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Chapter6:SomeAlternativeInvestmentRules6.1a.PaybackperiodofProjectA=1+($7,500-$4,000)/$3,500=2yearsPaybackperiodofProjectB=2+($5,000-$2,500-$1,200)/$3,000=2.43yearsProjectAshouldbechosen.b.NPVA=-$7,500+$4,000/1.15+$3,500/1.152+$1,500/1.153=-$388.96NPVB=-$5,000+$2,500/1.15+$1,200/1.152+$3,000/1.153=$53.83ProjectBshouldbechosen.6.2a.Paybackperiod=6+{$1,000,000-($150,000(6)}/$150,000=6.67yearsYes,theprojectshouldbeadopted.b.$150,000=$974,259Thediscountedpaybackperiod=11+($1,000,000-$974,259)/($150,000/1.112)=11.54yearsc.NPV=-$1,000,000+$150,000/0.10=$500,0006.3a.AverageInvestment:($16,000+$12,000+$8,000+$4,000+0)/5=$8,000Averageaccountingreturn:$4,500/$8,000=0.5625=56.25%b.1.AARdoesnotconsiderthetimingofthecashflows,henceitdoesnotconsiderthetimevalueofmoney.2.AARusesanarbitraryfirmstandardasthedecisionrule.3.AARusesaccountingdataratherthannetcashflows.6.4AverageInvestment=($2,000,000+0)/2=$1,000,000Averagenetincome=[$100,000{(1+g)5-1}/g]/5={$100,000A(1.075-1}/0.07}/5=$115,014.78AAR=$115,014.78/$1,000,000=11.50%No,sincethemachine’sAARislessthanthefirm’scutoffAAR.6.5aPI=$40,000/$160,000=1.04SincethePIexceedsoneaccepttheproject.6.7TheIRRisthediscountrateatwhichtheNPV=0.-$3,000+$2,500/(1+IRRA)+$1,000/(1+IRRA)2=0Bytrialanderror,IRRA=12.87%SinceprojectB’scashflowsaretwotimesofthoseofprojectA,theIRRB=IRRA=12.87%6.8a.Solvexbytrialanderror:-$4,000+$2,000/(1+x)+$1,500/(1+x)2+$1,000/(1+x)3=0x=6.93%b.No,sincetheIRR(6.93%)islessthanthediscountrateof8%.6.9FindtheIRRsofprojectAanalytically.SincetheIRRisthediscountratethatmakestheNPVequaltozero,thefollowingequationmusthold.-$200+$200/(1+r)+$800/(1+r)2-$800/(1+r)3=0$200[-1+1/(1+r)]-{$800/(1+r)2}[-1+1/(1+r)]=0[-1+1/(1+r)][$200-$800/(1+r)2]=0Forthisequationtohold,either[-1+1/(1+r)]=0or[$200-$800/(1+r)2]=0.Solveeachofthesefactorsfortherthatwouldcausethefactortoequalzero.TheresultingratesarethetwoIRRsforprojectA.Theyareeitherr=0%orr=100%.Note:ByinspectionyoushouldhaveknownthatoneoftheIRRsofprojectAiszero.Noticethatthesumoftheun-discountedcashflowsforprojectAiszero.Thus,notdiscountingthecashflowswouldyieldazeroNPV.Thediscountratewhichistantamounttonotdiscountingiszero.HerearesomeoftheinteractionsusedtofindtheIRRbytrialanderror.Sophisticatedcalculatorscancomputethisratewithoutallofthetediuminvolvedinthetrial-and-errormethod.NPV=-$150+$50/1.3+$100/1.32+$150/1.33=$15.91NPV=-$150+$50/1.4+$100/1.42+$150/1.43=-$8.60NPV=-$150+$50/1.37+$100/1.372+$150/1.373=-$1.89NPV=-$150+$50/1.36+$100/1.362+$150/1.363=$0.46NPV=-$150+$50/1.36194+$100/1.361942+$150/1.361943=$0.0010NPV=-$150+$50/1.36195+$100/1.361952+$150/1.361953=-$0.0013NPV=-$150+$50/1.361944+$100/1.3619442+$150/1.3619443=$0.0000906Thus,theIRRisapproximately36.1944%.6.10a.Solverintheequation:$5,000-$2,500/(1+r)-$2,000/(1+r)2-$1,000/(1+r)3-$1,000/(1+r)4=0Bytrialanderror,IRR=r=13.99%b.Sincethisproblemisthecaseoffinancing,accepttheprojectiftheIRRislessthantherequiredrateofreturn.IRR=13.99%10%Rejecttheoffer.c.IRR=13.99%20%Accepttheoffer.d.Whenr=10%:NPV=$5,000-$2,500/1.1-$2,000/1.12-$1,000/1.13-$1,000/1.14=-$359.95Whenr=20%:NPV=$5,000-$2,500/1.2-$2,000/1.22-$1,000/1.23-$1,000/1.24=$466.82Yes,theyareconsistentwiththechoicesoftheIRRrulesincethesignsofthecashflowschangeonlyonce.6.11a.ProjectA:NPV=-$5,000+$3,500/(1+r)+$3,500/(1+r)2=0IRR=r=25.69%ProjectB:NPV=-$100,000+$65,000/(1+r)+$65,000/(1+r)2=0IRR=r=19.43%b.ChooseprojectAbecauseithasahigherIRR.c.Thedifferenceinscaleisignored.d.ApplytheincrementalIRRmethod.e.C0C1C2B-A-$95,000$61,500$61,500NPV=-$95,000+$61,500/(1+r)+$61,500/(1+r)2=0IncrementalIRR=r=19.09%f.Ifthediscountrateislessthan19.09%,chooseprojectB.Otherwise,chooseprojectA.g.NPVA=-$5,000+$3,500/1.15+$3,500/1.152=$689.98NPVB=-$100,000+$65,000/1.15+$65,000/1.152=$5,671.08ChooseprojectB.6.12a.PVA={$5,000/(0.12-0.04)}/1.122=$49,824.61PVB=(-$6,000/0.12)/1.12=-$44,642.86b.TheIRRforprojectCmustsolve{$5,000/(x-0.04)}/(1+x)2+(-$6,000/x)/(1+x)=0$5,000/(x-0.04)-$6,000(1+x)/x=025x2+3.17x-1=0x={-3.17-(110.0489)0.5}/50or{-3.17+(110.0489)0.5}/50TherelevantpositiverootisIRR=x=0.1464=14.64%c.Toarriveattheappropriatedecisionrule,wemustgraphtheNPVasafunctionofthediscountrate.Atadiscountrateof14.64%theNPViszero.Todetermineifthegraphisupwardordownwardsloping,checktheNPVatanotherdiscountrate.Atadiscountrateof10%theNPVis$14,325.07[=$68,870.52-$54,545.54].Thus,thegraphoftheNPVisdownwardsloping.Fromthediscussioninthetext,ifanNPVgraphisdownwardsloping,theprojectisaninvestingproject.Thecorrectdecisionruleforaninvestingprojectistoaccepttheprojectifthediscountrateisbelow14.64%.6.13Generally,thestatementisfalse.IfthecashflowsofprojectBoccurearlyandthecashflowsofprojectAoccurlate,thenforalowdiscountratetheNPVofAcanexceedtheNPVofB.Examplesareeasytoconstruct.C0C1C2IRRNPV@0%A:-$1,000,000$0$1,440,0000.20$440,000B:-2,000,0002,400,00000.20400,000Inoneparticularcase,thestatementistrueforequallyriskyprojects.IfthelivesofthetwoprojectsareequalandineverytimeperiodthecashflowsoftheprojectBaretwicethecashflowsofprojectA,thentheNPVofprojectBwillbetwiceasgreatastheNPVofprojectAforan