part compares evolution of the Ukrainian insurance market with other nations. Finally, the thesis points out the main problems of the life insurance market in Ukraine and their probable solutions.

This diploma work deal with the charakteristics of risks imperilling the cynological organisation and their members. The diploma work contains theory, risk analysis, offer and comparison insurance products of commercial insurance company. On the end is the offer of a concrete insurance product.

Die Einführung des GKV-FQWG sorgt für einen verstärkten Wettbewerbsdruck innerhalb des Systems der gesetzlichen Krankenkassen. Bestehen hohe Anreize zur Risikoselektion, so kann dieser Druck in einen vermehrten Kampf um vermeintlich vorteilhafte Versichertengruppen führen. Die Studie stellt heraus, welche Anreize zur regionalen Risikoselektion unter einem differenzierten Risikostrukturausgleichssystem vorliegen und gibt einen Einblick in die Bedeutung des Problemfeldes. Hierfür werden regionale Versichertenstrukturen gegenüber ihrem geographischen Risikopotential mittels räumlicher Autokorrelationsanalyse untersucht.

This Master's Thesis project set out with the objective to propose a machine learning model for predicting insurance risk at the level of an individual coverage, and compare it towards the existing models used by the project provider Gjensidige Försäkring. Due to interpretability constraints, it was found that this problem can be translated into a standard tabular regression task, with well defined target distributions. However, it was early identified that the set of feasible models do not contain pure black box models such as XGBoost, LightGBM and CatBoost which are typical choices for tabular data regression. In the report, we explicitly formulate the interpretability constraints in sharp mathematical language. It is concluded that interpretability can be ensured by enforcing a particular structure on the Hilbert space across which we are looking for the model.  Using this formalism, we consider two different approaches for fitting high performing models that maintain interpretability, where we conclude that gradient boosted regression tree based Generalized Additive Models in general, and the Explainable Boosting Machine in particular, is a promising model candidate consisting of functions within the Hilbert space of interest. The other approach considered is the basis expansion approach, which is currently used at the project provider. We make the argument that the gradient boosted regression tree approach used by the Explainable Boosting Machine is a more suitable model type for an automated, data driven modelling approach which is likely to generalize well outside of the training set. Finally, we perform an empirical study on three different internal datasets, where the Explainable Boosting Machine is compared towards the current production models. We find that the Explainable Boosting Machine systematically outperforms the current models on unseen test data. There are many potential ways to explain this, but the main hypothesis brought forward in the report is that the sequential model fitting procedure allowed by the regression tree approach allows us to effectively explore a larger portion of the Hilbert space which contains all permitted models in comparison to the basis expansion approach. / Detta mastersexamensarbete utgår från målsättningen att föreslå en maskinlärningsmodell för att förutspå försäkringsrisk, på nivån av enskilda försäkringar. Denna modell ska sedan jämföras mot nuvarande modeller som används hos Gjensidige Försäkring, som tillhandahåller projektet. Detta problem kan formuleras som ett traditionellt regressionsproblem på tabulär data, med väldefinerade målfördelningar. På grund av begränsningar kring krav på modellens förståbarhet identifierades det tidigt i projektet att mängden av tillåtna modeller inte innehåller ren black box modeller som XGBoost, LightGBM eller CatBoost, vilket är typiska förstahandsval för den här problemklassen. I rapporten formulerar vi förståbarhetskraven i skarpt, matematiskt språk, och drar slutsatsen att önskad förståbarhet kan uppnås genom en specifik struktur på det Hilbertrum där vi letar efter den optimala modellen. Utifrån denna formalism evaluerar vi två olika metoder för att anpassa modeller med god prestanda som uppnår önskade förståbarhetskrav. Vi drar slutsatsen att Generalized Additive Models anpassade till datat genom gradientboostade regressionsträd i allmänhet, och Explainable Boosting Machine i synnerhet är en lovande modellkandidat bestående av funktioner i vårt Hilbertrum av intresse. Vi utvärderar dessutom ett tillvägagångssätt för att anpassa Generalized Additive Models till datat genom basexpansioner, vilket är den metod som primärt används idag hos Gjensidige Försäkring. Vi argumenterar för att metoder som bygger på gradientboostade regressionsträd, såsom Explainable Boosting Machine, är mer lämplig för ett automatiserbart, datadrivet arbetssätt till att bygga modeller som generaliserar väl utanför träningsdatat.  Slutligen genomför vi en empirisk studie på tre olika interna dataset, där Explainable Boosting Machine jämförs mot nuvarande produktionsmodeller, vilka bygger på den tidigare nämnda basexpansionsmetodiken. Vi finner att Explainable Boosting Machine systematiskt överpresterar kontra nuvarande modeller på osedd testdata. Det finns många potentiella förklaringar till detta, men den huvudsakliga hypotsen som diskuteras i denna rapport är att den gradientboostade regressionsträdsmetodiken gör det möjligt att effektivt utforska en större delmängd av det Hilbertrum som innehåller alla tillåtna modeller i jämförelse med basexpansionsmetodiken.

The aim of this thesis is to develop algorithmic methods for computing particular performance measures of interest for a general class of discrete-based insurance risk models. We build upon and generalize the insurance risk models considered by Drekic and Mera (2011) and Alfa and Drekic (2007), by incorporating a threshold-based dividend system in which dividends only get paid provided some period of good financial health is sustained above a pre-specified threshold level. We employ two fundamental methods for calculating the performance measures under the more general framework. The first method adopts the matrix-analytic approach originally used by Alfa and Drekic (2007) to calculate various ruin-related probabilities of interest such as the trivariate distribution of the time of ruin, the surplus prior to ruin, and the deficit at ruin. Specifically, we begin by introducing a particular trivariate Markov process and then expressing its transition probability matrix in a block-matrix form. From this characterization, we next identify an initial probability vector for the process, from which certain important conditional probability vectors are defined. For these vectors to be computed efficiently, we derive recursive expressions for each of them. Subsequently, using these probability vectors, we derive expressions which enable the calculation of conditional ruin probabilities and, from which, their unconditional counterparts naturally follow. The second method used involves the first claim conditioning approach (i.e., condition on knowing the time the first claim occurs and its size) employed in many ruin theoretic articles including Drekic and Mera (2011). We derive expressions for the finite-ruin time based Gerber-Shiu function as well as the moments of the total dividends paid by a finite time horizon or before ruin occurs, whichever happens first. It turns out that both functions can be expressed in elegant, albeit long, recursive formulas. With the algorithmic derivations obtained from the two fundamental methods, we next focus on computational aspects of the model class by comparing six different types of models belonging to this class and providing numerical calculations for several parametric examples, highlighting the robustness and versatility of our model class. Finally, we identify several potential areas for future research and possible ways to optimize numerical calculations.

The aim of this thesis is to develop algorithmic methods for computing particular performance measures of interest for a general class of discrete-based insurance risk models. We build upon and generalize the insurance risk models considered by Drekic and Mera (2011) and Alfa and Drekic (2007), by incorporating a threshold-based dividend system in which dividends only get paid provided some period of good financial health is sustained above a pre-specified threshold level. We employ two fundamental methods for calculating the performance measures under the more general framework. The first method adopts the matrix-analytic approach originally used by Alfa and Drekic (2007) to calculate various ruin-related probabilities of interest such as the trivariate distribution of the time of ruin, the surplus prior to ruin, and the deficit at ruin. Specifically, we begin by introducing a particular trivariate Markov process and then expressing its transition probability matrix in a block-matrix form. From this characterization, we next identify an initial probability vector for the process, from which certain important conditional probability vectors are defined. For these vectors to be computed efficiently, we derive recursive expressions for each of them. Subsequently, using these probability vectors, we derive expressions which enable the calculation of conditional ruin probabilities and, from which, their unconditional counterparts naturally follow. The second method used involves the first claim conditioning approach (i.e., condition on knowing the time the first claim occurs and its size) employed in many ruin theoretic articles including Drekic and Mera (2011). We derive expressions for the finite-ruin time based Gerber-Shiu function as well as the moments of the total dividends paid by a finite time horizon or before ruin occurs, whichever happens first. It turns out that both functions can be expressed in elegant, albeit long, recursive formulas. With the algorithmic derivations obtained from the two fundamental methods, we next focus on computational aspects of the model class by comparing six different types of models belonging to this class and providing numerical calculations for several parametric examples, highlighting the robustness and versatility of our model class. Finally, we identify several potential areas for future research and possible ways to optimize numerical calculations.

University, Diss.--Bonn, 2004.