《财务管理》(英文版)-10

Chapter3TimeValueofMoneyTheTimeValueofMoney

TheInterestRateSimpleInterestCompoundInterestAmortizingaLoanObviously,$10,000today.YoualreadyrecognizethatthereisTIMEVALUETOMONEY!!TheInterestRateWhichwouldyouprefer--$10,000todayor$10,000in5years?TIMEallowsyoutheopportunitytopostponeconsumptionandearnINTEREST.WhyTIME?WhyisTIMEsuchanimportantelementinyourdecision?TypesofInterestCompoundInterestInterestpaid(earned)onanypreviousinterestearned,aswellasontheprincipalborrowed(lent).SimpleInterestInterestpaid(earned)ononlytheoriginalamount,orprincipalborrowed(lent).SimpleInterestFormulaFormula

SI=P0(i)(n)

SI: SimpleInterest

P0: Deposittoday(t=0)

i: InterestRateperPeriod n: NumberofTimePeriodsSI =P0(i)(n)

=$1,000(.07)(2) =$140SimpleInterestExampleAssumethatyoudeposit$1,000inanaccountearning7%simpleinterestfor2years.Whatistheaccumulatedinterestattheendofthe2ndyear?

FV =P0+SI =$1,000

+$140 =

$1,140FutureValue

isthevalueatsomefuturetimeofapresentamountofmoney,oraseriesofpayments,evaluatedatagiveninterestrate.SimpleInterest(FV)WhatistheFutureValue(FV)ofthedeposit?

ThePresentValueissimplythe $1,000youoriginallydeposited. Thatisthevaluetoday!PresentValue

isthecurrentvalueofafutureamountofmoney,oraseriesofpayments,evaluatedatagiveninterestrate.SimpleInterest(PV)WhatisthePresentValue(PV)ofthepreviousproblem?WhyCompoundInterest?FutureValue(U.S.Dollars)

Assumethatyoudeposit$1,000atacompoundinterestrateof7%for2years.FutureValue

SingleDeposit(Graphic)

0

1

2$1,000FV27%FV1 =P0(1+i)1 =$1,000

(1.07) =$1,070CompoundInterest Youearned$70interestonyour$1,000depositoverthefirstyear. Thisisthesameamountofinterestyouwouldearnundersimpleinterest.FutureValue

SingleDeposit(Formula)FV1 =P0

(1+i)1 =$1,000(1.07) =$1,070FV2 =FV1(1+i)1 =P0(1+i)(1+i) =$1,000(1.07)(1.07) =P0

(1+i)2 =$1,000(1.07)2 =$1,144.90YouearnedanEXTRA

$4.90inYear2withcompoundoversimpleinterest.FutureValue SingleDeposit(Formula)

FV1 =P0(1+i)1

FV2 =P0(1+i)2GeneralFutureValueFormula:

FVn

=P0(1+i)n

or FVn=P0(FVIFi,n)--SeeTableIGeneralFutureValueFormulaetc.FVIFi,n

isfoundonTableIattheendofthebookoronthecardinsert.ValuationUsingTableI

FV2 =$1,000(FVIF7%,2) =$1,000(1.145) =$1,145

[DuetoRounding]UsingFutureValueTables

JulieMillerwantstoknowhowlargeherdepositof$10,000todaywillbecomeatacompoundannualinterestrateof10%for5years.StoryProblemExample

012345$10,000FV510%CalculationbasedonTableI:

FV5

=$10,000

(FVIF10%,5)

=$10,000

(1.611) =$16,110 [DuetoRounding]StoryProblemSolutionCalculationbasedongeneralformula:

FVn

=P0(1+i)n

FV5

=$10,000(1+0.10)5 =$16,105.10Wewillusethe“Rule-of-72”.DoubleYourMoney!!!Quick!Howlongdoesittaketodouble$5,000atacompoundrateof12%peryear(approx.)?Approx.YearstoDouble=72

/i%72/12%=6Years[ActualTimeis6.12Years]The“Rule-of-72”Quick!Howlongdoesittaketodouble$5,000atacompoundrateof12%peryear(approx.)?Assumethatyouneed$1,000

in2years.Let’sexaminetheprocesstodeterminehowmuchyouneedtodeposittodayatadiscountrateof7%compoundedannually.

0

1

2$1,0007%PV1PV0PresentValue SingleDeposit(Graphic)

PV0=FV2/(1+i)2 =$1,000

/(1.07)2 =FV2/(1+i)2

=$873.44PresentValue

SingleDeposit(Formula)

0

1

2$1,0007%PV0

PV0

=FV1/(1+i)1

PV0=FV2/(1+i)2GeneralPresentValueFormula:

PV0 =FVn/(1+i)n

or PV0=FVn(PVIFi,n)--SeeTableIIGeneralPresentValueFormulaetc.PVIFi,n

isfoundonTableIIattheendofthebookoronthecardinsert.ValuationUsingTableII

PV2 =$1,000(PVIF7%,2) =$1,000(.873) =$873

[DuetoRounding]UsingPresentValueTables

JulieMillerwantstoknowhowlargeofadeposittomakesothatthemoneywillgrowto$10,000

in5yearsatadiscountrateof10%.StoryProblemExample

012345$10,000PV010%

Calculationbasedongeneralformula:

PV0 =FVn/(1+i)n

PV0

=$10,000

/(1+0.10)5 =$6,209.21

CalculationbasedonTableI:

PV0

=$10,000

(PVIF10%,5)

=$10,000

(.621) =$6,210.00

[DuetoRounding]StoryProblemSolutionTypesofAnnuitiesOrdinaryAnnuity:Paymentsorreceiptsoccurattheendofeachperiod.AnnuityDue:Paymentsorreceiptsoccuratthe beginningofeachperiod.AnAnnuityrepresentsaseriesofequalpayments(orreceipts)occurringoveraspecifiednumberofequidistantperiods.ExamplesofAnnuities

StudentLoanPaymentsCarLoanPaymentsInsurancePremiumsMortgagePaymentsRetirementSavingsPartsofanAnnuity0123

$100$100$100(OrdinaryAnnuity)EndofPeriod1EndofPeriod2TodayEqualCashFlowsEach1PeriodApartEndofPeriod3PartsofanAnnuity0123$100$100$100(AnnuityDue)BeginningofPeriod1BeginningofPeriod2TodayEqualCashFlowsEach1PeriodApartBeginningofPeriod3FVAn=R(1+i)n-1+R(1+i)n-2+ ...+R(1+i)1

+R(1+i)0Overviewofan

OrdinaryAnnuity--FVA

RRR012nn+1FVAnR

=PeriodicCashFlowCashflowsoccurattheendoftheperiodi%...

FVA3=$1,000(1.07)2+ $1,000(1.07)1+$1,000(1.07)0

=$1,145

+

$1,070

+

$1,000

=

$3,215Exampleofan

OrdinaryAnnuity--FVA$1,000$1,000$1,00001234$3,215=FVA37%$1,070$1,145CashflowsoccurattheendoftheperiodHintonAnnuityValuationThefuturevalueofanordinaryannuitycanbeviewedasoccurringattheendofthelastcashflowperiod,whereasthefuturevalueofanannuityduecanbeviewedasoccurringatthebeginningofthelastcashflowperiod.

FVAn

=R(FVIFAi%,n) FVA3 =$1,000(FVIFA7%,3) =$1,000(3.215)=$3,215ValuationUsingTableIIIFVADn=R(1+i)n+R(1+i)n-1+ ...+R(1+i)2

+R(1+i)1

=FVAn

(1+i)OverviewViewofan

AnnuityDue--FVAD

RRRRR0123n-1

nFVADni%...Cashflowsoccuratthebeginningoftheperiod

FVAD3=$1,000(1.07)3+ $1,000(1.07)2+$1,000(1.07)1

=$1,225

+

$1,145

+

$1,070

=

$3,440Exampleofan

AnnuityDue--FVAD$1,000$1,000$1,000$1,07001234$3,440=FVAD37%$1,225$1,145CashflowsoccuratthebeginningoftheperiodFVADn

=R(FVIFAi%,n)(1+i) FVAD3 =$1,000(FVIFA7%,3)(1.07) =$1,000(3.215)(1.07)=$3,440ValuationUsingTableIIIPVAn=R/(1+i)1+R/(1+i)2 +...+R/(1+i)nOverviewofan

OrdinaryAnnuity--PVA

RRR012nn+1PVAnR

=PeriodicCashFlowi%...Cashflowsoccurattheendoftheperiod

PVA3= $1,000/(1.07)1+ $1,000/(1.07)2+ $1,000/(1.07)3

=$934.58+$873.44+$816.30 =

$2,624.32Exampleofan

OrdinaryAnnuity--PVA$1,000$1,000$1,00001234$2,624.32=PVA37%$934.58$873.44$816.30CashflowsoccurattheendoftheperiodHintonAnnuityValuationThepresentvalueofanordinaryannuitycanbeviewedasoccurringatthebeginningofthefirstcashflowperiod,whereasthepresentvalueofanannuityduecanbeviewedasoccurringattheendofthefirstcashflowperiod.

PVAn

=R(PVIFAi%,n) PVA3 =$1,000(PVIFA7%,3) =$1,000(2.624)=$2,624ValuationUsingTableIVPVADn=R/(1+i)0+R/(1+i)1+...+R/(1+i)n-1

=PVAn

(1+i)Overviewofan

AnnuityDue--PVAD

RRRR012n-1

nPVADnR:PeriodicCashFlowi%...CashflowsoccuratthebeginningoftheperiodPVADn=$1,000/(1.07)0+$1,000/(1.07)1+ $1,000/(1.07)2=$2,808.02Exampleofan

AnnuityDue--PVAD$1,000.00$1,000$1,0000123

4$2,808.02=

PVADn7%$934.58$873.44CashflowsoccuratthebeginningoftheperiodPVADn=R(PVIFAi%,n)(1+i) PVAD3 =$1,000(PVIFA7%,3)(1.07) =$1,000(2.624)(1.07)=$2,808ValuationUsingTableIV1.Readproblemthoroughly2.DetermineifitisaPVorFVproblem3.Createatimeline4.Putcashflowsandarrowsontimeline5.Determineifsolutioninvolvesasingle CF,annuitystream(s),ormixedflow6.Solvetheproblem7.Checkwithfinancialcalculator(optional)StepstoSolveTimeValueofMoneyProblems

JulieMillerwillreceivethesetofcashflowsbelow.WhatisthePresentValueatadiscountrateof10%?MixedFlowsExample

012345

$600$600$400$400$100PV010%

1. Solvea“piece-at-a-time”by discountingeachpiecebacktot=0. 2. Solvea“group-at-a-time”byfirst breakingproblemintogroups

of

annuity

streamsandanysingle

cash

flow

group.Thendiscount eachgroupbacktot=0.HowtoSolve?“Piece-At-A-Time”

012345

$600$600$400$400$10010%$545.45$495.87$300.53$273.21$62.09$1677.15=PV0

oftheMixedFlow“Group-At-A-Time”(#1)

012345

$600$600$400$400$10010%$1,041.60$573.57$62.10$1,677.27

=PV0

ofMixedFlow[UsingTables]$600(PVIFA10%,2)=$600(1.736)=$1,041.60$400(PVIFA10%,2)(PVIF10%,2)=$400(1.736)(0.826)=$573.57$100(PVIF10%,5)=$100(0.621)=$62.10“Group-At-A-Time”(#2)

01234

$400$400$400$400PV0

equals$1677.30.

012

$200$200

012345

$100$1,268.00$347.20$62.10PlusPlusGeneralFormula:FVn =PV0(1+[i/m])mn

n: NumberofYears m: CompoundingPeriodsperYear i: AnnualInterestRate FVn,m:FVattheendofYearn

PV0: PVoftheCashFlowtodayFrequencyofCompoundingJulieMillerhas$1,000toinvestfor2yearsatanannualinterestrateof12%.Annual FV2 =1,000(1+[.12/1])(1)(2) =1,254.40Semi FV2 =1,000(1+[.12/2])(2)(2) =1,262.48ImpactofFrequencyQrtly FV2 =1,000(1+[.12/4])(4)(2) =1,266.77MonthlyFV2

=1,000(1+[.12/12])(12)(2) =1,269.73Daily FV2

=1,000(1+[.12/365])(365)(2) =1,271.20ImpactofFrequencyEffectiveAnnualInterestRateTheactualrateofinterestearned(paid)afteradjustingthenomina