公司理财第4章

DiscountedCashFlowValuationChapter4KeyConceptsandSkillsBeabletocomputethefuturevalueand/orpresentvalueofasinglecashfloworseriesofcashflowsBeabletocomputethereturnonaninvestmentBeabletouseafinancialcalculatorand/orspreadsheettosolvetimevalueproblemsUnderstandperpetuitiesandannuitiesChapterOutline4.1Valuation:TheOne-PeriodCase4.2TheMultiperiodCase4.3CompoundingPeriods4.4Simplifications4.5WhatIsaFirmWorth?4.1TheOne-PeriodCaseIfyouweretoinvest$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).FutureValueIntheone-periodcase,theformulaforFVcanbewrittenas:FV=C0×(1+r)TWhereC0iscashflowtoday(timezero),andristheappropriateinterestrate.PresentValueIfyouweretobepromised$10,000dueinoneyearwheninterestratesare5-percent,yourinvestmentwouldbeworth$9,523.81intoday’sdollars.Theamountthataborrowerwouldneedtosetasidetodaytobeabletomeetthepromisedpaymentof$10,000inoneyeariscalledthePresentValue(PV).Notethat$10,000=$9,523.81×(1.05).PresentValueIntheone-periodcase,theformulaforPVcanbewrittenas:WhereC1iscashflowatdate1,andristheappropriateinterestrate.NetPresentValueTheNetPresentValue(NPV)ofaninvestmentisthepresentvalueoftheexpectedcashflows,lessthecostoftheinvestment.Supposeaninvestmentthatpromisestopay$10,000inoneyearisofferedforsalefor$9,500.Yourinterestrateis5%.Shouldyoubuy?NetPresentValueThepresentvalueofthecashinflowisgreaterthanthecost.Inotherwords,theNetPresentValueispositive,sotheinvestmentshouldbepurchased.NetPresentValueIntheone-periodcase,theformulaforNPVcanbewrittenas:NPV=–Cost+PVIfwehadnotundertakenthepositiveNPVprojectconsideredonthelastslide,andinsteadinvestedour$9,500elsewhereat5percent,ourFVwouldbelessthanthe$10,000theinvestmentpromised,andwewouldbeworseoffinFVterms:$9,500×(1.05)=$9,975

<$10,0004.2TheMultiperiodCaseThegeneralformulaforthefuturevalueofaninvestmentovermanyperiodscanbewrittenas:FV=C0×(1+r)TWhere C0iscashflowatdate0,ristheappropriateinterestrate,andTisthenumberofperiodsoverwhichthecashisinvested.FutureValueSupposeastockcurrentlypaysadividendof$1.10,whichisexpectedtogrowat40%peryearforthenextfiveyears.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.FutureValueandCompounding012345PresentValueandDiscountingHowmuchwouldaninvestorhavetosetasidetodayinordertohave$20,000fiveyearsfromnowifthecurrentrateis15%?012345$20,000PVHowLongistheWait?Ifwedeposit$5,000todayinanaccountpaying10%,howlongdoesittaketogrowto$10,000?Assumethetotalcostofacollegeeducationwillbe$50,000whenyourchildenterscollegein12years.Youhave$5,000toinvesttoday.Whatrateofinterestmustyouearnonyourinvestmenttocoverthecostofyourchild’seducation? WhatRateIsEnough?About21.15%.CalculatorKeysTexasInstrumentsBA-IIPlusFV=futurevaluePV=presentvalueI/Y=periodicinterestrateP/Ymustequal1fortheI/YtobetheperiodicrateInterestisenteredasapercent,notadecimalN=numberofperiodsRemembertocleartheregisters(CLRTVM)aftereachproblemOthercalculatorsaresimilarinformatMultipleCashFlows Consideraninvestmentthatpays$200oneyearfromnow,withcashflowsincreasingby$200peryearthroughyear4.Iftheinterestrateis12%,whatisthepresentvalueofthisstreamofcashflows?Iftheissueroffersthisinvestmentfor$1,500,shouldyoupurchaseit?MultipleCashFlows01234200400600800178.57318.88427.07508.411,432.93PresentValue<Cost→DoNotPurchaseValuing“Lumpy”CashFlowsFirst,setyourcalculatorto1paymentperyear.Then,usethecashflowmenu:CF2CF1F2F1CF0120011,432.930400INPV12CF4CF3F4F3160018004.3CompoundingPeriodsCompoundinganinvestmentmtimesayearforTyearsprovidesforfuturevalueofwealth:CompoundingPeriodsForexample,ifyouinvest$50for3yearsat12%compoundedsemi-annually,yourinvestmentwillgrowtoEffectiveAnnualRatesofInterestAreasonablequestiontoaskintheaboveexampleis“whatistheeffectiveannualrateofinterestonthatinvestment?”TheEffectiveAnnualRate(EAR)ofinterestistheannualratethatwouldgiveusthesameend-of-investmentwealthafter3years:EffectiveAnnualRatesofInterestSo,investingat12.36%compoundedannuallyisthesameasinvestingat12%compoundedsemi-annually.EffectiveAnnualRatesofInterestFindtheEffectiveAnnualRate(EAR)ofan18%APRloanthatiscompoundedmonthly.Whatwehaveisaloanwithamonthlyinterestraterateof1½%.Thisisequivalenttoaloanwithanannualinterestrateof19.56%.EARonaFinancialCalculatorkeys:description:[2nd][ICONV]Opensinterestrateconversionmenu[↓][EFF=][CPT]19.56TexasInstrumentsBAIIPlus

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[ENTER]Sets18APR.[↑][C/Y=]12[ENTER]Sets12paymentsperyearContinuousCompoundingThegeneralformulaforthefuturevalueofaninvestmentcompoundedcontinuouslyovermanyperiodscanbewrittenas:FV=C0×erTWhere C0iscashflowatdate0,risthestatedannualinterestrate,Tisthenumberofyears,andeisatranscendentalnumberapproximatelyequalto2.718.exisakeyonyourcalculator.4.4SimplificationsPerpetuityAconstantstreamofcashflowsthatlastsforeverGrowingperpetuityAstreamofcashflowsthatgrowsataconstantrateforeverAnnuityAstreamofconstantcashflowsthatlastsforafixednumberofperiodsGrowingannuityAstreamofcashflowsthatgrowsataconstantrateforafixednumberofperiodsPerpetuityAconstantstreamofcashflowsthatlastsforever0…1C2C3CPerpetuity:ExampleWhatisthevalueofaBritishconsolthatpromisestopay£15everyyearforever?Theinterestrateis10-percent.0…1£152£153£15GrowingPerpetuityAgrowingstreamofcashflowsthatlastsforever0…1C2C×(1+g)3C×(1+g)2GrowingPerpetuity:ExampleTheexpecteddividendnextyearis$1.30,anddividendsareexpectedtogrowat5%forever.Ifthediscountrateis10%,whatisthevalueofthispromiseddividendstream?0…1$1.302$1.30×(1.05)3$1.30×(1.05)2AnnuityAconstantstreamofcashflowswithafixedmaturity01C2C3CTCAnnuity:ExampleIfyoucanafforda$400monthlycarpayment,howmuchcarcanyouaffordifinterestratesare7%on36-monthloans?01$4002$4003$40036$400 Whatisthepresentvalueofafour-yearannuityof$100peryearthatmakesitsfirstpaymenttwoyearsfromtodayifthediscountrateis9%?

0 1 2 345$100 $100 $100$100$323.97$297.22GrowingAnnuityAgrowingstreamofcashflowswithafixedmaturity01C2C×(1+g)3C×(1+g)2TC×(1+g)T-1Growin