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CHAPTER4DISCOUNTEDCASHFLOWVALUATIONAnswerstoConceptsReviewandCriticalThinkingQuestions1.Assumingpositivecashflowsandinterestrates,thefuturevalueincreasesandthepresentvaluedecreases.2.Assumingpositivecashflowsandinterestrates,thepresentvaluewillfallandthefuturevaluewillrise.3.Thebetterdealistheonewithequalinstallments.4.Yes,theyshould.APRsgenerallydon’tprovidetherelevantrate.Theonlyadvantageisthattheyareeasiertocompute,but,withmoderncomputingequipment,thatadvantageisnotveryimportant.5.Afreshmandoes.Thereasonisthatthefreshmangetstousethemoneyformuchlongerbeforeintereststartstoaccrue.6.It’sareflectionofthetimevalueofmoney.GMACgetstousethe$500immediately.IfGMACusesitwisely,itwillbeworthmorethan$10,000inthirtyyears.7.Oddlyenough,itactuallymakesitmoredesirablesinceGMAConlyhastherighttopaythefull$10,000beforeitisdue.Thisisanexampleofa“call”feature.Suchfeaturesarediscussedatlengthinalaterchapter.8.Thekeyconsiderationswouldbe:(1)Istherateofreturnimplicitintheofferattractiverelativetoother,similarriskinvestments?and(2)Howriskyistheinvestment;i.e.,howcertainarewethatwewillactuallygetthe$10,000?Thus,ouranswerdoesdependonwhoismakingthepromisetorepay.9.TheTreasurysecuritywouldhaveasomewhathigherpricebecausetheTreasuryisthestrongestofallborrowers.10.Thepricewouldbehigherbecause,astimepasses,thepriceofthesecuritywilltendtorisetoward$10,000.Thisriseisjustareflectionofthetimevalueofmoney.Astimepasses,thetimeuntilreceiptofthe$10,000growsshorter,andthepresentvaluerises.In2008,thepricewillprobablybehigherforthesamereason.Wecannotbesure,however,becauseinterestratescouldbemuchhigher,orGMAC’sfinancialpositioncoulddeteriorate.Eithereventwouldtendtodepressthesecurity’sprice.SolutionstoQuestionsandProblemsNOTE:All-end-ofchapterproblemsweresolvedusingaspreadsheet.Manyproblemsrequiremultiplesteps.Duetospaceandreadabilityconstraints,whentheseintermediatestepsareincludedinthissolutionsmanual,roundingmayappeartohaveoccurred.However,thefinalanswerforeachproblemisfoundwithoutroundingduringanystepintheproblem.Basic1.Thesimpleinterestperyearis:Rs.5,000,000×.07=Rs.350,000So,after10years,youwillhave:Rs.350,000×10=Rs.3,500,000ininterest.ThetotalbalancewillbeRs.5,000,000+3,500,000=Rs.8,500,000Withcompoundinterest,weusethefuturevalueformula:FV=PV(1+r)tFV=Rs.5,000,000(1.07)10=Rs.9,835,760Thedifferenceis:Rs.9,835,760–8,500,000=Rs.1,335,7602.TofindtheFVofalumpsum,weuse:FV=PV(1+r)ta.FV=£1,000(1.05)10=£1,628.89b.FV=£1,000(1.07)10=£1,967.15c.FV=£1,000(1.05)20=£2,653.30d.Becauseinterestcompoundsontheinterestalreadyearned,thefuturevalueinpartcismorethantwicethefuturevalueinparta.Withcompoundinterest,futurevaluesgrowexponentially.3.TofindthePVofalumpsum,weuse:PV=FV/(1+r)tPV=¥15,451,000/(1.05)6=¥11,529,774PV=¥51,557,500/(1.11)9=¥20,155,104PV=¥886,073,100/(1.12)23=¥65,381,523PV=¥550,164,200/(1.19)18=¥24,024,0944.Toanswerthisquestion,wecanuseeithertheFVorthePVformula.Bothwillgivethesameanswersincetheyaretheinverseofeachother.WewillusetheFVformula,thatis:FV=PV(1+r)tSolvingforr,weget:r=(FV/PV)1/t–1FV=₩307=₩265(1+r)2;r=(₩307/₩265)1/2–1=7.63%FV=₩896=₩360(1+r)9;r=(₩896/₩360)1/9–1=10.66%FV=₩162,181=₩39,000(1+r)15;r=(₩162,181/₩39,000)1/15–1=9.97%FV=₩483,500=₩46,523(1+r)30;r=(₩483,500/₩46,523)1/30–1=8.12%5.Toanswerthisquestion,wecanuseeithertheFVorthePVformula.Bothwillgivethesameanswersincetheyaretheinverseofeachother.WewillusetheFVformula,thatis:FV=PV(1+r)tSolvingfort,weget:t=ln(FV/PV)/ln(1+r)FV=Z1,284=Z625(1.08)t;t=ln(Z1,284/Z625)/ln1.08=9.36yrsFV=Z4,341=Z810(1.07)t;t=ln(Z4,341/Z810)/ln1.07=24.81yrsFV=Z402,662=Z18,400(1.21)t;t=ln(Z402,662/Z18,400)/ln1.21=16.19yrsFV=Z173,439=Z21,500(1.29)t;t=ln(Z173,439/Z21,500)/ln1.29=8.20yrs6.Tofindthelengthoftimeformoneytodouble,triple,etc.,thepresentvalueandfuturevalueareirrelevantaslongasthefuturevalueistwicethepresentvaluefordoubling,threetimesaslargefortripling,etc.Toanswerthisquestion,wecanuseeithertheFVorthePVformula.Bothwillgivethesameanswersincetheyaretheinverseofeachother.WewillusetheFVformula,thatis:FV=PV(1+r)tSolvingfort,weget:t=ln(FV/PV)/ln(1+r)Thelengthoftimetodoubleyourmoneyis:FV=$2=$1(1.08)tt=ln2/ln1.08=9.01yearsThelengthoftimetoquadrupleyourmoneyis:FV=$4=$1(1.08)tt=ln4/ln1.08=18.01yearsNoticethatthelengthoftimetoquadrupleyourmoneyistwiceaslongasthetimeneededtodoubleyourmoney(thedifferenceintheseanswersisduetorounding).Thisisanimportantconceptoftimevalueofmoney.7.TofindthePVofalumpsum,weuse:PV=FV/(1+r)tPV=฿800,000,000/(1.095)25=฿7,734,690,9658.Toanswerthisquestion,wecanuseeithertheFVorthePVformula.Bothwillgivethesameanswersincetheyaretheinverseofeachother.WewillusetheFVformula,thatis:FV=PV(1+r)tSolvingforr,weget:r=(FV/PV)1/t–1r=($10,311,500/$12,377,500)1/4–1=–4.46%Noticethattheinterestrateisnegative.ThisoccurswhentheFVislessthanthePV.9.Aconsolisaperpetuity.TofindthePVofaperpetuity,weusetheequation:PV=C/rPV=£120/.15PV=£800.0010.Tofindthefuturevaluewithcontinuouscompounding,weusetheequation:FV=PVeRta.FV=€1,000e.12(5)=€1,822.12b.FV=€1,000e.10(3)=€1,349.86c.FV=€1,000e.05(10)=€1,648.72d.FV=€1,000e.07(9)=€1,828.0411.Toso