Financial Modeling, Actuarial Valuation and Solvency in Insurance

Risk management for financial institutions is one of the key topics the financial industry has to deal with. The present volume is a mathematically rigorous text on solvency modeling. Currently, there are many new developments in this area in the financial and insurance industry (Basel III and Solvency II), but none of these developments provides a fully consistent and comprehensive framework for the analysis of solvency questions. Merz and Wüthrich combine ideas from financial mathematics (no-arbitrage theory, equivalent martingale measure), actuarial sciences (insurance claims modeling, cash flow valuation) and economic theory (risk aversion, probability distortion) to provide a fully consistent framework. Within this framework they then study solvency questions in incomplete markets, analyze hedging risks, and study asset-and-liability management questions, as well as issues like the limited liability options, dividend to shareholder questions, the role of re-insurance, etc. This work embeds the solvency discussion (and long-term liabilities) into a scientific framework and is intended for researchers as well as practitioners in the financial and actuarial industry, especially those in charge of internal risk management systems. Readers should have a good background in probability theory and statistics, and should be familiar with popular distributions, stochastic processes, martingales, etc. Table of Contents Cover Financial Modeling, Actuarial Valuation and Solvency in Insurance ISBN 9783642313912 ISBN 9783642313929 Acknowledgements Contents Notation Chapter 1 Introduction 1.1 Full Balance Sheet Approach 1.2 Solvency Considerations 1.3 Further Modeling Issues 1.4 Outline of This Book Part I Chapter 2 State Price Deflator and Stochastic Discounting 2.1 Zero Coupon Bonds and Term Structure of Interest Rates o 2.1.1 Motivation for Discounting o 2.1.2 Spot Rates and Term Structure of Interest Rates o 2.1.3 Estimating the Yield Curve 2.2 Basic Discrete Time Stochastic Model o 2.2.1 Valuation at Time 0 o 2.2.2 Interpretation of State Price Deflator o 2.2.3 Valuation at Time t>0 2.3 Equivalent Martingale Measure o 2.3.1 Bank Account Numeraire o 2.3.2 Martingale Measure and the FTAP 2.4 Market Price of Risk Chapter 3 Spot Rate Models 3.1 General Gaussian Spot Rate Models 3.2 One-Factor Gaussian Affin Term Structure Models 3.3 Discrete Time One-Factor Vasicek Model o 3.3.1 Spot Rate Dynamics on a Yearly Grid o 3.3.2 Spot Rate Dynamics on a Monthly Grid o 3.3.3 Parameter Calibration in the One-Factor Vasicek Model 3.4 Conditionally Heteroscedastic Spot Rate Models 3.5 Auto-Regressive Moving Average (ARMA) Spot Rate Models o 3.5.1 AR(1) Spot Rate Model o 3.5.2 AR(p) Spot Rate Model o 3.5.3 General ARMA Spot Rate Models o 3.5.4 Parameter Calibration in ARMA Models 3.6 Discrete Time Multifactor Vasicek Model 3.6.1 Motivation for Multifactor Spot Rate Models o 3.6.2 Multifactor Vasicek Model (with Independent Factors) o 3.6.3 Parameter Estimation and the Kalman Filter 3.7 One-Factor Gamma Spot Rate Model o 3.7.1 Gamma Affin Term Structure Model o 3.7.2 Parameter Calibration in the Gamma Spot Rate Model 3.8 Discrete Time Black-Karasinski Model o 3.8.1 Log-Normal Spot Rate Dynamics o 3.8.2 Parameter Calibration in the Black-Karasinski Model o 3.8.3 ARMA Extended Black-Karasinski Model Chapter 4 Stochastic Forward Rate and Yield Curve Modeling 4.1 General Discrete Time HJM Framework 4.2 Gaussian Discrete Time HJM Framework 4.2.1 General Gaussian Discrete Time HJM Framework o 4.2.2 Two-Factor Gaussian HJM Model o 4.2.3 Nelson-Siegel and Svensson HJM Framework 4.3 Yield Curve Modeling 4.3.1 Derivations from the Forward Rate Framework o 4.3.2 Stochastic Yield Curve Modeling Chapter 5 Pricing of Financial Assets 5.1 Pricing of Cash Flows o 5.1.1 General Cash Flow Valuation in the Vasicek Model o 5.1.2 Defaultable Coupon Bonds 5.2 Financial Market o 5.2.1 A Log-Normal Example in the Vasicek Model o 5.2.2 A First Asset-and-Liability Management Problem 5.3 Pricing of Derivative Instruments Part II Chapter 6 Actuarial and Financial Modeling 6.1 Financial Market and Financial Filtration 6.2 Basic Actuarial Model 6.3 Improved Actuarial Model Chapter 7 Valuation Portfolio 7.1 Construction of the Valuation Portfolio o 7.1.1 Financial Portfolios and Cash Flows o 7.1.2 Construction of the VaPo o 7.1.3 Best-Estimate Reserves 7.2 Examples o 7.2.1 Examples in Life Insurance o 7.2.2 Example in Non-life Insurance 7.3 Claims Development Result and ALM o 7.3.1 Claims Development Result o 7.3.2 Hedgeable Filtration and ALM o 7.3.3 Examples Revisited 7.4 Approximate Valuation Portfolio Chapter 8 Protected Valuation Portfolio 8.1 Construction of the Protected Valuation Portfolio 8.2 Market-Value Margin 8.2.1 Risk-Adjusted Reserves o 8.2.2 Claims Development Result of Risk-Adjusted Reserves o 8.2.3 Fortuin-Kasteleyn-Ginibre (FKG) Inequality o 8.2.4 Examples in Life Insurance o 8.2.5 Example in Non-life Insurance o 8.2.6 Further Probability Distortion Examples 8.3 Numerical Examples o 8.3.1 Non-life Insurance Run-Off o 8.3.2 Life Insurance Examples Chapter 9 Solvency 9.1 Risk Measures 9.1.1 Definitio of (Conditional) Risk Measures o 9.1.2 Examples of Risk Measures 9.2 Solvency and Acceptability 9.2.1 Definitio of Solvency and Acceptability o 9.2.2 Free Capital and Solvency Terminology o 9.2.3 Insolvency 9.3 No Insurance Technical Risk o 9.3.1 Theoretical ALM Solution and Free Capital o 9.3.2 General Asset Allocations o 9.3.3 Limited Liability Option o 9.3.4 Margrabe Option o 9.3.5 Hedging Margrabe Options 9.4 Inclusion of Insurance Technical Risk o 9.4.1 Insurance Technical and Financial Result o 9.4.2 Theoretical ALM Solution and Solvency o 9.4.3 General ALM Problem and Insurance Technical Risk o 9.4.4 Cost-of-Capital Loading and Dividend Payments o 9.4.5 Risk Spreading and Law of Large Numbers o 9.4.6 Limitations of the Vasicek Financial Model 9.5 Portfolio Optimization o 9.5.1 Standard Deviation Based Risk Measure o 9.5.2 Estimation of the Covariance Matrix Chapter 10 Selected Topics and Examples 10.1 Extreme Value Distributions and Copulas 10.2 Parameter Uncertainty o 10.2.1 Parameter Uncertainty for a Non-life Run-Off o 10.2.2 Modeling of Longevity Risk 10.3 Cost-of-Capital Loading in Practice 10.3.1 General Considerations o 10.3.2 Cost-of-Capital Loading Example 10.4 Accounting Year Factors in Run-Off Triangles 10.4.1 Model Assumptions o 10.4.2 Predictive Distribution 10.5 Premium Liability Modeling o 10.5.1 Modeling Attritional Claims o 10.5.2 Modeling Large Claims o 10.5.3 Reinsurance 10.6 Risk Measurement and Solvency Modeling o 10.6.1 Insurance Liabilities o 10.6.2 Asset Portfolio and Premium Income o 10.6.3 Cost Process and Other Risk Factors o 10.6.4 Accounting Condition and Acceptability o 10.6.5 Solvency Toy Model in Action 10.7 Concluding Remarks Part III Chapter 11 Auxiliary Considerations 11.1 Helpful Results with Gaussian Distributions 11.2 Change of Numeraire Technique 11.2.1 General Changes of Numeraire o 11.2.2 Forward Measures and European Options on ZCBs o 11.2.3 European Options with Log-Normal Asset Prices References Index