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

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1)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.2)Aportfoliomanagerisconsideringbuyingtwobonds.BondAmaturesinthreeyearsandhasacouponrateof10%payablesemiannually.BondB,ofthesamecreditquality,maturesin10yearsandhasacouponrateof12%payablesemiannually.Bothbondsarepricedatpar.(a)Supposethattheportfoliomanagerplanstoholdthebondthatispurchasedforthreeyears.Whichwouldbethebestbondfortheportfoliomanagertopurchase?Answer:Theshortertermbondwillpayalowercouponratebutitwilllikelycostlessforagivenmarketrate.Sincethebondsareofequalriskintermsofcreitquality(Thematuritypremiumforthelongertermbondshouldbegreater),thequestionwhencomparingthetwobondinvestmentsis:Whatinvestmentwillbeexpectetogivethehighestcashflowperdollarinvested?Inotherwords,whichinvestmentwillbeexpectedtogivethehighesteffectiveannualrateofreturn.Ingeneral,holdingthelongertermbondshouldcompensatetheinvestorintheformofamaturitypremiumandahigherexpectedreturn.However,asseeninthediscussionbelow,theactualrealizedreturnforeitherinvestmentisnotknownwithcertainty.Tobeginwith,aninvestorwhopurchasesabondcanexpecttoreceiveadollarreturnfrom(i)theperiodiccouponinterestpaymentsmadebetheissuer,(ii)ancapitalgainwhenthebondmatures,iscalled,orissold;and(iii)interestincomegeneratedfromreinvestmentoftheperiodiccashflows.Thelastcomponentofthepotentialdollarreturnisreferredtoasreinvestmentincome.Forastandardbond(oursituation)thatmakesonlycouponpaymentsandnoperiodicprincipalpaymentspriortothematuritydate,theinterimcashflowsaresimplythecouponpayments.Consequently,forsuchbondsthereinvestmentincomeissimplyinterestearnedfromreinvestingthecouponinterestpayments.Forthesebonds,thethirdcomponentofthepotentialsourceofdollarreturnisreferredtoastheinterest-on-interestcomponents.Ifwearegoingtocouputeapotentialyieldtomakeadecision,weshouldbeawareofthefactthatanymeasureofabond’spotentialyieldshouldtakeintoconsiderationeachofthethreecomponentsdescribedabove.Thecurrentyieldconsidersonlythecouponinterestpayments.Noconsiderationisgiventoanycapitalgainorinterestoninterest.Theyieldtomaturitytakesintoaccountcouponinterestandanycapitalgain.Italsoconsiderstheinterest-on-interestcomponent.Additionally,implicitintheyield-to-maturitycomputationistheassumptionthatthecouponpaymentscanbereinvestedatthecomputedyieldtomaturity.Theyieldtomaturityisapromisedyieldandwillberealizedonlyifthebondisheldtomaturityandthecouponinterestpaymentsarereinvestedattheyieldtomaturity.Ifthebondisnotheldtomaturityandthecouponpaymentsarereinvestedattheyieldtomaturity,thentheactualyieldrealizedbyaninvestorcanbegreaterthanorlessthantheyieldtomaturity.Giventhefactsthat(i)onebond,ifbought,willnotbeheldtomaturity,and(ii)thecouponinterestpaymentswillbereinvestedatanunknownrate,wecannotdeterminewhichbondmightgivethehighestactualrealizedrate.Thus,wecannotcomparethembaseduponthiscriterion.However,iftheportfoliomanagerisriskinverseinthesensethatsheorhedoesn’twanttobuyalongertermbond,whichwilllikelhavemorevariabilityinitsreturn,thenthemanagermightprefertheshortertermbond(bondA)ofthresyears.Thisbondalsomatureswhenthemanagerwantstocashinthebond.Thus,themanagerwouldnothavetoworryaboutanypotentialcapitallossinsellingthelongertermbond(bondB).Themanagerwouldknowwithcertaintywhatthecashflowsare.IfThesecashflowsarespentwhenreceived,themanagerwouldknowexactlyhowmuchmoneycouldbespentatcertainpointsintime.Finally,amanagercantrytoprojectthetotalreturnperformanceofabondonthebasisofthepannedinvestmenthorizonandexpectationsconcerningreinvestmentratesandfuturemarketyields.Thisermitstheportfoliomanagertoevaluatethichofseveralpotentialbondsconsideredforacquisitionwillperformbestovertheplannedinvestmenthorizon.Aswejustrgued,thiscannotbedoneusingtheyieldtomaturityasameasureofrelativevalue.Usingtotalreturntoassessperformanceoversomeinvestmenthorizoniscalledhorizonanalysis.Whenatotalreturniscalculatedovenaninvestmenthorizon,itisreferredtoasahorizonreturn.Thehorizonanalysisframworenabledtheportfoliomanagertoanalyzetheperformanceofabondunderdifferentinterest-ratescenariosforreinvestmentratesandfuturemarketyields.Onlybyinvestigatingmultiplescenarioscantheportfoliomanagerseehowsensitivethebond’sperformancewillbetoeachscenario.Thiscanhelpthemanagerchoosebetweenthetwobondchoices.(b)Supposethattheportfoliomanagerplanstoholdthebondthatispurchasedforsixyearsinsteadofthreeyears.Inthiscase,whichwouldbethebestbondfortheportfoliomanagertopurchase?Answer:Simileartoourdiscussioninpart(a),wedonotknowwhichinvestmentwouldgivethehighestactualrelizedreturninsixyearswhenweconsiderreinvestingallcashflows.Ifthemanagerbuysathree-yearbond,thentherewouldbetheadditionaluncertaintyofnowknowingwhatthree-yearbondrateswouldbeinthreeyears.Thepurchaseoftheten-yearbondwouldbeheldlongerthanpreviously(sixyearscomparedtothreeyears)andrendercouponpaymentsforasix-yearperiodthatareknown.Ifthesecashflowsarespentwhenreceived,themanagerwillknowexactlyhowmuchmoneycouldbespentatcertainpointsintimeNotknowingwhichbondinvestmentwouldgivethehighestrealizedreturn,theportfoliomanagerwouldchoosethebondthatfitsthefirm’sgoalsintermsofmaturity.3)AnswerthebelowquestionsforbondsAandB.BondABondBCouponYieldtomaturityMaturity25Par$100.00$100.00Price$100.00$104.055(a)Calculatetheactualpriceofthebondsfora100-basis-pointincreaseininterestrates.Answer:ForBondA,wegetabondquoteof$100forourinitialpriceifwehavean8%couponrateandan8%yield.Ifwechangetheyield100basispointsotheyieldis9%,thenthevalueofthebond(P)isthepresentvalueofthecouponpaymentsplusthepresentvalueoftheparvalue.WehaveC=$40,y=4.5%,n=4,andM=$1,000.Insertingthesenumbersintoourpresentvalueofcouponbondformula,weget:Thepresentvalueoftheparormaturityvalueof$1,000is:Thus,thevalueofbondAwithayieldof9%,acouponrateof8%,andamaturityof2yearsis:P=$143.501+$838.561=$982.062.Thus,wegetabondquoteof$98.2062.WealreadyknowthatbondBwillgiveabondvalueof$1,000andabondquoteof$100sinceachangeof100basispointswillmaketheyieldandcouponratethesame,Forexample,insertingThus,thevalueofbondAwithayieldof9%,acouponrateof8%,andamaturityof2yearsis:P=$143.501+$838.561=$982.062.Thus,wegetabondquoteof$98.2062.WealreadyknowthatbondBwillgiveabondvalueof$1,000andabondquoteof$100sinceachangeof100basispointswillmaketheyieldandcouponratethesame,Forexample,inserting(b)Usingduration,estimatethepriceofthebondsfora100-basis-pointincreaseininterestrates.Answer:ToestimatethepriceofbondA,webeginbyfirstcomputingthemodifiedduration.WecanuseanalternativeformulathatdoesnotrequiretheextensivecalculationsrequiredbytheMacaulayprocedure.Theformulais:Puttingallapplicablevariablesintermsof$100,wehaveC=$4,n=4,y=0.045,andP=$98.2062.Insertingthesevalues,inthemodifieddurationformulagives:($1,975.308642[0.161439]+$35.664491)/$98.2062=($318.89117+$35.664491)/$98.2062=$354.555664/$98.2062=3.6103185orabout3.61.Convertingtoannualnumberbydividingbytwogivesamodifieddurationof1.805159(beforetheincreasein100basispointsitwas1.814948).Wenextsolveforthechangeinpriceusingthemodifieddurationof1.805159anddy=100basispoints=0.01.Wehave:WecannowsolveforthenewpriceofbondAasshownbelow:Thisisslightlylessthantheactualpriceof$982.062.Thedifferenceis$982.062–$981.948=$0.114.ToestimatethepriceofbondB,wefollowthesameprocedurejustshownforbondA.UsingthealternativeformulaformodifieddurationthatdoesnotrequiretheextensivecalculationsrequiredbytheMacaulayprocedureandnotingthatC=$45,n=10,y=0.045,andP=$100,weget:($791.27182+$0)/$100=7.912718orabout7.91(beforetheincreasein100basispointsitwas7.988834orabout7.99).Convertingtoanannualnumberbydividingbytwogivesamodifieddurationof3.956359(beforetheincreasein100basispointsitwas3.994417).WewillnowestimatethepriceofbondBusingthemodifieddurationmeasure.With100basispointsgivingdy=0.01andanapproximatedurationof3.956359,wehave:Thus,thenewpriceis(1–0.0395635)$1,040.55=(0.9604364)$1,040.55=$999.382.Thisisslightlylessthantheactualpriceof$1,000.Thedifferenceis$1,000–$999.382=$0.618.(c)Usingbothdurationandconvexitymeasures,estimatethepriceofthebondsfora100-basis-pointincreaseininterestrates.Answer:ForbondA,weusethedurationandconvexitymeasuresasgivenbelow.First,weusethedurationmeasure.Weadd100basispointsandgetayieldof9%.WenowhaveC=$40,y=4.5%,n=4,andM=$1,000.NOTE.Inpart(a)wecomputedtheactualbondpriceandgotP=$982.062.Priortothat,thepricesoldatpar(P=$1,000)sincethecouponrateandyieldwerethenequal.Theactualchangeinpriceis:($982.062–$1,000)=$17.938andtheactualpercentagechangeinpriceis:$17.938/$1,000=0.017938%.Wewillnowestimatethepricebyfirstapproximatingthedollarpricechange.With100basispointsgivingdy=0.01andamodifieddurationcomputedinpart(b)of1.805159,wehave:Thisisslightlymorenegativethantheactualpercentagedecreaseinpriceof1.7938%.Thedifferenceis1.7938%–(1.805159%)=1.7938%+1.805159%=0.011359%.Usingthe1.805159%justgivenbythedurationmeasure,thenewpriceforbondAis:Thisisslightlylessthantheactualpriceof$982.062.Thedifferenceis$982.062–$981.948=$0.114.Next,weusetheconvexitymeasuretoseeifwecanaccountforthedifferenceof0.011359%.Wehave:convexitymeasure(halfyears)ForbondA,weadd100basispointsandgetayieldof9%.WenowhaveC=$40,y=4.5%,n=4,andM=$1,000.NOTE.Inpart(a)wecomputedtheactualbondpriceandgotP=$982.062.Priortothat,thepricesoldatpar(P=$1,000)sincethecouponrateandyieldwerethenequal.Expressingnumbersintermsofa$100bondquote,wehave:C=$4,y=0.045,n=4,andP=$98.2062.Insertingthesenumbersintoourconvexitymeasureformulagives:convexitymeasure(halfyears)=Addingthedurationmeasureandtheconvexitymeasure,weget1.805159%+0.021166%=1.783994%.Recalltheactualchangeinpriceis:($982.062–$1,000)=$17.938andtheactualpercentagechangeinpriceis:$17.938/$1,000=0.017938orapproximately1.7938%.Usingthe1.783994%resultingfromboththedurationandconvexitymeasures,wecanestimatethenewpriceforbondA.Wehave:Addingthedurationmeasureandtheconvexitymeasure,weget1.805159%+0.021166%=1.783994%.Recalltheactualchangeinpriceis:($982.062–$1,000)=$17.938andtheactualpercentagechangeinpriceis:$17.938/$1,000=0.017938orapproximately1.7938%.Usingthe1.783994%resultingfromboththedurationandconvexitymeasures,wecanestimatethenewpriceforbondA.Wehave:Thisisslightlymorenegativethantheactualpercentagedecreaseinpriceof-3.896978%.Thedifferenceis(-3.896978%)-(-3.95635%)=0.059382%Usingthe-3.95635%justgivenbythedurationmeasure,thenewpriceforBondBis:Thisisslightlylessthantheactualpriceof$1,000.Thisdifferenceis$1,000-$999.382=$0.618Weusetheconvexitymeasuretoseeifwecanaccountforthedifferenceof00594%.Wehave:ForBondB,100basispointsareaddedandgetayieldof9%.WenowhaveC=$45,y=4.5%,n=10,andM=$1,000.Noteinpart(a),wecomputedtheactualbondpriceandgotP=$1,000sincethecouponrateandyieldwerethenequal.Priortothat,thepricesoldatP=$1,040.55.Expressingnumbersintermsofa$100bondquote,wehaveC=$4.5,y-0.045,n=10andP=$100.Insertingthesenumbersintoourconvexitymeasureformulagives:Theconvexitymeasure(inyears)=Note.DollarConvexityMeasure=ConvexityMeasure(years)timesP=19.452564($100)=$1,945.2564.ThepercentagepricechangeduetoconvexityisP2P2Thus,wehave0.097463%increaseinpricewhenweadjustforconvexitymeasure.Addingthedurationmeasureandconvexitymeasure,weget-3.9563659%+0.097263%equals-3.859096%.Recalltheactualchangeinpriceis($1,000-$1,040.55)=-$40.55andtheactualnewpriceisForBondA.Thisisaboutthesameastheactualpriceof$1,000.Thedifferenceis$1,000.394-$1,000=$0.394.Thus,usingtheconvexitymeasurealongwiththedurationmeasurehasnarrowedtheestimatedpricefromadifferenceof-$0.618to$0.394.(d)Commentontheaccuracyofyourresultsinpartsbandc,andstatewhyoneapproximationisclosertotheactualpricethantheother.Answer:ForbondA,theactualpriceis$982.062.Whenweusethedurationmeasure,wegetabondpriceof$981.948thatis$0.114lessthantheactualprice.Whenweusedurationandconvexmeasurestogether,wegetabondpriceof$982.160.Thisisslightlymorethantheactualpriceof$982.062.Thedifferenceis$982.160–$982.062=$0.098.Thus,usingtheconvexitymeasurealongwiththedurationmeasurehasnarrowedtheestimatedpricefromadifferenceof$0.114to$0.0981.ForbondB,theactualpriceis$1,000.Whenweusethedurationmeasure,wegetabondpriceof$999.382thatis$0.618lessthantheactualprice.Whenweusedurationandconvexmeasurestogether,wegetabondpriceof$1,000.394.Thisisslightlymorethantheactualpriceof$1,000.Thedifferenceis$1,000.394–$1,000=$0.394.Thus,usingtheconvexitymeasurealongwiththedurationmeasurehasnarrowedtheestimatedpricefromadifferenceof$0.618to$0.394Aswesee,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.Ceterisparibus,thehighertheyieldlevel,thelowerthepricevolatility.Thesamepropertyholdsformodifiedduration.Thus,a10%yieldtomaturitywillhavebothlessvolatilitythanan8%yieldtomaturityandalsoasmallerduration.Thereisconsistencybetweenthepropertiesofbondpricevolatilityandthepropertiesofmodifiedduration.Whenallotherfactorsareconstant,abondwithalongermaturitywillhavegreaterpricevolatility.Apropertyofmodifieddurationisthatwhenallotherfactorsareconstant,abondwithalongermaturitywillhaveagreatermodifiedduration.Also,allotherfactorsbeingconstant,abondwithalowercouponratewillhavegreaterbondpricevolatility.Also,generally,abondwithalowercouponratewillhaveagreatermodifiedduration.Thus,bondswithgreaterdurationswillgreaterpricevolatilities.44)Supposeaclientobservesthefollowingtwobenchmarkspreadsfortwotwobonds:BondissueUratedA:150basispointsBondissueVratedBBB:135basispointsYourclientisconfusedbecausehethoughtthelower-ratedbond(bondV)shouldofferahigherbenchmarkspreadthanthehigher-ratedbond(bondU).ExplainwhythebenchmarkspreadmaybelowerforbondU.5)ThebidandaskyieldsforaTreasurybillwerequotedbyadealeras5.91%and5.89%,respectively.Shouldn’tthebidyieldbelessthantheaskyield,becausethebidyieldindicateshowmuchthedealeriswillingtopayandtheaskyieldiswhatthedealeriswillingtoselltheTreasurybillfor?Answer:Thehigherbidmeansalowerprice.Sothedealeriswillingtopaylessthanwouldbepaidfortheloweraskprice.Weillustratethisbelow.Giventheyieldonabankdiscountbasis(Yd),thepriceofaTreasurybillisfoundbyfirstsolvingtheformulaforthedollardiscount(D),asfollows:ThepriceisthenPrice=F-DForthe100-dayTreasurybillwithafacevalue(F)of$100,000,iftheyieldonabankdiscountbasis(Yd)isquotedas5.91%,DisequalTherefore,price=$100,000–$1,641.67=$98,358.33.Forthe100-dayTreasurybillwithafacevalue(F)of$100,000,iftheyieldonabankdiscountbasis(Yd)isquotedas5.89%,Disequalto:Therefore,priceis:P=F–D=$100,000–$1,636.11=$98,363.89.Thus,thehigherbidquoteof5.91%(comparedtoloweraskquote5.89%)givesalowersellingpriceof$98,358.33(comparedto$98,363.89).The0.02%higheryieldtranslatesintoasellingpricethatis$5.56lower.Ingeneral,thequotedyieldonabankdiscountbasisisnotameaningfulmeasureofthereturnfromholdingaTreasurybill,fortworeasons.First,themeasureisbasedonaface-valueinvestmentratherthanontheactualdollaramountinvested.Second,theyieldisannualizedaccordingtoa360-dayratherthana365-dayyear,makingitdifficulttocompareTreasurybillyieldswithTreasurynotesandbonds,whichpayinterestona365-daybasis.Theuseof360daysforayearisamoneymarketconventionforsomemoneymarketinstruments,however.Despiteitsshortcomingsasameasureofreturn,thisisthemethodthatdealershaveadoptedtoquoteTreasurybills.Manydealerquotesheets,andsome