對(duì)外經(jīng)濟(jì)貿(mào)易大學(xué)國(guó)際經(jīng)濟(jì)貿(mào)易學(xué)院《固定收益證券》部分答案

國(guó)際經(jīng)濟(jì)貿(mào)易學(xué)院研究生課程班《固定收益證券》試題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.3aAnswerthebelowquestionsforbondsAandB.

BondABondBCoupon8%9%Yieldtomaturity8%8%Maturity(years)25Par$100.00$100.00Price$100.00$104.055Calculatetheactualpriceofthebondsfora100-basis-pointincreaseininterestrates.Answer:ForBondA,wegetabondquoteof$100forourinitialpriceifwehavean8%couponrateandan8%yield.Ifwechangetheyield100basispointsotheyieldis9%,thenthevalueofthebond(P)isthepresentvalueofthecouponpaymentsplusthepresentvalueoftheparvalue.WehaveC=$40,y=4.5%,n=4,andM=$1,000.Insertingthesenumbersintoourpresentvalueofcouponbondformula,weget:1--1—

1--1—

(1+r)nr二$4011(1+0.045)40045二$143.501Thepresentvalueoftheparormaturityvalueof$1,000is:(1+(1+r)n—(1.045)4$1,000=$838561Thus,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,insertingUsingduration,estimatethepriceofthebondsfora100-basis-pointincreaseininterestrates.Answer:ToestimatethepriceofbondA,webeginbyfirstcomputingthemodifiedduration.WecanuseanalternativeformulathatdoesnotrequiretheextensivecalculationsrequiredbytheMacaulayprocedure.Theformulais:c"，]+n(100-C/y)ModifiedDuration二二一(1，y)n[。+y)n+1Puttingallapplicablevariablesintermsof$100,wehaveC=$4,n=4,y=0.045,andP=$98.2062.Insertingthesevalues,inthemodifieddurationformulagives:$4「，1]4($100-$4/0.045)1+^0.04521(1.045)4」(1.045)598.206271--%n$4「，1]4($100-$4/0.045)1+^0.04521(1.045)4」(1.045)598.2062($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:dP.…_—=-{ModifiedDuration\dy)=-1.805159(0.01)=-0.0180515WecannowsolveforthenewpriceofbondAasshownbelow:dP一—(1+—)P=(1-0.0180515)$1,000=$981.948Thisisslightlylessthantheactualpriceof$982.062.Thedifferenceis$982.062$981.948=$0.114.ToestimatethepriceofbondB,wefollowthesameprocedurejustshownforbondA.UsingthealternativeformulaformodifieddurationthatdoesnotrequiretheextensivecalculationsrequiredbytheMacaulayprocedureandnotingthatC=$45,n=10,y=0.045,andP=$100,weget:斗--斗--]+〃(100一°/丁)ModifiedDuration=尸I(1十丁)n:。十丁)n”$4.5P1-1]+10($100-$4.5/0.045)004521_-(1.045)10」+(1.045)11$100($791.27182+$0)/$100=7.912718orabout7.91(beforetheincreasein100basispointsitwas7.988834orabout7.99).Convertingtoanannualnumberbydividingbytwogivesamodifieddurationof3.956359(beforetheincreasein100basispointsitwas3.994417).WewillnowestimatethepriceofbondBusingthemodifieddurationmeasure.With100basispointsgivingdy=0.01andanapproximatedurationof3.956359,wehave:dP.…_—=-(ModifiedDuration)(dy)=-3.956359(0.01)=-0.0395635Thus,thenewpriceis(10.0395635)$1,040.55=(0.9604364)$1,040.55=$999.382.Thisisslightlylessthantheactualpriceof$1,000.Thedifferenceis$1,000$999.382=$0.618.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:dP,……P=-(ModifiedDuration)(dy)=-1.805159(0.01)=-0.01805159Thisisslightlymorenegativethantheactualpercentagedecreaseinpriceof1.7938%.Thedifferenceis1.7938%(1.805159%)=1.7938%+1.805159%=0.011359%.Usingthe1.805159%justgivenbythedurationmeasure,thenewpriceforbondAis:dP__.」(1+—)P=(1-0.01805159)$1,000二$981.948Thisisslightlylessthantheactualpriceof$982,062.Thedifferenceis$982,062$981,948二$0.114.Next,weusetheconvexitymeasuretoseeifwecanaccountforthedifferenceof0.011359%.Wehave:convexitymeasure(halfyears)二"1=\2C[1—，]—2c+n(n+1)(100-C/y)[口dy2Py3(1+y)ny2(1+y)n+1(1+y)n+2|LP」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)==16.93252$4\1_1]_2($4)4+4(5)(100-$4/y=0.045)]\=16.93250.0453L(1.045)4_|0.0452(1.045)5(1.045)6_|[$98.2062ConvexityMeasureinmperiodperyear16.9325TheConvexityMeasure(inyears)===4.233125m222Addingthedurationmeasureandtheconvexitymeasure,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:dP」，，一NewPrice=(1+p)P=(1+-0.01783994)$1,000=(0.9819484)$1,000=$982.160Addingthedurationmeasureandtheconvexitymeasure,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:dP,、一P=-(ModifiedDuration)(dy)=—3.056359(0.01)=—0.0395635Thisisslightlymorenegativethantheactualpercentagedecreaseinpriceof-3.896978%.Thedifferenceis(-3.896978%)-(-3.95635%)=0.059382%Usingthe-3.95635%justgivenbythedurationmeasure,thenewpriceforBondBis:dP__」(1+獷)P=(1—0.0395635)$1,04055=$999.382Thisisslightlylessthantheactualpriceof$1,000.Thisdifferenceis$1,000-$999.382=$0.618Weusetheconvexitymeasuretoseeifwecanaccountforthedifferenceof00594%.Wehave:ConvexityMeasure(halfyears)=d2P1dy2P2cConvexityMeasure(halfyears)=d2P1dy2P2cy31----(1+y)n2Cn+n(n+1)(100-C/y)1y2(1+y)n+i(1+y)n+2ForBondB,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:ConvexityMeasure(halfyears)=2($4.5)ConvexityMeasure(halfyears)=2($4.5)

(0.045)32($4.5)4+10(11)(100-$4.5/0.045)(0.045)2(1.045)11(1.045)121

$100=7,781.03[0.01000]=77.8103Theconvexitymeasure(inyears)=convexitymeasureinmperiodperyear=77.8103=1945256422Note.DollarConvexityMeasure=ConvexityMeasure(years)timesP=19.452564($100)=$1,945.2564.E.?dP1/7、Thepercentagepricechangeduetoconvexityis=—convexitymeasure(dy)2^2dP1_Insertinginthevalues,weget——=—(77.8103)(0.01)2=0.00097463^2Thus,wehave0.097463%increaseinpricewhenweadjustforconvexitymeasure.Addingthedurationmeasureandconvexitymeasure,weget-3.9563659%+0.097263%equals-3.859096%.Recalltheactualchangeinpriceis($1,000-$1,040.55)=-$40.55andtheactualnewpriceisdP—一人?(1-下)P=(1-0.03859096)$1,04055=(0.9614091)$1,04055=$1,000.394ForBondA.Thisisaboutthesameastheactualpriceof$1,000.Thedifferenceis$1,000.394-$1,000=$0.394.Thus,usingtheconvexitymeasurealongwiththedurationmeasurehasnarrowedtheestimatedpricefromadifferenceof-$0.618to$0.394.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$0.394

Aswesee,usingthedurationandconvexitymeasurestogetherismoreaccurate.Thereasonisthataddingtheconvexitymeasuretoourestimateenablesustoincludethesecondderivativethatcorrectsfortheconvexityoftheprice-yieldrelationship.Moredetailsareofferedbelow.Duration(modifiedordollar)attemptstoestimateaconvexrelationshipwithastraightline(thetangentline).Wecanspecifyamathematicalrelationshipthatprovidesabetterapproximationtothepricechangeofthebondiftherequiredyieldchanges.WedothisbyusingthefirsttwotermsofaTaylorseriestoapproximatethepricechangeasfollows:dPdP=——dPdP=——dy+

dy1d2P2dy2+(dy)2+error(1)DividingbothsidesofthisequationbyPtogetthepercentagepricechangegivesus:dPdP11d2Perror(2)——=dy++(dy)2+(2)PdyP2dy2'PThefirsttermontheright-handsideofequation(1)isequationforthedollarpricechangebasedondollardurationandisourapproximationofthepricechangebasedonduration.Inequation(2),thefirsttermontheright-handsideistheapproximatepercentagechangeinpricebasedonmodifiedduration.Thesecondterminequations(1)and(2)includesthesecondderivativeofthepricefunctionforcomputingthevalueofabond.Itisthesecondderivativethatisusedasaproxymeasuretocorrectfortheconvexityoftheprice-yieldrelationship.Marketparticipantsrefertothesecondderivativeofbondpricefunctionasthedollarconvexitymeasureofthebond.Thesecondderivativedividedbypriceisameasureofthepercentagechangeinthepriceofthebondduetoconvexityandisreferredtosimplyastheconvexitymeasure.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.4aSupposeaclientobservesthefollowingtwobenchmarkspreadsfortwobonds:BondissueUratedA:150basispointsBondissueVratedBBB:135basispoints

Yourclientisconfusedbecausehethoughtthelower-ratedbond(bondV)shouldofferahigherbenchmarkspreadthanthehigher-ratedbond(bondU).ExplainwhythebenchmarkspreadmaybelowerforbondU.5)ThebidandaskyieldsforaTreasurybillwerequotedbyadealeras5.91%and5.89%,respectively.Shouldntthebidyieldbelessthantheaskyield,becausethebidyieldindicateshowmuchthedealeriswillingtopayandtheaskyieldiswhatthedealeriswillingtoselltheTreasurybillfor?Answer:Thehigherbidmeansalowerprice.Sothedealeriswillingtopaylessthanwouldbepaidfortheloweraskprice.Weillustratethisbelow.Giventheyieldonabankdiscountbasis(Yd),thepriceofaTreasurybillisfoundbyfirstsolvingtheformulaforthedollardiscount(D),asfollows:t360ThepriceisthenPrice=F-DForthe100-dayTreasurybillwithafacevalue(F)of$100,000,iftheyieldonabankdiscountbasis(Yd)isquotedas5.91%,Disequalto:D二）（fD二）（f）（t360)=0.0591($100,000)(100360):$1,641.67Therefore,price=$100,000$1,641.67=$98,358.33.Forthe100-dayTreasurybillwithafacevalue(F)of$100,000,iftheyieldonabankdiscountbasis(Yd)isquotedas5.89%,Disequalto:D二乙(fD二乙(f)(t360)=0.0589($100,000)(100360):$1,636.11Therefore,priceis:P=FD=$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.Theuseof360daysforayearisamone