3 Time value of money.docx

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

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12.Effective（equivalent）AnnualRate（EFForEAR）

TheannualratethatcausesPVtogrowtothesameFVasundermulti-periodcompounding.

EAR=（1+rnom/m）m-1

Excel:

Effect（nominalrate,npery）

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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

____________________________________

____________

_________

_________

_____________