TheTimeValueofMoney

ChapterOutlineTimevalueassociatedwithmoneyDeterminingfuturevalueatgiveninterestratePresentvaluebasedoncurrentvalueoffundstobereceivedDeterminingYieldonanInvestment.CompoundingordiscountingoccurringonalessthanannualbasisRelationshipto

TheCapitalOutlayDecisionThetimevalueofmoneyisusedtodeterminewhetherfuturebenefitsaresufficientlylargetojustifycurrentoutlaysMathematicaltoolsofthetimevalueofmoneyareusedinmakingcapitalallocationdecisionsFutureValue–SingleAmountMeasuringvalueofanamountthatisallowedtogrowatagiveninterestoveraperiodoftimeAssumingthattheworthof$1,000needstobecalculatedafter4yearsata10%interestperyear,wehave: 1styear……$1,000X1.10=$1,100 2ndyear…...$1,100X1.10=$1,210 3rdyear……$1,210X1.10=$1,331 4thyear……$1,331X1.10=$1,464FutureValue–SingleAmount(Cont’d)AgeneralizedformulaforFutureValue:

WhereFV=FuturevaluePV=Presentvaluei=Interestraten=Numberofperiods;Inthepreviouscase,PV=$1,000,i=10%,n=4,hence;FutureValueof$1(FVIF)Table9–1FutureValue–SingleAmount(Cont’d)Indeterminingfuturevalue,thefollowingcanbeused: Where=theinterestfactorIf$10,000wereinvestedfor10yearsat8%,thefuturevaluewouldbe:PresentValue–SingleAmountAsumpayableinthefutureisworthlesstodaythanthestatedamountTheformulaforthepresentvalueisderivedfromtheoriginalformulaforfuturevalue:Thepresentvaluecanbedeterminedbysolvingforamathematicalsolutiontotheformulaabove,thusrestatingtheformulaas:AssumingPresentValueof$1(PVIF)Table9–2RelationshipofPresent

andFutureValueFutureValue–AnnuityAnnuity:AseriesofconsecutivepaymentsorreceiptsofequalamountFutureValueofanAnnuity:CalculatedbycompoundingeachindividualpaymentintothefutureandthenaddingupallofthesepaymentsFutureValue–Annuity(cont’d)AgeneralizedformulaforFutureValueofAnnuity: FVA=A×FVIFA

Where:FVA=FuturevalueoftheAnnuityFVIFA=AnnuityFactor={[(1+i)n–1]÷i}A=Annuityvaluei=Interestraten=Numberofperiods;Assuming,A=$1,000,n=4,andi=10%CompoundingProcessforAnnuityFutureValue

ofanAnnuityof$1(FVIFA)Table9–3PresentValue–AnnuityCalculatedbydiscountingeachindividualpaymentbacktothepresentandthenaddingupallofthesepaymentsAgeneralizedformulaforPresentValueofAnnuity: PVA=A×PVIFA

Where:PVA=PresentvalueoftheAnnuityPVIFA=AnnuityFactor={1–[1÷(1+i)n]÷i}A=Annuityvaluei=Interestraten=NumberofperiodsPresentValue

ofanAnnuityof$1(PVIFA)AssumingthatA=$1,000,n=4,i=10%,wehave:Table9–4TimeValueRelationshipsComparisonsinclude:TherelationshipbetweenpresentvalueandfuturevalueInverserelationshipexistsbetweenthepresentvalueandfuturevalueofasingleamountTherelationshipbetweenthePresentValueofasingleamountandthePresentValueofanAnnuityThePresentValueofanAnnuityisthesumofthepresentvaluesofsingleamountspayableattheendofeachperiodTherelationshipbetweentheFutureValueandFutureValueofAnnuityTheFutureValueofanAnnuityisthesumofthefuturevaluesofsingleamountsreceivableattheendofeachperiodDeterminingtheAnnuityValueAre-lookatthevariablesinvolvedintimevalueofmoney:FV/PV:Future/PresentvalueofmoneyN:no.ofyearsI:InterestorYIELDA:AnnuityValue/paymentperperiodinanannuityGiventhefirstthreevariables,anddeterminingthefourthvariable“A”(unknown).AnnuityEqualingaFutureValueAssumingthatata10%interestrate,after4years,anamountof$4,641needstoaccumulated:Forn=4,andi=10%,is4.641.Thus,Aequals$1,000asbelow:AnnuityEqualingaPresentValueDeterminingwhatsizeofanannuitycanbeequatedtoagivenamount:Assumingn=4,i=6%:RelationshipofPresent

ValuetoAnnuityAnnualinterestisbasedonthebeginningbalanceforeachyearasshowninthefollowingtablethatshowsflowoffunds:Table9–5LoanAmortization Amortgageloantoberepaidover20years at8%interest:LoanAmortizationTableInsuchacasethepartofthepaymentstothemortgagecompanywillgotowardthepaymentofinterest,withtheremainderappliedtodebtreduction,asindicatedinthefollowingtable:Table9–6SixFormulasDeterminingtheYieldonInvestmentDeterminingtheunknownvariable“i“,giventhefollowingvariables:FV/PV:Future/PresentvalueofmoneyN:no.ofyearsA:AnnuityValue/paymentperperiodinanannuityYield–PresentValue

ofaSingleAmountTocalculatetheyieldonaninvestmentproducing$1,464after4yearshavingapresentvalueof$1,000:Weseethatforn=4and=0.683,theinterestrateoryieldis10%Yield–PresentValue

ofaSingleAmount(Cont’d)InterpolationmayalsobeusedtofindamorepreciseanswerDifferencebetweenthevalueatthelowestinterestrateandthedesignatedvalueTheexactvaluecanbedeterminedas:Yield–PresentValueofanAnnuityTocalculatetheyieldonaninvestmentof$10,000,producing$1,490perannumfor10years:Hence:Yield–PresentValueofanAnnuity(Cont’d)FlipbacktothetablecontainingthePresentValue-AnnuityfactorsonSlide9-16Readacrossthecolumnsforn=10periods,onecanseethattheyieldis8percentInterpolationappliedtoasingleamountcanalsobeappliedhereforamorepreciseanswerSpecialConsiderations

inTimeValueAnalysisCompoundingfrequencyCertaincontractualagreementsmayrequiresemiannual,quarterly,ormonthlycompoundingperiodsInsuchcases, N=No.ofyears

×No.ofcompoundingperiods duringtheyear I=Quotedannualinterest

/No.of compoundingperiodsduringtheyearSpecialConsiderations

inTimeValueAnalysisPatternsofPaymentProblemsmayevolvearoundanumberofdifferentpaymentorreceiptpatternsNoteverysituationinvolvesasingleamountoranannuityAcontractmaycallforthepaymentofadifferentamounteachyearoverthestatedperiodorperiodofannuityCompoundingfrequency:CasesCase1:Determinethefuturevalueofa$1,000investmentafter5yearsat8%annualinterestcompoundedsemiannuallyWhere,n=5×2=10;i=8%/2=4%(usingTable9–1FVIF=1.480)Case2:Determinethepresentvalueof20quarterlypaymentsof$2,000eachtobereceivedoverthenext5years,wherei=8%perannumWhere,n=20;i=2%PatternsofPayment:CasesAssumeacontractinvolvingpaymentsofdifferentamountseachyearforathree-yearperiodTodeterminethepresentvalue,eachpaymentisdiscountedtothepresentandthentotaled (Assuming8%discountrate)DeferredAnnuitySituationsinvolvingacombinationofsingleamountsandanannuity.WhenannuityispaidsometimeinthefutureDeferredAnnuity:CaseAssumingacontractinvolvingpaymentsofdifferentamountseachyearforathreeyearperiod:Anannuityof$1,000ispaidattheendofeachyearfromthefourththroughtheeighthyearTodeterminethepresentvalueofthecashflowsat8%discountrateTodeterminetheannuityDeferredAnnuity:Case(Cont’d)Todiscountthe$3,993backtothepresent,whichfallsatthebeginningofthefourthperiod,ineffec