rwj 7th edition solutions - colby college four-year liberal 第七版的解决方案- rwj寇比学院四年的自由

CHAPTER9NETPRESENTVALUEANDOTHERINVESTMENTCRITERIAANSWERSTOCONCEPTSREVIEWANDCRITICALTHINKINGQUESTIONS1APAYBACKPERIODLESSTHANTHEPROJECTSLIFEMEANSTHATTHENPVISPOSITIVEFORAZERODISCOUNTRATE,BUTNOTHINGMOREDEFINITIVECANBESAIDFORDISCOUNTRATESGREATERTHANZERO,THEPAYBACKPERIODWILLSTILLBELESSTHANTHEPROJECTSLIFE,BUTTHENPVMAYBEPOSITIVE,ZERO,ORNEGATIVE,DEPENDINGONWHETHERTHEDISCOUNTRATEISLESSTHAN,EQUALTO,ORGREATERTHANTHEIRRTHEDISCOUNTEDPAYBACKINCLUDESTHEEFFECTOFTHERELEVANTDISCOUNTRATEIFAPROJECTSDISCOUNTEDPAYBACKPERIODISLESSTHANTHEPROJECTSLIFE,ITMUSTBETHECASETHATNPVISPOSITIVE2IFAPROJECTHASAPOSITIVENPVFORACERTAINDISCOUNTRATE,THENITWILLALSOHAVEAPOSITIVENPVFORAZERODISCOUNTRATETHUS,THEPAYBACKPERIODMUSTBELESSTHANTHEPROJECTLIFESINCEDISCOUNTEDPAYBACKISCALCULATEDATTHESAMEDISCOUNTRATEASISNPV,IFNPVISPOSITIVE,THEDISCOUNTEDPAYBACKPERIODMUSTBELESSTHANTHEPROJECTSLIFEIFNPVISPOSITIVE,THENTHEPRESENTVALUEOFFUTURECASHINFLOWSISGREATERTHANTHEINITIALINVESTMENTCOSTTHUSPIMUSTBEGREATERTHAN1IFNPVISPOSITIVEFORACERTAINDISCOUNTRATER,THENITWILLBEZEROFORSOMELARGERDISCOUNTRATERTHUSTHEIRRMUSTBEGREATERTHANTHEREQUIREDRETURN3APAYBACKPERIODISSIMPLYTHEACCOUNTINGBREAKEVENPOINTOFASERIESOFCASHFLOWSTOACTUALLYCOMPUTETHEPAYBACKPERIOD,ITISASSUMEDTHATANYCASHFLOWOCCURRINGDURINGAGIVENPERIODISREALIZEDCONTINUOUSLYTHROUGHOUTTHEPERIOD,ANDNOTATASINGLEPOINTINTIMETHEPAYBACKISTHENTHEPOINTINTIMEFORTHESERIESOFCASHFLOWSWHENTHEINITIALCASHOUTLAYSAREFULLYRECOVEREDGIVENSOMEPREDETERMINEDCUTOFFFORTHEPAYBACKPERIOD,THEDECISIONRULEISTOACCEPTPROJECTSTHATPAYBACKBEFORETHISCUTOFF,ANDREJECTPROJECTSTHATTAKELONGERTOPAYBACKBTHEWORSTPROBLEMASSOCIATEDWITHPAYBACKPERIODISTHATITIGNORESTHETIMEVALUEOFMONEYINADDITION,THESELECTIONOFAHURDLEPOINTFORPAYBACKPERIODISANARBITRARYEXERCISETHATLACKSANYSTEADFASTRULEORMETHODTHEPAYBACKPERIODISBIASEDTOWARDSSHORTTERMPROJECTSITFULLYIGNORESANYCASHFLOWSTHATOCCURAFTERTHECUTOFFPOINTCDESPITEITSSHORTCOMINGS,PAYBACKISOFTENUSEDBECAUSE1THEANALYSISISSTRAIGHTFORWARDANDSIMPLEAND2ACCOUNTINGNUMBERSANDESTIMATESAREREADILYAVAILABLEMATERIALITYCONSIDERATIONSOFTENWARRANTAPAYBACKANALYSISASSUFFICIENTMAINTENANCEPROJECTSAREANOTHEREXAMPLEWHERETHEDETAILEDANALYSISOFOTHERMETHODSISOFTENNOTNEEDEDSINCEPAYBACKISBIASEDTOWARDSLIQUIDITY,ITMAYBEAUSEFULANDAPPROPRIATEANALYSISMETHODFORSHORTTERMPROJECTSWHERECASHMANAGEMENTISMOSTIMPORTANT4ATHEDISCOUNTEDPAYBACKISCALCULATEDTHESAMEASISREGULARPAYBACK,WITHTHEEXCEPTIONTHATEACHCASHFLOWINTHESERIESISFIRSTCONVERTEDTOITSPRESENTVALUETHUSDISCOUNTEDPAYBACKPROVIDESAMEASUREOFFINANCIAL/ECONOMICBREAKEVENBECAUSEOFTHISDISCOUNTING,JUSTASREGULARPAYBACKPROVIDESAMEASUREOFACCOUNTINGBREAKEVENBECAUSEITDOESNOTDISCOUNTTHECASHFLOWSGIVENSOMEPREDETERMINEDCUTOFFFORTHEDISCOUNTEDPAYBACKPERIOD,THEDECISIONRULEISTOACCEPTPROJECTSWHOSEDISCOUNTEDCASHFLOWSPAYBACKBEFORETHISCUTOFFPERIOD,ANDTOREJECTALLOTHERPROJECTSBTHEPRIMARYDISADVANTAGETOUSINGTHEDISCOUNTEDPAYBACKMETHODISTHATITIGNORESALLCASHFLOWSTHATOCCURAFTERTHECUTOFFDATE,THUSBIASINGTHISCRITERIONTOWARDSSHORTTERMPROJECTSASARESULT,THEMETHODMAYREJECTPROJECTSTHATINFACTHAVEPOSITIVENPVS,ORITMAYACCEPTPROJECTSWITHLARGEFUTURECASHOUTLAYSRESULTINGINNEGATIVENPVSINADDITION,THESELECTIONOFACUTOFFPOINTISAGAINANARBITRARYEXERCISECDISCOUNTEDPAYBACKISANIMPROVEMENTONREGULARPAYBACKBECAUSEITTAKESINTOACCOUNTTHETIMEVALUEOFMONEYFORCONVENTIONALCASHFLOWSANDSTRICTLYPOSITIVEDISCOUNTRATES,THEDISCOUNTEDPAYBACKWILLALWAYSBEGREATERTHANTHEREGULARPAYBACKPERIOD5ATHEAVERAGEACCOUNTINGRETURNISINTERPRETEDASANAVERAGEMEASUREOFTHEACCOUNTINGPERFORMANCEOFAPROJECTOVERTIME,COMPUTEDASSOMEAVERAGEPROFITMEASUREATTRIBUTABLETOTHEPROJECTDIVIDEDBYSOMEAVERAGEBALANCESHEETVALUEFORTHEPROJECTTHISTEXTCOMPUTESAARASAVERAGENETINCOMEWITHRESPECTTOAVERAGETOTALBOOKVALUEGIVENSOMEPREDETERMINEDCUTOFFFORAAR,THEDECISIONRULEISTOACCEPTPROJECTSWITHANAARINEXCESSOFTHETARGETMEASURE,ANDREJECTALLOTHERPROJECTSBAARISNOTAMEASUREOFCASHFLOWSANDMARKETVALUE,BUTAMEASUREOFFINANCIALSTATEMENTACCOUNTSTHATOFTENBEARLITTLERESEMBLANCETOTHERELEVANTVALUEOFAPROJECTINADDITION,THESELECTIONOFACUTOFFISARBITRARY,ANDTHETIMEVALUEOFMONEYISIGNOREDFORAFINANCIALMANAGER,BOTHTHERELIANCEONACCOUNTINGNUMBERSRATHERTHANRELEVANTMARKETDATAANDTHEEXCLUSIONOFTIMEVALUEOFMONEYCONSIDERATIONSARETROUBLINGDESPITETHESEPROBLEMS,AARCONTINUESTOBEUSEDINPRACTICEBECAUSE1THEACCOUNTINGINFORMATIONISUSUALLYAVAILABLE,2ANALYSTSOFTENUSEACCOUNTINGRATIOSTOANALYZEFIRMPERFORMANCE,AND3MANAGERIALCOMPENSATIONISOFTENTIEDTOTHEATTAINMENTOFCERTAINTARGETACCOUNTINGRATIOGOALS6ANPVISSIMPLYTHEPRESENTVALUEOFAPROJECTSCASHFLOWSNPVSPECIFICALLYMEASURES,AFTERCONSIDERINGTHETIMEVALUEOFMONEY,THENETINCREASEORDECREASEINFIRMWEALTHDUETOTHEPROJECTTHEDECISIONRULEISTOACCEPTPROJECTSTHATHAVEAPOSITIVENPV,ANDREJECTPROJECTSWITHANEGATIVENPVBNPVISSUPERIORTOTHEOTHERMETHODSOFANALYSISPRESENTEDINTHETEXTBECAUSEITHASNOSERIOUSFLAWSTHEMETHODUNAMBIGUOUSLYRANKSMUTUALLYEXCLUSIVEPROJECTS,ANDCANDIFFERENTIATEBETWEENPROJECTSOFDIFFERENTSCALEANDTIMEHORIZONTHEONLYDRAWBACKTONPVISTHATITRELIESONCASHFLOWANDDISCOUNTRATEVALUESTHATAREOFTENESTIMATESANDNOTCERTAIN,BUTTHISISAPROBLEMSHAREDBYTHEOTHERPERFORMANCECRITERIAASWELLAPROJECTWITHNPV2,500IMPLIESTHATTHETOTALSHAREHOLDERWEALTHOFTHEFIRMWILLINCREASEBY2,500IFTHEPROJECTISACCEPTED7ATHEIRRISTHEDISCOUNTRATETHATCAUSESTHENPVOFASERIESOFCASHFLOWSTOBEIDENTICALLYZEROIRRCANTHUSBEINTERPRETEDASAFINANCIALBREAKEVENRATEOFRETURNATTHEIRRDISCOUNTRATE,THENETVALUEOFTHEPROJECTISZEROTHEIRRDECISIONRULEISTOACCEPTPROJECTSWITHIRRSGREATERTHANTHEDISCOUNTRATE,ANDTOREJECTPROJECTSWITHIRRSLESSTHANTHEDISCOUNTRATEBIRRISTHEINTERESTRATETHATCAUSESNPVFORASERIESOFCASHFLOWSTOBEZERONPVISPREFERREDINALLSITUATIONSTOIRRIRRCANLEADTOAMBIGUOUSRESULTSIFTHEREARENONCONVENTIONALCASHFLOWS,ANDITALSOAMBIGUOUSLYRANKSSOMEMUTUALLYEXCLUSIVEPROJECTSHOWEVER,FORSTANDALONEPROJECTSWITHCONVENTIONALCASHFLOWS,IRRANDNPVAREINTERCHANGEABLETECHNIQUESCIRRISFREQUENTLYUSEDBECAUSEITISEASIERFORMANYFINANCIALMANAGERSANDANALYSTSTORATEPERFORMANCEINRELATIVETERMS,SUCHAS“12”,THANINABSOLUTETERMS,SUCHAS“46,000”IRRMAYBEAPREFERREDMETHODTONPVINSITUATIONSWHEREANAPPROPRIATEDISCOUNTRATEISUNKNOWNAREUNCERTAININTHISSITUATION,IRRWOULDPROVIDEMOREINFORMATIONABOUTTHEPROJECTTHANWOULDNPV8ATHEPROFITABILITYINDEXISTHEPRESENTVALUEOFCASHINFLOWSRELATIVETOTHEPROJECTCOSTASSUCH,ITISABENEFIT/COSTRATIO,PROVIDINGAMEASUREOFTHERELATIVEPROFITABILITYOFAPROJECTTHEPROFITABILITYINDEXDECISIONRULEISTOACCEPTPROJECTSWITHAPIGREATERTHANONE,ANDTOREJECTPROJECTSWITHAPILESSTHANONEBPINPVCOST/COST1NPV/COSTIFAFIRMHASABASKETOFPOSITIVENPVPROJECTSANDISSUBJECTTOCAPITALRATIONING,PIMAYPROVIDEAGOODRANKINGMEASUREOFTHEPROJECTS,INDICATINGTHE“BANGFORTHEBUCK”OFEACHPARTICULARPROJECT9FORAPROJECTWITHFUTURECASHFLOWSTHATAREANANNUITYPAYBACKI/CANDTHEIRRIS0IC/IRRSOLVINGTHEIRREQUATIONFORIRR,WEGETIRRC/INOTICETHISISJUSTTHERECIPROCALOFTHEPAYBACKSOIRR1/PBFORLONGLIVEDPROJECTSWITHRELATIVELYCONSTANTCASHFLOWS,THESOONERTHEPROJECTPAYSBACK,THEGREATERISTHEIRR10THEREAREANUMBEROFREASONSTWOOFTHEMOSTIMPORTANTHAVETODOWITHTRANSPORTATIONCOSTSANDEXCHANGERATESMANUFACTURINGINTHEUSPLACESTHEFINISHEDPRODUCTMUCHCLOSERTOTHEPOINTOFSALE,RESULTINGINSIGNIFICANTSAVINGSINTRANSPORTATIONCOSTSITALSOREDUCESINVENTORIESBECAUSEGOODSSPENDLESSTIMEINTRANSITHIGHERLABORCOSTSTENDTOOFFSETTHESESAVINGSTOSOMEDEGREE,ATLEASTCOMPAREDTOOTHERPOSSIBLEMANUFACTURINGLOCATIONSOFGREATIMPORTANCEISTHEFACTTHATMANUFACTURINGINTHEUSMEANSTHATAMUCHHIGHERPROPORTIONOFTHECOSTSAREPAIDINDOLLARSSINCESALESAREINDOLLARS,THENETEFFECTISTOIMMUNIZEPROFITSTOALARGEEXTENTAGAINSTFLUCTUATIONSINEXCHANGERATESTHISISSUEISDISCUSSEDINGREATERDETAILINTHECHAPTERONINTERNATIONALFINANCE11THESINGLEBIGGESTDIFFICULTY,BYFAR,ISCOMINGUPWITHRELIABLECASHFLOWESTIMATESDETERMININGANAPPROPRIATEDISCOUNTRATEISALSONOTASIMPLETASKTHESEISSUESAREDISCUSSEDINGREATERDEPTHINTHENEXTSEVERALCHAPTERSTHEPAYBACKAPPROACHISPROBABLYTHESIMPLEST,FOLLOWEDBYTHEAAR,BUTEVENTHESEREQUIREREVENUEANDCOSTPROJECTIONSTHEDISCOUNTEDCASHFLOWMEASURESDISCOUNTEDPAYBACK,NPV,IRR,ANDPROFITABILITYINDEXAREREALLYONLYSLIGHTLYMOREDIFFICULTINPRACTICE12YES,THEYARESUCHENTITIESGENERALLYNEEDTOALLOCATEAVAILABLECAPITALEFFICIENTLY,JUSTASFORPROFITSDOHOWEVER,ITISFREQUENTLYTHECASETHATTHE“REVENUES”FROMNOTFORPROFITVENTURESARENOTTANGIBLEFOREXAMPLE,CHARITABLEGIVINGHASREALOPPORTUNITYCOSTS,BUTTHEBENEFITSAREGENERALLYHARDTOMEASURETOTHEEXTENTTHATBENEFITSAREMEASURABLE,THEQUESTIONOFANAPPROPRIATEREQUIREDRETURNREMAINSPAYBACKRULESARECOMMONLYUSEDINSUCHCASESFINALLY,REALISTICCOST/BENEFITANALYSISALONGTHELINESINDICATEDSHOULDDEFINITELYBEUSEDBYTHEUSGOVERNMENTANDWOULDGOALONGWAYTOWARDBALANCINGTHEBUDGETSOLUTIONSTOQUESTIONSANDPROBLEMSNOTEALLENDOFCHAPTERPROBLEMSWERESOLVEDUSINGASPREADSHEETMANYPROBLEMSREQUIREMULTIPLESTEPSDUETOSPACEANDREADABILITYCONSTRAINTS,WHENTHESEINTERMEDIATESTEPSAREINCLUDEDINTHISSOLUTIONSMANUAL,ROUNDINGMAYAPPEARTOHAVEOCCURREDHOWEVER,THEFINALANSWERFOREACHPROBLEMISFOUNDWITHOUTROUNDINGDURINGANYSTEPINTHEPROBLEMBASIC1TOCALCULATETHEPAYBACKPERIOD,WENEEDTOFINDTHETIMETHATTHEPROJECTHASRECOVEREDITSINITIALINVESTMENTAFTERTWOYEARS,THEPROJECTHASCREATED1,2002,5003,700INCASHFLOWSTHEPROJECTSTILLNEEDSTOCREATEANOTHER4,8003,7001,100INCASHFLOWSDURINGTHETHIRDYEAR,THECASHFLOWSFROMTHEPROJECTWILLBE3,400SO,THEPAYBACKPERIODWILLBE2YEARS,PLUSWHATWESTILLNEEDTOMAKEDIVIDEDBYWHATWEWILLMAKEDURINGTHETHIRDYEARTHEPAYBACKPERIODISPAYBACK21,100/3,400232YEARS2TOCALCULATETHEPAYBACKPERIOD,WENEEDTOFINDTHETIMETHATTHEPROJECTHASRECOVEREDITSINITIALINVESTMENTTHECASHFLOWSINTHISPROBLEMAREANANNUITY,SOTHECALCULATIONISSIMPLERIFTHEINITIALCOSTIS3,000,THEPAYBACKPERIODISPAYBACK3480/840357YEARSTHEREISASHORTCUTTOCALCULATETHEFUTURECASHFLOWSAREANANNUITYJUSTDIVIDETHEINITIALCOSTBYTHEANNUALCASHFLOWFORTHE3,000COST,THEPAYBACKPERIODISPAYBACK3,000/840357YEARSFORANINITIALCOSTOF5,000,THEPAYBACKPERIODISPAYBACK5800/840595YEARSTHEPAYBACKPERIODFORANINITIALCOSTOF7,000ISALITTLETRICKIERNOTICETHATTHETOTALCASHINFLOWSAFTEREIGHTYEARSWILLBETOTALCASHINFLOWS88406,720IFTHEINITIALCOSTIS7,000,THEPROJECTNEVERPAYSBACKNOTICETHATIFYOUUSETHESHORTCUTFORANNUITYCASHFLOWS,YOUGETPAYBACK7,000/840833YEARSTHISANSWERDOESNOTMAKESENSESINCETHECASHFLOWSSTOPAFTEREIGHTYEARS,SOAGAIN,WEMUSTCONCLUDETHEPAYBACKPERIODISNEVER3PROJECTAHASCASHFLOWSOFCASHFLOWS30,00018,00048,000DURINGTHISFIRSTTWOYEARSTHECASHFLOWSARESTILLSHORTBY2,000OFRECAPTURINGTHEINITIALINVESTMENT,SOTHEPAYBACKFORPROJECTAISPAYBACK22,000/10,000220YEARSPROJECTBHASCASHFLOWSOFCASHFLOWS9,00025,00035,00069,000DURINGTHISFIRSTTWOYEARSTHECASHFLOWSARESTILLSHORTBY1,000OFRECAPTURINGTHEINITIALINVESTMENT,SOTHEPAYBACKFORPROJECTBISBPAYBACK31,000/425,0003002YEARSUSINGTHEPAYBACKCRITERIONANDACUTOFFOF3YEARS,ACCEPTPROJECTAANDREJECTPROJECTB4WHENWEUSEDISCOUNTEDPAYBACK,WENEEDTOFINDTHEVALUEOFALLCASHFLOWSTODAYTHEVALUETODAYOFTHEPROJECTCASHFLOWSFORTHEFIRSTFOURYEARSISVALUETODAYOFYEAR1CASHFLOW7,000/1146,14035VALUETODAYOFYEAR2CASHFLOW7,500/11425,77101VALUETODAYOFYEAR3CASHFLOW8,000/11435,39977VALUETODAYOFYEAR4CASHFLOW8,500/11445,03268TOFINDTHEDISCOUNTEDPAYBACK,WEUSETHESEVALUESTOFINDTHEPAYBACKPERIODTHEDISCOUNTEDFIRSTYEARCASHFLOWIS6,14035,SOTHEDISCOUNTEDPAYBACKFORAN8,000INITIALCOSTISDISCOUNTEDPAYBACK18,0006,14035/5,77101132YEARSFORANINITIALCOSTOF13,000,THEDISCOUNTEDPAYBACKISDISCOUNTEDPAYBACK213,0006,140355,77101/5,39977220YEARSNOTICETHECALCULATIONOFDISCOUNTEDPAYBACKWEKNOWTHEPAYBACKPERIODISBETWEENTWOANDTHREEYEARS,SOWESUBTRACTTHEDISCOUNTEDVALUESOFTHEYEAR1ANDYEAR2CASHFLOWSFROMTHEINITIALCOSTTHISISTHENUMERATOR,WHICHISTHEDISCOUNTEDAMOUNTWESTILLNEEDTOMAKETORECOVEROURINITIALINVESTMENTWEDIVIDETHISAMOUNTBYTHEDISCOUNTEDAMOUNTWEWILLEARNINYEAR3TOGETTHEFRACTIONALPORTIONOFTHEDISCOUNTEDPAYBACKIFTHEINITIALCOSTIS18,000,THEDISCOUNTEDPAYBACKISDISCOUNTEDPAYBACK318,0006,140355,771015,39977/5,03268314YEARS5R041,600/2,100476YEARSDISCOUNTEDPAYBACKREGULARPAYBACK476YEARSR52,100/1052,100/10522,100/10532,100/10542,100/10559,091902,100/10561,56705DISCOUNTEDPAYBACK510,0009,09190/1,56705558YEARSR152,100/1152,100/11522,100/11532,100/11542,100/11552,100/11567,94741THEPROJECTNEVERPAYSBACK6OURDEFINITIONOFAARISTHEAVERAGENETINCOMEDIVIDEDBYTHEAVERAGEBOOKVALUETHEAVERAGENETINCOMEFORTHISPROJECTISAVERAGENETINCOME1,416,0001,868,0001,562,000985,000/41,457,750ANDTHEAVERAGEBOOKVALUEISAVERAGEBOOKVALUE15M0/275MSO,THEAARFORTHISPROJECTISAARAVERAGENETINCOME/AVERAGEBOOKVALUE1,457,750/7,500,00019447THEIRRISTHEINTERESTRATETHATMAKESTHENPVOFTHEPROJECTEQUALTOZEROSO,THEEQUATIONTHATDEFINESTHEIRRFORTHISPROJECTIS030,00020,000/1IRR14,000/1IRR211,000/1IRR3USINGASPREADSHEET,FINANCIALCALCULATOR,ORTRIALANDERRORTOFINDTHEROOTOFTHEEQUATION,WEFINDTHATIRR2648SINCETHEIRRISGREATERTHANTHEREQUIREDRETURNWEWOULDACCEPTTHEPROJECT8THENPVOFAPROJECTISTHEPVOFTHEOUTFLOWSMINUSBYTHEPVOFTHEINFLOWSTHEEQUATIONFORTHENPVOFTHISPROJECTATAN11PERCENTREQUIREDRETURNISNPV30,00020,000/11114,000/111211,000/11137,42384ATAN11PERCENTREQUIREDRETURN,THENPVISPOSITIVE,SOWEWOULDACCEPTTHEPROJECTTHEEQUATIONFORTHENPVOFTHEPROJECTATA30PERCENTREQUIREDRETURNISNPV30,00020,000/13014,000/130211,000/13031,32453ATA30PERCENTREQUIREDRETURN,THENPVISNEGATIVE,SOWEWOULDREJECTTHEPROJECT9THENPVOFAPROJECTISTHEPVOFTHEOUTFLOWSMINUSBYTHEPVOFTHEINFLOWSSINCETHECASHINFLOWSAREANANNUITY,THEEQUATIONFORTHENPVOFTHISPROJECTATAN8PERCENTREQUIREDRETURNISNPV70,00014,000PVIFA8,917,45643ATAN8PERCENTREQUIREDRETURN,THENPVISPOSITIVE,SOWEWOULDACCEPTTHEPROJECTTHEEQUATIONFORTHENPVOFTHEPROJECTATA16PERCENTREQUIREDRETURNISNPV70,00014,000PVIFA16,95,50839ATA16PERCENTREQUIREDRETURN,THENPVISNEGATIVE,SOWEWOULDREJECTTHEPROJECTWEWOULDBEINDIFFERENTTOTHEPROJECTIFTHEREQUIREDRETURNWASEQUALTOTHEIRROFTHEPROJECT,SINCEATTHATREQUIREDRETURNTHENPVISZEROTHEIRROFTHEPROJECTIS040,00014,000PVIFAIRR,9IRR137010THEIRRISTHEINTERESTRATETHATMAKESTHENPVOFTHEPROJECTEQUALTOZEROSO,THEEQUATIONTHATDEFINESTHEIRRFORTHISPROJECTIS08,0003,200/1IRR4,000/1IRR26,100/1IRR3USINGASPREADSHEET,FINANCIALCALCULATOR,ORTRIALANDERRORTOFINDTHEROOTOFTHEEQUATION,WEFINDTHATIRR268311THENPVOFAPROJECTISTHEPVOFTHEOUTFLOWSMINUSBYTHEPVOFTHEINFLOWSATAZERODISCOUNTRATEANDONLYATAZERODISCOUNTRATE,THECASHFLOWSCANBEADDEDTOGETHERACROSSTIMESO,THENPVOFTHEPROJECTATAZEROPERCENTREQUIREDRETURNISNPV8,0003,2004,0006,1005,300THENPVATA10PERCENTREQUIREDRETURNISNPV8,0003,200/114,000/1126,100/1132,79790THENPVATA20PERCENTREQUIREDRETURNISNPV8,0003,200/124,000/1226,100/12397454ANDTHENPVATA30PERCENTREQUIREDRETURNISNPV8,0003,200/134,000/1326,100/13339508NOTICETHATASTHEREQUIREDRETURNINCREASES,THENPVOFTHEPROJECTDECREASESTHISWILLALWAYSBETRUEFORPROJECTSWITHCONVENTIONALCASHFLOWSCONVENTIONALCASHFLOWSARENEGATIVEATTHEBEGINNINGOFTHEPROJECTANDPOSITIVETHROUGHOUTTHERESTOFTHEPROJECT12ATHEIRRISTHEINTERESTRATETHATMAKESTHENPVOFTHEPROJECTEQUALTOZEROTHEEQUATIONFORTHEIRROFPROJECTAIS034,00016,500/1IRR14,000/1IRR210,000/1IRR36,000/1IRR4USINGASPREADSHEET,FINANCIALCALCULATOR,ORTRIALANDERRORTOFINDTHEROOTOFTHEEQUATION,WEFINDTHATIRR1660THEEQUATIONFORTHEIRROFPROJECTBIS034,0005,000/1IRR10,000/1IRR218,000/1IRR319,000/1IRR4USINGASPREADSHEET,FINANCIALCALCULATOR,ORTRIALANDERRORTOFINDTHEROOTOFTHEEQUATION,WEFINDTHATIRR1572EXAMININGTHEIRRSOFTHEPROJECTS,WESEETHATTHEIRRAISGREATERTHANTHEIRRB,SOIRRDECISIONRULEIMPLIESACCEPTINGPROJECTATHISMAYNOTBEACORRECTDECISIONHOWEVER,BECAUSETHEIRRCRITERIONHASARANKINGPROBLEMFORMUTUALLYEXCLUSIVEPROJECTSTOSEEIFTHEIRRDECISIONRULEISCORRECTORNOT,WENEEDTOEVALUATETHEPROJECTNPVSBTHENPVOFPROJECTAISNPVA34,00016,500/11114,000/111210,000/11136,000/1114NPVA3,49188ANDTHENPVOFPROJECTBISNPVB34,0005,000/11110,000/111218,000/111319,000/1114NPVB4,29806THENPVBISGREATERTHANTHENPVA,SOWESHOULDACCEPTPROJECTBCTOFINDTHECROSSOVERRATE,WESUBTRACTTHECASHFLOWSFROMONEPROJECTFROMTHECASHFLOWSOFTHEOTHERPROJECTHERE,WEWILLSUBTRACTTHECASHFLOWSFORPROJECTBFROMTHECASHFLOWSOFPROJECTAONCEWEFINDTHESEDIFFERENTIALCASHFLOWS,WEFINDTHEIRRTHEEQUATIONFORTHECROSSOVERRATEISCROSSOVERRATE011,500/1R4,000/1R28,000/1R313,000/1R4USINGASPREADSHEET,FINANCIALCALCULATOR,ORTRIALANDERRORTOFINDTHEROOTOFTHEEQUATION,WEFINDTHATR1375ATDISCOUNTRATESABOVE1375CHOOSEPROJECTAFORDISCOUNTRATESBELOW1375CHOOSEPROJECTBINDIFFERENTBETWEENAANDBATADISCOUNTRATEOF137513THEIRRISTHEINTERESTRATETHATMAKESTHENPVOFTHEPROJECTEQUALTOZEROTHEEQUATIONTOCALCULATETHEIRROFPROJECTXIS05,0002,700/1IRR1,700/1IRR22,300/1IRR3USINGASPREADSHEET,FINANCIALCALCULATOR,ORTRIALANDERRORTOFINDTHEROOTOFTHEEQUATION,WEFINDTHATIRR1682FORPROJECTY,THEEQUATIONTOFINDTHEIRRIS05,0002,300/1IRR1,800/1IRR22,700/1IRR3USINGASPREADSHEET,FINANCIALCALCULATOR,ORTRIALANDERRORTOFINDTHEROOTOFTHEEQUATION,WEFINDTHATIRR1660TOFINDTHECROSSOVERRATE,WESUBTRACTTHECASHFLOWSFROMONEPROJECTFROMTHECASHFLOWSOFTHEOTHERPROJECT,ANDFINDTHEIRROFTHEDIFFERENTIALCASHFLOWSWEWILLSUBTRACTTHECASHFLOWSFROMPROJECTYFROMTHECASHFLOWSFROMPROJECTXITISIRRELEVANTWHICHCASHFLOWSWESUBTRACTFROMTHEOTHERSUBTRACTINGTHECASHFLOWS,THEEQUATIONTOCALCULATETHEIRRFORTHESEDIFFERENTIALCASHFLOWSISCROSSOVERRATE0400/1R100/1R2400/1R3USINGASPREADSHEET,FINANCIALCALCULATOR,ORTRIALANDERRORTOFINDTHEROOTOFTHEEQUATION,WEFINDTHATR1328THETABLEBELOWSHOWSTHENPVOFEACHPROJECTFORDIFFERENTREQUIREDRETURNSNOTICETHATPROJECTYALWAYSHASAHIGHERNPVFORDISCOUNTRATESBELOW1328PERCENT,ANDALWAYSHASALOWERNPVFORDISCOUNTRATESABOVE1328PERCENTRNPVXNPVY01,700001,8000051,100211,1554910587536070615145561363520238432708325574406256014ATHEEQUATIONFORTHENPVOFTHEPROJECTISNPV28M53M/118M/11213,570,24793THENPVISGREATERTHAN0,SOWEWOULDACCEPTTHEPROJECTBTHEEQUATIONFORTHEIRROFTHEPROJECTIS028M53M/1IRR8M/1IRR2FROMDESCARTESRULEOFSIGNS,WEKNOWTHEREARETWOIRRSSINCETHECASHFLOWSCHANGESIGNSTWICEFROMTRIALANDERROR,THETWOIRRSAREIRR7275,8346WHENTHEREAREMULTIPLEIRRS,THEIRRDECISIONRULEISAMBIGUOUSBOTHIRRSARECORRECT,THATIS,BOTHINTERESTRATESMAKETHENPVOFTHEPROJECTEQUALTOZEROIFWEAREEVALUATINGWHETHERORNOTTOACCEPTTHISPROJECT,WEWOULDNOTWANTTOUSETHEIRRTOMAKEOURDECISION15THEPROFITABILITYINDEXISDEFINEDASTHEPVOFTHECASHINFLOWSDIVIDEDBYTHEPVOFTHECASHOUTFLOWSTHEEQUATIONFORTHEPROFITABILITYINDEXATAREQUIREDRETURNOF10PERCENTISPI3,200/113,900/1122,600/113/7,0001155THEEQUATIONFORTHEPROFITABILITYINDEXATAREQUIREDRETURNOF15PERCENTISPI3,200/1153,900/11522,600/1153/7,0001063THEEQUATIONFORTHEPROFITABILITYINDEXATAREQUIREDRETURNOF22PERCENTISPI3,200/1223,900/12222,600/1223/7,0000954WEWOULDACCEPTTHEPROJECTIFTHEREQUIREDRETURNWERE10PERCENTOR15PERCENTSINCETHEPIISGREATERTHANONEWEWOULDREJECTTHEPROJECTIFTHEREQUIREDRETURNWERE22PERCENTSINCETHEPIISLESSTHANONE16ATHEPROFITABILITYINDEXISTHEPVOFTHEFUTURECASHFLOWSDIVIDEDBYTHEINITIALINVESTMENTTHECASHFLOWSFORBOTHPROJECTSAREAN