corporate finance Chap005new.docx

corporate finance Chap005new.docx

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corporatefinanceChap005new

SolutionstoChapter5

TheTimeValueofMoney

1.a.$100/（1.08）10=$46.32

b.$100/（1.08）20=$21.45

c.$100/（1.04）10=$67.56

d.$100/（1.04）20=$45.64

Esttime:

01–05

2.a.$100（1.08）10=$215.89

b.$100（1.08）20=$466.10

c.$100（1.04）10=$148.02

d.$100（1.04）20=$219.11

Esttime:

01–05

3.$100（1.04）113=$8,409.45

$100（1.08）113=$598,252.29

Esttime:

01–05

4.Withsimpleinterest,youearn4%of$1,000,or$40eachyear.Thereisnointerestoninterest.After10years,youearntotalinterestof$400,andyouraccountaccumulatesto$1,400.Withcompoundinterest,youraccountgrowsto$1,000（1.04）10=$1,480.24

Therefore$80.24isinterestoninterest.

Esttime:

01–05

5.PV=$700/（1.05）5=$548.47

Esttime:

01–05

6.

PresentValue

Years

FutureValue

InterestRate

a.

$400

11

$684

b.

$183

4

$249

c.

$300

7

$300

Tofindtheinterestrate,werearrangethebasicfuturevalueequationasfollows:

FV=PV（1+r）tr=

Esttime:

06–10

7.Youshouldcomparethepresentvaluesofthetwoannuities.

a.

b.

c.Whentheinterestrateislow,asinpart（a）,thelonger（i.e.,15-year）butsmallerannuityismorevaluablebecausetheimpactofdiscountingonthepresentvalueoffuturepaymentsislesssignificant.

Esttime:

01–05

8.$100（1+r）3=$115.76r=5.00%

$200（1+r）4=$262.16r=7.00%

$100（1+r）5=$110.41r=2.00%

Esttime:

06–10

9.PV=（$200/1.06）+（$400/1.062）+（$300/1.063）=$188.68+$356.00+$251.89=$796.57

Esttime:

01–05

10.Intheseproblems,youeithercansolvetheequationprovideddirectlyorcanuseyourfinancialcalculator,settingPV=（）400,FV=1,000,PMT=0,iasspecifiedbytheproblem.Thencomputenonthecalculator.

a.$400（1.04）t=$1,000t=23.36periods

b.$400（1.08）t=$1,000t=11.91periods

c.$400（1.16）t=$1,000t=6.17periods

Esttime:

01–05

11.

APR

Compounding

Period

Effective

AnnualRate

a.

12%

1month

（m=12/yr）

1.01121=0.1268=12.68%

b.

8%

3months

（m=4/yr）

1.0241=0.0824=8.24%

c.

10%

6months

（m=2/yr）

1.0521=0.1025=10.25%

Esttime:

01–05

12.

EffectiveRate

CompoundingPeriod

Per-PeriodRate

APR

a.

10.00%

1month

（m=12/yr）

1.10（1/12）1=0.0080

0.096=9.6%

b.

6.09%

6months

（m=2/yr）

1.0609（1/2）1=0.0300

0.060=6.0%

c.

8.24%

3months

（m=4/yr）

1.0824（1/4）1=0.0200

0.080=8.0%

Esttime:

06–10

13.Solvethefollowingfort:

1.08t=2t=11.9years

Onafinancialcalculator,enterPV=（）1,FV=2,PMT=0,i=6;thencomputen.

Esttime:

01–05

14.Semiannualcompoundingmeansthatthe8.6%loanreallycarriesinterestof4.3percentperhalfyear.Similarly,the8.4%loanhasamonthlyrateof0.7%.

APR

Compoundingperiod

Effectiveannualrate

8.6%

6months

（m=2/yr）

1.04321=0.0878=8.78%

8.4%

1month

（m=12/yr）

1.007121=0.0873=8.73%

Choosethe8.4%loanforitsslightlylowereffectiverate.

Esttime:

01–05

15.APR=1%×52=52%

EAR=（1.01）521=0.6777=67.77%

Esttime:

01–05

16.Sinceweareassumingthatitiscurrently2012,112yearshavepassedsince1900.

a.$1,000（1.05）112=$236,157.40

b.PV（1.05）112=$1,000,000PV=$4,234.47

Esttime:

01–05

17.$1,0001.04=$1,040.00interest=$40

$1,0401.04=$1,081.60interest=$1,081.60$1,040=$41.60

After9years,youraccounthasgrownto$1,000（1.04）9=$1,423.31.

After10years,youraccounthasgrownto$1,000（1.04）10=$1,480.24.

Interestearnedin10thyear=$1,480.24$1,423.31=$56.93

Esttime:

01–05

18.Ifyouearnedsimpleinterest（withoutcompounding）,thenthetotalgrowthinyouraccountafter25yearswouldbe4%peryear25years=100%.

Therefore,yourmoneywoulddouble.Withcompoundinterest,yourmoneywouldgrowfasterthanitwouldwithsimpleinterestandthereforewouldrequirelessthan25yearstodouble.

Esttime:

01–05

19.Wesolvethefollowingequationforr:

422.41（1+r）10=1000r=9.00%

[Onafinancialcalculator,enterPV=（）422.41,FV=1,000,n=10,PMT=0;computetheinterestrate.]

Esttime:

01–05

20.ThePVforthequarterbackisthepresentvalueofa5-year,$3millionannuity:

$3millionannuityfactor（10%,5years）=

Thereceivergets$4millionnowplusa5-year,$2millionannuity.Thepresentvalueoftheannuityis:

Withthe$4millionimmediatepayment,thereceiver’scontractisworth:

$4million+$7.58million=$11.58million

Thereceiver’scontractisworthmorethanthequarterback’seventhoughthereceiver’sundiscountedtotalpaymentsarelessthanthequarterback’s.

Esttime:

06–10

21.Rateofgrowthforapples:

$0.92（1+r）10=$1.26r=3.195%

Rateofgrowthfororanges:

$0.70（1+r）10=$1.10r=4.62%

Priceofapplesin2030:

$1.26（1.03195）20=$2.36

Priceoforangesin2030:

$1.10（1.0462）20=$2.72

Esttime:

01–05

22.IfthepaymentisdenotedC,then:

Themonthlyinterestrateis0.10/12=0.008333=0.8333%.

Therefore,theeffectiveannualinterestrateontheloanis:

（1.008333）121=0.1047=10.47%

Esttime:

01–05

23.a.PV=100annuityfactor（6%,3periods）=100

b.Ifthepaymentstreamisdeferredbyanadditionalyear,theneachpaymentisdiscountedbyanadditionalfactorof1.06.Therefore,thepresentvalueisreducedbyafactorof1.06to$267.30/1.06=$252.17.

Esttime:

01–05

24.a.Thisisanannuityproblem;usetrialanderrortosolveforrinthefollowingequation:

Usingafinancialcalculator,enterPV=（–）80,000,n=2012=240months,

FV=0,PMT=600;computei.TocomputeEAR:

EAR=（1+0.00548）121=0.0678=6.78%

b.ComputethepaymentbysolvingforCinthefollowingequation:

Usingafinancialcalculator,entern=240,i=0.5%,FV=0,PV=（）80,000;computePMT=$573.14.

Esttime:

01–05

25.a.Yourmonthlypaymentsof$400cansupportaloanof$15,189.58.

Thisiscomputedasfollows:

PV=

Usingafinancialcalculator,entern=48,i=12%/12=1%,FV=0,PMT=400;computePV=$15,189.58.

Withadownpaymentof$2,000,youcanpayatmost$17,189.58forthecar.

b.Inthiscase,nincreasesfrom48to60.Youcantakeoutaloanof$17,982.02basedonthispaymentschedule.Thisiscomputedasfollows:

PV=

Thus,youcanpay$19,982.02forthecar.

Esttime:

01–05

26.a.WithPV=$9,000andFV=$10,000,theannualinterestrateisdeterminedbysolvingthefollowingequationforr:

$9,000（1+r）=$10,000r=11.11%

b.Thepresentvalueis$10,000（1d）.

Thefuturevaluetobepaidbackis$10,000.

Therefore,theannualinterestrateisdeterminedasfollows:

PV（1+r）=FV

[$10,000（1–d）]（1+r）=$10,000

c.Thediscountiscalculatedasafractionofthefuturevalueoftheloan.Infact,theproperwaytocomputetheinterestrateisasafractionofthefundsborrowed.SincePVislessthanFV,theinterestpaymentisasmallerfractionofthefuturevalueoftheloanthanitisofthepresentvalue.Thus,thetrueinterestrateexceedsthestateddiscountfactoroftheloan.

Esttime:

06–10

27.a.Ifweassumecashflowscomeattheendofeachperiod（ordinaryannuity）wheninfacttheyactuallycomeatthebeginning（annuitydue）,wediscounteachcashflowbyoneperiodtoomany.Therefore,wecanobtainthePVofanannuityduebymultiplyingthePVofanordinaryannuityby（1+r）.

b.Similarly,theFVofanannuitydueequalstheFVofanordinaryannuitytimes（1+r）.Becauseeachcashflowcomesatthebeginningoftheperiod,ithasanextraperiodtoearninterestcomparedtoanordinaryannuity.

Esttime:

06–10

28.Usetrialanderrortosolvethefollowingequationforr:

Usingafinancialcalculator,enterPV=（）8,000;n=48;PMT=240,FV=0;thencomputer=1.599%permonth.

APR=1.599%12=19.188%

Theeffectiveannualrateis（1.01599）121=0.2097=20.97%.

Esttime:

06–10

29.Theannualpaymentovera4-yearperiodthathasapresentvalueof$8,000iscomputedbysolvingthefollowingequationforC:

[Usingafinancialcalculator,enterPV=（）8,000,n=4,FV=0,i=20.97;computePMT.]Withmonthlypayments,youwouldpayonly$24012=$2,880peryear.Thisvalueislowerbecausethemonthlypaymentscomebeforeyear-endandthereforehaveahigherPV.

Esttime:

01–05

30.Leasingthetruckmeansthatthefirmmustmakeaseriesofpaymentsintheformofanannuity.Calculatethepresentvalueasfollows:

PV=

Usingafinancialcalculator,enterPMT=8,000,n=6,i=7%,FV=0;computePV=$38,132.32.

Since$38,132.32<$40,000（thecostofbuyingatruck）,itislessexpensivetoleasethantobuy.

Esttime:

01–05

31.PVofanannuitydue=PVofordinaryannuity（1+r）

（SeeProblem27foradiscussionofthevalueofanordinaryannuityversusanannuitydue.）Therefore,withimmediatepayment,thevalueoftheleasepaymentsincreasesfrom$38,132.32（asshowninthepreviousproblem）to:

$38,132.321.07=$40,801.58

Sincethisisgreaterthan$40,000（thecostofbuyingatruck）,weconcludethat,ifthefirstpaymentontheleaseisdueimmediately,itislessexpensivetobuythetruckthantoleaseit.

Esttime:

01–05

32.Comparethepresentvalueofthepayments.Assumetheproductsellsfor$100.

Installmentplan:

PV=$25+[$25annuityfactor（5%,3years）]

PV=

Payinfull:

Paymentnetofdiscount=$90

Choosethesecondpaymentplanforitslowerpresentvalueofpayments.

Esttime:

01–05

33.Installmentplan:

PV=$25annuityfactor（5%,4years）

PV=