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Stochastic Methods for Pension Funds

Stochastic Methods for Pension Funds

Pierre Devolder, Jacques Janssen, Raimondo Manca

ISBN: 978-1-118-56203-1 March 2013 Wiley-ISTE 320 Pages

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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.

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. Solution of DTHSMP and DTNHSMP in the transient case: a transportation example 284

10.4. Discrete time reward processes 294

10.5. General algorithms for DTSMRWP 304

Chapter 11. Generalized Semi-Markov Non-homogeneous Models for Pension Funds and Manpower Management 307

11.1. Application to pension funds evolution 307

11.2. Generalized non-homogeneous semi-Markov model for manpower management 338

11.3. Algorithms 347


Appendix 1. Basic Probabilistic Tools for Stochastic Modeling 361

Appendix 2. Itô Calculus and Diffusion Processes 397

Bibliography 437

Index 449