3 Time value of money.docx
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3 Time value of money.docx
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3Timevalueofmoney
TimeValueofMoney
Chapter5and6
Topics:
1.FutureValueofaSingleSum
2.PresentValueofaSingleSum
3.NetPresentValue
4.RateofReturn
5.Perpetuity
6.GrowingPerpetuities
7.PresentValueofanOrdinaryAnnuity(andAnnuityDue)
8.FutureValueofanOrdinaryAnnuity(andAnnuityDue)
9.Growingannuity
10.PeriodicRate
11.AnnualPercentageRate(APR)
12.Effective(equivalent)AnnualRate(EFForEAR)
13.ContinuousCompounding
14.InflationandInterestRates
15.AmortizingLoan
CashflowsarediscountedfortworeasonsTimeandrisk:
1.Adollartodayisworthmorethanadollartomorrow.
2.Asafedollarisworthmorethanariskyone.
1.FutureValueofaSingleSum
EXCEL
FV(Rate,NPER,PMT,PV)
________________________________________________________________________
2.PresentValueofaSingleSum
Youwillrequire$700in5years,ifyouearn5%interestonyourfunds,howmuchwillyouneedtoinvesttodayinordertoreachyoursavingsgoal?
EXCEL
PV(Rate,NPER,PMT,FV)
EXCEL:
rate(nper,pmt,pv,fv)
__________________________________________
nper(rate,pmt,pv,fv)
_______________________________________________________________________
3.NetPresentValue
NPV=PV(cashflow)–PV(cashoutflow)
EXCEL:
C0+npv(rate,value1,value2,..)
________________________________________________________________________
4.RateofReturn
________________________________________________________________________
5.Perpetuity
Thesearebondsthatthegovernmentisundernoobligationtorepaybutthatofferafixedincome(C)foreachyeartoperpetuity
________________________________________________________________________6.GrowingPerpetuities
[youcanusethisformulaonlywheng ________________________________________________________________________ Youownanoilpipelinewhichwillgeneratea$2millioncashreturnoverthecomingyear.Thepipeline’soperatingcostsarenegligible,anditisexpectedtolastforaverylongtime.Unfortunately,thevolumeofoilshippedisdeclining,andcashflowsareexpectedtodeclineby4%peryear.Thediscountrateis10%. WhatisthePVofthepipeline’scashflowsifitscashflowsareassumedtolastforever? 7.PresentValueofanOrdinaryAnnuity(andAnnuityDue) Anassetthatpaysafixedsumofeachyearforaspecificnumberofyears.Itsvalueisthedifferencebetweenthevaluesoftwoperpetuities ___________________________ ________________________ EXCEL PV(Rate,NPER,PMT,FV,type) ________________________________________________________________________ 8.FutureValueofanOrdinaryAnnuity(andAnnuityDue) EXCEL FV(Rate,NPER,PMT,PV,type) ________________________________________________________________________ 9.Growingannuity Youownanoilpipelinewhichwillgeneratea$2millioncashreturnoverthecomingyear.Thepipeline’soperatingcostsarenegligible,anditisexpectedtolastforaverylongtime.Unfortunately,thevolumeofoilshippedisdeclining,andcashflowsareexpectedtodeclineby4%peryear.Thediscountrateis10%. a.WhatisthePVofthepipeline’scashflowsifitscashflowsareassumedtolastforever? b.WhatisthePVofthecashflowsifthepipelineisscrappedafter20years? ________________________________________________________________________ 10.PeriodicRate wheremisnumberofcompoundingperiodsperyear. ________________________________________________________________________ 11.AnnualPercentageRate(APR) Interestratethatisannualizedusingsimpleinterestrate. APR=(rPer)(m),Simplecompounding ________________________________________________________________________ 12.Effective(equivalent)AnnualRate(EFForEAR) TheannualratethatcausesPVtogrowtothesameFVasundermulti-periodcompounding. EAR=(1+rnom/m)m-1 Excel: Effect(nominalrate,npery) _______________________________________________________________________ 13.ContinuousCompounding limm→ [1+(1/m)]m=e=2.718 limm→ [1+(r/m)]m=er limm→ [1+(r/m)]mt=ert eistheyear-endvaluetowhichaprincipleof$1willgrowifinterestattherateof100perannumiscompoundedcontinuously. Ifabankpays6percentinterestwithcontinuouscompounding.Whatistheeffectiveannualrate? ________________________________________________________________________ 14.InflationandInterestRates Currentdollarcashflowsmustbediscountedbynominalinterestrate;realcashflowsmustbediscountedbytherealinterestrate. Bydiscountingrealcashflowsattherealinterestrate,yougetthesamePVthatyougetwhenyoudiscountthenominalcashflowsatthenominalinterestrate. 15.AmortizingLoan 1.Youareofferedanotethatpays$1,000in15months(or456days)for$850.Youhave$850inabankthatpaysa7.0%nominalrate,with365dailycompounding,whichisadailyrateof0.019178%andanEARof7.25%.Youplantoleavethemoneyinthebankifyoudon’tbuythenote.Thenoteisriskless.Shouldyoubuyit? 2.Assumethatyourfatherisnow50yearsold,thatheplanstoretirein10years,andthatheexpectstolivefor25yearsafterheretires,thatis,untilheis85.Hewantsafixedretirementincomethathasthesamepurchasingpoweratthetimeheretiresa$40,000hastoday(herealizesthattherealvalueofhisretirementincomewilldeclineyearbyyearafterheretires).Hisretirementincomewillbeginthedayheretires,10yearsfromtoday,andhewillthenget24additionalannualpayments.Inflationisexpectedtobe5percentperyearfromtodayforward;hecurrentlyhas$100,000savedup;andheexpectstoearnareturnonhissavingof8%peryear,annualcompounding.Tothenearestdollar,howmuchmusthesaveduringeachofthenext10years(withdepositbeingmadeattheendofeachyear)tomeethisretirementgoal? 1.FutureValue(SingleSum) r= 0.1 nper=t= 3 Period0 Period1 period2 period3 75.13148 FV(r,nper,pmt,pv) FV(.1,3,0,100) $100.00 FVIF=(1+r)^t 1.331 FV=C(FVIF) 100.00 ____________________________________ ____________ _________ _________ _____________ 2.PresentValue(SingleSum) r= 0.1 nper=t= 3 Period0 Period1 period2 period3 100 PV(rate,nper,pmt,fv) PV(.1,3,0,100) ($75.13) PVIF=1/(1+r)^t= 0.751314801 PV=C(PVIF) 75.13148009 ____________________________________ ____________ _________ _________ _____________ 3.NetPresentValue r= 0.1 nper=t= 3 Period0 Period1 period2 period3 -200 100 100 100 C0+npv(rate,value1,value2,..)= $48.69 ____________________________________ ____________ _________ _________ _____________ 7.PresentValueofanAnnuity r= 0.1 t= 3 Period0 Period1 period2 period3 100 100 100 PV(rate,nper,pmt,fv) PV(.1,3,100,0) ($248.69) PVIFA=1/r-[(1/r)*(1/(1+r)^t]= 2.4869 PV=CPVIFA 248.69 __________________________________ Period0 Period1 period2 period3 100 100 100 1000 PV(.1,3,100,1000) ($1,000.00) ____________________________________ ____________ _________ _________ _____________ 8.FutureValueofanAnnuity r= 0.1 t= 3 Period0 Period1 period2 period3 Annuity -100 -100 -100 FV(.1,3,-100,0) $331.00 FVIFA=[(1+r)^t-1]/r= 3.31 FV=C(FVIFA) 331 __________________________________ Period0 Period1 period2 period3 -100 -100 -100 -1000 FV(.1,3,-100,-1000)= $1,662.00 ____________________________________ ____________ _________ _________ _____________ r= 0.1 g= 0.05 t= 3 a= 0.954545 C= 100 5.Perpetuity PV=C/r= 1000 6.GrowthPerpetuity PV=C/(r-g) 2000 9.Growingannuity PV=C/(1+r)*(1-a^t)/(1-a) 260.5184072 ____________________________________ ____________ _________ _________ _____________ r= 0.1 m= 12 10.PeriodicRate rper=r/m 0.008333333 11.EffectiveRate(EAR) ((1+r/m)^m)-1= 0.1047 12.AnnualPercentageRate(APR) 0.1 APR=(m)(rper)= ____________________________________ ____________ _________ _________ _____________ 13.ContinuousCompounding r= 0.1 t= 5 C= 100 (ContinuousCompounding)FV=Ce^(rct) 164.8720716 (Compounding)FV=C(1+r)^t 161.051 ____________________________________ ____________ _________ _________ _____________ OtherFunctions Assumerisunknown C0 C1 C2 C3 rate(nper,pmt,pv,fv) 100 100 100 rate(3,100,-248.69,0) -248.69 10% AssumeNPERisunknown nper(rate,pmt,pv,fv) C0 C1 C2 C? ? ? ? nper(.1,100,-248.69,0) 100 100 100 3.00 -248.69 AssumeNPERisunknown nper(rate,pmt,pv,fv) C0 C? ? ? ? nper(.04,0,-400,1000) -400 1000 23.36241894 ____________________________________ ____________ _________ _________ _____________
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