An Introduction to the Mathematics of Money: Saving and Investing

308 Pages·2006·1.25 MB·English

by  David Lovelock

An Introduction to the
Mathematics of Money

David Lovelock Marilou Mendel A. Larry Wright
An Introduction to the
Mathematics of Money
Saving and Investing
David Lovelock Marilou Mendel
Department of Mathematics Department of Mathematics
University of Arizona University of Arizona
Tucson, AZ 85721 Tucson, AZ 85721
USA USA
[email protected] [email protected]
A. Larry Wright
Department of Mathematics
University of Arizona
Tucson, AZ 85721
USA
[email protected]
Mathematics Subject Classification (2000): 91B82
Library of Congress Control Number: 2006931194
ISBN-10: 0-387-34432-2
ISBN-13: 978-0387-34432-4
Printed on acid-free paper.
© 2007 Springer Science+Business Media, LLC
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Preface
Introduction
Some people distinguish between savings and investments, where savings are
moniesplacedinrelativelyrisk-freeaccountswithmodestrewards,andwhere
investments involve more risk and the potential for greater rewards. In this
book we do not distinguish between these ideas. We treat them both under
the umbrella of investing.
In general, income falls into two categories: earned income—which is
theincomederivedfromyoureverydayjob—andunearned income—which
is income derived from investing. You attend college to strengthen your
prospects for earned income, so why do you need to worry about unearned
income, namely, investment income?
There are many reasons to invest and to learn about investing. Perhaps
the primary one is to take charge of your own financial future. You need
money for short-term goals (such as living expenses, emergencies) and for
long-term goals (such as buying a car, buying a house, educating children,
paying catastrophic medical bills, funding retirement).
Investing involves borrowing and lending, and buying and selling.
• borrowing and lending. When you put money into a bank savings
account,youarelendingyourmoneyandthebankisborrowingit.Youcan
lend money to a bank, a business, a government, or a person. In exchange
forthis,theborrowerpromisestopayyouinterestandtoreturnyourinitial
investmentatafuturedate.Whywouldtheborrowerdothis?Becausethe
borrower anticipates using this money in a way that earns more than the
interest promised to you. Examples of borrowing and lending are savings
accounts, certificates of deposits, money-market accounts, and bonds.
• buying and selling. Whenyoubuysomethingforinvestmentpurposes,
you are buying an asset from a seller. You expect that this asset will
generate a profit or will increase in value, part of which will be returned
to you. Examples of this are owning real estate or stocks in companies.
vi Preface
There are two ways that you can make (lose) money buying stocks: stock-
priceappreciation(depreciation)—whichdependsontheexpectationsand
opinionsofthepublic—anddividendspaidtoyoubythecompany—which
depend on the company sharing its profits with you, a shareholder.
Wheninvesting,therearethreethingsthatcanimpactyourprofit—taxes,
inflation,andrisk.Thefirst,taxes,shouldconcerneveryone.Thesecond,infla-
tion, should concern you if you make a profit. The third, risk, should concern
youbeforeyoumakeaninvestmentbecauseriskinfluencestheprofitabilityof
the investment. Generally, if you expect a high return on your money, then
youshouldalsoexpectahighrisk.Inthesameway,lowrisksareusuallyasso-
ciated with low returns. The larger the risk the greater the chance of actually
losingmoney.Therearevarioustypesofrisk:inflation,market,currencyfluc-
tuations, political, interest-rate, liquidity, economic, default, business, etc.
Objectives and Background
We wrote this book with two objectives in mind:
• To use investing as a vehicle to introduce you, the student, to ideas, tech-
niques,andapplicationsthatyoumightnotencounterinyourothermath-
ematics courses. These include proofs by induction, recurrence relations,
inequalities(inparticular,theArithmetic-GeometricMeaninequalityand
theCauchy-Schwarzinequality),andelementsofprobabilityandstatistics.
• Tointroduceyou,thestudent,toelementsofinvestingthatareoflife-long
practicaluse.Ifyouhavenotyetdoneso,thenasyouadvancethroughlife,
you are forced to deal with such things as credit cards, student loans, car-
loans, savings accounts, certificates of deposit, money-market accounts,
mortgage payments, buying and selling bonds, and buying and selling
stocks.
Thisbooktargetsstudentsatthesophomore/juniorlevel,withoutassum-
ing a background or any experience in investing. We assume knowledge of
a two-semester calculus course as well as some mathematical sophistication.
Specificallyweuseinequalities,log,exp,differentiation,theMeanValueThe-
orem,integration,Newton’smethod,limitsofsequences,geometricseries,the
binomial expansion, and Taylor series.
There are problems at the end of each chapter. Some of these problems
requirethatyouhaveaccesstoaspreadsheetprogramandthatyouknowhow
touseit.Asimplescientificorfinancialcalculator(withfunctionssuchaslog,
exp, and the ability to calculate yx) is all that is required for the remainder
oftheproblemsthatinvolvearithmeticalcalculations.Someproblemsrequire
you to obtain data from the World Wide Web (WWW), so access to the
WWW, and familiarity with a browser, is a prerequisite.1
1 The web page www.mathematics-of-money.com is dedicated to this book.
Preface vii
Comments
The following numbering system is used throughout the book: Example 2.3
refers to the third example in Chapter 2, Theorem 4.1 refers to the first
theorem in Chapter 4, Figure 4.2 refers to the second figure in Chapter 4,
Table1.3 refersto thethird tablein Chapter 1, and Problem 1.5 refersto the
fifth problem in Chapter 1.
Thesymbol(cid:1)indicatestheendofanexample,andthesymbol(cid:1)indicates
the end of a proof.
Many of the theorems in the book are given names (for example, The
Compound Interest Theorem). This is done for ease of navigation for the
student.
The problems are divided into two groups: “Walking”, which involve rou-
tine, straight-forward calculations, and “Running”, which are more challeng-
ing problems.
There are two appendices. Appendix A covers mathematical induction,
recurrence relations, and inequalities. This material should be introduced at
the beginning of the course. Appendix B covers elements of probability and
statistics. It is not needed until the latter part of the course and can be
introducedasneeded.Manystudentsmayhaveseenthismaterialinprevious
classes.
Unlessindicatedotherwise,allnumericalresultsareroundedtothreedec-
imal places, and all dollar amounts are rounded to cents. Because of this
convention,whenthesamecalculationisperformedintwodifferentways,the
answers may differ slightly.
In most, but not all, cases in this book the interest rate is assumed to be
positive. It is interesting to note that there are instances when the interest
rateisnegative.See,forexample,[21].Agoodreferenceoninvestmentsis[4].
A more advanced treatment is [18].
Theinformationcontainedinthisbookisnotintendedtobeconstruedas
investment, legal, or accounting advice.
The Family
In order to try to personalize the investment examples and problems in this
book, we have introduced a fictional family, the Kendricks. Helen (48) and
Hugh (50) Kendrick, are husband and wife. They have three children, twins
Wendy(25)andTom(25),andAmanda(20),acollegefreshman.JanaCarmel
(35) is one of Hugh’s coworkers.
viii Preface
Acknowledgments
This book originated from classes taught in the Department of Mathematics
at The University of Arizona, and in the Industrial Engineering and Opera-
tionsResearchDepartmentatColumbiaUniversity.Severalstudentsprovided
valuable comments and corrected errors in the original notes. In particular,
we thank Tom Wilkening and Michael Urbancic.
We also thank Wayne Hacker, David Lomen, and Doug Ulmer of the De-
partment of Mathematics at The University of Arizona, who reviewed large
portions of the manuscript and corrected several errors.
A guest lecture series, where professionals from both inside and outside
academia are invited to discuss their specialities, is an integral part of the
classandintroducesthestudentstoreal-worldapplicationsofthemathemat-
ics of money. We thank the following guest lecturers: Dennis Bartlett, Sue
Burroughs, Steve Kou, Steve Przewlocki, and Lauren Wright.
Special thanks go to Murray Teitelbaum and some of the many people at
the New York Stock Exchange who are involved with the Teachers Workshop
Program.
We thank the many other people who have assisted in the preparation of
themanuscript:PatBrockett,HanGao,JoeHarwood,PavanKorada,Robert
Maier, Charles Newman, Keith Schlottman, Michael Sobel, and Aramian
Wasielak.
Wealsothankthereviewersofthemanuscriptfortheirinvaluablesugges-
tions.
Finally, we thank our contacts at Springer—Achi Dosanjh, Yana Mermel,
and Frank Ganz, for their indispensable support and advice.
Tucson, Arizona, David Lovelock
July, 2006 Marilou Mendel
A. Larry Wright
Contents
Preface ........................................................ v
1 Simple Interest ............................................ 1
1.1 The Simple Interest Theorem ............................ 1
1.2 Ambiguities When Interest Period is Measured in Days ..... 8
1.3 Problems.............................................. 10
2 Compound Interest........................................ 13
2.1 The Compound Interest Theorem ........................ 13
2.2 Time Diagrams and Cash Flows.......................... 23
2.3 Internal Rate of Return ................................. 26
2.4 The Rule of 72......................................... 36
2.5 Problems.............................................. 37
3 Inflation and Taxes ........................................ 45
3.1 Inflation............................................... 45
3.2 Consumer Price Index (CPI)............................. 48
3.3 Personal Taxes......................................... 50
3.4 Problems.............................................. 51
4 Annuities.................................................. 55
4.1 An Ordinary Annuity ................................... 55
4.2 An Annuity Due ....................................... 66
4.3 Perpetuities ........................................... 70
4.4 Problems.............................................. 71
5 Loans and Risks ........................................... 75
5.1 Problems.............................................. 79