管理學財務管理第三章課件

Chapter3TheTimeValueofMoneyChapter3TheTimeValueofMonAfterstudyingChapter3,youshouldbeableto:Understandwhatismeantby"thetimevalueofmoney."Understandtherelationshipbetweenpresentandfuturevalue.Describehowtheinterestratecanbeusedtoadjustthevalueofcashflows–bothforwardandbackward–toasinglepointintime.Calculateboththefutureandpresentvalueof:(a)anamountinvestedtoday;(b)astreamofequalcashflows(anannuity);and(c)astreamofmixedcashflows.Distinguishbetweenan“ordinaryannuity”andan“annuitydue.”Useinterestfactortablesandunderstandhowtheyprovideashortcuttocalculatingpresentandfuturevalues.Useinterestfactortablestofindanunknowninterestrateorgrowthratewhenthenumberoftimeperiodsandfutureandpresentvaluesareknown.Buildan“amortizationschedule”foraninstallment-styleloan.AfterstudyingChapter3,youTheTimeValueofMoney

TheInterestRateSimpleInterestCompoundInterestAmortizingaLoanCompoundingMoreThanOnceperYearTheTimeValueofMoneyTheInObviously,$10,000today.YoualreadyrecognizethatthereisTIMEVALUETOMONEY!!TheInterestRateWhichwouldyouprefer–$10,000todayor$10,000in5years?Obviously,$10,000today.TheITIMEallowsyoutheopportunitytopostponeconsumptionandearnINTEREST.WhyTIME?WhyisTIMEsuchanimportantelementinyourdecision?TIMEallowsyoutheopportunitTypesofInterestCompoundInterestInterestpaid(earned)onanypreviousinterestearned,aswellasontheprincipalborrowed(lent).SimpleInterestInterestpaid(earned)ononlytheoriginalamount,orprincipal,borrowed(lent).TypesofInterestCompoundInteSimpleInterestFormulaFormula

SI=P0(i)(n)

SI: SimpleInterest

P0: Deposittoday(t=0)

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

=$1,000(0.07)(2) =$140SimpleInterestExampleAssumethatyoudeposit$1,000inanaccountearning7%simpleinterestfor2years.Whatistheaccumulatedinterestattheendofthe2ndyear?SI =P0(i)(n) =$1,000(

FV =P0+SI =$1,000

+$140 =

$1,140FutureValue

isthevalueatsomefuturetimeofapresentamountofmoney,oraseriesofpayments,evaluatedatagiveninterestrate.SimpleInterest(FV)WhatistheFutureValue(FV)ofthedeposit? FV =P0+SI =$1,0

ThePresentValueissimplythe $1,000youoriginallydeposited. Thatisthevaluetoday!PresentValue

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

Assumethatyoudeposit$1,000atacompoundinterestrateof7%for2years.FutureValue

SingleDeposit(Graphic)

0

1

2$1,000FV27% Assumethatyoudeposit$1,00FV1 =P0(1+i)1 =$1,000

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

SingleDeposit(Formula)FV1 =P0(1+i)1 =$1,000FV1 =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 =$1,0

FV1 =P0(1+i)1

FV2 =P0(1+i)2GeneralFutureValueFormula:

FVn =P0(1+i)n

or FVn=P0(FVIFi,n)–SeeTableIGeneralFutureValueFormulaetc. FV1 =P0(1+i)1GeneralFuFVIFi,n

isfoundonTableIattheendofthebook.ValuationUsingTableIFVIFi,nisfoundonTableIVa

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

[DuetoRounding]UsingFutureValueTables FV2 =$1,000(FVIF7%,2) =

JulieMillerwantstoknowhowlargeherdepositof$10,000todaywillbecomeatacompoundannualinterestrateof10%for5years.StoryProblemExample

012345$10,000FV510% JulieMillerwantstoknowhoCalculationbasedonTableI:

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 =$16,105.10CalculationbasedonTableI: Wewillusethe“Rule-of-72”.DoubleYourMoney!!!Quick!Howlongdoesittaketodouble$5,000atacompoundrateof12%peryear(approx.)?Wewillusethe“Rule-of-72”.DApprox.YearstoDouble=72

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

in2years.Let’sexaminetheprocesstodeterminehowmuchyouneedtodeposittodayatadiscountrateof7%compoundedannually.

0

1

2$1,0007%PV1PV0PresentValue SingleDeposit(Graphic)Assumethatyouneed$1,000in

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

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

=$873.44

0

1

2$1,0007%PV0PresentValue

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

PV0

=FV1/(1+i)1

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

PV0 =FVn/(1+i)n

or PV0=FVn(PVIFi,n)–SeeTableIIetc.GeneralPresentValueFormula PV0=FV1/(1+i)1etc.GenPVIFi,n

isfoundonTableIIattheendofthebook.ValuationUsingTableIIPVIFi,nisfoundonTableIIV

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

[DuetoRounding]UsingPresentValueTables PV2 =$1,000(PVIF7%,2) =

JulieMillerwantstoknowhowlargeofadeposittomakesothatthemoneywillgrowto$10,000

in5yearsatadiscountrateof10%.

012345$10,000PV010%StoryProblemExample JulieMillerwantstoknowho

Calculationbasedongeneralformula:

PV0 =FVn/(1+i)n

PV0

=$10,000

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

CalculationbasedonTableI:

PV0

=$10,000

(PVIF10%,5)

=$10,000

(0.621) =$6,210.00

[DuetoRounding]StoryProblemSolution CalculationbasedongeneralOrdinaryAnnuity:Paymentsorreceiptsoccurattheendofeachperiod.AnnuityDue:Paymentsorreceiptsoccuratthe beginningofeachperiod.AnAnnuityrepresentsaseriesofequalpayments(orreceipts)occurringoveraspecifiednumberofequidistantperiods.TypesofAnnuitiesOrdinaryAnnuity:Paymentsor

StudentLoanPaymentsCarLoanPaymentsInsurancePremiumsMortgagePaymentsRetirementSavingsExamplesofAnnuitiesStudentLoanPaymentsExample0123

$100$100$100(OrdinaryAnnuity)EndofPeriod1EndofPeriod2TodayEqualCashFlowsEach1PeriodApartEndofPeriod3PartsofanAnnuity010123$100$100$100(AnnuityDue)BeginningofPeriod1BeginningofPeriod2TodayEqualCashFlowsEach1PeriodApartBeginningofPeriod3PartsofanAnnuity01FVAn=R(1+i)n-1+R(1+i)n-2+ ...+R(1+i)1

+R(1+i)0

RRR012nn+1FVAnR

=PeriodicCashFlowCashflowsoccurattheendoftheperiodi%...Overviewofan

OrdinaryAnnuity–FVAFVAn=R(1+i)n-1+R(1+i)n

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

=$1,145

+

$1,070

+

$1,000

=

$3,215$1,000$1,000$1,0000123

4$3,215=FVA37%$1,070$1,145CashflowsoccurattheendoftheperiodExampleofan

OrdinaryAnnuity–FVA FVA3=$1,000(1.07)2+ Thefuturevalueofanordinaryannuitycanbeviewedasoccurringattheendofthelastcashflowperiod,whereasthefuturevalueofanannuityduecanbeviewedasoccurringatthebeginningofthelastcashflowperiod.HintonAnnuityValuationThefuturevalueofanordinar

FVAn =R(FVIFAi%,n) FVA3 =$1,000(FVIFA7%,3) =$1,000(3.215)=$3,215ValuationUsingTableIII FVAn =R(FVIFAi%,n) FVA3FVADn=R(1+i)n+R(1+i)n-1+ ...+R(1+i)2

+R(1+i)1 =FVAn(1+i)

RRRRR0123n–1

nFVADni%...OverviewViewofan

AnnuityDue–FVADCashflowsoccuratthebeginningoftheperiodFVADn=R(1+i)n+R(1+i)

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

=$1,225

+

$1,145

+

$1,070

=

$3,440$1,000$1,000$1,000$1,07001234$3,440=FVAD37%$1,225$1,145Exampleofan

AnnuityDue–FVADCashflowsoccuratthebeginningoftheperiod FVAD3=$1,000(1.07)3+ FVADn =R(FVIFAi%,n)(1+i) FVAD3 =$1,000(FVIFA7%,3)(1.07) =$1,000(3.215)(1.07)=$3,440ValuationUsingTableIIIFVADn =R(FVIFAi%,n)(1+i) PVAn=R/(1+i)1+R/(1+i)2 +...+R/(1+i)n

RRR012nn+1PVAnR

=PeriodicCashFlowi%...Overviewofan

OrdinaryAnnuity–PVACashflowsoccurattheendoftheperiodPVAn=R/(1+i)1+R/(1+i)2

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

=$934.58+$873.44+$816.30 =

$2,624.32$1,000$1,000$1,00001234$2,624.32=PVA37%$934.58$873.44$816.30Exampleofan

OrdinaryAnnuity–PVACashflowsoccurattheendoftheperiod PVA3= $1,000/(1.07)1+ Thepresentvalueofanordinaryannuitycanbeviewedasoccurringatthebeginningofthefirstcashflowperiod,whereasthefuturevalueofanannuityduecanbeviewedasoccurringattheendofthefirstcashflowperiod.HintonAnnuityValuationThepresentvalueofanordina

PVAn =R(PVIFAi%,n) PVA3 =$1,000(PVIFA7%,3) =$1,000(2.624)=$2,624ValuationUsingTableIV PVAn =R(PVIFAi%,n) PVA3PVADn=R/(1+i)0+R/(1+i)1+...+R/(1+i)n–1

=PVAn(1+i)

RRRR012n–1

nPVADnR:PeriodicCashFlowi%...Overviewofan

AnnuityDue–PVADCashflowsoccuratthebeginningoftheperiodPVADn=R/(1+i)0+R/(1+iPVADn=$1,000/(1.07)0+$1,000/(1.07)1+ $1,000/(1.07)2=$2,808.02$1,000.00$1,000$1,0000123

4$2,808.02=PVADn7%$934.58$873.44Exampleofan

AnnuityDue–PVADCashflowsoccuratthebeginningoftheperiodPVADn=$1,000/(1.07)0+$1,00PVADn=R(PVIFAi%,n)(1+i) PVAD3 =$1,000(PVIFA7%,3)(1.07) =$1,000(2.624)(1.07)=$2,808ValuationUsingTableIVPVADn=R(PVIFAi%,n)(1+i1.Readproblemthoroughly2.Createatimeline3.Putcashflowsandarrowsontimeline4.DetermineifitisaPVorFVproblem5.Determineifsolutioninvolvesasingle CF,annuitystream(s),ormixedflow6.Solvetheproblem7.Checkwithfinancialcalculator(optional)StepstoSolveTimeValueofMoneyProblems1.ReadproblemthoroughlySte

JulieMillerwillreceivethesetofcashflowsbelow.WhatisthePresentValueatadiscountrateof10%.

012345

$600$600$400$400$100PV010%MixedFlowsExample JulieMillerwillreceivethe

012345

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

oftheMixedFlow012

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)=$62.10012GeneralFormula:FVn =PV0(1+[i/m])mn

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

PV0: PVoftheCashFlowtodayFrequencyofCompoundingGeneralFormula:FrequencyofCJulieMillerhas$1,000toinvestfor2Yearsatanannualinterestrateof12%.Annual FV2 =1,000(1+[0.12/1])(1)(2) =1,254.40Semi FV2 =1,000(1+[0.12/2])(2)(2) =1,262.48ImpactofFrequencyJulieMillerhas$1,000toinvQrtly FV2 =1,000(1+[0.12/4])(4)(2) =1,266.77MonthlyFV2

=1,000(1+[0.12/12])(12)(2) =1,269.73Daily FV2

=1,000(1+[0.12/365])(365)(2) =1,271.20ImpactofFrequencyQrtly FV2 =1,000(1+[0.EffectiveAnnualInterestRateTheactualrateofinterestearned(paid)afteradjustingthenominalrate

forfactorssuchasthenumberofcompoundingperiodsperyear.(1+[i

/m])m

–1EffectiveAnnual

InterestRateEffectiveAnnualInterestRateBasketWonders(BW)hasa$1,000CDatthebank.Theinterestrateis6%

compoundedquarterlyfor1year.WhatistheEffectiveAnnualInterestRate(EAR)?

EAR =(1+0.06/4)4–1 =1.0614-1=0.0614or6.14%!BWsEffective

AnnualInterestRateBasketWonders(BW)hasa$1,01. Calculatethepaymentperperiod.2. DeterminetheinterestinPeriodt. (LoanBalanceatt–1)x(i%/m)3. ComputeprincipalpaymentinPeriodt. (Payment-InterestfromStep2)4. DetermineendingbalanceinPeriodt. (Balance-principalpaymentfromStep3)5. StartagainatStep2andrepeat.StepstoAmortizingaLoan1. CalculatethepaymentperJulieMillerisborrowing$10,000atacompoundannualinterestrateof12%.Amortizetheloanifannualpaymentsaremadefor5years.Step1: Payment

PV0 =R(PVIFAi%,n)

$10,000 =R(PVIFA12%,5)

$10,000 =R(3.605)

R=$10,000/3.605=$2,774AmortizingaLoanExampleJulieMillerisborrowing$10,EndofYearPaymentInterestPrincipalEndingBalance0———$10,0001$2,774$1,200$1,5748,42622,7741,0111,7636,66332,7748001,9744,68942,7745632,2112,47852,7752972,4780$13,871$3,871$10,000[LastPaymentSlightlyHigherDuetoRounding]AmortizingaLoanExampleEndofYearPaymentInterestPrinc2. CalculateDebtOutstanding–Thequantityofoutstandingdebtmaybeusedinfinancingtheday-to-dayactivitiesofthefirm.1. DetermineInterestExpense– Interestexpensesmayreduce taxableincomeofthefirm.UsefulnessofAmortization2. CalculateDebtOutstandingChapter3TheTimeValueofMoneyChapter3TheTimeValueofMonAfterstudyingChapter3,youshouldbeableto:Understandwhatismeantby"thetimevalueofmoney."Understandtherelationshipbetweenpresentandfuturevalue.Describehowtheinterestratecanbeusedtoadjustthevalueofcashflows–bothforwardandbackward–toasinglepointintime.Calculateboththefutureandpresentvalueof:(a)anamountinvestedtoday;(b)astreamofequalcashflows(anannuity);and(c)astreamofmixedcashflows.Distinguishbetweenan“ordinaryannuity”andan“annuitydue.”Useinterestfactortablesandunderstandhowtheyprovideashortcuttocalculatingpresentandfuturevalues.Useinterestfactortablestofindanunknowninterestrateorgrowthratewhenthenumberoftimeperiodsandfutureandpresentvaluesareknown.Buildan“amortizationschedule”foraninstallment-styleloan.AfterstudyingChapter3,youTheTimeValueofMoney

TheInterestRateSimpleInterestCompoundInterestAmortizingaLoanCompoundingMoreThanOnceperYearTheTimeValueofMoneyTheInObviously,$10,000today.YoualreadyrecognizethatthereisTIMEVALUETOMONEY!!TheInterestRateWhichwouldyouprefer–$10,000todayor$10,000in5years?Obviously,$10,000today.TheITIMEallowsyoutheopportunitytopostponeconsumptionandearnINTEREST.WhyTIME?WhyisTIMEsuchanimportantelementinyourdecision?TIMEallowsyoutheopportunitTypesofInterestCompoundInterestInterestpaid(earned)onanypreviousinterestearned,aswellasontheprincipalborrowed(lent).SimpleInterestInterestpaid(earned)ononlytheoriginalamount,orprincipal,borrowed(lent).TypesofInterestCompoundInteSimpleInterestFormulaFormula

SI=P0(i)(n)

SI: SimpleInterest

P0: Deposittoday(t=0)

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

=$1,000(0.07)(2) =$140SimpleInterestExampleAssumethatyoudeposit$1,000inanaccountearning7%simpleinterestfor2years.Whatistheaccumulatedinterestattheendofthe2ndyear?SI =P0(i)(n) =$1,000(

FV =P0+SI =$1,000

+$140 =

$1,140FutureValue

isthevalueatsomefuturetimeofapresentamountofmoney,oraseriesofpayments,evaluatedatagiveninterestrate.SimpleInterest(FV)WhatistheFutureValue(FV)ofthedeposit? FV =P0+SI =$1,0

ThePresentValueissimplythe $1,000youoriginallydeposited. Thatisthevaluetoday!PresentValue

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

Assumethatyoudeposit$1,000atacompoundinterestrateof7%for2years.FutureValue

SingleDeposit(Graphic)

0

1

2$1,000FV27% Assumethatyoudeposit$1,00FV1 =P0(1+i)1 =$1,000

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

SingleDeposit(Formula)FV1 =P0(1+i)1 =$1,000FV1 =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 =$1,0

FV1 =P0(1+i)1

FV2 =P0(1+i)2GeneralFutureValueFormula:

FVn =P0(1+i)n

or FVn=P0(FVIFi,n)–SeeTableIGeneralFutureValueFormulaetc. FV1 =P0(1+i)1GeneralFuFVIFi,n

isfoundonTableIattheendofthebook.ValuationUsingTableIFVIFi,nisfoundonTableIVa

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

[DuetoRounding]UsingFutureValueTables FV2 =$1,000(FVIF7%,2) =

JulieMillerwantstoknowhowlargeherdepositof$10,000todaywillbecomeatacompoundannualinterestrateof10%for5years.StoryProblemExample

012345$10,000FV510% JulieMillerwantstoknowhoCalculationbasedonTableI:

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 =$16,105.10CalculationbasedonTableI: Wewillusethe“Rule-of-72”.DoubleYourMoney!!!Quick!Howlongdoesittaketodouble$5,000atacompoundrateof12%peryear(approx.)?Wewillusethe“Rule-of-72”.DApprox.YearstoDouble=72

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

in2years.Let’sexaminetheprocesstodeterminehowmuchyouneedtodeposittodayatadiscountrateof7%compoundedannually.

0

1

2$1,0007%PV1PV0PresentValue SingleDeposit(Graphic)Assumethatyouneed$1,000in

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

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

=$873.44

0

1

2$1,0007%PV0PresentValue

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

PV0

=FV1/(1+i)1

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

PV0 =FVn/(1+i)n

or PV0=FVn(PVIFi,n)–SeeTableIIetc.GeneralPresentValueFormula PV0=FV1/(1+i)1etc.GenPVIFi,n

isfoundonTableIIattheendofthebook.ValuationUsingTableIIPVIFi,nisfoundonTableIIV

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

[DuetoRounding]UsingPresentValueTables PV2 =$1,000(PVIF7%,2) =

JulieMillerwantstoknowhowlargeofadeposittomakesothatthemoneywillgrowto$10,000

in5yearsatadiscountrateof10%.

012345$10,000PV010%StoryProblemExample JulieMillerwantstoknowho

Calculationbasedongeneralformula:

PV0 =FVn/(1+i)n

PV0

=$10,000

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

CalculationbasedonTableI:

PV0

=$10,000

(PVIF10%,5)

=$10,000

(0.621) =$6,210.00

[DuetoRounding]StoryProblemSolution CalculationbasedongeneralOrdinaryAnnuity:Paymentsorreceiptsoccurattheendofeachperiod.AnnuityDue:Paymentsorreceiptsoccuratthe beginningofeachperiod.AnAnnuityrepresentsaseriesofequalpayments(orreceipts)occurringoveraspecifiednumberofequidistantperiods.TypesofAnnuitiesOrdinaryAnnuity:Paymentsor

StudentLoanPaymentsCarLoanPaymentsInsurancePremiumsMortgagePaymentsRetirementSavingsExamplesofAnnuitiesStudentLoanPaymentsExample0123

$100$100$100(OrdinaryAnnuity)EndofPeriod1EndofPeriod2TodayEqualCashFlowsEach1PeriodApartEndofPeriod3PartsofanAnnuity010123$100$100$100(AnnuityDue)BeginningofPeriod1BeginningofPeriod2TodayEqualCashFlowsEach1PeriodApartBeginningofPeriod3PartsofanAnnuity01FVAn=R(1+i)n-1+R(1+i)n-2+ ...+R(1+i)1

+R(1+i)0

RRR012nn+1FVAnR

=PeriodicCashFlowCashflowsoccurattheendoftheperiodi%...Overviewofan

OrdinaryAnnuity–FVAFVAn=R(1+i)n-1+R(1+i)n

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

=$1,145

+

$1,070

+

$1,000

=

$3,215$1,000$1,000$1,0000123

4$3,215=FVA37%$1,070$1,145CashflowsoccurattheendoftheperiodExampleofan

OrdinaryAnnuity–FVA FVA3=$1,000(1.07)2+ Thefuturevalueofanordinaryannuitycanbeviewedasoccurringattheendofthelastcashflowperiod,whereasthefuturevalueofanannuityduecanbeviewedasoccurringatthebeginningofthelastcashflowperiod.HintonAnnuityValuationThefuturevalueofanordinar

FVAn =R(FVIFAi%,n) FVA3 =$1,000(FVIFA7%,3) =$1,000(3.215)=$3,215ValuationUsingTableIII FVAn =R(FVIFAi%,n) FVA3FVADn=R(1+i)n+R(1+i)n-1+ ...+R(1+i)2

+R(1+i)1 =FVAn(1+i)

RRRRR0123n–1

nFVADni%...OverviewViewofan

AnnuityDue–FVADCashflowsoccuratthebeginningoftheperiodFVADn=R(1+i)n+R(1+i)

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

=$1,225

+

$1,145

+

$1,070

=

$3,440$1,000$1,000$1,000$1,07001234$3,440=FVAD37%$1,225$1,145Exampleofan

AnnuityDue–FVADCash