Introduction to the Mathematics of Finance. A Deterministic Approach

446 Pages·2013·4.6 MB·English

by  Stephen Garrett (Auth.)

An Introduction to the
Mathematics of Finance
A Deterministic Approach
Second Edition
S. J. Garrett
Published for the Institute
and Faculty of Actuaries (RC000243)
http://www.actuaries.org.uk
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Dedication
DedicatedtoAdamandMatthewGarrett,mytwogreatest
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Preface
This book is a revision of the original An Introduction to the Mathematics of
FinancebyJ.J.McCutcheonandW.F.Scott.Thesubjectoffinancialmathematics
hasexpandedimmenselysincethepublicationofthatfirsteditioninthe1980s,
and the aim of this second edition is to update the content for the modern
audience. Despite the recent advances in stochastic models within financial
mathematics, the book remains concerned almost entirely with deterministic
approaches.Thereasonforthisistwofold.Firstly,manyreaderswillfindasolid
understanding of deterministic methods within the classical theory of
compound interest entirely sufficient for their needs. This group of readers is
likely to include economists, accountants, and general business practitioners.
Secondly, readers intending to study towards an advanced understanding
of financial mathematics need to start with the fundamental concept of
compoundinterest.Suchreadersshouldtreatthisasanintroductorytext.Care
hasbeentakentopointtowardsareaswherestochasticconceptswilllikelybe
developedinlaterstudies;indeed,Chapters10,11,and12areintendedasan
introduction to the fundamentals and application of modern financial math-
ematics in the broader sense.
The book is primarily aimed at readers who are preparing for university or
professional examinations. The material presented here now covers the entire
CT1syllabusoftheInstituteandFacultyofActuaries(asat2013)andalsosome
materialrelevanttotheCT8andST5syllabuses.Thiscombinationofmaterial
correspondstotheFM-FinancialMathematicssyllabusoftheSocietyofActuaries.
Furthermore,studentsoftheCFAInstitutewillfindthisbookusefulinsupport
ofvariousaspectsoftheirstudies.Withexampreparationinmind,thissecond
editionincludesmanypastexaminationquestionsfromtheInstituteandFaculty
of Actuaries andthe CFAInstitute, with worked solutions.
The book is necessarily mathematical, but I hope not too mathematical. It is
expectedthatreadershaveasolidunderstandingofcalculus,linearalgebra,and
probability,buttoalevelnohigherthanwouldbeexpectedfromastrongfirst
yearundergraduateinanumeratesubject.Thatisnottosaythematerialiseasy,
xi
xii Preface
ratherthedifficultyarisesfromthesheerbreathofapplicationandtheperhaps
unfamiliarreal-world contexts.
Whereappropriate,additionalmaterialinthiseditionhasbeenbasedoncore
readingmaterialfromtheInstituteandFacultyofActuaries,andIamgratefulto
Dr.TrevorWatkinsforpermissiontousethis.IamalsogratefultoLauraClarke
andSallyCalderoftheInstituteandFacultyofActuariesfortheirhelp,notleastin
sourcing relevant past examination questions from their archives. I am also
gratefultoKathleenPaoniandDr.J.ScottBentleyofElsevierforsupportingme
inmyfirstventureintotheworldoftextbooks.Ialsowishtoacknowledgethe
entertainingcompanyofmygoodfriendandcolleagueDr.AndrewMcMullan
of the University of Leicester on the numerous coffee breaks between writing.
Thiseditionhasbenefittedhugelyfromcommentsmadebyundergraduateand
postgraduate students enrolled on my modules An Introduction to Actuarial
MathematicsandTheoryofInterestattheUniversityofLeicesterin2012.Particular
mention should be given to the eagle eyes of Fern Dyer, George Hodgson-
Abbott, Hitesh Gohel, Prashray Khaire, Yueh-Chin Lin, Jian Li, and Jianjian
Shao, who pointed out numerous typos in previous drafts. Any errors that
remain are of course entirely my fault.
Thislistofacknowledgementswouldnotbecompletewithoutspecialmention
of my wife, Yvette, who puts up with my constant working and occasional
grumpiness.Yvette is aconstant supporter ofeverything Ido, andI could not
havedonethis, or indeed much else, without her.
Dr.Stephen J.Garrett
DepartmentofMathematics,UniversityofLeicester
January2013
CHAPTER 1
Introduction
1.1 THE CONCEPT OF INTEREST CONTENTS
Interestmayberegardedasarewardpaidbyonepersonororganization(the 1.1 TheConceptof
Interest..........1
borrower) for the use of an asset, referred to as capital, belonging to
another person or organization (the lender). The precise conditions of any 1.2 Simple
transactionwillbemutuallyagreed.Forexample,afterastatedperiodoftime, Interest..........2
thecapitalmaybereturnedtothelenderwiththeinterestdue.Alternatively, 1.3 Compound
severalinterestpaymentsmaybemadebeforetheborrowerfinallyreturnsthe Interest..........4
asset.
1.4 Some Practical
Illustrations...6
Capitalandinterestneednotbemeasuredintermsofthesamecommodity,but
throughoutthisbook,whichrelatesprimarilytoproblemsofafinancialnature, Summary..............9
weshallassumethatbotharemeasuredinthemonetaryunitsofagivencurrency.
Whenexpressedinmonetaryterms,capitalisalsoreferredtoasprincipal.
Ifthereissomeriskofdefault(i.e.,lossofcapitalornon-paymentofinterest),
alenderwouldexpecttobepaidahigherrateofinterestthanwouldotherwise
bethecase;thisadditionalinterestisknownastheriskpremium.Theadditional
interest in such a situation may be considered as a further reward for the
lender’s acceptance of the increased risk. For example, a person who uses his
moneytofinancethedrillingforoilinapreviouslyunexplored regionwould
expectarelativelyhighreturnonhisinvestmentifthedrillingissuccessful,but
mighthavetoacceptthelossofhiscapitalifnooilweretobefound.Afurther
factorthatmayinfluencetherateofinterestonanytransactionisanallowance
for the possible depreciation or appreciation in the value of the currency in
whichthetransactioniscarriedout.Thisfactorisobviouslyveryimportantin
times of high inflation.
Itisconvenienttodescribetheoperationofinterestwithinthefamiliarcontext
of a savings account, held in a bank, building society, or other similar orga-
nization.Aninvestorwhohadopenedsuchanaccountsometimeagowithan
initial depositof£100,andwho hadmadeno otherpaymentstoor fromthe
account,wouldexpecttowithdrawmorethan£100ifhewerenowtoclosethe
account. Suppose, for example, that he receives £106 on closing his account. 1
AnIntroductiontotheMathematicsofFinance.http://dx.doi.org/10.1016/B978-0-08-098240-3.00001-1
(cid:1)2013InstituteandFacultyofActuaries(RC000243).PublishedbyElsevierLtd.Allrightsreserved.
2 CHAPTER 1: Introduction
This sum may be regarded as consisting of £100 as the return of the initial
depositand£6asinterest.Theinterestisapaymentbythebanktotheinvestor
for the use ofhis capital over the durationof the account.
The most elementary concept is that of simple interest. This naturally leads to
the idea of compound interest, which is much more commonly found in
practice in relation to all but short-term investments. Both concepts are easily
described within the framework of a savings account, as described in the
following sections.
1.2 SIMPLE INTEREST
Supposethataninvestoropensasavingsaccount,whichpayssimpleinterestat
the rate of 9% per annum, with a single deposit of £100. The account will be
credited with £9 of interest for each complete year the money remains on
deposit.Iftheaccountisclosedafter1year,theinvestorwillreceive£109;ifthe
account is closed after 2 years, he will receive £118, and so on. This may be
summarizedmore generallyas follows.
IfanamountCisdepositedinanaccountthatpayssimpleinterestattherateof
iperannumandtheaccountisclosedafternyears(therebeingnointervening
paymentstoorfromtheaccount),thentheamountpaidtotheinvestorwhen
the account is closed willbe
Cð1þniÞ (1.2.1)
ThispaymentconsistsofareturnoftheinitialdepositC,togetherwithinterest
of amount
niC (1.2.2)
Inourdiscussionsofar,wehaveimplicitlyassumedthat,ineachoftheselast
two expressions, n is an integer. However, the normal commercial practice in
relationtofractionalperiodsofayearistopayinterestonaproratabasis,so
that Eqs 1.2.1 and 1.2.2 may be considered as applying for all non-negative
values ofn.
Notethatiftheannualrateofinterestis12%,theni¼0.12perannum;ifthe
annualrate ofinterest is 9%, then i¼0.09 per annum; andso on.
Note that in the solution to Example1.2.1,wehaveassumedthat6months
and 10 months are periods of 1/2 and 10/12 of 1 year, respectively. For
accounts of duration less than 1 year, it is usual to allow for the actual
numberofdaysanaccountisheld,so,forexample,two6-monthperiodsare
not necessarily regarded as being of equal length. In this case Eq. 1.2.1
becomes
1.2 Simple Interest 3
EEEXXXAAAMMMPPPLLLEEE 111...222...111
Suppose that £860 is deposited in a savings account that 1 in Eq. 1.2.1 with C¼860 and i¼0.05375, we obtain the
pays simple interest at the rate of 5:375% per annum. answers
Assuming that there are no subsequent payments to or
(a) £883.11,
from the account,find the amountfinally withdrawn if the
(b) £898.52,
accountisclosedafter
(c) £906.23.
(a) 6months,
Ineachcasewehavegiventheanswertotwodecimalplaces
(b) 10months,
of one pound, rounded down. This is quite common in
(c) 1year.
commercialpractice.
Solution
Theinterestrateisgivenasaperannumvalue;therefore,n
mustbemeasuredinyears.Bylettingn¼6/12,10/12,and
EEEXXXAAAMMMPPPLLLEEE 111...222...222
(cid:1) (cid:3)
30
Calculatethepriceofa30-day£2,000treasurybillissued £2;000 1(cid:2) (cid:3)0:05 ¼ £1;991:78
365
by the government at a simple rate of discount of 5% per
annum. The investor has received interest of £8.22 under this
Solution transaction.
Byissuingthetreasurybill,thegovernmentisborrowingan
amountequaltothepriceofthebill.Inreturn,itpays£2,000
after30days.Thepriceisgivenby
(cid:1) (cid:3)
mi
C 1þ (1.2.3)
365
where misthe duration ofthe account, measuredin days, andiisthe annual
rate of interest.
Theessentialfeatureofsimpleinterest,asexpressedalgebraicallybyEq.1.2.1,
isthatinterest,oncecreditedtoanaccount,doesnotitselfearnfurtherinterest.
This leads to inconsistencies that are avoided by the application of compound
interest theory, as discussed in Section 1.3.
Asaresultoftheseinconsistencies,simpleinteresthaslimitedpracticaluse,and
this book will, necessarily, focus on compound interest. However, an impor-
tant commercial application of simple interest is simple discount, which is
commonly used for short-term loan transactions, i.e., up to 1 year. Under
4 CHAPTER 1: Introduction
simplediscount,theamountlentisdeterminedbysubtractingadiscountfrom
theamountdueatthelaterdate.Ifalenderbaseshisshort-termtransactionson
asimplerateofdiscountd,then,inreturnforarepaymentofXafteraperiodt
(typically t<1), he will lend X(1(cid:2)td) at the start of the period. In this situa-
tion,d is also knownas arate of commercial discount.
1.3 COMPOUND INTEREST
Supposenowthatacertaintypeofsavingsaccountpayssimpleinterestatthe
rate of i per annum. Suppose further that this rate is guaranteed to apply
throughoutthenext2yearsandthataccountsmaybeopenedandclosedatany
time. Consider an investor who opens an account at the present time (t¼0)
with an initial deposit of C. The investor may close this account after 1 year
(t¼1),atwhichtime hewillwithdraw C(1þi)(seeEq.1.2.1).Hemaythen
placethissumondepositinanewaccountandclosethissecondaccountafter
onefurtheryear(t¼2).Whenthislatteraccountisclosed,thesumwithdrawn
(again see Eq. 1.2.1)willbe
½Cð1þiÞ(cid:4)(cid:3)ð1þiÞ ¼ Cð1þiÞ2 ¼ Cð1þ2iþi2Þ
If,however,theinvestorchoosesnottoswitchaccountsafter1yearandleaveshis
moneyintheoriginalaccount,onclosingthisaccountafter2years,hewillreceive
C(1 þ 2i). Therefore, simply by switching accounts in the middle of the 2-year
period, the investor will receive an additional amount i2C at the end of the
period.Thisextrapaymentis,ofcourse,equaltoi(iC)andarisesasinterestpaid
(att¼2)ontheinterestcreditedtotheoriginalaccountattheendofthefirstyear.
From a practical viewpoint, it would be difficult to prevent an investor
switching accounts in the manner described here (or with even greater
frequency). Furthermore, the investor, having closed his second account after
1year,couldthendeposittheentireamountwithdrawninyetanotheraccount.
Any bank would find it administratively very inconvenient to have to keep
opening and closing accounts in the manner just described. Moreover, on
closingoneaccount,theinvestormightchoosetodeposithismoneyelsewhere.
Therefore, partly to encourage long-term investment and partly for other
practical reasons, it is common commercial practice (at least in relation to
investments of duration greater than 1 year) to pay compound interest on
savings accounts. Moreover, the concepts of compound interest are used in
the assessment and evaluation of investments as discussed throughout this
book.
Theessentialfeatureofcompoundinterestisthatinterestitselfearnsinterest.The
operation of compound interest may be described as follows: consider
a savings account, which pays compound interest at rate i per annum, into
1.3 Compound Interest 5
whichisplacedaninitialdepositCattimet¼0.(Weassumethatthereareno
furtherpayments to or from the account.) If the account is closed after 1 year
(t¼1)theinvestorwillreceiveC(1þi).Moregenerally,letA betheamount
n
thatwillbereceivedbytheinvestorifheclosestheaccountafternyears(t¼n).
ItisclearthatA ¼C(1þi).Bydefinition,theamountreceivedbythe investor
1
onclosingtheaccountattheendofanyyearisequaltotheamounthewould
havereceivedifhehadclosedtheaccount1yearpreviouslyplusfurtherinterest
ofitimesthisamount.Theinterestcreditedtotheaccountuptothestartofthe
finalyearitselfearnsinterest(atrateiperannum)overthefinalyear.Expressed
algebraically, this definition becomes
Anþ1 ¼ AnþiAn
or
Anþ1 ¼ ð1þiÞAn n(cid:5)1 (1.3.1)
Since, by definition, A ¼C(1þi),Eq.1.3.1 implies that,for n¼1, 2, . . .,
1
A ¼ Cð1þiÞn (1.3.2)
n
Therefore, if the investorcloses the account after nyears, he willreceive
Cð1þiÞn (1.3.3)
This payment consists of a return of the initial deposit C, together with accu-
mulatedinterest(i.e.,interestwhich,ifn>1,hasitselfearnedfurtherinterest)
ofamount
C½ð1þiÞn(cid:2)1(cid:4) (1.3.4)
Inourdiscussionsofar,wehaveassumedthatinboththeselastexpressionsnis
aninteger.However,inChapter2wewillwidenthediscussionandshowthat,
under very general conditions, Eqs 1.3.3 and 1.3.4 remain valid for all non-
negativevaluesof n.
Since
½Cð1þiÞt1(cid:4)ð1þiÞt2 ¼ Cð1þiÞt1þt2
an investor who is able to switch his money between two accounts, both of
which pay compound interest at the same rate, is not able to profit by such
action.Thisisincontrastwiththesomewhatanomaloussituation,describedat
the beginning ofthis section, which may occur if simple interest is paid.
Equations 1.3.3 and 1.3.4 should be compared with the corresponding
expressionsundertheoperationofsimpleinterest(i.e.,Eqs1.2.1and1.2.2).If
interestcompounds(i.e.,earnsfurtherinterest),theeffectontheaccumulationof
anaccountcan be verysignificant,especially if the duration of the account or