Stochastic Methods for Pension Funds

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.

Quantitative finance has become these last years a extraordinaryfield of research and interest as well from an academic point ofview as for practical applications.At the same time, pension issue is clearly a major economicaland financial topic for the next decades in the context of thewell-known longevity risk. Surprisingly few books are devoted toapplication of modern stochastic calculus to pension analysis.The aim of this book is to fill this gap and to show how recentmethods of stochastic finance can be useful for to the riskmanagement of pension funds. Methods of optimal control will beespecially developed and applied to fundamental problems such asthe optimal asset allocation of the fund or the cost spreading of apension scheme. In these various problems, financial as wellas demographic risks will be addressed and modelled.

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 Funding152.1. General equilibrium equation of a pension scheme 152.2. General principles of funding mechanisms for DB Schemes212.3. Particular funding methods 22Chapter 3. Deterministic and Stochastic Optimal Control313.1. 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 control453.6. Examples 52Chapter 4. Defined Contribution and Defined Benefit PensionPlans 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 RateStochastic 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 bond995.5. Dynamic deterministic continuous time model for aninstantaneous interest rate 1005.6. Stochastic continuous time dynamic model for aninstantaneous interest rate 1045.7. Zero-coupon pricing under the assumption of no arbitrage1145.8. Market evaluation of financial flows 1305.9. Stochastic continuous time dynamic model for asset values1325.10. VaR of one asset 136Chapter 6. Risk Modeling and Solvency for Pension Funds1496.1. 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 PensionScheme 1817.1. Introduction 1817.2. A first discrete time approach: stochastic amortizationstrategy 1817.3. Optimal control of a pension fund in continuous time194Chapter 8. Optimal Control of a Defined Contribution PensionScheme 2078.1. Introduction 2078.2. Stochastic optimal control of annuity contracts 2088.3. Stochastic optimal control of DC schemes with guaranteesand 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) andReward SMP 27710.1. Discrete time semi-Markov processes 27710.2. DTSMP numerical solutions 28010.3. Solution of DTHSMP and DTNHSMP in the transient case: atransportation example 28410.4. Discrete time reward processes 29410.5. General algorithms for DTSMRWP 304Chapter 11. Generalized Semi-Markov Non-homogeneous Modelsfor Pension Funds and Manpower Management 30711.1. Application to pension funds evolution 30711.2. Generalized non-homogeneous semi-Markov model for manpowermanagement 33811.3. Algorithms 347APPENDICES 359Appendix 1. Basic Probabilistic Tools for Stochastic Modeling361Appendix 2. Itô Calculus and Diffusion Processes 397Bibliography 437Index 449