公司理財(cái)?shù)?章(英文)課件

CHAPTER4NetPresent

Value0CHAPTER4NetPresent

Value0ChapterOutline4.1TheOne-PeriodCase4.2TheMultiperiodCase4.3CompoundingPeriods4.4Simplifications4.5WhatIsaFirmWorth?4.6SummaryandConclusions1ChapterOutline4.1TheOne-Per4.1TheOne-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).24.1TheOne-PeriodCase:

Futur4.1TheOne-PeriodCase:

FutureValueIntheone-periodcase,theformulaforFVcanbewrittenas:FV=C0×(1+r)TWhereC0iscashflowtoday(timezero)andristheappropriateinterestrate.T(Time)=1.34.1TheOne-PeriodCase:

Futur4.1TheOne-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).44.1TheOne-PeriodCase:Prese4.1TheOne-PeriodCase:

PresentValueIntheone-periodcase,theformulaforPVcanbewrittenas:WhereC1iscashflowatdate1andristheappropriateinterestrate.54.1TheOne-PeriodCase:

Prese4.1TheOne-PeriodCase:

NetPresentValueTheNetPresentValue(NPV)ofaninvestmentisthepresentvalueoftheexpectedcashflows,lessthecostoftheinvestment.Supposeaninvestmentthatpromisestopay$10,000inoneyearisofferedforsalefor$9,500.Yourinterestrateis5%.Shouldyoubuy?Yes!64.1TheOne-PeriodCase:

NetP4.1TheOne-PeriodCase:NetPresentValueIntheone-periodcase,theformulaforNPVcanbewrittenas:NPV=–Cost+PVIfwehadnotundertakenthepositiveNPVprojectconsideredonthelastslide,andinsteadinvestedour$9,500elsewhereat5-percent,ourFVwouldbelessthanthe$10,000theinvestmentpromisedandwewouldbeunambiguouslyworseoffinFVtermsaswell:$9,500×(1.05)=$9,975<$10,000.74.1TheOne-PeriodCase:NetP4.2TheMultiperiodCase:

FutureValueThegeneralformulaforthefuturevalueofaninvestmentovermanyperiodscanbewrittenas:FV=C0×(1+r)TWhere C0iscashflowatdate0,ristheappropriateinterestrate,andTisthenumberofperiodsoverwhichthecashisinvested.(1+r)Tisthefuturevaluefactor84.2TheMultiperiodCase:

Futu4.2TheMultiperiodCase:

FutureValueSupposethatJayRitterinvestedintheinitialpublicofferingoftheModiglianicompany.Modiglianipaysacurrentdividendof$1.10,whichisexpectedtogrowat40-percentperyearforthenextfiveyears.Whatwillthedividendbeinfiveyears?FV=C0×(1+r)T$5.92=$1.10×(1.40)594.2TheMultiperiodCase:

FutuFutureValueandCompoundingNoticethatthedividendinyearfive,$5.92,isconsiderablyhigherthanthesumoftheoriginaldividendplusfiveincreasesof40-percentontheoriginal$1.10dividend:$5.92>$1.10+5×[$1.10×.40]=$3.30Thisisduetocompounding.Theprocessofleavingthemoneyinthecapitalmarketandlendingitforanotheryeariscalledcompounding.10FutureValueandCompoundingNoFutureValueandCompounding01234511FutureValueandCompounding01PresentValueandDiscounting(p68)Howmuchwouldaninvestorhavetosetasidetodayinordertohave$20,000fiveyearsfromnowifthecurrentrateis15%?012345$20,000PV12PresentValueandDiscountingPresentValueandDiscounting(p68)Theprocessofcalculatingthepresentvalueofafuturecashflowiscalleddiscounting.Itistheoppositeofcompounding.13PresentValueandDiscountingIfwedeposit$5,000todayinanaccountpaying10%,howlongdoesittaketogrowto$10,000?HowLongistheWait?14Ifwedeposit$5,000todayinAssumethetotalcostofacollegeeducationwillbe$50,000whenyourchildenterscollegein12years.Youhave$5,000toinvesttoday.Whatrateofinterestmustyouearnonyourinvestmenttocoverthecostofyourchild’seducation? About21.15%.WhatRateIsEnough?15Assumethetotalcostofacol4.3CompoundingPeriods(p72)CompoundinganinvestmentmtimesayearforTyearsprovidesforfuturevalueofwealth:Forexample,ifyouinvest$50for3yearsat12%compoundedsemi-annually,yourinvestmentwillgrowto164.3CompoundingPeriods(p72)CoEffectiveAnnualInterestRatesAreasonablequestiontoaskintheaboveexampleiswhatistheeffectiveannualrateofinterestonthatinvestment?TheEffectiveAnnualInterestRate(EAIR)istheannualratethatwouldgiveusthesameend-of-investmentwealthafter3years:17EffectiveAnnualInterestRate1818EffectiveAnnualInterestRates(continued)So,investingat12.36%compoundedannuallyisthesameasinvestingat12%compoundedsemiannually.19EffectiveAnnualInterestRateEffectiveAnnualInterestRates(continued)FindtheEffectiveAnnualRate(EAIR)ofan18%APR(Annualpercentagerate)loanthatiscompoundedmonthly.Whatwehaveisaloanwithamonthlyinterestraterateof1?percent.Thisisequivalenttoaloanwithanannualinterestof19.56percent20EffectiveAnnualInterestRate2121ContinuousCompounding(Advancedp74)Thegeneralformulaforthefuturevalueofaninvestmentcompoundedcontinuouslyovermanyperiodscanbewrittenas:FV=C0×erTPV=

C0/erTWhere C0iscashflowatdate0,risthestatedannualinterestrate,Tisthenumberofperiodsoverwhichthecashisinvested,andeisatranscendentalnumberapproximatelyequalto2.718.exisakeyonyourcalculator.Readexampleonp74,7522ContinuousCompounding(Advanc4.4Simplifications(p75)PerpetuityAconstantstreamofcashflowsthatlastsforever.GrowingperpetuityAstreamofcashflowsthatgrowsataconstantrateforever.AnnuityAstreamofconstantcashflowsthatlastsforafixednumberofperiods.GrowingannuityAstreamofcashflowsthatgrowsataconstantrateforafixednumberofperiods.234.4Simplifications(p75)PerpetPerpetuityAconstantstreamofcashflowsthatlastsforever.0…1C2C3CTheformulaforthepresentvalueofaperpetuityis:24PerpetuityAconstantstreamofPerpetuity:ExampleWhatisthevalueofaBritishconsolthatpromisestopay￡15eachyear,everyyearuntilthesunturnsintoaredgiantandburnstheplanettoacrisp?Theinterestrateis10-percent.0…1￡152￡153￡1525Perpetuity:ExampleWhatistheGrowingPerpetuityAgrowingstreamofcashflowsthatlastsforever.0…1C2C×(1+g)3C×(1+g)2Theformulaforthepresentvalueofagrowingperpetuityis:26GrowingPerpetuityAgrowingstGrowingPerpetuity:ExampleTheexpecteddividendnextyearis$1.30anddividendsareexpectedtogrowat5%forever.Ifthediscountrateis10%,whatisthevalueofthispromiseddividendstream?0…1$1.302$1.30×(1.05)3$1.30×(1.05)227GrowingPerpetuity:ExampleTheReadtheexampleonpage78,finddifferencebetweenitandtheexampleonlastslide.Ifthedividendistobereceivedimmediately,PV=C+C(1+g)/(r-g)2C×(1+g)23C×(1+g)3C(1+g)C0128Readtheexampleonpage78,finAnnuity(p79)Aconstantstreamofcashflowswithafixedmaturity.Theformulaforthepresentvalueofanannuityis:01C2C3CTC29Annuity(p79)AconstantstreamIscalledanannuityfactor,itisexpressedasArT,itstandsforthepresentvalueof$1ayearforTyearsataninterestrateofr.30Iscalledanannuityfactor,iAnnuityIntuitionAnannuityisvaluedasthedifferencebetweentwoperpetuities:oneperpetuitythatstartsattime1

lessaperpetuitythatstartsattimeT+101C2C3CTC31AnnuityIntuitionAnannuityiAnnuity:ExampleIfyoucanafforda$400monthlycarpayment,howmuchcarcanyouaffordifinterestratesare7%on36-monthloans?01$4002$4003$40036$40032Annuity:ExampleIfyoucanaff

Trick1:ADelayedAnnuityWhatisthepresentvalueofafour-yearannuityof$100peryearthatmakesitsfirstpaymenttwoyearsfromtodayifthediscountrateis9%?

0 1 2 345$100 $100 $100$100$327.97$297.2233 Trick1:ADelayedAnnuityWhTrick2:AnnuityinAdvance

Anannuitywithanimmediateinitialpaymentiscalledan“annuityinadvance”01$500002$500003$5000020$5000034Trick2:AnnuityinAdvance

AnTrick3:TheInfrequencyAnnuityAnannuitywithpaymentsoccurringlessfrequentlythanonceayearForexampleonp82,012$4504$45020$45035Trick3:TheInfrequencyAnnuiTrick4:Equatingpresentvalueoftwoannuities01deposit2deposit3deposit18tuitionForexampleon83,211736Trick4:EquatingpresentvaluGrowingAnnuityAgrowingstreamofcashflowswithafixedmaturity.Theformulaforthepresentvalueofagrowingannuity:01C2C×(1+g)3C×(1+g)2T

C×(1+g)T-137GrowingAnnuityAgrowingstreaGrowingAnnuityAdefined-benefitretirementplanofferstopay$20,000peryearfor40yearsandincreasetheannualpaymentbythree-percenteachyear.Whatisthepresentvalueatretirementifthediscountrateis10percent?01$20,0002$20,000×(1.03)40$20,000×(1.03)3938GrowingAnnuityAdefined-benefPVofadelayedgrowingannuityYourfirmisabouttomakeitsinitialpublicofferingofstockandyourjobistoestimatethecorrectofferingprice.Forecastdividendsareasfollows.Year:1234Dividendspershare$1.50$1.65$1.825%growththereafterIfinvestorsdemanda10%returnoninvestmentsofthisrisklevel,whatpricewilltheybewillingtopay?39PVofadelayedgrowingannuitPVofadelayedgrowingannuityYear0123Cashflow$1.50$1.65$1.824$1.82×1.05…Thefirststepistodrawatimeline.Thesecondstepistodecideonwhatweknowandwhatitiswearetryingtofind.40PVofadelayedgrowingannuitPVofadelayedgrowingannuityYear0123Cashflow$1.50$1.65$1.82dividend+P3PVofcashflow$32.81=$1.82+$38.2241PVofadelayedgrowingannuit4.5WhatIsaFirmWorth?(p86)Conceptually,afirmshouldbeworththepresentvalueofthefirm’scashflows.Seetheexampleon87Thetrickypartisdeterminingthesize,timingandriskofthosecashflows.424.5WhatIsaFirmWorth?(p86)4.6SummaryandConclusionsTwobasicconcepts,futurevalueandpresentvalueareintroducedinthischapter.Interestratesarecommonlyexpressedonanannualbasis,butsemi-annual,quarterly,monthlyandevencontinuouslycompoundedinterestratearrangementsexist.Theformulaforthenetpresentvalueofaninvestmentthatpays$CforNperiodsis:434.6SummaryandConclusionsTwo4.6SummaryandConclusions(continued)Wepresentedfoursimplifyingformulae:444.6SummaryandConclusions(cMultipleChoice1.Thecompoundvalueisdefinedas A) thevalueofadollarreceivedyesterday. B) thevalueofasumafterinvestingoveroneormoreperiods. C) thevalueofasumtobereceivedtoday. D) therateofgrowthinasumtoday. E) Noneoftheabove.45MultipleChoice1.ThecompounAnswer:B46Answer:B462.Presentvaluemaybedefinedas

A) futurecashflowsdiscountedtothepresent. B) estimatedfuturecashflowinyearn+1. C) presentcashflowscompoundedintothefuture. D) theaverageofthebidandaskedprice. E) Noneoftheabove.472.Presentvaluemaybedefine3.Thepresentvalueoffuturecashflowsminusinitialcostiscalled A) thefuturevalueoftheproject.

B) thenetpresentvalueoftheproject. C) theequivalentsumoftheinvestment. D) theinitialinvestmentriskequivalentvalue. E) Noneoftheabove.483.Thepresentvalueoffuture4.Ifyouhaveachoicetoearnsimpleintereston$10,000forthreeyearsat8%orannuallycompoundinterestat7.5%forthreeyearswhichonewillpaymoreandbyhowmuch? A) Simpleinterestby$50.00 B) Compoundinterestby$22.97 C) Compoundinterestby$150.75 D) Compoundinterestby$150.00 E) Noneoftheabove.494.IfyouhaveachoicetoearAnswer:B Rationale: SimpleInterest=$10,000(.08)(3)=$2,400; CompoundInterest=$10,000((1.075)3-1)=$2,422.97; Difference=$2,422.97-$2,400=$22.9750Answer:B505.Theequation[1+(r/m)]m-1givesthe

A) effectiveannualinterestrate. B) quotedrate. C) statedannualinterestrate. D) semiannualrate. E) Noneoftheabove.515.Theequation[1+(r/m)]m-16.Whatisthefuturevalueofinvesting$9,000for7yearsatacontinuouslycompoundedrateof11%? A) $15,930.00 B) $18,685.44 C) $19,369.83 D) $19,437.90 E) Noneoftheabove.526.WhatisthefuturevalueofAnswer:D Rationale: $9,000(e.11(7))=$9,000(2.15976625)=$19,437.9053Answer:D537.Whichofthefollowingamountsisclosesttotheendvalueofinvesting$5,000for14monthsatastatedannualinterestrateof6percentcompoundedmonthly? A) $5,293 B) $5,345 C) $5,352 D) $5,362 E) $6,183547.WhichofthefollowingamouAnswer:D Rationale: EAR=(1+r/m)m-1=[1+(0.06/12)]12-1=0.06167781 Endvalue=$5,000(1+0.06167781)14/12=$5,361.6155Answer:D558.Ifthecompoundperiodisgreaterthanone A) theeffectiveannualinterestrateisalwaysequaltotheannualpercentagerate. B) theeffectiveannualinterestrateisalwayslessthantheannualpercentagerate.

C) theeffectiveannualinterestrateisalwaysgreaterthantheannualpercentagerate. D) theeffectiveannualinterestrateisnevergreaterthantheannualpercentagerate. E) Noneoftheabove.568.Ifthecompoundperiodisg9.Whatisthenetpresentvalueofaprojectthatcontributes$25,000attheendofthefirstyearand$12,000attheendofthesecondyear?Theinitialcostis$33,000.Theappropriateinterestrateis7%forthefirstyearand10%forthesecondyear. A) $281.85 B) $559.90 C) $1,579.44 D) $-2,148.26 E) Noneoftheaboveiswithin$10ofthecorrectanswer.579.WhatisthenetpresentvalAnswer:B Rationale: NPV=$-33,000+($25,000/1.07)+[$12,000/(1.07)(1.10)]=$559.9058Answer:B5810.AuntClarissehaspromisedtoleaveyouanannuitythatwillpay$60nextyearandgrowatanannualrateof4%.Thepaymentsareexpectedtogoonindefinitelyandtheinterestrateis9%.Whatisthevalueofthegrowingperpetuity? A) $667 B) $693 C) $1,200 D) $1,248 E) Noneoftheabove.5910.AuntClarissehaspromisedAnswer:C Rationale: PVgrowingperpetuity=C/(r-g)=$60/(.09-.04)=$1,20060Answer:C Rationale:60CHAPTER4NetPresent

Value61CHAPTER4NetPresent

Value0ChapterOutline4.1TheOne-PeriodCase4.2TheMultiperiodCase4.3CompoundingPeriods4.4Simplifications4.5WhatIsaFirmWorth?4.6SummaryandConclusions62ChapterOutline4.1TheOne-Per4.1TheOne-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).634.1TheOne-PeriodCase:

Futur4.1TheOne-PeriodCase:

FutureValueIntheone-periodcase,theformulaforFVcanbewrittenas:FV=C0×(1+r)TWhereC0iscashflowtoday(timezero)andristheappropriateinterestrate.T(Time)=1.644.1TheOne-PeriodCase:

Futur4.1TheOne-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).654.1TheOne-PeriodCase:Prese4.1TheOne-PeriodCase:

PresentValueIntheone-periodcase,theformulaforPVcanbewrittenas:WhereC1iscashflowatdate1andristheappropriateinterestrate.664.1TheOne-PeriodCase:

Prese4.1TheOne-PeriodCase:

NetPresentValueTheNetPresentValue(NPV)ofaninvestmentisthepresentvalueoftheexpectedcashflows,lessthecostoftheinvestment.Supposeaninvestmentthatpromisestopay$10,000inoneyearisofferedforsalefor$9,500.Yourinterestrateis5%.Shouldyoubuy?Yes!674.1TheOne-PeriodCase:

NetP4.1TheOne-PeriodCase:NetPresentValueIntheone-periodcase,theformulaforNPVcanbewrittenas:NPV=–Cost+PVIfwehadnotundertakenthepositiveNPVprojectconsideredonthelastslide,andinsteadinvestedour$9,500elsewhereat5-percent,ourFVwouldbelessthanthe$10,000theinvestmentpromisedandwewouldbeunambiguouslyworseoffinFVtermsaswell:$9,500×(1.05)=$9,975<$10,000.684.1TheOne-PeriodCase:NetP4.2TheMultiperiodCase:

FutureValueThegeneralformulaforthefuturevalueofaninvestmentovermanyperiodscanbewrittenas:FV=C0×(1+r)TWhere C0iscashflowatdate0,ristheappropriateinterestrate,andTisthenumberofperiodsoverwhichthecashisinvested.(1+r)Tisthefuturevaluefactor694.2TheMultiperiodCase:

Futu4.2TheMultiperiodCase:

FutureValueSupposethatJayRitterinvestedintheinitialpublicofferingoftheModiglianicompany.Modiglianipaysacurrentdividendof$1.10,whichisexpectedtogrowat40-percentperyearforthenextfiveyears.Whatwillthedividendbeinfiveyears?FV=C0×(1+r)T$5.92=$1.10×(1.40)5704.2TheMultiperiodCase:

FutuFutureValueandCompoundingNoticethatthedividendinyearfive,$5.92,isconsiderablyhigherthanthesumoftheoriginaldividendplusfiveincreasesof40-percentontheoriginal$1.10dividend:$5.92>$1.10+5×[$1.10×.40]=$3.30Thisisduetocompounding.Theprocessofleavingthemoneyinthecapitalmarketandlendingitforanotheryeariscalledcompounding.71FutureValueandCompoundingNoFutureValueandCompounding01234572FutureValueandCompounding01PresentValueandDiscounting(p68)Howmuchwouldaninvestorhavetosetasidetodayinordertohave$20,000fiveyearsfromnowifthecurrentrateis15%?012345$20,000PV73PresentValueandDiscountingPresentValueandDiscounting(p68)Theprocessofcalculatingthepresentvalueofafuturecashflowiscalleddiscounting.Itistheoppositeofcompounding.74PresentValueandDiscountingIfwedeposit$5,000todayinanaccountpaying10%,howlongdoesittaketogrowto$10,000?HowLongistheWait?75Ifwedeposit$5,000todayinAssumethetotalcostofacollegeeducationwillbe$50,000whenyourchildenterscollegein12years.Youhave$5,000toinvesttoday.Whatrateofinterestmustyouearnonyourinvestmenttocoverthecostofyourchild’seducation? About21.15%.WhatRateIsEnough?76Assumethetotalcostofacol4.3CompoundingPeriods(p72)CompoundinganinvestmentmtimesayearforTyearsprovidesforfuturevalueofwealth:Forexample,ifyouinvest$50for3yearsat12%compoundedsemi-annually,yourinvestmentwillgrowto774.3CompoundingPeriods(p72)CoEffectiveAnnualInterestRatesAreasonablequestiontoaskintheaboveexampleiswhatistheeffectiveannualrateofinterestonthatinvestment?TheEffectiveAnnualInterestRate(EAIR)istheannualratethatwouldgiveusthesameend-of-investmentwealthafter3years:78EffectiveAnnualInterestRate7918EffectiveAnnualInterestRates(continued)So,investingat12.36%compoundedannuallyisthesameasinvestingat12%compoundedsemiannually.80EffectiveAnnualInterestRateEffectiveAnnualInterestRates(continued)FindtheEffectiveAnnualRate(EAIR)ofan18%APR(Annualpercentagerate)loanthatiscompoundedmonthly.Whatwehaveisaloanwithamonthlyinterestraterateof1?percent.Thisisequivalenttoaloanwithanannualinterestof19.56percent81EffectiveAnnualInterestRate8221ContinuousCompounding(Advancedp74)Thegeneralformulaforthefuturevalueofaninvestmentcompoundedcontinuouslyovermanyperiodscanbewrittenas:FV=C0×erTPV=

C0/erTWhere C0iscashflowatdate0,risthestatedannualinterestrate,Tisthenumberofperiodsoverwhichthecashisinvested,andeisatranscendentalnumberapproximatelyequalto2.718.exisakeyonyourcalculator.Readexampleonp74,7583ContinuousCompounding(Advanc4.4Simplifications(p75)PerpetuityAconstantstreamofcashflowsthatlastsforever.GrowingperpetuityAstreamofcashflowsthatgrowsataconstantrateforever.AnnuityAstreamofconstantcashflowsthatlastsforafixednumberofperiods.GrowingannuityAstreamofcashflowsthatgrowsataconstantrateforafixednumberofperiods.844.4Simplifications(p75)PerpetPerpetuityAconstantstreamofcashflowsthatlastsforever.0…1C2C3CTheformulaforthepresentvalueofaperpetuityis:85PerpetuityAconstantstreamofPerpetuity:ExampleWhatisthevalueofaBritishconsolthatpromisestopay￡15eachyear,everyyearuntilthesunturnsintoaredgiantandburnstheplanettoacrisp?Theinterestrateis10-percent.0…1￡152￡153￡1586Perpetuity:ExampleWhatistheGrowingPerpetuityAgrowingstreamofcashflowsthatlastsforever.0…1C2C×(1+g)3C×(1+g)2Theformulaforthepresentvalueofagrowingperpetuityis:87GrowingPerpetuityAgrowingstGrowingPerpetuity:ExampleTheexpecteddividendnextyearis$1.30anddividendsareexpectedtogrowat5%forever.Ifthediscountrateis10%,whatisthevalueofthispromiseddividendstream?0…1$1.302$1.30×(1.05)3$1.30×(1.05)288GrowingPerpetuity:ExampleTheReadtheexampleonpage78,finddifferencebetweenitandtheexampleonlastslide.Ifthedividendistobereceivedimmediately,PV=C+C(1+g)/(r-g)2C×(1+g)23C×(1+g)3C(1+g)C0189Readtheexampleonpage78,finAnnuity(p79)Aconstantstreamofcashflowswithafixedmaturity.Theformulaforthepresentvalueofanannuityis:01C2C3CTC90Annuity(p79)AconstantstreamIscalledanannuityfactor,itisexpressedasArT,itstandsforthepresentvalueof$1ayearforTyearsataninterestrateofr.91Iscalledanannuityfactor,iAnnuityIntuitionAnannuityisvaluedasthedifferencebetweentwoperpetuities:oneperpetuitythatstartsattime1

lessaperpetuitythatstartsattimeT+101C2C3CTC92AnnuityIntuitionAnannuityiAnnuity:ExampleIfyoucanafforda$400monthlycarpayment,howmuchcarcanyouaffordifinterestratesare7%on36-monthloans?01$4002$4003$40036$40093Annuity:ExampleIfyoucanaff

Trick1:ADelayedAnnuityWhatisthepresentvalueofafour-yearannuityof$100peryearthatmakesitsfirstpaymenttwoyearsfromtodayifthediscountrateis9%?

0 1 2 345$100 $100 $100$100$327.97$297.2294 Trick1:ADelayedAnnuityWhTrick2:AnnuityinAdvance

Anannuitywithanimmediateinitialpaymentiscalledan“annuityinadvance”01$500002$500003$5000020$5000095Trick2:AnnuityinAdvance

AnTrick3:TheInfrequencyAnnuityAnannuitywithpaymentsoccurringlessfrequentlythanonceayearForexampleonp82,012$4504$45020$45096Trick3:TheInfrequencyAnnuiTrick4:Equatingpresentvalueoftwoannuities01deposit2deposit3deposit18tuitionForexampleon83,211797Trick4:EquatingpresentvaluGrowingAnnuityAgrowingstreamofcashflowswithafixedmaturity.Theformulaforthepresentvalueofagrowingannuity:01C2C×(1+g)3C×(1+g)2T

C×(1+g)T-198GrowingAnnuityAgrowingstreaGrowingAnnuityAdefined-benefitretirementplanofferstopay$20,000peryearfor40yearsandincreasetheannualpaymentbythree-percenteachyear.Whatisthepresentvalueatretirementifthediscountrateis10percent?01$20,0002$20,000×(1.03)40$20,000×(1.03)3999GrowingAnnuityAdefined-benefPVofadelayedgrowingannuityYourfirmisabouttomakeitsinitialpublicofferingofstockandyourjobistoestimatethecorrectofferingprice.Forecastdividendsareasfollows.Year:1234Dividendspershare$1.50$1.65$1.825%growththereafterIfinvestorsdemanda10%returnoninvestmentsofthisrisklevel,whatpricewilltheybewillingtopay?100PVofadelayedgrowingannuitPVofadelayedgrowingannuityYear0123Cashflow$1.50$1.65$1.824$1.82×1.05…Thefirststepistodrawatimeline.Thesecondstepistodecideonwhatweknowandwhatitiswearetryingtofind.101PVofadelayedgrowingannuitPVofadelayedgrowingannuityYear0123Cashflow$1.50$1.65$1.82dividend+P3PVofcashflow$32.81=$1.82+$38.22102PVofadelayedgrowingannuit4.5WhatIsaFirmWorth?(p86)Conceptually,afirmshouldbeworththepresentvalueofthefirm’scashflows.Seetheexampleon87Thetrickypartisdeterminingthesize,timingandriskofthosecashflows.1034.5WhatIsaFirmWorth?(p86)4.6SummaryandConclusionsTwobasicconcepts,futurevalueandpresentvalueareintroducedinthischapter.Interestratesarecommonlyexpressedonanannualbasis,butsemi-annual,quarterly,monthlyandevencontinuouslycompoundedinterestratearrangementsexist.Theformulaforthenetpresentvalueofaninvestmentthatpays$CforNperiodsis:1044.6SummaryandConclusionsTwo4.6SummaryandConclusions(continued)Wepresentedfoursimplifyingformulae:1054.6SummaryandConclusions(cMultipleChoice1.Thecompoundvalueisdefinedas A) thevalueofadollarreceivedyesterday. B) thevalueofasumafterinvestingoveroneormoreperiods. C) thevalueofasumtobereceivedtoday. D) therateofgrowthinasumtoday. E) Noneoftheabove.106MultipleChoice1.ThecompounAnswer:B107Answer:B462.Presentvaluemaybedefinedas

A) futurecashflowsdisc