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DiscountedCashFlowValuationChapter4DiscountedCashFlowValuationKeyConceptsandSkillsBeabletocomputethefuturevalueand/orpresentvalueofasinglecashfloworseriesofcashflowsBeabletocomputethereturnonaninvestmentBeabletouseafinancialcalculatorand/orspreadsheettosolvetimevalueproblemsUnderstandperpetuitiesandannuitiesKeyConceptsandSkillsBeableChapterOutline4.1Valuation:TheOne-PeriodCase4.2TheMultiperiodCase4.3CompoundingPeriods4.4Simplifications4.5WhatIsaFirmWorth?ChapterOutline4.1Valuation: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).4.1TheOne-PeriodCaseIfyouFutureValueIntheone-periodcase,theformulaforFVcanbewrittenas:FV=C0×(1+r)TWhereC0iscashflowtoday(timezero),andristheappropriateinterestrate.FutureValueIntheone-periodPresentValueIfyouweretobepromised$10,000dueinoneyearwheninterestratesare5-percent,yourinvestmentwouldbeworth$9,523.81intoday’sdollars.Theamountthataborrowerwouldneedtosetasidetodaytobeabletomeetthepromisedpaymentof$10,000inoneyeariscalledthePresentValue(PV).Notethat$10,000=$9,523.81×(1.05).PresentValueIfyouweretobePresentValueIntheone-periodcase,theformulaforPVcanbewrittenas:WhereC1iscashflowatdate1,andristheappropriateinterestrate.PresentValueIntheone-periodNetPresentValueTheNetPresentValue(NPV)ofaninvestmentisthepresentvalueoftheexpectedcashflows,lessthecostoftheinvestment.Supposeaninvestmentthatpromisestopay$10,000inoneyearisofferedforsalefor$9,500.Yourinterestrateis5%.Shouldyoubuy?NetPresentValueTheNetPreseNetPresentValueThepresentvalueofthecashinflowisgreaterthanthecost.Inotherwords,theNetPresentValueispositive,sotheinvestmentshouldbepurchased.NetPresentValueThepresentvNetPresentValueIntheone-periodcase,theformulaforNPVcanbewrittenas:NPV=–Cost+PVIfwehadnotundertakenthepositiveNPVprojectconsideredonthelastslide,andinsteadinvestedour$9,500elsewhereat5percent,ourFVwouldbelessthanthe$10,000theinvestmentpromised,andwewouldbeworseoffinFVterms:$9,500×(1.05)=$9,975

<$10,000NetPresentValueIntheone-pe4.2TheMultiperiodCaseThegeneralformulaforthefuturevalueofaninvestmentovermanyperiodscanbewrittenas:FV=C0×(1+r)TWhere C0iscashflowatdate0,ristheappropriateinterestrate,andTisthenumberofperiodsoverwhichthecashisinvested.4.2TheMultiperiodCaseThegeFutureValueSupposeastockcurrentlypaysadividendof$1.10,whichisexpectedtogrowat40%peryearforthenextfiveyears.Whatwillthedividendbeinfiveyears?FV=C0×(1+r)T$5.92=$1.10×(1.40)5FutureValueSupposeastockcuFutureValueandCompoundingNoticethatthedividendinyearfive,$5.92,isconsiderablyhigherthanthesumoftheoriginaldividendplusfiveincreasesof40-percentontheoriginal$1.10dividend:$5.92>$1.10+5×[$1.10×.40]=$3.30Thisisduetocompounding.FutureValueandCompoundingNoFutureValueandCompounding012345FutureValueandCompounding01PresentValueandDiscountingHowmuchwouldaninvestorhavetosetasidetodayinordertohave$20,000fiveyearsfromnowifthecurrentrateis15%?012345$20,000PVPresentValueandDiscountingHHowLongistheWait?Ifwedeposit$5,000todayinanaccountpaying10%,howlongdoesittaketogrowto$10,000?HowLongistheWait?IfwedepAssumethetotalcostofacollegeeducationwillbe$50,000whenyourchildenterscollegein12years.Youhave$5,000toinvesttoday.Whatrateofinterestmustyouearnonyourinvestmenttocoverthecostofyourchild’seducation? WhatRateIsEnough?About21.15%.AssumethetotalcostofacolCalculatorKeysTexasInstrumentsBA-IIPlusFV=futurevaluePV=presentvalueI/Y=periodicinterestrateP/Ymustequal1fortheI/YtobetheperiodicrateInterestisenteredasapercent,notadecimalN=numberofperiodsRemembertocleartheregisters(CLRTVM)aftereachproblemOthercalculatorsaresimilarinformatCalculatorKeysTexasInstrumenMultipleCashFlows Consideraninvestmentthatpays$200oneyearfromnow,withcashflowsincreasingby$200peryearthroughyear4.Iftheinterestrateis12%,whatisthepresentvalueofthisstreamofcashflows?Iftheissueroffersthisinvestmentfor$1,500,shouldyoupurchaseit?MultipleCashFlows ConsideraMultipleCashFlows01234200400600800178.57318.88427.07508.411,432.93PresentValue<Cost→DoNotPurchaseMultipleCashFlows01234200400Valuing“Lumpy”CashFlowsFirst,setyourcalculatorto1paymentperyear.Then,usethecashflowmenu:CF2CF1F2F1CF0120011,432.930400INPV12CF4CF3F4F316001800Valuing“Lumpy”CashFlowsFirs4.3CompoundingPeriodsCompoundinganinvestmentmtimesayearforTyearsprovidesforfuturevalueofwealth:4.3CompoundingPeriodsCompounCompoundingPeriodsForexample,ifyouinvest$50for3yearsat12%compoundedsemi-annually,yourinvestmentwillgrowtoCompoundingPeriodsForexampleEffectiveAnnualRatesofInterestAreasonablequestiontoaskintheaboveexampleis“whatistheeffectiveannualrateofinterestonthatinvestment?”TheEffectiveAnnualRate(EAR)ofinterestistheannualratethatwouldgiveusthesameend-of-investmentwealthafter3years:EffectiveAnnualRatesofInteEffectiveAnnualRatesofInterestSo,investingat12.36%compoundedannuallyisthesameasinvestingat12%compoundedsemi-annually.EffectiveAnnualRatesofInteEffectiveAnnualRatesofInterestFindtheEffectiveAnnualRate(EAR)ofan18%APRloanthatiscompoundedmonthly.Whatwehaveisaloanwithamonthlyinterestraterateof1?%.Thisisequivalenttoaloanwithanannualinterestrateof19.56%.EffectiveAnnualRatesofInteEARonaFinancialCalculatorkeys:description:[2nd][ICONV]Opensinterestrateconversionmenu[↓][EFF=][CPT]19.56TexasInstrumentsBAIIPlus

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[ENTER]Sets18APR.[↑][C/Y=]12[ENTER]Sets12paymentsperyearEARonaFinancialCalculatorkContinuousCompoundingThegeneralformulaforthefuturevalueofaninvestmentcompoundedcontinuouslyovermanyperiodscanbewrittenas:FV=C0×erTWhere C0iscashflowatdate0,risthestatedannualinterestrate,Tisthenumberofyears,andeisatranscendentalnumberapproximatelyequalto2.718.exisakeyonyourcalculator.ContinuousCompoundingThegene4.4SimplificationsPerpetuityAconstantstreamofcashflowsthatlastsforeverGrowingperpetuityAstreamofcashflowsthatgrowsataconstantrateforeverAnnuityAstreamofconstantcashflowsthatlastsforafixednumberofperiodsGrowingannuityAstreamofcashflowsthatgrowsataconstantrateforafixednumberofperiods4.4SimplificationsPerpetuityPerpetuityAconstantstreamofcashflowsthatlastsforever0…1C2C3CPerpetuityAconstantstreamofPerpetuity:ExampleWhatisthevalueofaBritishconsolthatpromisestopay￡15everyyearforever?Theinterestrateis10-percent.0…1￡152￡153￡15Perpetuity:ExampleWhatistheGrowingPerpetuityAgrowingstreamofcashflowsthatlastsforever0…1C2C×(1+g)3C×(1+g)2GrowingPerpetuityAgrowingstGrowingPerpetuity:ExampleTheexpecteddividendnextyearis$1.30,anddividendsareexpectedtogrowat5%forever.Ifthediscountrateis10%,whatisthevalueofthispromiseddividendstream?0…1$1.302$1.30×(1.05)3$1.30×(1.05)2GrowingPerpetuity:ExampleTheAnnuityAconstantstreamofcashflowswithafixedmaturity01C2C3CTCAnnuityAconstantstreamofcaAnnuity:ExampleIfyoucanafforda$400monthlycarpayment,howmuchcarcanyouaffordifinterestratesare7%on36-monthloans?01$4002$4003$40036$400Annuity:ExampleIfyoucanaff Whatisthepresentvalueofafour-yearannuityof$100peryearthatmakesitsfirstpaymenttwoyearsfromtodayifthediscountrateis9%?

0 1 2 345$100 $100 $100$100$323.97$297.22 WhatisthepresentvalueofGrowingAnnuityAgrowingstreamofcashflowswithafixedmaturity01C2C×(1+g)3C×(1+g)2TC×(1+g)T-1GrowingAnnuityAgrowingstreaGrowingAnnuity:ExampleAdefined-benefitretirementplanofferstopay$20,000peryearfor40yearsandincreasetheannualpaymentbythree-percenteachyear.Whatisthepresentvalueatretirementifthediscountrateis10percent?01$20,0002$20,000×(1.03)40$20,000×(1.03)39GrowingAnnuity:ExampleAdefiGrowingAnnuity:ExampleYouareevaluatinganincomegeneratingproperty.Netrentisreceivedattheendofeachyear.Thefirstyear'srentisexpectedtobe$8,500,andrentisexpectedtoincrease7%eachyear.Whatisthepresentvalueoftheestimatedincomestreamoverthefirst5yearsifthediscountrateis12%?0 1 2 345$34,706.26GrowingAnnuity:ExampleYouar4.5WhatIsaFirmWorth?Conceptually,afirmshouldbeworththepresentvalueofthefirm’scashflows.Thetrickypartisdeterminingthesize,timingandriskofthosecashflows.4.5WhatIsaFirmWorth?ConceQuickQuizHowisthefuturevalueofasinglecashflowcomputed?Howisthepresentvalueofaseriesofcashflowscomputed.WhatistheNetPresentValueofaninvestment?WhatisanEAR,andhowisitcomputed?Whatisaperpetuity?Anannuity?QuickQuizHowisthefuturevaDiscountedCashFlowValuationChapter4DiscountedCashFlowValuationKeyConceptsandSkillsBeabletocomputethefuturevalueand/orpresentvalueofasinglecashfloworseriesofcashflowsBeabletocomputethereturnonaninvestmentBeabletouseafinancialcalculatorand/orspreadsheettosolvetimevalueproblemsUnderstandperpetuitiesandannuitiesKeyConceptsandSkillsBeableChapterOutline4.1Valuation:TheOne-PeriodCase4.2TheMultiperiodCase4.3CompoundingPeriods4.4Simplifications4.5WhatIsaFirmWorth?ChapterOutline4.1Valuation: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).4.1TheOne-PeriodCaseIfyouFutureValueIntheone-periodcase,theformulaforFVcanbewrittenas:FV=C0×(1+r)TWhereC0iscashflowtoday(timezero),andristheappropriateinterestrate.FutureValueIntheone-periodPresentValueIfyouweretobepromised$10,000dueinoneyearwheninterestratesare5-percent,yourinvestmentwouldbeworth$9,523.81intoday’sdollars.Theamountthataborrowerwouldneedtosetasidetodaytobeabletomeetthepromisedpaymentof$10,000inoneyeariscalledthePresentValue(PV).Notethat$10,000=$9,523.81×(1.05).PresentValueIfyouweretobePresentValueIntheone-periodcase,theformulaforPVcanbewrittenas:WhereC1iscashflowatdate1,andristheappropriateinterestrate.PresentValueIntheone-periodNetPresentValueTheNetPresentValue(NPV)ofaninvestmentisthepresentvalueoftheexpectedcashflows,lessthecostoftheinvestment.Supposeaninvestmentthatpromisestopay$10,000inoneyearisofferedforsalefor$9,500.Yourinterestrateis5%.Shouldyoubuy?NetPresentValueTheNetPreseNetPresentValueThepresentvalueofthecashinflowisgreaterthanthecost.Inotherwords,theNetPresentValueispositive,sotheinvestmentshouldbepurchased.NetPresentValueThepresentvNetPresentValueIntheone-periodcase,theformulaforNPVcanbewrittenas:NPV=–Cost+PVIfwehadnotundertakenthepositiveNPVprojectconsideredonthelastslide,andinsteadinvestedour$9,500elsewhereat5percent,ourFVwouldbelessthanthe$10,000theinvestmentpromised,andwewouldbeworseoffinFVterms:$9,500×(1.05)=$9,975

<$10,000NetPresentValueIntheone-pe4.2TheMultiperiodCaseThegeneralformulaforthefuturevalueofaninvestmentovermanyperiodscanbewrittenas:FV=C0×(1+r)TWhere C0iscashflowatdate0,ristheappropriateinterestrate,andTisthenumberofperiodsoverwhichthecashisinvested.4.2TheMultiperiodCaseThegeFutureValueSupposeastockcurrentlypaysadividendof$1.10,whichisexpectedtogrowat40%peryearforthenextfiveyears.Whatwillthedividendbeinfiveyears?FV=C0×(1+r)T$5.92=$1.10×(1.40)5FutureValueSupposeastockcuFutureValueandCompoundingNoticethatthedividendinyearfive,$5.92,isconsiderablyhigherthanthesumoftheoriginaldividendplusfiveincreasesof40-percentontheoriginal$1.10dividend:$5.92>$1.10+5×[$1.10×.40]=$3.30Thisisduetocompounding.FutureValueandCompoundingNoFutureValueandCompounding012345FutureValueandCompounding01PresentValueandDiscountingHowmuchwouldaninvestorhavetosetasidetodayinordertohave$20,000fiveyearsfromnowifthecurrentrateis15%?012345$20,000PVPresentValueandDiscountingHHowLongistheWait?Ifwedeposit$5,000todayinanaccountpaying10%,howlongdoesittaketogrowto$10,000?HowLongistheWait?IfwedepAssumethetotalcostofacollegeeducationwillbe$50,000whenyourchildenterscollegein12years.Youhave$5,000toinvesttoday.Whatrateofinterestmustyouearnonyourinvestmenttocoverthecostofyourchild’seducation? WhatRateIsEnough?About21.15%.AssumethetotalcostofacolCalculatorKeysTexasInstrumentsBA-IIPlusFV=futurevaluePV=presentvalueI/Y=periodicinterestrateP/Ymustequal1fortheI/YtobetheperiodicrateInterestisenteredasapercent,notadecimalN=numberofperiodsRemembertocleartheregisters(CLRTVM)aftereachproblemOthercalculatorsaresimilarinformatCalculatorKeysTexasInstrumenMultipleCashFlows Consideraninvestmentthatpays$200oneyearfromnow,withcashflowsincreasingby$200peryearthroughyear4.Iftheinterestrateis12%,whatisthepresentvalueofthisstreamofcashflows?Iftheissueroffersthisinvestmentfor$1,500,shouldyoupurchaseit?MultipleCashFlows ConsideraMultipleCashFlows01234200400600800178.57318.88427.07508.411,432.93PresentValue<Cost→DoNotPurchaseMultipleCashFlows01234200400Valuing“Lumpy”CashFlowsFirst,setyourcalculatorto1paymentperyear.Then,usethecashflowmenu:CF2CF1F2F1CF0120011,432.930400INPV12CF4CF3F4F316001800Valuing“Lumpy”CashFlowsFirs4.3CompoundingPeriodsCompoundinganinvestmentmtimesayearforTyearsprovidesforfuturevalueofwealth:4.3CompoundingPeriodsCompounCompoundingPeriodsForexample,ifyouinvest$50for3yearsat12%compoundedsemi-annually,yourinvestmentwillgrowtoCompoundingPeriodsForexampleEffectiveAnnualRatesofInterestAreasonablequestiontoaskintheaboveexampleis“whatistheeffectiveannualrateofinterestonthatinvestment?”TheEffectiveAnnualRate(EAR)ofinterestistheannualratethatwouldgiveusthesameend-of-investmentwealthafter3years:EffectiveAnnualRatesofInteEffectiveAnnualRatesofInterestSo,investingat12.36%compoundedannuallyisthesameasinvestingat12%compoundedsemi-annually.EffectiveAnnualRatesofInteEffectiveAnnualRatesofInterestFindtheEffectiveAnnualRate(EAR)ofan18%APRloanthatiscompoundedmonthly.Whatwehaveisaloanwithamonthlyinterestraterateof1?%.Thisisequivalenttoaloanwithanannualinterestrateof19.56%.EffectiveAnnualRatesofInteEARonaFinancialCalculatorkeys:description:[2nd][ICONV]Opensinterestrateconversionmenu[↓][EFF=][CPT]19.56TexasInstrumentsBAIIPlus

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[ENTER]Sets18APR.[↑][C/Y=]12[ENTER]Sets12paymentsperyearEARonaFinancialCalculatorkContinuousCompoundingThegeneralformulaforthefuturevalueofaninvestmentcompoundedcontinuouslyovermanyperiodscanbewrittenas:FV=C0×erTWhere C0iscashflowatdate0,risthestatedannualinterestrate,Tisthenumberofyears,andeisatranscendentalnumberapproximatelyequalto2.718.exisakeyonyourcalculator.ContinuousCompoundingThegene4.4SimplificationsPerpetuityAconstantstreamofcashflowsthatlastsforeverGrowingperpetuityAstreamofcashflowsthatgrowsataconstantrateforeverAnnuityAstreamofconstantcashflowsthatlastsforafixednumberofperiodsGrowingannuityAstreamofcashflowsthatgrowsataconstantrateforafixednumberofperiods4.4SimplificationsPerpetuityPerpetuityAconstantstreamofcashflowsthatlastsforever0…1C2C3CPerpetuityAconstantstreamofPerpetuity:ExampleWhatisthevalueofaBritishconsolthatpromisestopay￡15everyyearforever?Theinterestrateis10-percent.0…1￡152￡153￡15Perpetuity:ExampleWhatistheGrowingPerpetuityAgrowingstreamofcashflowsthatlastsforever0…1C2C×(1+g)3C×(1+g)2GrowingPerpetuityAgrowingstGrowingPerpetuity:ExampleTheexpecteddividendnextyearis$1.30,anddividendsareexpectedtogrowat5%forever.Ifthediscountrateis10%,whatisthevalueofthispromiseddividendstream?0…1$1.302$1.30×(1.05)3$1.30×(1.05)2GrowingPer