Practical Financial Optimization E-Book

Soren S Nielson

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Serie: Wiley Finance Series
• Sprache: Englisch

Seitenzahl: 202

Veröffentlichungsjahr: 2010

Beschreibung

In Practical Financial Optimization: A Library of GAMS Models, the authors provide a diverse set of models for portfolio optimization, based on the General Algebraic Modelling System. 'GAMS' consists of a language which allows a high-level, algebraic representation of mathematical models and a set of solvers – numerical algorithms – to solve them. The system was developed in response to the need for powerful and flexible front-end tools to manage large, real-life models.

The work begins with an overview of the structure of the GAMS language, and discusses issues relating to the management of data in GAMS models. The authors provide models for mean-variance portfolio optimization which address the question of trading off the portfolio expected return against its risk. Fixed income portfolio optimization models perform standard calculations and allow the user to bootstrap a yield curve from bond prices. Dedication models allow for standard portfolio dedication with borrowing and re-investment decisions, and are extended to deal with maximisation of horizon return and to incorporate various practical considerations on the portfolio tradeability. Immunization models provide for the factor immunization of portfolios of treasury and corporate bonds.

The scenario-based portfolio optimization problem is addressed with mean absolute deviation models, tracking models, regret models, conditional VaR models, expected utility maximization models and put/call efficient frontier models. The authors employ stochastic programming for dynamic portfolio optimization, developing stochastic dedication models as stochastic extensions of the fixed income models discussed in chapter 4. Two-stage and multi-stage stochastic programs extend the scenario models analysed in Chapter 5 to allow dynamic rebalancing of portfolios as time evolves and new information becomes known. Models for structuring index funds and hedging interest rate risk on international portfolios are also provided.

The final chapter provides a set of 'case studies': models for large-scale applications of portfolio optimization, which can be used as the basis for the development of business support systems to suit any special requirements, including models for the management of participating insurance policies and personal asset allocation.

The title will be a valuable guide for quantitative developers and analysts, portfolio and asset managers, investment strategists and advanced students of finance.

Leseprobe

Table of Contents

Title Page

Copyright Page

Dedication

Preface

Acknowledgements

Notation

Sets and Indices

Variables and Parameters

Glossary of Symbols

Abbreviations

Chapter 1 - An Introduction to the GAMS Modeling System

1.1 Preview

1.2 Basics of Modeling

1.3 The GAMS Language

1.4 Getting Started

Notes and References

Chapter 2 - Data Management

2.1 Preview

2.2 Basics of Data Handling

2.3 Data Generation

2.4 A Complete Example: Portfolio Dedication

Chapter 3 - Mean-Variance Portfolio Optimization

3.1 Preview

3.2 Basics of Mean-Variance Models

3.3 Sharpe Ratio Model

3.4 Diversification Limits and Transaction Costs

3.5 International Portfolio Management

Chapter 4 - Portfolio Models for Fixed Income

4.1 Preview

4.2 Basics of Fixed-Income Modeling

4.3 Dedication Models

4.4 Immunization Models

4.5 Factor Immunization Model

4.6 Factor Immunization for Corporate Bonds

Chapter 5 - Scenario Optimization

5.1 Preview

5.2 Data sets

5.3 Mean Absolute Deviation Models

5.4 Regret Models

5.5 Conditional Value-at-Risk Models

5.6 Utility Maximization Models

5.7 Put/Call Efficient Frontier Models

Chapter 6 - Dynamic Portfolio Optimization with Stochastic Programming

6.1 Preview

6.2 Dynamic Optimization for Fixed-Income Securities

6.3 Formulating Two-Stage Stochastic Programs

6.4 Single Premium Deferred Annuities: A Multi-stage Stochastic Program

Chapter 7 - Index Funds

7.1 Preview

7.2 Models for Index Funds

Chapter 8 - Case Studies in Financial Optimization

8.1 Preview

8.2 Application I: International Asset Allocation

8.3 Application II: Corporate Bond Portfolio Management

8.4 Application III: Insurance Policies with Guarantees

8.5 Application IV: Personal Financial Planning

Bibliography

Index

This edition first published 2009

© 2009 Stavros Zenios, Andrea Consiglio and Søren S. Nielsen†

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Preface

Where the spirit does not work with the hand, there is no art.

Leonardo Da Vinci

This is a book about art: the art of modeling financial decision making using optimization. The science of financial optimization models has been introduced in the companion volume by S.A. Zenios, Practical Financial Optimization: Decision Making for Financial Engineers (Blackwell Publishing, Cambridge, MA, 2007), henceforth abbreviated as PFO.

In this book the reader’s spirit works closely with his hand to create models. The reader is expected to have an understanding of creativity and to possess the skills and tools necessary to illustrate his or her ideas of business reality, finance concepts, and market expectations. The reader who has an unorthodox understanding of the problem is able to create a model that can act as a portal representing the realities of his or her own specific problem. This portal enables the user to perceive things from a decision maker’s perspective, rather than from a broad and abstract perspective. In any event, the model does not teach a precise way of perception. The decision maker must maintain open-mindedness in order to question what has been created by the model, and how that interacts and relates with the decision making problem at hand. The decision maker comes to realize the great number of possibilities that are layered within any model.

Of course these possibilities must be well grounded in currently accepted theories of financial economics, while at the same time they should work well in practice and provide all the ‘bells and whistles’ required by the user. To paraphrase another Da Vinci quote, a good model must be a masterpiece of engineering and a work of art. But these issues were addressed in PFO. The current book leads the reader from the discussion of PFO to functional models implemented in the high-level algebraic modeling language GAMS - a General Algebraic Modeling System - of Brooke, Kendrick and Meeraus (1992). All models discussed in this book are available through the library FINLIB, which can be accessed from http://www.gams.com/finlib. The library provides instantiations of all the models discussed here, complete with market data. Users can readily substitute their own data in a model, or can use FINLIB to find the building blocks for their own models.

All the models in FINLIB can be run on any computer that supports GAMS and, therefore, are absolutely independent from the machine or the operating system. A student version of GAMS can also be downloaded free of charge from the web site. This version provides the full GAMS functionality, so that FINLIB models can be compiled by it. However, size restrictions apply to the number of variables and equations that can be solved. Therefore, large-scale instances of FINLIB models, with many asset classes, time-periods, scenarios, and so on, cannot be solved with the student version.

An explanation of the precise link between PFO, the current book, and FINLIB is in order. PFO introduces the general concepts and theories of financial optimization models that are used by financial engineers; it can be used as an introduction to the subject of financial modeling or as a stand-alone reference. The current book describes technical concepts relating to the implementation of GAMS models and discusses the implementation of several models from PFO, either verbatim or with small technical modifications. However, not all models in PFO are implemented here, although sufficient information is provided so that any additional models can be easily implemented by the reader. It is expected that this book will be used in conjunction with PFO, and references are made herein to PFO using the notation PFO-m.n.p where m.n.p refers to PFO labels. For instance, “Section PFO-6.3” refers to Section 3 of Chapter 6, and “Model PFO-6.4.2” refers to the second model of the fourth section of Chapter 6; readers will easily locate the cited material. Finally, FINLIB provides the software and the data required to instantiate all the models discussed in this book. There is a one-to-one correspondence between the models in the current book and FINLIB, and a strong relationship of all the models in FINLIB with PFO.

Organization of the Book

The organization of the chapters in this book closely follows the structure of PFO, with two introductory chapters, five chapters on optimization models and a chapter with several case studies.

The introductory part consists of Chapters 1 and 2, and gives an introduction to the GAMS modeling system and discusses data management. These chapters can be skipped by readers familiar with GAMS.

The bulk of the optimization models are given in the five chapters corresponding to the PFO chapters on portfolio optimization models; here we also develop one or more GAMS models for each model class. This part is the core of the book. It develops GAMS models for supporting the classic mean-variance analysis in Chapter 3, for fixed-income portfolios in Chapter 4, for scenario optimization in Chapter 5, for dynamic portfolio optimization using stochastic programming in Chapter 6, and for creating indexed portfolios in Chapter 7.

Finally, Chapter 8 deals with case studies and develops GAMS models for diverse real-world applications. The four sections in this chapter develop complete models for all the applications discussed in PFO: international asset allocation, corporate bond portfolio management, insurance policies with guarantees and personal financial planning. The models in this part capture policy restrictions, regulatory requirements, business objectives, and similar practical considerations, and they come complete with real-world data.

Acknowledgments

Some of the models developed in this book are based on our own published research, and we wish to express our appreciation to all colleagues and students from whom we have learned a great deal. Their contributions are gratefully acknowledged without implicating them in the final product.

Alex Meeraus, unquestionably the father of algebraic modeling languages and the founder of GAMS Development Corporation, has been not only a constant source of encouragement and a solid sounding board for ideas, but he also added a helping hand in compiling the library that accompanies this volume.

Over the years we have benefited from interaction with numerous collaborators. They all have had a significant impact on our work with financial modeling, and we thank them in strictly alphabetical order: Andrea Beltratti, Marida Bertocchi, Flavio Cocco, Rita D’Ecclesia, Rosella Giacometti, Ben Golub, Martin Holmer, Norbert Jobst, Roy Kouwenberg, Franz Nelissen, David Saunders, Nicholas Topaloglou, Hercules Vladimirou.

Sadly, during the writing of this book our colleague, student, and good friend Søren S. Nielsen passed away. His fingerprints are everywhere in this book, from the writing of the introduction to the GAMS modeling system, from the discussion of the models in several chapters, to the GAMS implementation of many models. The long delay in completing this volume is a sign of the great value that Søren added to the team, and his name features deservedly as a coauthor. But what was felt most was the absence of his great sense of humor during the long hours we toiled, occasionally together with Alex Meeraus, to complete this book and the accompanying FINLIB library.

This work was funded in part through European Commission contract ICA1-CT-2000- 70015 establishing the HERMES Center of Excellence on Computational Finance and Economics , and also through research grants from the National Science Foundation (USA), Consiglio Nazionale delle Ricerche (Italy) and the Cyprus Research Promotion Foundation. Andrea Consiglio was partially supported by the research grant PRIN 2007TKLTSR.

Sincere appreciation is due to the Wiley editor Caitlin Cornish and assistant Aimee Dibbens who saw this project through with the utmost professionalism and good humor. Thanks for editing go to Kathy Stephanides; she did a superb job, thus allowing us to take the full credit for any remaining errors.

Palermo and Nicosia

Andrea [email protected] A. [email protected]

Notation

We use throughout this book the notation introduced in S.A. Zenios, Practical Financial Optimization: Decision Making for Financial Engineers, (Blackwell Publishing, Cambridge, MA, 2007), abbreviated as PFO. To the extent possible the same notation is adopted in the accompanying software. Cross-references to chapters, sections, models, and so on from PFO are given using the notation PFO-m.n.p where m.n.p refers to PFO labels. For instance, “Section PFO-6.3” refers to Section 3 of Chapter 6, and “Model PFO-6.4.2” refers to the second model of the fourth section of Chapter 6. Readers will easily locate the cited material in PFO.

The current GAMS system, which includes the FINLIB, is available at: http://www.gams.com/download/.

The FINLIB is available at: http://www.gams.com/finlib/.

Sets and Indices

i index of instrument or asset class from the set U .

t index of time periods from the set T .

j index of risk factor from the set K.

l index of scenario from the set Ω.

Variables and Parameters

xn-dimensional vector of investments in assets, with elementsxi . The units are in percentages of the total asset value or amounts in face value; the choice of units depends on the model and is made clear in the text.

b0n-dimensional vector of initial holdings in assets, with elements b0i .

cash invested in short-term deposits at period t .

cash borrowed at short-term rates at period t .

v0 initial holdings in risk-free asset (cash).

pl statistical probability assigned to scenario l.

n-dimensional random vector of asset returns, with elementsi.

rln-dimensional vector of asset returns in scenario l, with elements ri ..

tn-dimensional random vector of asset returns at period t , with elementsti .

t n-dimensional vector of asset returns at period t in scenario l, with elements

rft spot rate of return of the risk-free asset at period t.

n-dimensional random vector of cashflows from assets, with elementsi.

Fln-dimensional vector of cashflows from assets in scenario l, with elements

Ftn-dimensional random vector of cashflows at period t , with elementsti.

n-dimensional vector of cashflows from the assets at period t in scenario l, with elements

n-dimensional random vector of prices of assets, with elementsi.

Pln-dimensional vector of prices of assets in scenario l, with elements

tn-dimensional random vector of prices at period t , with elementsti

n-dimensional vector of prices of assets at period t in scenario l, with elements

n-dimensional random vector of ask prices at period t, with elementsIn order to buy an instrument the buyer has to pay the price asked by traders.

n-dimensional random vector of bid prices at period t, with elementsIn order to sell an instrument the owner must accept the price at which traders are bidding.

n-dimensional vector of ask prices at period t in scenario l, with elements

n-dimensional vector of bid prices at period t in scenario l, with elements

random variable of the total return of a benchmark portfolio or a market index.

Il total return of a benchmark portfolio or a market index in scenario l.

t random variable liability due at period t.

value of the liability in scenario l.

Q a conformable covariance matrix.

σii’ covariance of random variables indexed by i and i’.

ρii’ correlation of random variables indexed by i and i’.

maximum holdings in asset i.

xi minimum holdings in asset i.

Glossary of Symbols

ε[] expectation of the random variable or vectorwith respect to the statistical probabilities pl assigned to scenarios l ∈ Ω.

EΡ [] or Eλ[] expectation of the random variable or vectorwith respect to the probability distribution Ρ or the probabilities λ ∈ Ρ .

U(a) utility function with arguments over the real numbers a.

mean value of a random variable or vector.

R(x;) portfolio return as a function of x with parameters.

V (x;) portfolio value as a function of x with parameters. max[a,b] the maximum of a and b.

Prob (= r) the probability that the random variable argumenttakes the certain value r .

I a conformable identity matrix.

1 conformable vector with all components equal to 1.

Abbreviations

ALM Asset and liability management.

APT Arbitrage Pricing Theory.

CAPM Capital Asset Pricing Model.

CBO Collateralized bond obligation.

CEexROE Certainty equivalent excess return on equity.

CEROE Certainty equivalent return on equity.

CLO Collateralized loan obligation.

CRO Chief risk officer.

CVaR Conditional Value-at-Risk.

EWRM Enterprise-wide risk management.

FHA Federal Housing Association.

LTCM Long Term Capital Management.

MAD Mean absolute deviation.

MBS Mortgage-backed security.

OAP Option adjusted premium.

OAS Option adjusted spread.

PFOPractical Financial Optimization: Decision Making for Financial Engineers, Blackwell Publishing, Cambridge, MA, 2007, by S.A. Zenios.

PSA Public Securities Association.

ROE Return on equity.

SPDA Single premium deferred annuities.

VaR Value-at-Risk.

GAMS General Algebraic Modeling System

IDE Integrated Development Environment.

Chapter 1

An Introduction to the GAMS Modeling System

1.1 Preview

In this chapter we introduce the high-level algebraic modeling language that will be used in the rest of this book to build financial optimization models. The basic elements of the language are given first, together with details on getting started with the language, and the FINLIB library of models are also discussed here.

1.2 Basics of Modeling

Optimization is concerned with the representation and solution of models. Models can be represented in a number of ways, and they can be solved using a number of methods or algorithms. The General Algebraic Modeling System, GAMS, is a system for formulating and solving optimization models. It consists of a language that allows a high-level, algebraic representation of mathematical models, and provides a set of solvers, i.e., numerical algorithms, to solve them.

Why use algebraic modeling? Small models are easy to formulate and solve. They have a simple structure and one can simply edit a file containing the model’s coefficients, and then call a standard linear programming solver to solve it. In fact, in the early days of optimization, models were solved using specialized matrix generators that provided the necessary input files for solvers. However, as models grow larger and become more complex, they become difficult to manage and to verify using this approach. GAMS was developed in response to the need for powerful and flexible front-end tools to manage large, real-life models. Large collections of data and models are only manageable when they possess structure, and algebra provides this structure in a well-known mathematical notation.

Conceptually, a model consists of two parts: The algebraic structure and the associated data instance. The formal linear programming model has the associated data A, b, c; see Appendix PFO-A for optimization basics. GAMS provides an algebraic framework for defining and manipulating data as well as building the models that use them. In addition to being concise and easily readable, the GAMS statement of a model is machine-independent and allows interaction with a number of solvers. Hence, it is not dependent on any particular optimizer or computer system.

(1.1)

(1.2)

(1.3)

The GAMS System consists of the GAMS compiler and a number of solvers. The compiler is responsible for user-interaction, by compiling and executing user commands given in a GAMS source file. A source file can implement a simple textbook problem, or it can represent a large-scale system consisting of several interrelated optimization models. The solvers are stand-alone programs that implement optimization algorithms for the different GAMS model types. The GAMS system can be called from the command line (for instance, a Unix prompt), or through the Integrated Development Environment (IDE), a windows-based environment that facilitates modeling by integrating editors with the GAMS system.

Section 1.3 gives an introduction to the GAMS language and Section 1.4 is a guide to quickly become accustomed to using GAMS. Readers who want a quick overview of GAMS modeling may start by reading Section 2.4, which contains a complete example drawn from financial planning.

1.3 The GAMS Language

Optimization models and their associated data are communicated to the GAMS system using a general-purpose language with elements of both ordinary programming languages (data declarations, control structures), and high-level modeling structures such as sets, equations, and models.

GAMS models are typically structured with the following building blocks:

1. Sets, which form the basis for indexation and serve as building blocks for data and model definitions.

2. Data, which are specified, either through direct statements (perhaps included in external files), or by calculating derived data.

3. Variables and constraints, which are used to define models, equations, and an objective.

4. An output section is sometimes used where the final results are calculated and presented.

In the remainder of this section we give an introduction to the elements of the language. In addition to the above-mentioned items, the main topics are expressions, which are used in assignment statements and in constraint declarations, and control structures, which lend programming language capabilities to GAMS. Readers who want to see a larger, complete modeling exercise may skip ahead to Section 2.4, and then refer back to this section for coverage of advanced features of the language when ready to embark on more substantial models.

1.3.1 Lexical conventions

A GAMS source file is an ordinary text file. The first character position on each line indicates how the line should be interpreted:

* (asterisk): A comment line, ignored by GAMS.

$ (dollar sign): Indicates a compiler directive or option. A list of the most common $-controls is given in Table 1.1. Many more exist than can be covered here; consult the User’s Guide (see Notes and References at end of chapter) for complete information.

Table 1.1: The most common $-control commands. See Section 1.3.10 for examples of the use of $SET, $IF, $LABEL and $GOTO for conditional compilation.

Any other character indicates a model source line. Customarily such lines start with a space character.

The main lexical elements of a GAMS statement are keywords, identifiers, and operators. The example model shows keywords in all capital letters, and identifiers use a mixture of upper and lower case, but the language is not case sensitive. Identifiers consist of letters, digits, or the underscore character, “-”, and must begin with a letter (in early versions of GAMS, identifiers were limited to at most 10 characters).

The GAMS examples that follow use comments of the standard kind (“*” in position 1), but in addition assume that text that starts with the sequence “#” to the end of the current line is regarded as a comment. Hence, the command is assumed to be in effect in all examples. C++ and Java programmers might prefer to use the more familiar

1.3.2 Sets

The primary tool for structuring large-scale models is the set. In any nontrivial model we will need to use data, variables, and constraints that are indexed, and sets form the basis for such indexing.

The simplest set declarations have the form: These lines declare the two sets Time and Bonds. The set Time contains the elements from 2002 through 2006 (the asterisk indicates filling out the intervening elements; one could have written this: SET Time /2002, 2003, 2004, 2005, 2006/;). Similarly, the Bonds set contains bonds named GOVT_1 through GOVT_4.

The ALIAS statement is a convenient way to declare indices to be used in connection with sets. The code above binds the name t to the set Time, and the names i and j to the set Bonds. These names can henceforth be used as indices into their associated sets (and only those sets).

The text “Bonds universe” in the Bonds declaration is an explanatory text that GAMS outputs in the listing whenever it lists the Bonds set, as a help in documentation. Such texts can occur in all declarations and can be a great help when reading GAMS listings. They need not be enclosed in quotation marks, but if they aren’t then they cannot contain certain characters, which can lead to quite subtle syntax errors.

Indices and indexation

Most GAMS modeling elements (data, variables, etc.) can be indexed, with up to 10 indices. For instance, a two-dimensional parameter F can be defined over the sets declared above as (more on data declarations in Section 2.2.1):