會(huì)計(jì)學(xué)：企業(yè)決策的基礎(chǔ)（管理會(huì)計(jì)分冊(cè)）（原書第19版）課件 AppB- The Time Value of Money Future Amounts and Present Values

AppendixBTheTimeValueofMoney:FutureAmountsandPresentValuesConceptofTimeValueofMoneyOneofthemostbasic—andimportant—conceptsofinvestingisthetimevalueofmoney.Thisconceptisbasedontheideathatanamountofmoneyavailabletodaycanbeinvestedtoaccumulatetoalargeramountinthefuture.Asaresult,anamountofmoneyavailabletodayisconsideredtobeequivalentinvaluetoalargersumavailableatafuturedate.Wewillrefertoanamountofmoneyavailabletodayasapresentvalue.

Incontrast,anamountreceivableorpayableatafuturedatewillbedescribedasafutureamount.Illustration:ValueofMoneyoverTimeAssumethatyouplace$500inaninvestmentaccountthatearnsinterestattherateof8percentperyear.Thebalanceofyouraccountattheendofeachofthenextfouryearsisasfollows:RelationshipsbetweenPresentValuesandFutureAmountsThedifferencebetweenapresentvalueandanyfutureamountistheinterestthatisincludedinthefutureamount.Wehaveseenthatinterestaccruesovertime.Therefore,thedifferencebetweenthepresentvalueandafutureamountdependsontwofactors:Therateofinterestatwhichthepresentvalueincreases.Thelengthoftimeoverwhichinterestaccumulates.PresentValuesChangeoverTimeNoticethatthepresentvalueofyourinvestmentaccountisalwayslessthanitsfutureamounts.Thisisthebasicideaunderlyingthetimevalueofmoney.Butthisideaoftenisexpressedindifferentways,includingthefollowing:Apresentvalueisalwayslessthanafutureamount.Afutureamountisalwaysgreaterthanapresentvalue.Adollaravailabletodayisalwaysworthmorethanadollarthatdoesnotbecomeavailableuntilafuturedate.Adollaravailableatafuturedateisalwaysworthlessthanadollarthatisavailabletoday.CompoundInterestTherelationshipsbetweenpresentvaluesandfutureamountsassumethattheinterestearnedontheinvestmentisreinvested,ratherthanwithdrawn.Thisconceptofteniscalledcompoundingtheinterest.

Compoundinghasaninterestingeffect.Reinvestingtheinterestcausestheamountinvestedtoincreaseeachperiod.This,inturn,causesmoreinteresttobeearnedineachsuccessiveperiod.Overalongperiodoftime,aninvestmentinwhichinterestiscompoundedcontinuouslywillincreasetosurprisinglylargeamounts.CaseinPointIn1626,PeterMinuitissaidtohavepurchasedManhattanIslandfromagroupofIndiansfor$24worthof“beads,cloth,andtrinkets.”Thisepisodeoftenisportrayedasanincrediblebargain—evenasteal.ButiftheIndianshadinvestedthis$24toearninterestatacompoundinterestrateof8percent,theywouldhavemorethanenoughmoneytodaytobuytheislandback—alongwitheverythingonit.ApplicationsoftheTimeValueofMoneyInvestors,accountants,andotherdecisionmakersapplythetimevalueofmoneyinthreebasicways.Theseapplicationsaresummarizedasfollows:Todeterminetheamounttowhichaninvestmentwillaccumulateovertime.Todeterminetheamountthatmustbeinvestedeveryperiodtoaccumulatearequiredfutureamount.Todeterminethepresentvalueofcashflowsexpectedtooccurinthefuture.FutureAmountsAfutureamountissimplythedollaramounttowhichapresentvaluewillaccumulateovertime.Thedifferencebetweenapresentvalueandarelatedfutureamountdependson:Theinterestrate.Theperiodoftimeoverwhichthepresentvalueaccumulates.Startingwiththepresentvalue,wemaycomputefutureamountsthroughaseriesofmultiplications.FutureValueof$1afternPeriodsTheTablesApproachAtableoffutureamountsshowsthefutureamounttowhich$1willaccumulateoveragivennumberofperiods,assumingthatithasbeeninvestedtoearnanyoftheillustratedinterestrates.Wewillrefertotheamountsshowninthebodyofthesetablesasfactors,ratherthanasdollaramounts.Tofindthefutureamountofapresentvaluegreaterthan$1,simplymultiplythepresentvaluebythefactorobtainedfromthetable.TheformulaforusingthisapproachisasfollowsFutureAmount=PresentValue×Factor

(TableFA-1inpreviousslide)ComputingtheRequiredInvestmentAtthebeginningofyear1,MetroRecyclingagreestocreateafullyfundedpensionplanforitsemployeesbytheendofyear5.Itisestimatedthat$5millionwillberequiredtofullyfundthepensionplan.HowmuchmustMetroinvestinthisplantodaytoaccumulatethepromised$5millionbytheendofyear5,assumingthatpaymentstothefundwillbeinvestedtoearnanannualreturnof8percent?

FutureAmount=PresentValue×FactorPresentValue=FutureAmount÷Factor(TableFA-1)$3,403,676=$5million÷1.469TheFutureAmountofaSingleInvestmentTheFutureAmountofanAnnuityInmanysituations,aninvestorwillmakeaseriesofinvestmentpaymentsratherthanasinglepayment.Tofindthefutureamountofanordinaryannuityofpaymentsgreaterthan$1,wesimplymultiplytheamountoftheperiodicpaymentbythefactorappearinginthetable.FutureAmountofanAnnuity=PeriodicPayment×Factor(TableFA-2)FutureAmountof$PaidPeriodicallyfornPeriodsIllustration:TheFutureAmountofanAnnuityAsanexample,assumethatyouplantodeposit$500intoyourinvestmentaccountattheendofeachofthenextfiveyears.Iftheaccountpaysannualinterestof8percent,whatwillbethebalanceinyouraccountattheendofthefifthyear?FutureAmountofanAnnuity=PeriodicPayment×Factor(TableFA-2)$2,934=$500×5.867ComputingtheRequiredPeriodicPaymentsAssumethatUltraTechCompanyisrequiredtoaccumulate$10millioninabondsinkingfundtoretirebondspayablefiveyearsfromnow.ThebondindenturerequiresUltraTechtomakeequalpaymentstothefundattheendofeachofthenextfiveyears.Whatistheamountoftherequiredperiodicpayment,assumingthatthefundwillearn10percentannualinterest?FutureAmountofanAnnuity=PeriodicPayment×Factor(TableFA-2)PeriodicPayment=FutureAmountofanAnnuity÷Factor(TableFA-2)$1,638,000=$10million÷6.105FutureAmountofaSeriesofInvestmentsInterestPeriodsofLessthanOneYearInourcomputationsoffutureamounts,wehaveassumedthatinterestispaid(compounded)orpaymentsaremadeannually.Therefore,inusingthetables,weusedannualperiodsandanannualinterestrate.Investmentpaymentsorinterestpaymentsmaybemadeonamorefrequentbasis,suchasmonthly,quarterly,orsemiannually.Presentvaluetablesmaybeusedwithanyofthesepaymentperiods,buttherateofinterestmustrepresenttheinterestrateforthatperiod.Example:MonthlyPaymentsAssumethat24monthlypaymentsaretobemadetoaninvestmentfundthatpaysa12percentannualinterestrate.Todeterminethefutureamountofthisinvestment,wewouldmultiplytheamountofthemonthlypaymentsbythefactorfromTableFA–2for24periods,usingamonthlyinterestrateof1percent—the12percentannualratedividedby12months.PresentValuesThepresentvalueistoday’svalueoffundstobereceivedinthefuture.Whilepresentvaluehasmanyapplicationsinbusinessandaccounting,itismosteasilyexplainedinthecontextofevaluatinginvestmentopportunities.Inthiscontext,thepresentvalueistheamountthataknowledgeableinvestorwouldpaytodayfortherighttoreceiveanexpectedfutureamountofcash.Thepresentvalueisalwayslessthanthefutureamount,becausetheinvestorwillexpecttoearnareturnontheinvestment.PresentValues(cont.)Theamountbywhichthefuturecashreceiptexceedsitspresentvaluerepresentstheinvestor’sprofit.Theamountoftheprofitonaparticularinvestmentdependsontwofactors:Therateofreturn(calledthediscountrate)requiredbytheinvestor.Thelengthoftimeuntilthefutureamountwillbereceived.Theprocessofdeterminingthepresentvalueofafuturecashreceiptiscalleddiscountingthefutureamount.Illustration:PresentValueAssumethataninvestmentisexpectedtoresultina$1,000cashreceiptattheendofoneyearandthataninvestorrequiresa10percentreturnonthisinvestment.Thedifferencebetweenapresentvalueandafutureamountisthereturn(interest)ontheinvestment.Inourexample,thefutureamountwouldbeequalto110percentoftheoriginalinvestment,becausetheinvestorexpects100percentoftheinvestmentbackplusa10percentreturnontheinvestment.Thus,theinvestorwouldbewillingtopay$909($1,000

÷

1.10)forthisinvestment.Ifthe$1,000istobereceivedtwoyearsinthefuture,theinvestorwouldpayonly$826fortheinvestmenttoday[($1,000

÷

1.10)

÷

1.10].Illustration:PresentValue(cont.)WhatistheAppropriateDiscountRate?Thediscountratemaybeviewedastheinvestor’srequiredrateofreturn.Allinvestmentsinvolvesomedegreeofriskthatactualfuturecashflowsmayturnouttobelessthanexpected.Investorswillrequirearateofreturnthatjustifiestakingthisrisk.Whenahigherdiscountrateisused,thepresentvalueoftheinvestmentwillbelower.Inotherwords,astheriskofaninvestmentincreases,itsvaluetoinvestorsdecreases.ThePresentValueofanAnnuityManyinvestmentopportunitiesareexpectedtoproduceannualcashflowsforanumberofyears,insteadofonesinglefuturecashflow.Ifthecashflowsareuniform,tofindthepresentvalueofanordinaryannuitywesimplymultiplytheamountoftheperiodicpaymentbythefactorappearinginthetable.PresentValueofanAnnuity=PeriodicCashFlows×Factor(TablePV-2)PresentValuesof$toBeReceivedPeriodicallyfornPeriodsIllustration:TheFutureAmountofanAnnuityAsanexample,assumethataninvestmentwasexpectedtoreturn$10,000peryearforthenextthreeyears,andthecompany’srequiredrateofreturnwas12percentperyear.Whatisthepresentvalueoftheinvestment?PresentValueofanAnnuity=PeriodicCashFlows×Factor(TablePV-2)$24,020=$10,000×

2.402DiscountPeriodsofLessthanOneYearTheintervalbetweenregularperiodiccashflowsiscalledthediscountperiod.

Inourprecedingexamples,wehaveassumedcashflowsonceayear.Often,cashflowsoccuronamorefrequentbasis,suchasmonthly,quarterly,orsemiannually.Thepresentvaluetablescanbeusedwithdiscountperiodsofanylength,butthediscountratemustbeforthatlengthoftime.ValuationofFinancialInstrumentsCashEquityContractsAccountantsusethephrasefinancialinstrumentstodescribecash,equityinvestmentinanotherbusiness,andanycontractsthatcallforreceiptsorpaymentsofcash.Wheneverthepresentvalueofafinancialinstrumentdifferssignificantlyfromthesumoftheexpectedfuturecashflows,theinstrumentisrecordedintheaccountingrecordsatitspresentvalue—notattheexpectedamountofthefuturecashreceiptsorpayments.ValuationofBalanceSheetAccountsMarketableSecuritiesAccountsReceivableAccountsPayableAppearinthebalancesheetattheircurrentmarketvalues,whichrepresentstheirpresentvalue.Appearinthebalancesheetattheamountsexpectedtobecollectedorpaidinthenearfuture.Technically,thesearefutureamountsbuttheyareusuallyreceivedorpaidwithin30or60days,sothedifferencesbetweenthesefutureamountsandtheirpresentvaluessimplyarenotmaterial.Interest-BearingReceivablesandPayablesWhenafinancialinstrumentcallsforthereceiptorpaymentofinterest,thedifferencebetweenthepresentvalueandthefutureamountsdoesbecomematerial.Thus,interest-bearingreceivablesandpayablesinitiallyarerecordedinaccountingrecordsatthepresentvalueofthefuturecashflows—alsocalledthe“principalamount”oftheobligation.Thispresentvalueoftenissubstantiallylessthanthesumoftheexpectedfutureamounts.Illustration:BondsPayableConsider,forexample,$100millionin30-year,9percentbondspayableissuedatpar.Attheissuancedate,thepresentvalueofthisbondissueis$100million—theamountofcashreceived.Butthefuturepaymentstobondholdersareexpectedtototal$370million,computedasfollows:Thusthe$100millionissuancepricerepresentsthepresentvalueof$370millioninfuturecashpaymentstobemadeoveraperiodof30years.“Non-Interest-Bearing”NotesOnoccasion,companiesmayissueoracceptnotesthatmakenomentionofinterest,orinwhichthestatedinterestratesareunreasonablylow.Ifthedifferencebetweenthepresentvalueofsuchanoteanditsfaceamountismaterial,thenoteinitiallyisrecordedatitspresentvalue.“Non-Interest-Bearing”Notes(cont.1)AssumethatonJanuary1,2021,ElronCorporationpurchaseslandfromU.S.DevelopmentCo.Asfullpaymentforthisland,Elronissuesa$300,000installmentnotepayable,dueinthreeannualinstallmentsof$100,000,beginningDecember31,2021.Thisnotemakesnomentionofinterestcharges.Elronshouldusethepresentvalueofthisnote—notthefaceamount—indeterminingthecostofthelandandreportingitsliability.Assumethatarealisticinterestrateforfinancinglandoverathree-yearperiodcurrentlyis10percent.PresentValueofanAnnuity=PeriodicCash