Pierre Devolder & Jacques Janssen 
Stochastic Methods for Pension Funds [PDF ebook] 

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. Va R 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
APPENDICES 359
Appendix 1. Basic Probabilistic Tools for Stochastic Modeling
361
Appendix 2. Itô Calculus and Diffusion Processes 397
Bibliography 437
Index 449

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.