Chapter05IM10thEd1Finance.docx

Chapter05IM10thEd1Finance.docx

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Chapter05IM10thEd1Finance

CHAPTER5

TheTimeValueofMoney

CHAPTERORIENTATION

Inthischaptertheconceptofatimevalueofmoneyisintroduced,thatis,adollartodayisworthmorethanadollarreceivedayearfromnow.Thusifwearetologicallycompareprojectsandfinancialstrategies,wemusteithermovealldollarflowsbacktothepresentorouttosomecommonfuturedate.

CHAPTEROUTLINE

I.Compoundinterestresultswhentheinterestpaidontheinvestmentduringthefirstperiodisaddedtotheprincipalandduringthesecondperiodtheinterestisearnedontheoriginalprincipalplustheinterestearnedduringthefirstperiod.

A.Mathematically,thefuturevalueofaninvestmentifcompoundedannuallyatarateofifornyearswillbe

FVn=PV（l+i）n

wheren=thenumberofyearsduringwhichthecompoundingoccurs

i=theannualinterest（ordiscount）rate

PV=thepresentvalueororiginalamountinvestedatthebeginningofthefirstperiod

FVn=thefuturevalueoftheinvestmentattheendofnyears

1.Thefuturevalueofaninvestmentcanbeincreasedbyeitherincreasingthenumberofyearsweletitcompoundorbycompoundingitatahigherrate.

2.Ifthecompoundedperiodislessthanoneyear,thefuturevalueofaninvestmentcanbedeterminedasfollows:

FVn=PV

mn

wherem=thenumberoftimescompoundingoccursduringtheyear

II.Determiningthepresentvalue,thatis,thevalueintoday'sdollarsofasumofmoneytobereceivedinthefuture,involvesnothingotherthaninversecompounding.Thedifferencesinthesetechniquescomeaboutmerelyfromtheinvestor'spointofview.

A.Mathematically,thepresentvalueofasumofmoneytobereceivedinthefuturecanbedeterminedwiththefollowingequation:

PV=FVn

where:

n=thenumberofyearsuntilpaymentwillbereceived,

i=theinterestrateordiscountrate

PV=thepresentvalueofthefuturesumofmoney

FVn=thefuturevalueoftheinvestmentattheendofnyears

1.Thepresentvalueofafuturesumofmoneyisinverselyrelatedtoboththenumberofyearsuntilthepaymentwillbereceivedandtheinterestrate.

III.Anannuityisaseriesofequaldollarpaymentsforaspecifiednumberofyears.Becauseannuitiesoccurfrequentlyinfinance,forexample,bondinterestpayments,wetreatthemspecially.

A.Acompoundannuityinvolvesdepositingorinvestinganequalsumofmoneyattheendofeachyearforacertainnumberofyearsandallowingittogrow.

1.Thiscanbedonebyusingourcompoundingequation,andcompoundingeachoneoftheindividualdepositstothefutureorbyusingthefollowingcompoundannuityequation:

FVn=PMT

where:

PMT=theannuityvaluedepositedattheendofeachyear

i=theannualinterest（ordiscount）rate

n=thenumberofyearsforwhichtheannuitywilllast

FVn=thefuturevalueoftheannuityattheendofthenthyear

B.Pensionfunds,insuranceobligations,andinterestreceivedfrombondsallinvolveannuities.Tocomparethesefinancialinstrumentswewouldliketoknowthepresentvalueofeachoftheseannuities.

1.Thiscanbedonebyusingourpresentvalueequationanddiscountingeachoneoftheindividualcashflowsbacktothepresentorbyusingthefollowingpresentvalueofanannuityequation:

PV=PMT

where:

PMT=theannuitydepositedorwithdrawnattheendofeachyear

i=theannualinterestordiscountrate

PV=thepresentvalueofthefutureannuity

n=thenumberofyearsforwhichtheannuitywilllast

C.ThisprocedureofsolvingforPMT,theannuityvaluewheni,n,andPVareknown,isalsotheprocedureusedtodeterminewhatpaymentsareassociatedwithpayingoffaloaninequalinstallments.Loanspaidoffinthisway,inperiodicpayments,arecalledamortizedloans.HereagainweknowthreeofthefourvaluesintheannuityequationandaresolvingforavalueofPMT,theannualannuity.

IV.Annuitiesduearereallyjustordinaryannuitieswherealltheannuitypaymentshavebeenshiftedforwardbyoneyear.Compoundingthemanddeterminingtheirpresentvalueisactuallyquitesimple.Becauseanannuity,duemerelyshiftsthepaymentsfromtheendoftheyeartothebeginningoftheyear,wenowcompoundthecashflowsforoneadditionalyear.Therefore,thecompoundsumofanannuitydueis

FVn（annuitydue）=PMT（FVIFAi,n）（1+i）

A.Likewise,withthepresentvalueofanannuitydue,wesimplyreceiveeachcashflowoneyearearlier–thatis,wereceiveitatthebeginningofeachyearratherthanattheendofeachyear.Thusthepresentvalueofanannuitydueis

PV（annuitydue）=PMT（PVIFAi,n）（1+i）

V.Aperpetuityisanannuitythatcontinuesforever,thatiseveryyearfromnowonthisinvestmentpaysthesamedollaramount.

A.Anexampleofaperpetuityispreferredstockwhichyieldsaconstantdollardividendinfinitely.

B.Thefollowingequationcanbeusedtodeterminethepresentvalueofaperpetuity:

PV=

where:

PV=thepresentvalueoftheperpetuity

pp=theconstantdollaramountprovidedbytheperpetuity

i=theannualinterestordiscountrate

VI.Toaidinthecalculationsofpresentandfuturevalues,tablesareprovidedatthebackofFinancialManagement（FM）.

A.ToaidindeterminingthevalueofFVninthecompoundingformula

FVn=PV（1+i）n=PV（FVIFi,n）

tableshavebeencompiledforvaluesofFVIFi,nor（i+1）ninAppendixB,"CompoundSumof$1,"inFM.

B.Toaidinthecomputationofpresentvalues

PV=FVn

=FVn（PVIFi,n）

tableshavebeencompiledforvaluesof

orPVIFi,n

andappearinAppendixCinthebackofFM.

C.Becauseofthetime-consumingnatureofcompoundinganannuity,

FVn=PMT

=PMT（FVIFAi,n）

TablesareprovidedinAppendixDofFMfor

orFVIFAi,n

forvariouscombinationsofnandi.

D.Tosimplifytheprocessofdeterminingthepresentvalueofanannuity

PV=PMT

=PMT（PVIFAi,n）

tablesareprovidedinAppendixEofFMforvariouscombinationsofnandiforthevalue

orPVIFAi,n

V.SpreadsheetsandtheTimeValueofMoney.

A.Whilethereareseveralcompetingspreadsheets,themostpopularoneisMicrosoftExcel.Justaswiththekeystrokecalculationsonafinancialcalculator,aspreadsheetcanmakeeasyworkofmostcommonfinancialcalculations.ListedbelowaresomeofthemostcommonfunctionsusedwithExcelwhenmovingmoneythroughtime:

Calculation:

Formula:

PresentValue=PV（rate,numberofperiods,payment,futurevalue,type）

FutureValue=FV（rate,numberofperiods,payment,presentvalue,type）

Payment=PMT（rate,numberofperiods,presentvalue,futurevalue,type）

NumberofPeriods=NPER（rate,payment,presentvalue,futurevalue,type）

InterestRate=RATE（numberofperiods,payment,presentvalue,futurevalue,type,guess）

where:

rate=i,theinterestrateordiscountrate

numberofperiods=n,thenumberofyearsorperiods

payment=PMT,theannuitypaymentdepositedorreceivedattheendofeachperiod

futurevalue=FV,thefuturevalueoftheinvestmentattheendofnperiodsoryears

presentvalue=PV,thepresentvalueofthefuturesumofmoney

type=whenthepaymentismade,（0ifomitted）

0=atendofperiod

1=atbeginningofperiod

guess=astartingpointwhencalculatingtheinterestrate,ifomitted,thecalculationsbeginwithavalueof0.1or10%

ANSWERSTO

END-OF-CHAPTERQUESTIONS

5-1.Theconceptoftimevalueofmoneyisrecognitionthatadollarreceivedtodayisworthmorethanadollarreceivedayearfromnoworatanyfuturedate.Itexistsbecausethereareinvestmentopportunitiesonmoney,thatis,wecanplaceourdollarreceivedtodayinasavingsaccountandoneyearfromnowhavemorethanadollar.

5-2.Compoundinganddiscountingareinverseprocessesofeachother.Incompounding,moneyismovedforwardintime,whileindiscountingmoneyismovedbackintime.Thiscanbeshownmathematicallyinthe

compoundingequation:

FVn=PV（1+i）n

Wecanderivethediscountingequationbymultiplyingeachsideof

thisequationby

andweget:

PV=FVn

5-3.Weknowthat

FVn=PV（1+i）n

Thus,anincreaseiniwillincreaseFVnandadecreaseinnwill

decreaseFVn.

5-4.BankCwhichcompoundsdailypaysthehighestinterest.Thisoccursbecause,whileallbankspaythesameinterest,5percent,bankCcompoundsthe5percentdaily.Dailycompoundingallowsinteresttobeearnedmorefrequentlythantheothercompoundingperiods.

5-5.Thevaluesinthepresentvalueofanannuitytable（Table5-8）areactuallyderivedfromthevaluesinthepresentvaluetable（Table5-4）.Thiscanbeseen,byexaminingthevaluesrepresentedineachtable.Thepresentvaluetablegivesvaluesof

forvariousvaluesofiandn,whilethepresentvalueofanannuitytablegivesvaluesof

forvariousvaluesofiandn.Thusthevalueinthepresentvalueofannuitytableforann-yearannuityforanydiscountrateiismerelythesumofthefirstnvaluesinthepresentvaluetable.PVIFA10%,10yrs=6.145.

PVIF10%,n=6.144=0.909+0.826+0.751+0.683+0.621+0.564+0.513+0.467+0.424+0.386

5-6.Anannuityisaseriesofequaldollarpaymentsforaspecifiednumberofyears.Examplesofannuitiesincludemortgagepayments,interestpaymentsonbonds,fixedleasepayments,andanyfixedcontractualpayment.Aperpetuityisanannuitythatcontinuesforever,thatis,everyyearfromnowonthisinvestmentpaysthesamedollaramount.Thedifferencebetweenanannuityandaperpetuityisthataperpetuityhasnoterminationdatewhereasanannuitydoes.

SOLUTIONSTO

END-OF-CHAPTERPROBLEMS

SolutionstoProblemSetA

5-1A.（a）FVn=PV（1+i）n

FV10=$5,000（1+0.10）10

FV10=$5,000（2.594）

FV10=$12,970