Stochastic Methods for Pension Funds

Pierre Devolder is Professor of quantitative finance and actuarial sciences. He is associate editor of the ASTIN Bulletin and a member of the board of the AFIR section of the International Actuarial Association. His main research interests are pension funding, the application of stochastic processes to finance and insurance, fair valuation and solvency of insurance liabilities.Jacques Janssen is Honorary Professor at the Solvay Business School in Brussels, Belgium. He is a member of many scientific and actuarial associations (Belgium, France, and Switzerland) and chairman of the International ASMDA Steering Committee. His research interests include stochastic processes, financial and actuarial mathematics, operations research and data mining.Raimondo Manca is Professor of mathematical methods applied to economics, finance and actuarial science. He is associate editor of the journal Methodology and Computing in Applied Probability. His main research interests are multidimensional linear algebra, computational probability, the application of stochastic processes to economics, finance and insurance and simulation models.

• Preface xiiiChapter 1. Introduction: Pensions in Perspective 11.1. Pension issues 11.2. Pension scheme 71.3. Pension and risks 111.4. The multi-pillar philosophy 14Chapter 2. Classical Actuarial Theory of Pension Funding 152.1. General equilibrium equation of a pension scheme 152.2. General principles of funding mechanisms for DB Schemes 212.3. Particular funding methods 22Chapter 3. Deterministic and Stochastic Optimal Control 313.1. Introduction 313.2. Deterministic optimal control 313.3. Necessary conditions for optimality 333.4. The maximum principle 423.5. Extension to the one-dimensional stochastic optimal control 453.6. Examples 52Chapter 4. Defined Contribution and Defined Benefit Pension Plans 554.1. Introduction 554.2. The defined benefit method 564.3. The defined contribution method 574.4. The notional defined contribution (NDC) method 584.5. Conclusions 93Chapter 5. Fair and Market Values and Interest Rate Stochastic Models 955.1. Fair value 955.2. Market value of financial flows 965.3. Yield curve 975.4. Yield to maturity for a financial investment and for a bond 995.5. Dynamic deterministic continuous time model for an instantaneous interest rate 1005.6. Stochastic continuous time dynamic model for an instantaneous interest rate 1045.7. Zero-coupon pricing under the assumption of no arbitrage 1145.8. Market evaluation of financial flows 1305.9. Stochastic continuous time dynamic model for asset values 1325.10. VaR of one asset 136Chapter 6. Risk Modeling and Solvency for Pension Funds 1496.1. Introduction 1496.2. Risks in defined contribution 1496.3. Solvency modeling for a DC pension scheme 1506.4. Risks in defined benefit 1706.5. Solvency modeling for a DB pension scheme 171Chapter 7. Optimal Control of a Defined Benefit Pension Scheme 1817.1. Introduction 1817.2. A first discrete time approach: stochastic amortization strategy 1817.3. Optimal control of a pension fund in continuous time 194Chapter 8. Optimal Control of a Defined Contribution Pension Scheme 2078.1. Introduction 2078.2. Stochastic optimal control of annuity contracts 2088.3. Stochastic optimal control of DC schemes with guarantees and under stochastic interest rates 223Chapter 9. Simulation Models 2319.1. Introduction2319.2. The direct method 2339.3. The Monte Carlo models 2509.4. Salary lines construction 252Chapter 10. Discrete Time Semi-Markov Processes (SMP) and Reward SMP 27710.1. Discrete time semi-Markov processes 27710.2. DTSMP numerical solutions 28010.3. Solution of DTHSMP and DTNHSMP in the transient case: a transportation example 28410.4. Discrete time reward processes 29410.5. General algorithms for DTSMRWP 304Chapter 11. Generalized Semi-Markov Non-homogeneous Models for Pension Funds and Manpower Management 30711.1. Application to pension funds evolution 30711.2. Generalized non-homogeneous semi-Markov model for manpower management 33811.3. Algorithms 347APPENDICES 359Appendix 1. Basic Probabilistic Tools for Stochastic Modeling 361Appendix 2. Itô Calculus and Diffusion Processes 397Bibliography 437Index 449