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.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
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