Part1BondValuation

Whatisabond?DifferentissuersDifferentmaturitiesDifferentstylesBondratingTerminologyBondpricingZerocouponbondOrdinarycouponbondExamplesConversequestionSomeconceptsYieldtomaturityTermstructureFuturespotrateForwardratePart1BondValuation

Whati1Whatisabond?Bondsareloansthatinvestorsmaketocorporationsandgovernments.Theborrowersgetthecashtheyneedwhilethelendersearninterest.(TheWallStreetJournal–Guidetounderstandingmoney&investing)a. Differentissuers–Government(Treasurysecurities:T-bills,T-notes,T-bonds,Foreign-targetedT-notes),states,cities,counties,andtowns(Municipalbonds),corporations(corporatebonds)Whatisabond?Bondsareloans2Whatisabond?Differentmaturities–shortterm(lessthan1years,e.g.T-bills)vs.longterm(T-notes,T-bonds,etc)Differentstyles–Couponbonds(mostofcorporatebonds)vs.zero-couponbonds(e.g.STRIPS–SeparateTradingofRegisteredInterestandPrincipalofSecurities) →WhyareSTRIPSpopularespeciallyforanagentwhocandefertax?Whatisabond?Differentmatur3Whatisabond?d.Bondrating–HardtoevaluatethedefaultriskinabondAgencies:S&P(StandardandPoor’s)andMoody’sInvestmentgrade:AtleastBBB(S&P)orBaa(Moody’s)Junkbonds:AtmostCC(S&P)orCa(Moody’s)Whatisabond?d.Bondrating4Whatisabond?Moody’sLong-TermRatingDefinitions-ExampleAaa ObligationsratedAaaarejudgedtobeofthehighestquality,withminimalcreditrisk.Aa ObligationsratedAaarejudgedtobeofhighqualityandaresubjecttoverylowcreditrisk.A ObligationsratedAareconsideredupper-mediumgradeandaresubjecttolowcreditrisk.Baa ObligationsratedBaaaresubjecttomoderatecreditrisk.Theyareconsideredmedium-gradeandassuchmaypossesscertainspeculativecharacteristics.Whatisabond?Moody’sLong-Te5Whatisabond?Ba ObligationsratedBaarejudgedtohavespeculativeelementsandaresubjecttosubstantialcreditrisk.B ObligationsratedBareconsideredspeculativeandaresubjecttohighcreditrisk.Caa ObligationsratedCaaarejudgedtobeofpoorstandingandaresubjecttoveryhighcreditrisk.Ca ObligationsratedCaarehighlyspeculativeandarelikelyin,orverynear,default,withsomeprospectofrecoveryofprincipalandinterest.C ObligationsratedCarethelowestratedclassofbondsandaretypicallyindefault,withlittleprospectforrecoveryofprincipalorinterest.Whatisabond?Ba6Whatisabond?e.TerminologyParvalue(facevalue):Dollaramountofthebondatthetimeitisissued.Itappearsseveraltimesonthefaceofbond.Usually,multipliesof$1,000.Interestrate(couponrate):Percentageofparvaluethatispaidtothebond-holderonaregularbasis.

Whatisabond?e.Terminology7Bondpricinga. Zerocouponbond Price=PV(Facevalue)=b. Ordinarycouponbond Price=PV(Coupons)+PV(Facevalue)

=Bondpricinga. Zerocouponbon8Bondpricingc. ExamplesSupposeyounowownaT-bondthathasacouponrateof5%,facevalueof$1,000,andmaturityof5years.Acomparableinvestmenthasrateofreturnof6%.AssumeannualcouponsBondpricingc. Examples9BondpricingAssumesemi-annualcoupons

Whyistheredifferenceofbondspricesbetweenannualcouponsandsemi-annualcoupons? 2conflictingfactors:Moreimmediatepaymentand

higherdiscountrate.→cannotbedeterminedsimply.BondpricingAssumesemi-annual10Bondpricingd. Conversequestion Sincethemarketdecidesthebondprice,thenwecanbackout“r”.So,this“r”andthepriceofabondhaveone-to-onerelationship.Example:Supposeabondwithfacevalueof$1,000,coupon rateof11.5%,andmaturityof5yearsisbeingtraded at$1161.84.Then,whatis“r”forthisbond?

1161.84=115/(1+r)+115/(1+r)2+… +115/(1+r)5+1000/(1+r)5. r=0.075,butusuallythroughtrialanderrormethod.Bondpricingd. Converseques11Someconcepta.Yieldtomaturity(YTM)–“r”inournotation.

IRRforabond.Financialnewspapersreportthisnumber. IRR:InternalrateofreturnSomeconcepta.Yieldtomatur12SomeconceptExample:WSJ5/15/02 MaturityASKRATEMO/YRBIDASKEDCHGYLD6.500May05107:6107:763.90Someconcept13Someconcept“6.500”meanscouponrate(%)“May05”meansmaturity“107:6”isbidpriceanditisnot$107.6,but(107and6/32)%ofFV(usually,$1,000)→$1,071.875“107:7”isaskprice→(107and7/32)%ofFV,$1,072.1875 Notice:Spreadisreallysmall,1/32(about0.03)%of FV,Spreadofstockprice–around1-2%Someconcept“6.500”meanscoup14Someconcept“6”meansthatthebondpriceisincreasedby$6/32fromthepreviousday.“3.90”isYTM:Thisisbackedoutbyusing 1072.19=65/(1+r)+65/(1+r)2

+65/(1+r)3+1000/(1+r)3 →r≈0.039ProblemsofYTM–Theassumptionofthesamediscountratefromt=1toT.Itisveryunusual. →ThinkthejobofMr.Greenspan.Someconcept“6”meansthatthe15Someconceptb.Changeofinterestrateandbondprice

Inourexample(priceis$1072.19),ifinterestrate(notcouponrate)isincreasedinto15%,thenthebondpricewillbedecreased. PVat15%=65/(1.15)+65/(1.15)2

+65/(1.15)3+1000/(1.15)3 =805.93 So,priceisdecreasedwithhigherinterestrate.Someconceptb.Changeofinte16Someconceptc.RateofreturnWhatisthedifferencebetweenrateofreturnandYTMforabond?Fromtheaboveexample,after1year,thebondpriceis1077.10.Then,whatwouldberateofreturnandYTM?First,rateofreturnis($65+$1,077.10-$1,072.19)/$1,072.19=6.52%Someconceptc.Rateofreturn17SomeconceptForYTM,1077.10(PVatyear1)=65/(1+r)+65/(1+r)2+1000/(1+r)2 Then,YTM=2.5%WhatistherelationshipbetweenYTMandtherateofreturn?→Ifinterestratesdonotchange,thebondpricechangeswithtimesothatthetotalreturnonthebondisequaltotheYTM.SomeconceptForYTM,1077.10(18Someconceptd.Spotrateandforwardrate1Yearr1=0f1r1=0f1r2=0r2representedby1yearrate1f2att=0Someconceptd.Spotrateand19Someconcepti.Futurespotrate

Theactualinterestrateinthefuture,e.g.theactual interestratefromyear1toyear2atyear1.YTMonzerocouponbond. ii.ForwardrateSinceafuturespotrateisunknownnow,wehavetoinferitfromvariousbondsprices.Ingeneral,(1+t-1ft)(1+1rt-1)(t-1)=(1+1rt)tSomeconcepti.Futurespotra20SomeconceptExampleforspotrateandforwardrate:Considerfollowing4bondswithfacevalueof$100andannualcoupons.MaturityCouponPriceYTM1094.346.0026.5100.566.1936.098.986.38311.0112.286.37SomeconceptExampleforspotr21SomeconceptWhatis2yearspotrate(r2)?

100.56=6.5/(1.06)+6.5/(1+r2)2

+100/(1+r2)2 So,r2=6.2%SomeconceptWhatis2yearspo22SomeconceptThen,r3? 98.98=6.0/(1.06)+6.0/(1+0.062)2

+106/(1+r3)3

So,r3=6.4% SomeconceptThen,r3?23SomeconceptWhatare1f2?

(1+r2)2=(1+r1)*(1+1f2) Intheaboveproblem,weknowthatr2is 6.2%andr1is6%. So,1f2is6.4%And2f3? Withthesamelogic,2f3=6.8%SomeconceptWhatare1f2?24Someconcepte.TermstructureTherelationshipbetweenshort-andlong-terminterestratesataparticulartimeonbondsthatarefundamentallysimilarexceptmaturity.Therelationshipcanbepresentedgraphicallyasayieldcurve.Usually,theyieldcurveisupwardsloping.Inourexample,r1<r2<r3.Advancedpoints:Termstructurecanberepresentedbytherelationbetweenspotrate(e.g.r2)andforwardrate(1f2).Ifr2>r1,then1f2>r2,andifr2<r1,then1f2<r2.Someconcepte.Termstructure25Part2BONDRISKDefaultRiskCallRiskMarketRiskPart2BONDRISKDefaultRis26DefaultRisk(1)DefaultRisk:Uncertaintythattherealizedreturnwilldeviatefromtheexpectedreturnbecausetheissuerwillfailtomeetthecontractualobligationsspecifiedintheindenture.Themajorconcernisfailuretomeetinterestandprincipalpayments.DefaultRisk(1)DefaultRisk:27DefaultRisk(2)Mostinvestorsdonotdirectlyaccessabond’sdefaultrisk,butinsteadusethequalityratingsprovidedbyMoody’s,S&P,andFetchtoevaluatethedegreeofrisk.AAA,AA,A,BBB,……,C,……DefaultRates:.12%peryearforallbondssinceWWII.1980s:3.27%peryearonjunkbonds.1990-1991:9%onjunkbondsDefaultRisk(2)Mostinvestors28DefaultRisk(3)Empiricalstudiesthatlookedattherelationbetweendefaultriskpremium(RP)andthestateoftheeconomy.DefaultRisk(3)Empiricalstudi29DefaultRisk(4)Studylookedattheyieldcurvesfordifferentqualitybonds.FoundthattheYCforlowerqualitybondstendedonaveragetobenegativelysloped.Reason:Greaterconcernovertherepaymentofprincipalonlowqualitybonds.DefaultRisk(4)Studylookedat30DefaultRisk(5)AppliedtheEvansandArchermethodologytobondportfoliosgroupedintermsoftheirqualityratings.FoundthatlowerqualitybondportfolioshadlessriskbecauseoftheirlowercorrelationsDefaultRisk(5)AppliedtheEva31CallRisk(1)CallRisk:Uncertaintythattherealizedreturnwilldeviatefromtheexpectedreturnbecausetheissuercallsthebond,forcingtheinvestortoreinvestinamarketwithlowerrates.Note:Whenabondiscalledtheholderreceivesthecallprice(CP).SincetheCPusuallyexceedstheprincipal,thereturntheinvestorreceivesoverthecallperiodisoftengreaterthantheinitialYTM.Theinvestor,though,usuallyhastoreinvestinamarketwithlowerrateswhichoftencauseshisreturnfortheinvestmentperiodtobelessthantheinitialYTM.CallRisk(1)CallRisk:Uncert32CallRisk(2)Example:ComparetheARRforthecallperiodwiththeARRfortheinvestmentperiodforabondthatiscalled.Buy:10-year,10%annualcouponbondatpar($1000);callableat110:CP=$1100.Assume:HD=10years.FlatYCat10%.YCstaysat10%untiltheendofyear3.Year3,theYCshiftsdowntoaflat8%andthebondiscalled.Investorreinvestat8%forthenext7years.CallRisk(2)Example:Compare33CallRisk(3)CallRisk(3)34CallRisk(4)CallRiskPremiumPriceCompression:Callfeaturesputlimitationsontheprice-yieldcurve.Attheratewherethebondcouldbecalled(YTM*),theYCflattens,withthepriceequaltotheCP.CallRisk(4)CallRiskPremium35CallRisk(5)Points:NeedforavaluationmodelforcallablebondsdifferentfromthePVmodel.A10-yearcallablebondmaybemorelikea3-yearbond.BinomialTreeModeloroptionpricingmodel.CallRisk(5)Points:36MarketRisk(1)MarketRisk:Uncertaintythattherealizedreturnwilldeviatefromtheexpectedreturnbecauseinterestrateschange.Recall,thereturnonabondcomesfromcoupons,theinterestearnedfromreinvestingcoupons(interestoninterest),andcapitalgains.Achangeinratesaffectsinterestoninterestandcapitalgainsorlosses.MarketRisk(1)MarketRisk:Un37MarketRisk(2)Note:OnewaytoeliminatemarketriskistobuyaPDBwithM=HD.MarketRisk(2)Note:Onewayt38MarketRisk(3)Example:HD=3.5yearsBuyNoncallable10-year,10%annualbondatpar(F=$1000)atYTMof10%.YCflatat10%.IfthereisnochangeintheYC,yourARRfor3.5yearswillbe10%:MarketRisk(3)Example:39BondValuation債券定價(jià)現(xiàn)代金融市場(chǎng)概論教學(xué)課件40MarketRisk(4)Example:IftheYCshifteduptoaflat12%shortlyafteryouboughtthebondandremainedthereuntilyourHD,thenyourARRfor3.5yearswouldbe8.23%:MarketRisk(4)Example:41MarketRisk(5)Example:IftheYCshifteddowntoaflat8%shortlyafteryouboughtthebondandremainedthereuntilyourHD,thenyourARRfor3.5yearswouldbe11.95%:MarketRisk(5)Example:42MarketRisk(6)Relation:Ratechangehavetwooppositeeffects:InversepriceeffectDirectinterestoninteresteffectWhethertheARRvariesdirectlyorinverselytoratechangesdependsonwhicheffectdominates.Inthiscase(10-year,10%couponbondwithHD=3.5yrs),theinversepriceeffectdominates.ThiscausestheARRtovaryinverselywithratechanges.MarketRisk(6)Relation:43MarketRisk(7)WithanHDof3.5years,a4year,20%bondhasaninterestoninteresteffectwhichdominatesthepriceeffect.Thisbond’sARRwouldvarydirectlywithratechanges.MarketRisk(7)WithanHDof344MarketRisk(8)Note:Itispossibletobuyabondinwhichthepriceandinterestoninteresteffectexactlyoffseteachother.Inthiscase,theARRwillbeinvarianttochangesinrates.Forexample,a4-year,9%couponbondpurchasedwhentheYCisat10%(Po=968.30)wouldyieldanHDvalueatyear3.5of1351andanARRof10%,regardlessofrates.MarketRisk(8)Note:Itispos45MarketRisk(9)BondImmunization:Whatisdistinctiveabouta4-year,9%couponbondwhentheYTMis10%isthatithasaduration(D)equalto3.5years--duration=HD.Durationcanbedefinedastheweightedaverageofthetimeperiods.BondimmunizationisbondstrategyofminimizingmarketriskbybuyingabondoraportfolioofbondswithadurationequaltotheHD.MarketRisk(9)BondImmunizati46MarketRisk(9)Durationof4-year,9%couponBond:MarketRisk(9)Durationof4-y47Duration(1)Durationisalsoameasureofabond’spricesensitivitytointerestratechanges.Thismeasureofdurationisknownasthemodifiedduration;themeasureofdurationasaweightedaverageofthetimeperiodsisknownasMacauley’sduration.Duration(1)Durationisalsoa48Duration(2)Duration(2)49Duration(3)Amodifieddurationmeasureforabondwhichpayscouponseachperiodanditsprincipalatmaturity(note:lety=YTM):Duration(3)Amodifiedduratio50Duration(4)AnnualizedDuration:Durationisdefinedintermsofthelengthoftheperiodbetweenpayments.IftheCFsaredistributedannually,thendurationisinyears;ifCFsaresemi-annual,thendurationismeasuredinhalfyears.Theconventionistoexpresseddurationasanannualmeasure.Theannualizeddurationisobtainedbydividingdurationbythenumberofpaymentsperyear(n):Duration(4)AnnualizedDuratio51Duration(5)Example:Thedurationinhalf-yearsfora10-year,9%couponbondsellingatpar(F=100)andpayingcouponssemiannuallyis-13anditsannualizeddurationis-6.5:Duration(5)Example:Thedura52Duration(6)DescriptiveParameter:Measureofabond’spricesensitivitytointerestratechanges--ameasureofabond’svolatility.Note,durationisconsistentwithourbondpricerelationDuration(6)DescriptiveParame53Duration(7)DefineStrategies:Usedurationtodefineactive(speculative)andpassivebondstrategies.Examples:Rate-AnticipationSwap:Ratesexpectedtodecreaseacrossallmaturities,golonginhighdurationbonds.Ratesexpectedtoincreaseacrossallmaturities,changebondportfoliocompositionsothatithaslowerdurationbonds.BondImmunizationStrategy.Duration(7)DefineStrategies:54Duration(8)Estimatethepercentagechangeinabond’spriceforagivenchangeinrates:For10-year,9%bond,anincreaseintheannualizedyieldby10BP(.09to.0910)wouldleadtoa.65%decreaseinprice(theactualis.6476%).A200BPincrease(9%to11%)wouldleadtoanestimatedpricedecreaseof13%;theactualdecrease,though,isonly12%.Duration(8)Estimatetheperce55Convexity(1)Durationisameasureoftheslopeoftheprice-yieldcurveatagivenpoint--first-order

derivative.Convexityisameasureofthechangeintheslopeoftheprice-yieldcurve--second-orderderivative.

Convexitymeasureshowbowed-shapedtheprice-yieldcurveis.Convexity(1)Durationisamea56Convexity(2)Property:Thegreaterabond’sconvexity,thegreateritscapitalgainsandthesmalleritscapitallossesforgivenabsolutechangesinyields.Convexity(2)Property:Thegre57Convexity(3)Measure:Convexity(3)Measure:58Convexity(4)Theconvexitymeasureforabondwhichpayscouponseachperiodanditsprincipalatmaturity:Convexity(4)Theconvexityme59Convexity(5)Example:Theconvexityinhalf-yearsfora10-year,9%couponbondsellingatpar(F=100)andpayingcouponssemiannuallyis225.43anditsannualizedconvexityis56.36:Convexity(5)Example:Thecon60Convexity(6)DescriptiveParameter:Greaterk-gainsandsmallerk-lossesthegreaterabond’sconvexity.Estimationof:UsingTaylorExpansion,abetterestimateofpricechangestodiscretechangesinyieldthanthedurationmeasurecanbeobtainedbycombiningdurationandconvexitymeasures.Convexity(6)DescriptiveParam61Convexity(7)TaylorExpansion:For10-year,9%bond,anincreaseintheannualizedyieldby200BP(9%to11%)wouldleadtoanestimated11.87%decreaseinpriceusingTaylorExpansion(theactualis12%):Convexity(7)TaylorExpansion:62Convexity(8)Note:UsingTaylorExpansionthepercentageincreasesinpricearenotsymmetricalwiththepercentagedecreasesforgivenabsolutechangesinyields.Convexity(8)Note:UsingTaylo63DerivationofDurationandConvexity(1)Duration:Takederivativewithrespecttoy:DerivationofDurationandCon64DerivationofDurationandConvexity(2)

DerivationofDurationandCon65DerivationofDurationandConvexity(3)

DerivationofDurationandCon66DerivationofDurationandConvexity(4)Convexity:TakethederivativeofDerivationofDurationandCon67Part1BondValuation

Whatisabond?DifferentissuersDifferentmaturitiesDifferentstylesBondratingTerminologyBondpricingZerocouponbondOrdinarycouponbondExamplesConversequestionSomeconceptsYieldtomaturityTermstructureFuturespotrateForwardratePart1BondValuation

Whati68Whatisabond?Bondsareloansthatinvestorsmaketocorporationsandgovernments.Theborrowersgetthecashtheyneedwhilethelendersearninterest.(TheWallStreetJournal–Guidetounderstandingmoney&investing)a. Differentissuers–Government(Treasurysecurities:T-bills,T-notes,T-bonds,Foreign-targetedT-notes),states,cities,counties,andtowns(Municipalbonds),corporations(corporatebonds)Whatisabond?Bondsareloans69Whatisabond?Differentmaturities–shortterm(lessthan1years,e.g.T-bills)vs.longterm(T-notes,T-bonds,etc)Differentstyles–Couponbonds(mostofcorporatebonds)vs.zero-couponbonds(e.g.STRIPS–SeparateTradingofRegisteredInterestandPrincipalofSecurities) →WhyareSTRIPSpopularespeciallyforanagentwhocandefertax?Whatisabond?Differentmatur70Whatisabond?d.Bondrating–HardtoevaluatethedefaultriskinabondAgencies:S&P(StandardandPoor’s)andMoody’sInvestmentgrade:AtleastBBB(S&P)orBaa(Moody’s)Junkbonds:AtmostCC(S&P)orCa(Moody’s)Whatisabond?d.Bondrating71Whatisabond?Moody’sLong-TermRatingDefinitions-ExampleAaa ObligationsratedAaaarejudgedtobeofthehighestquality,withminimalcreditrisk.Aa ObligationsratedAaarejudgedtobeofhighqualityandaresubjecttoverylowcreditrisk.A ObligationsratedAareconsideredupper-mediumgradeandaresubjecttolowcreditrisk.Baa ObligationsratedBaaaresubjecttomoderatecreditrisk.Theyareconsideredmedium-gradeandassuchmaypossesscertainspeculativecharacteristics.Whatisabond?Moody’sLong-Te72Whatisabond?Ba ObligationsratedBaarejudgedtohavespeculativeelementsandaresubjecttosubstantialcreditrisk.B ObligationsratedBareconsideredspeculativeandaresubjecttohighcreditrisk.Caa ObligationsratedCaaarejudgedtobeofpoorstandingandaresubjecttoveryhighcreditrisk.Ca ObligationsratedCaarehighlyspeculativeandarelikelyin,orverynear,default,withsomeprospectofrecoveryofprincipalandinterest.C ObligationsratedCarethelowestratedclassofbondsandaretypicallyindefault,withlittleprospectforrecoveryofprincipalorinterest.Whatisabond?Ba73Whatisabond?e.TerminologyParvalue(facevalue):Dollaramountofthebondatthetimeitisissued.Itappearsseveraltimesonthefaceofbond.Usually,multipliesof$1,000.Interestrate(couponrate):Percentageofparvaluethatispaidtothebond-holderonaregularbasis.

Whatisabond?e.Terminology74Bondpricinga. Zerocouponbond Price=PV(Facevalue)=b. Ordinarycouponbond Price=PV(Coupons)+PV(Facevalue)

=Bondpricinga. Zerocouponbon75Bondpricingc. ExamplesSupposeyounowownaT-bondthathasacouponrateof5%,facevalueof$1,000,andmaturityof5years.Acomparableinvestmenthasrateofreturnof6%.AssumeannualcouponsBondpricingc. Examples76BondpricingAssumesemi-annualcoupons

Whyistheredifferenceofbondspricesbetweenannualcouponsandsemi-annualcoupons? 2conflictingfactors:Moreimmediatepaymentand

higherdiscountrate.→cannotbedeterminedsimply.BondpricingAssumesemi-annual77Bondpricingd. Conversequestion Sincethemarketdecidesthebondprice,thenwecanbackout“r”.So,this“r”andthepriceofabondhaveone-to-onerelationship.Example:Supposeabondwithfacevalueof$1,000,coupon rateof11.5%,andmaturityof5yearsisbeingtraded at$1161.84.Then,whatis“r”forthisbond?

1161.84=115/(1+r)+115/(1+r)2+… +115/(1+r)5+1000/(1+r)5. r=0.075,butusuallythroughtrialanderrormethod.Bondpricingd. Converseques78Someconcepta.Yieldtomaturity(YTM)–“r”inournotation.

IRRforabond.Financialnewspapersreportthisnumber. IRR:InternalrateofreturnSomeconcepta.Yieldtomatur79SomeconceptExample:WSJ5/15/02 MaturityASKRATEMO/YRBIDASKEDCHGYLD6.500May05107:6107:763.90Someconcept80Someconcept“6.500”meanscouponrate(%)“May05”meansmaturity“107:6”isbidpriceanditisnot$107.6,but(107and6/32)%ofFV(usually,$1,000)→$1,071.875“107:7”isaskprice→(107and7/32)%ofFV,$1,072.1875 Notice:Spreadisreallysmall,1/32(about0.03)%of FV,Spreadofstockprice–around1-2%Someconcept“6.500”meanscoup81Someconcept“6”meansthatthebondpriceisincreasedby$6/32fromthepreviousday.“3.90”isYTM:Thisisbackedoutbyusing 1072.19=65/(1+r)+65/(1+r)2

+65/(1+r)3+1000/(1+r)3 →r≈0.039ProblemsofYTM–Theassumptionofthesamediscountratefromt=1toT.Itisveryunusual. →ThinkthejobofMr.Greenspan.Someconcept“6”meansthatthe82Someconceptb.Changeofinterestrateandbondprice

Inourexample(priceis$1072.19),ifinterestrate(notcouponrate)isincreasedinto15%,thenthebondpricewillbedecreased. PVat15%=65/(1.15)+65/(1.15)2

+65/(1.15)3+1000/(1.15)3 =805.93 So,priceisdecreasedwithhigherinterestrate.Someconceptb.Changeofinte83Someconceptc.RateofreturnWhatisthedifferencebetweenrateofreturnandYTMforabond?Fromtheaboveexample,after1year,thebondpriceis1077.10.Then,whatwouldberateofreturnandYTM?First,rateofreturnis($65+$1,077.10-$1,072.19)/$1,072.19=6.52%Someconceptc.Rateofreturn84SomeconceptForYTM,1077.10(PVatyear1)=65/(1+r)+65/(1+r)2+1000/(1+r)2 Then,YTM=2.5%WhatistherelationshipbetweenYTMandtherateofreturn?→Ifinterestratesdonotchange,thebondpricechangeswithtimesothatthetotalreturnonthebondisequaltotheYTM.SomeconceptForYTM,1077.10(85Someconceptd.Spotrateandforwardrate1Yearr1=0f1r1=0f1r2=0r2representedby1yearrate1f2att=0Someconceptd.Spotrateand86Someconcepti.Futurespotrate

Theactualinterestrateinthefuture,e.g.theactual interestratefromyear1toyear2atyear1.YTMonzerocouponbond. ii.ForwardrateSinceafuturespotrateisunknownnow,wehavetoinferitfromvariousbondsprices.Ingeneral,(1+t-1ft)(1+1rt-1)(t-1)=(1+1rt)tSomeconcepti.Futurespotra87SomeconceptExampleforspotrateandforwardrate:Considerfollowing4bondswithfacevalueof$100andannualcoupons.MaturityCouponPriceYTM1094.346.0026.5100.566.1936.098.986.38311.0112.286.37SomeconceptExampleforspotr88SomeconceptWhatis2yearspotrate(r2)?

100.56=6.5/(1.06)+6.5/(1+r2)2

+100/(1+r2)2 So,r2=6.2%SomeconceptWhatis2yearspo89SomeconceptThen,r3? 98.98=6.0/(1.06)+6.0/(1+0.062)2

+106/(1+r3)3

So,r3=6.4% SomeconceptThen,r3?90SomeconceptWhatare1f2?

(1+r2)2=(1+r1)*(1+1f2) Intheaboveproblem,weknowthatr2is 6.2%andr1is6%. So,1f2is6.4%And2f3? Withthesamelogic,2f3=6.8%SomeconceptWhatare1f2?91Someconcepte.TermstructureTherelationshipbetweenshort-andlong-terminterestratesataparticulartimeonbondsthatarefundamentallysimilarexceptmaturity.Therelationshipcanbepresentedgraphicallyasayieldcurve.Usually,theyieldcurveisupwardsloping.Inourexample,r1<r2<r3.Advancedpoints:Termstructurecanberepresentedbytherelationbetweenspotrate(e.g.r2)andforwardrate(1f2).Ifr2>r1,then1f2>r2,andifr2<r1,then1f2<r2.Someconcepte.Termstructure92Part2BONDRISKDefaultRiskCallRiskMarketRiskPart2BONDRISKDefaultRis93DefaultRisk(1)DefaultRisk:Uncertaintythattherealizedreturnwilldeviatefromtheexpectedreturnbecausetheissuerwillfailtomeetthecontractualobligationsspecifiedintheindenture.Themajorconcernisfailuretomeetinterestandprincipalpayments.DefaultRisk(1)DefaultRisk:94DefaultRisk(2)Mostinvestorsdonotdirectlyaccessabond’sdefaultrisk,butinsteadusethequalityratingsprovidedbyMoody’s,S&P,andFetchtoevaluatethedegreeofrisk.AAA,