CorporateFinanceFifthEditionChapter4TheTimeValueofMoneyCopyright?2020,2017,2014PearsonEducation,Inc.

AllRightsReservedChapterOutline(1of2)4.1

TheTimeline4.2

TheThreeRulesofTimeTravel4.3

ValuingaStreamofCashFlows4.4

CalculatingtheNetPresentValue4.5

PerpetuitiesandAnnuitiesChapterOutline(2of2)4.6

UsinganAnnuitySpreadsheetorCalculator4.7

Non-AnnualCashFlows4.8

SolvingfortheCashPayments4.9TheInternalRateofReturnLearningObjectives(1of4)Drawatimelineillustratingagivensetofcashflows.Listanddefinethreerulesoftimetravel.Calculatethefuturevalueofthefollowing:AsinglesumAnunevenstreamofcashflows,startingeithernoworsometimeinthefutureAnannuity,startingeithernoworsometimeinthefutureSeveralcashflowsoccurringatregularintervals,whichgrowataconstantrateeachperiodLearningObjectives(2of4)CalculatethepresentvalueofthefollowingAsinglesumAnunevenstreamofcashflows,startingeithernoworsometimeinthefutureAninfinitestreamofidenticalcashflowsAnannuity,startingeithernoworsometimeinthefutureAninfinitestreamofcashflowsthatgrowataconstantrateeachperiodSeveralcashflowsoccurringatregularintervals,whichgrowataconstantrateeachperiodLearningObjectives(3of4)Givenfouroutofthefollowingfiveinputsforanannuity,computethefifth:(a)presentvalue,(b)futurevalue,(c)numberofperiods,(d)periodicinterestrate,(e)periodicpayment.Giventhreeoutofthefollowingfourinputsforasinglesum,computethefourth:(a)presentvalue,(b)futurevalue,(c)numberofperiods,(d)periodicinterestrate.LearningObjectives(4of4)Givencashflowsandpresentorfuturevalue,computetheinternalrateofreturnforaseriesofcashflows.4.1TheTimeline(1of4)Atimelineisalinearrepresentationofthetimingofpotentialcashflows.Drawingatimelineofthecashflowswillhelpyouvisualizethefinancialproblem.4.1TheTimeline(2of4)Assumethatyoumadealoantoafriend.Youwillberepaidintwopayments,oneattheendofeachyearoverthenexttwoyears.4.1TheTimeline(3of4)DifferentiatebetweentwotypesofcashflowsInflowsarepositivecashflows.Outflowsarenegativecashflows,whichareindicatedwithasign.4.1TheTimeline(4of4)Assumethatyouarelending$10,000todayandthattheloanwillberepaidintwoannual$6,000payments.Thefirstcashflowatdate0(today)isrepresentedasanegativesumbecauseitisanoutflow.Timelinescanrepresentcashflowsthattakeplaceattheendofanytimeperiod—amonth,aweek,aday,etc.TextbookExample4.1(1of2)ConstructingaTimelineProblemSupposeyoumustpaytuitionof$10,000peryearforthenexttwoyears.Yourtuitionpaymentsmustbemadeinequalinstallmentsatthestartofeachsemester.Whatisthetimelineofyourtuitionpayments?TextbookExample4.1(2of2)SolutionAssumingtodayisthestartofthefirstsemester,yourfirstpaymentoccursatdate0(today).Theremainingpaymentsoccuratsemesterintervals.Usingonesemesterastheperiodlength,wecanconstructatimelineasfollows:AlternativeExample4.1(1of2)ProblemSupposeyoucanpurchaseathree-yearbondwitha$1,000facevalueandacouponrateof4.5%,paidsemi-annually.Drawatimelineforthecashinflowsofthebond.AlternativeExample4.1(2of2)Solution4.2TheThreeRulesofTimeTravelFinancialdecisionsoftenrequirecombiningcashflowsorcomparingvalues.Threerulesgoverntheseprocesses.Table4.1

TheThreeRulesofTimeTravelRule1OnlyvaluesatthesamepointintimecanbecomparedorcombinedBlankRule2Tomoveacashflowforwardintime,youmustcompoundit.FuturevalueofaCashflowFVsubn=CtimesLeftparenthesis1+rrightparenthesistothenpower.Rule3Tomoveacashflowbackwardintime,youmustdiscountit.PresentvalueofaCashflowPV=Cdividedbyleftparenthesis1+rrightparenthesistothenpower=startfractionCoverleftparenthesis1+rrightparenthesistothenpower.Rule1:ComparingandCombiningValuesAdollartodayandadollarinoneyeararenotequivalent.Itisonlypossibletocompareorcombinevaluesatthesamepointintime.Whichwouldyouprefer:Agiftof$1,000todayor$1,210atalaterdate?Toanswerthis,youwillhavetocomparethealternativestodecidewhichisworthmore.Onefactortoconsider:Howlongis“l(fā)ater?”Rule2:MovingCashFlowsForwardinTime(1of2)Tomoveacashflowforwardintime,youmustcompoundit.Supposeyouhaveachoicebetweenreceiving$1,000todayor$1,210intwoyears.Youbelieveyoucanearn10%onthe$1,000todaybutwanttoknowwhatthe$1,000willbeworthintwoyears.Rule2:MovingCashFlowsForwardinTime(2of2)FutureValueofaCashFlowUsingaFinancialCalculator:TheBasics(1of2)H

P10

BFutureValuePresentValueI/YInterestRateperYearInterestisenteredasapercent,notadecimalFor10%,enter10,NOT.10UsingaFinancialCalculator:TheBasics(2of2)HP10BNumberofPeriodsPeriodsperYearGoldClearsoutallTVMregistersShoulddobetweenallproblemsUsingaFinancialCalculator:SettingtheKeysHP10BGoldCAll(Holddown[C]button)CheckP/YR

SetsPeriodsperYearto

Goldand[=]buttonSetsdisplayto#decimalplacesUsingaFinancialCalculatorHP10BIICashflowsmovinginoppositedirectionsmusthaveoppositesigns.FinancialCalculatorSolutionInputs:N=2I

=10PV=1,000Output:

Figure4.1TheCompositionofInterestoverTimeTextbookExample4.2(1of2)ThePowerofCompoundingProblemSupposeyouinvest$1,000inanaccountpaying10%interestperyear.Howmuchwillyouhaveintheaccountinsevenyears?In20years?In75years?TextbookExample4.2(2of2)SolutionYoucanapplyEq.4.1tocalculatethefuturevalueineachcaseNotethatat10%interest,yourmoneywillnearlydoublein7years.After20years,itwillincreasealmostsevenfold.Andifyouinvestfor75years,youwillbeamillionaire!TextbookExample4.2:FinancialCalculatorSolutionforN=7yearsInputs:N=7I

=10PV

=1,000Output:

AlternativeExample4.2(1of2)ProblemSupposeyouhaveachoicebetweenreceiving$5,000todayor$10,000infiveyears.Youbelieveyoucanearn10%onthe$5,000today,butwanttoknowwhatthe$5,000willbeworthinfiveyears.AlternativeExample4.2(2of2)SolutionThetimelinelookslikethis:Infiveyears,the$5,000willgrowto:Thefuturevalueof$5,000at10%forfiveyears

is$8,053.Youwouldbebetteroffforgoingthegiftof$5,000todayandtakingthe$10,000infiveyears.AlternativeExample4.2:FinancialCalculatorSolutionInputs:N=5I=10PV=5,000Output:

Rule3:MovingCashFlowsBackinTimeTomoveacashflowbackwardintime,wemustdiscountit.PresentValueofaCashFlowTextbookExample4.3(1of2)PresentValueofaSingleFutureCashFlowProblemYouareconsideringinvestinginasavingsbondthatwillpay$15,000in10years.Ifthecompetitivemarketinterestrateisfixedat6%peryear,whatisthebondworthtoday?TextbookExample4.3(2of2)SolutionThecashflowsforthisbondarerepresentedbythefollowingtimeline:Thus,thebondisworth$15,000in10years.Todeterminethevaluetoday,wecomputethepresentvalue:Thebondisworthmuchlesstodaythanitsfinalpayoffbecauseofthetimevalueofmoney.TextbookExample4.3:FinancialCalculatorSolutionInputs:N=10I=6FV=15,000Output:

AlternativeExample4.3(1of2)ProblemSupposeyouareofferedaninvestmentthatpays$10,000infiveyears.Ifyouexpecttoearna10%return,whatisthevalueofthisinvestmenttoday?AlternativeExample4.3(2of2)SolutionThe$10,000isworth:

AlternativeExample4.3:FinancialCalculatorSolutionInputs:N=5I=10FV=10,000Output:

ApplyingtheRulesofTimeTravel(1of5)Recallthefirstrule:Itisonlypossibletocompareorcombinevaluesatthesamepointintime.Sofarwe’veonlylookedatcomparing.Supposeweplantosave$1,000today,and$1,000attheendofeachofthenexttwoyears.Ifwecanearnafixed10%interestrateonoursavings,howmuchwillwehavethreeyearsfromtoday?ApplyingtheRulesofTimeTravel(2of5)Thetimelinewouldlooklikethis:ApplyingtheRulesofTimeTravel(3of5)ApplyingtheRulesofTimeTravel(4of5)ApplyingtheRulesofTimeTravel(5of5)Table4.1TheThreeRulesofTimeTravelRule1Onlyvaluesatthesamepointintimecanbecomparedorcombined.BlankRule2Tomoveacashflowforwardintime,youmustcompoundit.FutureValueofaCashFlowFVsubn=Ctimesleftparenthesis1+rrightparenthesistothenpowerRule3Tomoveacashflowbackwardintime,youmustdiscountit.PresentValueofaCashFlowPV=Cdividedbyleftparenthesis1+rrightparenthesistothenpower=startfractionCoverleftparenthesis1+rrightparenthesistothenpower.TextbookExample4.4(1of3)ComputingtheFutureValueProblemLet’srevisitthesavingsplanweconsideredearlier:weplantosave$1,000todayandattheendofeachofthenexttwoyears.Atafixed10%interestrate,howmuchwillwehaveinthebankthreeyearsfromtoday?TextbookExample4.4(2of3)SolutionLet’ssolvethisprobleminadifferentwaythanwedidearlier.First,computethepresentvalueofthecashflows.Thereareseveralwaystoperformthiscalculation.Herewetreateachcashflowseparatelyandthencombinethepresentvalues.TextbookExample4.4(3of3)Saving$2,735.54todayisequivalenttosaving$1,000peryearforthreeyears.Nowlet’scomputeitsfuturevalueinyear3:Thisanswerof$3,641ispreciselythesameresultwefoundearlier.Aslongasweapplythethreerulesoftimetravel,wewillalwaysgetthecorrectanswer.TextbookExample4.4:FinancialCalculatorSolutionAlternativeExample4.4(1of4)ProblemAssumethataninvestmentwillpayyou$5,000nowand$10,000infiveyears.Thetimelinewouldlikethis:AlternativeExample4.4(2of4)SolutionYoucancalculatethepresentvalueofthecombinedcashflowsbyaddingtheirvaluestoday.Thepresentvalueofbothcashflowsis$11,209.Thepresentvalueofbothcashflowsis$11,209.AlternativeExample4.4(3of4)SolutionYoucancalculatethefuturevalueofthecombinedcashflowsbyaddingtheirvalues

inYear5.Thefuturevalueofbothcashflowsis$18,053.AlternativeExample4.4(4of4)Solution4.3ValuingaStreamofCashFlows

(1of2)Basedonthefirstruleoftimetravelwecanderiveageneralformulaforvaluingastreamofcashflows:ifwewanttofindthepresentvalueofastreamofcashflows,wesimplyaddupthepresentvaluesofeach.4.3ValuingaStreamofCashFlows(2of2)PresentValueofaCashFlowStreamTextbookExample4.5(1of4)PresentValueofaStreamofCashFlowsProblemYouhavejustgraduatedandneedmoneytobuyanewcar.YourrichuncleHenrywilllendyouthemoneysolongasyouagreetopaybackwithinfouryears,andyouoffertopayhimtherateofinterestthathewouldotherwisegetbyputtinghismoneyinasavingsaccount.Basedonyourearningsandlivingexpenses,youthinkyouwillbeabletopayhim$5,000inoneyear,andthen$8000eachyearforthenextthreeyears.IfuncleHenrywouldotherwiseearn6%peryearonhissavings,howmuchcanyouborrowfromhim?TextbookExample4.5(2of4)SolutionThecashflowsyoucanpromiseUncleHenryareasfollows:HowmuchmoneyshouldUncleHenrybewillingtogiveyoutodayinreturnforyourpromiseofthesepayments?Heshouldbewillingtogiveyouanamountthatisequivalenttothesepaymentsinpresentvalueterms.Thisistheamountofmoneythatitwouldtakehimtoproducethesesamecashflows,whichwecalculateasfollows:TextbookExample4.5(3of4)Thus,UncleHenryshouldbewillingtolendyou$24,890.65inexchangeforyourpromisedpayments.Thisamountislessthanthetotalyouwillpayhimduetothetimevalueofmoney.Let’sverifyouranswer.Ifyourunclekepthis$24,890.65inthebanktodayearning6%interest,infouryearshewouldhaveNowsupposethatUncleHenrygivesyouthemoney,andthendepositsyourpaymentstohiminthebankeachyear.Howmuchwillhehavefouryearsfromnow?Weneedtocomputethefuturevalueoftheannualdeposits.Onewaytodosoistocomputethebankbalanceeachyear:TextbookExample4.5(4of4)Wegetthesameanswerbothways(withinapenny,whichisbecauseofrounding).TextbookExample4.5:FinancialCalculatorSolutionAlternativeExample4.5(1of2)ProblemWhatisthefuturevalueinthreeyearsofthefollowingcashflowsifthecompoundingrateis5%?AlternativeExample4.5(2of2)SolutionOrFutureValueofCashFlowStream FutureValueofaCashFlowStreamwithaPresentValueofP

V4.4CalculatingtheNetPresentValueCalculatingtheN

P

V

offuturecashflowsallowsustoevaluateaninvestmentdecision.NetPresentValuecomparesthepresentvalueofcashinflows(benefits)tothepresentvalueofcashoutflows(costs).TextbookExample4.6(1of4)NetPresentValueofanInvestmentOpportunityProblem

Youhavebeenofferedthefollowinginvestmentopportunity:ifyouinvest$1,000today,youwillreceive$500attheendofeachofthenextthreeyears.Ifyoucouldotherwiseearn10%peryearonyourmoney,shouldyouundertaketheinvestmentopportunity?TextbookExample4.6(2of4)SolutionAsalways,westartwithatimeline.Wedenotetheupfrontinvestmentasanegativecashflow(becauseitismoneyweneedtospend)andthemoneywereceiveasapositivecashflow.Todecidewhetherweshouldacceptthisopportunity,wecomputetheN

P

Vbycomputingthepresentvalueofthestream:TextbookExample4.6(3of4)BecausetheN

P

Vispositive,thebenefitsexceedthecostsandweshouldmaketheinvestment.Indeed,theN

P

Vtellsusthattakingthisopportunityislikegettinganextra$243.43thatyoucanspendtoday.Toillustrate,supposeyouborrow$1,000toinvestintheopportunityandanextra$243.43tospendtoday.Howmuchwouldyouoweonthe$1,243.43loaninthreeyears?At10%interest,theamountyouwouldowewouldbeTextbookExample4.6(4of4)Atthesametime,theinvestmentopportunitygeneratescashflows.Ifyouputthesecashflowsintoabankaccount,howmuchwillyouhavesavedthreeyearsfromnow?ThefuturevalueofthesavingsisAsyousee,youcanuseyourbanksavingstorepaytheloan.Takingtheopportunitythereforeallowsyoutospend$243.43todayatnoextracost.TextbookExample4.6:FinancialCalculatorSolutionAlternativeExample4.6(1of2)ProblemWouldyoubewillingtopay$5,000forthefollowingstreamofcashflowsifthediscountrateis7%?AlternativeExample4.6(2of2)SolutionThepresentvalueofthebenefitsis:Thepresentvalueofthecostis$5000,becauseitoccursnow.

AlternativeExample4.6:FinancialCalculatorSolutionOnapresentvaluebasis,thebenefitsexceedthecostsby$366.91.4.5PerpetuitiesandAnnuities(1of2)PerpetuitiesWhenaconstantcashflowwilloccuratregularintervalsforeveritiscalledaperpetuity.4.5PerpetuitiesandAnnuities(2of2)Thevalueofaperpetuityissimplythecashflowdividedbytheinterestrate.PresentValueofaPerpetuityTextbookExample4.7(1of2)EndowingaPerpetuityProblemYouwanttoendowanannualM

B

Agraduationpartyatyouralmamater.Youwanttheeventtobeamemorableone,soyoubudget$30,000peryearforeverfortheparty.Iftheuniversityearns8%peryearonitsinvestments,andifthefirstpartyisinoneyear’stime,howmuchwillyouneedtodonatetoendowtheparty?TextbookExample4.7(2of2)SolutionThetimelineofthecashflowsyouwanttoprovideisThisisastandardperpetuityof$30,000peryear.Thefundingyouwouldneedtogivetheuniversityinperpetuityisthepresentvalueofthiscashflowstream.Fromtheformula,Ifyoudonate$375,000today,andiftheuniversityinvestsitat8%peryearforever,thentheM

B

Aswillhave$30,000everyyearfortheirgraduationparty.AlternativeExample4.7(1of2)ProblemYouwanttoendowachairforafemaleprofessoroffinanceatyouralmamater.You’dliketoattractaprestigiousfacultymember,soyou’dliketheendowmenttoadd$100,000peryeartothefacultymember’sresources(salary,travel,databases,etc.).Ifyouexpecttoearnarateofreturnof4%annuallyontheendowment,howmuchwillyouneedtodonatetofundthechair?AlternativeExample4.7(2of2)SolutionThetimelineofthecashflowslookslikethis:Thisisaperpetuityof$100,000peryear.Thefundingyouwouldneedtogiveisthepresentvalueofthatperpetuity.Fromtheformula:Youwouldneedtodonate$2.5milliontoendowthechair.4.5PerpetuitiesandAnnuitiesAnnuitiesWhenaconstantcashflowwilloccuratregularintervalsforafinitenumberofNperiods,itiscalledanannuity.PresentValueofanAnnuityPresentValueofanAnnuity(1of3)Tofindasimplerformula,supposeyouinvest$100inabankaccountpaying5%interest.Aswiththeperpetuity,supposeyouwithdrawtheinteresteachyear.Insteadofleavingthe$100inforever,youclosetheaccountandwithdrawtheprincipalin20years.PresentValueofanAnnuity(2of3)Youhavecreateda20-yearannuityof$5peryear,plusyouwillreceiveyour$100backin20years.SoRe-arrangingtermsPresentValueofanAnnuity(3of3)Forthegeneralformula,substitutePfortheprincipalvalueandTextbookExample4.8(1of3)PresentValueofaLotteryPrizeAnnuityProblemYouaretheluckywinnerofthe$30millionstatelottery.Youcantakeyourprizemoneyeitheras(a)30paymentsof$1millionperyear(startingtoday),or(b)$15millionpaidtoday.Iftheinterestrateis8%,whichoptionshouldyoutake?TextbookExample4.8(2of3)SolutionOption(a)provides$30millionofprizemoneybutpaidannually.Inthiscase,thecashflowsareanannuityinwhichthefirstpaymentbeginsimmediately,sometimescalledanannuitydue.Becausethefirstpaymentispaidtoday,thelastpaymentwilloccurin29years(foratotalof30payments).Wecancomputethepresentvalueofthefinal29paymentsasastandardannuityof$1millionperyearusingtheannuityformula:Addingthe$1millionwereceiveupfront,thisoptionhasapresentvalueof$12.16million:TextbookExample4.8(3of3)Addingthe$1millionwereceiveupfront,thisoptionhasapresentvalueof$12.16million:Therefore,thepresentvalueofoption(a)isonly$12.16million,andsoitismorevaluabletotakeoption(b)andreceive$15millionupfront—eventhoughwereceiveonlyhalfthetotalcashamount.Thedifference,ofcourse,isduetothetimevalueofmoney.Toseethat(b)reallyisbetter,ifyouhavethe$15milliontoday,youcanuse$1millionimmediatelyandinvesttheremaining$14millionatan8%interestrate.Thisstrategywillgiveyouinperpetuity!Alternatively,youcanspendmilliontoday,andinvesttheremaining$11.16million,whichwillstillallowyoutowithdraw$1millioneachyearforthenext29yearsbeforeyouraccountisdepleted.TextbookExample4.8:FinancialCalculatorSolution(1of2)Sincethepaymentsbegintoday,thisisanAnnuityDue.FirstTextbookExample4.8:FinancialCalculatorSolution(2of2)Then$15million>$12.16million,sotakethelumpsum.FutureValueofanAnnuityFutureValueofanAnnuityTextbookExample4.9(1of3)RetirementSavingsPlanAnnuityProblemEllenis35yearsold,andshehasdecideditistimetoplanseriouslyforherretirement.Attheendofeachyearuntilsheis65,shewillsave$10,000inaretirementaccount.Iftheaccountearns10%peryear,howmuchwillEllenhavesavedatage65?TextbookExample4.9(2of3)SolutionAsalways,webeginwithatimeline.Inthiscase,itishelpfultokeeptrackofboththedatesandEllen’sage:Ellen’ssavingsplanlookslikeanannuityof$10,000peryearfor30years.(Hint:Itiseasytobecomeconfusedwhenyoujustlookatage,ratherthanatbothdatesandage.AcommonerroristothinkthereareonlyWritingdownbothdatesandageavoidsthisproblem.)TextbookExample4.9(3of3)SolutionTodeterminetheamountEllenwillhaveinthebankatage65,wecomputethefuturevalueofthisannuity:TextbookExample4.9:FinancialCalculatorSolution(1of2)Sincethepaymentsbegininoneyear,thisisanOrdinaryAnnuity.Besuretoputthecalculatorbackon“End”mode:TextbookExample4.9:FinancialCalculatorSolution(2of2)ThenGrowingCashFlows(1of2)GrowingPerpetuityAssumeyouexpecttheamountofyourperpetualpaymenttoincreaseataconstantrate,g.PresentValueofaGrowingPerpetuityTextbookExample4.10(1of2)EndowingaGrowingPerpetuityProblemInexample4.7,youplannedtodonatemoneytoyouralmamatertofundanannual$30,000MBAgraduationparty.Givenaninterestrateof8%peryear,therequireddonationwasthepresentvalueofBeforeacceptingthemoney,however,theMBAstudentassociationhasaskedthatyouincreasethedonationtoaccountfortheeffectofinflationonthecostofthepartyinfutureyears.Although$30,000isadequatefornextyear’sparty,thestudentsestimatethattheparty’scostwillriseby4%peryearthereafter.Tosatisfytheirrequest,howmuchdoyouneedtodonatenow?TextbookExample4.10(2of2)SolutionThecostofthepartynextyearis$30,000,andthecostthenincreases4%peryearforever.Fromthetimeline,werecognizetheformofagrowingperpetuity.Tofinancethegrowingcost,youneedtoprovidethepresentvaluetodayofYouneedtodoublethesizeofyourgift!AlternativeExample4.10(1of2)ProblemInAlternativeExample4.7,youplannedtodonatemoneytoendowachairatyouralmamatertosupplementthesalaryofaqualifiedindividualby$100,000peryear.Givenaninterestrateof4%peryear,therequireddonationwas$2.5million.Theuniversityhasaskedyoutoincreasethedonationtoaccountfortheeffectofinflation,whichisexpectedtobe2%peryear.Howmuchwillyouneedtodonatetosatisfythatrequest?AlternativeExample4.10(2of2)SolutionThetimelineofthecashflowslookslikethis:Thecostoftheendowmentwillstartat$100,000,andincreaseby2%eachyear.Thisisagrowingperpetuity.Fromtheformula:Youwouldneedtodonate$5.0milliontoendowthechair.GrowingCashFlows(2of2)GrowingAnnuityThepresentvalueofagrowingannuitywiththeinitialcashflowc,growthrateg,andinterestraterisdefinedas:PresentValueofaGrowingAnnuityTextbookExample4.11(1of3)RetirementSavingswithaGrowingAnnuityProblemInExample4.9,Ellenconsideredsaving$10,000peryearforherretirement.Although$10,000isthemostshecansaveinthefirstyear,sheexpectshersalarytoincreaseeachyearsothatshewillabletoincreasehersavingsby5%peryear.Withthisplan,ifsheearns10%peryearonhersavings,howmuchwillEllenhavesavedatage65?TextbookExample4.11(2of3)SolutionHernewsavingsplanisrepresentedbythefollowingtimeline:Thisexampleinvolvesa30-yeargrowingannuity,withagrowthrateof5%,andaninitialcashflowof$10,000.ThepresentvalueofthisgrowingannuityisgivenbyTextbookExample4.11(3of3)Ellen’sproposedsavingsplanisequivalenttohaving$150,463inthebanktoday.Todeterminetheamountshewillhaveatage65,weneedtomovethisamountforward30years:Ellenwillhavesaved$2.625millionatage65usingthenewsavingsplan.Thissumisalmost$1millionmorethanshehadwithouttheadditionalannualincreasesinsavings.AlternativeExample4.11(1of2)ProblemYouwanttobeginsavingforyourretirement.Youplantocontribute$12,000totheaccountattheendofthisyear.Youanticipateyouwillbeabletoincreaseyourannualcontributionsby3%eachyearforthenext45years.Ifyourexpectedannualreturnis8%,howmuchdoyouexpecttohaveinyourretirementaccountwhenyouretirein45years?AlternativeExample4.11(2of2)SolutionThepresentvalueoftheseriesofdepositsis:Thefuturevalueoftheseriesofdepositsis:4.6UsinganAnnuitySpreadsheetorCalculatorSpreadsheetssimplifythecalculationsofT

V

MproblemsN

P

E

RR

A

T

EP

VP

M

TF

VThesefunctionsallsolvetheproblem:TextbookExample4.12ComputingtheFutureValueinExcelProblemSupposeyouplantoinvest$20,000inanaccountpaying8%interest.Howmuchwillyouhaveintheaccountin15years?TextbookExample4.12(2of4)SolutionWerepresentthisproblemwiththefollowingtimeline:TextbookExample4.12(3of4)SolutionTocomputethesolution,weenterthefourvariablesweknowandsolvefortheonewewanttodetermine(F

V

)usingtheExcelfunctionF

V(R

A

T

E,N

P

E

R,P

M

T,P

V

).Thespreadsheetherecalculatesafuturevalueof$63,443.BlankNPERRATEPVPMTFVExcelFormulaGiven158.00%Negative20,000.0BlankBlankSolveforFVBlankBlankBlankBlank63,443=FVleftparenthesis0.08,15,0,negative20,000rightparenthesis.TextbookExample4.12(4of4)NotethatweenteredP

Vasanegativenumber(theamountweareputtinginto

thebank),andF

Visshownasapositivenumber(theamountwecantakeout

ofthebank).Itisimportanttousesignscorrectlytoindicatethedirectioninwhichthemoneyisflowingwhenusingthespreadsheetfunctions.Tochecktheresult,wecansolvethisproblemdirectly:TextbookExample4.13(1of4)UsingtheAnnuitySpreadsheetProblemSupposethatyouinvest$20,000inanaccountpaying8%interest.Youplantowithdraw$2,000attheendofeachyearfor15years.Howmuchmoneywillbeleftintheaccountafter15years?TextbookExample4.13(2of4)SolutionAgain,westartwiththetimelineshowingourinitialdepositandsubsequentwithdrawals:TextbookExample4.13(3of4)SolutionNotethatP

Visnegative(moneyinto

thebank),whileP

M

T

ispositive(moneyout

ofthebank).Wesolveforthefinalbalanceintheaccount