annually

.

.

CHAPTER14:BONDPRICESANDYIELDS

PROBLEMSETS

rophebond—Abondthatallowstheissuertotransfer

“catastropherisk”orsin

thebondsreceiveacompensationfortakingontheriskintheformof

ventofacatastrophe,thebondholderswill

receiveonlypartorperhapsnoneoftheprincipalpaymentduetothem

ercanbedefinedbytotalinsuredlossorbycriteria

suchaswindspeedinahurricaneorRichterlevelinanearthquake.

nd—Abondthatisdenominatedinonecurrency,usuallythat

oftheissuer,butsoldinothernationalmarkets.

-couponbond—ors

receiveparvalueatthematuritydatebutreceivenointerestpayments

ondsareissuedatpricesbelowparvalue,andthe

investor’sreturncomesfromthedifferencebetweenissuepriceandthe

paymentofparvalueatmaturity(capitalgain).

ibond—Yen-dominatedbondssoldinJapanbynon-Japane

issuers.

nd—Abondwithalowcreditratingduetoitshighdefaultrisk;

alsoknownashigh-yieldbonds.

tiblebond—Abondthatgivesthebondholdersanoptionto

exchangethebondforaspecifiednumberofsharesofcommonstockof

thefirm.

bonds—s

maturequentially,theprincipalrepaymentburdenforthefirmis

spreadovertime.

entobligationbond—Acollateralizedbondforwhichthe

irmdefaultsonthe

bond,thebondholderswouldreceivetheequipment.

alissuediscountbond—Abondissuedatadiscounttotheface

value.

dbond—Abondthatmakespaymentsthataretiedtoa

generalpriceindexorthepriceofaparticularcommodity.

.

.

lebond—Abondthatgivestheissuertheoptiontorepurcha

thebondataspecifiedcallpricebeforethematuritydate.

.

.

lebond—Abondthatgivesthebondholdertheoptiontollback

thebondataspecifiedputpricebeforethematuritydate.

dcallableat105shouldllatalowerpricebecauthecall

ore,itsyieldtomaturity

shouldbehigher.

ore,the

investor'sproceedsfromthebondareindependentoftherateatwhich

couponscouldbereinvested(iftheywerepaid).Thereisnoreinvestment

rateuncertaintywithzeros.

’scouponinterestpaymentsandprincipalrepaymentarenot

uently,ifmarketrates

increa,bondinvestorsinthecondarymarketsarenotwillingtopay

asmuchforaclaimonagivenbond’sfixedinterestandprincipal

lationshipis

apparentfromtheinverrelationshipbetweeninterestratesand

eainthediscountrate(i.e.,themarketrate.

decreastheprentvalueofthefuturecashflows.

couponrate:4.80%$48Couponpayments

Currentyield:

$48

4.95%

$970









iveannualratefor3-monthT-bill:

%0.10100.0102412.11

645,97

000,100

4

4















iveannualinterestrateforcouponbondpaying5%

miannually:

(1.05.2—1=0.1025or10.25%

Thereforethecouponbondhasthehighereffectiveannualinterestrate.

ectiveannualyieldonthemiannualcouponbondsis8.16%.If

theannualcouponbondsaretollatpartheymustofferthesame

yield,whichrequiresanannualcouponrateof8.16%.

.

.

pass,thebondprice,whichis

nowaboveparvalue,willapproachpar.

omaturity:Usingafinancialcalculator,enterthefollowing:

n=3;PV=953.10;FV=1000;PMT=80;COMPi

Thisresultsin:YTM=9.88%

Realizedcompoundyield:First,findthefuturevalue(vested

couponsandprincipal:

FV=($80*1.10*1.12.+($80*1.12.+$1,080=$1,268.16

Thenfindtherate(y

realized

.thatmakestheFVofthepurchapriceequalto

$1,268.16:

$953.10(1+y

realized

.3=$1,268.16y

realized

=9.99%orapproximately10%

Usingafinancialcalculator,enterthefollowing:N=3;PV=953.10;FV=

1,268.16;PMT=0;is9.99%.

10.

upon8%

cou

po

n

10%coupon

Currentprices$463.19$1,000.00$1,134.20

1yearfromnow$500.25$1,000.00$1,124.94

Priceincrea$37.06$0.00−$9.26

Couponincome$0.00$80.00$100.00

Pretaxincome$37.06$80.00$90.74

Pretaxrateofreturn8.00%8.00%8.00%

Taxes*$11.12$24.00$28.15

After-taxincome$25.94$56.00$62.59

After-taxrateofreturn5.60%5.60%5.52%

1yearfromnow$543.93$1,065.15$1,195.46

Priceincrea$80.74$65.15$61.26

Couponincome$0.00$80.00$100.00

Pretaxincome$80.74$145.15$161.26

Pretaxrateofreturn17.43%14.52%14.22%

Taxes†$19.86$37.03$42.25

After-taxincome$60.88$108.12$119.01

After-taxrateofreturn13.14%10.81%10.49%

.

.

*Incomputingtaxes,weassumethatthe10%couponbondwasissuedat

parandthatthedecreainpricewhenthebondissoldatyear-endis

treatedasacapitallossandthereforeisnottreatedasanofftto

ordinaryincome.

†Incomputingtaxesforthezerocouponbond,$37.06istaxedas

ordinaryincome(epart(b);theremainderofthepriceincreais

taxedasacapitalgain.

.

.

ancialcalculator,enterthefollowing:

n=40;FV=1000;PV=–950;PMT=40

Youwillfindthattheyieldtomaturityonamiannualbasisis4.26%.

Thisimpliesabondequivalentyieldtomaturityequalto:4.26%*2=

8.52%

Effectiveannualyieldtomaturity=(1.0426)2–1=0.0870=8.70%

hebondisllingatpar,theyieldtomaturityonamiannual

basisisthesameasthemiannualcouponrate,i.e.,4%.Thebond

equivalentyieldtomaturityis8%.

Effectiveannualyieldtomaturity=(1.04)2–1=0.0816=8.16%

gotherinputsunchangedbutttingPV=–1050,wefinda

bondequivalentyieldtomaturityof7.52%,or3.76%onamiannual

basis.

Effectiveannualyieldtomaturity=(1.0376)2–1=0.0766=7.66%

hebondpaymentsarenowmadeannuallyinsteadofmiannually,

thebondequivalentyieldtomaturityisthesameastheeffectiveannual

yieldtomaturity.[Onafinancialcalculator,n=20;FV=1000;PV=–

price;PMT=80]

Theresultingyieldsforthethreebondsare:

BondPrice

BondEquivalentYield

=

EffectiveAnnualYield

$9508.53%

1,0008.00

1,0507.51

Theyieldscomputedinthiscaarelowerthantheyieldscalculated

eequal,bondswithannualpayments

arelessattractivetoinvestorsbecaumoretimeelapsbefore

ondpriceisthesamewithannual

payments,thenthebond'syieldtomaturityislower.

13.

Price

Maturity

(years.

Bond

Equivale

nt

YTM

$400.0020.004.688%

500.0020.003.526

500.0010.007.177

.

.

385.5410.0010.000

463.1910.008.000

400.0011.918.000

.

.

dpays$rentpriceis:

[$50×Annuityfactor(4%,6)]+[$1,000×PVfactor(4%,6)]=$1,052.42

Alternatively,PMT=$50;FV=$1,000;I=4;N=orPV=

$1,052.42.

Ifthemarketinterestrateremains4%perhalfyear,pricesixmonths

fromnowis:

[$50×Annuityfactor(4%,5)]+[$1,000×PVfactor(4%,5)]=$1,044.52

Alternatively,PMT=$50;FV=$1,000;I=4;N=orPV=

$1,044.52.

return

$50($1,044.52$1,052.42)$50$7.90

4.0%

$1,052.42$1,052.42





ortedbondpriceis:$1,001.250

However,15dayshavepasdsincethelastmiannualcouponwaspaid,

so:

Accruedinterest=$35*(15/182)=$2.885

Theinvoicepriceisthereportedpriceplusaccruedinterest:$1,004.14

ieldtomaturityisgreaterthanthecurrentyield,thenthebond

offerstheprospectofpriceappreciationasitapproachesitsmaturity

ore,thebondmustbellingbelowparvalue.

ponrateislessthan9%.Ifcoupondividedbypriceequals9%,

andpriceislessthanpar,thenpricedividedbyparislessthan9%.

18.

Time

Inflation

inYear

Just

Ended

ParValue

Coupon

Payment

Principal

Repayment

0$1,000.00

12%1,020.00$40.80$0.00

23%$1,050.60$42.02$0.00

31%$1,061.11$42.44$1,061.11

Thenominalrateofreturnandrealrateofreturnonthebondineach

yeararecomputedasfollows:

.

.

Nominalrateofreturn=

interest+priceappreciation

initialprice

Realrateofreturn=

1+nominalreturn

1+inflation

1

SecondYearThirdYear

Nominalreturn071196.0

020,1$

60.30$02.42$





050400.0

60.050,1$

51.10$44.42$





Realreturn

%0.4040.01

03.1

071196.1

%0.4040.01

01.1

050400.1



Therealrateofreturnineachyearisprecilythe4%realyieldonthe

bond.

cescheduleisasfollows:

Year

Remaining

Maturity(T).

ConstantYieldValue

$1,000/(1.08)T

ImputedInterest

(increain

constant

yieldvalue)

0(now)20years$214.55

119231.71$17.16

218250.2518.54

191925.93

2001,000.0074.07

disissuedatapriceof$ore,itsyieldtomaturityis:

6.8245%

Therefore,usingtheconstantyieldmethod,wefindthatthepriceinone

year(whenmaturityfallsto9years)willbe(atanunchangedyield.

$814.60,reprentinganincreaof$axableincomeis:

$40.00+$14.60=$54.60

dllsfor$1,124.72badonthe3.5%yieldtomaturity.

[n=60;i=3.5;FV=1000;PMT=40]

Therefore,yieldtocallis3.368%miannually,6.736%annually.

[n=10miannualperiods;PV=–1124.72;FV=1100;PMT=40]

allpricewere$1,050,wewouldtFV=1,050andredo

part(a)tofindthatyieldtocallis2.976%miannually,5.952%

owercallprice,theyieldtocallislower.

ocallis3.031%miannually,6.062%annually.

[n=4;PV=−1124.72;FV=1100;PMT=40]

.

.

tedyieldtomaturity,badonpromidpayments,equals16.075%.

[n=10;PV=–900;FV=1000;PMT=140]

Badonexpectedreducedcouponpaymentsof$70annually,the

expectedyieldtomaturityis8.526%.

ldtomaturityequalsthecoupon

rate,10%.Ifthefirst-yearcouponisreinvestedataninterestrateofr

percent,thentotalproceedsattheendofthecondyearwillbe:[$100*

(1+r)]+$1,100

Therefore,realizedcompoundyieldtomaturityisafunctionofr,asshown

inthefollowingtable:

rTotalproceedsRealizedYTM=Proceeds/1000–1

8%$1,208

1208/1000–1=0.0991=9.91%

10%$1,210

1210/1000–1=0.1000=10.00%

12%$1,212

1212/1000–1=0.1009=10.09%

ore,

theinvoicepricewillbehigherthanthestatedaskpricebyanamount

priceis101.25

percentofpar,sotheinvoicepriceis:

$1,012.50+(½*$50)=$1,037.50

sthatmightmaketheABCdebtmoreattractivetoinvestors,

thereforejustifyingalowercouponrateandyieldtomaturity,are:

debtisalargerissueandthereforemayllwithgreater

liquidity.

ontoextendthetermfrom10yearsto20yearsisfavorableif

interestrates10yearsfromnowarelowerthantoday’

contrast,ifinterestratesincrea,theinvestorcanprentthebondfor

paymentandreinvestthemoneyforahigherreturn.

ventoftrouble,

moreunderlyingcurityintheformofafirstclaimagainstreal

property.

lfeatureontheXYZbondsmakestheABCbondsrelatively

moreattractivesinceABCbondscannotbecalledfromtheinvestor.

bondhasasinkingfundrequiringXYZtoretirepartofthe

ostsinkingfundsgivethefirmtheoptionto

.

.

retirethisamountatthelowerofparormarketvalue,thesinkingfund

canbedetrimentalforbondholders.

vestorbelievesthefirm’screditprospectsarepoorinthenear

termandwishestocapitalizeonthis,theinvestorshouldbuyacredit

ghashortsaleofabondcouldaccomplishthesame

objective,liquidityisoftengreaterintheswapmarketthanitisinthe

estorcouldpickaswapwithamaturity

ngthe

swap,theinvestorwouldreceivecompensationifthebondexperiences

anincreaincreditrisk.

editriskincreas,creditdefaultswapsincreainvalue

default

swapsdonotprovideprotectionagainstinterestrateriskhowever.

eainthefirm’stimesinterest-earnedratiodecreasthe

defaultriskofthefirmincreasthebond’spricedecreastheYTM.

eaintheissuingfirm’sdebt-equityratioincreasthe

defaultriskofthefirmdecreasthebond’spriceincreasYTM.

eaintheissuingfirm’squickratioincreasshort-run

liquidity,implyingadecreaindefaultriskofthefirmincreas

thebond’spricedecreasYTM.

atingratenotepaysacouponthatadjuststomarketlevels.

Therefore,itwillnotexperiencedramaticpricechangesasmarket

edratenotewillthereforehaveagreaterprice

range.

ngratenotesmaynotllatparforanyofveralreasons:

(i)Theyieldspreadbetweenone-yearTreasurybillsandother

moneymarketinstrumentsofcomparablematuritycouldbewider

(enthebondwasissued.

(ii)Thecreditstandingofthefirmmayhaveeroded(orimproved.

relativetoTreasurycurities,whichhavenocreditrisk.

Therefore,the2%premiumwouldbecomeinsufficienttosustain

theissueatpar.

(iii)Thecouponincreasareimplementedwithalag,i.e.,once

aperiodofchanginginterestrates,eventhis

brieflagwillbereflectedinthepriceofthecurity.

.

.

ethebondwillalmostsurelynotllfor

muchaboveparvalue(givenitsadjustablecouponrate),itisunlikely

thatthebondwilleverbecalled.

ed-ratenotecurrentlyllsatonly88%ofthecallprice,so

skis

currentlylow,sinceyieldswouldneedtofallsubstantiallyforthefirm

touitsoptiontocallthebond.

9%couponnotescurrentlyhavearemainingmaturityof15years

andllatayieldtomaturityof9.9%.Thisisthecouponratethat

wouldbeneededforanewlyissued15-yearmaturitybondtollatpar.

ethefloatingratenotepaysavariablestreamofinterest

paymentstomaturity,theeffectivematurityforcomparativepurpos

withotherdebtcuritiesisclortothenextcouponretdatethan

ore,yield-to-maturityisan

indeterminablecalculationforafloatingratenote,with“yield-to-

recoupondate”amoremeaningfulmeasureofreturn.

ldtomaturityontheparbondequalsitscouponrate,8.75%.

Allelequal,the4%couponbondwouldbemoreattractivebecauits

couponrateisfarbelowcurrentmarketyields,anditspriceisfarbelow

ore,ifyieldsfall,capitalgainsonthebondwillnot

rast,the8¾%couponbondcan

increainvaluetoatmost$1,050,offeringamaximumpossiblegainof

only0.5%.Thedisadvantageofthe8¾%couponbond,intermsof

vulnerabilitytobeingcalled,showsupinitshigherpromidyieldto

maturity.

vestorexpectsyieldstofallsubstantially,the4%bondoffersa

greaterexpectedreturn.

itcallprotectionisofferedinthenthatanylikelyfall

inyieldswouldnotbenearlyenoughtomakethefirmconsider

n,thecallfeatureisalmostirrelevant.

lpriceP

0

=$705.46[n=20;PMT=50;FV=1000;i=8]

Nextyear'spriceP

1

=$793.29[n=19;PMT=50;FV=1000;i=7]

HPR

%54.191954.0

46.705$

)46.705$29.793($50$







.

.

IDtaxrules,thecostbasisandimputedinterestunderthe

constantyieldmethodareobtainedbydiscountingbondpaymentsatthe

original8%yieldandsimplyreducingmaturitybyoneyearatatime:

Constantyieldprices(comparethetoactualpricestocomputecapital

gains.:

P

0

=$705.46

P

1

=$711.89implicitinterestoverfirstyear=$6.43

P

2

=$718.84implicitinterestovercondyear=$6.95

Taxonexplicitinterestplusimplicitinterestinfirstyear=

0.40*($50+$6.43)=$22.57

Capitalgaininfirstyear=Actualpriceat7%YTM—constantyieldprice=

$793.29—$711.89=$81.40

Taxoncapitalgain=0.30*$81.40=$24.42

Totaltaxes=$22.57+$24.42=$46.99

axHPR=

%88.121288.0

46.705$

99.46$)46.705$29.793($50$





fbondaftertwoyears=$798.82[usingn=18;i=7%;PMT=$50;

FV=$1,000]

Reinvestedincomefromthecouponinterestpayments=$50*1.03+$50=

$101.50

Totalfundsaftertwoyears=$798.82+$101.50=$900.32

Therefore,theinvestmentof$705.46growsto$900.32intwoyears:

$705.46(1+r)2=$900.32r=0.1297=12.97%

interestreceivedinfirstyear:$50.00

Less:taxoncouponinterest40%:–20.00

Less:taxonimputedinterest(0.40*$6.43):–2.57

Netcashflowinfirstyear:$27.43

Theyear-1cashflowcanbeinvestedatanafter-taxrateof:

3%×(1–0.40)=1.8%

Byyear2,thisinvestmentwillgrowto:$27.43×1.018=$27.92

Intwoyears,llthebondfor:$798.82[n=18;i=7%%;PMT=

$50;FV=$1,000]

Less:taxonimputedinterestincondyear:–2.78[0.40×$6.95]

Add:after-taxcouponinterestreceived

incondyear:+30.00[$50×(1–0.40)]

.

.

Less:Capitalgainstaxon

(salesprice–constantyieldvalue):–23.99[0.30×(798.82–718.84)]

Add:CFfromfirstyear'scoupon(reinvested):+27.92[fromabove]

Total$829.97

$705.46(1+r)2=$829.97r=0.0847=8.47%

CFAPROBLEMS

ngfundprovisionrequirestheearlyredemptionofabondissue.

Theprovisionmaybeforaspecificnumberofbondsorapercentageof

kingfundcanretire

alloraportionofanissueoverthelifeoftheissue.

b.(i)Comparedtoabondwithoutasinkingfund,thesinkingfund

reducestheaveragelifeoftheoverallissuebecausomeofthe

bondsareretiredpriortothestatedmaturity.

(ii)Thecompanywillmakethesametotalprincipalpaymentsover

thelifeoftheissue,althoughthetimingofthepaymentswillbe

alinterestpaymentsassociatedwiththeissuewill

bereducedgiventheearlyredemptionofprincipal.

einvestor’spointofview,thekeyreasonfordemandinga

triskisreducedbythe

orderlyretirementoftheissue.

2.a.(i)Currentyield=Coupon/Price=$70/$960=0.0729=7.29%

(ii)YTM=3.993%miannually,or7.986%annualbondequivalentyield.

Onafinancialcalculator,enter:n=10;PV=–960;FV=1000;PMT=35

Computetheinterestrate.

(iii)Realizedcompoundyieldis4.166%(miannually),or8.332%annual

inthisvalue,firstfindthefuturevalue

(FV)illbesixpaymentsof

$35each,reinvestedmiannuallyat3%ancial

calculator,enter:

PV=0;PMT=35;n=6;i=3%.Compute:FV=226.39

Threeyearsfromnow,thebondwillbellingattheparvalueof$1,000

becautheyieldtomaturityisforecasttoequalthecouponrate.

Therefore,totalproceedsinthreeyearswillbe:$226.39+$1,000=

$1,226.39

.

.

Thenfindtherate(y

realized

.thatmakestheFVofthepurcha

priceequalto$1,226.39:

$960×(1+y

realized

.6=$1,226.39y

realized

=4.166%(miannual.

Alternatively,PV=−$960;FV=$1,226.39;N=6;PMT=$orI=

4.16%.

omingsofeachmeasure:

(i)Currentyielddoesnotaccountforcapitalgainsorlossonbonds

doesnotaccountfor

reinvestmentincomeoncouponpayments.

(ii)Yieldtomaturityassumesthebondishelduntilmaturityandthatall

couponincomecanbereinvestedatarateequaltotheyieldtomaturity.

(iii)Realizedcompoundyieldisaffectedbytheforecastof

reinvestmentrates,holdingperiod,andyieldofthebondattheend

oftheinvestor'sholdingperiod.

urityofeachbondis10years,andweassumethatcoupons

othbondsarellingatparvalue,the

currentyieldforeachbondiqualtoitscouponrate.

Iftheyielddeclinesby1%to5%(2.5%miannualyield.,theSentinal

bondwillincreainvalueto$107.79[n=20;i=2.5%;FV=100;PMT=3].

ThepriceoftheColinabondwillincrea,butonlytothecallpriceof

ntvalueofscheduledpaymentsisgreaterthan102,but

thecallpriceputsaceilingontheactualbondprice.

sareexpectedtofall,theSentinalbondismoreattractive:

sinceitisnotsubjecttocall,itspotentialcapitalgainsaregreater.

Ifratesareexpectedtori,

highercoupon(whichpresumablyiscompensationtoinvestorsforthe

ovideahigherrateofreturnthanthe

Sentinalbond.

eainthevolatilityofrateswillincreathevalueofthe

firm’sgodown,thefirmcan

callthebond,,greater

volatilitymakestheoptiontocallbackthebondmorevaluabletothe

kesthebondlessattractivetotheinvestor.

conversionvalue=Valueifconvertedintostock=20.83×$28=

$583.24

.

.

Conversionpremium=Bondprice–Marketconversionvalue

=$775.00–$583.24=$191.76

lfeaturerequiresthefirmtoofferahighercoupon(orhigher

promidyieldtomaturity)onthebondinordertocompensatethe

investorforthefirm'soptiontocallbackthebondataspecifiedprice

orsarewillingtograntthis

valuableoptiontotheissuer,butonlyforapricethatreflectsthe

iceisthehigher

promidyieldatwhichtheyarewillingtobuythebond.

rest

ratesfallsubstantiallysothatthelikelihoodofacallincreas,

investorswilltreatthebondasifitwill"mature"andbepaidoffatthe

calldate,therhand,ifrates

ri,thebondmustbepaidoffatthematuritydate,

asymmetrymeansthattheexpectedlifeofthebondislessthanthe

statedmaturity.

antageofacallablebondisthehighercoupon(andhigher

promidyieldtomaturity)ondisnever

called,thenaninvestorearnsahigherrealizedcompoundyieldona

callablebondissuedatparthananoncallablebondissuedatparonthe

ratesfallandthebondiscalled,thentheinvestorreceivesthecallprice

andthenhastoreinvesttheproceedsatinterestratesthatarelower

thivent,thefirm'ssavingsininterestpaymentsistheinvestor'sloss.

.

.

6.a.(iii)

b.(iii)Theyieldtomaturityonthecallablebondmustcompensate

theinvestorfortheriskofcall.

Choice(i)iswrongbecau,althoughtheownerofacallable

bondreceivesapremiumplustheprincipalintheeventofacall,

interestratethatmakesitprofitablefortheissuertocallthe

bondalsomakesitabaddealforthebond’sholder.

Choice(ii)iswrongbecauabondismoreapttobecalledwhen

ratesarelowwilltherebean

interestsavingfortheissuer.

c.(iii)

d.(ii)