THE TIME VALUE OF MONEY.docx
- 文档编号:9259670
- 上传时间:2023-05-17
- 格式:DOCX
- 页数:35
- 大小:132.62KB
THE TIME VALUE OF MONEY.docx
《THE TIME VALUE OF MONEY.docx》由会员分享,可在线阅读,更多相关《THE TIME VALUE OF MONEY.docx(35页珍藏版)》请在冰点文库上搜索。
THETIMEVALUEOFMONEY
CHAPTER3
THETIMEVALUEOFMONEY
OVERVIEW
Adollarinthehandtodayisworthmorethanadollartobereceivedinthefuturebecause,ifyouhaditnow,youcouldinvestthatdollarandearninterest.Ofallthetechniquesusedinfinance,noneismoreimportantthantheconceptofthetimevalueofmoney,ordiscountedcashflow(DCF)analysis.Theprinciplesoftimevalueanalysisthataredevelopedinthischapterhavemanyapplications,rangingfromsettingupschedulesforpayingoffloanstodecisionsaboutwhethertoacquirenewequipment.
Futurevalueandpresentvaluetechniquescanbeappliedtoasinglecashflow(lumpsum),ordinaryannuities,annuitiesdue,andunevencashflowstreams.Futureandpresentvaluescanbecalculatedusingaregularcalculatororacalculatorwithfinancialfunctions.Whencompoundingoccursmorefrequentlythanonceayear,theeffectiverateofinterestisgreaterthanthequotedrate.
OUTLINE
Thecashflowtimelineisoneofthemostimportanttoolsintimevalueofmoneyanalysis.Cashflowtimelineshelptovisualizewhatishappeninginaparticularproblem.Cashflowsareplaceddirectlybelowthetickmarks,andinterestratesareshowndirectlyabovethetimeline;unknowncashflowsareindicatedbyquestionmarks.Thus,tofindthefuturevalueof$100after5yearsat5percentinterest,thefollowingcashflowtimelinecanbesetup:
5%
Time:
012345
||||||
Cashflows:
-100FV5=?
◆Acashoutflowisapayment,ordisbursement,ofcashforexpenses,investments,andsoon.
◆Acashinflowisareceiptofcashfromaninvestment,anemployer,orothersources.
Compoundingistheprocessofdeterminingthevalueofacashfloworseriesofcashflowssometimeinthefuturewhencompoundinterestisapplied.Thefuturevalueistheamounttowhichacashfloworseriesofcashflowswillgrowoveragivenperiodoftimewhencompoundedatagiveninterestrate.Thefuturevaluecanbecalculatedas
FVn=PV(1+k)n,
wherePV=presentvalue,orbeginningamount;k=interestrateperperiod;andn=numberofperiodsinvolvedintheanalysis.Thisequationcanbesolvedinoneoftwoways:
numericallyorwithafinancialcalculator.Forcalculations,assumethefollowingdatathatwerepresentedinthetimelineabove:
presentvalue(PV)=$100,interestrate(k)=5%,andnumberofyears(n)=5.
◆Compoundedinterestisinterestearnedoninterest.
◆Tosolvenumerically,usearegularcalculatortofind1+k=1.05raisedtothefifthpower,whichequals1.2763.MultiplythisfigurebyPV=$100togetthefinalanswerofFV5=$127.63.
◆Withafinancialcalculator,thefuturevaluecanbefoundbyusingthetimevalueofmoneyinputkeys,whereN=numberofperiods,I=interestrateperperiod,PV=presentvalue,PMT=annuitypayment,andFV=futurevalue.ByenteringN=5,I=5,PV=-100,andPMT=0,andthenpressingtheFVkey,theanswer127.63isdisplayed.
Somefinancialcalculatorsrequirethatallcashflowsbedesignatedaseitherinflowsoroutflows,thusanoutflowmustbeenteredasanegativenumber(forexample,PV=100insteadofPV=100).
Somecalculatorsrequireyoutopressa“Compute”keybeforepressingtheFVkey.
◆Agraphofthecompoundingprocessshowshowanysumgrowsovertimeatvariousinterestrates.Thegreatertherateofinterest,thefasteristherateofgrowth.
Theinterestrateis,infact,agrowthrate.
Thetimevalueconceptscanbeappliedtoanythingthatisgrowing.
Findingthepresentvalueofacashfloworseriesofcashflowsiscalleddiscounting,anditissimplythereverseofcompounding.Ingeneral,thepresentvalueisthevaluetodayofafuturecashfloworseriesofcashflows.BysolvingforPVinthefuturevalueequation,thepresentvalue,ordiscounting,equationcanbedevelopedandwritteninseveralforms:
PV=
.
◆Tosolveforthepresentvalueof$127.63discountedback5yearsata5%opportunitycostrate,onecanutilizeeitherofthetwosolutionmethods:
Numericalsolution:
Divide$127.63by1.05fivetimestogetPV=$100.
Financialcalculatorsolution:
EnterN=5,I=5,PMT=0,andFV=127.63,andthenpressthePVkeytogetPV=-100.
◆Theopportunitycostrateistherateofreturnonthebestavailablealternativeinvestmentofequalrisk.
◆Agraphofthediscountingprocessshowshowthepresentvalueofanysumtobereceivedinthefuturediminishesandapproacheszeroasthepaymentdateisextendedfartherintothefuture.Atrelativelyhighinterestrates,fundsdueinthefutureareworthverylittletoday,andevenatarelativelylowdiscountrate,thepresentvalueofasumdueintheverydistantfutureisquitesmall.
Thecompoundinganddiscountingprocessesarereciprocals,orinverses,ofoneanother.Inaddition,therearefourvariablesinthetimevalueofmoneyequations:
PV,FV,k,andn.Ifthreeofthefourvariablesareknown,youcanfindthevalueofthefourth.
◆IfwearegivenPV,FV,andn,wecandeterminekbysubstitutingtheknownvaluesintoeitherthepresentvalueorfuturevalueequations,andthensolvingfork.Thus,ifyoucanbuyasecurityatapriceof$78.35whichwillpayyou$100after5years,whatistheinterestrateearnedontheinvestment?
Numericalsolution:
Useatrialanderrorprocesstoreachthe5%valuefork.Thisisatediousandinefficientprocess.Alternatively,youcouldusealgebratosolvethetimevalueequation.
Financialcalculatorsolution:
EnterN=5,PV=-78.35,PMT=0,andFV=100,thenpresstheIkey,andI=5isdisplayed.
◆Likewise,ifwearegivenPV,FV,andk,wecandeterminenbysubstitutingtheknownvaluesintoeitherthepresentvalueorfuturevalueequations,andthensolvingforn.Thus,ifyoucanbuyasecuritywitha5percentinterestrateatapriceof$78.35today,howlongwillittakeforyourinvestmenttoreturn$100?
Numericalsolution:
Useatrialanderrorprocesstoreachthevalueof5forn.Thisisatediousandinefficientprocess.Theequationcanalsobesolvedalgebraically.
Financialcalculatorsolution:
EnterI=5,PV=-78.35,PMT=0,andFV=100,thenpresstheNkey,andN=5isdisplayed.
Anannuityisaseriesofequalpaymentsmadeatfixedintervalsforaspecifiednumberofperiods.Ifthepaymentsoccurattheendofeachperiod,astheytypicallydo,theannuityisanordinary,ordeferred,annuity.Ifthepaymentsoccuratthebeginningofeachperiod,itiscalledanannuitydue.
◆Thefuturevalueofanordinaryannuity,FVAn,isthetotalamountonewouldhaveattheendoftheannuityperiodifeachpaymentwereinvestedatagiveninterestrateandheldtotheendoftheannuityperiod.
DefiningFVAnasthefuturevalueofanordinaryannuityofnyears,andPMTastheperiodicpayment,wecanwrite
FVAn=PMT
=PMT
=PMT
.
Usingafinancialcalculator,enterN=3,I=5,PV=0,andPMT=-100.ThenpresstheFVkey,and315.25isdisplayed.
Foranannuitydue,eachpaymentiscompoundedforoneadditionalperiod,sothefuturevalueoftheentireannuityisequaltothefuturevalueofanordinaryannuitycompoundedforoneadditionalperiod.Thus:
FVA(DUE)n=PMT
.
Mostfinancialcalculatorshaveaswitch,orkey,marked“DUE”or“BEG”thatpermitsyoutoswitchfromend-of-periodpayments(anordinaryannuity)tobeginning-of-periodpayments(anannuitydue).Switchyourcalculatorto“BEG”mode,andcalculateasyouwouldforanordinaryannuity.Donotforgettoswitchyourcalculatorbackto“END”modewhenyouarefinished.
◆Thepresentvalueofanordinaryannuity,PVAn,isthesingle(lumpsum)paymenttodaythatwouldbeequivalenttotheannuitypaymentsspreadovertheannuityperiod.Itistheamounttodaythatwouldpermitwithdrawalsofanequalamount(PMT)attheend(orbeginningforanannuitydue)ofeachperiodfornperiods.
DefiningPVAnasthepresentvalueofanordinaryannuityofnyearsandPMTastheperiodicpayment,wecanwrite
PVAn=PMT
=PMT
=PMT
.
Usingafinancialcalculator,enterN=3,I=5,PMT=-100,andFV=0,andthenpressthePVkey,forananswerof$272.32.
Oneespeciallyimportantapplicationoftheannuityconceptrelatestoloanswithconstantpayments,suchasmortgagesandautoloans.Withtheseamortizedloanstheamountborrowedisthepresentvalueofanordinaryannuity,andthepaymentsconstitutetheannuitystream.
◆Thepresentvalueforanannuitydueis
PVA(DUE)n=PMT
.
Usingafinancialcalculator,switchtothe“BEG”mode,andthenenterN=3,I=5,PMT=-100,andFV=0,andthenpressPVtogettheanswer,$285.94.Again,donotforgettoswitchyourcalculatorbackto“END”modewhenyouarefinished.
◆Youcansolvefortheinterestrate(rateofreturn)earnedonanannuity.
Tosolvenumerically,youmustusethetrial-and-errorprocessandplugindifferentvaluesforkintheannuityequationtosolvefortheinterestrate.
YoucanusethefinancialcalculatorbyenteringtheappropriatevaluesforN,PMT,andeitherFVorPV,andthenpressingItosolvefortheinterestrate.
◆Youcansolveforthenumberofperiods(N)inanannuity.
Tosolvenumerically,youmustusethetrial-and-errorprocessandplugindifferentvaluesforNintheannuityequationtosolveforthenumberofperiods.
YoucanusethefinancialcalculatorbyenteringtheappropriatevaluesforI,PMT,andeitherFVorPV,andthenpressingNtosolveforthenumberofperiods.
Aperpetuityisastreamofequalpaymentsexpectedtocontinueforever.
◆Thepresentvalueofaperpetuityis:
PVP=
.
Forexample,iftheinterestratewere12percent,aperpetuityof$1,000ayearwouldhaveapresentvalueof$1,000/0.12=$8,333.33.
- 配套讲稿:
如PPT文件的首页显示word图标,表示该PPT已包含配套word讲稿。双击word图标可打开word文档。
- 特殊限制:
部分文档作品中含有的国旗、国徽等图片,仅作为作品整体效果示例展示,禁止商用。设计者仅对作品中独创性部分享有著作权。
- 关 键 词:
- THE TIME VALUE OF MONEY