Stochastic Methods for Pension Funds

Preface xiii Chapter 1. Introduction: Pensions in Perspective 1 1.1. Pension issues 1 1.2. Pension scheme 7 1.3. Pension and risks 11 1.4. The multi-pillar philosophy 14 Chapter 2. Classical Actuarial Theory of Pension Funding15 2.1. General equilibrium equation of a pension scheme 15 2.2. General principles of funding mechanisms for DB Schemes21 2.3. Particular funding methods 22 Chapter 3. Deterministic and Stochastic Optimal Control31 3.1. Introduction 31 3.2. Deterministic optimal control 31 3.3. Necessary conditions for optimality 33 3.4. The maximum principle 42 3.5. Extension to the one-dimensional stochastic optimal control45 3.6. Examples 52 Chapter 4. Defined Contribution and Defined Benefit PensionPlans 55 4.1. Introduction 55 4.2. The defined benefit method 56 4.3. The defined contribution method 57 4.4. The notional defined contribution (NDC) method 58 4.5. Conclusions 93 Chapter 5. Fair and Market Values and Interest RateStochastic Models 95 5.1. Fair value 95 5.2. Market value of financial flows 96 5.3. Yield curve 97 5.4. Yield to maturity for a financial investment and for a bond99 5.5. Dynamic deterministic continuous time model for aninstantaneous interest rate 100 5.6. Stochastic continuous time dynamic model for aninstantaneous interest rate 104 5.7. Zero-coupon pricing under the assumption of no arbitrage114 5.8. Market evaluation of financial flows 130 5.9. Stochastic continuous time dynamic model for asset values132 5.10. VaR of one asset 136 Chapter 6. Risk Modeling and Solvency for Pension Funds149 6.1. Introduction 149 6.2. Risks in defined contribution 149 6.3. Solvency modeling for a DC pension scheme 150 6.4. Risks in defined benefit 170 6.5. Solvency modeling for a DB pension scheme 171 Chapter 7. Optimal Control of a Defined Benefit PensionScheme 181 7.1. Introduction 181 7.2. A first discrete time approach: stochastic amortizationstrategy 181 7.3. Optimal control of a pension fund in continuous time194 Chapter 8. Optimal Control of a Defined Contribution PensionScheme 207 8.1. Introduction 207 8.2. Stochastic optimal control of annuity contracts 208 8.3. Stochastic optimal control of DC schemes with guaranteesand under stochastic interest rates 223 Chapter 9. Simulation Models 231 9.1. Introduction231 9.2. The direct method 233 9.3. The Monte Carlo models 250 9.4. Salary lines construction 252 Chapter 10. Discrete Time Semi-Markov Processes (SMP) andReward SMP 277 10.1. Discrete time semi-Markov processes 277 10.2. DTSMP numerical solutions 280 10.3. Solution of DTHSMP and DTNHSMP in the transient case: atransportation example 284 10.4. Discrete time reward processes 294 10.5. General algorithms for DTSMRWP 304 Chapter 11. Generalized Semi-Markov Non-homogeneous Modelsfor Pension Funds and Manpower Management 307 11.1. Application to pension funds evolution 307 11.2. Generalized non-homogeneous semi-Markov model for manpowermanagement 338 11.3. Algorithms 347 APPENDICES 359 Appendix 1. Basic Probabilistic Tools for Stochastic Modeling361 Appendix 2. Itô Calculus and Diffusion Processes 397 Bibliography 437 Index 449