TheTimeValueofMoneyWhatisTimeValue?WesaythatmoneyhasatimevaluebecausethatmoneycanbeinvestedwiththeexpectationofearningapositiverateofreturnInotherwords,“adollarreceivedtodayisworthmorethanadollartobereceivedtomorrow”Thatisbecausetoday’sdollarcanbeinvestedsothatwehavemorethanonedollartomorrowTheTerminologyofTimeValuePresentValue-Anamountofmoneytoday,orthecurrentvalueofafuturecashflowFutureValue-AnamountofmoneyatsomefuturetimeperiodPeriod-Alengthoftime(oftenayear,butcanbeamonth,week,day,hour,etc.)InterestRate-Thecompensationpaidtoalender(orsaver)fortheuseoffundsexpressedasapercentageforaperiod(normallyexpressedasanannualrate)AbbreviationsPV-PresentvalueFV-FuturevaluePmt-PerperiodpaymentamountN-Eitherthetotalnumberofcashflowsor thenumberofaspecificperiodi-TheinterestrateperperiodTimelines012345PVFVTodayAtimelineisagraphicaldeviceusedtoclarifythetimingofthecashflowsforaninvestmentEachtickrepresentsonetimeperiodCalculatingtheFutureValueSupposethatyouhaveanextra$100todaythatyouwishtoinvestforoneyear.Ifyoucanearn10%peryearonyourinvestment,howmuchwillyouhaveinoneyear?012345-100?CalculatingtheFutureValue(cont.)Supposethatattheendofyear1youdecidetoextendtheinvestmentforasecondyear.Howmuchwillyouhaveaccumulatedattheendofyear2?012345-110?GeneralizingtheFutureValueRecognizingthepatternthatisdeveloping,wecangeneralizethefuturevaluecalculationsasfollows:Ifyouextendedtheinvestmentforathirdyear,youwouldhave:CompoundInterestNotefromtheexamplethatthefuturevalueisincreasingatanincreasingrateInotherwords,theamountofinterestearnedeachyearisincreasingYear1:$10Year2:$11Year3:$12.10ThereasonfortheincreaseisthateachyearyouareearninginterestontheinterestthatwasearnedinpreviousyearsinadditiontotheinterestontheoriginalprincipleamountCompoundInterestGraphicallyTheMagicofCompoundingOnNov.25,1626PeterMinuit,aDutchman,reportedlypurchasedManhattanfromtheIndiansfor$24worthofbeadsandothertrinkets(珠子和其他飾品).WasthisagooddealfortheIndians?Thishappenedabout371yearsago,soiftheycouldearn5%peryeartheywouldnow(in1997)have:Iftheycouldhaveearned10%peryear,theywouldnowhave:That’sabout54,563Trillion(萬億)dollars!TheMagicofCompounding(cont.)TheWallStreetJournal(17Jan.92)saysthatallofNewYorkcityrealestateisworthabout$324billion.Ofthisamount,Manhattanisabout30%,whichis$97.2billionAt10%,thisis$54,562trillion!OurU.S.GNPisonlyaround$6trillionperyear.Sothisamountrepresentsabout9,094yearsworthofthetotaleconomicoutputoftheUSA!.CalculatingthePresentValueSofar,wehaveseenhowtocalculatethefuturevalueofaninvestmentButwecanturnthisaroundtofindtheamountthatneedstobeinvestedtoachievesomedesiredfuturevalue:PresentValue:AnExampleSupposethatyourfive-yearolddaughterhasjustannouncedherdesiretoattendcollege.Aftersomeresearch,youdeterminethatyouwillneedabout$100,000onher18thbirthdaytopayforfouryearsofcollege.Ifyoucanearn8%peryearonyourinvestments,howmuchdoyouneedtoinvesttodaytoachieveyourgoal?AnnuitiesAnannuityisaseriesofnominallyequalpaymentsequallyspacedintime(等時(shí)間間隔)Annuitiesareverycommon:RentMortgagepaymentsCarpaymentPensionincomeThetimelineshowsanexampleofa5-year,$100annuity012345100100100100100ThePrincipleofValueAdditivityHowdowefindthevalue(PVorFV)ofanannuity?First,youmustunderstandtheprincipleofvalueadditivity:ThevalueofanystreamofcashflowsisequaltothesumofthevaluesofthecomponentsInotherwords,ifwecanmovethecashflowstothesametimeperiodwecansimplyaddthemalltogethertogetthetotalvalue價(jià)值相加PresentValueofanAnnuityWecanusetheprincipleofvalueadditivitytofindthepresentvalueofanannuity,bysimplysummingthepresentvaluesofeachofthecomponents:PresentValueofanAnnuity(cont.)Usingtheexample,andassumingadiscountrateof10%peryear,wefindthatthepresentvalueis:01234510010010010010062.0968.3075.1382.6490.91379.08PresentValueofanAnnuity(cont.)Actually,thereisnoneedtotakethepresentvalueofeachcashflowseparatelyWecanuseaclosed-formofthePVAequationinstead:PresentValueofanAnnuity(cont.)Wecanusethisequationtofindthepresentvalueofourexampleannuityasfollows:Thisequationworksforallregularannuities,regardlessofthenumberofpaymentsTheFutureValueofanAnnuityWecanalsousetheprincipleofvalueadditivitytofindthefuturevalueofanannuity,bysimplysummingthefuturevaluesofeachofthecomponents:TheFutureValueofanAnnuity(cont.)Usingtheexample,andassumingadiscountrateof10%peryear,wefindthatthefuturevalueis:100100100100100012345146.41133.10121.00110.00}=610.51atyear5TheFutureValueofanAnnuity(cont.)JustaswedidforthePVAequation,wecouldinsteaduseaclosed-formoftheFVAequation:Thisequationworksforallregularannuities,regardlessofthenumberofpaymentsTheFutureValueofanAnnuity(cont.)Wecanusethisequationtofindthefuturevalueoftheexampleannuity:AnnuitiesDue

預(yù)付年金Thusfar,theannuitiesthatwehavelookedatbegintheirpaymentsattheendofperiod1;thesearereferredtoasregularannuitiesAannuitydueisthesameasaregularannuity,exceptthatitscashflowsoccuratthebeginningoftheperiodratherthanattheend0123451001001001001001001001001001005-periodAnnuityDue5-periodRegularAnnuityPresentValueofanAnnuityDueWecanfindthepresentvalueofanannuitydueinthesamewayaswedidforaregularannuity,withoneexceptionNotefromthetimelinethat,ifweignorethefirstcashflow,theannuityduelooksjustlikeafour-periodregularannuityTherefore,wecanvalueanannuityduewith:PresentValueofanAnnuityDue(cont.)Therefore,thepresentvalueofourexampleannuitydueis:NotethatthisishigherthanthePVofthe,otherwiseequivalent,regularannuityFutureValueofanAnnuityDueTocalculatetheFVofanannuitydue,wecantreatitasregularannuity,andthentakeitonemoreperiodforward:012345PmtPmtPmtPmtPmtFutureValueofanAnnuityDue(cont.)Thefuturevalueofourexampleannuityis:Notethatthisishigherthanthefuturevalueofthe,otherwiseequivalent,regularannuityDeferredAnnuities

遞延年金Adeferredannuityisthesameasanyotherannuity,exceptthatitspaymentsdonotbeginuntilsomelaterperiodThetimelineshowsafive-perioddeferredannuity01234510010010010010067PVofaDeferredAnnuityWecanfindthepresentvalueofadeferredannuityinthesamewayasanyotherannuity,withanextrasteprequiredBeforewecandothishowever,thereisanimportantruletounderstand: WhenusingthePVAequation,theresultingPVisalwaysoneperiodbeforethefirstpaymentoccursPVofaDeferredAnnuity(cont.)TofindthePVofadeferredannuity,wefirstfindusethePVAequation,andthendiscountthatresultbacktoperiod0Hereweareusinga10%discountrate0123450010010010010010067PV2=379.08PV0=313.29PVofaDeferredAnnuity(cont.)Step1:Step2:FVofaDeferredAnnuityThefuturevalueofadeferredannuityiscalculatedinexactlythesamewayasanyotherannuityTherearenoextrastepsatallUnevenCashFlowsVeryoftenaninvestmentoffersastreamofcashflowswhicharenoteitheralumpsumoranannuityWecanfindthepresentorfuturevalueofsuchastreambyusingtheprincipleofvalueadditivityUnevenCashFlows:AnExample(1)Assumethataninvestmentoffersthefollowingcashflows.Ifyourrequiredreturnis7%,whatisthemaximumpricethatyouwouldpayforthisinvestment?012345100200300UnevenCashFlows:AnExample(2)Supposethatyouweretodepositthefollowingamountsinanaccountpaying5%peryear.Whatwouldthebalanceoftheaccountbeattheendofthethirdyear?012345300500700Non-annualCompoundingSofarwehaveassumedthatthetimeperiodisequaltoayearHowever,thereisnoreasonthatatimeperiodcan’tbeanyotherlengthoftimeWecouldassumethatinterestisearnedsemi-annually,quarterly,monthly,daily,oranyotherlengthoftimeTheonlychangethatmustbemadeistomakesurethattherateofinterestisadjustedtotheperiodlengthNon-annualCompounding(cont.)Supposethatyouhave$1,000availableforinvestment.Afterinvestigatingthelocalbanks,youhavecompiledthefollowingtableforcomparison.Inwhichbankshouldyoudeposityourfunds?Non-annualCompounding(cont.)Tosolvethisproblem,youneedtodeterminewhichbankwillpayyouthemostinterestInotherwords,atwhichbankwillyouhavethehighestfuturevalue?Tofindout,let’schangeourbasicFVequationslightly:Inthisversionoftheequation‘m’isthenumberofcompoundingperiodsperyearNon-annualCompounding(cont.)WecanfindtheFVforeachbankas