《债券市场：分析与策略（第七版）》Ch03

Chapter3

MeasuringYield

1Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallLearningObjectivesAfterreadingthischapter,youwillunderstandhowtocalculatetheyieldonanyinvestmenthowtocalculatethecurrentyield,yieldtomaturity,yieldtocall,yieldtoput,andcashflowyieldhowtocalculatetheyieldforaportfoliohowtocalculatethediscountmarginforafloating-ratesecuritythethreepotentialsourcesofabond’sreturn2Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallLearningObjectives

(continued)Afterreadingthischapter,youwillunderstandhowtocalculatetheyieldonanyinvestmentwhatreinvestmentriskisthelimitationsofconventionalyieldmeasureshowtocalculatethetotalreturnforabondwhythetotalreturnissuperiortoconventionalyieldmeasureshowtousehorizonanalysistoassessthepotentialreturn3Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallComputingtheYieldorInternalRateofReturnonanyInvestmentTheyieldonanyinvestmentistheinterestratethatwillmakethepresentvalueofthecashflowsfromtheinvestmentequaltotheprice(orcost)oftheinvestment.Mathematically,theyieldonanyinvestment,y,istheinterestratethatsatisfiestheequation.

whereP

=priceoftheinvestment,CF

=cashflowinyeart=1,2,3,…N,y=yieldcalculatedfromthisrelationship(andalsocalledtheinternalrateofreturn),

N

=numberofyears.4Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallComputingtheYieldorInternalRateofReturnonanyInvestment(continued)Absentafinancialcalculatororcomputersoftware,solvingfortheyield(y)requiresatrial-and-error(iterative)procedure.Theobjectiveistofindtheyieldthatwillmakethepresentvalueofthecashflowsequaltotheprice.Keepinmindthattheyieldcomputedistheyieldfortheperiod.Thatis,ifthecashflowsaresemiannual,theyieldisasemiannualyield.Ifthecashflowsaremonthly,theyieldisamonthlyyield.Tocomputethesimpleannualinterestrate,theyieldfortheperiodismultipliedbythenumberofperiodsintheyear.5Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallComputingtheYieldorInternalRateofReturnonanyInvestment(continued)SpecialCase:InvestmentwithOnlyOneFutureCashFlowWhenthecasewherethereisonlyonefuturecashflow,itisnotnecessarytogothroughthetime-consumingtrial-and-errorproceduretodeterminetheyield.Wecanusethebelowequation:6Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallComputingtheYieldorInternalRateofReturnonanyInvestment(continued)AnnualizingYieldsToobtainaneffectiveannualyieldassociatedwithaperiodicinterestrate,thefollowingformulaisused:effectiveannualyield=(1+periodicinterestrate)m

–1

wherem

isthefrequencyofpaymentsperyear.Toillustrate,ifinterestispaidquarterlyandtheperiodicinterestrateis0.08/4=0.02,thenwehave:effectiveannualyield=(1.02)4–1=1.0824–1=0.0824or8.24%7Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallComputingtheYieldorInternalRateofReturnonanyInvestment(Continued)AnnualizingYieldsWecanalsodeterminetheperiodicinterestratethatwillproduceagivenannualinterestratebysolvingtheeffectiveannualyieldequationfortheperiodicinterestrate.Solving,wefindthatperiodicinterestrate=(1+effectiveannualyield)1/m

–1Toillustrate,iftheperiodicquarterlyinterestratethatwouldproduceaneffectiveannualyieldof12%,thenwehave:periodicinterestrate=(1.12)1/4–1=1.0287–1=0.0287or2.87%8Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallComputingtheYieldorInternalRateofReturnonanyInvestment(continued)AnnualizingYieldsWecanalsodeterminetheperiodicinterestratethatwillproduceagivenannualinterestratebysolvingtheeffectiveannualyieldequationfortheperiodicinterestrate.Solving,wefindthatperiodicinterestrate=(1+effectiveannualyield)1/m

–1Toillustrate,iftheperiodicquarterlyinterestratethatwouldproduceaneffectiveannualyieldof12%,thenwehave:periodicinterestrate=(1.12)1/4–1=1.0287–1=0.0287or2.87%9Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallConventionalYieldMeasuresBondyieldmeasurescommonlyquotedbydealersandusedbyportfoliomanagersare:CurrentYieldYieldToMaturityYieldToCallYieldToPutYieldToWorstCashFlowYieldYield(InternalRateofReturn)foraPortfolioYieldSpreadMeasuresforFloating-RateSecurities10Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallConventionalYieldMeasures(continued)1)CurrentYieldCurrentyieldrelatestheannualcouponinteresttothemarketprice.Theformulaforthecurrentyieldis:currentyield=annualdollarcouponinterest/priceThecurrentyieldcalculationtakesintoaccountonlythecouponinterestandnoothersourceofreturnthatwillaffectaninvestor’syield.Thetimevalueofmoneyisalsoignored.11Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallConventionalYieldMeasures(continued)2)YieldToMaturityTheyieldtomaturity

istheinterestratethatwillmakethepresentvalueofthecashflowsequaltotheprice(orinitialinvestment).Forasemiannualpaybond,theyieldtomaturityisfoundbyfirstcomputingtheperiodicinterestrate,y,whichsatisfiestherelationship:whereP

=priceofthebond,C

=semiannualcouponinterest(indollars),M

=maturityvalue(indollars),andn

=numberofperiods(numberofyearsmultipliedby2).12Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallConventionalYieldMeasures(Continued)2)YieldToMaturity(continued)Forasemiannualpaybond,doublingtheperiodicinterestrateordiscountrate(y)givestheyieldtomaturity.Theyieldtomaturitycomputedonthebasisofthismarketconventioniscalledthebond-equivalentyield:whereM

=maturityvalue(indollars),P

=priceofthebond,andn

=numberofperiods(numberofyearsmultipliedby2).Theyield-to-maturitycalculationtakesintoaccount(i)thecurrentcouponincome,(ii)anycapitalgainorlossrealizedbyholdingthebondtomaturity,and(iii)thetimingofthecashflows.13Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallConventionalYieldMeasures(Continued)3)YieldToCallThecallpriceisthepriceatwhichthebondmaybecalled.Thereisacallschedulethatspecifiesacallpriceforeachcalldate.Theyieldtocallassumesthattheissuerwillcallthebondatsomeassumedcalldateandthecallpriceisspecifiedinthecallschedule.Theyieldtocallcanbeexpressedasfollows:whereM

*=callprice(indollars)andn*=numberofperiodsuntiltheassumedcalldate(numberofyearstimes2)Forasemiannualpaybond,doublingtheperiodicinterestrate(y)givestheyieldtocallonabond-equivalentbasis.14Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallConventionalYieldMeasures(Continued)4)YieldToPutIfanissueisputable,itmeansthatthebondholdercanforcetheissuertobuytheissueataspecifiedprice.Theputschedulespecifieswhentheissuecanbeputandtheputprice.

Whenanissueisputable,ayieldtoputiscalculated.Theyieldtoputistheinterestratethatmakesthepresentvalueofthecashflowstotheassumedputdateplustheputpriceonthatdateassetforthintheputscheduleequaltothebond’sprice.Theformulafortheyieldtoputisthesameasfortheyieldtocall,butM

*isnowdefinedastheputpriceandn*isthenumberofperiodsuntiltheassumedputdate.Theprocedureisthesameascalculatingtheyieldtomaturityandtheyieldtocall.15Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallConventionalYieldMeasures(Continued)5)YieldToWorstApracticeintheindustryisforaninvestortocalculatetheyieldtomaturity,theyieldtoeverypossiblecalldate,andtheyieldtoeverypossibleputdate.Theminimumofalloftheseyieldsiscalledtheyieldtoworst.6)CashFlowYield

Amortizingsecuritiesinvolvecashflowsthatincludeinterestplusprincipalrepaymentandthecashfloweachperiodconsistsofthreecomponents:(i)couponinterest,(ii)scheduledprincipalrepayment,and(iii)prepayments.Foramortizingsecurities,marketparticipantscalculateacashflowyield,whichistheinterestratethatwillmakethepresentvalueoftheprojectedcashflowsequaltothemarketprice.Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallConventionalYieldMeasures(Continued)7)Yield(InternalRateofReturn)foraPortfolioTheyieldforaportfolioofbondsisnotsimplytheaverageorweightedaverageoftheyieldtomaturityoftheindividualbondissuesintheportfolio.Itiscomputedbydeterminingthecashflowsfortheportfolioanddeterminingtheinterestratethatwillmakethepresentvalueofthecashflowsequaltothemarketvalueoftheportfolio.17Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallConventionalYieldMeasures(Continued)8)YieldSpreadMeasuresforFloating-RateSecuritiesThecouponrateforafloating-ratesecuritychangesperiodicallybasedonthecouponresetformula.Thisformulaconsistsofthereferencerateandthequotedmargin.Sincethefuturevalueforthereferencerateisunknown,itisnotpossibletodeterminethecashflows.Thismeansthatayieldtomaturitycannotbecomputed.18Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallConventionalYieldMeasures(Continued)8)YieldSpreadMeasuresforFloating-RateSecuritiesInstead,thereareseveralconventionalmeasuresusedasmarginorspreadmeasurescitedbymarketparticipantsforfloaters.Theseincludespreadforlife(orsimplemargin),adjustedsimplemargin,adjustedtotalmargin,anddiscountmargin.Themostpopularofthesemeasuresisthediscountmargin,whichestimatestheaveragemarginoverthereferenceratethattheinvestorcanexpecttoearnoverthelifeofthesecurity.Exhibit3-1showsthecalculationofthediscountmarginforasix-yearfloating-ratesecurity.(SeeOverhead3-20.)19Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallExhibit3-1CalculationoftheDiscountMarginforaFloating-RateSecurity

Floating-ratesecurity

Maturity:sixyears

Couponrate:referencerate+80basispoints

ReseteverysixmonthsPresentValueofCashFlowatAssumedAnnualMargin(basispoints)PeriodReferenceRateCashFlowa

80848896100110%5.45.12335.12245.12145.11955.11852105.44.86094.85904.85724.85354.85163105.44.61184.60924.60664.60134.59874105.44.37554.37224.36894.36234.35905105.44.15144.14744.14354.13564.13176105.43.93873.93423.92973.92083.91637105.43.73693.73193.72703.71713.71228105.43.54543.54013.53473.52403.51869105.43.36383.35803.35233.34093.335210105.43.19143.18543.17943.16733.161311105.43.02793.02163.01533.00282.99651210105.456.072955.945455.818255.564755.4385PresentValue=100.000099.826999.654199.309899.1381aForperiods1–11:cashflow=100(referencerate+assumedmargin)(0.5);forperiod12:cashflow=100(referencerate+assumedmargin)(0.5)+100.20Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallPotentialSourcesofaBond’sDollarReturnAninvestorwhopurchasesabondcanexpecttoreceiveadollarreturnfromoneormoreofthesesources:theperiodiccouponinterestpaymentsmadebytheissueranycapitalgain(orcapitalloss—negativedollarreturn)whenthebondmatures,iscalled,orissoldinterestincomegeneratedfromreinvestmentoftheperiodiccashflowsThecurrentyieldconsidersonlythecouponinterestpayments.Theyieldtomaturity,yieldtocall,andcashflowyieldalltakeintoaccountthethreecomponents.21Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallPotentialSourcesofaBond’sDollarReturn

(continued)DeterminingtheInterest-On-InterestDollarReturnTheinterest-on-interestcomponentcanrepresentasubstantialportionofabond’spotentialreturn.Thecouponinterestplusinterestoninterestcanbefoundbyusingthefollowingequation:whereCisthecouponinterestr

isthesemiannualreinvestmentratenisthenumberofperiods22Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallPotentialSourcesofaBond’sDollarReturn

(continued)DeterminingtheInterest-On-InterestDollarReturnThetotaldollaramountofcouponinterestisfoundbymultiplyingthesemiannualcouponinterestbythenumberofperiods:totalcouponinterest=nCwhereCisthecouponinterest,r

isthesemiannualreinvestmentrate,andnisthenumberofperiods.Theinterest-on-interestcomponentisthenthedifferencebetweenthecouponinterestplusinterestoninterestandthetotaldollarcouponinterest,asexpressedbytheformula23Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallPotentialSourcesofaBond’sDollarReturn

(continued)DeterminingtheInterest-On-InterestDollarReturnExample.Assumethatthecouponinterest(C)is$50,thesemiannualreinvestmentrate(r)is4.5%,andthenumberofperiods(n)is40.Whatistheinterest-on-interest?Usingourequationforinterestoninterestandinsertinginourgivenvaluesweget:24Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallPotentialSourcesofaBond’sDollarReturn

(continued)YieldToMaturityandReinvestmentRiskTheinvestorrealizestheyieldtomaturityonlyifthebondisheldtomaturityandthecouponpaymentscanbereinvestedatthecomputedyieldtomaturity.Reinvestmentriskistherisk

thatfuturereinvestmentrateswillbelessthantheyieldtomaturityatthetimethebondispurchased.Therearetwocharacteristicsofabondthatdeterminetheimportanceoftheinterest-on-interestcomponentandthereforethedegreeofreinvestmentrisk:maturityandcoupon.25PotentialSourcesofaBond’sDollarReturn

(continued)YieldToMaturityandReinvestmentRiskForagivenyieldtomaturityandagivencouponrate,thelongerthematurity,themoredependentthebond’stotaldollarreturnisontheinterest-on-interestcomponentinordertorealizetheyieldtomaturityatthetimeofpurchase.Foragivenmaturityandagivenyieldtomaturity,highercouponrateswillmakethebond’stotaldollarreturnmoredependentonthereinvestmentofthecouponpaymentsinordertoproducetheyieldtomaturityanticipatedatthetimeofpurchase.26PotentialSourcesofaBond’sDollarReturn

(continued)CashFlowYieldandReinvestmentRiskForamortizingsecurities,reinvestmentriskisevengreaterthanfornonamortizingsecurities.Thereasonisthattheinvestormustnowreinvesttheperiodicprincipalrepaymentsinadditiontotheperiodiccouponinterestpayments.Typically,fornonamortizingsecuritiestheborrowercanacceleratetheperiodicprincipalrepayment,inparticular,aborrowerwilltendtoprepaywheninterestratesdecline.Ifaborrowerprepayswheninterestratesdecline,theinvestorfacesgreaterreinvestmentriskbecauseheorshemustreinvesttheprepaidprincipalatalowerinterestrate.27Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallTotalReturnTheyieldtomaturityisapromisedyieldbecauseatthetimeofpurchaseaninvestorispromisedayield,asmeasuredbytheyieldtomaturity,ifbothofthefollowingconditionsaresatisfied:thebondisheldtomaturityallcouponinterestpaymentsarereinvestedattheyieldtomaturityThetotalreturnisameasureofyieldthatincorporatesanexplicitassumptionaboutthereinvestmentrate.Theyield-to-callmeasureissubjecttothesameproblemsastheyieldtomaturitybecauseitassumesthatthe:bondwillbehelduntilthefirstcalldatecouponinterestpaymentswillbereinvestedattheyieldtocall28Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallTotalReturn

(continued)ComputingtheTotalR