Stochastic Methods for Pension Funds

Klappentext Quantitative finance has become these last years a extraordinary field of research and interest as well from an academic point of view as for practical applications.At the same time, pension issue is clearly a major economical and financial topic for the next decades in the context of the well-known longevity risk. Surprisingly few books are devoted to application of modern stochastic calculus to pension analysis.The aim of this book is to fill this gap and to show how recent methods of stochastic finance can be useful for to the risk management of pension funds. Methods of optimal control will be especially developed and applied to fundamental problems such as the optimal asset allocation of the fund or the cost spreading of a pension scheme. In these various problems, financial as well as demographic risks will be addressed and modelled. Zusammenfassung The aim of this book is to fill this gap and to show how recent methods of stochastic finance can be useful for to the risk management of pension funds. Methods of optimal control will be especially developed and applied to fundamental problems such as the optimal asset allocation of the fund or the cost spreading of a pension scheme. Inhaltsverzeichnis 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 Funding 15 2.1. General equilibrium equation of a pension scheme 15 2.2. General principles of funding mechanisms for DB Schemes 21 2.3. Particular funding methods 22 Chapter 3. Deterministic and Stochastic Optimal Control 31 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 control 45 3.6. Examples 52 Chapter 4. Defined Contribution and Defined Benefit Pension Plans 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 Rate Stochastic 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 bond 99 5.5. Dynamic deterministic continuous time model for an instantaneous interest rate 100 5.6. Stochastic continuous time dynamic model for an instantaneous interest rate 104 5.7. Zero-coupon pricing under the assumption of no arbitrage 114 5.8. Market evaluation of financial flows 130 5.9. Stochastic continuous time dynamic model for asset values 132 5.10. VaR of one asset 136 Chapter 6. Risk Modeling and Solvency for Pension Funds 149 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 Pension Scheme 181 7.1. Introduction 181 7.2. A first discrete time approach: stochastic amortization strategy 181 7.3. Optimal control of a pension fund in continuous time 194 Chapter 8. Optimal Control of a Defined Contribution Pension Scheme 207 8.1. Introduction 207 8.2. Stochastic optimal control of annuity contracts 208 8.3. Stochastic optimal control of DC schemes with guarantees and 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) and Reward SMP 277 10.1. Discrete time semi-Markov processes 277 10.2. DTSMP numerical solutions 280 10.3. Solutio...

List of contents

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

About the author

Pierre De Volder, Full-time Professor, UCL; President of the Institut des Sciences Actuarielles, UCL; Member of The Royal Association of Belgian Actuaries (ARAB / KVBA). Jacques Janssen, Universite Libre de Bruxelles. Raimondo Manca, Università degli Studi di Roma La Sapienza.

Summary

The aim of this book is to fill this gap and to show how recent methods of stochastic finance can be useful for to the risk management of pension funds. Methods of optimal control will be especially developed and applied to fundamental problems such as the optimal asset allocation of the fund or the cost spreading of a pension scheme.

Product details

• Authors: DE VOLDER, Pierr De Volder, Pierre De Volder, Pierre/ Janssen De Volder, De_volder, Pierre Devolder, Jacque Janssen, Jacques Janssen, Raimondo Manca, Pierre de Volder
• Publisher: Wiley, John and Sons Ltd
• Languages: English
• Product format: Hardback
• Released: 26.07.2010
• EAN: 9781848212046
• ISBN: 978-1-84821-204-6
• No. of pages: 320
• Series: ISTE; ISTE
• Subjects: Natural sciences, medicine, IT, technology > Mathematics; Social sciences, law, business > Business > General, dictionaries; Statistik, Mathematik, Statistics, Altersversorgung, Angewandte Mathematik, Mathematics, Applied mathematics, Institutionelle Finanzplanung, Statistics for Finance, Business & Economics, Finanz- u. Wirtschaftsstatistik