Stochastic Methods for Pension Funds

by Devolder, Pierre; Janssen, Jacques; Manca, Raimondo

• ISBN13: 9781848212046
• ISBN10: 1848212046
• Edition: 1st
• Format: Hardcover
• Copyright: 2012-02-13
• Publisher: Wiley-ISTE

Summary

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.

Author Biography

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.

Table of Contents

Preface	p. xiii
Introduction: Pensions in Perspective	p. 1
Pension issues	p. 1
The challenge	p. 1
Some figures	p. 2
Pension scheme	p. 7
Definition	p. 7
The four dimensions of a pension scheme	p. 8
Pension and risks	p. 11
Demographic risks	p. 11
Financial risks	p. 12
Impact of the risks on various kinds of pension schemes	p. 12
The time horizon of a pension scheme	p. 13
The multi-pillar philosophy	p. 14
Classical Actuarial Theory of Pension Funding	p. 15
General equilibrium equation of a pension scheme	p. 15
Principles	p. 15
The retrospective reserve	p. 16
The prospective reserve	p. 18
Equilibrated pension funding	p. 18
Decomposition of the reserve	p. 20
Classification of the methods	p. 20
General principles of funding mechanisms for DB Schemes	p. 21
Particular funding methods	p. 22
Unit credit cost methods	p. 23
Level premium methods	p. 25
Aggregate cost methods	p. 28
Deterministic and Stochastic Optimal Control	p. 31
Introduction	p. 31
Deterministic optimal-control	p. 31
Formulation of the optimal control problem	p. 31
Necessary conditions for optimality	p. 33
Bellman function	p. 33
Bellman optimality equation	p. 34
Hamilton-Jacobi equation	p. 37
The synthesis function	p. 38
Other types of optimal controls	p. 39
Example: the classical quadratic/linear control problem	p. 41
The maximum principle	p. 42
The maximum principle from the dynamic programming approach	p. 42
Extension to the one-dimensional stochastic optimal control	p. 45
Formulation of the one-dimensional stochastic optimal control problem	p. 46
Necessary conditions for one-dimensional stochastic optimality	p. 46
Extension to the multi-dimensional stochastic optimal control	p. 48
Dynamic programming principle	p. 50
The Hamilton-Jacobi-Bellman equation	p. 50
Examples	p. 52
Merton portfolio allocation problem	p. 52
Defined Contribution and Defined Benefit Pension Plans	p. 55
Introduction	p. 55
The defined benefit method	p. 56
The defined contribution method	p. 57
The model	p. 57
The capitalization system	p. 58
The notional defined contribution (NDC) method	p. 58
Historical preliminaries	p. 58
The Dini reform transformation coefficients	p. 60
Theoretical preliminaries	p. 63
The construction of a unitary pension present value	p. 65
Numerical example and results comparison	p. 78
Conclusions	p. 93
Fair and Market Values and Interest Rate Stochastic Models	p. 95
Fair value	p. 95
Market value of financial flows	p. 96
Yield curve	p. 97
Yield to maturity for a financial investment and for a bond	p. 99
Dynamic deterministic continuous time model for an instantaneous interest rate	p. 100
Instantaneous interest rate	p. 100
Particular cases	p. 101
Yield curve associated with an instantaneous interest rate	p. 101
Examples of theoretical models	p. 102
Stochastic continuous time dynamic model for an instantaneous interest rate	p. 104
The OUV stochastic model	p. 105
The CIR model (1985)	p. 111
Zero-coupon pricing under the assumption of no arbitrage	p. 114
Stochastic dynamics of zero-coupons	p. 114
Application of the no arbitrage principle and risk premium	p. 116
Partial differential equation for the structure of zero coupons	p. 117
Values of zero coupons without arbitrage opportunity for particular cases	p. 118
Market evaluation of financial flows	p. 130
Stochastic continuous time dynamic model for asset values	p. 132
The Black-Scholes continuous time model	p. 132
The solution of the Black-Scholes-Samuelson model	p. 132
Prediction	p. 135
VaR of one asset	p. 136
Motivation	p. 136
Definition of VaR for one asset	p. 137
Normal distribution case	p. 138
Lognormal distribution case	p. 140
Trajectory simulation	p. 143
VaR extensions: TVaR and conditional VaR	p. 144
Risk Modeling and Solvency for Pension Funds	p. 149
Introduction	p. 149
Risks in defined contribution	p. 149
Solvency modeling for a DC pension scheme	p. 150
The model	p. 150
Maturity risk	p. 151
Liquidity risk	p. 156
Lifecycle strategy in DC schemes	p. 163
Introduction of the longevity risk	p. 166
Risks in defined benefit	p. 170
Solvency modeling for a DB pension scheme	p. 171
The model	p. 171
Maturity risk	p. 173
Liquidity risk	p. 177
Introduction of longevity risk	p. 180
Optimal Control of a Defined Benefit Pension Scheme	p. 181
Introduction	p. 181
A first discrete time approach: stochastic amortization strategy	p. 181
The problem	p. 181
Stochastic evolution of the fund	p. 182
Asymptotic evolution of the fund and the contribution	p. 184
Optimal amortization period	p. 191
Optimal control of a pension fund in continuous time	p. 194
The problem	p. 194
The model	p. 195
Optimal Control of a Defined Contribution Pension Scheme	p. 207
Introduction	p. 207
Stochastic optimal control of annuity contracts	p. 208
The problem	p. 208
The general model	p. 209
Case with single contribution and no annuitization	p. 215
Case with regular contributions and no annuitization	p. 216
Case with single contribution and annuitization	p. 216
Case with regular premiums and annuitization	p. 218
Extension: model with several risky assets	p. 219
Stochastic optimal control of DC schemes with guarantees and under stochastic interest rates	p. 223
The problem	p. 223
The financial market	p. 223
The pension scheme	p. 226
The optimal control formulation	p. 226
The solution	p. 228
Simulation Models	p. 231
Introduction	p. 231
The direct method	p. 233
The model	p. 233
A real life example	p. 244
The Monte Carlo models	p. 250
The MAGIS model (individual as operational variable)	p. 250
Time as an operational variable	p. 251
Salary lines construction	p. 252
A direct generalization of the Bernoulli process	p. 253
The salary line construction by means of the generalized Bernoulli process	p. 257
A real data application	p. 264
The studied cases	p. 266
Discrete Time Semi-Markov Processes (SMP) and Reward SMP	p. 277
Discrete time semi-Markov processes	p. 277
Purpose	p. 277
DTSMP Definition	p. 278
DTSMP numerical solutions	p. 280
Solution of DTHSMP and DTNHSMP in the transient case: a transportation example	p. 284
Principle of the solution	p. 284
Semi-Markov transportation example	p. 286
Discrete time reward processes	p. 294
Classification and notation	p. 294
Undiscounted SMRWP	p. 297
Discounted SMRWP	p. 301
General algorithms for DTSMRWP	p. 304
Generalized Semi-Markov Non-homogeneous Models for Pension Funds and Manpower Management	p. 307
Application to pension funds evolution	p. 307
Introduction	p. 308
The non-homogeneous semi-Markov pension fund model	p. 310
The reserve structure	p. 317
The impact of inflation and interest variability	p. 319
Solving evolution equations	p. 322
The dynamic population evolution of the pension funds	p. 327
Financial equilibrium of the pension funds	p. 330
Scenario and data	p. 333
The usefulness of the NHSMPFM	p. 337
Generalized non-homogeneous semi-Markov model for manpower management	p. 338
Introduction	p. 338
GDTNHSMP salary lines construction	p. 339
GDTNHSMR WP for a reserve structure	p. 342
Reserve structure with stochastic interest rate	p. 343
The dynamics of population evolution	p. 344
The computation of salary cost present value	p. 346
Algorithms	p. 347
The algorithm for the GNHSMP with a 2 time random variable	p. 347
The algorithm for the pension model	p. 352
Appendices	p. 359
Basic Probabilistic Tools for Stochastic Modeling	p. 361
Probability space and random variables	p. 361
Expectation and independence	p. 364
Main distribution probabilities	p. 367
Binomial distribution	p. 367
Negative exponential distribution	p. 368
Normal (or Laplace Gauss) distribution	p. 369
Poisson distribution	p. 371
Lognormal distribution	p. 372
Gamma distribution	p. 372
Pareto distribution	p. 373
Uniform distribution	p. 374
Gumbel distribution	p. 375
Weibull distribution	p. 375
Multidimensional normal distribution	p. 375
Conditioning	p. 378
Stochastic processes	p. 386
Martingales	p. 390
Brownian motion	p. 394
Itô Calculus and Diffusion Processes	p. 397
Problem of stochastic integration	p. 397
Stochastic integration of simple predictable processes and semi-martingales	p. 399
General definition of the stochastic integral	p. 403
Itô's formula	p. 410
Quadratic variation of a semi-martingale	p. 410
Itô's formula	p. 412
Stochastic integral with a standard Brownian motion as the integrator process	p. 413
Case of predictable simple processes	p. 414
Extension to general integrator processes	p. 416
Stochastic differentiation	p. 417
Definition	p. 417
Examples	p. 417
Back to the itô's formula	p. 419
Stochastic differential of a product	p. 419
Examples	p. 419
The Ito's formula with time dependence	p. 420
Interpretation of the Ito's formula	p. 421
Other extensions of the Ito's formula	p. 422
Stochastic differential equations	p. 425
Existence and unicity general theorem [GDC 68]	p. 425
Solution of stochastic differential equations	p. 429
Diffusion processes	p. 429
Multidimensional diffusion processes	p. 432
Bibliography	p. 437
Index	p. 449
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