The capital requirements for insurance companies in the Solvency I framework are based on the premium and claim expenditure. This approach does not take the individual risk of the insurer into consideration and give policy holder little assur- ance. Therefore a framework called Solvency II is under development by EU and its members. The capital requirements in Solvency II are based on risk management and is related to the specific risks of the insurer. Moreover, the insurer must make disclosures both to the supervising authority and to the market. This puts pressure on the insurance companies to use better risk and capital management, which gives the policy holders better assurance. In this thesis we present a stochastic model that describes the development of assets and liabilities. We consider the following risks: Stock market, bond market, interest rate and mortality intensity. These risks are modeled by stochastic processes that are aggregated to describe the change in the insurers Risk Bearing Capital. The capital requirement, Solvency Capital Requirement, is calculated using Conditional Value-at-Risk at a 99% confidence level and Monte Carlo simulation. The results from this model is compared to the Swiss Solvency Test model for three different types of life insurance policies. We can conclude that for large portfolios, the model presented in this thesis gives a lower solvency capital requirement than the Swiss model for all three policies. For small portfolios, the capital requirement is larger due to the stochastic mortality risk which is not included in the Swiss model.

The diploma thesis focuses on the topic of the future regulatory regime of the insurance and reinsurance market of the European Union called Solvency II. Currently the most discussed issue without a final structure is an extensive set of legislative and technical changes not only in the area of solvency treatment. Primarily, the work focuses on the standard formula calculation of capital requirements that reflect the solvency position of companies on the market. The first part deals with the theoretical description of the calculating methods of the required capital levels under current and future rules on the basis of available official documents. Further the general overview of the Solvency II is presented, a more detailed description of the valuation techniques of balance sheet items for the purposes of Solvency II, dealing with company's own funds and possible ways to optimize the asset portfolio are also included. Some theoretical descriptions of computational procedures applied on a fictitious life insurance company are presented in the second, more practical part of the diploma thesis.

This thesis studies project Solvency II, which is focused on the integrated regulation of insurance market in the European Union. It presents basic division and capital requirements arising from it. It describes division of the project into the three areas, refered to as pillars in practice. The thesis summarizes the basic methods for measuring the risk (Value at Risk, Tail Value at Risk), necessary in the calculation of the solvency capital requirements. The thesis studies the method of calculation of the solvency capital requirement SCR and the minimum capital requirement MCR. The calculation of the SCR is focused mainly on the method of the calculation of the capital requirement using the standard formula. Lastly, capital requirements are calculated using concrete data set.

Title: Capital requirements imposed on insurance companies in Solveny II and their quantification Author: Bc. Martin Kožár Department: Department of probability and mathematical statistics Supervisor: Mgr. Martin Pleška Abstract: This thesis studies project Solvency II, which is focused on the integrated regulation of insurance market in the European Union. It pre- sents basic division and capital requirements arising from it. It describes division of the project into the three areas, refered to as pillars in practice. The thesis summarizes the basic methods for measuring the risk (Value at Risk, Tail Value at Risk), necessary in the calculation of the solvency capital requirements. The thesis studies the method of calculation of the solvency capital requirement SCR and the minimum capital requirement MCR. The calculation of the SCR is focused mainly on the method of the calculation of the capital requirement using the standard formula. Lastly, capital requi- rements are calculated using concrete data set. Keywords: Solvency II, solvency capital requirement SCR, minimum capital requirement MCR 1

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

Initialement, la modélisation des risques en assurance non vie, supposait l'indépendance entre les différentes variables des modèles actuariels. De nos jours, cette hypothèse d'indépendance est souvent relâchée afin de tenir compte de possibles interactions entre les différents éléments. Cette thèse a pour but de contribuer à la littérature existante de la modélisation de la dépendance en assurance non vie. Concrètement, nous introduisons une nouvelle méthodologie d'analyse des risques en assurance à travers le développement des modèles de dépendance, principalement dans un cadre dynamique. Dans le premier chapitre de la thèse nous introduisons le contexte actuel de solvabilité, ainsi que la modélisation de la dépendance en assurance, avec une présentation des principaux résultats. Le deuxième chapitre est essentiellement constitué d'un article coécrit avec Christian de Peretti et Lotfi Belkacem, intitulé "Modelling Temporal Dependence of Claims In Insurance Using Autoregressive Conditional Amount Models" (voir Araichi et al. (2013)). Dans ce chapitre nous montrons l'existence d'une forme de dépendance temporelle (dynamique) entre les montants de sinistres d'une même branche d'assurance. Nous proposons un nouveau modèle nommé Autoregressive Conditional Amount Model (ACA), qui permet de capturer le comportement dynamique des sinistres. Également, nous développons un nouveau modèle nommé Generalized Extreme Value ACA model (GEVACA), afin d'analyser la dépendance dynamique des montants élevés, au niveau des queues de distribution. Enfin, nous donnons une nouvelle expression pour la Value at Risk (VaR) paramétrique adaptée pour des risques à dépendance temporelle. Des applications sur des données réelles et des techniques de backtesting sont ensuite effectuées afin de montrer la pertinence des modèles proposés. Le troisième chapitre est constitué d'un article coécrit avec Christian de Peretti et Lotfi Belkacem, intitulé "Generalized Autoregressive Conditional Sinistrality Model : A novel model for claims reserving in Non life insurance", (voir Araichi et al. (2015)). Dans ce chapitre, nous abordons d'abord le problème de l'évaluation des réserves dans un cadre dynamique. Nous montrons l'existence d'une forme de dépendance dynamique dans un triangle de liquidation. En particulier, nous nous intéressons à l'analyse de la dépendance temporelle entre les sinistres, ainsi qu'entre les années de développement. Nous proposons un nouveau modèle nommé "Generalized Autoregressive Conditional Sinistrality Model (GACSM), qui constitue une extension du modèle linéaire généralisé classique. Ensuite, nous fournissons une méthode de simulation bootstrap basée sur le modèle GACSM, qui permet d'évaluer les réserves en tenant compte du caractère dynamique des sinistres. Enfin, afin de montrer l'impact du modèle proposé sur l'évaluation des réserves et du capital, nous effectuons une comparaison des résultats obtenus avec ceux obtenus des modèles classiques (Chain Ladder et modèle linéaire généralisé). Dans le quatrième chapitre de la thèse, qui est constitué d'un article, coécrit avec Christian de Peretti et Lotfi Belkacem, intitulé "Time Varying Copula Model for claims reserving in Non life insurance". Nous intéressons à évaluer le montant agrégé des sinistres, en analysant conjointement la dépendance dynamique inter-sinistres ainsi qu'entre les sinistres de deux branches. Nous proposons un modèle basé sur le modèle GACSM et les copules conditionnelles, qui permettent de suivre l'évolution de la dépendance au cours du temps. Enfin, nous effectuons des applications sur des données réelles, ainsi que des méthodes de simulation sont considérées. En comparant les résultats obtenus, nous avons pu illustrer l'impact de la dépendance dynamique sur les réserves et le besoin en capital / In this thesis a different aspects of dependence modeling are considered. Indeed, temporal dependence structures between claims amounts and between lines of business are analyzed. In the first chapter, a general introduction on modeling dependence in insurance is provided. The second chapter is essentially constituted by the article "Modeling Temporal Dependence of Claims In Insurance Using Autoregressive Conditional Amount Models", written with Christian de Peretti and Lotfi Belkacem, (see Araichi et al. (2013)) It deals with the problem of existing a temporal dependence structure between claims amounts of one line of business. To this end, we propose a new model for handling the dynamic behaviour of claims amounts in insurance companies using an Autoregressive Conditional Amount (ACA) framework. This model will be named Autoregressive Conditional Amount Model (ACA). A Gamma ACA model and a Generalized Extreme Value ACA model are proposed. It is shown that these models are more appropriate to describe and to forecast the process of claims of the lines Auto Damage and Auto Liability than traditional models. Furthermore, a parametric Value at Risk based on ACA framework (VaR ACA) is proposed for evaluating a coverage amount of these claims. Using backtesting techniques, the VaR ACA provides an accurate estimation of risk. The third chapter of this thesis is based on the article "Generalized Autoregressive Conditional Sinistrality Model: A novel model for claims reserving in Non life insurance", written with Christian de Peretti and Lotfi Belkacem, (see Araichi et al. (2015)). In this chapter, a Generalized Autoregressive Conditional Sinistrality Model (GACSM) for claims is proposed. We extend the Generalized Linear Model (GLM) by incorporating temporal dependence between claims amounts of one triangle. The GACSM is used for model parameter estimation and consistency of such estimate is proved. Bootstrap procedure is implemented for prediction reserves and prediction errors. Results show that taking into account the temporal dependence between losses improves the precision of the reserve distribution estimate, and thus evaluates an accurate SCR. Finally the fourth chapter is based on the article "Time Varying Copula Model for claims reserving in Non life insurance", written with Christian de Peretti and LotfiBelkacem. In this chapter, a time varying copula models to understand the behavior of claims amounts of two lines of business. Time varying copula functions with a Generalized Autoregressive Conditional Sinistrality model are used to analyze the evolution in time of dependence between two lines and the temporal dependence between claims of each line. Simulation study is performed to highlight the impact on reserves and Solvency Capital Requirement. Results show that our approach provides a diversification effect between claims amounts

Detta examensarbetes syfte var att undersöka möjligheter till förbättring av hantering av lagernivåer för Bygg Ole Saltsjö-Boo. En kombination av aspekter från både systemteknik och industriell ekonomi har använts. I rapporten applicerades Guaranteed Service-Level modellen baserad på historisk försäljning i kombination relevanta teorier om lagerkostnad. Rapporten var begränsad till att behandla utvalda produkter med hög omsättning från två utvalda leverantörer till Bygg Ole. Efterfrågan för alla produkter i rapporten utom en är icke säsongsberoende. Särskild hänsyn har dessutom tagits till servicenivå, kapitalkostnader och variation i efterfråga. Resultatet gav att en implementering av modellen skulle ge lägre lagernivåer och därmed lägre lagerkostnader. Slutsatsen från rapporten var att modellen skulle kunna implementeras, eventuellt med höga administrativa kostnader i början. Bygg Ole har också en möjlighet att använda ett ordersystem baserat på den matematiska GSL-modellen (Guaranteed Service-Level) i kombination med prognoser över efterfrågan producerade av försäljningsavdelningen på Bygg Ole. Detta skulle potentiellt kunna öka precisionen i lagerhanteringen. Den nuvarande lagerräntan är relativt lågt bestämd och därför minskas de beräknade besparingarna från implementering av modellen. Om lagerräntan skulle vara högre skulle den ekonomiska fördelen med implementeringen vara tydligare. Rekommendationen till Bygg Ole är att tillämpa den rekommenderade GSL-modellen i kombination med ett system för prognos över efterfrågan på några utvalda produkter och sedan utvärdera resultatet. / The aim of this thesis was to examine possible improvements in the inventory management and procedure of ordering at Bygg Ole Saltsjö-Boo. A combination of aspects from both Systems Engineering and Industrial Engineering and Management has been used. In the report, a Guaranteed Service-Level model based on historical data of sales in combination with relevant theories about inventory carrying cost has been applied. The study was limited to specific chosen products with high sales from two selected suppliers of Bygg Ole. All these products in the study except one experienced low seasonal variety in demand. Furthermore special consideration was taken to service level, cost of capital and variability of demand. The result was that an implementation of the model would yield lower inventory levels and therefore lower carrying costs of inventory. The conclusion from the report was that the model could be implemented, although with possibly high administrative costs in the beginning. Bygg Ole also has a possibility of using an ordering system based on the mathematical GSL-model (Guaranteed Service-Level) in combination with forecasts of demand conducted by the sales department of Bygg Ole. This could potentially increase precision in the inventory management. The current inventory carrying charge is compounded relatively low and therefore decreases the calculated savings from implementing the model. If the carrying charge would be higher, the benefits of implementation would be more evident. The recommendation for Bygg Ole is to apply the recommended GSL-model in combination with a demand forecast planning system on a few selected products and then evaluate the result.

In the present work we study the economic capital of pension funds and their possible extension into the new concept of Solvency II. The main task is to examine the risks that are characteristic for pension fund activity. We use several modified stress simulations, which we model using a virtual model of pension fund. Primarily we focus on changes in net asset value (NAV) which is used in standard formula for calculation of the solvency capital requirement (SCR). In conclusion, we evaluate the possible impact of applications Solvency II to pension funds, the resulting economic capital and solvency of modeled pension fund.

歐盟保險業新監理架構Solvency II 於第5 次量化衝擊研究完成後計畫將在近年正式施行，我國保險業監理制度是否朝採用Solvency II 架構的方向前進仍未為定論，但必頇先行瞭解採行此制度可能對業界造成的影響。 本研究以2010 年8 月時CEIOPS 對Solvency II 所進行的第5 次量化衝擊研究QIS5 設立的標準與原則，對公司的資產與負債做假設後，以公帄價值法衡量壽險公司各部位資產位與負債，包括準備金的公帄價值衡量，並利用QIS5 所提供之計算工具標準法計算四家台灣壽險公司在2009 年底時的清償資本要求SCR。而QIS5 是在金融風暴後不久，當時環境使得利率極低，為了估算在利率環境較正常的情況下，本研究以2007 年底之利率做敏感度分析，重新計算各公司之資產與負債狀況與清償資本要求SCR。 研究結果發現在本研究假設下，負債面的準備金提存不足，保險公司以經濟 資本角度來衡量已經屬於破產狀態。投資型分離帳戶以外的準備金計算與目前準備金計提的方式除了頇以公帄價值衡量保險責任的最佳估計外，另外需要計提風險邊際，此數額約為最佳估計總額的12.4%至30.2%，保險公司自有資本不足有很大的因素是由於此部分準備金的計提。 也發現所計算出的SCR 中所最大的比率為利率風險或匯率風險，在假設以較 高利率環境做敏感度分析後發現壽險公司淨值仍然為負數，且所需要的SCR 與之前所得結果相差不大，顯示壽險公司負債部位對利率敏感度相當高，即使曝險部位變少，對於未來的利率變動仍需要準備相當大的資本以防範虧損。 / After the completion of the Fifth Quantitative Impact Study (QIS5) for the new insurance industrial regulation framework- Solvency II, European Union planned to implement the project in few years. No matter that the regulatory system of insurance industry in Taiwan will follow the trend or will not, it is a must that we should estimate the impacts on the whole industry before making the decisions. This study have an aim to estimate the Solvency Capital Requirements of 4 life insurance companies in Taiwan in the same principles with QIS5, which were took place in August 2010 by CEIOPS. In order to calculate the SCR, we made a lot of hypotheses and then estimated the fair value of the company assets and liabilities, including the fair value of technical provision. By means of the calculating helpers provided by CEIOPS used in QIS5, we found out the SCRs of these companies when they were on 31 December 2009. Then we performed the sensitivity analysis by the different interest rate which is based on the data on 31 December 2007, and recalculated the SCRs of the companies. This study had conclusions that the technical provisions were not sufficient to fulfill the obligations in aspect of the economic value. The surplus of companies were exhausted, because the technical provisions increased by fair valuation. Also, the heavy loadings of risk margins as 12.4% to 30.2% of the best estimates were the important reason of the negative own fund. We found that the capital requirements of interest risk and currency risk took great percentages of total SCRs. And the SCRs will not reduce in great amount caused by technical provisions reduced in the situation that interest rate come back to the level in 2007. It showed that the SCRs had great sensitivity to the interest risk and insurance companies should prepare sufficient own fund to prevent financial crisis caused by interest rate shock.

The purpose of this project is to validate the adequacy of the Standard formula, used to calculate the Solvency Capital Requirement (SCR), with respect to a Swedish insurance company. The sub-modules evaluated are Equity risk (type 1) and Interest rate risk. The validation uses a quantitative assessment and the concept of Value at Risk (VaR). Additionally, investment strategies for risk free assets are evaluated through a scenario based analysis. The findings support that the Equity shock of 39%, as proposed in the Standard formula, is appropriate for a diversified portfolio of global equities. Furthermore, to some extent; the Equity shock is also sufficient for a diversified global portfolio with an overweight of Swedish equities. Additionally, the findings shows that the Standard formula for Interest rate risks occasionally underestimates the true Interest rate risk. Furthermore, it’s shown that there are some advantage of selecting an investment strategy that stabilizes the Own fund of an insurance company rather than a strategy that minimizes the SCR. / Syftet med detta arbete är att utvärdera Standardformeln, som används för att beräkna solvenskapitalkravet (SCR) under Solvens II, med avseende på dess lämplighet för ett svensk försäkringsbolag. Modulerna som utvärderas är aktierisk (typ 1) och ränterisk. Utvärderingen genomförs med kvantitativa metoder och utifrån konceptet Value at Risk (VaR). Dessutom utvärderas investeringsstrategier för riskfria tillgångar genom en scenariobaserad analys. Resultaten stödjer att den av Standardformeln föreskrivna aktiechocken på -39 % är tillräcklig för en diversifierad global aktieportfölj. Dessutom är aktiechocken även tillräcklig för en diversifierad global portfölj med en viss övervikt mot svenska aktier. Vidare visar resultaten att Standardformeln under vissa omständigheter underskattar ränterisken. Slutligen visar den scenariobaserade analysen att det är fördelaktigt att välja en investeringsstrategi som stabiliserar Own fund, hellre än en strategi som minimerar SCR.