对外经济贸易大学国际经济贸易学院《固定收益证券》部分答案

国际经济贸易学院研究生课程班《固定收益证券》试题Explainwhyyouagreeordisagreewiththefollowingstatement: “Thepriceofafloaterwillalwaystradeatitsparvalue. ”Answer:Idisagreewiththestatement: “Thepriceofafloaterwillalwaystradeatitsparvalue.”First,thecouponrateofafloating-ratesecurity(orfloater)isequaltoareferencerateplussomespreadormargin.Forexample,thecouponrateofafloatercanresetattherateonathree-monthTreasurybill(thereferencerate)plus50basispoints(thespread).Next,thepriceofafloaterdependsontwofactors:(1)thespreadoverthereferencerateand(2)anyrestrictionsthatmaybeimposedontheresettingofthecouponrate.Forexample,afloatermayhaveamaximumcouponratecalledacaporaminimumcouponratecalledafloor.Thepriceofafloaterwilltradeclosetoitsparvalueaslongas(1)thespreadabovethereferenceratethatthemarketrequiresisunchangedand(2)neitherthecapnorthefloorisreached.However,ifthemarketrequiresalarger(smaller)spread,thepriceofafloaterwilltradebelow(above)par.Ifthecouponrateisrestrictedfromchangingtothereferencerateplusthespreadbecauseofthecap,thenthepriceofafloaterwilltradebelowpar.Aportfoliomanagerisconsideringbuyingtwobonds.BondAmaturesinthreeyearsandhasacouponrateof10%payablesemiannually.BondB,ofthesamecreditquality,maturesin10yearsandhasacouponrateof12%payablesemiannually.Bothbondsarepricedatpar.Supposethattheportfoliomanagerplanstoholdthebondthatispurchasedforthreeyears.Whichwouldbethebestbondfortheportfoliomanagertopurchase?Answer:Theshortertermbondwillpayalowercouponratebutitwilllikelycostlessforagivenmarketthebondsareofequalriskintermsofcreitquality(Thematuritypremiumforthelongertermbondshouldbegreater),thequestionwhencomparingthetwobondinvestmentsis:Whatinvestmentwillbeexpectetogivethehighestcashflowperdollarinvested?Inotherwords,whichinvestmentwillbeexpectedtogivethehighesteffectiveannualrateofgeneral,holdingthelongertermbondshouldcompensatetheinvestorintheformofamaturitypremiumandahigherexpected,asseeninthediscussionbelow,theactualrealizedreturnforeitherinvestmentisnotknownwithcertainty.Tobeginwith,aninvestorwhopurchasesabondcanexpecttoreceiveadollarreturnfrom(i)theperiodiccouponinterestpaymentsmadebetheissuer,(ii)ancapitalgainwhenthebondmatures,iscalled,orissold;and(iii)interestincomegeneratedfromreinvestmentoftheperiodiccashlastcomponentofthepotentialdollarreturnisreferredtoasreinvestmentastandardbond(oursituation)thatmakesonlycouponpaymentsandnoperiodicprincipalpaymentspriortothematuritydate,theinterimcashflowsaresimplythecoupon,forsuchbondsthereinvestmentincomeissimplyinterestearnedfromreinvestingthecouponinterestthesebonds,thethirdcomponentofthepotentialsourceofdollarreturnisreferredtoastheinterest-on-interestcomponents.Ifwearegoingtocouputeapotentialyieldtomakeadecision,weshouldbeawareofthefactthatanymeasureofabond’spotentialyieldshouldtakeintoconsiderationeachofthethreecomponentsdescribedcurrentyieldconsidersonlythecouponinterestconsiderationisgiventoanycapitalgainorinterestonyieldtomaturitytakesintoaccountcouponinterestandanycapitalalsoconsiderstheinterest-on-interest,implicitintheyield-to-maturitycomputationistheassumptionthatthecouponpaymentscanbereinvestedatthecomputedyieldtoyieldtomaturityisapromisedyieldandwillberealizedonlyifthebondisheldtomaturityandthecouponinterestpaymentsarereinvestedattheyieldtothebondisnotheldtomaturityandthecouponpaymentsarereinvestedattheyieldtomaturity,thentheactualyieldrealizedbyaninvestorcanbegreaterthanorlessthantheyieldtomaturity.Giventhefactsthat(i)onebond,ifbought,willnotbeheldtomaturity,and(ii)thecouponinterestpaymentswillbereinvestedatanunknownrate,wecannotdeterminewhichbondmightgivethehighestactualrealized,wecannotcomparethembaseduponthis,iftheportfoliomanagerisriskinverseinthesensethatsheorhedoesn ’twanttobuyalongertermbond,whichwilllikelhavemorevariabilityinitsreturn,thenthemanagermightprefertheshortertermbond(bondA)ofthresbondalsomatureswhenthemanagerwantstocashinthe,themanagerwouldnothavetoworryaboutanypotentialcapitallossinsellingthelongertermbond(bondB).ThemanagerwouldknowwithcertaintywhatthecashflowsThesecashflowsarespentwhenreceived,themanagerwouldknowexactlyhowmuchmoneycouldbespentatcertainpointsintime.Finally,amanagercantrytoprojectthetotalreturnperformanceofabondonthebasisofthepannedinvestmenthorizonandexpectationsconcerningreinvestmentratesandfuture marketermits the portfolio managertoevaluate thichof severalpotential bondsconsidered for acquisitionwillperformbest overthe plannedinvestmentwejustrgued,thiscannotbedoneusingtheyieldtomaturityasameasureofrelativetotalreturntoassessperformanceoversomeinvestmenthorizoniscalledhorizonatotalreturniscalculatedovenaninvestmenthorizon,itisreferredtoasahorizonhorizonanalysis framworenabledtheportfoliomanagertoanalyzetheperformanceofabondunderdifferentinterest-ratescenariosforreinvestmentratesandfuturemarketbyinvestigatingmultiplescenarioscantheportfoliomanagerseehowsensitivethebond’sperformancewillbetoeachcanhelpthemanagerchoosebetweenthetwobondchoices.Supposethattheportfoliomanagerplanstoholdthebondthatispurchasedforsixyearsinsteadofthreeyears.Inthiscase,whichwouldbethebestbondfortheportfoliomanagertopurchase?Answer:Simileartoourdiscussioninpart(a),wedonotknowwhichinvestmentwouldgivethehighestactualrelizedreturninsixyearswhenweconsiderreinvestingallcashthemanagerbuysathree-yearbond,thentherewouldbetheadditionaluncertaintyofnowknowingwhatthree-yearbondrateswouldbeinthreepurchaseoftheten-yearbondwouldbeheldlongerthanpreviously(sixyearscomparedtothreeyears)andrendercouponpaymentsforasix-yearperiodthatarethesecashflowsarespentwhenreceived,themanagerwillknowexactlyhowmuchmoneycouldbespentatcertainpointsintimeNotknowingwhichbondinvestmentwouldgivethehighestrealizedreturn,theportfoliomanagerwouldchoosethebondthatfitsthefirm 'sgoalsintermsofmaturity.AnswerthebelowquestionsforbondsAandB.BondABondBCoupon8%9%Yieldtomaturity8%8%Maturity(years)25Par$$Price$$Calculatetheactualpriceofthebondsfora100-basis-pointincreaseininterestrates.Answer:ForBondA,wegetabondquoteof$100forourinitialpriceifwehavean8%couponrateandan8%yield.Ifwechangetheyield100basispointsotheyieldis9%,thenthevalueofthebond(P)isthepresentvalueofthecouponpaymentsplusthepresentvalueoftheparvalue.WehaveC=$40,y=%,n=4,andM=$1,000.Insertingthesenumbersintoourpresentvalueofcouponbondformula,weget:Thepresentvalueoftheparormaturityvalueof$1,000is:Thus,thevalueofbondAwithayieldof9%,acouponrateof8%,andamaturityof2yearsis:P=$+$=$.Thus,wegetabondquoteof$.WealreadyknowthatbondBwillgiveabondvalueof$1,000andabondquoteof$100sinceachangeof100basispointswillmaketheyieldandcouponratethesame,Forexample,insertingThus,thevalueofbondAwithayieldof9%,acouponrateof8%,andamaturityof2yearsis:P=$+$=$.Thus,wegetabondquoteof$.WealreadyknowthatbondBwillgiveabondvalueof$1,000andabondquoteof$100sinceachangeof100basispointswillmaketheyieldandcouponratethesame,Forexample,inserting(b)Usingduration,estimatethepriceofthebondsfora100-basis-pointincreaseininterestrates.Answer:ToestimatethepriceofbondA,webeginbyfirstcomputingthemodifiedduration.WecanuseanalternativeformulathatdoesnotrequiretheextensivecalculationsrequiredbytheMacaulayprocedure.Theformulais:Puttingallapplicablevariablesintermsof$100,wehaveC=$4,n=4,y=,andP=$.Insertingthesevalues,inthemodifieddurationformulagives:($1,口+$/$=($+$/$=$/$=orabout.Convertingtoannualnumberbydividingbytwogivesamodifieddurationof(beforetheincreasein100basispointsitwas.Wenextsolveforthechangeinpriceusingthemodifieddurationofanddy=100basispoints=.Wehave:WecannowsolveforthenewpriceofbondAasshownbelow:Thisisslightlylessthantheactualpriceof$.Thedifferenceis$ 一$=$.ToestimatethepriceofbondB,wefollowthesameprocedurejustshownforbondA.UsingthealternativeformulaformodifieddurationthatdoesnotrequiretheextensivecalculationsrequiredbytheMacaulayprocedureandnotingthatC=$45,n=10,y=,andP=$100,weget:($+$0)/$100=orabout(beforetheincreasein100basispointsitwasorabout.Convertingtoanannualnumberbydividingbytwogivesamodifieddurationof(beforetheincreasein100basispointsitwas.Wewillnowestimatethepriceof

bondBusingthemodifieddurationmeasure.With100basispointsgivingdy=andanapproximatedurationof,wehave:Thus,thenewpriceis(1-$1,=$1,=$.Thisisslightlylessthantheactualpriceof$1,000.Thedifferenceis$1,000 —$=$.Usingbothdurationandconvexitymeasures,estimatethepriceofthebondsfora100-basis-pointincreaseininterestrates.Answer:ForbondA,weusethedurationandconvexitymeasuresasgivenbelow.First,weusethedurationmeasure.Weadd100basispointsandgetayieldof9%.WenowhaveC=$40,y=%,n=4,andM=$1,000.NOTE.Inpart(a)wecomputedtheactualbondpriceandgotP=$.Priortothat,thepricesoldatpar(P=$1,000)sincethecouponrateandyieldwerethenequal.Theactualchangeinpriceis:($-$1,000)=?$andtheactualpercentagechangeinpriceis:?$/$1,000=?%.Wewillnowestimatethepricebyfirstapproximatingthedollarpricechange.With100basispointsgivingdy=andamodifieddurationcomputedinpart(b)of,wehave:Thisisslightlymorenegativethantheactualpercentagedecreaseinpriceof?%.Thedifferenceis?% -(?%)=?%+%=%.Usingthe?%justgivenbythedurationmeasure,thenewpriceforbondAis:Thisisslightlylessthantheactualpriceof$.Thedifferenceis$-$=$.Next,weusetheconvexitymeasuretoseeifwecanaccountforthedifferenceof%.Wehave:convexitymeasure(halfyears)2_d2P12Cd1 2Cnn(n1)(100C/y)11dy2P y3 (1y)n y2(1y)n1(1y)n2PForbondA,weadd100basispointsandgetayieldof9%.WenowhaveC=$40,y=%,n=4,andM=$1,000.NOTE.Inpart(a)wecomputedtheactualbondpriceandgotP=$.Priortothat,thepricesoldatpar(P=$1,000)sincethecouponrateandyieldwerethenequal.Expressingnumbersintermsofa$100bondquote,wehave:C=$4,y=,n=4,andP=$.Insertingthesenumbersintoourconvexitymeasureformulagives:convexitymeasure(halfyears)=2$4/ 12$4/ 1 31 40.0453 (1.045)416.93250.0452(1.045)5 (1.045)6 $98.2062Addingthedurationmeasureandtheconvexitymeasure,weget?%+%=?%.Recalltheactualchangeinpriceis:($ -$1,000)=?$andtheactualpercentagechangeinpriceis:?$/$1,000=?orapproximately?%.Usingthe?%resultingfromboththedurationandconvexitymeasures,wecanestimatethenewpriceforbondA.Wehave:Addingthedurationmeasureandtheconvexitymeasure,weget?%+%=?%.Recalltheactualchangeinpriceis:($ -$1,000)=?$andtheactualpercentagechangeinpriceis:?$/$1,000=?orapproximately?%.Usingthe?%resultingfromboththedurationandconvexitymeasures,wecanestimatethenewpriceforbondA.Wehave:Thisisslightlymorenegativethantheactualpercentagedecreaseinpriceof%.Thedifferenceis%)-%)=%Usingthe%justgivenbythedurationmeasure,thenewpriceforBondBis:Thisisslightlylessthantheactualpriceof$1,000.Thisdifferenceis$1,000-$=$Weusetheconvexitymeasuretoseeifwecanaccountforthedifferenceof00594%.Wehave:ForBondB,100basispointsareaddedandgetayieldof9%.WenowhaveC=$45,y=%,n=10,andM=$1,000.Noteinpart(a),wecomputedtheactualbondpriceandgotP=$1,000sincethecouponrateandyieldwerethenequal.Priortothat,thepricesoldatP=$1,.Expressingnumbersintermsofa$100bondquote,wehaveC=$,, n=10andP=$100.Insertingthesenumbersintoourconvexitymeasureformulagives:Theconvexitymeasure(inyears)=Note.DollarConvexityMeasure=ConvexityMeasure(years)timesP=($100)=$1,.ThepercentagepricechangeduetoconvexityisdP1convexitymeasure(dy)2ThepercentagepricechangeduetoconvexityisdP1 o Insertinginthevalues,weget (77.8103)(0.01)20.00097463P2Thus,wehave%increaseinpricewhenweadjustforconvexitymeasure.Addingthedurationmeasureandconvexitymeasure,weget%+%equals%.Recalltheactualchangeinpriceis($1,000-$1,=-$andtheactualnewpriceisForBondA.Thisisaboutthesameastheactualpriceof$1,000.Thedifferenceis$1,$1,000=$.Thus,usingtheconvexitymeasurealongwiththedurationmeasurehasnarrowedtheestimatedpricefromadifferenceof-$to$.Commentontheaccuracyofyourresultsinpartsbandc,andstatewhyoneapproximationisclosertotheactualpricethantheother.Answer:ForbondA,theactualpriceis$.Whenweusethedurationmeasure,wegetabondpriceof$thatis$lessthantheactualprice.Wherweusedurationandconvexmeasurestogether,wegetabondpriceof$.Thisisslightlymorethantheactualpriceof$.Thedifferenceis$ - $=$.Thus,usingtheconvexitymeasurealongwiththedurationmeasurehasnarrowedtheestimatedpricefromadifferenceof?$to$.ForbondB,theactualpriceis$1,000.Whenweusethedurationmeasure,wegetabondpriceof$thatis$lessthantheactualprice.Whenweusedurationandconvexmeasurestogether,wegetabondpriceof$1,.Thisisslightlymorethan theactualpriceof$1,000.Thedifferenceis$1,-$1,000=$.Thus,usingtheconvexitymeasurealongwiththedurationmeasurehasnarrowedtheestimatedpricefromadifferenceof?$to$Aswesee,usingthedurationandconvexitymeasurestogetherismoreaccurate.Thereasonisthataddingtheconvexitymeasuretoourestimateenablesustoincludethesecondderivativethatcorrectsfortheconvexityoftheprice-yieldrelationship.Moredetailsareofferedbelow.Duration(modifiedordollar)attemptstoestimateaconvexrelationshipwithastraightline(thetangentline).Wecanspecifyamathematicalrelationshipthatprovidesabetterapproximationtothepricechangeofthebondiftherequiredyieldchanges.WedothisbyusingthefirsttwotermsofaTaylorseriestoapproximatethepricechangeasfollows:DividingbothsidesofthisequationbyPtogetthepercentagepricechangegivesus:Thefirsttermontheright-handsideofequation(1)isequationforthedollarpricechangebasedondollardurationandisourapproximationofthepricechangebasedonduration.Inequation(2),thefirsttermontheright-handsideistheapproximatepercentagechangeinpricebasedonmodifiedduration.Thesecondterminequations(1)and(2)includesthesecondderivativeofthepricefunctionforcomputingthevalueofabond.Itisthesecondderivativethatisusedasaproxymeasuretocorrectfortheconvexityoftheprice-yieldrelationship.Marketparticipantsrefertothesecondderivativeofbondpricefunctionasthedollarconvexitymeasureofthebond.Thesecondderivativedividedbypriceisameasureofthepercentagechangeinthepriceofthebondduetoconvexityandisreferredtosimplyastheconvexitymeasure.(e)Withoutworkingthroughcalculations,indicatewhetherthedurationofthetwobondswouldbehigherorloweriftheyieldtomaturityis10%ratherthan8%.Answer:Liketermtomaturityandcouponrate,theyieldtomaturityisafactorthatinfluencespricevolatility.Ceteris paribus,thehighertheyieldlevel,the lowerthepricevolatility.Thesamepropertyholdsformodifiedduration.Thus,a10%yieldtomaturitywillhavebothlessvolatilitythanan8%yieldtomaturityandalsoasmallerduration.Thereisconsistencybetweenthepropertiesofbondpricevolatilityandthepropertiesofmodifiedduration.Whenallotherfactorsareconstant,abondwithalongermaturitywillhavegreaterpricevolatility.Apropertyofmodifieddurationisthatwhenallotherfactorsareconstant,abondwithalongermaturitywillhaveagreatermodifiedduration.Also,allotherfactorsbeingconstant,abondwithalowercouponratewillhavegreaterbondpricevolatility.Also,generally,abondwithalowercouponratewillhaveagreatermodifiedduration.Thus,bondswithgreaterdurationswillgreaterpricevolatilities.Supposeaclientobservesthefollowingtwobenchmarkspreadsfortwobonds:BondissueUratedA:150basispointsBondissueVratedBBB:135basispointsYourclientisconfusedbecausehethoughtthelower-ratedbond(bondV)shouldofferahigherbenchmarkspreadthanthehigher-ratedbond(bondU).ExplainwhythebenchmarkspreadmaybelowerforbondU.ThebidandaskyieldsforaTreasurybillwerequotedbyadealeras%and%,respectively.Shouldn'tthebidyieldbelessthantheaskyield,becausethebidyieldindicateshowmuchthedealeriswillingtopayandtheaskyieldiswhatthedealeriswillingtoselltheTreasurybillfor?Answer:Thehigherbidmeansalowerprice.Sothedealeriswillingtopaylessthanwouldbepaidfortheloweraskprice.Weillustratethisbelow.Giventheyieldonabankdiscountbasis(Yd),thepriceofaTreasurybillisfoundbyfirstsolvingtheformulaforthedollardiscount(D),asfollows:ThepriceisthenPrice=F-DForthe100-dayTreasurybillwithafacevalue(F)of$100,000,iftheyieldonabankdiscountbasis(Yd)isquotedas%,Disequalto:Therefore,price=$100,000 -$1,=$98,.Forthe100-dayTreasurybillwithafacevalue(F)of$100,000,iftheyieldonabankdiscountbasis(Yd)isquotedas%,Disequalto:Therefore,priceis:P=F -D=$100,000 -$1,=$98,.Thus,thehigherbidquoteof%(comparedtoloweraskquote%)givesalowersellingpriceof$98,(comparedto$98,.The%higheryieldtranslatesintoasellingpricethatis$lower.Ingeneral,thequotedyieldonabankdiscountbasisisnotameaningfulmeasureofthereturnfromholdingaTreasurybill,fortworeasons.First,themeasureisbasedonaface-valueinvestmentratherthanontheactualdollaramountinvested.Second,theyieldisannualizedaccordingtoa360-dayratherthana365-dayyear,makingitdifficulttocompareTreasurybillyieldswithTreasurynotesandbonds,whichpayinterestona365-daybasis.Theuseof360daysforayearisamoneymarketconventionforsomemoneymarketinstruments,however.Despiteitsshortcomingsasameasureofreturn,thisisthemethodthatdealershaveadoptedtoquoteTreasurybills.Manydealerquote