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Financial Management

Principles & Practice 7e

By Timothy Gallagher Colorado State University

Changes to the new Seventh Edition:

Updating of all time sensitive material and some new chapter openers that refer to recent major financial events

Many fictional company and people names in end-of-chapter material changes to appeal to today’s college age students

Brand-new test item file and test disk Improved PowerPoint Lecture Slides--all new design All new problems in Chapter 8 – Time Value of Money plus many new problems

in other chapters. Benefit corporations – including those chartered by those states that allow

corporate charters that permit the firm to consider the interests of society and non-owners along with those of the owners and those corporations certified by B Lab as having similarly broad goals

Aftermath of the 2008 Financial Crisis including updates on regulatory changes inspired by this crisis and the implementation of the Dodd Frank law and its Volker Rule

Activist investors – how they influence companies and how they pressure companies into certain actions that might not otherwise be undertaken

Spreadsheet solutions and descriptions, including Excel screen shots for time value of money and capital budgeting material

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Instructor’s Manual

Solutions Manual

Computerized Test Bank

PowerPoint Lecture Slides

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Table of Contents

1 Finance and the Firm

2 Financial Markets and Interest Rates

3 Financial Institutions

4 Review of Accounting

5 Analysis of Financial Statements

6 Forecasting for Financial Planning

7 Risk and Return

8 The Time Value of Money

9 The Cost of Capital

10 Capital Budgeting Decision Methods

11 Estimating Incremental Cash Flows

12 Business Valuation

13 Capital Structures Basics

14 Corporate Bonds, Preferred Stock & Leasing

15 Common Stock

16 Dividend Policy

17 Working Capital Policy

18 Managing Cash

19 Accounts Receivable and Inventory

20 Short-Term Financing

21 International Finance

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their mastery by entering data into the spreadsheet. A macro causes the spreadsheet to "reply," and the student responds with the

next indicated step, and so on until the problem is solved. While useful for testing mastery for all concepts, this application is a great tool for help-

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163

Get a Free $1,000 Bond with Every Car Bought This Week! Thereisacardealerwhoappearsontelevisionregularly.Hedoeshisowncommercials.Heisquiteloudandisalsoveryaggressive.Accordingtohimyouwillpaywaytoomuchifyoubuyyourcarfromanyoneelseintown.Youmighthaveacardealerlikethisinyourhometown.Theauthorofthisbookusedtowatchandlistentothetelevision

commercialsforaparticularcardealer.Onepromotionstruckhimasbeingparticularlyinteresting.Theautomobilemanufacturershadbeenofferingcashrebatestobuyersofparticularcarsbutthispromotionhadrecentlyended.Thislocalcardealerseemedtobepickingupwherethemanufacturershadleftoff.Hewasoffering“afree$1,000bondwitheverynewcarpurchased”duringaparticularweek.Thissoundedprettygood.Thefineprintofthisdealwasnotrevealeduntilthebuyerwas

signingthefinalsalespapers.Eventhenyouhadtolookclosesincetheprintwassmall.Itturnsoutthe“$1,000bond”offeredtoeachcarbuyerwaswhatisknownasa“zerocouponbond.”Thismeansthattherearenoperiodicinterestpayments.Thebuyerofthebondpaysapricelessthanthefacevalueof$1,000andthenatmaturitytheissuerpays$1,000totheholderofthebond.Theinvestor’sreturnisentirelyequaltothedifferencebetweenthelowerpricepaidforthebondandthe$1,000receivedatmaturity.Howmuchlessthan$1,000didthedealerhavetopaytogetthisbondhewasgivingaway?Theamountpaidbythedealeriswhatthebondwouldbeworth(less

afterpayingcommissions)ifthecarbuyerwantedtosellthisbondnow.Itturnsoutthatthisbondhadamaturityof30years.Thisishowlongthecarbuyerwouldhavetowaittoreceivethe$1,000.Thevalueofthebondatthetimethecarwaspurchasedwasabout$57.Thisiswhatthecardealerpaidforeachofthesebondsandistheamountthecarbuyerwouldgetfromsellingthebond.It’saprettyshrewdmarketinggimmickwhenacardealercanbuysomethingfor$57andgetthecustomerstobelievetheyarereceivingsomethingworth$1,000.

Learning ObjectivesAfterreadingthischapter,youshouldbeableto:

1. Explainthetimevalueofmoneyanditsimportanceinthebusinessworld.

2. Calculatethefuturevalueandpresentvalueofasingleamount.

3. Findthefutureandpresentvaluesofanannuity.

4. Solvetimevalueofmoneyproblemswithunevencashflows.

5. Solvefortheinterestrate,numberoramountofpayments,orthenumberofperiodsinafutureorpresentvalueproblem.

8The Time Value of Money “The importance of money flows from it being a link between the present and the future.” —John Maynard Keynes

CHAPTER

164 Part2 EssentialConceptsinFinance

Inthischapteryouwillbecomearmedagainstsuchdeceptions.It’sallinthetimevalueofmoney.

Source: Thisisinspiredbyanactualmarketingpromotion.Someofthedetailshavebeenchanged,andallidentitieshavebeenhidden,sotheauthordoesnotgetsued.

Chapter Overview Adollarinhandtodayisworthmorethanapromiseofadollartomorrow.Thisisoneofthebasicprinciplesoffinancialdecisionmaking.Timevalueanalysisisacrucialpartoffinancialdecisions.Ithelpsanswerquestionsabouthowmuchmoneyaninvestmentwillmakeovertimeandhowmuchafirmmustinvestnowtoearnanexpectedpayofflater.Inthischapterwewillinvestigatewhymoneyhastimevalue,aswellaslearnhowto

calculatethefuturevalueofcashinvestedtodayandthepresentvalueofcashtobereceivedinthefuture.Wewillalsodiscussthepresentandfuturevaluesofanannuity—aseriesofequalcashpaymentsatregulartimeintervals.Finally,wewillexaminespecialtimevalueofmoneyproblems,suchashowtofindtherateofreturnonaninvestmentandhowtodealwithaseriesofunevencashpayments.

Why Money Has Time Value Thetimevalueofmoneymeansthatmoneyyouholdinyourhandtodayisworthmorethanthesameamountofmoneyyouexpecttoreceiveinthefuture.Similarly,agivenamountofmoneyyoumustpayouttodayisagreaterburdenthanthesameamountpaidinthefuture.InChapter2welearnedthatinterestratesarepositiveinpartbecausepeoplepreferto

consumenowratherthanlater.Positiveinterestratesindicate,then,thatmoneyhastimevalue.Whenonepersonletsanotherborrowmoney,thefirstpersonrequirescompensa-tioninexchangeforreducingcurrentconsumption.Thepersonwhoborrowsthemoneyiswillingtopaytoincreasecurrentconsumption.Thecostpaidbytheborrowertothelenderforreducingconsumption,knownasanopportunity cost,istherealrateofinterest.Therealrateofinterestreflectscompensationforthepuretimevalueofmoney.The

realrateofinterestdoesnotincludeinterestchargedforexpectedinflationortheotherriskfactorsdiscussedinChapter2.RecallfromtheinterestratediscussioninChapter2thatmanyfactors—includingthepuretimevalueofmoney,inflationrisk,defaultrisk,illiquidityrisk,andmaturityrisk—determinemarketinterestrates.Therequiredrateofreturnonaninvestmentreflectsthepuretimevalueofmoney,an

adjustmentforexpectedinflation,andanyotherriskpremiumspresent.

Measuring the Time Value of Money Financialmanagersadjustforthetimevalueofmoneybycalculatingthefuturevalueandthepresentvalue.Futurevalueandpresentvaluearemirrorimagesofeachother.Futurevalueisthevalueofastartingamountatafuturepointintime,giventherateofgrowthperperiodandthenumberofperiodsuntilthatfuturetime.Howmuchwill$1,000investedtodayata10percentinterestrategrowtoin15years?Presentvalueisthevalueofafutureamounttoday,assumingaspecificrequiredinterestrateforanumberofperiodsuntilthatfutureamountisrealized.Howmuchshouldwepaytodaytoobtainapromisedpaymentof$1,000in15yearsifinvestingmoneytodaywouldyielda10percentrateofreturnperyear?

The Future Value of a Single Amount Tocalculatethefuturevalueofasingleamount,wemustfirstunderstandhowmoneygrowsovertime.Oncemoneyisinvested,itearnsaninterestratethatcompensatesforthetimevalueofmoneyand,aswelearnedinChapter2,fordefaultrisk,inflation,andother

Chapter8 TheTimeValueofMoney 165

factors.Often,theinterestearnedoninvestmentsiscompoundinterest—interestearnedoninterestandontheoriginalprincipal.Incontrast,simple interestisinterestearnedonlyontheoriginalprincipal.Toillustratecompoundinterest,assumethefinancialmanagerofSaveComdecidedto

invest$100ofthefirm’sexcesscashinanaccountthatearnsanannualinterestrateof5percent.Inoneyear,SaveComwillearn$5ininterest,calculatedasfollows:

Balance at the end of year 1 Principal Interest

$100 (100 .05)

$100 (1 .05)

$100 1.05

$105

= +

= + ×

= × +

= ×

=

Thetotalamountintheaccountattheendofyear1,then,is$105.Butlookwhathappensinyears2and3.Inyear2,SaveComwillearn5percentof105.

The$105istheoriginalprincipalof$100plusthefirstyear’sinterest—sotheinterestearnedinyear2is$5.25,ratherthan$5.00.Theendofyear2balanceis$110.25—$100inoriginalprincipaland$10.25ininterest.Inyear3,SaveComwillearn5percentof$110.25,or$5.51,foranendingbalanceof$115.76,shownasfollows:

Beginning = Ending Balance × (1 + Interest Rate) Balance Interest

Year 1 $100.00 × 1.05 = $105.00 $5.00 Year 2 $105.00 × 1.05 = $110.25 $5.25 Year 3 $110.25 × 1.05 = $115.76 $5.51

Inourexample,SaveComearned$5ininterestinyear1,$5.25ininterestinyear2($110.25–$105.00),and$5.51inyear3($115.76–$110.25)becauseofthecompound-ingeffect.IftheSaveComdepositearnedinterestonlyontheoriginalprincipal,ratherthanontheprincipalandinterest,thebalanceintheaccountattheendofyear3wouldbe$115($100+($5×3)=$115).Inourcasethecompoundingeffectaccountsforanextra$.76($115.76–$115.00=.76).Thesimplestwaytofindthebalanceattheendofyear3istomultiplytheoriginal

principalby1plustheinterestrateperperiod(expressedasadecimal),1+k,raisedtothepowerofthenumberofcompoundingperiods,n.Here’stheformulaforfindingthefuturevalue—orendingbalance—giventheoriginalprincipal, interestrateperperiod,andnumberofcompoundingperiods:

FutureValueforaSingleAmountAlgebraicMethod

FV=PV×(1+k)n (8-1a)

where: FV=FutureValue,theendingamount

PV=PresentValue,thestartingamount,ororiginalprincipal

k=Rateofinterestperperiod(expressedasadecimal)

n=Numberoftimeperiods1

1.Thecompoundingperiodsareusuallyyearsbutnotalways.Asyouwillseelaterinthechapter,compoundingperiodscanbemonths,weeks,days,oranyspecifiedperiodoftime.


InourSaveComexample,PVistheoriginaldepositof$100,kis5percent,andnis3.Tosolvefortheendingbalance,orFV,weapplyEquation8-1aasfollows:

FV=PV×(1+k)n

=$100×(1.05)3

=$100×1.1576

=$115.76

Wemayalsosolveforfuturevalueusingafinancialtable.Financialtablesareacom-pilationofvalues,knownasinterestfactors,thatrepresentaterm,(1+k)ninthiscase,intimevalueofmoneyformulas.TableIintheAppendixattheendofthebookisdevelopedbycalculating(1+k)nformanycombinationsofkandn.TableIintheAppendixattheendofthebookisthefuturevalueinterestfactor(FVIF)

table.TheformulaforfuturevalueusingtheFVIFtablefollows:

FutureValueforaSingleAmountTableMethod

FV=PV×( )FVIFk, n (8-1b)


PV=PresentValue,thestartingamount

FVIFk,n=FutureValueInterestFactorgiveninterestrate,k,andnumberofperiods,n,fromTableI

InourSaveComexample,inwhich$100isdepositedinanaccountat5percentinter-estforthreeyears,theendingbalance,orFV,accordingtoEquation8-1b,isasfollows:

FV=PV×( )FVIFk, n

=$100×( )FVIF5%, 3

=$100×1.1576(fromtheFVIFtable)

=$115.76

TosolveforFVusingafinancialcalculator,weenterthenumbersforPV,n,andk(kisdepictedasI/YontheTIBAIIPLUScalculator;onothercalculatorsitmaybesymbol-izedbyiorI),andaskthecalculatortocomputeFV.Thekeystrokesfollow.

OlderTIBAIIPLUSfinancialcalculatorsweresetatthefactorytoadefaultvalueof12forP/Y.Changethisdefaultvalueto1asshownhereifyouhaveoneoftheseoldercalculators.ForthepastseveralyearsTIhasmadethedefaultvalue1forP/Y.Ifafterpressingyouseethevalue1inthedisplaywindowyoumaypresstwicetobackoutofthis.NochangestotheP/Yvalueareneededifyourcalculatorisalreadysetto

Take Note Because future value interest factors are rounded to four decimal places in Table I, you may get a slightly different solution compared to a problem solved by the algebraic method. Press 1 , . This sets the calculator to the

mode where one payment per year is expected, which is the assumption for the problems in this chapter.

Older TI BAII PLUS financial calculators were set at the factory to a default value of 12 for P/Y. Change this default value to 1 as shown here if you have one of these older calculators. For the past several years TI has made the default value 1 for P/Y. If after pressing you see the value 1 in the display window you may press twice to back out of this. No changes to the P/Y value are needed if your calculator is already set to the default value of 1. You may similarly skip this adjustment throughout this chapter where the financial calculator solutions are shown if your calculator already has its P/Y value set to 1. If your calculator is an older one that requires you to change the default value of P/Y from 12 to 1, you do not need to make that change again. The default value of P/Y will stay set to 1 unless you specifically change it.

Input values for principal (PV), interest rate (k or I/Y on the calculator), and number of periods (n).

TI BAII PLUS Financial Calculator Solution

Step 1: First press . This clears all the time value of money keys of allpreviously calculated or entered values.

Step 2:

Step 3:

100 5 3 Answer: 115.76


thedefaultvalueof1.YoumaysimilarlyskipthisadjustmentthroughoutthischapterwherethefinancialcalculatorsolutionsareshownifyourcalculatoralreadyhasitsP/Yvaluesetto1.IfyourcalculatorisanolderonethatrequiresyoutochangethedefaultvalueofP/Yfrom12to1,youdonotneedtomakethatchangeagain.ThedefaultvalueofP/Ywillstaysetto1unlessyouspecificallychangeit.

IntheSaveComexample,weinput–100forthepresentvalue(PV),3fornumberofperiods(N),and5fortheinterestrateperyear(I/Y).Thenweaskthecalculatortocom-putethefuturevalue,FV.Theresultis$115.76.OurTIBAIIPLUSissettodisplaytwodecimalplaces.Youmaychooseagreaterorlessernumberifyouwish.ThisfuturevalueproblemcanalsobesolvedusingtheFVfunctioninthespreadsheetbelow.

Wehave learned fourways tocalculate the futurevalueof a singleamount: thealge-braicmethod,thefinancialtablemethod,thefinancialcalculatormethod,andthespreadsheetmethod.Inthenextsection,weseehowfuturevaluesarerelatedtochangesintheinterestrate,k,andthenumberoftimeperiods,n.Wealsoinput–100forthepvvalueinthespreadsheet.

The Sensitivity of Future Values to Changes in Interest Rates or the Number of Compounding Periods Futurevaluehasapositiverelationshipwiththeinterestrate,k,andwiththenumberofperiods,n.Thatis,astheinterestrateincreases,futurevalueincreases.Similarly,asthenumberofperiodsincreases,sodoesfuturevalue.Incontrast,futurevaluedecreaseswithdecreasesinkandnvalues.Itisimportanttounderstandthesensitivityoffuturevaluetokandnbecauseincreases

areexponential,asshownbythe(1+k)nterminthefuturevalueformula.Considerthis:Abusinessthatinvests$10,000inasavingsaccountat5percentfor20yearswillhaveafuturevalueof$26,532.98.Iftheinterestrateis8percentforthesame20years,thefuturevalueoftheinvestmentis$46,609.57.Weseethatthefuturevalueoftheinvest-mentincreasesaskincreases.Figure8-1ashowsthisgraphically.Nowlet’ssaythatthebusinessdeposits$10,000for10yearsata5percentannualinter-

estrate.Thefuturevalueofthatsumis$16,288.95.Anotherbusinessdeposits$10,000for20yearsatthesame5percentannualinterestrate.Thefuturevalueofthat$10,000investment

Press 1 , . This sets the calculator to the mode where one payment per year is expected, which is the assumption for the problems in this chapter.

Older TI BAII PLUS financial calculators were set at the factory to a default value of 12 for P/Y. Change this default value to 1 as shown here if you have one of these older calculators. For the past several years TI has made the default value 1 for P/Y. If after pressing you see the value 1 in the display window you may press twice to back out of this. No changes to the P/Y value are needed if your calculator is already set to the default value of 1. You may similarly skip this adjustment throughout this chapter where the financial calculator solutions are shown if your calculator already has its P/Y value set to 1. If your calculator is an older one that requires you to change the default value of P/Y from 12 to 1, you do not need to make that change again. The default value of P/Y will stay set to 1 unless you specifically change it.

Input values for principal (PV), interest rate (k or I/Y on the calculator), and number of periods (n).


Step 1: First press . This clears all the time value of money keys of allpreviously calculated or entered values.

Step 2:

Step 3:

100 5 3 Answer: 115.76

E10

$115.76


is$26,532.98.Justaswiththeinterestrate,thehigherthenumberofperiods,thehigherthefuturevalue.Figure8-1bshowsthisgraphically.

The Present Value of a Single Amount Presentvalueistoday’sdollarvalueofaspecificfutureamount.Withabond,forinstance,theissuerpromisestheinvestorfuturecashpaymentsatspecifiedpointsintime.With

FIGURE 8-1A Future Value at Different Interest Rates

Figure 8-1a shows the future value of $10,000 after 20 years at interest rates of 5 percent and 8 percent.

FIGURE 8-1B Future Value at Different Times

Figure 8-1b shows the future value of $10,000 after 10 years and 20 years at an interest rate of 5 percent.

$50,000

$45,000

$40,000

$35,000

$30,000

$25,000

$20,000

$15,000

$10,000

$5,000

$00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Number of Years, n

Futu

re V

alue

, FV

Future Value at 5%Future Value at 8%

Future Value of $10,000 After 20 Years at 5% and 8%

$30,000

$25,000

$20,000

$15,000

$10,000

$5,000

$00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Number of Years, n

Futu

re V

alue

, FV

Future Value after 10 yearsFuture Value after 20 years

Future Value of $10,000 after 10 Years and 20 Years at 5%


aninvestmentinnewplantorequipment,certaincashreceiptsareexpected.Whenwecalculatethepresentvalueofafuturepromisedorexpectedcashpayment,wediscountit(markitdowninvalue)becauseitisworthlessifitistobereceivedlaterratherthannow.Similarly,futurecashoutflowsarelessburdensomethanpresentcashoutflowsofthesameamount.Futurecashoutflowsaresimilarlydiscounted(madelessnegative).Inpresentvalueanalysis,then,theinterestrateusedinthisdiscountingprocessisknownasthediscountrate.Thediscountrateistherequiredrateofreturnonaninvestment.Itreflectsthelostopportunitytospendorinvestnow(theopportunitycost)andthevariousrisksassumedbecausewemustwaitforthefunds.Discountingis theinverseofcom-pounding.Compoundinterestcausesthevalueofabeginningamounttoincreaseatanincreasingrate.Discountingcausesthepresentvalueofafutureamounttodecreaseatadecreasing

rate.Todemonstrate,imaginetheSaveComfinancialmanagerneededtoknowhowmuchtoinvestnowtogenerate$115.76inthreeyears,givenaninterestrateof5percent.Givenwhatweknowsofar,thecalculationwouldlooklikethis:

FV PV (1 k)

$115.76 PV 1.05

$115.76 PV 1.157625

PV $100.00

n

3

= × +

= ×

= ×

=

Tosimplifysolvingpresentvalueproblems,wemodifythefuturevalueforasingleamountequationbymultiplyingbothsidesby1/(1+k)ntoisolatePVononesideoftheequalsign.Thepresentvalueformulaforasingleamountfollows:

ThePresentValueofaSingleAmountFormulaAlgebraicMethod

PV FV 1

(1 k)n= ×+ (8-2a)

where: PV=PresentValue,thestartingamount

FV=FutureValue,theendingamount

k=Discountrateofinterestperperiod(expressedasadecimal)

n=Numberoftimeperiods

ApplyingthisformulatotheSaveComexample,inwhichitsfinancialmanagerwantedtoknowhowmuchthefirmshouldpaytodaytoreceive$115.76attheendofthreeyears,assuminga5percentdiscountratestartingtoday,thefollowingisthepresentvalueoftheinvestment:

PV FV 1(1 k)

$115.76 1(1 .05)

$115.76 .86384

$100.00

n

3

= ×+

= ×+

= ×

=


SaveComshouldbewillingtopay$100todaytoreceive$115.76threeyearsfromnowata5percentdiscountrate.TosolveforPV,wemayalsousethePresentValueInterestFactorTableinTableIIin

theAppendixattheendofthebook.Apresentvalueinterestfactor,orPVIF,iscalculatedandshowninTableII.Itequals1/(1+k)nforgivencombinationsofkandn.Thetablemethodformula,Equation8-2b,follows:

ThePresentValueofaSingleAmountFormulaTableMethod

PV=FV×(PVIFk,n) (8-2b)

where: PV=PresentValue

FV=FutureValue

PVIFk,n=PresentValueInterestFactorgivendiscountrate,k,andnumberofperiods,n,fromTableII

Inourexample,SaveCom’sfinancialmanagerwantedtosolvefortheamountthatmustbeinvestedtodayata5percentinterestratetoaccumulate$115.76withinthreeyears.Applyingthepresentvaluetableformula,wefindthefollowingsolution:

PV FV PVIFk, n( )= ×

$115.76 PVIF 5%, 3( )= ×

=$115.76×.8638(fromthePVIFtable)

=$99.99(slightlylowerthan$100duetotheroundingtofourplacesinthetable)

Thepresentvalueof$115.76,discountedbackthreeyearsata5percentdiscountrate,is$100.Tosolveforpresentvalueusingafinancialcalculator,enterthenumbersforfuture

value,FV,thenumberofperiods,n,andtheinterestrate,k—symbolizedasI/Yonthecalculator—then hit theCPT (compute) andPV (present value) keys.The sequencefollows:

Thefinancialcalculatorresultisdisplayedasanegativenumbertoshowthatthepres-entvaluesumisacashoutflow—thatis,thatsumwillhavetobeinvestedtoearn$115.76inthreeyearsata5percentannualinterestrate.ThispresentvalueproblemcanalsobesolvedusingthePVfunctioninthespreadsheetthatfollows.


Step 1: Press to clear previous values.

Step 2: Press 1 , to ensure the calculator is in the mode for annual interest payments.

Step 3: Input the values for future value, the interest rate, and number of periods, and compute PV.

115.76 5 3 Answer: –100.00


Wehaveexaminedhowtofindpresentvalueusingthealgebraic,table,financialcal-culator,andspreadsheetmethods.Next,weseehowpresentvalueanalysisisaffectedbythediscountrate,k,andthenumberofcompoundingperiods,n.

The Sensitivity of Present Values to Changes in the Interest Rate or the Number of Compounding Periods Incontrastwithfuturevalue,presentvalueisinverselyrelatedtokandnvalues.Inotherwords, presentvaluemoves in theoppositedirectionofk andn. If k increases, pres-entvaluedecreases;ifkdecreases,presentvalueincreases.Ifnincreases,presentvaluedecreases;ifndecreases,presentvalueincreases.Considerthis:Abusinessthatexpectsa5percentannualreturnonitsinvestment(k=

5%)shouldbewillingtopay$3,768.89today(thepresentvalue)for$10,000tobereceived20yearsfromnow.Iftheexpectedannualreturnis8percentforthesame20years,thepres-entvalueoftheinvestmentisonly$2,145.48.Weseethatthepresentvalueoftheinvest-mentdecreasesasthevalueofkincreases.Thewaythepresentvalueofthe$10,000varieswithchangesintherequiredrateofreturnisshowngraphicallyinFigure8-2a.Nowlet’ssaythatabusinessexpectstoreceive$10,000tenyearsfromnow.Ifits

requiredrateofreturnfortheinvestmentis5percentannually,thenitshouldbewillingtopay$6,139fortheinvestmenttoday(thepresentvalueis$6,139).Ifanotherbusinessexpectstoreceive$10,000twentyyearsfromnowandithasthesame5percentannualrequiredrateofreturn,thenitshouldbewillingtopay$3,769fortheinvestment(thepresentvalueis$3,769).Justaswiththeinterestrate,thegreaterthenumberofperiods,thelowerthepresentvalue.Figure8-2bshowshowitworks.Inthissectionwehavelearnedhowtofindthefuturevalueandthepresentvalueof

asingleamount.Next,wewillexaminehowtofindthefuturevalueandpresentvalueofseveralamounts.

Working with Annuities Financialmanagersoftenneedtoassessaseriesofcashflowsratherthanjustone.Onecommontypeofcashflowseriesistheannuity—aseriesofequalcashflows,spacedevenlyovertime.Professionalathletesoftensigncontractsthatprovideannuitiesforthemaftertheyretire,

inadditiontothesigningbonusandregularsalarytheymayreceiveduringtheirplayingyears.Consumerscanpurchaseannuitiesfrominsurancecompaniesasameansofproviding

E10

–$100.00

This Excel spreadsheet shows you how to compute the present value of $115.76 discounted back 3 years with a 5% annual discount rate. Note that the computed PV has a negative value indicating that you should be willing to pay that amount now for the right to receive $115.76 in three years with a 5% discount rate.


retirementincome.Theinvestorpaystheinsurancecompanyalumpsumnowinordertoreceivefuturepaymentsofequalsizeatregularlyspacedtimeintervals(usuallymonthly).Anotherexampleofanannuityistheinterestonabond.Theinterestpaymentsareusuallyequaldollaramountspaideitherannuallyorsemiannuallyduringthelifeofthebond.Annuitiesareasignificantpartofmanyfinancialproblems.Youshouldlearntorecognize

annuitiesanddeterminetheirvalue,futureorpresent.Inthissectionwewillexplainhowto

FIGURE 8-2A Present Value at Different Interest Rates

Figure 8-2a shows the present value of $10,000 to be received in 20 years at interest rates of 5 percent and 8 percent.

$12,000

$10,000

$8,000

$6,000

$4,000

$2,000

$00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Number of Years, n

Pres

ent V

alue

, PV

Present Value at 5%Present Value at 8%

Present Value of $10,000 Expected in 20 Years at 5% and 8%

FIGURE 8-2B Present Value at Different Times

Figure 8-2b shows the present value of $10,000 to be received in 10 years and 20 years at an interest rate of 5 percent.

$12,000

$10,000

$8,000

$6,000

$4,000

$2,000

$00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Number of Years, n

Pres

ent

Valu

e,PV

Present Value for 10-Year DealPresent Value for 20-Year Deal

Present Value of $10,000 to Be Receivedin 10 Years and 20 Years at 5%


calculatethefuturevalueandpresentvalueofannuitiesinwhichcashflowsoccurattheendofthespecifiedtimeperiods.Annuitiesinwhichthecashflowsoccurattheendofeachofthespecifiedtimeperiodsareknownasordinary annuities.Annuitiesinwhichthecashflowsoccuratthebeginningofeachofthespecifiedtimeperiodsareknownasannuities due.

Future Value of an Ordinary Annuity Financialmanagersoftenplanforthefuture.Whentheydo,theyoftenneedtoknowhowmuchmoneytosaveonaregularbasis toaccumulateagivenamountofcashataspecifiedfuturetime.Thefuturevalueofanannuityistheamountthatagivennumberofannuitypayments,n,willgrow toat a futuredate, foragivenperiodicinterestrate,k.Forinstance,supposetheSaveComCompanyplanstoinvest$500inamoneymarket

accountattheendofeachyearforthenextfouryears,beginningoneyearfromtoday.Thebusinessexpects toearna5percentannual rateof returnon its investment.HowmuchmoneywillSaveComhaveintheaccountattheendoffouryears?TheproblemisillustratedinthetimelineinFigure8-3.Thetvaluesinthetimelinerepresenttheendofeachtimeperiod.Thus,t1istheendofthefirstyear,t2istheendofthesecondyear,andsoon.Thesymbolt0isnow,thepresentpointintime.Becausethe$500paymentsareeachsingleamounts,wecansolvethisproblemone

stepatatime.LookingatFigure8-4,weseethatthefirststepistocalculatethefuturevalueofthecashflowsthatoccuratt1,t2,t3,andt4usingthefuturevalueformulaforasingleamount.Thenextstepistoaddthefourvaluestogether.Thesumofthosevaluesistheannuity’sfuturevalue.AsshowninFigure8-4, thesumof the futurevaluesof the foursingleamounts is

the annuity’s future value, $2,155.05.However, the step-by-stepprocess illustrated inFigure8-4istime-consumingeveninthissimpleexample.Calculatingthefuturevalueofa20-or30-yearannuity,suchaswouldbethecasewithmanybonds,wouldtakeanenormousamountoftime.Instead,wecancalculatethefuturevalueofanannuityeasilybyusingthefollowingformula:

FutureValueofanAnnuityAlgebraicMethod

FVA PMT (1 k) 1

k

n

= ×+ −

(8-3a)

where: FVA=FutureValueofanAnnuity

PMT=Amountofeachannuitypayment

k=Interestratepertimeperiodexpressedasadecimal

n=Numberofannuitypayments

t0(Today)

t1 t2 t3 t4

$500 $500 $500 $500

Σ = FVA = ?

FV at 5% interest for

1 year

FV at5% interest for

3 years

FV at 5% interest for

2 years

FIGURE 8-3 SaveCom Annuity Timeline


UsingEquation8-3ainourSaveComexample,wesolveforthefuturevalueoftheannuityat5percentinterest(k=5%)withfour$500end-of-yearpayments(n=4andPMT=$500),asfollows:

FVA = 500 × (1 + .05)4 − 1.05

⎡

⎣⎢⎢

⎤

⎦⎥⎥

= 500 × 4.3101

= $2,155.05

Fora$500annuitywitha5percentinterestrateandfourannuitypayments,weseethatthefuturevalueoftheSaveComannuityis$2,155.05.Tofindthefuturevalueofanannuitywiththetablemethod,wemustfindthefuturevalue

interestfactorforanannuity(FVIFA),foundinTableIIIintheAppendixattheendofthebook.TheFVIFAk,nisthevalueof[(1+k)

n–1]÷kfordifferentcombinationsofkandn.

FutureValueofanAnnuityFormulaTableMethod

FVA=PMT×FVIFAk,n (8-3b)

where: FVA=FutureValueofanAnnuity


FVIFAk,n=FutureValueInterestFactorforanAnnuityfromTableIII

k=Interestrateperperiod


InourSaveComexample,then,weneedtofindtheFVIFAforadiscountrateof5per-centwithfourannuitypayments.TableIIIintheAppendixshowsthattheFVIFAk=5%,n=4is4.3101.Usingthetablemethod,wefindthefollowingfuturevalueoftheSaveComannuity:

FVA=500×FVIFA5%,4

=500×4.3101(fromtheFVIFAtable)

=$2,155.05

FIGURE 8-4 Future Value of the SaveCom Annuity

t0(Today)

t1 t2 t3 t4

$500 $500 $500 $500FVA

Σ = FVA = $2,155.05

FV = 500 × (1 + .05)3

= 500 × 1.1576= 578.80

FV = 500 × (1 + .05)2

= 500 × 1.1025= 551.25

FV = 500 × (1 + .05)1

= 500 × 1.05= 525

FV = 500525

551.25

578.80


Tofindthefuturevalueofanannuityusingafinancialcalculator,keyinthevaluesfortheannuitypayment(PMT),n,andk(rememberthatthenotationfortheinterestrateontheTIBAIIPLUScalculatorisI/Y,notk).Thencomputethefuturevalueoftheannuity(FVonthecalculator).Foraseriesoffour$500end-of-year(ordinaryannuity)paymentswheren=4andk=5percent,thecomputationisasfollows:

ThespreadsheetthatfollowssolvesthissamefuturevalueproblemusingtheFVfunction.

Inthefinancialcalculatorandspreadsheetinputs,notethatthepaymentiskeyedinasanegativenumbertoindicatethatthepaymentsarecashoutflows—thepaymentsflowoutfromthecompanyintoaninvestment.

The Present Value of an Ordinary Annuity Becauseannuitypaymentsareoftenpromised(aswithinterestonabondinvestment)orexpected(aswithcashinflowsfromaninvestmentinnewplantorequipment),itisimpor-tanttoknowhowmuchtheseinvestmentsareworthtoustoday.Forexample,assumethatthefinancialmanagerofBuy4Later,Inc.learnsofanannuitythatpromisestomakefourannualpaymentsof$500,beginningoneyearfromtoday.Howmuchshouldthecompanybewillingtopaytoobtainthatannuity?Theansweristhepresentvalueoftheannuity.Becauseanannuityisnothingmorethanaseriesofequalsingleamounts,wecould

calculate the present value of an annuitywith the present value formula for a singleamountandsumthetotals,butthatwouldbeacumbersomeprocess.Imaginecalculatingthepresentvalueofa50-yearannuity!Wewouldhavetofindthepresentvalueforeachofthe50annuitypaymentsandtotalthem.

Take Note Note that slight differences occur between the table solution and the algebraic, financial calculator, and spreadsheet solutions. This is because our financial tables round interest factors to four decimal places, whereas the other methods generally use many more significant figures for greater accuracy.


Step 1: sserP to clear previous values.

Step 2: sserP 1 , . Repeat

litnu END shows in the display to set the annual interest ratemode and to set the annuity payment to end of period mode.

Step 3: tupnI the values and compute.

0 5 4 500 Answer: 2,155.06

E9

$2,155.06


Fortunately,wecancalculatethepresentvalueofanannuityinonestepwiththefol-lowingformula:

ThePresentValueofanAnnuityFormulaAlgebraicMethod

PVA = PMT × 1 − 1(1 + k)n

k

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

(8-4a)

where: PVA=PresentValueofanAnnuity


k=Discountrateperperiodexpressedasadecimal


Usingourexampleofafour-yearordinaryannuitywithpaymentsof$500peryearanda5percentdiscountrate,wesolveforthepresentvalueoftheannuityasfollows:

PVA = 500 × 1 − 1(1 + .05)4

.05

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

= 500 × 3.54595

= $1,772.97

Thepresent value of the four-year ordinary annuitywith equal yearly payments of$500ata5percentdiscountrateis$1,772.97.Wecanalsousethefinancialtableforthepresentvalueinterestfactorforanannuity

(PVIFA)tosolvepresentvalueofannuityproblems.ThePVIFAtableisfoundinTableIVintheAppendixattheendofthebook.Theformulaforthetablemethodfollows:

ThePresentValueofanAnnuityFormulaTableMethod

PVA=PMT×PVIFAk,n (8-4b)

where: PVA=PresentValueofanAnnuity


PVIFAk,n=PresentValueInterestFactorforanAnnuityfromTableIV

k=Discountrateperperiod


ApplyingEquation8-4b,wefindthatthepresentvalueofthefour-yearannuitywith$500equalpaymentsanda5percentdiscountrateisasfollows:2

PVA=500×PVIFA5%,4

=500×3.5460

=$1,773.00

2.The$.03differencebetweenthealgebraicresultandthetableformulasolutionisduetodifferencesinrounding.


Wemayalso solve for thepresent valueof an annuitywith a financial calculator orspreadsheet.Forthefinancialcalculator,simplykeyinthevaluesforthepayment,PMT,numberofpaymentperiods,n,andtheinterestrate,k—symbolizedbyI/YontheTIBAIIPLUScalculator—andaskthecalculatortocomputePVA(PVonthecalculator).Fortheseriesoffour$500paymentswheren=4andk=5percent,thecomputationfollows:

ThePVfunctioninthespreadsheetbelowalsosolvesthispresentvalueofanordinaryannuityproblem.

Thefinancialcalculatorandspreadsheetpresentvalueresultsaredisplayedasnegativenumbers.Aninvestment(acashoutflow)of$1,772.98wouldearna5percentannualrateofreturnif$500isreceivedattheendofeachyearforthenextfouryears.

Future and Present Values of Annuities Due Sometimeswemustdealwithannuities inwhich theannuitypaymentsoccur at thebeginningof eachperiod.Theseareknownasannuitiesdue, in contrast toordinaryannuitiesinwhichthepaymentsoccurredattheendofeachperiod,asdescribedintheprecedingsection.Annuities due aremore likely to occurwhen doing future value of annuity (FVA)

problemsthanwhendoingpresentvalueofannuity(PVA)problems.Today,forinstance,youmaystartaretirementprogram,investingregularequalamountseachmonthoryear.Calculating the amount youwould accumulatewhen you reach retirement agewould




litnu END shows in the display to set the annual interest ratemode and to set the annuity payment to end of period mode.


5 4 500 Answer: –1,772.98

E10

–$1,772.98


beafuturevalueofanannuitydueproblem.Evaluatingthepresentvalueofapromisedorexpectedseriesofannuitypaymentsthatbegantodaywouldbeapresentvalueofanannuitydueproblem.Thisislesscommonbecausecarandmortgagepaymentsalmostalwaysstartattheendofthefirstperiodmakingthemordinaryannuities.WheneveryourunintoanFVAoraPVAofanannuitydueproblem,theadjustment

neededisthesameinbothcases.UsetheFVAorPVAofanordinaryannuityformulashownearlier,thenmultiplyyouranswerby(1+k).WemultiplytheFVAorPVAfor-mulaby(1+k)becauseannuitiesduehaveannuitypaymentsearninginterestoneperiodsooner.So,higherFVAandPVAvalues resultwithanannuitydue.Thefirstpaymentoccurssoonerinthecaseofafuturevalueofanannuitydue.Inpresentvalueofannu-itydueproblems,eachannuitypaymentoccursoneperiodsooner,sothepaymentsarediscountedlessseverely.InourSaveComexample,thefuturevalueofa$500ordinaryannuity,withk=5%

andn=4,was$2,155.06.Ifthe$500paymentsoccurredatthebeginningofeachperiodinsteadofattheend,wewouldmultiply$2,155.06by1.05(1+k=1+.05).Theproduct,$2,262.81,isthefuturevalueoftheannuitydue.InourearlierBuy4Later,Inc.,example,wefoundthatthepresentvalueofa$500ordinaryannuity,withk=5%andn=4,was$1,772.97.Ifthe$500paymentsoccurredatthebeginningofeachperiodinsteadofattheend,wewouldmultiply$1,772.97by1.05(1+k=1+.05)andfindthatthepresentvalueofBuy4Later’sannuitydueis$1,861.62.Thefinancialcalculatorsolutionsfortheseannuitydueproblemsareshownnext.

Hereisthespreadsheetsolution.Notehowthe[type]valueissetto“1”toindicatethatthisisanannuitydueproblem.

Future Value of a Four-Year, $500 Annuity Due, k = 5%




command until the display shows BGN, to set to the annualinterest rate mode and to set the annuity payment to beginning of period mode.

Step 3: tupnI the values for the annuity due and compute.

5 4 500 Answer: 2,262.82

Present Value of a Four-Year, $500 Annuity Due, k = 5%






5 4 500 Answer: –1,861.62

E10

$2,262.82

This Excel spreadsheet computes the future value of a 4 year annual annuity due of $500 using a 5% annual discount rate. Note that the FV function inputs the $500 annuity payment as negative because you would pay out that amount at the beginning of each year to have $2,262.82 in 5 years.


Hereisthespreadsheetsolution.Notehowthe[type]valueisagainsetto“1”toindi-catethatthisisanannuitydueproblem.

Inthissectionwediscussedordinaryannuitiesandannuitiesdueandlearnedhowtocomputethepresentandfuturevaluesoftheannuities.Next,wewilllearnwhataperpe-tuityisandhowtosolveforitspresentvalue.

Perpetuities Anannuitythatgoesonforeveriscalledaperpetualannuityoraperpetuity.Perpetuitiescontainaninfinitenumberofannuitypayments.Anexampleofaperpetuityisthedivi-dendstypicallypaidonapreferredstockissue.Thefuturevalueofperpetuitiescannotbecalculated,butthepresentvaluecanbe.We

startwiththepresentvalueofanannuityformula,Equation8-3a.

PVA = PMT × 1 − 1(1 + k)n

k

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

Nowimaginewhathappensintheequationasthenumberofpayments(n)getslargerandlarger.The(1+k)ntermwillgetlargerandlarger,andasitdoes,itwillcausethe1/(1+k)nfractiontobecomesmallerandsmaller.Asnapproachesinfinity,the(1+k)n

Future Value of a Four-Year, $500 Annuity Due, k = 5%






5 4 500 Answer: 2,262.82

Present Value of a Four-Year, $500 Annuity Due, k = 5%






5 4 500 Answer: –1,861.62

E10

–$1,861.62

This Excel spreadsheet computes the present value of a 4 year annual annuity due of $500 using a 5% annual discount rate. Note that the PV function displays $1,861.62 as a negative value because you would be willing to pay out that amount now to receive $500 at the beginning of each year for 4 years with a 5% annual discount rate.


termbecomesinfinitelylarge,andthe1/(1+k)ntermapproacheszero.Theentireformulareducestothefollowingequation:

PresentValueofPerpetuity

PVP = PMT × 1 − 0k

⎡⎣⎢

⎤⎦⎥

or

PVP = PMT × 1k

⎛⎝⎜

⎞⎠⎟ (8-5)

where: PVP=PresentValueofaPerpetuity

k=Discountrateexpressedasadecimal

Neitherthetablemethodnorthefinancialcalculatorcansolveforthepresentvalueofaperpetuity.ThisisbecausethePVIFAtabledoesnotcontainvaluesforinfinityandthefinancialcalculatordoesnothaveaninfinitykey.Suppose you had the opportunity to buy a share of preferred stock that pays $70

peryearforever.Ifyourrequiredrateofreturnis8percent,whatisthepresentvalueofthepromiseddividendstoyou?Inotherwords,givenyourrequiredrateofreturn,howmuchshouldyoubewillingtopayforthepreferredstock?Theanswer,foundbyapplyingEquation8-5,follows:

PVP = PMT × 1k

⎛⎝⎜

⎞⎠⎟

= $70 × 1.08

⎛⎝⎜

⎞⎠⎟

= $875

Thepresentvalueofthepreferredstockdividends,withakof8percentandapaymentof$70peryearforever,is$875.

Present Value of an Investment with Uneven Cash Flows Unlikeannuitiesthathaveequalpaymentsovertime,manyinvestmentshavepaymentsthatareunequalovertime.Thatis,someinvestmentshavepaymentsthatvaryovertime.When theperiodicpaymentsvary,we say that the cash flowstreamsareuneven.Forinstance,aprofessionalathletemaysignacontractthatprovidesforanimmediate$7mil-lionsigningbonus,followedbyasalaryof$2millioninyear1,$4millioninyear2,then$6millioninyears3and4.Whatisthepresentvalueofthepromisedpaymentsthattotal$25million?Assumeadiscountrateof8percent.ThepresentvaluecalculationsareshowninTable8-1.AsweseefromTable8-1,wecalculatethepresentvalueofanunevenseriesof

cashflowsbyfindingthepresentvalueofasingleamountforeachseriesandsum-mingthetotals.Wecanalsouseafinancialcalculatortofindthepresentvalueofthisunevenseries

ofcashflows.TheworksheetmodeoftheTIBAIIPLUScalculatorisespeciallyhelpfulinsolvingproblemswithunevencashflows.TheCdisplayshowseachcashpaymentfollowingCF0, the initialcashflow.TheFdisplaykey indicates thefrequencyof thatpayment.Thekeystrokesfollow.


Weseefromthecalculatorkeystrokesthatwearesolvingforthepresentvalueofasingleamountforeachpaymentintheseriesexceptforthelasttwopayments,whicharethesame.ThevalueofF03,thefrequencyofthethirdcashflowaftertheinitialcashflow,was2insteadof1becausethe$6millionpaymentoccurredtwiceintheseries(inyears3and4).Wehaveseenhowtocalculatethefuturevalueandpresentvalueofannuities,thepresent

valueofaperpetuity,andthepresentvalueofaninvestmentwithunevencashflows.Nowweturntotimevalueofmoneyproblemsinwhichwesolvefork,n,ortheannuitypayment.

Take Note We used the NPV (net present value) key on our calculator to solve this problem. NPV will be discussed in Chapter 10.

TABLE 8-1 The Present Value of an Uneven Stream of Cash Flows

Time Cash Flow PV of Cash Flow

t0 $7,000,000 × =$7,000,000 11.08

$7,000,0000

t1 $2,000,000 × =$2,000,000 11.08

$1,851,8521

t2 $4,000,000 × =$4,000,000 11.08

$3,429,3552

t3 $6,000,000 × =$6,000,000 11.08

$4,762,9933

t4 $6,000,000 × =$6,000,000 11.08

$4,410,1794

Sum of the PVs = $21,454,380

TI BAII PLUS Financial Calculator PV SolutionUneven Series of Cash Flows

Keystrokes Display

CF0 = old contents

CF0 = 0.00

7000000 7,000,000.00

2000000 C01 = 2,000,000.00

F01 = 1.00

4000000 C02 = 4,000,000.00

F02 = 1.00

6000000 C03 = 6,000,000.00

2 F03 = 2.00

I = 0.00

8 I = 8.00

NPV = 21,454,379.70


Future Value Interest Factors, Compounded at k percent for n Periods, Part of Table I Number of Periods, Interest Rate, k n 0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 18 1.0000 1.1961 1.4282 1.7024 2.0258 2.4066 2.8543 3.3799 3.9960 4.7171 5.5599 19 1.0000 1.2081 1.4568 1.7535 2.1068 2.5270 3.0256 3.6165 4.3157 5.1417 6.1159 20 1.0000 1.2202 1.4859 1.8061 2.1911 2.6533 3.2071 3.8697 4.6610 5.6044 6.7275

25 1.0000 1.2824 1.6406 2.0938 2.6658 3.3864 4.2919 5.4274 6.8485 8.6231 10.8347

Special Time Value of Money Problems Financialmanagersoften face timevalueofmoneyproblemsevenwhen theyknowboththepresentvalueandthefuturevalueofaninvestment.Inthosecases,financialmanagersmaybeaskedtofindoutwhatreturnaninvestmentmade—thatis,whattheinterestrateisontheinvestment.Stillothertimesfinancialmanagersmustfindeitherthenumberofpaymentperiodsortheamountofanannuitypayment.Inthenextsec-tion,wewilllearnhowtosolveforkandn.Wewillalsolearnhowtofindtheannuitypayment(PMT).

Finding the Interest Rate Financialmanagersfrequentlyhavetosolvefortheinterestrate,k,whenfirmsmakealong-terminvestment.Themethodofsolvingforkdependsonwhethertheinvestmentisasingleamountoranannuity.

Finding k for a Single-Amount Investment Financialmanagersmayneedtodeterminehowmuchperiodicreturnaninvestmentgen-eratedovertime.Forexample,imaginethatyouareheadofthefinancedepartmentofGrabLand, Inc.Say thatGrabLandpurchasedahouseonprime land20yearsago for$40,000.Recently,GrabLandsoldthepropertyfor$106,131.Whataverageannualrateofreturndidthefirmearnonits20-yearinvestment?First,thefuturevalue—orendingamount—ofthepropertyis$106,131.Thepresent

value—thestartingamount—is$40,000.Thenumberofperiods,n, is20.Armedwiththosefacts,youcouldsolvethisproblemusingthetableversionofthefuturevalueofasingleamountformula,Equation8-1b,asfollows:

FV PV FVIF

$106,131 $40,000 FVIF

$106,131 $40,000 FVIF

2.6533 FVIF

k, n

k ?, n 20

k ?, n 20

k ?, n 20

( )

( )

= ×

= ×

÷ =

=

= =

= =

= =

NowfindtheFVIFvalueinTableI,showninpartbelow.ThewholetableisintheAppendixattheendofthebook.Youknown=20,sofindthen=20rowontheleft-handsideofthetable.YoualsoknowthattheFVIFvalueis2.6533,somoveacrossthen=20rowuntilyoufind(orcomecloseto)thevalue2.6533.Youfindthe2.6533valueinthek=5%column.Youdiscover,then,thatGrabLand’spropertyinvestmenthadaninterestrateof5percent.


SolvingforkusingtheFVIFtableworkswellwhentheinterestrateisawholenumber,butitdoesnotworkwellwhentheinterestrateisnotawholenumber.Tosolvefortheinterestrate,k,werewritethealgebraicversionofthefuturevalueofasingleamountformula,Equation8-1a,tosolvefork:

TheRateofReturn,k

k = FV

PV⎛⎝⎜

⎞⎠⎟

1n

− 1 (8-6)

where: k=Rateofreturnexpressedasadecimal

FV=FutureValue

PV=PresentValue

n=Numberofcompoundingperiods

Let’suseEquation8-6tofindtheaverageannualrateofreturnonGrabLand’shouseinvestment.Recallthatthecompanyboughtit20yearsagofor$40,000andsolditrecentlyfor$106,131.WesolveforkapplyingEquation8-6asfollows:

k = FVPV

⎛⎝⎜

⎞⎠⎟

1n− 1

= $106,131$40,000

⎛⎝⎜

⎞⎠⎟

120− 1

= 2.653275.05 − 1

= 1.05 − 1

= .05, or 5%

Equation8-6willfindanyrateofreturnorinterestrategivenastartingvalue,PV,anendingvalue,FV,andanumberofcompoundingperiods,n.Tosolveforkwithafinancialcalculator,keyinalltheothervariablesandaskthecal-

culatortocomputek(depictedasI/Yonyourcalculator).ForGrabLand’shouse-buyingexample,thecalculatorsolutionfollows:

ThespreadsheetRATEfunctioncanalsofindthisvalue.



Step 2: sserP 1 , .


40000 106131 20 Answer: 5.00


Rememberwhenusingthefinancialcalculatororspreadsheettosolvefortherateofreturn,youmustentercashoutflowsasanegativenumber.Inourexample,the$40,000PVisenteredasanegativenumberbecauseGrabLandspentthatamounttoinvestinthehouse.

Finding k for an Annuity Investment Financialmanagersmayneedtofindtheinterestrateforanannuityinvestmentwhenthey know the starting amount (PVA), n, and the annuity payment (PMT), but theydonotknow the interest rate,k.Forexample, supposeGrabLandwanteda15-year,$100,000amortizedloanfromabank.Anamortizedloanisaloanthatispaidoffinequalamountsthatincludeprincipalaswellasinterest.3Accordingtothebank,Grab-Land’spaymentswillbe$12,405.89peryearfor15years.Whatinterestrateisthebankchargingonthisloan?TosolveforkwhentheknownvaluesarePVA(the$100,000loanproceeds),n(15),

andPMT(theloanpayments$12,405.89),westartwiththepresentvalueofanannuityformula,Equation8-3b,asfollows:

PVA PMT PVIFA

$100,000 $12,405.89 PVIFA

8.0607 PVIFA

k, n

k ?, n 15

k ?, n 15

= × ( )

= ×

=

= =

= =

( )

NowrefertothePVIFAvaluesinTableIV,showninpartonthenextpage.ThewholetableisintheAppendixattheendofthebook.Youknown=15,sofindthen=15rowontheleft-handsideofthetable.YouhavealsodeterminedthatthePVIFAvalueis8.0607($100,000/$12,405=8.0607),somoveacross then=15rowuntilyoufind(orcomecloseto)thevalueof8.0607.Inourexample,thelocationonthetablewheren=15andthePVIFAis8.0607isinthek=9%column,sotheinterestrateonGrabLand’sloanis9percent.

3.AmortizecomesfromtheLatinwordmortalis,whichmeans“death.”Youwillkillofftheentireloanaftermakingthescheduledpayments.

E10

5%


Tosolvethisproblemwithafinancialcalculator,keyinallthevariablesbutk,andaskthecalculatortocomputek(depictedasI/YontheTIcalculator)asfollows:

AgainherethespreadsheetRATEfunctioncancomputethisvalue.

Notehowinthefinancialcalculatorsolutionwetooktheperspectiveoftheborrower,makingthe$12,405.89PMTnegativeandthe$100,000loanamountPVpositive.Inthespreadsheetwetookthebank’sperspective,showingtheloanamountasnegativeandthepaymentsaspositive.Youcanseethatbothapproachesgavethesame9%answerfortheinterestrate.

Finding the Number of Periods Supposeyoufoundaninvestmentthatofferedyouareturnof6percentperyear.Howlongwouldittakeyoutodoubleyourmoney?Inthisproblemyouarelookingforn,thenumberofcompoundingperiodsitwilltakeforastartingamount,PV,todoubleinsize(FV=2×PV).

Present Value Interest Factors for an Annuity, Discounted at k percent for n Periods, Part of Table IV Number of Periods, Interest Rate, k n 0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 13 13.0000 12.1337 11.3484 10.6350 9.9856 9.3936 8.8527 8.3577 7.9038 7.4869 7.1034 14 14.0000 13.0037 12.1062 11.2961 10.5631 9.8986 9.2950 8.7455 8.2442 7.7862 7.3667 15 15.0000 13.8651 12.8493 11.9379 11.1184 10.3797 9.7122 9.1079 8.5595 8.0607 7.6061

16 16.0000 14.7179 13.5777 12.5611 11.6523 10.8378 10.1059 9.4466 8.8514 8.3126 7.8237



Step 2: Press 1 , . Repeat

Step 3: Input the values and compute.

100000 15 12405.89 Answer: 9.00

until END shows in the display. Press after you see END in the display.

E10

9.0%

Finding k for an Ordinary Annuity Investment


Thisproblemcanalsobesolvedonafinancialcalculatorquitequickly.Justkeyinalltheknownvariables(PV,FV,andI/Y)andaskthecalculatortocomputen.

ThespreadsheetNPERfunctionsolvesforthenumberofperiodsforthisproblem.

Tofindninourexample,startwiththeformulaforthefuturevalueofasingleamountandsolvefornasfollows:

FV PV FVIF

2 PV PV FVIF

2.0 FVIF

k, n

k 6%, n ?

k 6%, n ?

( )

( )

= ×

× = ×

=

= =

= =

NowrefertotheFVIFvalues,shownbelowinpartofTableI.Youknowk=6%,soscanacrossthetoprowtofindthek=6%column.KnowingthattheFVIFvalueis2.0,movedownthek=6%columnuntilyoufind(orcomecloseto)thevalue2.0.Notethatitoccursintherowinwhichn=12.Therefore,ninthisproblem,andthenumberofperiodsitwouldtakeforthevalueofaninvestmenttodoubleat6percentinterestperperiod,is12.

E10

11.90

Future Value Interest Factors, Compounded at k percent for n Periods, Part of Table I Number of Periods, Interest Rate, k n 0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11 1.0000 1.1157 1.2434 1.3842 1.5395 1.7103 1.8983 2.1049 2.3316 2.5804 2.8531 12 1.0000 1.1268 1.2682 1.4258 1.6010 1.7959 2.0122 2.2522 2.5182 2.8127 3.1384 13 1.0000 1.1381 1.2936 1.4685 1.6651 1.8856 2.1329 2.4098 2.7196 3.0658 3.4523

14 1.0000 1.1495 1.3195 1.5126 1.7317 1.9799 2.2609 2.5785 2.9372 3.3417 3.7975



Step 2: Press 1 , .


1 2 6 Answer: 11.90


Inourexamplen=12when$1ispaidoutand$2receivedwitharateofinterestof6per-cent.Thatis,ittakesapproximately12yearstodoubleyourmoneyata6percentannualrateofinterest.

Solving for the Payment Lendersandfinancialmanagersfrequentlyhavetodeterminehowmucheachpayment—orinstallment—willneedtobetorepayanamortizedloan.Forexample,supposeyouareabusinessownerandyouwanttobuyanofficebuildingforyourcompanythatcosts$200,000.Youhave$50,000foradownpaymentandthebankwilllendyouthe$150,000balanceata6percentannualinterestrate.Howmuchwilltheannualpaymentsbeifyouobtaina10-yearamortizedloan?Aswesawearlier,anamortizedloanisrepaidinequalpaymentsovertime.Theperiod

oftimemayvary.Let’sassumeinourexamplethatyourpaymentswilloccurannually,sothatattheendofthe10-yearperiodyouwillhavepaidoffallinterestandprincipalontheloan(FV=0).Becausethepaymentsareregularandequalinamount,thisisanannuityproblem.The

presentvalueoftheannuity(PVA)isthe$150,000loanamount,theannualinterestrate(k)is6percent,andnis10years.Thepaymentamount(PMT)istheonlyunknownvalue.BecauseallthevariablesbutPMTareknown,theproblemcanbesolvedbysolvingfor

PMTinthepresentvalueofanannuityformula,equation8-4a,asfollows:

PVA = PMT × 1 − 1(1 + k)n

k

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

$150,000 = PMT × 1 − 1(1.06)10

.06

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

$150,000 = PMT × 7.36009

$150,0007.36009

= PMT

$20,380.19 = PMT

Wesee,then,thatthepaymentforanannuitywitha6percentinterestrate,annof10,andapresentvalueof$150,000is$20,380.19.WecanalsosolveforPMTusingthetableformula,Equation8-4b,asfollows:

PVA PMT PVIFA

$150,000 PMT PVIFA

$150,000 PMT 7.3601 (look up PVIFA, Table IV)

$150,0007.3601

PMT

$20,380.16 PMT (note the $0.03 rounding error)

k, n

6%, 10 years( )

( )= ×

= ×

= ×

=

=

Thetableformulashowsthatthepaymentforaloanwiththepresentvalueof$150,000atanannualinterestrateof6percentandannof10is$20,380.16.


With the financialcalculator, simplykey inall thevariablesbutPMTandhave thecalculatorcomputePMTasfollows:

The financial calculator and the spreadsheet that followswill display the payment,$20,380.19, as a negative number because it is a cash outflow for the borrowerwhoreceivedthe$150,000loanamount.

Loan Amortization Aseachpaymentismadeonanamortizedloan,theinterestdueforthatperiodispaid,alongwith a repayment of someof the principal thatmust also be “killed off.”Afterthelastpaymentismade,alltheinterestandprincipalontheloanhavebeenpaid.Thisstep-by-steppaymentoftheinterestandprincipalowedisoftenshowninanamortiza-tion table.Theamortizationtablefortheten-year6percentannualinterestrateloanof$150,000thatwasdiscussedintheprevioussectionisshowninTable8-2.Theannualpayment,calculatedintheprevioussection,is$20,380.19.WeseefromTable8-2howthebalanceonthe$150,000loaniskilledoffalittle

each year until the balance at the end of year 10 is zero.The payments reflect anincreasing amount going toward principal and a decreasing amount going towardinterestovertime.

Compounding More Than Once per Year So far in this chapter,we have assumed that interest is compoundedannually. How-ever,thereisnothingmagicalaboutannualcompounding.Manyinvestmentspayinterestthatiscompoundedsemiannually,quarterly,orevendaily.Mostbanks,savingsandloan

F10

–$20,380.19



Step 2: Press 1 , . Repeat

command until the display shows END, to set to the annual interest rate mode and to set the annuity payment to end of period mode.

Step 3: Input the values for the ordinary annuity and compute.

150,000 6 10 Answer: –20,380.19


associations,andcreditunions,forexample,compoundinterestontheirdepositsmorefrequentlythanannually.Supposeyoudeposited$100inasavingsaccountthatpaid12percentannualinterest,

compoundedannually.Afteroneyearyouwouldhave$112inyouraccount($112=$100×1.121).Now,however,let’sassumethebankusedsemiannual compounding.Withsemiannual

compoundingyouwouldreceivehalfayear’sinterest(6percent)aftersixmonths.Inthesecondhalfoftheyear,youwouldearninterestbothontheinterestearnedinthefirstsixmonthsandontheoriginalprincipal.Thetotalinterestearnedduringtheyearona$100investmentat12percentannualinterestwouldbe:

$ 6.00 (interest for the first six months) + $ .36 (interest on the $6 interest during the second 6 months)4

+ $ 6.00 (interest on the principal during the second six months) = $ 12.36 total interest in year 1

Attheendoftheyear,youwillhaveabalanceof$112.36iftheinterestiscompoundedsemiannually,comparedwith$112.00withannualcompounding—adifferenceof$.36.Here’showtofindanswerstoproblemsinwhichthecompoundingperiodislessthana

year:Applytherelevantpresentvalueorfuturevalueequation,butadjustkandnsotheyreflecttheactualcompoundingperiods.Todemonstrate, let’s lookatourexampleofa$100deposit inasavingsaccountat

12percentforoneyearwithsemiannualcompoundedinterest.Becausewewanttofind

4.The$.36wascalculatedbymultiplying$6byhalfof12%:$6.00×.06=$.36.

TABLE 8-2 Amortization Table for a $150,000 Loan, 6 percent Annual Interest Rate, 10-Year Term

Loan Amortization Schedule

Amount Borrowed: $150,000 Interest Rate: 6.0% Term: 10 years Required Payments: $20,380.19 (found using Equation 8-4a) Col. 1 Col. 2 Col. 3 Col. 4 Col. 5 Col. 1 x .06 Col. 2 - Col. 3 Col. 1 - Col. 4 Beginning Total Payment Payment Ending Year Balance Payment of Interest of Principal Balance 1 $ 150,000.00 $ 20,380.19 $ 9,000.00 $ 11,380.19 $ 138,619.81 2 $ 138,619.81 $ 20,380.19 $ 8,317.19 $ 12,063.01 $ 126,556.80 3 $ 126,556.80 $ 20,380.19 $ 7,593.41 $ 12,786.79 $ 113,770.02 4 $ 113,770.02 $ 20,380.19 $ 6,826.20 $ 13,553.99 $ 100,216.02 5 $ 100,216.02 $ 20,380.19 $ 6,012.96 $ 14,367.23 $ 85,848.79 6 $ 85,848.79 $ 20,380.19 $ 5,150.93 $ 15,229.27 $ 70,619.52 7 $ 70,619.52 $ 20,380.19 $ 4,237.17 $ 16,143.02 $ 54,476.50 8 $ 54,476.50 $ 20,380.19 $ 3,268.59 $ 17,111.60 $ 37,364.90 9 $ 37,364.90 $ 20,380.19 $ 2,241.89 $ 18,138.30 $ 19,226.60 10 $ 19,226.60 $ 20,380.19 $ 1,153.60 $ 19,226.60 $ 0.00


outwhatthefuturevalueofasingleamountwillbe,weusethatformulatosolveforthefuturevalueoftheaccountafteroneyear.Next,wedividetheannualinterestrate,12per-cent,bytwobecauseinterestwillbecompoundedtwiceeachyear.Then,wemultiplythenumberofyearsn(oneinourcase)bytwobecausewithsemiannualinteresttherearetwocompoundingperiodsinayear.Thecalculationfollows:

FV PV (1 k/2)

$100 (1 .12/2)

$100 (1 .06)

$100 1.1236

$112.36

n 2

1 2

2

= × +

= × +

= × +

= ×

=

×

×

Thefuturevalueof$100afteroneyear,earning12percentannualinterestcompoundedsemiannually,is$112.36.Tousethetablemethodforfindingthefuturevalueofasingleamount,findtheFVIFk,n

inTableIintheAppendixattheendofthebook.Then,dividethekbytwoandmultiplythenbytwoasfollows:

FV PV FVIF

$100 FVIF

$100 FVIF

$100 1.1236

$112.36

k / 2, n 2

12% / 2, 1 2 periods

6%, 2 periods

( )

( )

( )= ×

= ×

= ×

= ×

=

×

×

Tosolvetheproblemusingafinancialcalculator,dividethek(representedasI/YontheTIBAIIPLUScalculator)bytwoandmultiplythenbytwo.Next,keyinthevariablesasfollows:

Thefuturevalueof$100investedfortwoperiodsat6percentperperiodis$112.36.5

5.Notethattheinterestrateandnumberofperiodsareexpressedinsemi-annualtermswhenusingtheTIBAIIPLUScalculatorandtheFVfunctionintheExcelspreadsheet.Weusedthesemiannualinterestrateof6percent,nottheannualinterestrateof12percent.Similarly,nwasexpressedasthenumberofsemiannualperiods,twoinoneyear.



Step 2: Press 1 , .


100 6 2 Answer: 112.36


Thisfuturevalueiscomputedwithaspreadsheethere.

Othercompoundingrates,suchasquarterlyormonthlyrates,canbefoundbymodi-fying the applicable formula to adjust for the compounding periods.With a quarterlycompoundingperiod,then,annualkshouldbedividedbyfourandannualnmultipliedbyfour.Formonthlycompounding,annualkshouldbedividedbytwelveandannualnmultipliedbytwelve.Similaradjustmentscouldbemadeforothercompoundingperiods.

Annuity Compounding Periods Manyannuityproblemsalsoinvolvecompoundingordiscountingperiodsoflessthanayear.Forinstance,supposeyouwanttobuyacarthatcosts$20,000.Youhave$5,000foradownpaymentandplantofinancetheremaining$15,000at6percentannualinterestforfouryears.Whatwouldyourmonthlyloanpaymentsbe?First,changethestatedannualrateofinterest,6percent,toamonthlyratebydividing

by12(6%/12=1/2%or.005).Second,multiplythefour-yearperiodby12toobtainthenumberofmonthsinvolved(4×12=48months).Nowsolvefortheannuitypaymentsizeusingtheannuityformula.Inourcase,weapplythepresentvalueofanannuityformula,equation8-4a,asfollows:

PVA = PMT 1 1(1 + k)n

k

$15,000 = PMT 1 1(1.005)48

.005

$15,000 = PMT 42.5803

$15,00042.5803

= PMT

$352.28 = PMT

Themonthly payment on a $15,000 car loanwith a 6 percent annual interest rate(.5percentpermonth)forfouryears(48months)is$352.28.

E11

$112.36

This Excel spreadsheet computes the future value of $100 invested for 1 year at a 12% annual interest rate (6% per semiannual period) for 1 year (2 semiannual periods). Note that the FV function interest rate is divided by 2 and the number of years multiplied by 2 for this semiannual compounding problem and that it inputs the $100 present value as a negative number since that amount is being paid out.


SolvingthisproblemwiththePVIFAtableinTableIVintheAppendixattheendofthebookwouldbedifficultbecausethe.5percentinterestrateisnotlistedinthePVIFAtable.IfthePVIFAwerelisted,wewouldapplythetableformula,maketheadjustmentstoreflectthemonthlyinterestrateandthenumberofperiods,andsolveforthepresentvalueoftheannuity.Onafinancialcalculator,wewouldfirstadjustthekandntoreflectthesametime

period—monthly,inourcase—andtheninputtheadjustedvariablestosolvetheproblemasfollows:

Thefollowingspreadsheetalsosolvesforthemonthlypaymentofthisamortizedloan.

The interest rate we entered was not the 6 percent rate per year but rather the.5percentratepermonth.Weenterednot thenumberofyears, four,butrather thenumberofmonths,48.Becausewewereconsistentinenteringthekandnvaluesinmonthlyterms,thecalculatorandspreadsheetgaveusthecorrectmonthlypaymentof–352.28(anoutflowof$352.28permonth).ThevaluesofRATEandNPERweresimilarlyadjustedinthespreadsheet.

Continuous Compounding Theeffectofincreasingthenumberofcompoundingperiodsperyearistoincreasethefuturevalueoftheinvestment.Themorefrequentlyinterestiscompounded,thegreaterthe future value.The smallest compounding period is usedwhenwe do continuouscompounding—compoundingthatoccurseverytinyunitoftime(thesmallestunitoftimeimaginable).




dnammoc until the display shows END, to set to the annualinterest rate mode and to set the annuity payment to end of period mode.

Step 3: IInput the values for the ordinary annuity and compute.

15,000 .5 48 Answer: –352.28

E10

–$352.28


Recallour$100deposit inanaccountat12percent foroneyearwithannualcom-pounding.Attheendofyear1,ourbalancewas$112.Withsemiannualcompounding,theamountincreasedto$112.36.Whencontinuouscompoundingisinvolved,wecannotdividekbyinfinityandmul-

tiplynbyinfinity.Instead,weusetheterme,whichyoumayrememberfromyourmathclass.Wedefineeasfollows:

e = lim 1 + 1h

⎡⎣⎢

⎤⎦⎥

h

≈ 2.71828

h →∞

Thevalueofeisthenaturalantilogof1andisapproximatelyequalto2.71828.Thisnumber isoneof those likepi (approximatelyequal to3.14159),whichwecanneverexpressexactlybutcanapproximate.Usinge, theformulaforfindingthefuturevalueofagivenamountofmoney,PV,investedatannualrate,k,fornyears,withcontinuouscompounding,isasfollows:

FutureValuewithContinuousCompounding

FV=PV×e(k×n) (8-7)

wherek(expressedasadecimal)andnareexpressedinannualterms.ApplyingEquation8-7toourexampleofa$100depositat12percentannualinter-

estwithcontinuouscompounding,attheendofoneyearwewouldhavethefollowingbalance:

FV=$100×2.71828(.12×1)

=$112.75

The future valueof $100, earning12percent annual interest compounded continu-ously,is$112.75.Asthissectiondemonstrates,thecompoundingfrequencycanimpactthevalueofan

investment.Investors,then,shouldlookcarefullyatthefrequencyofcompounding.Isitannual,semiannual,quarterly,daily,orcontinuous?Otherthingsbeingequal,themorefrequentlyinterestiscompounded,themoreinteresttheinvestmentwillearn.

What’s Next Inthischapterweinvestigatedtheimportanceofthetimevalueofmoneyinfinancialdecisionsandlearnedhowtocalculatepresentandfuturevaluesforasingleamount,forordinaryannuities,andforannuitiesdue.Wealsolearnedhowtosolvespecialtimevalueofmoneyproblems,suchasfindingtheinterestrateorthenumberofperiods.Theskillsacquiredinthischapterwillbeappliedinlaterchapters,asweevaluatepro-

posedprojects,bonds,andpreferredandcommonstock.Theywillalsobeusedwhenweestimatetherateofreturnexpectedbysuppliersofcapital.Inthenextchapter,wewillturntothecostofcapital.


Summary n 1.Explain the time value of money and its importance in the

business world. Moneygrowsovertimewhenitearnsinterest.Moneyexpectedorpromisedinthefutureisworthlessthanthesameamountofmoneyinhandtoday.Thisisbecausewelosetheopportunitytoearninterestwhenwehavetowaittoreceivemoney.Similarly,moneyweoweislessburdensomeifitistobepaidinthefutureratherthannow.Theseconceptsareattheheartofinvestmentandvaluationdecisionsofafirm.

2.Calculate the future value and present value of a single amount. Tocalculatethefuturevalueandthepresentvalueofasingledollaramount,wemayusethealgebraic,table,orcalculatormethods.Futurevalueandpresentvaluearemirrorimagesofeachother.Theyarecompoundinganddiscounting,respectively.Withfuturevalue,increasesinkandnresultinanexponentialincreaseinfuturevalue.Increasesinkandnresultinanexponentialdecreaseinpresentvalue.

3.Find the future and present values of an annuity.Annuitiesareaseriesofequalcashflows.Anannuitythathaspaymentsthatoccurattheendofeachperiodisanordinaryannuity.Anannuitythathaspaymentsthatoccuratthebeginningofeachperiodisanannuitydue.Aperpetuityisaperpetualannuity.Tofindthefutureand

presentvaluesofanordinaryannuity,wemayusethealgebraic,table,orfinancialcalculatormethod.Tofindthefutureandpresentvaluesofanannuitydue,multiplytheapplicableformulaby(1+k)toreflecttheearlierpayment.

4.Solve time value of money problems with uneven cash flows. Tosolvetimevalueofmoneyproblemswithunevencashflows,wefindthevalueofeachpayment(eachsingleamount)inthecashflowseriesandtotaleachsingleamount.Sometimestheserieshasseveralcashflowsofthesameamount.Ifso,calculatethepresentvalueofthosecashflowsasanannuityandaddthetotaltothesumofthepresentvaluesofthesingleamountstofindthetotalpresentvalueoftheunevencashflowseries.

5.Solve for the interest rate, number or amount of payments, or the number of periods in a future or present value problem. Tosolvespecialtimevalueofmoneyproblems,weusethepresentvalueandfuturevalueequationsandsolveforthemissingvariable,suchastheloanpayment,k,orn.Wemayalsosolveforthepresentandfuturevaluesofsingleamountsorannuitiesinwhichtheinterestrate,payments,andnumberoftimeperiodsareexpressedintermsotherthanayear.Themoreofteninterestiscompounded,thelargerthefuturevalue.

Equations Introduced in This Chapter nEquation8-1a. FutureValueofaSingleAmount—AlgebraicMethod:

FV=PV×(1+k)n


PV=PresentValue,thestartingamount,ororiginalprincipal

k=Rateofinterestperperiod(expressedasadecimal)


Equation8-1b. FutureValueofaSingleAmount—TableMethod:

FV=PV× ( )FVIFk, n


PV=PresentValue,thestartingamount

FVIFk,n=FutureValueInterestFactorgiveninterestrate,k,andnumberofperiods,n,fromTableI


Equation8-2a. PresentValueofaSingleAmount—AlgebraicMethod:

PV FV 1

(1 k)n= ×+

where: PV=PresentValue,thestartingamount

FV=FutureValue,theendingamount

k=Discountrateofinterestperperiod(expressedasadecimal)


Equation8-2b. PresentValueofaSingleAmount—TableMethod:

PV=FV×

PVIFk, n( ) where: PV=PresentValue

FV=FutureValue

PVIFk,n=PresentValueInterestFactorgivendiscountrate,k,andnumberofperiods,n,fromTableII

Equation8-3a. FutureValueofanAnnuity—AlgebraicMethod:

FVA = PMT × (1 + k)n− 1

k⎡

⎣⎢⎢

⎤

⎦⎥⎥

where:FVA=FutureValueofanAnnuity


k=Interestratepertimeperiodexpressedasadecimal


Equation8-3b. FutureValueofanAnnuity—TableMethod:

FVA=PMT×FVIFAk,n

where:FVA=FutureValueofanAnnuity


FVIFAk,n=FutureValueInterestFactorforanAnnuityfromTableIII

k=Interestrateperperiod


Equation8-4a. PresentValueofanAnnuity—AlgebraicMethod:

PVA = PMT × 1 − 1(1 + k)n

k

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

where:PVA=PresentValueofanAnnuity


k=Discountrateperperiodexpressedasadecimal



Equation8-4b. PresentValueofanAnnuity—TableMethod:

PVA=PMT×PVIFAk,n

where:PVA=PresentValueofanAnnuity


PVIFAk,n=PresentValueInterestFactorforanAnnuityfromTableIV

k=Discountrateperperiod


Equation8-5. PresentValueofaPerpetuity:

PVP = PMT 1

k

where:PVP=PresentValueofaPerpetuity

k=Discountrateexpressedasadecimal

Equation8-6. RateofReturn:

k = FV

PV⎛⎝⎜

⎞⎠⎟

1n

− 1

where: k=Rateofreturnexpressedasadecimal

FV=FutureValue

PV=PresentValue

n=Numberofcompoundingperiods

Equation8-7. FutureValuewithContinuousCompounding:

FV=PV×e(k×n)

where: FV=FutureValue

PV=PresentValue

e=Naturalantilogof1

k=Statedannualinterestrateexpressedasadecimal

n=Numberofyears

Self-Test nST-1. Jedisinvesting$5,000intoaneight-yearcertificateof

deposit(CD)thatpays6percentannualinterestwithannualcompounding.HowmuchwillhehavewhentheCDmatures?

ST-2. Timhasfounda2013Toyota4-Runneronsalefor$19,999.Thedealershipsaysthatitwillfinancetheentireamountwithaone-yearloan,andthemonthlypaymentswillbe$1,776.98.Whatistheannualizedinterestrateontheloan(themonthlyratetimes12)?

ST-3. Heidi’sgrandmotherdiedandprovidedinherwillthatHeidiwillreceive$100,000fromatrustwhenHeiditurns21yearsofage,10yearsfromnow.Iftheappropriatediscountrateis8percent,whatisthepresentvalueofthis$100,000toHeidi?

ST-4. ZackwantstobuyanewFordMustangautomobile.Hewillneedtoborrow$20,000togowithhisdownpaymentinordertoaffordthiscar.Ifcarloansareavailableata6percentannualinterestrate,whatwillZack’smonthlypaymentbeonafour-yearloan?

ST-5. Bridgetinvested$5,000inagrowthmutualfund,andin10yearsherinvestmenthadgrownto$15,529.24.WhatannualrateofreturndidBridgetearnoverthis10-yearperiod?

ST-6. IfTominvests$1,000ayearbeginningtodayintoaportfoliothatearnsa10percentreturnperyear,howmuchwillhehaveattheendof10years?(Hint:Recognizethatthisisanannuitydueproblem.)


Review Questions n1. Whatisthetimevalueofmoney?

2. Whydoesmoneyhavetimevalue?

3. Whatiscompoundinterest?Comparecompoundinteresttodiscounting.

4. Howispresentvalueaffectedbyachangeinthediscountrate?

5. Whatisanannuity?

6. Supposeyouareplanningtomakeregularcontributionsinequalpaymentstoaninvestmentfundforyourretirement.

Whichformulawouldyouusetofigureouthowmuchyourinvestmentswillbeworthatretirementtime,givenanassumedrateofreturnonyourinvestments?

7. Howdoescontinuouscompoundingbenefitaninvestor?

8. IfyouaredoingPVAandFVAproblems,whatdifferencedoesitmakeiftheannuitiesareordinaryannuitiesorannuitiesdue?

9. Whichformulawouldyouusetosolveforthepaymentrequiredforacarloanifyouknowtheinterestrate,lengthoftheloan,andtheborrowedamount?Explain.

Build Your Communication Skills nCS-1. Obtaininformationfromfourdifferentfinancial

institutionsaboutthetermsoftheirbasicsavingsaccounts.Comparetheinterestratespaidandthefrequencyofthecompounding.Foreachaccount,howmuchmoneywouldyouhavein10yearsifyoudeposited$100today?Writeaone-totwo-pagereportofyourfindings.

CS-2. Interviewamortgagelenderinyourcommunity.Writeabriefreportaboutthemortgageterms.Includeinyourdiscussioncommentsaboutwhatinterestratesarechargedondifferenttypesofloans,whyratesdiffer,andwhatfeesarechargedinadditiontotheinterestandprincipalthatmortgageesmustpaytothislender.Describetheadvantagesanddisadvantagesofsomeoftheloansoffered.

Problems nAlltheseproblemsmaybesolvedusingalgebra,tables,financialcalculator,orExcel.Thealgebra,table,andfinancialcalculatorapproachesarepresentedinthebodyofthischapter.ThosewhowishtosolvetheseproblemsusingExcelmaylearntheskillstodosoforeachtypeofproblempresentedherebygoingtowww.textbookmedia.comandgoingtotheareaofthewebsiteforthistextbook.

8-1. Whatisthefuturevalueof$1,000investedtodayifyouearn2percentannualinterestforfiveyears?

8-2. Calculatethefuturevalueof$20,000tenyearsfromnowiftheannualinterestrateis a. 0percent b. 2percent c. 5percent d. 10percent

8-3. Howmuchwillyouhavein10yearsifyoudeposit$5,000todayandearn3percentannualinterest?

8-4. Calculatethefuturevalueof$20,000investedtodayfifteenyearsfromnowbasedonthefollowingannualinterestrates:

a. 3percent b. 6percent c. 9percent d. 12percent

8-5. Calculatethefuturevaluesofthefollowingamountsat4percentfortwentyyears: a. $50,000 b. $75,000 c. $100,000 d. $125,000

Future Value

Future Value

Future Value

Future Value

Future Value


8-6. Calculatethefuturevalueof$40,000at5percentforthefollowingyears: a. 5years b. 10years c. 15years d. 20years

8-7. Whatisthepresentvalueof$90,000tobereceivedtenyearsfromnowusinga7percentannualdiscountrate?

8-8. Calculatethepresentvalueof$15,000tobereceivedtwentyyearsfromnowatanannualdiscountrateof:


8-9. JimboJonesisgoingtoreceiveagraduationpresentof$10,000fromhisgrandparentsinfouryears.Iftheannualdiscountrateis3percent,whatisthisgiftworthtoday?

8-10. Calculatethepresentvaluesof$25,000tobereceivedintenyearsusingthefollowingannualdiscountrates:


8-11. Calculatethepresentvaluesofthefollowingusinga4percentannualdiscountrateattheendof15years:

a. $20,000 b. $60,000 c. $90,000 d. $130,000

8-12. Calculatethepresentvalueof$100,000ata5percentannualdiscountratetobereceivedin: a. 5years b. 10years c. 15years d. 20years

8-13. Whatisthepresentvalueofa$2,000ten-yearannualordinaryannuityata4percentannualdiscountrate?

8-14. Calculatethepresentvalueofa$25,000thirty-yearannualordinaryannuityatanannualdiscountrateof:


8-15. Whatisthepresentvalueofaten-yearordinaryannuityof$40,000,usinga2percentannualdiscountrate?

Future Value

Present Value

Present Value

Present Value of an Ordinary Annuity

Present Value

Present Value

Present Value

Present Value




8-16. Findthepresentvalueofafour-yearordinaryannuityof$10,000,usingthefollowingannualdiscountrates:


8-17. Whatisthefuturevalueofafive-yearannualordinaryannuityof$900,usinga4percentannualinterestrate?

8-18.Calculatethefuturevalueofatwelve-year,$15,000annualordinaryannuity,usinganannualinterestrateof:


8-19. Whatisthefuturevalueofaforty-yearordinaryannuityof$5,000,usinganannualinterestrateof4percent?

8-20. Whatisthefuturevalueofaneight-yearordinaryannuityof$12,000,usinga7percentannualinterestrate?

8-21. Findthefuturevalueofthefollowingfive-yearordinaryannuities,usinga3percentannualinterestrate.

a. $1,000 b. $10,000 c. $75,000 d. $125,000

8-22. Startingtoday,youinvest$10,000ayearintoyourindividualretirementaccount(IRA).IfyourIRAearns8percentayear,howmuchwillbeavailableattheendof40years?

8-23. DolphStarbeamwilldeposit$5,000atthebeginningofeachyearfortwentyyearsintoanaccountthathasanannualinterestrateof4percent.HowmuchwillDolphhavetowithdrawintwentyyears?

8-24. Anaccountmanagerhasfoundthatthefuturevalueof$10,000,depositedattheendofeachyear,forfiveyearsatanannualinterestrateof2percentwillamountto$52,040.40.Whatisthefuturevalueofthisscenarioiftheaccountmanagerdepositsthemoneyatthebeginningofeachyear?

8-25. Ifyourrequiredannualrateofreturnis6percent,howmuchwillaninvestmentthatpays$50ayearatthebeginningofeachofthenext40yearsbeworthtoyoutoday?

8-26. AllisonTaylorhaswonthelotteryandisgoingtoreceive$100,000peryearfor25years;shereceivedherfirstchecktoday.Thecurrentannualdiscountrateis5percent.Findthepresentvalueofherwinnings.

8-27. LeonKompowskypaysadebtserviceof$1,300amonthandwillcontinuetomakethispaymentfor15years.Whatisthepresentvalueofthesepaymentsdiscountedat6percentifhemailshisfirstpaymentintoday?Besuretodomonthlydiscounting.

8-28. Youinvested$50,000,and10yearslaterthevalueofyourinvestmenthasgrownto$185,361.07.Whatisyourcompoundedannualrateofreturnoverthisperiod?

8-29. Youinvested$1,000fiveyearsago,andthevalueofyourinvestmenthasfallento$903.92.Whatisyourcompoundedannualrateofreturnoverthisperiod?


Future Value of an Ordinary Annuity





Future Value of an Annuity Due



Present Value of an Annuity Due



Solving for k

Solving for k


8-30.Whatistherateofreturnonaninvestmentthatgrowsfrom$50,000to$89,542.38in10years?

8-31. Whatisthepresentvalueofa$50annualperpetualannuityusingadiscountrateof2percent?

8-32. Apaymentof$70peryearforeverismadewithanannualdiscountrateof6percent.Whatisthepresentvalueofthesepayments?

8-33. Youarevaluingapreferredstockthatmakesadividendpaymentof$65peryear,andthecurrentdiscountrateis6.5percent.Whatisthevalueofthisshareofpreferredstock?

8-34. HermanHermannisfinancinganewboatwithanamortizingloanof$24,000,whichistoberepaidin10annualinstallmentsof$3,576.71each.WhatinterestrateisHermanpayingontheloan?

8-35. OnJune1,2012,SelmaBouvierpurchasedahomefor$220,000.Sheput$20,000downonthehouseandobtaineda30-yearfixed-ratemortgagefortheremaining$200,000.Underthetermsofthemortgage,Selmamustmakepaymentsof$898.09amonthforthenext30yearsstartingJune30.WhatistheeffectiveannualinterestrateonSelma’smortgage?

8-36. Whatistheamountyouhavetoinvesttodayat4percentannualinteresttobeabletoreceive$10,000after

a. 5years? b. 10years? c. 20years?

8-37. HowmuchmoneywouldAlexisWhitneyneedtodepositinhersavingsaccountatGreatWesternBanktodayinordertohave$26,850.58inheraccountafterfiveyears?Assumeshemakesnootherdepositsorwithdrawalsandthebankguaranteesa6percentannualinterestrate,compoundedannually.

8-38. Ifyouinvest$15,000today,howmuchwillyoureceiveafter a. 7yearsata2percentannualinterestrate? b. 10yearsata4percentannualinterestrate?

8-39. TheApplestockyoupurchasedtwelveyearsagofor$23ashareisnowworth$535.86.Whatisthecompoundedannualrateofreturnyouhaveearnedonthisinvestment?

8-40. JessicaLovejoydeposited$1,000inasavingsaccount.Theannualinterestrateis10percent,compoundedsemiannually.Howmanyyearswillittakeforhermoneytogrowto$4,321.94?

8-41. Beginningayearfromnow,ClancyWiggumwillreceive$80,000ayearfromhispensionfund.Therewillbefifteenoftheseannualpayments.Whatisthepresentvalueofthesepaymentsifa3percentannualinterestrateisappliedasthediscountrate?

8-42. Ifyouinvest$6,000peryearintoyourIndividualRetirementAccount(IRA),earning8percentannually,howmuchwillyouhaveattheendoftwentyyears?Youmakeyourfirstpaymentof$4,000today.

8-43. Whatwouldyouaccumulateifyouweretoinvest$500everyquarterfortenyearsintoanaccountthatreturned8percentannually?Yourfirstdepositwouldbemadeonequarterfromtoday.Interestiscompoundedquarterly.

8-44. Ifyouinvest$2,000peryearforthenextfifteenyearsata6percentannualinterestrate,beginningoneyearfromtodaycompoundedannually,howmuchareyougoingtohaveattheendofthefifteenthyear?

Solving for k

Present Value of a Perpetuity


Present Value of a Perpetuity (Preferred Stock)

Solving for a Loan Interest Rate

Present Value

Future Value

Solving for k



Solving for the Monthly Interest Rate on a Mortgage

Future Value

Solving for n




8-45. Itisthebeginningofthequarterandyouintendtoinvest$750intoyourretirementfundattheendofeveryquarterforthenextthirtyyears.Youarepromisedanannualinterestrateof8percent,compoundedquarterly.

a. Howmuchwillyouhaveafterthirtyyearsuponyourretirement? b. Howlongwillyourmoneylastifyoustartwithdrawing$9,000attheendofevery

quarterafteryouretire?

8-46. A$50,000loanobtainedtodayistoberepaidinequalannualinstallmentsoverthenextsevenyearsstartingattheendofthisyear.Iftheannualinterestrateis2percent,compoundedannually,howmuchistobepaideachyear?

8-47. JaneyPowellinvested$1,000inacertificateofdeposit(CD)thatpays4%annualinterest,compoundedcontinuously,fortwentyyears.HowmuchmoneywillshehavewhenthisCDmatures?

8-48. AristotleAmadopolisismovingtoCentralAmerica.Beforehepacksuphiswifeandsonhepurchasesanannuitythatguaranteespaymentsof$10,000ayearinperpetuity.Howmuchdidhepayiftheannualinterestrateis4percent?

8-49. NedandMaudFlandersdeposited$1,000intoasavingsaccountthedaytheirson,Todd,wasborn.TheirintentionwastousethismoneytohelppayforTodd’sweddingexpenseswhenandifhedecidedtogetmarried.Theaccountpays5percentannualinterestwithcontinuouscompounding.UponhisreturnfromagraduationtriptoHawaii,ToddsurpriseshisparentswiththesuddenannouncementofhisplannedmarriagetoFrancine.ThecouplesettheweddingdatetocoincidewithTodd’stwenty-thirdbirthday.HowmuchmoneywillbeinTodd’saccountonhisweddingday?

8-50. Youdeposit$2,000inanaccountthatpays8percentannualinterest,compoundedannually.Howlongwillittaketotripleyourmoney?

8-51. Uponreadingyourmostrecentcreditcardstatement,youareshockedtolearnthatthebalanceowedonyourpurchasesis$4,000.Resolvingtogetoutofdebtonceandforall,youdecidenottochargeanymorepurchasesandtomakeregularmonthlypaymentsuntilthebalanceiszero.Assumingthatthebank’screditcardannualinterestrateis19.5percentandthemostyoucanaffordtopayeachmonthis$350,howlongwillittakeyoutopayoffyourdebt?

8-52. WaylonSmithersborrows$17,729.75foranewcar.Heisrequiredtorepaytheamortizedloanwithfourannualpaymentsof$5,000each.Whatistheannualinterestrateonhisloan?

8-53. ErnstandGunterareplanningfortheireventualretirementfromthemagicbusiness.Theyplantomakequarterlydepositsof$2,000intoanIRAstartingthreemonthsfromtoday.Theguaranteedannualinterestrateis4percent,compoundedquarterly.Theyplantoretirein15years.

a. Howmuchmoneywillbeintheirretirementaccountwhentheyretire? b. Usingtheprecedinginterestrateandthetotalaccountbalancefromparta,forhowmany

yearswillErnstandGunterbeabletowithdraw$6,000attheendofeachquarter?

8-54. MoeSzyslakcomestoyouforfinancialadvice.Heisconsideringaddingvideogamestohistaverntoattractmorecustomers.ThecompanythatsellsthevideogameshasgivenMoeachoiceoffourdifferentpaymentoptions.WhichofthefollowingfouroptionswouldyourecommendthatMoechoose?Why?Option1.Pay$5,900cashimmediately.Option2.Pay$6,750cashinonelumpsumtwoyearsfromnow.Option3.Pay$800attheendofeachquarterfortwoyears.Option4.Pay$1,000immediatelyplus$5,250inonelumpsumtwoyearsfromnow.

Moetellsyouhecanearn6percentannualinterest,compoundedquarterly,onhismoney.Youhavenoreasontoquestionhisassumption.


(Challenge Problem)

Solving for a Loan Payment

Future Value


Time to Double Your Money

Time to Pay Off a Credit Card Balance

Solving for k

Future Value

Future Value of an Annuity

(Challenge Problem)

Challenge Problem


Answers to Self-Test n

8-55. Titaniawantstoborrow$225,000fromamortgagebankertopurchasea$300,000house.Themortgageloanistoberepaidinmonthlyinstallmentsoverathirty-yearperiod.Theannualinterestrateis6percent.HowmuchwillTitania’smonthlymortgagepaymentsbe?

8-56. CarlCarlson’sfamilyhasfoundthehouseoftheirdreams.Theyhave$50,000touseasadownpaymentandtheywanttoborrow$400,000fromthebank.Thecurrentmortgageannualinterestrateis6percent.Iftheymakeequalmonthlypaymentsfortwentyyears,howmuchwilltheirmonthlymortgagepaymentbe?

8-57. OttoMannhashisheartsetonanewMiatasportscar.Hewillneedtoborrow$28,000togetthecarhewants.ThebankwillloanOttothe$28,000atanannualinterestrateof3percent.

a. HowmuchwouldOtto’smonthlycarpaymentsbeforafour-yearloan? b. HowmuchwouldOtto’smonthlycarpaymentsbeifheobtainsasix-yearloanatthe

sameinterestrate?

8-58. Assumethefollowingsetofcashflows:

End of End of End of End of Year 1 Year 2 Year 3 Year 4 $100 $150 ? $100

Atanannualdiscountrateof10percent,thetotalpresentvalueofallthecashflowsabove,includingthemissingcashflow,is$452.22.Giventheseconditions,whatisthevalueofthemissingcashflow?

8-59. Youareconsideringfinancingthepurchaseofanautomobilethatcosts$22,000.Youintendtofinancetheentirepurchasepricewithafour-yearamortizedloanwitha6percentannualinterestrate.

a. Calculatetheamountofthemonthlypaymentsforthisloan. b. ConstructanamortizationtableforthisloanusingtheformatshowninTable8-2.Use

monthlypayments. c. Ifyouelectedtopayoffthebalanceoftheloanattheendofthethirty-sixthmonth

howmuchwouldyouhavetopay?

For more, see 7e Spreadsheet Templates for Microsoft Excel.

For more, see 7e Spreadsheet Templates for Microsoft Excel.

Solving for a Mortgage Loan Payment

Solving for Auto Loan Payments

Solving for a Mortgage Loan Payment

Challenge Problem (Value of Missing Cash Flow)

Loan Amortization Table

ST-1. FV=$5,000×1.068=$7,969.24=Jed’sbalancewhentheeight-yearCDmatures.

ST-2. $19,999 $1,1776.89 PVIFAk = ?, n = 12( )= ×

11.2551=PVIFAk=?,n=12

k=1%monthly(fromthePVIFATableattheendofthen=12row,

k=1%)

AnnualRate=1%×12=12%

ST-3. PV=$100,000×(1/1.0810)=$46,319.35

=thepresentvalueofHeidi’s$100,000

ST-4. $20,000=PMT×[1–(1/1.00548)]/.005

$20,000=PMT×42.5803

PMT=$469.70=Zack’scarloanpayment

ST-5. $5,000×FVIFk=?,n=10=$15,529.24

FVIFk=?,n=10=3.1058,thereforek=12%=Bridget’sannualrateofreturnonthemutualfund.

ST-6. F V PMT FVIFA 1 kk%, n( ) ( )= × × +

$1,000 FVIFA 1 .1010%, 10( ) ( )= × × +

=$1,000×15.9374×1.10

=$17,531.14

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