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QUANTITATIVEMETHODS FOR
FINANCE ANDINVESTMENTS
QRMA01 30/08/2005 12:30 PM Page i
QRMA01 30/08/2005 12:30 PM Page ii
John L. Teall and Iftekhar Hasan
QUANTITATIVEMETHODS FOR
FINANCE ANDINVESTMENTS
QRMA01 30/08/2005 12:30 PM Page iii
© 2002 by John L. Teall and Iftekhar HasanEditorial OfÞces:108 Cowley Road, Oxford OX4 1JF, UK
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The right of the John L. Teall and Iftekhar Hasan to be identiÞed as the Authors of this Work has been asserted in accordance with the UK Copyright, Designs and Paten ts Act 1988. All rights reserved. No part of this publication may be reproduced, stor ed in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, pho tocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Paten ts Act 1988, without the prior permission of the publisher. The Blackwell Publishing logo is a trade mark of Blackwell Publishing Lt d. First published 2002 by Blackwell Publishers Ltd, a Blackwell Publishing company Library of Congress Cataloging-in-Publication Data
Teall, John L., 1958Ð
Quantitative methods for Þnance and investments / John L. Teall and I ftekhar Hasan. p. cm.
Includes bibliographical references and index.
ISBN 0-631-22338-X (alk. paper) Ñ ISBN 0-631-22339-8 (pbk. : alk. paper)
1. FinanceÑMathematical models. 2. InvestmentsÑMathematical models. I. Hasan,
Iftekhar. II. Title.
HG106 .T4 2002
332.015118Ñdc21 2001043228
A catalogue record for this title is available from the British Library.
Set in 10 on 12 pt Photina
by Graphicraft Limited, Hong Kong Printed and bound in Great Britain by TJ International, Padstow, Cornwal l
For further information on
Blackwell Publishers, visit our website:
www.blackwellpublishers.co.uk
QRMA01 30/08/2005 12:30 PM Page iv
Dedicated to Polly Mitchell
QRMA01 30/08/2005 12:30 PM Page v
QRMA01 30/08/2005 12:30 PM Page vi
Prefacexii
Acknowledgmentsxiv
1 Introduction and Overview 1
1.1The importance of mathematics in Þnance 1
1.2Mathematical and computer modeling in Þnance 2
1.3Money, securities, and markets 3
1.4Time value, risk, arbitrage, and pricing 5
1.5The organization of this book 6
2 A Review of Elementary Mathematics:
Functions and Operations 7
2.1Introduction 7
2.2Variables, equations, and inequalities 7
2.3Exponents 8Application 2.1: Interest and future value9
2.4The order of arithmetic operations and the rules of algebra 10Application 2.2: Initial deposit amounts11
2.5The number e 11
2.6Logarithms 12Application 2.3: The time needed to double your money13
2.7Subscripts 14
2.8Summations 14Application 2.4: Mean values15
2.9Double summations 16
2.10Products 17Application 2.5: Geometric means17
Application 2.6: The term structure of interest rates18
CONTENTS
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viiiContents
2.11Factorial products 19Application 2.7: Deriving the numbere19
2.12Permutations and combinations 20
Exercises 21
Appendix 2.A An introduction to the Excelª spreadsheet 23
3 A Review of Elementary Mathematics:
Algebra and Solving Equations 25
3.1Algebraic manipulations 25Application 3.1: Purchase power parity27
Application 3.2: Finding break-even production levels28 Application 3.3: Solving for spot and forward interest rates29
3.2The quadratic formula 29Application 3.4: Finding break-even production levels30
Application 3.5: Finding the perfectly hedged portfolio31
3.3Solving systems of equations that contain multiple variables 32Application 3.6: Pricing factors35
Application 3.7: External Þnancing needs35
3.4Geometric expansions 38Application 3.8: Money multipliers40
3.5Functions and graphs 41Application 3.9: Utility of wealth43
Exercises 44
Appendix 3.A Solving systems of equations on a spreadsheet 48
4 The Time Value of Money 51
4.1Introduction and future value 51
4.2Simple interest 51
4.3Compound interest 52
4.4Fractional period compounding of interest 53Application 4.1: APY and bank account comparisons55
4.5Continuous compounding of interest 56
4.6Annuity future values 57Application 4.2: Planning for retirement59
4.7Discounting and present value 60
4.8The present value of a series of cash ßows 61
4.9Annuity present values 62Application 4.3: Planning for retirement, part II64
Application 4.4: Valuing a bond64
4.10Amortization 65Application 4.5: Determining the mortgage payment66
4.11Perpetuity models 67
4.12Single-stage growth models 68Application 4.6: Stock valuation models70
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Contentsix
4.13Multiple-stage growth models 72
Exercises 73
Appendix 4.A Time value spreadsheet applications 77
5 Return, Risk, and Co-movement 79
5.1Return on investment 79Application 5.1: Fund performance81
5.2Geometric mean return on investment 82Application 5.2: Fund performance, part II83
5.3Internal rate of return 84
5.4Bond yields 87
5.5An introduction to risk 88
5.6Expected return 88
5.7Variance and standard deviation 89
5.8Historical variance and standard deviation 91
5.9Covariance 93
5.10The coefÞcient of correlation and the coefÞcient of determination 94
Exercises 95
Appendix 5.A Return and risk spreadsheet applications 99
6 E lementary Portfolio Mathematics103
6.1An introduction to portfolio analysis 103
6.2Portfolio return 103
6.3Portfolio variance 104
6.4DiversiÞcation and efÞciency 106
6.5The market portfolio and beta 110
6.6Deriving the portfolio variance expression 111
Exercises 113
7 E lements of Matrix Mathematics115
7.1An introduction to matrices 115Application 7.1: Portfolio mathematics116
7.2Matrix arithmetic 117Application 7.2: Portfolio mathematics, part II120
Application 7.3: PutÐcall parity121
7.3Inverting matrices 123
7.4Solving systems of equations 125Application 7.4: External funding requirements126
Application 7.5: Coupon bonds and deriving yield curves127
Application 7.6: Arbitrage with riskless bonds130
Application 7.7: Fixed income portfolio dedication131
Application 7.8: Binomial option pricing132
7.5Spanning the state space 133Application 7.9: Using options to span the state space136
QRMA01 30/08/2005 12:30 PM Page ix
xContents
Exercises 137
Appendix 7.A Matrix mathematics on a spreadsheet 142
8 D ifferential Calculus145
8.1Functions and limits 145Application 8.1: The natural log146
8.2Slopes, derivatives, maxima, and minima 147
8.3Derivatives of polynomials 149Application 8.2: Marginal utility151
Application 8.3: Duration and immunization153
Application 8.4: Portfolio risk and diversiÞcation156
8.4Partial and total derivatives 157
8.5The chain rule, product rule, and quotient rule 158Application 8.5: Plotting the Capital Market Line159
8.6Logarithmic and exponential functions 165
8.7Taylor series expansions 166Application 8.6: Convexity and immunization167
8.8The method of Lagrange multipliers 168Application 8.7: Optimal portfolio selection170
Exercises 172
Appendix 8.A Derivatives of polynomials 176
Appendix 8.B A table of rules for Þnding derivatives 177 Appendix 8.C Portfolio risk minimization on a spreadsheet 178
9 Integral Calculus 180
9.1Antidifferentiation and the indeÞnite integral 180
9.2Riemann sums 181
9.3DeÞnite integrals and areas 185Application 9.1: Cumulative densities186
Application 9.2: Expected value and variance188
Application 9.3: Valuing continuous dividend payments189
Application 9.4: Expected option values191
9.4Differential equations 191Application 9.5: Security returns in continuous time193
Application 9.6: Annuities and growing annuities194
Exercises 195
Appendix 9.A Rules for Þnding integrals 198
Appendix 9.B Riemann sums on a spreadsheet 199
10 Elements of Options Mathematics 203
10.1An introduction to stock options 203
10.2Binomial option pricing: one time period 205
10.3Binomial option pricing: multiple time periods 207
10.4The BlackÐScholes option pricing model 210
10.5Puts and valuation 212
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Contentsxi
10.6BlackÐScholes model sensitivities 213
10.7Estimating implied volatilities 215
Exercises 219
References222
Appendix A Solutions to Exercises224
Appendix B The z-Table266
Appendix C Notation267
Appendix D Glossary270
Index274
QRMA01 30/08/2005 12:30 PM Page xi
One of the most intriguing and captivating aspects of Þnance is its a pplication of math-ematics. Unfortunately, most faculty in Þnance Þnd that this relia nce on mathematicsprompts too many business students to approach Þnance with great drea d. Even manyof our Þnance majors approach their studies with apprehension and fai l to comprehendmany of the more subtle aspects of the subject material. Fortunately, mo st universitystudents, despite their lack of conÞdence, really do have the quantit ative skills neces-sary to understand Þnance through the M.B.A. level. In many cases, on ly relativelyminor preparation or review is needed to provide the student with the gr oundwork andconÞdence required to study Þnance. The purpose of our text is to provide M.B.A. andundergraduate Þnance students this opportunity to update basic mathem atics skills forbasic business Þnance and investments courses. The level of mathematics reviewed in this text does not extend beyond Þ rst-year linear algebra, statistics, and calculus, with the possible exception of a brie f introduction to differential equations in chapter 9. This is consistent with the level o f mathematics required for Þnance concentrations in most M.B.A. and undergraduate business p rograms. Most students will not require an intense instruction in high-school lev el mathem- atics. Hence, we offer only brief reviews of mathematics at these lowest levels and only those that are most important for studying Þnance (chapters 2 and 3) . We then intro- duce three chapters on the mathematics of time value, return, and risk ( chapters 4Ð6). Chapters 7Ð9 provide more detailed reviews and Þnance applications of college-level linear mathematics and calculus. The organization of the text is general ly by mathem- atics topic followed by Þnance applications. Exceptions to this inclu de those chapters devoted to time value, return, and risk measures, portfolio analysis and elementary options mathematics. Spreadsheet applications are provided in many of the chapte r appendices, and exercises are provided for all but the Þrst chapter. The exercise s are solved in the end-of-text appendix A. This text can be used as prerequisite or parallel reading for most intro ductory busi- ness Þnance texts, such as Ross, WesterÞeld, and Jordan (2001),
Brealey and Myers
(2000), and Brealey, Myers, and Marcus (1995), and most basic invest ments texts, includ- ing Bodie, Kane, and Marcus (1999). It may also serve as foundation re ading for more
PREFACE
QRMA01 30/08/2005 12:30 PM Page xii
Prefacexiii
advanced texts, such as Teall (1999), Neftci (1996), Pliska (1997) , Hull (2000), and
Elton and Gruber (1995).
1 This book can also be used as a primary text for an under- graduate or M.B.A. course such as ÒQuantitative Methods for Finance,Ó intended to improve Þnance studentsÕ mathematics skills. Finally, the book may serve as a primary or secondary text for a prerequisite M.B.A. mathematics course covering elementary linear mathematics and calculus for business students. The authors welcome your comments and suggestions regarding this book. P rofes- sor Teall can be contacted by e-mail at jteall@pace.edu or jteall@juno.c om or by using contact information available on his web page at http://webpage.pace.edu /jteall. Professor Hasan can be contacted by e-mail at hasan@adm.njit.edu.
John L. Teall and Iftekhar Hasan
August 2001
1
References are provided on pages 222Ð3.
QRMA01 30/08/2005 12:30 PM Page xiii
We have been fortunate to have had a number of students, colleagues, and friends assistand provide guidance in the preparation of this book. Michael Dang, Hyan gbab Ku,and T. J. Wu all strengthened our book with useful comments and correcti ons on earl-ier versions of the manuscript. Al Bruckner, Elizabeth Wald, and especia lly GeoffreyPalmer, provided wonderful editorial support. Our old friends Ed Downe, Peter Knopf,and John Knopf contributed encouragement and advice throughout the vario us stagesof writing this book. Miriam Vasquez dispensed irrepressible needling an d stole draftsof the text manuscript to pass on to students, yet continues to be a goo d friend. Anne,who never forgets having been left out of the acknowledgments of an earl ier book andEmily, whose Pocahontas game shares disk space with preliminary drafts o f this book,ensured that the various publisher deadlines would not be met prematurel y. They haveearned eternal gratitude for the delightful manner in which they accompl ished this.
ACKNOWLEDGMENTS
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1.1 THEIMPORTANCE OFMATHEMATICS INFINANCE
Finance is an immensely exciting academic discipline and a most rewardin g professional endeavor. However, ever-increasing sophistication of Þnancial markets , design of Þnan- cial securities, and computational technology has heightened the busines s practitionerÕs need for understanding of mathematics. In order to understand and partic ipate in Þnan- cial decision-making, students and practitioners of Þnance need to be familiar with the basic tools and techniques used in more formal analysis of Þnanci al problems. Mathematics is among the most important of these tools. In addition to b eing crucial to the analysis of Þnancial problems, mathematics is a language that is most useful for communication of Þnancial concepts, techniques, and results. Yet many of us approach mathematics very tentatively, feeling overwhelmed by its appare nt complexity and insecure about our own preparation to understand and apply the varie ty of useful tools available to us. In many cases, our preparation for handling Þn ancial analysis merely requires a review of mathematics that we may have been quite comf ortable with in the past; in others, we may need to learn new tools and techniques in mathematics. This book aspires to facilitate review and presentation of more elementa ry mathem- atics concepts sufÞcient to understand many of the most interesting c oncepts in Þnancial analysis. The vast majority of students of Þnance require at least so me review of mathem- atics tools, and most will be surprised at how quickly they can be grasp ed with relat- ively little dedication and effort. A few students will question the need to understand mathematics and its applica- tion to Þnance. One challenge occasionally posed in the Þnance cla ssroom, ÒDonÕt they have computers that do this?Ó, can be difÞcult to confront convinc ingly. Nevertheless, computers cannot actually understand what types of problems can be solve d with what types of technique. Computers are capable only of performing unthinking operations exactly as they are programmed, without any ability to criticize the sui tability of solu- tion technique or to understand the implications of the results that the y generate. The computer user or Þnancial analyst must formulate the problem, determi ne the solu- tion technique, and interpret the computer output. This essentially rele gates the role
CHAPTER ONE
INTRODUCTION ANDOVERVIEW
QRMC01 30/08/2005 12:30 PM Page 1
of the computer to mindlessly performing routine and perhaps repetitive computationsin exactly the manner speciÞed by its user. The user should understan d all of the aspectsof the problem and its results, and must also be able to determine the u sefulness of theresults and apply them. Hence, an analyst, perhaps working in conjunctio n with otheranalysts and professionals, must understand the problem, formulate a sol ution, andinterpret the results. In a sense, the computer cannot solve mush of the problem itself;not only must someone know enough to communicate to it exactly what to d o, butmost of the analytical work must be left to thinking and understanding h umans. Financial analysis is performed in intensely competitive and uncertain e nviron- ments, bringing together many individuals, businesses, governments, and other insti- tutions. These participants in markets interact with one another over ma ny periods of time, which can be presumed to be inÞnitely divisible. Numerous sourc es of uncertainty abound and information derives from the unexpected as well as the expect ed. While this complex Þnancial environment is most fascinating and exciting, i ts analysis can be most confounding, requiring application of many branches of mathemati cs, rang- ing from simple arithmetic, algebra, and calculus to stochastic processe s, numerical methods, wave theory, and nonlinear dynamics. This book presents the mos t essential mathematical technique and its applications to Þnancial analysis. The level of presenta- tion of this book might be considered to be either prerequisite or paral lel to that of an introductory business Þnance or investments course.
1.2 MATHEMATICAL ANDCOMPUTERMODELING
IN
FINANCE
Financial analysis starts with the construction of Þnancial models. A model is an artiÞcial or idealized structure describing the relationships among v ariables or factors. All of the methodology in this book is intended to facilitate the develo pment, imple- mentation, and analysis of Þnancial models to solve Þnancial probl ems. For example, the elementary mathematics presented in chapters 2 and 3 is integral to almost any serious Þnancial analysis. The valuation models in chapter 4 provide a groundwork for making investment and budgeting decisions, while the more sophistica ted option pricing models in chapter 10 enable us to grasp the essentials for under standing derivative instruments. The use of models is important in Þnance beca use the direct analysis of actual markets is extraordinarily complex. For example, a th orough valu- ation of a stock should require us to understand everything about the ec onomy, polit- ical arena, human psychology, and so on that could have an impact on tha t stockÕs value. This thorough understanding is simply impossible; it is much more practical to construct a valuation model that accounts for only the most important fa ctors. Models provide the analyst the opportunity to simplify real-world circumstances to a construct that can be easily be manipulated and understood. Financial decision-mak ers fre- quently use existing models or construct new ones that relate to the typ es of decisions they wish to make. The aim of a Þnancial model is to simulate or behave like a real Þ nancial situation. Analysts who create Þnancial models exclude Òreal-worldÓ condit ions that have only minor impact on the results of their decisions. Instead, analysts focus on those factors
2Introduction and overview
QRMC01 30/08/2005 12:30 PM Page 2
Money, securities, and markets3
that are most relevant to their situations. The most important condition s retained by analysts for model-building along with those ignored for sake of simplic ity form the basis for the set of assumptions for the model. In most cases, analysts must m ake unrealistic assumptions in order to simplify their models and make them easier to wo rk with. The simplest models building on more restrictive simplifying assumptions can be adapted to more life-like scenarios by relaxing the most unrealistic assumptions . The best Þnancial models are those that appropriately account for the most sig niÞcant factors affecting Þnancial decisions, are simple enough be practical and easy to work with, and are useful for predicting actual Þnancial results. Models that predic t actual Þnancial outcomes most accurately are most useful. Unfortunately, accuracy and si mplicity in the construction of Þnancial models often conßict with each oth er. Some degree of inaccuracy in the model must usually be tolerated in order to maintain i ts ease of use and analysis. The appropriate tradeoff between accuracy and simplicity i s a problem constantly faced by the Þnancial analyst. This book is concerned with both theoretical and practitioner-oriented m odels. The bookÕs primary focus is the mathematics and quantitative technique re quired to cre- ate and analyze Þnancial models. Theoretical models are created to ex plain Þnancial markets and scenarios; practitioner-oriented empirical models are intend ed to bequotesdbs_dbs20.pdfusesText_26
Finance Quantitative - FSEGN