IV. the time value of money

Totalliabilitiesandowner’sequityAssetsLiabilities＋owner’sequityCurrentassetsCurrentliabilitiesFixedassetsLong-termliabilities(Securities)CommonstockTotalassetsCapitalgainReturnedearningsIV.thetimevalueofmoneySimpleinterestCompoundinterestandfuturevaluePresentvalueandnetpresentvalueAnnuitiesPerpetuitiesCompoundingperiodandeffectiveinterestratesIntertemporalConsumptionOpportunitySetandIndifferenceCurvesIntertemporalConsumptionOpportunitySet$0$20,000$40,000$60,000$80,000$100,000$120,000$0$20,000$40,000$60,000$80,000$100,000$120,000ConsumptiontodayConsumptionatt+1Ms.ImpatienceMs.PatienceTheEffectofDifferentInterestRatesonConsumptionOpportunities$0$20,000$40,000$60,000$80,000$100,000$120,000$140,000$0$20,000$40,000$60,000$80,000$100,000$120,000$140,000$160,000r=10%r=50%IntertemporalConsumptionOpportunitySetConsumptiontodayConsumptionatt+11.simpleinterest•－theinterestpaidorearnedontheprincipalonly.•I=PV0*i*nSample•Marieagreestoinvest$1000inaventurethatpromisestopay10percentsimpleinteresteachyearfor2years.Howmuchmoneywillshehaveattheendofthesecondyear?•FV2=PV0[1+(10%*2)]=$1000*1.2=$12002.compoundinterestandfuturevalue•compoundinterestisinterestthatispaidnotonlyontheprincipalbutalsoonanyinterestearnedbutnotwithdrawnduringearlierperiods.TheTime-Value-of-Money•TheBasicTime-Value-of-MoneyRelationship:Ct+T=Ctx(1+r)T•where–ristheinterestrateperperiod–Tisthedurationoftheinvestment,statedinthecompoundingtimeunit–Ct(PV)isthevalueatperiodt(beginningoftheinvestment)–Ct+T(FV)isthevalueatperiodt+T(endoftheinvestment)•Compoundingfrequency:howoftenisinterestcalculatedFutureValueandCompoundingCompounding:Howmuchwill$1investedtodayat9%beworthintwoyears?(TheTimeLine)Year012C$1$1.1881Futurevalue(C2)=$1x1.0922=$1.1881Sample:howmuchforthatisland?•In1626,Minuit,theIndiansboughtallofManhattanIslandforabout$24ingoodsandtrinkets.Thissoundscheap,buttheIndiansmayhadgottenthebetterendofthedeal.IftheIndianshadsoldthegoodsandinvestedthe$24at10%,howmuchwoulditbeworthtoday?(1991)sample•FV365＝$24×(1+10%)365•＝$24×1,300,000,000,000,000•(1+10%)365≈1,300,000,000,000,000sample•BenjaminFranklin,whodiedonApril17,1790.Inhiswill,hegave1000poundssterlingtoMassachusettsandthecityofBoston.HegavealikeamounttoPennsylvaniaandthecityofPhiladelphia.•Franklinoriginallyspecifiedthatthemoneyshouldbepaidout100yearsafterhisdeathandusedtotrainyoungpeople.Aftersomelegalwrangling,itwasagreedthatthemoneywouldbepaidoutin1990,200yearsafterFranklin’sdeath.Bythattime,thePennsylvaniabequesthadgrowntoabout$2million;theMassachusettsbequesthadgrownto$4.5million.ThemoneywasusedtofundtheFranklinInstitutesinBostonandPhiladelphia.sample•Assumingthat1000poundssterlingwasequivalentto1000dollars,whatrateofreturndidthetwostatesearn?(thedollardidnotbecometheofficialUScurrencyuntil1792)•Pennsylvaniabequest:•$1000＝$2million/(1+r)200•r＝3.87%•Massachusettsbequest:•$1000＝$4.5million/(1+r)200•r＝4.3%3.Presentvalueandnetpresentvalue•Presentvalue(PV)•Netpresentvalue(NPV)PresentValueandDiscountingDiscounting:Howmuchis$1thatwewillreceiveintwoyearsworthtoday(r=9%)?Year012C$0.842$1Presentvalue(C0)=$1/1.0922=$0.842Theinterestrate(9%)isalsocalledthediscountrate.Netpresentvalue(NPV)•SometipsforcomputingNPV:–Onlyadd(subtract)cashflowsfromthesametimeperiod–UsetheTimeLine–Specifyacashflowforeachtimeperiod(evenwhenitis$0)–Useanappropriatediscountrate•ThegeneralformulaforcalculatingNPV:NPV=-C0+C1/(1+r)+C2/(1+r)2+..+CT/(1+r)T4.Annuities•anannuityisthepaymentorreceiptofequalcashflowsperperiodforaspecifiedamountoftime.Ordinaryannuity•AnordinaryannuityisoneinwhichthepaymentsorreceiptsoccurattheendofeachperiodAnnuitydue•Anannuitydueisoneinwhichpaymentsorreceiptsoccuratthebeginningofeachperiod.5.futurevalueofanordinaryannuity•FVn,o=C∑(1+i)n-t•=C[(1+i)n–1]/i6.futurevalueofanannuitydue•FVn,d=C[[(1+i)n–1]/i](1+i)7.presentvalueofanordinaryannuity•PVn,o=C∑[1/(1+i)t]•=C[1–1/(1+i)n]/iSample•Supposeyouplantocontribute$2000everyyearintoaretirementaccountpaying8percent.Ifyouretirein30years,howmuchwillyouhavewhenyouretire?Sample•PV0＝2000/(1+8%)+2000/(1+8%)2+2000/(1+8%)3+…+2000/(1+8%)30•＝C[1/r–1/r(1+r)t]•＝$2000×11.2578＝$22515.57Sample•FV30＝PV0(1+r)30＝$22515.57×10.0627•＝＄226566.4•FV30＝C[(1+r)n–1]/r•＝$2000[(1+8%)30–1]/8%•＝＄226566.4ProvePV0＝C/(1+r)+C/(1+r)2+C/(1+r)3+…+C/(1+r)n•PV0(1+r)＝C+C/(1+r)+C/(1+r)2+C/(1+r)3+…+C/(1+r)n-1•PV0(1+r)–PV0＝PV0(r)＝C–C/(1+r)n•PV0＝C/r–C/r(1+r)n8.loanamortizationandcapitalrecoveryproblem•PV0=C/r[1–1/(1+r)n]•C=PV0/[1/r(1-1/(1+r)n]例已知小万按10％的年利率借入1000元，计划每年按相同的金额偿还本息，分4年还清。现应如何筹划？–这实际上是要计算出每年的还本付息额（即年金C）–由PV0＝C[1/r–1/r(1+r)t]–C＝PV0/[1/r–1/r(1+r)t]–＝1000元/[1/10%-1/10%(1+10%)4]–＝1000元/3.17＝315.46元9.presentvalueofanannuitydue•PV0,d=C[1/r–1/r(1+r)t](1+r)10.Perpetuities•aperpetuityisafinancialinstrumentthatpromisestopayanequalcashflowperperiodforever;thatis,aninfiniteseriesofpayments.•PV0=∑[C/(1+i)]•=C/i11.presentvalueofdeferredannuities•forexample,supposethatyouwishtoprovideforthecollegeeducationofyourdaughter.Shewillbegincollege5yearsfromnow,andyouwishtohave$15000availableforheratthebeginningofeachyearincol