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Chapter4IntroductiontoValuation:TheTimeValueofMoneyLearningobjectivesHowtodeterminethefuturevalueofaninvestmentmadetodayHowtodeterminethepresentvalueofcashtobereceivedatafuturedateHowtofindthereturnonaninvestmentHowlongittakesforaninvestmenttoreachadesiredvalue.4ChapterOutline4.1FutureValueandCompounding4.2PresentValueandDiscounting4.3MoreonPresentandFutureValues4.1futurevalueandcompoundingFuturevalue(FV)-futurevaluereferstotheamountofmoneyaninvestmentwillgrowtooversomeperiodoftimeatsomegiveninterestrate.Putanotherway,futurevalueisthecashvalueofinvestmentatsometimeinthefuture.6Figure4.17exampleSupposeyouinvest$1000foroneyearat5%peryear.Whatisthefuturevalueinoneyear?Interest=1000(.05)=50Valueinoneyear=principal+interest=1000+50=1050FutureValue(FV)=1000(1+.05)=1050Supposeyouleavethemoneyforanotheryear.Howmuchwillyouhavetwoyearsfromnow?FV=1000(1.05)(1.05)=1000(1.05)2=1102.508FutureValues:GeneralFormulaFV=PV(1+r)tFV=futurevaluePV=presentvaluer=periodinterestrate,expressedasadecimalT=numberofperiodsFuturevalueinterestfactor=(1+r)t9EffectsofCompoundingSimpleinterest（单利）Compoundinterest(复利)ConsiderthepreviousexampleFVwithsimpleinterest=1000+50+50=1100FVwithcompoundinterest=1102.50Theextra2.50comesfromtheinterestof.05(50)=2.50earnedonthefirstinterestpaymentEffectofcompoundingTheeffectofcompoundingissmallforasmallnumberofperiods,butincreasesasthenumberofperiodsincreases.11Figure4.212CalculatorKeysTexasInstrumentsBA-IIPlusFV=futurevaluePV=presentvalueI/Y=periodinterestrateInterestisenteredasapercent,notadecimalP/Y=paymentperyearC/Y=compoundingperiodperyearN=numberofperiodsRemembertocleartheregisters(CE/C)(2ND)(CLRTVM)beforesolvingeachproblemOthercalculatorsaresimilarinformat13SettingsDecimal(2):(2ND)(FORMAT)(2OR4)(ENTER)P/Y(Paymentsperyear)=1；C/Y(Compoundingperiodsperyear)=1:(2ND)(P/Y)(1)(ENTER)(shift

DOWN)(1)(ENTER)END-OF-PERIOD:(2ND)(BGN)(2ND)(SET)(CE/C)Howtoavoidmistakesinusingthecalculator(P94)14Usingafinancialcalculator(onP71)-100(keyin100andkeyin+/-)=PV5=N10(decimal)=I/YCPT(COMP) FVGotFV=161.0515FutureValues–Example2Supposeyouinvestthe$1000at5%for5years.Howmuchwouldyouhaveatmaturity?FV=1000(1.05)5=1276.28Simpleinterestwouldhaveafuturevalueof$1250(1000(1+0.05*5)=1250),foradifferenceof$26.2816FutureValues–Example3Supposeyouhadadeposit$10at5.5%interest200yearsago.Howmuchwouldtheinvestmentbeworthtoday?FV=10(1.055)200=447,189.84Whatistheeffectofcompounding?Simpleinterest=10(1+200*0.055)=120.00Compoundingadded$447,069.84tothevalueoftheinvestment17

Example4-thevalueofManhattanThedealofforManhanttan(P70)In1626,PeterMinuitboughtManhattanIslandforUSD24ingoodsandtrinkets(小装饰品

).IsitagooddealtoPeter?SupposePeterinvestedUSD24at10%,whatisthevalueat2023(total380years)?18

Example4-thevalueofManhattanWorkitwithcalculator,N=380,I/Y=10,PV=-24,CPTFV:wegetthesameresult.19

Example4-thevalueofManhattan10%istoohigh,whatabout3.25%？FV=4554431.42？！20怎么样看待国际间旳博弈？世界金融市场都受到罗斯切尔德家族旳控制吗？“据估计，1850年左右，罗斯切尔德家族总共积累了60亿美元旳财富，假如以6%旳回报率计算，到150数年后旳今日，他们旳家族资产将至少在30万亿美元之上。”——宋鸿兵《货币战争》21怎么样看待国际间旳博弈？“中国有句俗话，弟兄同心，其利断金。罗斯切尔德家族经过家族内部通婚严格预防财富稀释和外流。在１００数年里，家族内部通婚１８次，１６次是在第一表亲（堂兄妹）之间。”——宋鸿兵《货币战争》

多么奇怪旳逻辑！22怎么样看待国际间旳博弈？假如我们都相信，这个世界都由某一种超级大家族所操控，我们除了听天由命还有什么能做旳？234.2PresentValues现值HowmuchdoIhavetoinvesttodaytohavesomeamountinthefuture?FV=PV(1+r)tRearrangetosolveforPV=FV/(1+r)tWhenwetalkaboutdiscounting,wemeanfindingthepresentvalueofsomefutureamount.Whenwetalkaboutthe“value”ofsomething,wearetalkingaboutthepresentvalueunlesswespecificallyindicatethatwewantthefuturevalue.24Presentvalue现值Discountingfactor:25PV–OnePeriodExampleSupposeyouneed$10,000inoneyearforthedownpayment(首付)onanewcar.Ifyoucanearn7%annually,howmuchdoyouneedtoinvesttoday?PV=10,000/(1.07)1=9345.79Calculator1N7I/Y10,000FVCPTPV=-9345.7926PresentValues–Example2Youwanttobeginsavingforyoudaughter’scollegeeducationandyouestimatethatshewillneed$150,000in17years.Ifyoufeelconfidentthatyoucanearn8%peryear,howmuchdoyouneedtoinvesttoday?PV=150,000/(1.08)17=40,540.3427PresentValues–Example3Yourparentssetupatrustfundforyou10yearsagothatisnowworth$19,671.51.Ifthefundearned7%peryear,howmuchdidyourparentsinvest?PV=19,671.51/(1.07)10=10,00028PV–ImportantRelationshipIForagiveninterestrate–thelongerthetimeperiod,thelowerthepresentvalueWhatisthepresentvalueof$500tobereceivedin5years?10years?Thediscountrateis10%5years:PV=500/(1.1)5=310.4610years:PV=500/(1.1)10=192.7729PV–ImportantRelationshipIIForagiventimeperiod–thehighertheinterestrate,thesmallerthepresentvalueWhatisthepresentvalueof$500receivedin5yearsiftheinterestrateis10%?15%?Rate=10%:PV=500/(1.1)5=310.46Rate=15%;PV=500/(1.15)5=248.5830Figure4.34.3moreonpresentandfuturevaluesIfyoulookbackattheexpressionwecameupwithforpresentandfuturevalues,youwillseethereisaverysimplerelationshipbetweenthetwo.Weexplorethisrelationshipandsomerelatedissuesinthissection32TheBasicPVEquation-RefresherPV=FV/(1+r)tTherearefourpartstothisequationPV,FV,randtIfweknowanythree,wecansolveforthefourthIfyouareusingafinancialcalculator,besureandrememberthesignconvention（符号要求），e.g.(+/-)oryouwillreceiveanerrorwhensolvingforrort33DiscountRateOftenwewillwanttoknowwhattheimpliedinterestrateisinaninvestmentRearrangethebasicPVequationandsolveforrFV=PV(1+r)tr=(FV/PV)1/t–1Ifyouareusingformulas,youwillwanttomakeuseofboththeyxandthe1/xkeys34DiscountRate–Example1Youarelookingataninvestmentthatwillpay$1200in5yearsifyouinvest$1000today.Whatistheimpliedrateofinterest?r=(1200/1000)1/5–1=.03714=3.714%Calculator–thesignconventionmatters!!!N=5PV=-1000(youpay1000today)FV=1200(youreceive1200in5years)CPTI/Y=3.714%35DiscountRate–Example2Supposeyouareofferedaninvestmentthatwillallowyoutodoubleyourmoneyin6years.Youhave$10,000toinvest.Whatistheimpliedrateofinterest?r=(20,000/10,000)1/6–1=.122462=12.25%36DiscountRate–Example3Supposeyouhavea1-yearoldsonandyouwanttoprovide$75,000in17yearstowardshiscollegeeducation.Youcurrentlyhave$5000toinvest.Whatinterestratemustyouearntohavethe$75,000whenyouneedit?r=(75,000/5,000)1/17–1=.172688=17.27%37FindingtheNumberofPeriodsStartwithbasicequationandsolvefort(rememberyourlogs)FV=PV(1+r)tt=ln(FV/PV)/ln(1+r)Youcanusethefinancialkeysonthecalculatoraswell,justrememberthesignconvention.38NumberofPeriods–Example1Youwanttopurchaseanewcarandyouarewillingtopay$20,000.Ifyoucaninvestat10%peryearandyoucurrentlyhave$15,000,howlongwillitbebeforeyouhaveenoughmoneytopaycashforthecar?t=ln(20,000/15,000)/ln(1.1)=3.02years39NumberofPeriods–Example2Supposeyouwanttobuyanewhouse.Youcurrentlyhave$15,000andyoufigureyouneedtohavea10%downpaymentplusanadditional5%inclosingcosts(借款手续费).Ifthetypeofhouseyouwantcostsabout$150,000andyoucanearn7.5%peryear,howlongwillitbebeforeyouhaveenoughmoneyforthedownpaymentandclosingcosts?40Example2ContinuedHowmuch