第三章 货币的时间价值 chapter3- Time value of money

Doyouhavethesepuzzles?Buyamobilebylump-sumpaymentorbyinstallment?Arrangesavingsforafutureexpenditure?Whatkindofloanstoapplyfor?Buyorsellabondyouareholding?……2024/6/12Ch3TimeValueofMoneyCHAPTER3

TimeValueofMoneyCompoundingandDiscountingofSingleSumsAnnuitiesTypesofInterestRates2024/6/12Ch3TimeValueofMoneyCH3TheTimeValueofMoneyTofindtheanswer,you’llhavetoknow2024/6/12Ch3TimeValueofMoney3.1CompoundingandDiscountingSingleSums2024/6/12Ch3TimeValueofMoneyWeknowthatreceiving$1todayisworthmorethan$1inthefuture.Thisisdue

to

opportunitycosts.Theopportunitycostofreceiving$1inthefutureisthe

interest

wecouldhaveearnedifwehadreceivedthe$1sooner.TodayFuture2024/6/12Ch3TimeValueofMoneyIfwecanmeasurethisopportunitycost,wecan:Translate$1todayintoitsequivalentinthefuture

(compounding).Translate$1inthefutureintoitsequivalenttoday

(discounting).?TodayFutureToday?Future2024/6/12Ch3TimeValueofMoneyTimelinesShowthetimingofcashflows.Tickmarksoccurattheendofperiods,soTime0istoday;Time1istheendofthefirstperiod(year,month,etc.)orthebeginningofthesecondperiod.CF0CF1CF3CF20123i%2024/6/12Ch3TimeValueofMoneyDrawingtimelines:

1）$100lumpsumduein2years;100012i%$100lumpsumduein2years2024/6/12Ch3TimeValueofMoneyDrawingtimelines:

2)Unevencashflowstream;CF0=-$50,

CF1=$100,CF2=$75,andCF3=$5010050750123i%-50Unevencashflowstream2024/6/12Ch3TimeValueofMoneyCompounding:TofindtheFutureValue(FV)ofacashflow,wesupposeweearninterestonprincipalaswellasoninterestaccumulatedeachterm.Today?Future2024/6/12Ch3TimeValueofMoneyExampleWhatisthefuturevalue(FV)ofaninitial$100after3years,ifI/YR=10%?FVcanbesolvedbyusingthearithmetic,orTableA-3(pp.A-6)FV=?012310%1002024/6/12Ch3TimeValueofMoneySolvingforFV:

ThearithmeticmethodAfter1year:FV1=PV(1+i)=$100(1.10)

=$110.00After2years:FV2=PV(1+i)2=$100(1.10)2

=$121.00After3years:FV3=PV(1+i)3=$100(1.10)3

=$133.10Afternyears(generalcase):FVn=PV(1+i)n2024/6/12Ch3TimeValueofMoneySolvingforFV:

Thetable-checkingmethodFVn=PV*FVIF(i,n)FV3=$100*FVIF(10%,3)=$100*1.331=$133.102024/6/12Ch3TimeValueofMoneyDiscounting:FindingthePresentValue(PV)ofacashfloworseriesofcashflowswhencompoundinterestisapplied(thereverseofcompounding).?TodayFuture2024/6/12Ch3TimeValueofMoneyPV=?100ExampleWhatisthepresentvalue(PV)of$100duein3years,ifI/YR=10%?012310%2024/6/12Ch3TimeValueofMoneySolvingforPV:

ThearithmeticmethodSolvethegeneralFVequationforPV:PV=FVn/(1+i)nPV=FV3/(1+i)3 =$100/(1.10)3 =$75.132024/6/12Ch3TimeValueofMoneySolvingforPV:

Thetable-checkingmethodSolvethegeneralFVequationforPV:PV=FVn*PVIF(i,n)Ref.:TableA-1(PP.A-2)PV=FV3*PVIF(10%,3)=$100*0.7513 =$75.132024/6/12Ch3TimeValueofMoneySolvesthegeneralFVorPVequationforN.Sameaspreviousproblems,butnowsolvingforN.

Eg.Pp.104Otherapplication1:SolvingforN

WithI,PV,FVknowntoyou2024/6/12Ch3TimeValueofMoneySolvesthegeneralFVorPVequationfori.Sameaspreviousproblems,butnowsolvingforN.Eg.Pp.103Otherapplication2:SolvingforI

Withn,PV,FVknowntoyou2024/6/12Ch3TimeValueofMoneyAmagic72

ruleWhenthecompoundinterestrateislessthan20%,theyearsyouneedtodoubleyourtoday’swealthis72/theinterestratenumber.Conversely,ifyouwanttodoubleyourwealthinXyears,youhavetoensureyourinvestmentearn[72/X]%annually!2024/6/12Ch3TimeValueofMoneyThePowerofCompoundInterest

compounding—the8thwonder!Thepowerofcompoundinterestexceedsthenuclearbomb!2024/6/12Ch3TimeValueofMoneyStoryofanancientAthenianThisAthenianpocketedallthemoneyexceptasingleDrachma,whichheinvestedinAtheniangovernmentbondspaying3percentcompoundedannually.Hedidn'tlivelongenoughtoseetheresults,butafter2,000yearsthatDrachmaewoundupbeingworthmorethanalltheassetsontheEarth!2024/6/12Ch3TimeValueofMoneyTomakethe“nuclearbomb”work,investingearly,or,alongenoughinvestmentperiod

isnecessary!2024/6/12Ch3TimeValueofMoneySupposethemerchantinvestsonly3years,hewillhaveonly1*FVIF(3%,3)=1.0927Drachmae2024/6/12Ch3TimeValueofMoneyasequenceofequalcashflows,occurringatfixedintervalsforaspecifiednumberofperiods.012343.2Annuities2024/6/12Ch3TimeValueofMoneyabond’ssemi-annualcouponinterestpaymentsoverthelifeofthebond.Repayingbankloans:astreamofequalrepayments.ExamplesofAnnuities:2024/6/12Ch3TimeValueofMoneyTypesofannuities:

Ordinary(deferred)annuityAnnuityduePerpetuities2024/6/12Ch3TimeValueofMoneyDifferencebetweenanordinaryannuityandanannuitydueOrdinaryAnnuityPMTPMTPMT0123i%PMTPMT0123i%PMTAnnuityDue2024/6/12Ch3TimeValueofMoney3.2.1SolvingforFVandPV

ofordinaryannuities

2024/6/12Ch3TimeValueofMoneyFVAn=A*FVIFA(i,n)Eg.WhatistheFVof3-yearordinaryannuityof$100at10%?FVA3=$100*FVIFA(10%,3)=$100*3.310=$331.0SolvingforFVofordinaryannuity:

2024/6/12Ch3TimeValueofMoneyHowaboutthis?SolveforFV01234100100100FVA3=100*FVIFA(10%,3)=100*3.310=331.0010%012341001001002024/6/12Ch3TimeValueofMoneyPVAn=A*PVIFA(i,n)Eg.WhatisthePVof3-yearordinaryannuityof$100at10%?PVAn=$100*PVIFA(10%,3)=$100*2.487=$248.70exercises:P.1273-6SolvingforPVofordinaryannuity:

2024/6/12Ch3TimeValueofMoneyHowaboutthis?SolveforPV01234100100100100*PVIFA(10%,3)*PVIF(10%,1)=100*2.487*0.9091=226.0910%2024/6/12Ch3TimeValueofMoney3.2.2SolvingforFVandPVof

annuitiesdue2024/6/12Ch3TimeValueofMoneySolvingforFVofannuitydue:

ItcanbecalculatedbyadjustingtheFVequationforordinaryannuity:FVAn(annuitydue)=A*[FVIFA(i,n+1)-1]2024/6/12Ch3TimeValueofMoneyEg.WhatistheFVof3-yearannuitydueof$100at10%?FVA3(annuitydue)=100*[FVIFA(i,n+1)-1]=100*[FVIFA(10%,4)-1]=100*(4.641-1)=364.1Exercises：P.1273-72024/6/12Ch3TimeValueofMoneySolvingforPVofannuitydue:

ItcanbecalculatedbyadjustingthePVequationforordinaryannuity:PVAn(annuitydue)=A*[PVIFA(i,n-1)+1]2024/6/12Ch3TimeValueofMoneyEg.WhatisthePVof3-yearannuitydueof$100at10%?PVA3(annuitydue)

=A*[PVIFA(i,n-1)+1]=100*[PVIFA(10%,2)+1]=100*(1.736+1)=$273.60Exercises：P.127Whatisthepresentvalueofthecashflowin3-7?2024/6/12Ch3TimeValueofMoney3.2.3PerpetuitiesPerpetuityisastreamofequalpaymentsexpectedtocontinueforever.

01234AAAA….2024/6/12Ch3TimeValueofMoneyCanyoudecidetheFVofaperpetuity?PVofaperpetuity:PV(perpetuity)=A/IEg.Johnwantstoestablishafundtoaidthehandicapped$10000peryear.Supposetheannualinvestmentreturnofthefundisexpectedtobe8%forever,howmuchmoneyshouldheinvesttoday?PV(perpetuity)=A/I=10000/0.08=125,000.002024/6/12Ch3TimeValueofMoneyUnevencashflowstream010013002300310%-5042024/6/12Ch3TimeValueofMoneyWhatisthePVofthisunevencashflowstream?010013002300310%-50490.91247.93225.39-34.15530.08=PV2024/6/12Ch3TimeValueofMoneyExercises:P.1313-392024/6/12Ch3TimeValueofMoneyAgain:powerofcompounding

withthecaseofannuities1.A20-year-oldstudentwantstostartsavingforretirement.heplanstosave$3aday.Everyday,heputs$3inhisdrawer.Attheendoftheyear,heinveststheaccumulatedsavings($1,095)inanonlinestockaccount.Thestockaccounthasanexpectedannualreturnof12%.Howmuchmoneywillhehavewhenheis60yearsold?2024/6/12Ch3TimeValueofMoneyIfhebeginssavingtoday,andstickstoherplan,hewillhave$839,963.55whenheis60.FVA40=1095*FVIFA(12%,40)=839963.552024/6/12Ch3TimeValueofMoney2.Ifthestudentbeginssavingwhenheis40yearsold,hewillhaveFVA20=1095*FVIFA(12%,20)=$78896.94atage60.Thisis$761,066.61lessthanifstartingatage20.2024/6/12Ch3TimeValueofMoney3.Howmuchmustthe40-yearolddepositannuallytocatchthe20-yearold?thefinalgoal:$839,963.55Pmtperiod:202024/6/12Ch3TimeValueofMoneyFVA20=A*FVIFA(12%,20)=839963.55SolveforA:A=839963.55/FVIFA(12%,20)=$11657.74Thestudenthastosave$31.94dailytoachievethegoal.10timesofthe20years’old!2024/6/12Ch3TimeValueofMoneyLesson:

Itpaystostartsavingearly.2024/6/12Ch3TimeValueofMoneyCompoundingfrequenciesalsomatter!2024/6/12Ch3TimeValueofMoneyWilltheFVofalumpsumbelargerorsmallerifcompoundedmoreoften,holdingthestatedI%constant?LARGER,asthemorefrequentlycompoundingoccurs,interestisearnedoninterestmoreoften.Annually:FV3=$100(1.10)3=$133.10012310%100133.10Semiannually:FV6=$100(1.05)6=$134.0101235%456134.0112301002024/6/12Ch3TimeValueofMoney3.3TypesofinterestratesNominalratePeriodicrateEffectiveannualrateRefertop.1182024/6/12Ch3TimeValueofMoney1)Nominalrate(iNOM)–alsocalledthequotedorstatedrate.Anannualratethatignorescompoundingeffects.iNOMisstatedincontracts.Thenumberofcompoundingperiodsmustalsobegivene.g.8%,Quarterlyor8%,Daily.iNOMcanbecomparedwithoneanotheronlyiftheinstrumentshavethesamecompoundingperiodperyear.iNOMisnevershownonatimeline;neitherisitusedasaninputofcompoundingordiscountingcalculations.2024/6/12Ch3TimeValueofMoneyExampleForeignCurrencyTimeDepositInterestRates(%p.a.)ofHSBCCurrency1month3monthsAustralianDollar(AUD)5.09005.2000BritishPound(GBP)1.37501.7500CanadianDollar(CAD)1.12501.31252024/6/12Ch3TimeValueofMoney招商银行储蓄存款基准利率

利率单位为%/年

（2011.10）存期人民币活期0.5整存整取三个月3.1整存整取一年3.5整存整取二年4.42024/6/12Ch3TimeValueofMoney2)Periodicrate(iPER)–therateofinterestchargedeachcompounding/paymentperiod,e.g.monthlyorquarterly.iPER=iNOM/m,where:misthenumberofcompoundingperiodsperyear.Eg:m=4forquarterlyandm=12formonthlycompounding.2024/6/12Ch3TimeValueofMoney3)Effective(orequivalent)annualrate(EAR=EFF%)–theannualrateofinterestactuallybeingearned,takingintoaccountcompounding.EFF%for10%semiannualinvestment

EFF% =(1+iNOM/m)m-1 =(1+0.10/2)2–1=10.25%(iNOM/m—theperiodicrateM—thenumberofcompoundingperiodsperyear)2024/6/12Ch3TimeValueofMoneyAninvestorwouldbeindifferentbetweenaninvestmentofferinga10.25%annualreturnandoneofferinga10%annualreturn,compoundedsemiannually.2024/6/12Ch3TimeValueofMoneyEARisnotusedincalculations.Itisusedtocomparetheeffectivecostorrateofreturnwhenloans/investmenspaymentperiodsdiffer

2024/6/12Ch3TimeValueofMoneyExample: Effectiveratesofbankloanquotation10%,annually/quarterly/monthly/dailyare: EARANNUAL 10.00% EARQUARTERLY 10.38% EARMONTHLY 10.47% EARDAILY(365) 10.52%2024/6/12Ch3TimeValueofMoney招商银行三个月整存整取（3.1%），自动转存，一年的实际利率是多少？与一年期整存整取（3.5%）相比如何呢？EAR=3.14%2024/6/12Ch3TimeValueofMoneyCantheeffectiverateeverbeequaltothenominalrate?Yes,butonlyifannualcompoundingisused,i.e.,ifm=1.Ifm>1,EFF%willalwaysbegreaterthanthenominalrate.2024/6/12Ch3TimeValueofMoneyCreditcardDoyouhaveacreditcard?Doyouknowhowistheinterestcharged?2024/6/12Ch3TimeValueofMoney建行、工行等大部分银行:透支利息按月计收复利农业银行:透支利息按月计收取单利利率：大部分银行按日息万分之5计算，但也有例外，如：

中银长城人民币卡规定：自透支之日起15天内按日息万分之五计算，超过15天按日息万分之十计算，超过30天或透支金额超过规定限额的，按日息万分之十五计算2024/6/12Ch3TimeValueofMoneyIfacreditcardchargesDAILYinterestat0.05%,compoundedmonthly.WhatistheEFF%?Isitexpensive?EFF% =(1+1.5%)12-1

=20%2024/6/12Ch3TimeValueofMoney招商银行个人贷款参考利率

(自2011年7月生效)

贷款年限年利率6月-1年(含1年)6.1%2024/6/12Ch3TimeValueofMoneyAgain:Wheniseachrateused?iNOM writtenintocontracts,quotedby banksandbrokers.Notusedin calculationsorshownontimelines.iPER

Usedincalculationsandshownon timelines.Ifm=1,iNOM=iPER= EAR.EAR Usedtocomparereturnson investmentswithdifferentpayments peryear.2024/6/12Ch3TimeValueofMoneyWhatistheFVof$100after3yearsunder10%semiannualcompounding?Quarterlycompounding?2024/6/12Ch3TimeValueofMoneyInclassexercises:P.1313-37a;dP.1313-38a;dP.1323-40P.1323-41P.1323-442024/6/12Ch3TimeValueofMoneySummaryofCh3:p.123Keyconcepts:Compou