Time Value of Money

TimeValueofMoneyFuturevaluePresentvalueAnnuitiesRatesofreturnAmortizationTimelinesShowthetimingofcashflows.Tickmarksoccurattheendofperiods,soTime0istoday;Time1istheendofthefirstperiod(year,month,etc.)orthebeginningofthesecondperiod.CF0CF1CF3CF20123I%Drawingtimelines1001001000123I%3year$100ordinaryannuity100012I%$100lumpsumduein2yearsDrawingtimelines10050750123I%-50UnevencashflowstreamWhatisthefuturevalue(FV)ofaninitial$100after3years,ifI/YR=10%?FindingtheFVofacashfloworseriesofcashflowsiscalledcompounding.FVcanbesolvedbyusingthestep-by-step,financialcalculator,andspreadsheetmethods.FV=?012310%100SolvingforFV:Thestep-by-stepandformulamethodsAfter1year:FV1=PV(1+I)=$100(1.10)=$110.00After2years:FV2=PV(1+I)2=$100(1.10)2=$121.00After3years:FV3=PV(1+I)3=$100(1.10)3=$133.10AfterNyears(generalcase):FVN=PV(1+I)NSolvingforFV:ThecalculatormethodSolvesthegeneralFVequation.Requires4inputsintocalculator,andwillsolveforthefifth.(SettoP/YR=1andENDmode.)INPUTSOUTPUTNI/YRPMTPVFV3100133.10-100PV=?100Whatisthepresentvalue(PV)of$100duein3years,ifI/YR=10%?FindingthePVofacashfloworseriesofcashflowsiscalleddiscounting(thereverseofcompounding).ThePVshowsthevalueofcashflowsintermsoftoday’spurchasingpower.012310%SolvingforPV:TheformulamethodSolvethegeneralFVequationforPV:PV=FVN/(1+I)NPV=FV3/(1+I)3=$100/(1.10)3=$75.13SolvingforPV:ThecalculatormethodSolvesthegeneralFVequationforPV.ExactlylikesolvingforFV,exceptwehavedifferentinputinformationandaresolvingforadifferentvariable.INPUTSOUTPUTNI/YRPMTPVFV3100100-75.13SolvingforI:Whatinterestratewouldcause$100togrowto$125.97in3years?SolvesthegeneralFVequationforI.Hardtosolvewithoutafinancialcalculatororspreadsheet.INPUTSOUTPUTNI/YRPMTPVFV380125.97-100SolvingforN:Ifsalesgrowat20%peryear,howlongbeforesalesdouble?SolvesthegeneralFVequationforN.Hardtosolvewithoutafinancialcalculatororspreadsheet.INPUTSOUTPUTNI/YRPMTPVFV3.82002-1Whatisthedifferencebetweenanordinaryannuityandanannuitydue?OrdinaryAnnuityPMTPMTPMT0123i%PMTPMT0123i%PMTAnnuityDueSolvingforFV:3-yearordinaryannuityof$100at10%$100paymentsoccurattheendofeachperiod,butthereisnoPV.INPUTSOUTPUTNI/YRPMTPVFV310-1003310SolvingforPV:3-yearordinaryannuityof$100at10%$100paymentsstilloccurattheendofeachperiod,butnowthereisnoFV.INPUTSOUTPUTNI/YRPMTPVFV3101000-248.69SolvingforFV:3-yearannuitydueof$100at10%Now,$100paymentsoccuratthebeginningofeachperiod.FVAdue=FVAord(1+I)=$331(1.10)=$364.10.Alternatively,setcalculatorto“BEGIN”modeandsolvefortheFVoftheannuity:INPUTSOUTPUTNI/YRPMTPVFV310-100364.100BEGINSolvingforPV:3-yearannuitydueof$100at10%Again,$100paymentsoccuratthebeginningofeachperiod.PVAdue=PVAord(1+I)=$248.69(1.10)=$273.55.Alternatively,setcalculatorto“BEGIN”modeandsolveforthePVoftheannuity:INPUTSOUTPUTNI/YRPMTPVFV3101000-273.55BEGINWhatisthepresentvalueofa5-year$100ordinaryannuityat10%?BesureyourfinancialcalculatorissetbacktoENDmodeandsolveforPV:N=5,I/YR=10,PMT=100,FV=0.PV=$379.08Whatifitwerea10-yearannuity?A25-yearannuity?Aperpetuity?10-yearannuityN=10,I/YR=10,PMT=100,FV=0;solveforPV=$614.46.25-yearannuityN=25,I/YR=10,PMT=100,FV=0;solveforPV=$907.70.PerpetuityPV=PMT/I=$100/0.1=$1,000.ThePowerofCompoundInterestA20-year-oldstudentwantstosave$3adayforherretirement.Everydaysheplaces$3inadrawer.Attheendoftheyear,sheinveststheaccumulatedsavings($1,095)inabrokerageaccountwithanexpectedannualreturnof12%.Howmuchmoneywillshehavewhensheis65yearsold?SolvingforFV:Ifshebeginssavingtoday,howmuchwillshehavewhensheis65?Ifshestickstoherplan,shewillhave$1,487,261.89whensheis65.INPUTSOUTPUTNI/YRPMTPVFV4512-10951,487,2620SolvingforFV:Ifyoudon’tstartsavinguntilyouare40yearsold,howmuchwillyouhaveat65?Ifa40-year-oldinvestorbeginssavingtoday,andstickstotheplan,heorshewillhave$146,000.59atage65.Thisis$1.3millionlessthanifstartingatage20.Lesson:Itpaystostartsavingearly.INPUTSOUTPUTNI/YRPMTPVFV2512-1095146,0010SolvingforPMT:Howmuchmustthe40-yearolddepositannuallytocatchthe20-yearold?Tofindtherequiredannualcontribution,enterthenumberofyearsuntilretirementandthefinalgoalof$1,487,261.89,andsolveforPMT.INPUTSOUTPUTNI/YRPMTPVFV2512-11,154.421,487,2620WhatisthePVofthisunevencashflowstream?010013002300310%-50490.91247.93225.39-34.15530.08=PVSolvingforPV:UnevencashflowstreamInputcashflowsinthecalculator’s“CFLO”register:CF0=0CF1=100CF2=300CF3=300CF4=-50EnterI/YR=10,pressNPVbuttontogetNPV=$530.087.(HereNPV=PV.)WilltheFVofalumpsumbelargerorsmallerifcompoundedmoreoften,holdingthestatedI%constant?LARGER,asthemorefrequentlycompoundingoccurs,interestisearnedoninterestmoreoften.Annually:FV3=$100(1.10)3=$133.10012310%100133.10Semiannually:FV6=$100(1.05)6=$134.0101235%456134.011230100ClassificationsofinterestratesNominalrate(INOM)–alsocalledthequotedorstaterate.Anannualratethatignorescompoundingeffects.INOMisstatedincontracts.Periodsmustalsobegiven,e.g.8%Quarterlyor8%Dailyinterest.Periodicrate(IPER)–amountofinterestchargedeachperiod,e.g.monthlyorquarterly.IPER=INOM/M,whereMisthenumberofcompoundingperiodsperyear.M=4forquarterlya