CHAPTER6

TimeValueofMoneyFuturevaluePresentvalueAnnuitiesRatesofreturnAmortizationTimelinesShowthetimingofcashflows.Tickmarksoccurattheendofperiods,soTime0istoday;Time1istheendofthefirstperiod(year,month,etc.)orthebeginningofthesecondperiod.CF0CF1CF3CF20123i%Drawingtimelines:

$100lumpsumduein2years;

3-year$100ordinaryannuity1001001000123i%3year$100ordinaryannuity100012i%$100lumpsumduein2yearsDrawingtimelines:

Unevencashflowstream;CF0=-$50,

CF1=$100,CF2=$75,andCF3=$5010050750123i%-50UnevencashflowstreamWhatisthefuturevalue(FV)ofaninitial$100after3years,ifI/YR=10%?FindingtheFVofacashfloworseriesofcashflowswhencompoundinterestisappliediscalledcompounding.FVcanbesolvedbyusingthearithmetic,financialcalculator,andspreadsheetmethods.FV=?012310%100SolvingforFV:

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)nSolvingforFV:

ThecalculatormethodSolvesthegeneralFVequation.Requires4inputsintocalculator,andwillsolveforthefifth.(SettoP/YR=1andENDmode.)INPUTSOUTPUTNI/YRPMTPVFV3100133.10-100PV=?100Whatisthepresentvalue(PV)of$100duein3years,ifI/YR=10%?FindingthePVofacashfloworseriesofcashflowswhencompoundinterestisappliediscalleddiscounting(thereverseofcompounding).ThePVshowsthevalueofcashflowsintermsoftoday’spurchasingpower.012310%SolvingforPV:

ThearithmeticmethodSolvethegeneralFVequationforPV:PV=FVn/(1+i)nPV=FV3/(1+i)3 =$100/(1.10)3 =$75.13SolvingforPV:

ThecalculatormethodSolvesthegeneralFVequationforPV.ExactlylikesolvingforFV,exceptwehavedifferentinputinformationandaresolvingforadifferentvariable.INPUTSOUTPUTNI/YRPMTPVFV3100100-75.13SolvingforN:

Ifsalesgrowat20%peryear,howlongbeforesalesdouble?SolvesthegeneralFVequationforN.Sameaspreviousproblems,butnowsolvingforN.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.Setcalculatorto“BEGIN”mode.INPUTSOUTPUTNI/YRPMTPVFV310-100364.100SolvingforPV:

3yearannuitydueof$100at10%Again,$100paymentsoccuratthebeginningofeachperiod.Setcalculatorto“BEGIN”mode.INPUTSOUTPUTNI/YRPMTPVFV3101000-273.55WhatisthePVofthisunevencashflowstream?010013002300310%-50490.91247.93225.39-34.15530.08=PVSolvingforPV:

UnevencashflowstreamInputcashflowsinthecalculator’s“CFLO”register:CF0=0CF1=100CF2=300CF3=300CF4=-50EnterI/YR=10,pressNPVbuttontogetNPV=$530.09.(HereNPV=PV.)SolvingforI:

Whatinterestratewouldcause$100togrowto$125.97in3years?SolvesthegeneralFVequationforI.INPUTSOUTPUTNI/YRPMTPVFV380125.97-100ThePowerofCompoundInterestA20-year-oldstudentwantstostartsavingforretirement.Sheplanstosave$3aday.Everyday,sheputs$3inherdrawer.Attheendoftheyear,sheinveststheaccumulatedsavings($1,095)inanonlinestockaccount.Thestockaccounthasanexpectedannualreturnof12%.Howmuchmoneywillshehavewhensheis65yearsold?SolvingforFV:

SavingsproblemIfshebeginssavingtoday,andstickstoherplan,shewillhave$1,487,261.89whensheis65.INPUTSOUTPUTNI/YRPMTPVFV4512-10951,487,2620SolvingforFV:

Savingsproblem,ifyouwaituntilyouare40yearsoldtostartIfa40-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/YRPMTPVFV2512-11,154.421,487,2620WilltheFVofalumpsumbelargerorsmallerifcompoundedmoreoften,holdingthestatedI%constant?LARGER,asthemorefrequentlycompoundingoccurs,interestisearnedoninterestmoreoften.Annually:FV3=$100(1.10)3=$133.10012310%100133.10Semiannually:FV6=$100(1.05)6=$134.0101235%456134.011230100ClassificationsofinterestratesNominalrate(iNOM)–alsocalledthequotedorstaterate.Anannualratethatignorescompoundingeffects.iNOMisstatedincontracts.Periodsmustalsobegiven,e.g.8%Quarterlyor8%Dailyinterest.Periodicrate(iPER)–amountofinterestchargedeachperiod,e.g.monthlyorquarterly.iPER=iNOM/m,wheremisthenumberofcompoundingperiodsperyear.m=4forquarterlyandm=12formonthlycompounding.ClassificationsofinterestratesEffective(orequivalent)annualrate(EAR=EFF%)–theannualrateofinterestactuallybeingearned,takingintoaccountcompounding.EFF%for10%semiannualinvestment EFF% =(1+iNOM/m)m-1 =(1+0.10/2)2–1=10.25%Aninvestorwouldbeindifferentbetweenaninvestmentofferinga10.25%annualreturnandoneofferinga10%annualreturn,compoundedsemiannually.Whyisitimportanttoconsidereffectiveratesofreturn?Aninvestmentwithmonthlypaymentsisdifferentfromonewithquarterlypayments.MustputeachreturnonanEFF%basistocompareratesofreturn.MustuseEFF%forcomparisons.SeefollowingvaluesofEFF%ratesatvariouscompoundinglevels.

EARANNUAL 10.00% EARQUARTERLY 10.38% EARMONTHLY 10.47% EARDAILY(365) 10.52%Cantheeffectiverateeverbeequaltothenominalrate?Yes,butonlyifannualcompoundingisused,i.e.,ifm=1.Ifm>1,EFF%willalwaysbegreaterthanthenominalrate.Wheniseachrateused?iNOM writtenintocontracts,quotedby banksandbrokers.Notusedin calculationsorshownontimelines.iPER Usedincalculationsandshownon timelines.Ifm=1,iNOM=iPER= EAR.EAR Usedtocomparereturnson investmentswithdifferentpayments peryear.Usedincalculationswhen annuitypaymentsdon’tmatch compoundingperiods.WhatistheFVof$100after3yearsunder10%semiannualcompounding?Quarterlycompounding?What’stheFVofa3-year$100annuity,ifthequotedinterestrateis10%,compoundedsemiannually?Paymentsoccurannually,butcompoundingoccursevery6months.Cannotusenormalannuityvaluationtechniques.01100235%451001006123Method1:

Compoundeachcashflow110.25121.55331.80FV3=$100(1.05)4+$100(1.05)2+$100FV3=$331.8001100235%451006123100Method2:

FinancialcalculatorFindtheEARandtreatasanannuity.EAR=(1+0.10/2)2–1=10.25%.INPUTSOUTPUTNI/YRPMTPVFV310.25-100331.800FindthePVofthis3-yearordinaryannuity.Couldsolvebydiscountingeachcashflow,or…UsetheEARandtreatasanannuitytosolveforPV.INPUTSOUTPUTNI/YRPMTPVFV310.251000-247.59LoanamortizationAmortizationtablesarewidelyusedforhomemortgages,autoloans,businessloans,retirementplans,etc.Financialcalculatorsandspreadsheetsaregreatforsettingupamortizationtables.EXAMPLE:Constructanamortizationschedulefora$1,000,10%annualrateloanwith3equalpayments.Step1:

FindtherequiredannualpaymentAllinputinformationisalreadygiven,justrememberthattheFV=0becausethereasonforamortizingtheloanandmakingpaymentsistoretiretheloan.INPUTSOUTPUTNI/YRPMTPVFV310402.110-1000Step2:

FindtheinterestpaidinYear1Theborrowerwilloweinterestupontheinitialbalanceattheendofthefirstyear.Interesttobepaidinthefirstyearcanbefoundbymultiplyingthebeginningbalancebytheinterestrate.

INTt=Begbalt(i) INT1=$1,000(0.10)=$100Step3:

FindtheprincipalrepaidinYear1Ifapaymentof$402.11wasmadeattheendofthefirstyearand$100waspaidtowardinterest,theremainingvaluemustrepresenttheamountofprincipalrepaid. PRIN =PMT–INT =$402.11-$100=$302.11Step4:

FindtheendingbalanceafterYear1Tofindthebalanceattheendoftheperiod,subtracttheamountpaidtowardprincipalfromthebeginningbalance. ENDBAL =BEGBAL–PRIN =$1,000-$302.11 =$697.89Constructinganamortizationtable:

Repeatsteps1–4untilendofloanInterestpaiddeclineswitheachpaymentasthebalancedeclines.Whatarethetaximplicationsofthis?YearBEGBALPMTINTPRINENDBAL1$1,000$402$100$302$6982698402703323663