债券市场分析与策略第7版答案3.docx

债券市场分析与策略第7版答案3.docx

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债券市场分析与策略第7版答案3

CHAPTER3

MEASURINGYIELD

CHAPTERSUMMARY

InChapter2weshowedhowtodeterminethepriceofabond,andwedescribedtherelationshipbetweenpriceandyield.Inthischapterwediscussvariousyieldmeasuresandtheirmeaningforevaluatingtherelativeattractivenessofabond.Webeginwithanexplanationofhowtocomputetheyieldonanyinvestment.

COMPUTINGTHEYIELDORINTERNALRATEOFRETURN

ONANYINVESTMENT

Theyieldonanyinvestmentistheinterestratethatwillmakethepresentvalueofthecashflowsfromtheinvestmentequaltotheprice（orcost）oftheinvestment.

Mathematically,theyieldonanyinvestment,y,istheinterestratethatsatisfiestheequation.

P=

whereCFt=cashflowinyeart,P=priceoftheinvestment,N=numberofyears.Theyieldcalculatedfromthisrelationshipisalsocalledtheinternalrateofreturn.

Solvingfortheyield（y）requiresatrial-and-error（iterative）procedure.Theobjectiveistofindtheyieldthatwillmakethepresentvalueofthecashflowsequaltotheprice.Keepinmindthattheyieldcomputedistheyieldfortheperiod.Thatis,ifthecashflowsaresemiannual,theyieldisasemiannualyield.Ifthecashflowsaremonthly,theyieldisamonthlyyield.Tocomputethesimpleannualinterestrate,theyieldfortheperiodismultipliedbythenumberofperiodsintheyear.

SpecialCase:

InvestmentwithOnlyOneFutureCashFlow

Whenthecasewherethereisonlyonefuturecashflow,itisnotnecessarytogothroughthetime-consumingtrial-and-errorproceduretodeterminetheyield.Wecanusethebelowequation.

.

AnnualizingYields

Toobtainaneffectiveannualyieldassociatedwithaperiodicinterestrate,thefollowingformulaisused:

effectiveannualyield=（1+periodicinterestrate）m–1

wheremisthefrequencyofpaymentsperyear.Toillustrate,ifinterestispaidquarterlyandtheperiodicinterestrateis8%/4=2%）,thenwehave:

theeffectiveannualyield=（1.02）4–1=1.0824–1=0.0824or8.24%.

Wecanalsodeterminetheperiodicinterestratethatwillproduceagivenannualinterestratebysolvingtheeffectiveannualyieldequationfortheperiodicinterestrate.Solving,wefindthat:

periodicinterestrate=（1+effectiveannualyield）1/m–1.Toillustrate,iftheperiodicquarterlyinterestratethatwouldproduceaneffectiveannualyieldof12%,thenwehave:

periodicinterestrate=（1.12）1/4–1=1.0287–1=0.0287or2.87%.

CONVENTIONALYIELDMEASURES

Thereareseveralbondyieldmeasurescommonlyquotedbydealersandusedbyportfoliomanagers.Thesearedescribedbelow.

CurrentYield

Currentyieldrelatestheannualcouponinteresttothemarketprice.Theformulaforthecurrentyieldis:

currentyield=annualdollarcouponinterest/price.Thecurrentyieldcalculationtakesintoaccountonlythecouponinterestandnoothersourceofreturnthatwillaffectaninvestor’syield.Thetimevalueofmoneyisalsoignored.

YieldtoMaturity

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

P=

whereP=priceofthebond,C=semiannualcouponinterest（indollars）,M=maturityvalue（indollars）,andn=numberofperiods（numberofyearsx2）.

Forasemiannualpaybond,doublingtheperiodicinterestrateordiscountrate（y）givestheyieldtomaturity,whichunderstatestheeffectiveannualyield.Theyieldtomaturitycomputedonthebasisofthismarketconventioniscalledthebond-equivalentyield.

Itismucheasiertocomputetheyieldtomaturityforazero-couponbondbecausewecanuse:

.

Theyield-to-maturitycalculationtakesintoaccountnotonlythecurrentcouponincomebutalsoanycapitalgainorlossthattheinvestorwillrealizebyholdingthebondtomaturity.Inaddition,theyieldtomaturityconsidersthetimingofthecashflows.

YieldToCall

Thepriceatwhichthebondmaybecalledisreferredtoasthecallprice.Forsomeissues,thecallpriceisthesameregardlessofwhentheissueiscalled.Forothercallableissues,thecallpricedependsonwhentheissueiscalled.Thatis,thereisacallschedulethatspecifiesacallpriceforeachcalldate.

Forcallableissues,thepracticehasbeentocalculateayieldtocallaswellasayieldtomaturity.Theyieldtocallassumesthattheissuerwillcallthebondatsomeassumedcalldateandthecallpriceisthenthecallpricespecifiedinthecallschedule.Typically,investorscalculateayieldtofirstcalloryieldtonextcall,ayieldtofirstparcall,andyieldtorefunding.

Mathematically,theyieldtocallcanbeexpressedasfollows:

P=

whereM*=callprice（indollars）andn*=numberofperiodsuntiltheassumedcalldate（numberofyearstimes2）.Forasemiannualpaybond,doublingtheperiodicinterestrate（y）givestheyieldtocallonabond-equivalentbasis.

YieldToPut

Ifanissueisputable,itmeansthatthebondholdercanforcetheissuertobuytheissueataspecifiedprice.Aswithacallableissue,aputableissuecanhaveaputschedule.Theschedulespecifieswhentheissuecanbeputandtheprice,calledtheputprice.

Whenanissueisputable,ayieldtoputiscalculated.Theyieldtoputistheinterestratethatmakesthepresentvalueofthecashflowstotheassumedputdateplustheputpriceonthatdateassetforthintheputscheduleequaltothebond’sprice.Theformulaisthesameasfortheyieldtocall,butM*isnowdefinedastheputpriceandn*isthenumberofperiodsuntiltheassumedputdate.Theprocedureisthesameascalculatingyieldtomaturityandyieldtocall.

YieldToWorst

Apracticeintheindustryisforaninvestortocalculatetheyieldtomaturity,theyieldtoeverypossiblecalldate,andtheyieldtoeverypossibleputdate.Theminimumofalloftheseyieldsiscalledtheyieldtoworst.

CashFlowYield

Somefixedincomesecuritiesinvolvecashflowsthatincludeinterestplusprincipalrepayment.Suchsecuritiesarecalledamortizingsecurities.Foramortizingsecurities,thecashfloweachperiodconsistsofthreecomponents:

（i）couponinterest,（ii）scheduledprincipalrepayment,and（iii）prepayments.Foramortizingsecurities,marketparticipantscalculateacashflowyield.Itistheinterestratethatwillmakethepresentvalueoftheprojectedcashflowsequaltothemarketprice.

Yield（InternalRateofReturn）foraPortfolio

Theyieldforaportfolioofbondsisnotsimplytheaverageorweightedaverageoftheyieldtomaturityoftheindividualbondissuesintheportfolio.Itiscomputedbydeterminingthecashflowsfortheportfolioanddeterminingtheinterestratethatwillmakethepresentvalueofthecashflowsequaltothemarketvalueoftheportfolio.

YieldSpreadMeasuresforFloating-RateSecurities

Thecouponrateforafloating-ratesecuritychangesperiodicallybasedonthecouponresetformula.Thisformulaconsistsofthereferencerateandthequotedmargin.Sincethefuturevalueforthereferencerateisunknown,itisnotpossibletodeterminethecashflows.Thismeansthatayieldtomaturitycannotbecalculated.Instead,thereareseveralconventionalmeasuresusedasmarginorspreadmeasurescitedbymarketparticipantsforfloaters.Theseincludespreadforlife（orsimplemargin）,adjustedsimplemargin,adjustedtotalmargin,anddiscountmargin.

Themostpopularofthesemeasuresisdiscountmargin.Thismeasureestimatestheaveragemarginoverthereferenceratethattheinvestorcanexpecttoearnoverthelifeofthesecurity.

POTENTIALSOURCESOFABOND’SDOLLARRETURN

Aninvestorwhopurchasesabondcanexpecttoreceiveadollarreturnfromoneormoreofthesesources:

（i）theperiodiccouponinterestpaymentsmadebytheissuer,（ii）anycapitalgain（orcapitalloss—negativedollarreturn）whenthebondmatures,iscalled,orissold,and（iii）interestincomegeneratedfromreinvestmentoftheperiodiccashflows

Anymeasureofabond’spotentialyieldshouldtakeintoconsiderationeachofthesethreepotentialsourcesofreturn.Thecurrentyieldconsidersonlythecouponinterestpayments.Noconsiderationisgiventoanycapitalgain（orloss）orinterestoninterest.Theyieldtomaturitytakesintoaccountcouponinterestandanycapitalgain（orloss）.Italsoconsiderstheinterest-on-interestcomponent.Implicitintheyield-to-maturitycomputationistheassumptionthatthecouponpaymentscanbereinvestedatthecomputedyieldtomaturity.

Theyieldtocallalsotakesintoaccountallthreepotentialsourcesofreturn.Inthiscase,theassumptionisthatthecouponpaymentscanbereinvestedattheyieldtocall.Therefore,theyield-to-callmeasuresuffersfromthesamedrawbackastheyieldtomaturityinthatitassumescouponinterestpaymentsarereinvestedatthecomputedyieldtocall.Also,itpresupposesthattheissuerwillcallthebondonsomeassumedcalldate.

Thecashflowyieldalsotakesintoconsiderationallthreesourcesasisthecasewithyieldtomaturity,butitmakestwoadditionalassumptions.First,itassumesthattheperiodicprincipalrepaymentsarereinvestedatthecomputedcashflowyield.Second,itassumesthattheprepaymentsprojectedtoobtainthecashflowsareactuallyrealized.

DeterminingtheInterest-On-InterestDollarReturn

Theinterest-on-interestcomponentcanrepresentasubstantialportionofabond’spotentialreturn.Thecouponinterestplusinterestoninterestcanbefoundbyusingthefollowingequation:

whererdenotethesemiannualreinvestmentrate.

Thetotaldollaramountofcoupon