Chapter 3 answers.docx

Chapter 3 answers.docx

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Chapter3answers

CHAPTER3

MEASURINGYIELD

ANSWERSTOQUESTIONSFORCHAPTER3

（Questionsareinboldprintfollowedbyanswers.）

1.Adebtobligationoffersthefollowingpayments:

YearsfromNow

CashFlowtoInvestor

1

$2,000

2

$2,000

3

$2,500

4

$4,000

Supposethatthepriceofthisdebtobligationis$7,704.Whatistheyieldorinternalrateofreturnofferedbythisdebtobligation?

Theyieldonanyinvestmentistheinterestratethatwillmakethepresentvalueofthecashflowsfromtheinvestmentequaltotheprice（orcost）oftheinvestment.

Mathematically,theyieldonanyinvestment,y,istheinterestratethatsatisfiestheequation:

P=

whereCFt=cashflowinyeart,P=priceoftheinvestment,andN=numberofyears.Theyieldcalculatedfromthisrelationshipisalsocalledtheinternalrateofreturn.Tosolvefortheyield（y）,wecanuseatrial-and-error（iterative）procedure.Theobjectiveistofindtheinterestratethatwillmakethepresentvalueofthecashflowsequaltotheprice.Tocomputetheyieldforourproblem,differentinterestratesmustbetrieduntilthepresentvalueofthecashflowsisequalto$7,704（thepriceofthefinancialinstrument）.Tryinganannualinterestrateof10%givesthefollowingpresentvalue:

YearsfromNow

PromisedAnnualPayments

（CashFlowtoInvestor）

PresentValue

ofCashFlowat10%

1

$2,000

$1,818.18

2

$2,000

$1,652.89

3

$2,500

$1,878.29

4

$4,000

$2,732.05

Presentvalue=$8,081.41

Becausethepresentvalueof$8,081.41computedusinga10%interestrateexceedsthepriceof$7,704,ahigherinterestratemustbeused,toreducethepresentvalue.Tryinganannualinterestrateof13%givesthefollowingpresentvalue:

YearsfromNow

PromisedAnnualPayments

（CashFlowtoInvestor）

PresentValue

ofCashFlowat13%

1

$2,000

$1,769.91

2

$2,000

$1,566.29

3

$2,500

$1,732.63

4

$4,000

$2,453.27

Presentvalue=$7,522.10

Becausethepresentvalueof$7,522.10computedusinga13%interestrateisbelowthepriceof$7,704,alowerinterestratemustbeused,toreducethepresentvalue.Thus,toincreasethepresentvalue,alowerinterestratemustbetried.Tryinganannualinterestrateof12%givesthefollowingpresentvalue:

YearsfromNow

PromisedAnnualPayments

（CashFlowtoInvestor）

PresentValue

ofCashFlowat12%

1

$2,000

$1,785.71

2

$2,000

$1,594.39

3

$2,500

$1,779.45

4

$4,000

$2,542.07

Presentvalue=$7,701.62

Using12%,thepresentvalueofthecashflowis$7,701.62,whichisalmostequaltothepriceofthefinancialinstrumentof$7,704.Therefore,theyieldiscloseto12%.ThepreciseyieldusingExcelorafinancialcalculatoris11.987%.

Althoughtheformulafortheyieldisbasedonannualcashflows,itcanbegeneralizedtoanynumberofperiodicpaymentsinayear.Thegeneralizedformulafordeterminingtheyieldis

whereCFt=cashflowinperiodt,andn=numberofperiods.

Keepinmindthattheyieldcomputedistheyieldfortheperiod.Thatis,ifthecashflowsaresemiannual,theyieldisasemiannualyield.Ifthecashflowsaremonthly,theyieldisamonthlyyield.Tocomputethesimpleannualinterestrate,theyieldfortheperiodismultipliedbythenumberofperiodsintheyear.

2.Whatistheeffectiveannualyieldifthesemiannualperiodicinterestrateis4.3%?

Toobtainaneffectiveannualyieldassociatedwithaperiodicinterestrate,thefollowingformulaisused:

effectiveannualyield=（1+periodicinterestrate）m–1

wheremisthefrequencyofpaymentsperyear.Inourproblem,theperiodicinterestrateisasemiannualrateof4.3%andthefrequencyofpaymentsistwiceperyear.Insertingthesenumbers,wehave:

effectiveannualyield=（1.043）2–1=1.087849–1=0.087849orabout8.785%.

3.Whatistheyieldtomaturityofabond?

Theyieldtomaturityistheinterestratethatwillmakethepresentvalueofthecashflowsequaltotheprice（orinitialinvestment）.Forasemiannualpaybond,theyieldtomaturityisfoundbyfirstcomputingtheperiodicinterestrate,y,whichsatisfiestherelationship:

P=

whereP=priceofthebond,C=semiannualcouponinterest（indollars）,M=maturityvalue（indollars）,andn=numberofperiods（numberofyearstimes2）.

Itismucheasiertocomputetheyieldtomaturityforazero-couponbondbecausewecanuse:

.

Theyield-to-maturitycalculationtakesintoaccountnotonlythecurrentcouponincomebutalsoanycapitalgainorlossthattheinvestorwillrealizebyholdingthebondtomaturity.Inaddition,theyieldtomaturityconsidersthetimingofthecashflows.

4.Whatistheyieldtomaturitycalculatedonabond-equivalentbasis?

Forasemiannualpaybond,doublingtheperiodicinterestrateordiscountrate（y）givestheyieldtomaturity,whichunderstatestheeffectiveannualyield.The