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- ISBN: 9781848212046 | 1848212046
- Cover: Hardcover
- Copyright: 2/13/2012
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.
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 | 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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