169,99 €
Mehr erfahren.
- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
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.
Sie lesen das E-Book in den Legimi-Apps auf:
Seitenzahl: 450
Veröffentlichungsjahr: 2013
Ähnliche
Contents
Preface
Chapter 1. Introduction: Pensions in Perspective
1.1. Pension issues
1.2. Pension scheme
1.3. Pension and risks
1.4. The multi-pillar philosophy
Chapter 2. Classical Actuarial Theory of Pension Funding
2.1. General equilibrium equation of a pension scheme
2.2. General principles of funding mechanisms for DB Schemes
2.3. Particular funding methods
Chapter 3. Deterministic and Stochastic Optimal Control
3.1. Introduction
3.2. Deterministic optimal control
3.3. Necessary conditions for optimality
3.4. The maximum principle
3.5. Extension to the one-dimensional stochastic optimal control
3.6. Examples
Chapter 4. Defined Contribution and Defined Benefit Pension Plans
4.1. Introduction
4.2. The defined benefit method
4.3. The defined contribution method
4.4. The notional defined contribution (NDC) method
4.5. Conclusions
Chapter 5. Fair and Market Values and Interest Rate Stochastic Models
5.1. Fair value
5.2. Market value of financial flows
5.3. Yield curve
5.4. Yield to maturity for a financial investment and for a bond
5.5. Dynamic deterministic continuous time model for an instantaneous interest rate
5.6. Stochastic continuous time dynamic model for an instantaneous interest rate
5.7. Zero-coupon pricing under the assumption of no arbitrage
5.8. Market evaluation of financial flows
5.9. Stochastic continuous time dynamic model for asset values
5.10. VaR of one asset
Chapter 6. Risk Modeling and Solvency for Pension Funds
6.1. Introduction
6.2. Risks in defined contribution
6.3. Solvency modeling for a DC pension scheme
6.4. Risks in defined benefit
6.5. Solvency modeling for a DB pension scheme
Chapter 7. Optimal Control of a Defined Benefit Pension Scheme
7.1. Introduction
7.2. A first discrete time approach: stochastic amortization strategy
7.3. Optimal control of a pension fund in continuous time
Chapter 8. Optimal Control of a Defined Contribution Pension Scheme
8.1. Introduction
8.2. Stochastic optimal control of annuity contracts
8.3. Stochastic optimal control of DC schemes with guarantees and under stochastic interest rates
Chapter 9. Simulation Models
9.1. Introduction
9.2. The direct method
9.3. The Monte Carlo models
9.4. Salary lines construction
Chapter 10. Discrete Time Semi-Markov Processes (SMP) and Reward SMP
10.1. Discrete time semi-Markov processes
10.2. DTSMP numerical solutions
10.3. Solution of DTHSMP and DTNHSMP in the transient case: a transportation example
10.4. Discrete time reward processes
10.5. General algorithms for DTSMRWP
Chapter 11. Generalized Semi-Markov Non-homogeneous Models for Pension Funds and Manpower Management
11.1. Application to pension funds evolution
11.2. Generalized non-homogeneous semi-Markov model for manpower management
11.3. Algorithms
APPENDICES
Appendix 1. Basic Probabilistic Tools for Stochastic Modeling
A1.1. Probability space and random variables
A1.2. Expectation and independence
A1.3. Main distribution probabilities
A1.4. Conditioning
A1.5. Stochastic processes
A1.6. Martingales
A1.7. Brownian motion
Appendix 2. Itô Calculus and Diffusion Processes
A2.1. Problem of stochastic integration
A2.2. Stochastic integration of simple predictable processes and semi-martingales
A2.3. General definition of the stochastic integral
A2.4. Itô’s formula
A2.5. Stochastic integral with a standard Brownian motion as the integrator process
A2.6. Stochastic differentiation
A2.7. Back to the itô’s formula
A2.8. Stochastic differential equations
A2.9. Diffusion processes
A2.10. Multidimensional diffusion processes
Bibliography
Index
First published 2012 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:
ISTE Ltd27-37 St George’s RoadLondon SW19 4EUUK
John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USA
www.iste.co.uk
www.wiley.com
© ISTE Ltd 2012
The rights of Pierre Devolder, Jacques Janssen and RaimondoMancato be identified as the author of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.
Library of Congress Cataloging-in-Publication Data
Devolder, Pierre.
Stochastic methods for pension funds / Pierre Devolder, Jacques Janssen, Raimondo Manca.
p. cm.
Includes bibliographical references and index.
ISBN 978-1-84821-204-6
1. Pension trusts--Management. 2. Pension trusts--Mathematics. 3. Financial risk management--Mathematical models. 4. Stochastic models. I. Janssen, Jacques, 1939- II. Manca, Raimondo. III. Title.
HD7105.4.D48 2011
332.67’2540151923--dc23
2011048482
British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN: 978-1-84821-204-6
Preface
In recent years quantitative finance has become an extraordinary field of research and interest, from an academic point of view as well as for practical applications.
At the same time, the pension issue is clearly a major economic and financial topic for the coming decades, in the context of the well–known longevity risk. The emergence and development of pension schemes in our modern societies can essentially be explained by two factors:
– The individual approach to life: our modern world tends to substitute for large multi–generational families, for an individual model where each person is assumed to be self-supporting, before and after retirement age. As a consequence, personal pension planning, including social security pensions and other incomes, becomes a necessity for everyone
– The longevity syndrome: longevity has increased extraordinarily in recent decades, and nothing indicates a stopping or a reverse in this phenomenon could occur in the next few years. In the past it was usual to retire at 65 and then have a life expectancy of only a few more years. Nowadays, and in the near future, in many countries, it will be quite common for all of us to hope to survive until after 85!
The future of our pension systems is clearly one of the main economic challenges of the world for the coming decades. In the huge majority of countries, collective pension schemes have been created by states as well as by private companies or professional organizations. These systems can use very different tools, techniques, funding approaches or legal forms.
As described in Gronchi [GRO 99], pensions can be classified as a function of the calculation method of the benefits as:
– defined benefit (DB): the pension is a function of salaries and the number of years of service of the retiring person;
– defined contribution (DC): the pension is a function of the contribution that the retiring person paid during his life.
It can also be classified as a function of the financing technique as:
– pay as you go (PAYG): the pensions are paid by the current workers to the previous generation;
– funding: the pensions are generated by the savings accumulated by the contribution of the workers.
Surprisingly few books are devoted to the application of modern stochastic tools to pension analysis.
Therefore, the aim of this book is to fill this gap and to show how recent stochastic methods can be useful for the risk management of pension funds and the computation of market values. Optimal control methods 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 and economic risks will be addressed and modeled.
The layout of this book is as follows: Chapters 1 and 2 present the fundamental issues of the next decades and classical theory of pension funding.
In Chapter 3, we introduce the minimal basic results concerning control theory, both for deterministic and stochastic formulation in order to prepare the models presented in Chapters 7 and 8
Chapter 4 is devoted to concepts of defined contribution and defined benefit pension plans, while Chapter 5 presents some basic definitions on fair and market values for the evaluation of financial flows and stocks in the future – not only to improve the classical concept of present value, but also to fill up the constraints of new IFRS rules as well as Basel II and III rules for banks and Solvency II for insurance companies
Chapter 6 first develops stochastic models for DC and then for DB, in order to measure the various risks faced by the pension fund and to propose eventual solvency buffers in the philosophy of Basel II or Solvency II. In particular, it illustrates the importance of time in the risk assessment of a pension fund.
Chapters 7 and 8 give the main results of the pension theory as various models for the optimal control of the investment strategy and the contribution process for both defined contribution pension schemes and defined benefit pension schemes are presented
Chapter 7 presents – also for DB pension schemes in a stochastic environment – important asymptotic results of evolution of the fund and the contributions
Chapter 9 presents the construction of algorithms for the management of pension fund liabilities necessary for simulation models. Fundamentally we give two approaches to these models, one that is a semi–deterministic approach that we will call the direct method and the other based on the Monte Carlo method.
In short, with these last three chapters, it is possible to carefully study the evolution of a pension fund in the future; they constitute the core of this book.
The presentation, in Chapter 10, of discrete time homogeneous semi–Markov processes (DTHSMP), discrete time non–homogeneous (DTNHSMP) semi–Markov processes and the discrete time semi–Markov reward processes (SMRWP), is followed by Chapter 11, the last chapter of this book, which is devoted to another type of stochastic model of pension funds and manpower management study, called generalized non–homogeneous semi-Markov models.
This model is a general, rigorous and tractable stochastic evolution time model for pension funds, called the discrete time non–homogeneous semi-Markov pension fund model, taking into account economic, financial and demographic evolution factors so that it becomes a real–life model using important factors such as seniority, general age dependence, rate of inflation and salary lines. It can be particularly useful for the study of private pension fund evolution.
In conclusion, this book presents realistic stochastic models to carefully study the evolution of pension funds in the future, not only from a theoretical point of view but also from a practical point of view, as we present algorithms for the construction of simulation models.
We also present basic concepts in such a way that this book is relatively self-contained. Therefore, the book can be considered as the first textbook in the field of stochastic methods for pension funds and thus, it will be useful for graduate students in economics and actuarial science as well as for managers of pension funds and especially people involved in Solvency II for insurance companies and in Basel II and III for banks
Chapter 1
Introduction: Pensions in Perspective
1.1. Pension issues
1.1.1. The challenge
The future of our pension systems is clearly one of the main economic challenges for the world in the coming decades. In the vast majority of countries, collective pension schemes have been created by states as well as by private companies or professional organizations. These systems can use very different tools, techniques, funding approaches or legal forms.
From a historical point of view, the concept of an organized pension institution is relatively new in human evolution and can be seen essentially as a creation of the 20th Century. Of course for centuries, societies have accepted the idea of solidarity between generations and the fact that at a given age people must stop working and then receive other forms of income to survive. But in the traditional way of life, this solidarity was guaranteed by familial or tribal assistance without need of complicated collective systems. For instance, it was common, in this context, to have families with 3 (or even 4) different generations living together in the same house and sharing a global income. At the same time, the mean longevity was not as important and the number of retirement years was limited in general.
The emergence and development of pension schemes in our modern societies can essentially be explained by two factors:
– the “individual approach” of life: our modern world tends to replace large “multi–generational” families by an individual model where each person is assumed to be self–supporting, before and after retirement age. As a consequence, personal pension planning, including social security pensions and other incomes, becomes a necessity for everyone;
– the “longevity syndrome”: longevity has increased extraordinarily in recent decades and nothing indicates that a stopping or reversing of this phenomenon could occur in the next few years. In the past it was usual to retire at 65 and then have a life expectancy of only a few years. Nowadays, and in the near future, in a lot of countries, it will be quite common for all of us to hope to survive until after 85!
This longevity evolution is a major driving force in motivating the development of various pension schemes under different forms. But at the same time it induces a huge challenge in terms of financial sustainability. Indeed, for the first time in history, we can simultaneously observe two major demographic evolutions: a continuous increase in longevity leading to an expected increase in the proportion of retirees and a decrease of fertility rates leading to a decrease of active workers.
As a consequence, we will live in increasingly ageing societies, with fewer contributors and more beneficiaries.
1.1.2. Some figures
In order to give a global view of the importance of the pension challenge and its worldwide coverage, we present some international demographical and financial figures [OEC 11].
1.1.2.1. Longevity at 65
The following table compares in various countries, for men and women, the life expectancy at 65 in 2005 and that projected for 2050. It shows that everywhere our longevity is expected to continue increasing in the coming years by more than 3 years. For the OECD34, a man aged 65 in 2045 is expected to survive until 84 (88 for a woman).
Table 1.1.Life expectation at 65 in 2005 and 2045 (source: OECD 2011)
1.1.2.2. Fertility rates
The following table compares, in various countries, the evolution of the fertility rate between 1975 and 2050. It illustrates the persistent low level of this rate (less than 2.1, which is often seen as the necessary rate in order to guarantee the strict renewal of the population). For OECD34, this rate is expected to stay around 1.7.
Total fertility rates. 1975–2050
Table 1.2.Evolution of the fertility rates 1975–2050 (source: OECD 2011)
1.1.2.3. Public pension expenditure
The following table compares, in various countries, the evolution of the public pension expenditure between 1990 and 2007 (in % of the GDP). Absolute levels and evolutions can be quite different from one country to another, depending on the importance of social security and the eventual financial measures already taken by some states (for instance Sweden or Luxemburg). But from a global point of view, the increasing trend seems to be clear. For OECD34, nowadays public pension expenditures represent 7% of the GDP.
Table 1.3.Evolution of the public pension expenditure in % of GDP (source: OECD 2011)
1.1.2.4. Assets of pension funds
The following table gives the pension assets (in % of the GDP and in USD) for 2009, in various countries. These figures illustrate the importance of the pension fund assets invested in the financial markets. For OECD34, these assets represent two–thirds of the GDP.
Table 1.4.Importance of the assets of the pension funds in % of GDP and in US dollars (source: OECD 2011)
Pension funds
% of GDP
USD
OECD members
Australia
82.3
808,224
Austria
4.9
18,987
Belgium
3.3
16,677
Canada
62.9
806,350
Chile
65.1
106,596
Czech Republic
4.6
11,332
Denmark
43.3
133,980
Estonia
6.9
1,371
Finland
76.8
182,286
France
0.8
21,930
Germany
5.2
173,810
Greece
0.0
63
Hungary
13.1
16,886
Iceland
118.3
14,351
Ireland
44.1
100,278
Israel
46.9
95,257
Italy
4.1
86,818
Japan
25.2
1,042,770
Korea
2.2
29,632
Luxembourg
2.2
1,171
Mexico
7.5
107,135
Netherlands
129.8
1,028,077
New Zealand
11.8
13,755
Norway
7.3
27,852
Poland
13.5
58,143
Portugal
13.4
30,441
Slovak Republic
4.7
4,640
Slovenia
2.6
1,266
Spain
8.1
118,056
Sweden
7.4
35,307
Switzerland
101.2
496,957
Turkey
2.3
14,017
United Kingdom
73.0
1,589,409
United States
67.6
9,583,968
OECD 34
67.6
16,777,792
1.2. Pension scheme
After having seen the worldwide importance of the ageing trend and the unavoidable pension challenge for the coming years we present, in this section, general concepts used in pension theory.
1.2.1. Definition
A pension scheme can be defined as a systematic and organized mechanism, prescribed by law or by a convention, in order to provide after retirement regular incomes to a well defined category of people. This broad definition is based on the following keywords:
– Systematic: the benefits offered by a pension scheme are supposed to be defined by an objective rule and are not given “à la carte”. Two people in exactly the same situation will receive the same pension amount.
– Organized: a pension scheme must be initiated by a sponsor (a state, a private company, etc.) who is the guarantor of the continuity of the system.
– By law: when the sponsor is a state or an assimilated national or international institution (social security systems).
– By convention: when the sponsor is a private company or employer (occupational pension schemes).
– Regular incomes: the typical form of the benefits offered by a pension scheme is a lifetime annuity but sometimes other forms of benefits may exist (for instance, a lump sum paid at retirement age or temporary annuity).
– Category: the affiliates to the scheme are generally members of an objective category.
A pension scheme is therefore by definition a collective vehicle with precise rules, even if it may concern only a few people. Apart from this concept, individuals can also buy, during their active life, individual financial products or life insurances in order to increase their standard of living after retirement. This idea to accumulate different forms of pension arrangements, from the social security system to individual saving accounts, is a basic concept of the modern pension theory, sometimes known as the multi–pillar philosophy.
1.2.2. The four dimensions of a pension scheme
When considering setting up a pension scheme, various fundamental choices arise. In order to choose a specific pension scheme, we must first understand the alternatives that arise and measure the benefits and disadvantages of different systems.
We can roughly summarize the strategy of choosing a pension scheme in four basic questions to be answered:
– Dimension 1: for whom? Affiliates: who is affected by the plan?
– Dimension 2: by whom? Sponsor: who organizes and manages the system?
– Dimension 3: how much? Benefits: how much to provide?
– Dimension 4: how? Funding: how can we finance these pension benefits?
1.2.2.1. The dimension of the affiliates
The first question to ask is who is the population to be covered by the pension plan.
The table below shows some community types encountered in practice, showing the large variety in terms of magnitude:
POPULATION
EXAMPLE
1
The entire population of a country
Minimum pension income for all the citizens of a country
2
One category of workers of a country
Scheme for the civil servants
3
Members of a profession
Pension fund for accountants
4
One category of workers from one economic sector
Blue collar workers of the chemical industry
5
One category of workers of one private company
Workers of a company aged 25 or more
6
A key person of one company
CEO of the company
1.2.2.2. The dimension of the sponsor: public or private
The sponsor is the legal entity at the origin of the scheme. The table below shows some classical sponsors encountered in practice. Clearly the dimensions of the affiliates and of the sponsor are correlated.
SPONSOR
EXAMPLE
1
A State
General social security scheme
2
A region
Scheme for the public servants of one region
3
A public company
Scheme for a national electricity company
4
A group of private companies
Common scheme for the chemical industry
5
A company
Company XY
6
A professional body
Association of Notaries
1.2.2.3. The dimension of the benefits: DB or DC
There are two main logics in designing the benefits of a pension scheme:
– defined benefits pension scheme (DB schemes): “we say what we want to receive as benefits at retirement age”;
– defined contributions pension scheme (DC schemes): “we say what we want to pay as contributions during the active life”.
In a DB pension scheme , the pension incomes to receive after retirement are explicitly defined by the rules of the plan and are generally related to the salaries and the years of service of the affiliate. For instance, a pension plan could provide, after retirement age, fixed at 65, a pension income corresponding to 65% of the average of the last 5 salaries for 40 years of service in the company.
In a DC pension scheme , the contributions to be paid by the employer and/or the employee are explicitly defined by the rules of the plan and constitute the only liability of the sponsor. The benefits will be generated just by the accumulation of these contributions. For instance, a pension plan could provide a contribution of 5% of the salaries paid each year by the employer and 1% paid by the employee.
Clearly the sharing of risks between the affiliates and the sponsor are quite different in the two philosophies. This point will be illustrated in section 1.3.
1.2.2.4. The dimension of the funding: pay as you go or funding
In a DB pension scheme where benefits have been defined, we must finance these liabilities. In particular, we must compute a level of contributions in order to have an actuarial equilibrium between incomes and outcomes; this equilibrium is based on a long–term approach and not on a pure accounting point of view.
More precisely, the actuary will check on a certain time horizon and on a given population, the actuarial equivalence between the present value of the contributions and the present value of the benefits.
There are a lot of different actuarial techniques in order to finance a DB scheme depending on the way these discounted values are computed (definition of the time horizon and the population concerned). A major distinction is generally made between the pay as you go mechanism and funding techniques.
In a pay as you go mechanism (PAYG), the contributions paid one year by the active affiliates are directly used in order to pay the benefits to the retired people. Time horizon is typically one year. As a consequence, everybody is paying for somebody else! In particular, there are no (or nearly no) reserves in such a scheme.
The first condition in order to apply this technique is of course to be sure of the continuity of the sponsor. If one day, for one reason or another, the system must stop, clearly there is a big problem for the active affiliates! So in practice this technique can only be used for social security purposes. The ageing of the population, which was previously mentioned, is clearly a major concern for PAYG schemes.
On the other hand, PAYG systems are relatively protected against financial and inflation risks.
In a funding approach , the contributions paid by one generation are invested on the financial market and will be used by the same generation later upon retirement.
The time horizon can be very long and there are significant reserves. This technique can be used for any kind of sponsor. In case of stopping of the system, reserves exist in order to protect the rights of the affiliates. This method seems to be better adjusted against demographic risk (even if the longevity risk is also present as it will be explained in section 1.3). But inflation and market risks will also greatly affect the scheme. In particular, the search for a good investment strategy for the accumulated reserves is crucial.
This distinction between PAYG and funding is particularly meaningful for a DB pension scheme. But it can be also applied to a DC plan. When looking at a DC plan, it seems natural to associate a classical saving mechanism of individual accounts and therefore a funding approach. But DC schemes can also be based on pure PAYG techniques like notional accounts or point systems for social security purposes.
1.3. Pension and risks
Pension liabilities are exposed to a large variety of risks and the influence of these risks on the level of the benefits and on the funding mechanism may vary greatly from one scheme to another. For a given pension scheme, we must, for each kind of risk, answer two basic questions:
– Is the risk affecting the pension scheme and how much?
– If so, who is affected (the sponsor or/and the affiliate)?
We propose breaking the various risks down into two main categories: demographic risks and financial risks.
1.3.1. Demographic risks
By definition, a pension scheme is a collective vehicle covering a population.
Not surprisingly, the evolution of the affiliates and the beneficiaries is quite important. Among the demographic risks affecting the population of a pension fund, we have:
– the longevity risk: risk caused by the (unexpected) increase in the duration of human life; in a pension scheme this risk will typically increase the number of years of service of the benefits;
– the renewal risk: risk caused by a decrease in the number of entrances in the population; in a pension scheme this risk will typically decrease the amount of contributions;
– the lapse risk: risk caused by the number of people going out of the pension fund before retirement age (disability, dismissal, etc.).
1.3.2. Financial risks
In a pension scheme, we can also observe different financial risks:
– the market risk: risk caused by the variation of the market value of the financial assets underlying the pension benefits (stocks, investment funds, real estate, etc.);
– the interest rate risk: risk caused by the effect of a variation of the interest rates on the valuation of the assets and the liabilities of the pension scheme;
– the credit risk: risk caused by the change of rating of a counterparty or by a partial or total default of a debtor;
– the inflation risk: risk caused by the effect of inflation on the level of benefits to be paid by the fund.
1.3.3. Impact of the risks on various kinds of pension schemes
We will shortly analyze the effect of all these risks on 4 important kinds of pension schemes illustrated by the following table:
DB
DC
FUNDING
1
3
PAYG
2
4
1.3.3.1. DB funding scheme
If we consider a DB scheme using a funding technique where contributions are (mainly) paid by the sponsor (for instance an occupational scheme organized by a private company for its employees), clearly a lot of risks are supported by this sponsor (the company) and the affiliates (the employees) have nearly no risk. The main risks for the sponsor are the different financial risks and the longevity risk. Market and inflation risks are particularly crucial in this combination. The contributions computed actuarially and to be paid by the sponsor can largely fluctuate depending on the level of indexation of the benefits (inflation risk) and on the financial return achieved on assets (market and interest rate risks). For a pension scheme where benefits are paid by annuities after retirement, longevity risk has also a big influence on the actuarial valuation.
1.3.3.2. DB PAYG scheme
If we consider a DB scheme using a PAYG technique (for instance a social security scheme organized by a State for its civil servants), where contributions are (mainly) paid by the sponsor (the State), the main risk taken by the sponsor is now the demographic renewal risk. In a pure PAYG system without any assets, market, inflation and interest rates have no effect.
1.3.3.3. DC funding scheme
If we consider a DC scheme using a funding technique where contributions are paid by the sponsor and/or the affiliates (for instance an occupational scheme based on pension individual saving accounts), a lot of risks are now supported by the affiliates themselves. The only liability for the sponsor is to pay the fixed level of contributions. The main risks for the affiliates are all the financial risks (bad returns on the saving account of the affiliate or a significant inflation) and the longevity risk. To switch from a DB scheme to a DC scheme does not imply a removal of the risks but just their transfer from the sponsor to the affiliates!
1.3.3.4. DC PAYG scheme
The logic is the same if we consider a DC scheme using a PAYG mechanism (for instance a social security system based on notional accounts). The level of contributions to be paid is fixed but the level of benefits to pay to the retired people can fluctuate due to the influence of all the demographic risks (longevity and renewal risks).
1.3.4. The time horizon of a pension scheme
Apart from the typology of risks, another main characteristic of pension liabilities is the long time horizon. A young worker aged 20 and entering into a pension fund where benefits are paid by annuities after retirement age, can stay in this fund for more than 60 years!!! Pension governance is definitively a long term business, by essence very different from short term finance. Risk measures applied to pension funds must definitively take into account this stylized fact. One year for instance (as used for solvency of insurance companies or banks) is not a consistent time horizon for a pension fund.
1.4. The multi–pillar philosophy
Diversification is probably one of the most popular concepts in finance. Here we can apply the familiar sentence: do not put all your eggs in the same basket. The same kind of concern can be found in pension risk management.
In many countries, the pension landscape is based not only on one kind of system but rather on a combination of various schemes presenting different characteristics and different typologies of risk.
From an historical point of view, the origin of this philosophy can be found in the well known theory of the three pillars.
In this framework, a well organized system of pension in a country should be based on the superposition of three pillars; everybody should obtain at retirement age incomes from these three sources in order to achieve a good standard of living:
– First pillar: a social security scheme organized by the state for all the workers and based classically on a DB PAYG system.
– Second pillar: occupational schemes organized by companies for their employees, always based on a funding technique and using a DB or a DC design.
– Third pillar: individual pension schemes decided by the individual (life insurance, saving products, etc.).
The relative importance of these three pillars can of course vary from one country to another depending on the political importance given to the social security versus the private initiative. The final purpose is to mix DB and DC, funding and PAYG, collective and individual, state and private, assistance and insurance in order to have, at the same time, a good level of pension and a decrease of the risks.
New forms of “pillar theory” have emerged in recent years, sometimes with more than three pillars, introducing for instance a “pillar zero” of public assistance not only for workers but for all citizens. This multi–pillar philosophy has been especially advocated by the World Bank.
Chapter 2
Classical Actuarial Theory of Pension Funding
2.1. General equilibrium equation of a pension scheme
2.1.1. Principles
Any contributory pension scheme is based on two kinds of financial cash flows:
– contributions generally paid by active workers (or by their employer);
– benefits paid to retired people.
In a contributory mechanism the aim of the actuary is to find a global equilibrium between these two global quantities. This general principle is closed to classical static accounting where assets must correspond to liabilities. But the dimension of time must be taken into account and induce an infinite number of possible equilibrated techniques as will be shown later.
More precisely, pension fund equilibrium must incorporate two actuarial effects:
1) The discount effect : contributions and benefits of a pension scheme are by definition spread over a long time horizon; we have to consider some equivalence between the present value of contributions and the present value of benefits.
2) The population effect: a pension fund is not an individual saving account but a collective vehicle. In order to obtain an actuarial equivalence, we must describe a way to group the entire population into sub–groups where people are joined together and are completely interdependent in the financing of their own pension. These subgroups will be called the “community of risk”.
We are now able to rephrase the classical accounting equilibrium applied to a pension fund:
“A pension fund mechanism is based on the equivalence for each community of risk between the present value of their contributions and the present value of their benefits”.
The general equilibrium equation will be as follows:
This last equality must be fulfilled for each community of risk.
The different funding techniques will generate different community of risk forms.
2.1.2. The retrospective reserve
We consider a pension scheme characterized by the following simple assumptions.
(A1):Demographic assumptions
The population affiliated to the pension scheme is broken down into two categories assuming a single retirement age:
– the active population (contributing): people aged x0 , x0 +1,…., xr −1;
– the retired population (receiving benefits): people aged xr, xr +1,….,ω.
In general we will denote:
This population is said to be closed if a time horizon T exists, such that:
A non–closed population (perpetual entrances) is said to be open.
In first pillar pension schemes, because of the dynamic aspect of the population of a whole country, open populations are generally considered. On the contrary, closed populations are the rule for second pillars. In this section, formulas can be applied to both cases.
(A2):Financial assumptions
We will assume the same salary for every active worker (mean salary):
and the same contribution rate for people of the same age:
The discount rate will be assumed to be deterministic and constant, equal to i.
(A3): Pension plan elements:
We will denote by:
We can then define the following aggregate elements for the entire population:
– the total wages:
– the total contributions:
– the total benefits:
Then the retrospective reserve equation is given by:
[2.1]
2.1.3. The prospective reserve
The retrospective reserve is just looking at past elements.
On the contrary, the prospective reserve is the necessary amount in order to pay future benefits taking into account future contributions still to receive. The equation then becomes:
[2.2]
In a defined benefit pension scheme, the future value of the benefits can be determined; the future contributions have to be computed following an actuarial funding mechanism.
In a defined contribution pension scheme, the future value of the contributions is known and the benefits must be computed following an actuarial principle.
2.1.4. Equilibrated pension funding
An actuarial mechanism will be admissible if prospective and retrospective reserves are equal:
or taking into account [2.1] and [2.2]:
[2.3]
In a defined benefit pension scheme, this relation must help us to compute the vector of the future contributions C(s) (st,t+1,…). So we have an infinite number of possible equilibrated actuarial methods depending on the time structure of this vector.
In a defined contribution pension scheme, the future contributions are known in a deterministic model, and there are also many ways to compute benefits taking into account equilibrium condition [2.3].
Considering a defined benefit scheme, formula 2.3] generates, for instance, two extreme actuarial funding methods if we add some constraints.
Method 1: pure PAYG: the reserve V is always equal to zero.
Method 2: initial funding: the future contributions are equal to zero. In PAYG, condition [2.3] becomes:
Assuming a constant contribution rate (independent of age), we obtain:
[2.4]
which is an admissible solution to general equilibrium condition [2.3].
In this method, a community of risk is all the active and retired people alive in a given year.
In the initial funding method, we compute an initial reserve V(0), such that no future contributions are needed. This method can be considered as the opposite to PAYG because inducing now a maximum level of reserve.
The elements of this technique are given by:
[2.5]
In this method there is only one global community of risk: all the present and future active and retired people!
Between these two extreme techniques (pure PAYG with, no reserve; and initial funding with, maximum reserve)) a lot of intermediate actuarial methods exist, depending on the required level of reserve.
2.1.5. Decomposition of the reserve
The total reserve V can be shared between active and retired affiliates.
[2.6]
By definition, the reserve for retires is defined as the amount necessary to pay all the future pensions of the people already retired:
[2.7]
This reserve does not depend on the choice of the funding method.
Then the reserve for active people is just the difference between the total reserve and the reserve for retires:
[2.8]
This active reserve can become negative if the total reserve is not sufficient with respect to the existing liabilities for people already retired; this is, for instance, the case in pure PAYG. In fact the sign of this reserve can help us to classify the various actuarial techniques.
2.1.6. Classification of the methods
Depending on the sign of the reserve for active people, two important families of actuarial techniques can be defined:
1) PAYG family : methods with strictly negative active reserves.
For instance in pure PAYG (total reserve equal to zero), the active reserve is given by the following amount (lowest possible level!):
Other actuarial methods exist in this PAYG family, generating positive total reserves that are not sufficient to finance the reserve for retired people.
2) Funding family : methods with strictly positive active reserves.
In these techniques, the retirees have enough reserve to finance all their future pensions, and on top of this, active people have their own positive reserve.
The initial funding technique (see [2.5]) is a member of this family. In this case, the reserve for active people corresponds to the present value of all the future benefits to be paid (highest possible level!).
2.2. General principles of funding mechanisms for DB Schemes
As explained in section 2.1.6, funding mechanisms are a particular class of actuarial methods where the reserves for retired people are fully financed and where active people have their own reserve.
In a DB pension scheme, a funding method is a way to define, each year, the level of the contributions to be paid for the active affiliates.
In every funding mechanism, the following elements must be defined.
1) Actuarial Liability (AL(t))
The level of reserve to be detained by the pension fund and fixed by the chosen funding method (liability part). The actuarial liability is the result of an actuarial theoretical computation based on various assumptions (discount rate, demographic evolution, future salaries, etc.) and the use of a particular funding method as presented in the next section. It corresponds for a funding mechanism to the general concept of prospective reserve as presented in section 2.1.3.
2) Fund (F(t))
The market value of the assets covering the actuarial liability (asset part).
3) Normal cost (NC(t)):
The basic level of the contributions defined by the funding method used by the pension fund.
As the actuarial liability, the normal cost is computed by the actuary.
Actuarial liability and normal cost are related by the following general formula:
[2.9]
4) Unfunded Actuarial Liability (UAL(t))
The difference between the actuarial liability (reserve that we should have) and the fund (reserve that we really have):
[2.10]
This amount can be positive (deficit — underfunding) or negative (surplus — overfunding).
5) Adjustment (ADJ(t))
Supplemental contributions to be paid in order to finance the unfunded actuarial liability (amortization of surplus or deficit).
For instance, in the spread method a amortization period m is chosen and the adjustment is then given each year by:
[2.11]
6) Total contribution (C(t))
Sum of the normal cost and the adjustment:
[2.12]
2.3. Particular funding methods
We now present some basic funding methods generally used by pension funds.
From a theoretical point of view there are an infinite number of possible techniques, but in practice only some specific methods have been developed and really used.
We will consider the three most important actuarial techniques in practice: unit credit cost,level premium and aggregate cost.
2.3.1. Unit credit cost methods
Unit credit cost methods are individual funding techniques where normal costs and actuarial liabilities are computed separately for each affiliate and then combined. Here we develop the computation of the normal cost and the actuarial liability for one given affiliate.
We will use the following general notations for this affiliate:
So the pension benefit at retirement age will be given by:
This benefit is financed by normal costs that are assumed to be paid annually in advance:
Actuarial liabilities are computed at the end of the year:
The final actuarial liability at time tT is then given (for every actuarial method!) by:
The unit credit cost philosophy is sometimes called “liability driven” because the fundamental purpose of the method is to generate a well defined level of actuarial liability based on the complete funding of the part of the pension corresponding to the service accrued to date. The normal cost is then a consequence of this goal.
Inside this family, a distinction can be made between projected and unprojected methods.
In the unprojected version, we use the level of the salary as known at the time of valuation, without taking into account any future evolution. In the projected version, an estimation of the final salary is made using a salary scale and assumptions on future inflation.
a) Unprojected unit credit cost
Following the philosophy of the method, we first fix the required level of actuarial liability:
[2.13]
Then the normal cost NC(t) is the amount to pay in order to go from level AL(t) to level AL(t+1):
[2.14]
Substituting the value of the actuarial liabilities given by [2.13] in [2.14] we obtain:
[2.15]
The first part of the normal cost is the new unit of the year; the second part represents the cost of inflation on past services.
b) Projected unit credit cost
Instead of working with current salaries, the projected version is based at any moment on estimated future salaries. We will use the notation:
For instance using a constant increasing factor g till retirement, we obtain as an estimator:
Formulas [2.13] and [2.15] of actuarial liability and normal cost become respectively:
[2.16]
[2.17]
In particular, if the projections are not changed between t−1 and t (the same estimation of final salary), the normal cost has only one component (i.e. the unit of the year):
2.3.2. Level premium methods
Level premium methods are also individual funding techniques, but this time they are “contribution driven” in the sense that the constraint is now put on the structure of the contributions and the level of the actuarial liability is just a corollary. The main purpose of these methods is to obtain a stability of the contributions in some sense. Important similarities can be seen with classical life insurance contracts with annual premiums. Once again, this technique can be developed in an unprotected or a projected environment.
a) Unprojected level premium
In this technique we first compute the normal cost, using a recursive scheme.
[2.18]
Then each year this contribution must be adapted, taking into account the increase of salaries in one year:
[2.19]
The actuarial liability results directly from this form of normal cost and can be calculated using the general formula [2.9], giving here:
[2.20]
b) Projected level premium
In the projected level premium method, we use at any time an estimation of the final salary.
For instance, the initial normal cost [2.18] becomes:
[2.21]
The recursive scheme is given now by:
[2.22]
In particular, if the estimator of final salary is not changed between t–1 and t the normal constant remains constant and the method is then a real level technique:
The actuarial liability is a direct generalization of formula [2.20]:
[2.23]
c) Projected level percent
The stability of the contributions obtained in the last two methods is certainly not optimal even if it was initially the aim of the techniques!
In the unprojected method, formula [2.18] shows that the contribution is not constant at all as soon as the salary increases. The projected method seems better, especially when final salary estimations are not changed (see [2.22]) but constant normal costs are not really meaningful when salaries are increasing! It seems better to require stability of the individual contribution rate defined as the ratio between the individual normal cost and the salary:
[2.24]
In the projected level percent method, we try to achieve a stable contribution rate (stability of the normal cost in percentage of the salary). By definition, this method is projected.
Initially at time t=0, the first contribution rate is the solution of the following actuarial equivalence:
or:
[2.25]
with:
If we use a constant increasing factor for the future salaries:
then the contribution rate becomes:
[2.26]
2.3.3. Aggregate cost methods
Aggregate cost methods are collective funding techniques. No individual contribution or actuarial liability is computed. In this method there is also by definition no distinction between the actuarial liability and the fund. The aim of the method is to express the contribution rate as a constant fraction of the global payroll.
The aggregate cost method is based on the following actuarial equivalence:
The (collective) normal cost is then given by:
More precisely, working with the same standard pension plan as developed before, we introduce the following notations:
Sj(t|s)= projected salary at time t seen from time s ( s<t) for affiliate j
xr =retirement age
Then the actuarial equilibrium equation can be written:
where:
F(0)= initial available reserve
π(0)= initial contribution rate
giving the following formula of the contribution rate in aggregate cost:
[2.27]
If experience is coherent with assumptions, then by definition, the contribution rate can remain stable and the next normal costs become:
But in practice, there are always some differences between assumptions and reality!
Then a new contribution rate must be computed according to the following formula:
where the elements are defined as before but estimated now at time t.
In particular the fund F(t) can be defined recursively by:
We can observe in this method that no explicit actuarial liability is defined.
In fact and by definition, the actuarial liability in aggregate is equal to the fund level:
Chapter 3
Deterministic and Stochastic Optimal Control
3.1. Introduction
In this chapter, we introduce the minimal basic results concerning control theory, both in the deterministic and stochastic cases in such a way that this presentation will be comprehensive for non-specialists.
3.2. Deterministic optimal control
3.2.1. Formulation of the optimal control problem
Let us consider a dynamic economic or physical system S on the time interval [ta, tb ] and whose state is at any time t characterized by the column vector:
[3.1]
of n, called the state vector or simply the state of system S.
At any time t it is possible to control the system, and this control is mathematically represented by a vector u(t) of r:
[3.2]
The dynamic evolution of S is governed by the following differential system:
[3.3]
or with vector notations:
[3.4]
The column vector q being
[3.5] .
The interest and the principal difficulties of the optimal control theory come from the fact that a lot of constraints exist for the vector control u.
In the most general form, this vector must at any time be a set U, depending both of timet and state x:
[3.6]
the sets U(x,t) are defined a priori following the practical problem to be solved.
Every control vector satisfying relation [3.6] is called admissible.
The objective is to optimize, for example to minimize, a given functional J pending both of the vectors x and u of the general form:
[3.7] .
We must thus find, if possible, an admissible control vector u* such that:
min J(u)
[3.8]
The set U, called the admissibility set, giving all the admissible controls is:
[3.9] .
Finally, we must give an initial condition for the state at the initial time:
[3.10] .
A problem is well-posed if for every admissible control on [ta,tb ]a unique trajectory exists for S, in fact a solution of the Cauchy problem for the differential system [3.4] with [3.10] as the initial condition.
In this case, a unique optimal trajectory x* corresponds to the optimal control u*, which is a solution of the Cauchy problem:
[3.11]
with [3.10] as the initial condition.
To guarantee such a good situation, it is of course necessary to do some basic assumptions given below:
(i) the functions r and q are continuous diiierentiable in their respective domains of definitions contained in n × r × +;
(ii) the components of vectors u are piecewise continuous on [ta,tb,]i.e. of class D0[ta,tb] with an almost finite number of first class discontinuities.
Under such assumptions it is possible to find necessary conditions on u using dynamic programming [BEL 57] or the maximum principle [PON 62].
We choose the first approach as it seems more intuitive and by discretization it leads to good numerical approximations.
