1、CHAPTER5INTERESTRATESANDBONDVALUATIONAnswerstoConceptsReviewandCriticalThinkingQuestions1. No.Asinterestratesfluctuate,thevalueofaTreasurysecuritywillfluctuate.Long-termTreasurysecuritieshavesubstantialinterestraterisk.2. Allelsethesame,theTreasurysecuritywillhavelowercouponsbecauseofitslowerdefault
2、risk,soitwillhavegreaterinterestraterisk.3. No.Ifthebidwerehigherthantheask,theimplicationwouldbethatadealerwaswillingtosellabondandimmediatelybuyitbackatahigherprice.Howmanysuchtransactionswouldyouliketodo?4. Pricesandyieldsmoveinoppositedirections.Sincethebidpricemustbelower,thebidyieldmustbehighe
3、r.5. Therearetwobenefits.First,thecompanycantakeadvantageofinterestratedeclinesbycallinginanissueandreplacingitwithalowercouponissue.Second,acompanymightwishtoeliminateacovenantforsomereason.Callingtheissuedoesthis.Thecosttothecompanyisahighercoupon.Aputprovisionisdesirablefromaninvestor,sstandpoint
4、soithelpsthecompanybyreducingthecouponrateonthebond.Thecosttothecompanyisthatitmayhavetobuybackthebondatanunattractiveprice.6. Bondissuerslookatoutstandingbondsofsimilarmaturityandrisk.Theyieldsonsuchbondsareusedtoestablishthecouponratenecessaryforaparticularissuetoinitiallysellforparvalue.Bondissu
5、ersalsosimplyaskpotentialpurchaserswhatcouponratewouldbenecessarytoattractthem.Thecouponrateisfixedandsimplydetermineswhatthebond,scouponpaymentswillbe.Therequiredreturniswhatinvestorsactuallydemandontheissue,anditwillfluctuatethroughtime.Thecouponrateandrequiredreturnareequalonlyifthebondsellsforex
6、actlyatpar.7. Yes.Someinvestorshaveobligationsthataredenominatedindollars;i.e.,theyarenominal.Theirprimaryconcernisthataninvestmentprovidestheneedednominaldollaramounts.Pensionfunds,forexample,oftenmustplanforpensionpaymentsmanyyearsinthefuture.Ifthosepaymentsarefixedindollarterms,thenitisthenominal
7、returnonaninvestmentthatisimportant.Companiespaytohavetheirbondsratedsimplybecauseunratedbondscanbedifficulttosell;manylargeinvestorsareprohibitedfrominvestinginunratedissues.TreasurybondshavenocreditrisksinceitisbackedbytheU.S.government,soaratingisnotnecessary.Junkbondsoftenarenotratedbecausethere
8、wouldbenopointinanissuerpayingaratingagencytoassignitsbondsalowrating(it,slikepayingsomeonetokickyou!).10. Thetermstructureisbasedonpurediscountbonds.Theyieldcurveisbasedoncoupon-bearingissues.11. Bondratingshaveasubjectivefactortothem.Splitratingsreflectadifferenceofopinionamongcreditagencies.12. A
9、sageneralconstitutionalprinciple,thefederalgovernmentcannottaxthestateswithouttheirconsentifdoingsowouldinterferewithstategovernmentfunctions.Atonetime,thisprinciplewasthoughttoprovideforthetax-exemptstatusofmunicipalinterestpayments.However,moderncourtrulingsmakeitclearthatCongresscanrevokethemunic
10、ipalexemption,sotheonlybasisnowappearstobehistoricalprecedent.Thefactthatthestatesandthefederalgovernmentdonottaxeachother,ssecuritiesisreferredtoasreciprocalimmunity.,13. Lackoftransparencymeansthatabuyerorsellercan,tseerecenttransactions,soitismuchhardertodeterminewhatthebestbidandaskpricesareatan
11、ypointintime.14. Onemeasureofliquidityisthebid-askspread.Liquidinstrumentshaverelativelysmallspreads.LookingatFigure7.4,thebellwetherbondhasaspreadofonetick;itisoneofthemostliquidofallinvestments.Generally,liquiditydeclinesafterabondisissued.Someolderbonds,includingsomeofthecallableissues,havespread
12、saswideassixticks.15. Companieschargethatbondratingagenciesarepressuringthemtopayforbondratings.Whenacompanypaysforarating,ithastheopportunitytomakeitscaseforaparticularrating.Withanunsolicitedrating,thecompanyhasnoinput.16. A100-yearbondlookslikeashareofpreferredstock.Inparticular,itisaloanwithalif
13、ethatalmostcertainlyexceedsthelifeofthelender,assumingthatthelenderisanindividual.Withajunkbond,thecreditriskcanbesohighthattheborrowerisalmostcertaintodefault,meaningthatthecreditorsareverylikelytoendupaspartownersofthebusiness.Inbothcases,the“equityindisguisehasasignificanttaxadvantage.17. a.Thebo
14、ndpriceisthepresentvalueofthecashflowsfromabond.TheYTMistheinterestrateusedinvaluingthecashflowsfromabond.b. Ifthecouponrateishigherthantherequiredreturnonabond,thebondwillsellatapremium,sinceitprovidesperiodicincomeintheformofcouponpaymentsinexcessofthatrequiredbyinvestorsonothersimilarbonds.Ifthec
15、ouponrateislowerthantherequiredreturnonabond,thebondwillsellatadiscountsinceitprovidesinsufficientcouponpaymentscomparedtothatrequiredbyinvestorsonothersimilarbonds.Forpremiumbonds,thecouponrateexceedstheYTM;fordiscountbonds,theYTMexceedsthecouponrate,andforbondssellingatpar,theYTMisequaltothecoupon
16、rate.c. Currentyieldisdefinedastheannualcouponpaymentdividedbythecurrentbondprice.Forpremiumbonds,thecurrentyieldexceedstheYTM,fordiscountbondsthecurrentyieldislessthantheYTM,andforbondssellingatparvalue,thecurrentyieldisequaltotheYTM.Inallcases,thecurrentyieldplustheexpectedone-periodcapitalgainsyi
17、eldofthebondmustbeequaltotherequiredreturn.18. Along-termbondhasmoreinterestrateriskcomparedtoashort-termbond,allelsethesame.Alowcouponbondhasmoreinterestrateriskthanahighcouponbond,allelsethesame.Whencomparingahighcoupon,long-termbondtoalowcoupon,short-termbond,weareunsurewhichhasmoreinterestrateri
18、sk.Generally,thematurityofabondisamoreimportantdeterminantoftheinterestraterisk,sothelong-termhighcouponbondprobablyhasmoreinterestraterisk.Theexceptionwouldbeifthematuritiesareclose,andthecouponratesarevastlydifferent.SolutionstoQuestionsandProblemsNOTE:Allendofchapterproblemsweresolvedusingaspread
19、sheet.Manyproblemsrequiremultiplesteps.Duetospaceandreadabilityconstraints,whentheseintermediatestepsareincludedinthissolutionsmanual,roundingmayappeartohaveoccurred.However,thefinalanswerforeachproblemisfoundwithoutroundingduringanystepintheproblem.NOTE:Mostproblemsdonotexplicitlylistaparvalueforbo
20、nds.Eventhoughabondcanhaveanyparvalue,ingeneral,corporatebondsintheUnitedStateswillhaveaparvalueof$1,000.Wewillusethisparvalueinallproblemsunlessadifferentparvalueisexplicitlystated.BCISiC1. Thepriceofapurediscount(zerocoupon)bondisthepresentvalueofthepar.Eventhoughthebondmakesnocouponpayments,thepr
21、esentvalueisfoundusingsemiannualcompoundingperiods,consistentwithcouponbonds.Thisisabondpricingconvention.So,thepriceofthebondforeachYTMis:a.P=1,000/(1+.O3)20=553.68b.P=1,000/(1+.O5)20=376.89c.P=1,000/(1+.O7)20=258.422. ThepriceofanybondisthePVoftheinterestpayment,plusthePVoftheparvalue.Noticethispr
22、oblemassumesanannualcoupon.ThepriceofthebondateachYTMwillbe:a. P=40(l-1/(1+.O4)40/.04)+fl,000l/(1+.O4)40P=1,000.00WhentheYTMandthecouponrateareequal,thebondwillsellatpar.b. P=40(l-1/(1+.O5)40/.05)+fl,000l/(1+.O5)40P=828.41WhentheYTMisgreaterthanthecouponrate,thebondwillsellatadiscount.c. P=40(l-1/(1
23、O3)40/.03)+l,000l/(1+.O3)40P=1,231.15WhentheYTMislessthanthecouponrate,thebondwillsellatapremium.Wewouldliketointroduceshorthandnotationhere.Ratherthanwrite(ortype,asthecasemaybe)theentireequationforthePVofalumpsum,orthePVAequation,itiscommontoabbreviatetheequationsas:PVIF=l(l+r/whichstandsforPres
24、entValueInterestFactorPVIFA=(l-l(l+r)whichstandsforPresentValueInterestFactorofanAnnuityTheseabbreviationsareshorthandnotationfortheequationsinwhichtheinterestrateandthenumberofperiodsaresubstitutedintotheequationandsolved.Wewillusethisshorthandnotationinremainderofthesolutionskey.3. Herewearefindin
25、gtheYTMofasemiannualcouponbond.Thebondpriceequationis:P=元970=元43(PVIFAH%,2o)+元1,000(PVlFR%,2o)SincewecannotsolvetheequationdirectlyforR,usingaspreadsheet,afinancialcalculator,ortrialanderror,wefind:7?=4.531%Sincethecouponpaymentsaresemiannual,thisisthesemiannualinterestrate.TheYTMistheAPRofthebond,s
26、o:YTM=24.531%=9.06%4. Hereweneedtofindthecouponrateofthebond.Allweneedtodoistosetupthebondpricingequationandsolveforthecouponpaymentasfollows:P=$1,145=C(PVlFA3.75%,29)+$1,000(PVlF3.75%,29)Solvingforthecouponpayment,weget:C=$45.79Sincethisisthesemiannualpayment,theannualcouponpaymentis:2$45.79=$91.58
27、Andthecouponrateisthecouponratedividedbyparvalue,so:Couponrate=$91.58/$1,000=9.16%5. Theapproximaterelationshipbetweennominalinterestrates(R),realinterestrates(r),andinflation()is:R=r+hApproximater-.06-.045=.015or1.50%TheFisherequation,whichshowstheexactrelationshipbetweennominalinterestrates,realin
28、terestrates,andinflationis:(1+7?)=(1+r)(l+h)(1+.06)=(1+r)(l+.045)Exactr=(1+.06)/(1+.045)-1=.0144or1.44%6. TheFisherequation,whichshowstheexactrelationshipbetweennominalinterestrates,realinterestrates,andinflationis:(1+7?)=(1+r)(l+h)R=(1+.04)(1+.025)-1=.0660or6.60%7. TheFisherequation,whichshowstheex
29、actrelationshipbetweennominalinterestrates,realinterestrates,andinflationis:(1+R)=(1+r)(l+h)z=(l+.15)/(1+.09)-1=.0550or5.50%8. TheFisherequation,whichshowstheexactrelationshipbetweennominalinterestrates,realinterestrates,andinflationis:(1+/?)=(1+r)(l+h)r=(1+.134)/(1.045)-1=.0852or8.52%9. Thisisabond
30、sincethematurityisgreaterthan10years.Thecouponrate,locatedinthefirstcolumnofthequoteis6.125%.Thebidpriceis:Bidprice=116:05=1165/32=116.15625%$1,000=$1,161.5625Thepreviousday,saskpriceisfoundby:Previousday,saskedprice=Today,saskedprice-Change=1165/32-(-932)=11614/32Thepreviousday,spriceindollarswas:P
31、reviousday,sdollarprice=116.4375%$1,000=$1,164,37510. Thisisapremiumbondbecauseitsellsformorethan100%offacevalue.Thecurrentyieldis:Currentyield=Annualcouponpayment/Price=$75/$l,320.9375=5.68%TheYTMislocatedunderthe“ASKYLD,column,sotheYTMis4.93%.Thebid-askspreadisthedifferencebetweenthebidpriceandthe
32、askprice,so:Bid-Askspread=132:03-132:02=1/32ImernIediate11. HerewearefindingtheYTMofsemiannualcouponbondsforvariousmaturitylengths.Thebondpriceequationis:P=C(PVIFAE%)+1,000(PVlFK%,。MillerCorporationbond:Po=40(PVlFA3%,26)+1,000(PVIF3%,26)Pi=40(PVIFA3%,24)+1,000(PVlF3%,24)P3=40(PVIFA3%,20)+1,000(PVlF3
33、2。)P7=40(PVlFA3%,12)+1,000(PVIF3%,12)=1,178.77=1,169.36=1,148.77=1,099.54Pi2=f40(PVIFA3%,2)+l,000(PVIF3%,2)=1,019.13Pb=1,000ModiglianiCompanybond:Y:Po=30(PVlFA4%,26)+1,000(PVIF4%,26)Pi=30(PVlFA4%,24)+1,000(PVIF4%,24)P3=30(PVlFA4%,20)+1,000(PVlF4%,20)P7=30(PVlFA4%,12)+l,000(PVIF4oAi2)=840.17=847.53
34、864.10=906.15Pi2=30(PVIFA4%,2)+l,000(PVIF4%,2)=981.14Pb=1,000Allelseheldequal,thepremiumoverparvalueforapremiumbonddeclinesasmaturityapproaches,andthediscountfromparvalueforadiscountbonddeclinesasmaturityapproaches.Thisiscalledpulltopar.Inbothcases,thelargestpercentagepricechangesoccurattheshortest
35、maturitylengths.Also,noticethatthepriceofeachbondwhennotimeislefttomaturityistheparvalue,eventhoughthepurchaserwouldreceivetheparvalueplusthecouponpaymentimmediately.Thisisbecausewecalculatethecleanpriceofthebond.12. AnybondthatsellsatparhasaYTMequaltothecouponrate.Bothbondssellatpar,sotheinitialYTM
36、onbothbondsisthecouponrate,10percent.IftheYTMsuddenlyrisesto12percent:rans=50(PVlFA6%,4)+1,000(PVIF6%,4)=965.35Proei=50(PVIFA6oA3o)+1,000(PVlF6%,3。)=862.35Thepercentagechangeinpriceiscalculatedas:Percentagechangeinprice=(Newprice-Originalprice)/OriginalpricePEvans%=(965.35-1,000)/1,000=-3.47%Proei%=
37、862.35-1,000)/1,000=-13.76%IftheYTMsuddenlyfallsto8percent:PEvans=50(PVIFA4%,4)+1,000(PVIF4%,4)=1,036.30Proei=50(PVlFA4%,3。)+ei,000(PVIF4oA3)=1,172.92PEvans%=(1,036.30-1,000)/1,000=+3.63%Proei%=(1,172.92-1,000)/1,000=+17.29%Allelsethesame,thelongerthematurityofabond,thegreaterisitspricesensitivityt
38、ochangesininterestrates.13. Initially,ataYTMof7percent,thepricesofthetwobondsare:Pusan=b25(PVIFA3,5o16)+,OOO(PVIF3.5%,i6)=b879.06Piksan=b55(PVIFA3.5%,16)+,000(PVlF3.5%,16)=Bl,241.88IftheYTMrisesfrom7percentto9percent:PBusan=B25(PVIFA4.5,16)+B1,000(PVIF4.5%,16)=b775.32Piksan=B55(PVIFA4.5%,16)+B1,000(
39、PVIF45%,16)=Bl,112.34Thepercentagechangeinpriceiscalculatedas:Percentagechangeinprice=(Newprice-Originalprice)/OriginalpricePBusan%=(b775.32-879.06)/b879.06=-11.80%Pksan%=(Bl,112.34-1,241.88)/Bl,241.88=-10.43%IftheYTMdeclinesfrom7percentto5percent:Pusan=B25(PVIFA2.5%,16)+B1,OOO(PVIF2.5%,I6)=B1,000.0
40、00Plksan=B55(PVIFA2.5%,16)+,00(XPVlF2.5%,16)-三1,391.65PBusan%=(b1,000.00879.06)/b879.06=+13.76%Pksan%=(B1,391.65-1,241.88)/Bl,241.88=+12.06%Allelsethesame,thelowerthecouponrateonabond,thegreaterisitspricesensitivitytochangesininterestrates.14. Thebondpriceequationforthisbondis:Po=$1,040=$45(PVlFAA%,
41、18)+$1,000(PVIFR%,Usingaspreadsheet,financialcalculator,ortrialanderrorwefind:7?=4.179%Thisisthesemiannualinterestrate,sotheYTMis:YTM=24.179%=8.359%Thecurrentyieldis:Currentyield=Annualcouponpayment/Price=$90/$1,040=8.65%TheeffectiveannualyieldisthesameastheEAR,sousingtheEARequationfromthepreviousch
42、apter:Effectiveannualyield=(1+0.04179)2-1=8.532%15. Thecompanyshouldsetthecouponrateonitsnewbondsequaltotherequiredreturn.TherequiredreturncanbeobservedinthemarketbyfindingtheYTMonoutstandingbondsofthecompany.So,theYTMonthebondscurrentlysoldinthemarketis:P=元1,100二元40(PVIFAH%,4。)+元l,OOO(PV11%,4o)Usin
43、gaspreadsheet,financialcalculator,ortrialanderrorwefind:7?=3.5295%Thisisthesemiannualinterestrate,sotheYTMis:YTM=23.5295%=7.059%16. Accruedinterestisthecouponpaymentfortheperiodtimesthefractionoftheperiodthathaspassedsincethelastcouponpayment.Sincewehaveasemiannualcouponbond,thecouponpaymentpersixmo
44、nthsisone-halfoftheannualcouponpayment.Therearefourmonthsuntilthenextcouponpayment,soonemonthhaspassedsincethelastcouponpayment.Theaccruedinterestforthebondis:Accruedinterest=$72/22/6=$12Andwecalculatethecleanpriceas:Cleanprice=Dirtyprice-Accruedinterest=$1,140-12=$1,12817. Accruedinterestisthecoupo
45、npaymentfortheperiodtimesthefractionoftheperiodthathaspassedsincethelastcouponpayment.Sincewehaveasemiannualcouponbond,thecouponpaymentpersixmonthsisone-halfoftheannualcouponpayment.Therearethreemonthsuntilthenextcouponpayment,sothreemonthshavepassedsincethelastcouponpayment.Theaccruedinterestforthe
46、bondis:Accruedinterest=65/23/6=16.25Andwecalculatethedirtypriceas:Dirtyprice=Cleanprice+Accruedinterest=865+16.25=881.2518. Tofindthenumberofyearstomaturityforthebond,weneedtofindthepriceofthebond.Sincewealreadyhavethecouponrate,wecanusethebondpriceequation,andsolveforthenumberofyearstomaturity.Wear
47、egiventhecurrentyieldofthebond,sowecancalculatethepriceas:Currentyield=.0906=$11O/PoP0=$110/.0906=$1,214.13Nowthatwehavethepriceofthebond,thebondpriceequationis:P=$1,214.13=S110(l-(1/1.085/)/.085+$1,000/1.085Wecansolvethisequationfortasfollows:$1,214.13(1.085/=$1,294.12(1.085),-1,294.12+1,000294.12=
48、79.99(1.085/3.6769=1.085rt=log3.6769/log1.085=15.9616yearsThebondhas16yearstomaturity.19. Thebondhas10yearstomaturity,sothebondpriceequationis:P=$769,355=$36.875(PVIFAH%,2o)+$1,000(PVlFH%,20)Usingaspreadsheet,financialcalculator,ortrialanderrorwefind:R=5.64%Thisisthesemiannualinterestrate,sotheYTMis:YTM=25.64%=11.28%Thecurrentyieldistheannual
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