Financial Modeling, Actuarial Valuation and Solvency in Insurance

Transcription

1 Mario V. Wiithrich Michael Merz Financial Modeling, Actuarial Valuation and Solvency in Insurance 4y Springer

2 Contents 1 Introduction Full Balance Sheet Approach Solvency Considerations Further Modeling Issues Outline of This Book 6 Part I Financial Valuation Principles 2 State Price Deflators and Stochastic Discounting Zero Coupon Bonds and Term Structure of Interest Rates Motivation for Discounting Spot Rates and Term Structure of Interest Rates Estimating the Yield Curve Basic Discrete Time Stochastic Model Valuation at Time Interpretation of State Price Deflators Valuation at Time t > Equivalent Martingale Measure Bank Account Numeraire : Martingale Measure and the FTAP Market Price of Risk 31 3 Spot Rate Models General Gaussian Spot Rate Models One-Factor Gaussian Affine Term Structure Models Discrete Time One-Factor Vasicek Model Spot Rate Dynamics on a Yearly Grid Spot Rate Dynamics on a Monthly Grid Parameter Calibration in the One-Factor Vasicek Model Conditionally Heteroscedastic Spot Rate Models Auto-Regressive Moving Average (ARMA) Spot Rate Models AR(1) Spot Rate Model 61

3 viii Contents AR(p) Spot Rate Model General ARMA Spot Rate Models Parameter Calibration in ARMA Models Discrete Time Multifactor Vasicek Model Motivation for Multifactor Spot Rate Models Multifactor Vasicek Model (with Independent Factors) Parameter Estimation and the Kalman Filter One-Factor Gamma Spot Rate Model Gamma Affine Term Structure Model Parameter Calibration in the Gamma Spot Rate Model Discrete Time Black-Karasinski Model. -. -r Log-Normal Spot Rate Dynamics Parameter Calibration in the Black-Karasinski Model ARMA Extended Black-Karasinski Model 95 4 Stochastic Forward Rate and Yield Curve Modeling General Discrete Time HJM Framework Gaussian Discrete Time HJM Framework General Gaussian Discrete Time HJM Framework Two-Factor Gaussian HJM Model Nelson-Siegel and Svensson HJM Framework Yield Curve Modeling Derivations from the Forward Rate Framework Stochastic Yield Curve Modeling 109 Appendix Proofs of Chap Pricing of Financial Assets Pricing of Cash Flows General Cash Flow Valuation in the Vasicek Model Defaultable Coupon Bonds Financial Market A Log-Normal Example in the Vasicek Model A First Asset-and-Liability Management Problem Pricing of Derivative Instruments 146 Appendix Proofs of Chap. 5, 149 Part II Actuarial Valuation and Solvency 6 Actuarial and Financial Modeling Financial Market and Financial Filtration Basic Actuarial Model ' Improved Actuarial Model Valuation Portfolio Construction of the Valuation Portfolio Financial Portfolios and Cash Flows Construction of the VaPo 171

4 Contents Best-Estimate Reserves Examples Examples in Life Insurance Example in Non-life Insurance Claims Development Result and ALM Claims Development Result Hedgeable Filtration and ALM Examples Revisited Approximate Valuation Portfolio 197 Protected Valuation Portfolio ^ Construction of the Protected Valuation Portfolio Market-Value Margin Risk-Adjusted Reserves Claims Development Result of Risk-Adjusted Reserves Fortuin-Kasteleyn-Ginibre (FKG) Inequality Examples in Life Insurance Example in Non-life Insurance Further Probability Distortion Examples Numerical Examples Non-life Insurance Run-Off Life Insurance Examples 244 Solvency Risk Measures Definition of (Conditional) Risk Measures Examples of Risk Measures Solvency and Acceptability Definition of Solvency and Acceptability Free Capital and Solvency Terminology Insolvency No Insurance Technical Risk Theoretical ALM Solution and Free Capital General Asset Allocations Limited Liability Option Margrabe Option Hedging Margrabe Options Inclusion of Insurance Technical Risk Insurance Technical and Financial Result Theoretical ALM Solution and Solvency General ALM Problem and Insurance Technical Risk Cost-of-Capital Loading and Dividend Payments Risk Spreading and Law of Large Numbers Limitations of the Vasicek Financial Model Portfolio Optimization Standard Deviation Based Risk Measure Estimation of the Covariance Matrix 333

5 x Contents 10 Selected Topics and Examples Extreme Value Distributions and Copulas Parameter Uncertainty ~ Parameter Uncertainty for a Non-life Run-Off Modeling of Longevity Risk Cost-of-Capital Loading in Practice General Considerations Cost-of-Capital Loading Example Accounting Year Factors in Run-Off Triangles Model Assumptions Predictive Distribution...>...-, Premium Liability Modeling Modeling Attritional Claims Modeling Large Claims Reinsurance Risk Measurement and Solvency Modeling Insurance Liabilities Asset Portfolio and Premium Income Cost Process and Other Risk Factors Accounting Condition and Acceptability Solvency Toy Model in Action Concluding Remarks 402 Part III Appendix 11 Auxiliary Considerations Helpful Results with Gaussian Distributions Change of Numeraire Technique General Changes of Numeraire Forward Measures and European Options on ZCBs European Options with Log-Normal Asset Prices 415 References 419 Index 427