第四章凈現(xiàn)值

第四章凈現(xiàn)值4.1TheOne-PeriodCase:FutureValueIfyouweretoinvest$10,000at5-percentinterestforoneyear,yourinvestmentwouldgrowto$10,500$500wouldbeinterest($10,000×.05)$10,000istheprincipalrepayment($10,000×1)$10,500isthetotaldue.Itcanbecalculatedas:$10,500=$10,000×(1.05).ThetotalamountdueattheendoftheinvestmentiscalltheFutureValue(FV).4.1TheOne-PeriodCase:FutureValueIntheone-periodcase,theformulaforFVcanbewrittenas:FV=C1×(1+r)WhereC1iscashflowatdate1andristheappropriateinterestrate.4.1TheOne-PeriodCase:PresentValueIfyouweretobepromised$10,000dueinoneyearwheninterestratesareat5-percent,yourinvestmentbeworth$9,523.81intoday’sdollars.Theamountthataborrowerwouldneedtosetasidetodaytotoabletomeetthepromisedpaymentof$10,000inoneyeariscallthePresentValue(PV)of$10,000.Notethat$10,000=$9,523.81×(1.05).4.1TheOne-PeriodCase:PresentValueIntheone-periodcase,theformulaforPVcanbewrittenas:WhereC1iscashflowatdate1andristheappropriateinterestrate.4.1TheOne-PeriodCase:NetPresentValueTheNetPresentValue(NPV)ofaninvestmentisthepresentvalueoftheexpectedcashflows,lessthecostoftheinvestment.Supposeaninvestmentthatpromisestopay$10,000inoneyearisofferedforsalefor$9,500.Yourinterestrateis5%.Shouldyoubuy?Yes!4.1TheOne-PeriodCase:NetPresentValueIntheone-periodcase,theformulaforNPVcanbewrittenas:IfwehadnotundertakenthepositiveNPVprojectconsideredonthelastslide,andinsteadinvestedour$9,500elsewhereat5-percent,ourFVwouldbelessthanthe$10,000theinvestmentpromisedandwewouldbeunambiguouslyworseoffinFVtermsaswell:$9,500×(1.05)=$9,975<$10,000.4.2TheMultiperiodCase:FutureValueThegeneralformulaforthefuturevalueofaninvestmentovermanyperiodscanbewrittenas:FV=C0×(1+r)TWhere C0iscashflowatdate0,ristheappropriateinterestrate,andTisthenumberofperiodsoverwhichthecashisinvested.4.2TheMultiperiodCase:FutureValueSupposethatJayRitterinvestedintheinitialpublicofferingoftheModiglianicompany.Modiglianipaysacurrentdividendof$1.10,whichisexpectedtogrowat40-percentperyearforthenextfiveyears.Whatwillthedividendbeinfiveyears?FV=C0×(1+r)T$5.92=$1.10×(1.40)5FutureValueandCompoundingNoticethatthedividendinyearfive,$5.92,isconsiderablyhigherthanthesumoftheoriginaldividendplusfiveincreasesof40-percentontheoriginal$1.10dividend:$5.92>$1.10+5×[$1.10×.40]=$3.30Thisisduetocompounding.FutureValueandCompounding012345PresentValueandCompoundingHowmuchwouldaninvestorhavetosetasidetodayinordertohave$20,000fiveyearsfromnowifthecurrentrateis15%?012345$20,000PVHowLongistheWait?Ifwedeposit$5,000todayinanaccountpaying10%,howlongdoesittaketogrowto$10,000?Assumethetotalcostofacollegeeducationwillbe$50,000whenyourchildenterscollegein12years.Youhave$5,000toinvesttoday.Whatrateofinterestmustyouearnonyourinvestmenttocoverthecostofyourchild’seducation? About21.15%.WhatRateIsEnough?4.3CompoundingPeriodsCompoundinganinvestmentmtimesayearforTyearsprovidesforfuturevalueofwealth:Forexample,ifyouinvest$50for3yearsat12%compoundedsemi-annually,yourinvestmentwillgrowtoEffectiveAnnualInterestRatesAreasonablequestiontoaskintheaboveexampleiswhatistheeffectiveannualrateofinterestonthatinvestment?TheEffectiveAnnualInterestRate(EAR)istheannualratethatwouldgiveusthesameend-of-investmentwealthafter3years:EffectiveAnnualInterestRates(continued)So,investingat12.36%compoundedannuallyisthesameasinvestingat12%compoundedsemiannually.ContinuousCompounding(Advanced)Thegeneralformulaforthefuturevalueofaninvestmentcompoundedcontinuouslyovermanyperiodscanbewrittenas:FV=C0×erTWhere C0iscashflowatdate0,risthestatedannualinterestrate,Tisthenumberofperiodsoverwhichthecashisinvested,andeisatranscendentalnumberapproximatelyequalto2.718.exisakeyonyourcalculator.4.4SimplificationsPerpetuityAconstantstreamofcashflowsthatlastsforever.GrowingperpetuityAstreamofcashflowsthatgrowsataconstantrateforever.AnnuityAstreamofconstantcashflowsthatlastsforafixednumberofperiods.GrowingannuityAstreamofcashflowsthatgrowsataconstantrateforafixednumberofperiods.PerpetuityAconstantstreamofcashflowsthatlastsforever.0…1C2C3CTheformulaforthepresentvalueofaperpetuityis:Perpetuity:ExampleWhatisthevalueofaBritishconsolthatpromisestopay￡15eachyear,everyyearuntilthesunturnsintoaredgiantandburnstheplanettoacrisp?Theinterestrateis10-percent.0…1￡152￡153￡15GrowingPerpetuityAgrowingstreamofcashflowsthatlastsforever.0…1C2C×(1+g)3C×(1+g)2Theformulaforthepresentvalueofagrowingperpetuityis:GrowingPerpetuity:ExampleTheexpecteddividendnextyearis$1.30anddividendsareexpectedtogrowat5%forever.Ifthediscountrateis10%,whatisthevalueofthispromiseddividendstream?0…1$1.302$1.30×(1.05)3$1.30×(1.05)2AnnuityAconstantstreamofcashflowswithafixedmaturity.01C2C3CTheformulaforthepresentvalueofanannuityis:TCAnnuity:ExampleIfyoucanafforda$400monthlycarpayment,howmuchcarcanyouaffordifinterestratesare7%on36-monthloans?01$4002$4003$40036$400GrowingAnnuityAgrowingstreamofcashflowswithafixedmaturity.01CTheformulaforthepresentvalueofagrowingannuity:2C×(1+g)3C×(1+g)2TC×(1+g)T-1GrowingAnnuityAdefined-benefitretirementplanofferstopay$20,000peryearfor40yearsandincreasetheannualpaymentby3-percenteachyear.Whatisthepresentvalueatretirementifthediscount