技术经济学英文版演示文稿C32课件

Example:P=$500A=$140n=10NPV=?

i(％)01020253040∝ NPV900

360

87

0

-67

-162-500

Example:ROR的意义：从收益的观点看，ROR就是项目所能达到的最高收益水平。ROR的意义：ROR的意义：从收益的观点看，ROR就是项目所能达到的最高收益水平。ROR的意义：NPV是折现率ic的函数，ic连续，则NPV可导。一阶导数：NPV函数曲线单调递减。二阶导数：NPV函数曲线凸向原点。技术经济学英文版演示文稿C32课件当ic=0，当ic趋向于无穷大，NPV=NCF0技术经济学英文版演示文稿C32课件3.4RateofReturnAnalysisTherateofreturnanalysisisprobablythemostpopularcriterionineconomicanalysis.It'spopularitystemsfromtheeasewithwhichacommonpersoncanunderstandthemeaningofrateofreturn.Mostoftheinvestmentbrochureswilluserateofreturnonyourinvestmentasacriteriontoshowhowgoodagiveninvestmentopportunityis.Itismucheasiertounderstandthat"aprojectwillprovide20%returnonyourinvestment"than"theprojectwillresultinaNPVof$5,000."Unfortunately,althoughsimpletounderstand,thetechniquehassomemajordrawbacks.Inthissection,inadditiontoexplaininghowtocalculatetherateofreturn(ROR),wewilldiscusstheadvantagesanddisadvantagesofthistechnique.3.4RateofReturnAnalysisRateofreturnhastwodefinitions.Onedefinitioncanbestatedas“theinterestrateearnedontheunpaidbalanceofaloansuchthatthepaymentschedulemakestheunpaidbalanceequaltozerowhenthefinalpaymentismade.”Considerasimpleexampletoillustratethisdefinition.

1001,000$1,00010010012n-1nFigure3.9:ALoanof$1,000withaUniformPaymentofInterestRateofreturnhastwodefinitAssumethatyoutakealoanof$1,000fromabankataninterestrateof10%foraperiodoffouryears.Everyyear,includinglastyear,youpayaninterestof$100tothebank.Attheendoffouryears,youpaytheprincipalamountof$l,000.Therefore,attheendoffouryearstheunpaidbalanceiszero.Therateofreturnforthebankis(l,00/1000=)10%.Schematically,thecashflowisshowninFig.3.9.AssumethatyoutakealoanofThisdefinitioncanbeturnedaroundtostatethatthe"rateofreturnistheinterestrateearnedontheunrecoveredinvestmentsuchthatthepaymentschedulemakestheunrecoveredinvestmentequaltozeroattheendofthelifeoftheinvestment."Usingasimilarexampleasbefore,letusassumethatyouhaveinvested$10,000inthebankataninterestrateof6%forfiveyears.Attheendofeachyear,youwithdraw$600ininterestandattheendoffiveyears,youwithdraw$10,000.Theinvestmentinthebankattheendoffiveyearsis,therefore,zero.Youcanconsiderthattherateofreturnontheinvestmentis(600/10,000=)6%.Schematically,thecashflowprofileisshowninFig.3.10ThisdefinitioncanbeturnedFigure3.10:Investmentof$10,000withaUniformPaymentofInterest12n-1n60010,000$10,000600600Figure3.10:Investmentof$10Mathematically,therateofreturn(ROR)isdefinedastherateatwhichnetpresentworth(NPV)foragiveninvestmentisequaltozero.Inequationform,therateatwhich,(3.4)istherateofreturn.Inotherwords,therateatwhichNPV=03.5)IfweassumethatthecashflowforaparticularprojectisgivenbyAjwhereAjrepresentsthecashflowinyearj,wecanwritetheequationforNPVas,(3.6)IfwedefinetherateiRcorrespondingtotherateatwhichNPViszero,wecanwritetheequationforiRas,

(3.7)Mathematically,therateofreObservingEq.3.7,wenoticethattheequationrepresentsapolynomial(多项式)iniR

whichmayresultinnpossiblesolutionsforiR

whichwillsatisfyEq.3.7.Ineconomicanalysis,weareonlyinterestedinrealsolutions.Althoughnegativerateofreturnisarealvalue,wemaynotbeinterestedinaninvestmentofnegativerateofreturn.Asapracticalmatter,wearesearchingforpositive,realsolutionsofthisequation.Inmostinstances,wewillobtainonlyonepositive,realsolutionwhichrepresentstherateofreturn.Thisisshowninthefollowingexamples.ObservingEq.3.7,wenoticethExample3.17Calculatetherateofreturnforthefollowingcashflow.Year01234CashFlow-4,0002,5001,8001,300900SolutionUsingthecashflows,wecanwritetheequationforNPVas,Sincethisisapolynomialequationini,wewillhavetosolveitbytrialanderror,

Example3.17CalculatetheraSincethevalueofNPVchangesasignbetweeni=15%andi=35%,therateofreturnshouldfallinbetweenthetwovalues.Bylinearinterpolation(线形内插法),wecanwriteanapproximateequationfortherateofreturn(ROR)as,(3.8)wherei+andi-respectivelyrepresentthevialvalueswhichresultedinpositiveandnegativeNPVvalues,andNPV+andNPV_representthepositiveandthenegativeNPVvaluesrespectively.Inourexample,SincethevalueofNPVchangesTherefore,WecancalculatetheNPVat29.3%.NPV=-66.5Althoughclosetozero,wecantryonemoreinterpolationbetween15%and29.3%.NPVat28.3%=-10.2Wewouldassumethisvaluetobecloseenoughtozero.Youmaynotethathigheristhedifferencebetweenthei+andi-,biggerwillbethedeviation(背离)betweenthetrueRORandtheinterpolatedvalue.Therefore,theinterpolationmayhavetobecarriedoutmorethanoncetoobtainacorrectvalueoftheROR.Therefore,Example3.18Byinvesting$10,000inaproject,youarepromisedthatyouwillearn$2,700peryearforaperiodofsixyears.WhatistheRORforthisinvestment?SolutionFori=10%,Fori=20%,

UsingEq.3.8,

For16.3%,NPV=-129Ati=15.8%NPV=1.7≈0Therefore,therateofreturnis15.8%.Example3.18Byinvesting$10Fromtheaboveexamples,onecanseethattheRORcalculationhastobedonebytrialanderror.Manytimes,itisverydifficulttoassumetheinitialvalueofinterestrate.Onewaytoovercomethisproblemistousearatioofperiodicpaymenttoinitialinvestment.Wecanshowthatiftheinitialinvestmentisequaltothesalvagevalue,theRORcanbecalculatedas,Fromtheaboveexamples,onecUsingEq.3.9,ifthesalvagevalueislessthantheinitialinvestment,

Ontheotherhand,ifthesalvagevalueisgreaterthantheinitialinvestment,

Eq.3.9throughEq.3.11areapplicableonlyiftheinvestmentismadeatthebeginningoftheprojectandtheperiodicpaymentsareequaltoeachother.UsingEq.3.9,ifthesalvagevExample3.19Asaninvestment,youboughtahousefor$50,000.Ifyoucanrentthehousefor$800permonth,andcansellthehousefor$70,000attheendoftenyears,whatistheRORonyourinvestment?SolutionLetusassumetheRORtobe.017/month.

where120isthenumberofmonthsinwhichtherentiscollected.Therefore,theRORisl.7%/month.or20.4%/year.Example3.19AsaninvestmentAscanbeseenfromtheaboveexample,byusingthecorrectinitialguess.wedidnothavetousetoomanytrialanderrors.Asimilarequationcanbedevelopedforgeometricseriesasexplainedintheexamplebelow.Example3.20Aproposalcallsforaninvestmentof$25,000inanoilpropertywhichwillresultinaninitialincomeof$6,000peryeardecliningatarateof8%peryearoverthenexttwentyyears.Whatistherateofreturn?Assumethesalvagevaluetobezero.SolutionInthisexamplewehaveageometricseries.Given:A=$6,000,n=20years,g=-0.08Usingtheequationforgeometricseries,Ascanbeseenfromtheabove技术经济学英文版演示文稿C32课件Afteroneadditionaltrialanderror,theROR=15.7%.TheRORcanalsobecalculatedusingagraphicalprocedure.Foratypicalinvestmentscenario,wecallassumedifferentinterestratesandcalculatetheNPVasafunctionoftheinterestrate.AsshowninFig.3.ll,byconnectingthepoints,wecancalculatetheRORcorrespondingtoapointonthecurvewhereNPVisequaltozero.技术经济学英文版演示文稿C32课件Figure3.11:RORDeterminationFigure3.11:RORDeterminatio3.4.lEconomicCriteriaAsstatedbefore,theRORtechniqueisprobablythemostusedtechniqueineconomicanalysis.Itiseasytounderstand.Sinceeveryoneunderstandstheinterestrate,rateofreturnisequatedtoreturnoninvestmentintermsofaninterestratethatwouldbeearned.Intuitively(直观地讲),whencomparingtwoinvestments,onefetchingahigherRORisalwaysmoreattractive.Inacorporatestructure,toevaluatethefeasibilityofaproject,weneedtocomparetheRORtotheminimumrateofreturn(MROR).IftheROR>MROR,theprojectisselected;iftheROR<MROR,theprojectwillberejected.3.4.lEconomicCriteriaExample3.23Thefollowingtwoalternativesareconsideredforaproject,(a)(b)InitialInvestment$50,000$500,000AnnualBenefit$25,000$125,000Life,Years55SalvageValue$50,000$500,000IftheMRORis20%,whichalternativeshouldbeselected?Example3.23ThefollowingtSolutionThefirststepistoestimatetheRORoftheindividualalternativesandcomparetheRORwiththeMROR.IftheRORislessthantheMROR,thealternative(s)shouldberejected.Inthisexample,sincethesalvagevalueisequaltotheinitialinvestment,usingEq.3.9.SincetheRORa>MRORandRORb>MROR,bothalternativessatisfythefeasibilitycriterion.SolutionIntuitively,sinceRORa>RORb,onemaybeinclinedtoselect(a)over(b),butnoticethattheinitialinvestmentforbothalternativesisnotthesame.OneofthedrawbacksoftheRORanalysisisitsinability(无能)toaccountfortheinvestmentamount.Toproperly(完全)accountfortheinvestment,weneedtoconductincrementalanalysis.Thatis,tofindoutbyinvestingadditional(incremental)$450,000inalternative(b),whatincrementalbenefitarereceived?Subtractingvaluesrelatedtoalternative(a)fromalternative(b),weobtain,Intuitively,sinceRORa>RORbForincrementalinvestment,wecancalculatetheRORbyEq.3.9(sincetheinvestment=thesalvagevalue),a-bInitialInvestment$450,000AnnualBenefit$10,000Life,Years5SalvageValue$450,000Forincrementalinvestment,weThisnumberindicatesthattheRORonincrementalinvestmentis22.2%whichisgreaterthantheMROR.Inotherwords,byinvestinganadditional$450,000,wewillearnaRORof22.2%.Ontheotherhand,ifwedonotinvestanadditional$450,000inalternativeb,wewillearnonlyMRORonthatadditionalamount.Therefore,itismoreattractivetoinvesttheadditional$450,000inalternativeb.Thatis,toselectalternativebovera.ThisanalysiscanbeeasilyconfirmedbycalculatingtheNPVforboththealternativesatMROR.ThisnumberindicatesthattheForalternativea,Foralternativeb,ince(NPV)b>(NPV)a,alternativebshouldbechosen.Thisisconsistentwiththeanswerweobtainedfromtheincrementalanalysis.Foralternativea,Togeneralize,iftwoalternativesrequiringdifferentamountsofinvestmentneedtobecompared,weshouldcarryoutanincrementalanalysis.If△ROR≥MROR,weshouldselectanalternativerequiringalargerinvestment.If△

ROR<MROR,weshouldselectanalternativerequiringasmallerinvestment.Theprocedurecanbeeasilyextendedwhenconsideringmorethantwoalternatives.Briefly,thestepwiseprocedureforincrementalanalysiscanbestatedasa.CalculatetheRORforeachalternative.IfROR>MROR,assumethatthealternativeisfeasibleandretainitforfurtherincrementalanalysis.IftheROR<MROR,removethealternativefromfurtheranalysis.Togeneralize,iftwoalternatb.Taketwoalternativesrequiringthesmallestinvestments.CalculatetheRORontheincrementalinvestmentbysubtractingthesmallerinvestmentfromthelargerinvestment.WedenotetheRORonincrementalanalysisasROR.If△

ROR≥MROR,selectthealternativerequiringthelargerinvestment;if△

ROR<MROR,selectthealternativerequiringthesmallerinvestment.Removetherejectedalternativefromfurtheranalysis.c.Taketheremainingalternativeandcompareitwiththealternativerequiringthenextlargestinvestment.CalculatetheincrementalROR.If△

ROR≥MROR,selectthealternativerequiringthelargerinvestment;if△

ROR<MROR,selectthealternativerequiringthesmallerinvestment.Removetherejectedalternativefromfurtheranalysis.d.Repeatstep(c)tillonlyonealternativeremains.

b.TaketwoalternativesrequiExamp1e3.24Thefollowingthreealternativesareconsideredforaproject.IfMRORis15%,selecttheappropriatealternative.(a)(b)(c)InitialInvestment$1,000$3,000$6,000AnnualBenefit$300$1,000$1,800Life,Years101010SalvageValue$1,000$3,000$6,000Examp1e3.24ThefollowingthSolutionSinceRORforallthealternativesisgreaterthantheMROR,allarefeasible.Instep(b)takethetwoalternativesrequiringthesmallestinvestment.Inthisexample,wewillconsideralternatives(a)and(b)forincrementalanalysis,SolutionTherefore,Since△

RORb-a>MROR,select(b)over(a).Eliminatealternative(a)fromfurtheranalysis.Inthenextstep(stepc),compare(b)withtheremainingalternative(c).Forincrementalanalysis,(b)-(a)InitialInvestment$2,000AnnualBenefit$700Life,Years10SalvageValue$2,000Therefore,(b)-(a)InitialInvTherefore,SinceROR.c-b>MROR,select(c)over(b).Aftereliminating(b),weareleftwithonlyalternative(c).Thiswillbeourchoice.(c)-(b)InitialInvestment$3,000AnnualBenefit$800Life,Years10SalvageValue$3,000Therefore,(c)-(b)InitialTosummarize,theeconomiccriterionappliedfortherateofreturnanalysisforasingleproject:iftheROR>MROR,theprojectisselected;iftheROR<MROR,theprojectisrejected.Foraprojecthavingmultiplealternatives,anincrementalanalysisneedstobeconductedsolongasthereisadifferenceinthecashflowprofilesoftwoprojects.Onlyafterapplyingtheincrementalanalysis,thesolutionwillbeconsistentwiththeNPVanalysis.Tosummarize,theeconomiccri3.4.2MultipleRatesofReturnInadditiontotherequirementofincrementalanalysis,theRORanalysismethodalsohasanotherdrawback.Thismethodworkswellwhenagivenalternativerequiresaninitialinvestmentwhichisfollowedbyfuturebenefits.Forthistypeofalternative,thecashflowprofilecanbeshownasnegativecashflowinthefirstyearfollowedbypositivecashflowinthefutureyears.Forexample,ifweconsideraninvestmentof$1,000whichwillresultina$300annualbenefitforthenextsixyearswitha$500salvagevalueattheendofsixyears,thecashprofilecanbewrittenas.Year01234567CashFlow-1,000300300300300300300300+5003.4.2MultipleRatesofRetuInthisprofile,thereisonlyonesignchangeincashprofilebetweenYears0and1.SuchprofileisamenabletoconventionalRORanalysis.NotethattheRORcalculationrequiressolvingapolynomialofi.WecalculatethevalueofiforwhichtheNPViszero.Foreconomicanalysis,weareonlyinterestedinobtainingpositive,realvaluesofiforwhichtheNPVisequaltozero.Whenthereisonlyonesignchangeinthecashflowprofile,asshownabove,wecanonlyobtainoneorzeropositivesolutions.Inthisprofile,thereisonlyInsomeinstances,however,thesignchangesmorethanonceinacashflowprofile.Underthesecircumstances,wemayobtainmorethanonerealROR.Theruleofsignsforpolynomialsolutionstatesthatthenumberofrealsolutionsbetween-land∞isnevergreaterthanthenumberofsignchanges.Thatis,ifwehavetwosignchanges,wemayobtainamaximumoftworatesofreturnvaluesbetween-100%and∞.Thefollowingexampleillustratesthecalculationofthenumberoffeasiblesolutions.Insomeinstances,however,thExample3.25ForthefollowingfourcashRORbetween-100%and∞.PeriodABCD0-100-100300-500120120-200300220-3010030033050100-1004207020010053030-30-1006202010050Example3.25ForthefollowiSolutionTocalculatethemaximumnumberofpossiblerealsolutionsbetween-100%and∞,wecancalculatethenumberofsignchanges.ForcashflowA,thereisonlyonesignchangebetweenperiod0andl.ForB,therearethreesignchanges;betweenperiods0andl,periodsland2,and3.Similarly,forcashflowC,therearefoursignchanges,andforcashflowD,therearefivesignchanges.Asstatedbefore,thenumberofsignchangeswillindicatethemaximumnumberofpossiblerealsolutions.Thatis,forcashflowprofileC,thenumberofrealsolutionsbetween–100%and∞canbeeither4,3,2,l.orzero.SolutionThenumberofpossiblerealsolutionscanbenarroweddownevenfurtherbyapplyingcumulativecashflowsigntest.IfweassumeAjtobeacashflowinperiodj,thenwecandefinethecumulativecashflowCjas,IfCjstartswithanegativenumberandchangessignonlyonce,wewillobtainonlyonepositivesolution.ThiscumulativecashflowmethodmayallowustonarrowdownthenumberofpossiblesolutionsfortheROR.ThenumberofpossiblerealsoExample3.26ReconsiderthecashflowsprovidedinExample3.23.Applyingthecumulativecashflowsigntest,investigatethepossibilityofnarrowingthenumberofpositiveRORsolutions.SolutionWecancalculatethecumulativecashflowsforeachoftheprofilesasfollows:PeriodProjectAProjectBProjectCProjectDAjCjAjCjAjCjAjCj0-100-100-100-100300300-500-500120-8012020-200100300-200220-60-30-10100200300100330-305040100300-1000420-10701102005001001005301030140-30470-100062050201601005705050Example3.26ReconsiderthAsasamplecalculation,forperiod3forProjectA,wecancalculate.C3=-100+20+20+30=-30Forperiod6,C6=-100+20+20+30+20+30+30=50Lookingatthecashflowprofiles,forcashflowprofilesAandD,thereinonlyonesignchangeinthecumulativecashflowprofile.Thatis,wewillobtainonlyoneuniquepositivevalueoftheROR.ForcashflowprofilesBandC,theresultsofcumulativecashflowprofilesareinconclusive.Wecannotreducethepossiblenumberofsolutionsbyusingthecumulativecashflowprofileforthesetwoprofiles.Asasamplecalculation,forpExample3.27Anin-filldrillingprojectisbeingconsideredforanexistingoilfieldtoaccelerateoilrecovery.Thefollowingtwoscenarios,basedontwoalternatives(noin-filldrillingversusin-filldrilling)arepredicted.Whichalternativewouldyouselect?Thenumbersareinmillions.AssumethatMRORis20%.Year0123456A(nodrilling)030201814106B(in-filldrilling)-2060406420Example3.27Anin-filldrillSolutionThefirststepinRORanalysisistocompareindividualROR'sforeachalternativewiththeMROR.ForalternativeA,thereisnosignchangeinthecashflowprofile.Therefore,theRORforalternativeAis∞.ForalternativeB,RORcanbeshowntobegreaterthan20%(theRORforalternativeBis260%).Therefore,bothalternativessatisfytherequirementthattheRORbegreaterthantheMROR.Thenextstepistoconducttheincrementalanalysis.Thecashflowprofileforincrementalvaluescanbewrittenas,SolutionThecashflowprofileshowsmorethanonesignchange.Thecumulativecashflowprofilealsoshowsmorethanonesignchange.ThisindicatesthepossibilityofmorethanonepositiveRORsolution.NPVforanyinterestratecanbecalculatedas,Year0123456B-A-203020-12-10-8-6Cumulative(B-A)-2010301880-6ThecashflowprofileshowsmoFig.3.13showsaplotofNPVasafunctionofi.Figure3.13:PlotofNPVvs.iforExample3.27Asstatedbefore,theRORistherateatwhichtheNPVisequaltozero.BasedonFig.3.13,twoROR’sarepossible;11%and72%.IfweassumeRORtobe11%,thenalternativeA(analternativerequiringasmallerinvestment)shouldbeselected().IfweassumeRORtobe72%,thenalternativeB(analternativerequiringlargerinvestment)shouldbeselected().Obviously,ouranswerchangesdependingupontheselectedvalueofROR.Figure3.13:PlotofNPVvs.iforExample3.27Fig.3.13showsaplotofNPVAcorrectanswertotheproblemrequiresfurtheranalysis.BasedonFig.3.13,theNPVispositivebetween11%and72%;thatis,theadditionalinvestmentrequiredforalternativeBwillresultinpositiveNPV.Therefore,iftheMRORisanywherebetween11%and72%,weshouldselectalternativeB.Ontheotherhand,iftheMRORislessthan11%orgreaterthan72%(anunlikelyprospect),thenalternativeAshouldbeselected.Inthisexample,theMRORis20%;therefore,weshouldselectalternativeB.Togeneralize,iftheMRORfallsintherangeofinterestratewheretheNPVispositive,thealternativerequiringthelargerinvestmentshouldbeselected.IftheMRORfallsintherangewheretheNPVisnegative,thealternativerequiringasmallerinvestmentshouldbeselected.

AcorrectanswertotheprobleOneeasywaytoconfirmthisanalysisistocalculatetheNPVattheMROR(=20%)forincrementalcashflow.SinceNPVispositive,alternativeBshouldbeselected.Thisisthesamepredictedinthepreviousparagraph.Itisobviousthat,forsuchproblems(wheremorethanonesignchangeoccursincashflowanalysis),theRORanalysisisdifficulttoadopt.Abetteralternativewouldbetousethepresentworthanalysis.

OneeasywaytoconfirmthisaAAROR（IRR)的优缺点：

易理解；与基准点无关；

在项目寿命期内任意时刻，使项目收益换算值之和等于费用换算值之和的利率称为ROR(IRR)。

所以，ROR的计算，可以用NPV(i)=0,

NFV(i)=0,NAV(i)=0进行计算。

ROR（IRR)的优缺点：

易理解；与基准点无关；NPVIRR>0>ic=0=ic<0<icNPVIRR>0>ic=0=ic<0<icNPVIRR唯一值唯一值；无解；多解同一ic，具有可加性无可加性与基准点有关无关可用于互斥方案优选不可用NPVIRR唯一值唯一值；无解；多解同一ic，具有可加性无

ir—再投资利率if—融资成本

ir—再投资利率

3.5GrowthRateofReturnAnalysis

ArelatedtechniquetotheRORanalysisisthegrowthrateofreturn.UnlikeRORcalculations,whichareindependentofwhatwedowiththefuturebenefits,growthrateofreturndependsonthereinvestmentofthefuturebenefits.Itassumesthatthefuturebenefitsarereinvestedatcertaininterestratesandcalculatesthefuturevalueofallthebenefitsattheendoftheusefullifeoftheproject.

3.5GrowthRateofRet

Letusillustratethisschematically.InFig.3.14,wehaveacashflowprofileforcalculationoftherateofreturn.Weinvested$l,000inthebankataninterestrateof10%peryear,andwithdrewinterestof$100peryearfor10yearsfinallywithdrawingtheinitialinvestmentof$l,000.Inthissimplescenario,therateofreturnontheinvestmentis10%.TheRORvalueisindependentofwhatwedidwiththe$100wereceivedattheendofeachyear.Wecouldhavegambleditawayorcouldhavereinvesteditinbuyingstocks.TheRORwouldstillbe10%.ThisiswhytheRORissometimescalledinternalrateofreturn.Itonlydependsoninternallygeneratedrevenues,notonexternalrates.

Letusillustratethiss

Contrastthiscashflowprofilewithacasewherethe$100revenueperyearisreinvestedintreasurybillsata6%interestrate.TheschematicdiagramisshowninFig.3.15.Wewillassumethatassoonastheannualpaymentisreceived,itisreinvestedata6%interestrate.Attheendof10years,alltheaccumulatedsumiswithdrawnincludingtheinitialinvestmentof$1000,6%rateiscalledanexternalrateofreturn.1000

Contrastthiscashflow

Inthiscase,wecancalculatethefuturevalueofreinvestedamountbyknowingtherelationshipbetweenthefuturevalueandtheperiodicpayment.Forthisexample,Inadditionto$l,318,wewillalsoreceivetheoriginalinvestmentbackresultinginatotalfuturevalueof$2,318.Ifweknowthatthe$l,000investmenthasresultedinacumulativetheassetof$2,318,wecancalculatetherateatwhichourinvestmenthasgrownbyknowingtherelationshipbetweenthefutureandthepresentvalues.Therefore,i=8.8%

Inthiscase,wecancal

Example3.28

AssumethesamevaluesasgivenintheExample3.17.Assumefurtherthattheannualbenefitearnedisreinvestedatarateof10%.CalculatetheGROR.SolutionGiven:Initialinvestment=$1,000andannualbenefit=$2,700for6years.Wecancalculatethefuturevalueoftheannualbenefitsbyassumingthatthebenefitsareinvestedat10%.UsingEq.2.5,SolvingforGROR,GROR=13%

Example3.28Assumethe

Growthrateofreturnisausefulcalculationiftherateatwhichfuturebenefitsareinvestedisknown.OnepossibilityisthatthereinvestmentratecanbeassumedtobeequaltotheMROR.Anotherpossibilityistochosearatewhichreflectsthemostconservativeinvestment;i.e.,treasurybills.Sinceanycorporationisassumedtohaveperpetualexistence,theGRORisanaccurateindicatoroftherateatwhichthetreasuryofacorporationwillgrow.

Growthrateofreturn

OnemajoradvantageoftheGRORcalculationisthatiteliminatesthetrialanderrorprocedurerequiredfortheconventionalRORanalysis.TheprocedurealsoeliminatesthepossibilityofmultipleROR'sforagivencashflow.Ifthereismorethanonesignchangeinthecashflowprofile,allpositivecashflowsaretransferredattheendoftheusefullifebyassumingthatthepositivecashflowsareinvestedattherateofreinvestment.Thisstepwillresultineliminationofmultiplesignchangesandwillresultinonlyonesignchange.

Onemajoradvantageoft

Inapplyingthegrowthrateofreturn(GROR)asacriterion,weneedtocomparethecalculatedGRORwiththeMROR.IfGROR>MROR,weconsiderthealternativetobefeasible.IfGROR<MROR,wewillrejectthealternative.ItshouldbeunderstoodthattheGRORtechniquedoesnoteliminatetheneedforincrementalanalysis.L