Lecture 3 Time Value of Money

Lecture3TimeValueofMoneyWhichwouldyouprefer--$100,000todayor$100,000in5years?Obviously,$100,000today.YoualreadyrecognizethatthereisTIMEVALUEofMONEY!!Shouldyougetyourmasterdegree？Howshouldyouarrangeyourplansforgooddreams?(townhouse,BMW,spacewalk)OutlineofLecture3TimeValueofMoneyInterestRates(Simplevs.Compound)PresentandFutureValueMultipleCashFlowsAnnuitiesandPerpetuitiesGrowingAnnuitiesandPerpetuitiesEARandAPRLoanTypesandLoanAmortizationWhyTime?WhyisTIMEsuchanimportantelementinyourdecision?TIMEallowsyoutheopportunitytopostponeconsumptionandearnINTEREST.WhatWouldRepresenttheTimeValueofMoney?Norisk,noinflation.InterestrateofT-bills(one-year)(inAmerica)一年期基准存款利率（inChina)InterestRatesInChina()时间税前税后时间税前税后1993.5.1510.98%10.98%2004.10.302.25%1.80%1996.5.19.18%9.18%2006.8.202.52%2.02%1996.8.237.47%7.47%2007.3.192.79%2.23%1997.10.235.67%5.67%2007.05.193.062.4481998.3.255.22%5.22%2007.07.213.332.6641998.7.14.77%4.77%2007.08.15利息税降至5%3.333.16351998.12.73.78%3.78%2007.08.223.63.421999.6.102.25%2.25%2007.09.153.873.67651999.11.1征收20%利息税2.25%1.80%2007.12.214.143.9332002.2.211.98%1.58%CompoundInterestInterestpaid(earned)onanypreviousinterestearned,aswellasontheprincipalborrowed(lent).SimpleInterestInterestpaid(earned)ononlytheoriginalamount,orprincipalborrowed(lent).TypesofInterestRateSimpleInterestvs.CompoundInterestAssumethatyouhaveput$100inasavingsaccountat5%peryear:COMPOUNDINTERESTYearPrincipalSimpleInterestInterestonInterestInterestEarnedTotalInterestEarned1$100.00$5.00$0.00$5.00$5.002$105.00$5.00$0.25$5.25$10.253$110.25$5.00$0.51$5.51$15.764$115.76$5.00$0.79$5.79$21.555$121.55$5.00$1.08$6.08$27.63SIMPLEINTERESTYearPrincipalSimpleInterestInterestonInterestInterestEarnedTotalInterestEarned1$100.00$5.00$0.00$5.00$5.002$100.00$5.00$0.00$5.00$10.003$100.00$5.00$0.00$5.00$15.004$100.00$5.00$0.00$5.00$20.005$100.00$5.00$0.00$5.00$25.00TheEffectofCompoundingSimpleInterestCompoundInterestInitialInvestment$C$CTotalInterestEarned$Crt$C((1+r)t-1)FinalInvestment$C(1+rt)$C(1+r)tTheEffectofCompounding050001000015000200001stYear10thYear20thYear30thYearFutureValueofaSingle$1,000Deposit10%SimpleInterest7%CompoundInterest10%CompoundInterestTheEffectofCompounding1626年印地安人以24美元把纽约曼哈顿岛卖给英国移民，几乎所有人都认为卖得太便宜了！但若印地安人能把24美元投资年报酬率7%的股票或定存连续到2007年(381年后)则可得9,115,795,334,036美元(91157.95亿美元)！TimeLineforCashFlows0123n……CashoutflowCashinflowThetimingofcashflowisimportant:intheendoratthebeginningPresentValuevs.FutureValueofSingleCashFlowThebasicpresentvalueequationwherePVisthepresentvalue,FVisthefuturevalue,ristheinterestrate(indecimaltermsnotpercentageterms)andtisthenumberofperiods.ValuationusingTableA-1PVIFi,nisfoundonTableA-1attheendofthebook(复利现值系数表，一次支付现值系数）Period6%7%8%1.943.935.9262.890.873.8573.840.816.7944.792.763.7355.747.713.681PV2=1000×（P/F,i,n)=1000×（P/F,7%,2)=1000×0.873=873ValuationusingTableA-3FVIFi,nisfoundonTableA-3attheendofthebook(复利终值系数表，一次支付终值系数）Period6%7%8%11.06001.07001.080021.12361.14491.166431.19101.22501.259741.26251.31081.360551.33821.40261.4693FV2=1000×（F/P,i,n)=1000×（F/P,7%,2)=1000×1.1449=1144.9ApplicationtoPersonalFinance:AnExampleSupposeitisJanuary1st2007,youjustturned25andexpectingtoretireattheageof65onJanuary1st2047.Yourbiggestdreamistogoona1-monthyachttourintheMediterraneanthatwillcostyou$65,000in2047whenyouretire.Tobeabletopayforthisyachttourin2047,howmuchshouldyouputinasavingsaccountthathasaninterestrateof1%perquartercompoundedquarterly?TypesofCashFlowsFivetypesofcashflow:SimpleCashFlowAnnuity(年金）Aconstantlevelofcashflows($C)startingoneyearfromnowforafixedperiodoftime(T).GrowingAnnuityAnannuitythathasacashflowof$C(1+g)tattimet,i.e.anexponentiallygrowingcashflowstream.Perpetuity(永续年金)Anannuityinwhichthecashflows($C)continueforever.GrowingPerpetuityAnannuityinwhichthecashflowsgrowexponentially($C(1+g)tattimet).TypesofAnnuitiesAnAnnuityrepresentsaseriesofequalpayments(orreceipts)occurringoveraspecifiednumberofequidistantperiods.OrdinaryAnnuity:Paymentsorreceiptsoccurattheendofeachperiod.AnnuityDue:Paymentsorreceiptsoccuratthebeginningofeachperiod.ExamplesofAnnuityStudentLoanPaymentsCarLoanPaymentsInsurancePremiumsMortgagePaymentsRetirementSavingsThePresentValueofOrdinaryAnnuities0123n……P……A),,/(])1(11[)1()1()1()1(32nrAPArrrArArArArAPVnnTheFutureValueofOrdinaryAnnuities?0123n…………AF),,/(]11[)1()1(21nrAFArrAArArAFVnnnQuestions?What’thepresentvalueofannuitiesdue?What’sthepresentvalueofperpetuities?What’sthefuturevalueofperpetuities?PresentValueFormulaeCisthecashflow,ristheinterestrate,Tisthefixedtimeperiod,gisthegrowthrateofcashflowsandthecashflowsstartoneyearfromnowforannuitiesandperpetuities.DoestheCompoundingPeriodMatter?Assumethatafteranalyzingyouroptions,youdecidedtogetamasterdegree.However,youneedtogetastudentloantopayforthetuitionandyouhavefoundthefollowingthreeratesasquotedonthebank’swebsite:LoanOption1:2%permonthcompoundedmonthlyLoanOption2:5%perquartercompoundedquarterlyLoanOption3:25%peryearcompoundedannuallyEARandAPR