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”ANSWERIDISAGREEWITHTHESTATEMENT“THEPRICEOFAFLOATERWILLALWAYSTRADEATITSPARVALUE”FIRST,THECOUPONRATEOFAFLOATINGRATESECURITYORFLOATERISEQUALTOAREFERENCERATEPLUSSOMESPREADORMARGINFOREXAMPLE,THECOUPONRATEOFAFLOATERCANRESETATTHERATEONATHREEMONTHTREASURYBILLTHEREFERENCERATEPLUS50BASISPOINTSTHESPREADNEXT,THEPRICEOFAFLOATERDEPENDSONTWOFACTORS1THESPREADOVERTHEREFERENCERATEAND2ANYRESTRICTIONSTHATMAYBEIMPOSEDONTHERESETTINGOFTHECOUPONRATEFOREXAMPLE,AFLOATERMAYHAVEAMAXIMUMCOUPONRATECALLEDACAPORAMINIMUMCOUPONRATECALLEDAFLOORTHEPRICEOFAFLOATERWILLTRADECLOSETOITSPARVALUEASLONGAS1THESPREADABOVETHEREFERENCERATETHATTHEMARKETREQUIRESISUNCHANGEDAND2NEITHERTHECAPNORTHEFLOORISREACHEDHOWEVER,IFTHEMARKETREQUIRESALARGERSMALLERSPREAD,THEPRICEOFAFLOATERWILLTRADEBELOWABOVEPARIFTHECOUPONRATEISRESTRICTEDFROMCHANGINGTOTHEREFERENCERATEPLUSTHESPREADBECAUSEOFTHECAP,THENTHEPRICEOFAFLOATERWILLTRADEBELOWPAR2）APORTFOLIOMANAGERISCONSIDERINGBUYINGTWOBONDSBONDAMATURESINTHREEYEARSANDHASACOUPONRATEOF10PAYABLESEMIANNUALLYBONDB,OFTHESAMECREDITQUALITY,MATURESIN10YEARSANDHASACOUPONRATEOF12PAYABLESEMIANNUALLYBOTHBONDSAREPRICEDATPARASUPPOSETHATTHEPORTFOLIOMANAGERPLANSTOHOLDTHEBONDTHATISPURCHASEDFORTHREEYEARSWHICHWOULDBETHEBESTBONDFORTHEPORTFOLIOMANAGERTOPURCHASEANSWERTHESHORTERTERMBONDWILLPAYALOWERCOUPONRATEBUTITWILLLIKELYCOSTLESSFORAGIVENMARKETRATESINCETHEBONDSAREOFEQUALRISKINTERMSOFCREITQUALITYTHEMATURITYPREMIUMFORTHELONGERTERMBONDSHOULDBEGREATER,THEQUESTIONWHENCOMPARINGTHETWOBONDINVESTMENTSISWHATINVESTMENTWILLBEEXPECTETOGIVETHEHIGHESTCASHFLOWPERDOLLARINVESTEDINOTHERWORDS,WHICHINVESTMENTWILLBEEXPECTEDTOGIVETHEHIGHESTEFFECTIVEANNUALRATEOFRETURNINGENERAL,HOLDINGTHELONGERTERMBONDSHOULDCOMPENSATETHEINVESTORINTHEFORMOFAMATURITYPREMIUMANDAHIGHEREXPECTEDRETURNHOWEVER,ASSEENINTHEDISCUSSIONBELOW,THEACTUALREALIZEDRETURNFOREITHERINVESTMENTISNOTKNOWNWITHCERTAINTYTOBEGINWITH,ANINVESTORWHOPURCHASESABONDCANEXPECTTORECEIVEADOLLARRETURNFROMITHEPERIODICCOUPONINTERESTPAYMENTSMADEBETHEISSUER,IIANCAPITALGAINWHENTHEBONDMATURES,ISCALLED,ORISSOLDANDIIIINTERESTINCOMEGENERATEDFROMREINVESTMENTOFTHEPERIODICCASHFLOWSTHELASTCOMPONENTOFTHEPOTENTIALDOLLARRETURNISREFERREDTOASREINVESTMENTINCOMEFORASTANDARDBONDOURSITUATIONTHATMAKESONLYCOUPONPAYMENTSANDNOPERIODICPRINCIPALPAYMENTSPRIORTOTHEMATURITYDATE,THEINTERIMCASHFLOWSARESIMPLYTHECOUPONPAYMENTSCONSEQUENTLY,FORSUCHBONDSTHEREINVESTMENTINCOMEISSIMPLYINTERESTEARNEDFROMREINVESTINGTHECOUPONINTERESTPAYMENTSFORTHESEBONDS,THETHIRDCOMPONENTOFTHEPOTENTIALSOURCEOFDOLLARRETURNISREFERREDTOASTHEINTERESTONINTERESTCOMPONENTSIFWEAREGOINGTOCOUPUTEAPOTENTIALYIELDTOMAKEADECISION,WESHOULDBEAWAREOFTHEFACTTHATANYMEASUREOFABONDSPOTENTIALYIELDSHOULDTAKEINTOCONSIDERATIONEACHOFTHETHREECOMPONENTSDESCRIBEDABOVETHECURRENTYIELDCONSIDERSONLYTHECOUPONINTERESTPAYMENTSNOCONSIDERATIONISGIVENTOANYCAPITALGAINORINTERESTONINTERESTTHEYIELDTOMATURITYTAKESINTOACCOUNTCOUPONINTERESTANDANYCAPITALGAINITALSOCONSIDERSTHEINTERESTONINTERESTCOMPONENTADDITIONALLY,IMPLICITINTHEYIELDTOMATURITYCOMPUTATIONISTHEASSUMPTIONTHATTHECOUPONPAYMENTSCANBEREINVESTEDATTHECOMPUTEDYIELDTOMATURITYTHEYIELDTOMATURITYISAPROMISEDYIELDANDWILLBEREALIZEDONLYIFTHEBONDISHELDTOMATURITYANDTHECOUPONINTERESTPAYMENTSAREREINVESTEDATTHEYIELDTOMATURITYIFTHEBONDISNOTHELDTOMATURITYANDTHECOUPONPAYMENTSAREREINVESTEDATTHEYIELDTOMATURITY,THENTHEACTUALYIELDREALIZEDBYANINVESTORCANBEGREATERTHANORLESSTHANTHEYIELDTOMATURITYGIVENTHEFACTSTHATIONEBOND,IFBOUGHT,WILLNOTBEHELDTOMATURITY,ANDIITHECOUPONINTERESTPAYMENTSWILLBEREINVESTEDATANUNKNOWNRATE,WECANNOTDETERMINEWHICHBONDMIGHTGIVETHEHIGHESTACTUALREALIZEDRATETHUS,WECANNOTCOMPARETHEMBASEDUPONTHISCRITERIONHOWEVER,IFTHEPORTFOLIOMANAGERISRISKINVERSEINTHESENSETHATSHEORHEDOESNTWANTTOBUYALONGERTERMBOND,WHICHWILLLIKELHAVEMOREVARIABILITYINITSRETURN,THENTHEMANAGERMIGHTPREFERTHESHORTERTERMBONDBONDAOFTHRESYEARSTHISBONDALSOMATURESWHENTHEMANAGERWANTSTOCASHINTHEBONDTHUS,THEMANAGERWOULDNOTHAVETOWORRYABOUTANYPOTENTIALCAPITALLOSSINSELLINGTHELONGERTERMBONDBONDBTHEMANAGERWOULDKNOWWITHCERTAINTYWHATTHECASHFLOWSAREIFTHESECASHFLOWSARESPENTWHENRECEIVED,THEMANAGERWOULDKNOWEXACTLYHOWMUCHMONEYCOULDBESPENTATCERTAINPOINTSINTIMEFINALLY,AMANAGERCANTRYTOPROJECTTHETOTALRETURNPERFORMANCEOFABONDONTHEBASISOFTHEPANNEDINVESTMENTHORIZONANDEXPECTATIONSCONCERNINGREINVESTMENTRATESANDFUTUREMARKETYIELDSTHISERMITSTHEPORTFOLIOMANAGERTOEVALUATETHICHOFSEVERALPOTENTIALBONDSCONSIDEREDFORACQUISITIONWILLPERFORMBESTOVERTHEPLANNEDINVESTMENTHORIZONASWEJUSTRGUED,THISCANNOTBEDONEUSINGTHEYIELDTOMATURITYASAMEASUREOFRELATIVEVALUEUSINGTOTALRETURNTOASSESSPERFORMANCEOVERSOMEINVESTMENTHORIZONISCALLEDHORIZONANALYSISWHENATOTALRETURNISCALCULATEDOVENANINVESTMENTHORIZON,ITISREFERREDTOASAHORIZONRETURNTHEHORIZONANALYSISFRAMWORENABLEDTHEPORTFOLIOMANAGERTOANALYZETHEPERFORMANCEOFABONDUNDERDIFFERENTINTERESTRATESCENARIOSFORREINVESTMENTRATESANDFUTUREMARKETYIELDSONLYBYINVESTIGATINGMULTIPLESCENARIOSCANTHEPORTFOLIOMANAGERSEEHOWSENSITIVETHEBONDSPERFORMANCEWILLBETOEACHSCENARIOTHISCANHELPTHEMANAGERCHOOSEBETWEENTHETWOBONDCHOICESBSUPPOSETHATTHEPORTFOLIOMANAGERPLANSTOHOLDTHEBONDTHATISPURCHASEDFORSIXYEARSINSTEADOFTHREEYEARSINTHISCASE,WHICHWOULDBETHEBESTBONDFORTHEPORTFOLIOMANAGERTOPURCHASEANSWERSIMILEARTOOURDISCUSSIONINPARTA,WEDONOTKNOWWHICHINVESTMENTWOULDGIVETHEHIGHESTACTUALRELIZEDRETURNINSIXYEARSWHENWECONSIDERREINVESTINGALLCASHFLOWSIFTHEMANAGERBUYSATHREEYEARBOND,THENTHEREWOULDBETHEADDITIONALUNCERTAINTYOFNOWKNOWINGWHATTHREEYEARBONDRATESWOULDBEINTHREEYEARSTHEPURCHASEOFTHETENYEARBONDWOULDBEHELDLONGERTHANPREVIOUSLYSIXYEARSCOMPAREDTOTHREEYEARSANDRENDERCOUPONPAYMENTSFORASIXYEARPERIODTHATAREKNOWNIFTHESECASHFLOWSARESPENTWHENRECEIVED,THEMANAGERWILLKNOWEXACTLYHOWMUCHMONEYCOULDBESPENTATCERTAINPOINTSINTIMENOTKNOWINGWHICHBONDINVESTMENTWOULDGIVETHEHIGHESTREALIZEDRETURN,THEPORTFOLIOMANAGERWOULDCHOOSETHEBONDTHATFITSTHEFIRMSGOALSINTERMSOFMATURITY3）ANSWERTHEBELOWQUESTIONSFORBONDSAANDBBONDABONDBCOUPON89YIELDTOMATURITY88MATURITYYEARS25PAR1000010000PRICE10000104055ACALCULATETHEACTUALPRICEOFTHEBONDSFORA100BASISPOINTINCREASEININTERESTRATESANSWERFORBONDA,WEGETABONDQUOTEOF100FOROURINITIALPRICEIFWEHAVEAN8COUPONRATEANDAN8YIELDIFWECHANGETHEYIELD100BASISPOINTSOTHEYIELDIS9,THENTHEVALUEOFTHEBONDPISTHEPRESENTVALUEOFTHECOUPONPAYMENTSPLUSTHEPRESENTVALUEOFTHEPARVALUEWEHAVEC40,Y45,N4,ANDM1,000INSERTINGTHESENUMBERSINTOOURPRESENTVALUEOFCOUPONBONDFORMULA,WEGET411054301NRPTHEPRESENTVALUEOFTHEPARORMATURITYVALUEOF1,000IS41,083561NMRTHUS,THEVALUEOFBONDAWITHAYIELDOF9,ACOUPONRATEOF8,ANDAMATURITYOF2YEARSISP143501838561982062THUS,WEGETABONDQUOTEOF982062WEALREADYKNOWTHATBONDBWILLGIVEABONDVALUEOF1,000ANDABONDQUOTEOF100SINCEACHANGEOF100BASISPOINTSWILLMAKETHEYIELDANDCOUPONRATETHESAME,FOREXAMPLE,INSERTINGTHUS,THEVALUEOFBONDAWITHAYIELDOF9,ACOUPONRATEOF8,ANDAMATURITYOF2YEARSISP143501838561982062THUS,WEGETABONDQUOTEOF982062WEALREADYKNOWTHATBONDBWILLGIVEABONDVALUEOF1,000ANDABONDQUOTEOF100SINCEACHANGEOF100BASISPOINTSWILLMAKETHEYIELDANDCOUPONRATETHESAME,FOREXAMPLE,INSERTINGBUSINGDURATION,ESTIMATETHEPRICEOFTHEBONDSFORA100BASISPOINTINCREASEININTERESTRATESANSWERTOESTIMATETHEPRICEOFBONDA,WEBEGINBYFIRSTCOMPUTINGTHEMODIFIEDDURATIONWECANUSEANALTERNATIVEFORMULATHATDOESNOTREQUIRETHEEXTENSIVECALCULATIONSREQUIREDBYTHEMACAULAYPROCEDURETHEFORMULAIS2110/NNCCYYMODIFEDURATIONPPUTTINGALLAPPLICABLEVARIABLESINTERMSOF100,WEHAVEC4,N4,Y0045,ANDP982062INSERTINGTHESEVALUES,INTHEMODIFIEDDURATIONFORMULAGIVES2124510/10/459862NNCCYYODIFEURATINP1,975308642016143935664491/9820623188911735664491/982062354555664/98206236103185ORABOUT361CONVERTINGTOANNUALNUMBERBYDIVIDINGBYTWOGIVESAMODIFIEDDURATIONOF1805159BEFORETHEINCREASEIN100BASISPOINTSITWAS1814948WENEXTSOLVEFORTHECHANGEINPRICEUSINGTHEMODIFIEDDURATIONOF1805159ANDDY100BASISPOINTS001WEHAVE1805910851DPMOIFEDDURTIONYWECANNOWSOLVEFORTHENEWPRICEOFBONDAASSHOWNBELOW10185,4PTHISISSLIGHTLYLESSTHANTHEACTUALPRICEOF982062THEDIFFERENCEIS9820629819480114TOESTIMATETHEPRICEOFBONDB,WEFOLLOWTHESAMEPROCEDUREJUSTSHOWNFORBONDAUSINGTHEALTERNATIVEFORMULAFORMODIFIEDDURATIONTHATDOESNOTREQUIRETHEEXTENSIVECALCULATIONSREQUIREDBYTHEMACAULAYPROCEDUREANDNOTINGTHATC45,N10,Y0045,ANDP100,WEGET21210110/4545/0NNCYYMODIFEDURATINP791271820/1007912718ORABOUT791BEFORETHEINCREASEIN100BASISPOINTSITWAS7988834ORABOUT799CONVERTINGTOANANNUALNUMBERBYDIVIDINGBYTWOGIVESAMODIFIEDDURATIONOF3956359BEFORETHEINCREASEIN100BASISPOINTSITWAS3994417WEWILLNOWESTIMATETHEPRICEOFBONDBUSINGTHEMODIFIEDDURATIONMEASUREWITH100BASISPOINTSGIVINGDY001ANDANAPPROXIMATEDURATIONOF3956359,WEHAVE3956013956DPMOIFEDDURATIONYTHUS,THENEWPRICEIS1003956351,04055096043641,04055999382THISISSLIGHTLYLESSTHANTHEACTUALPRICEOF1,000THEDIFFERENCEIS1,0009993820618CUSINGBOTHDURATIONANDCONVEXITYMEASURES,ESTIMATETHEPRICEOFTHEBONDSFORA100BASISPOINTINCREASEININTERESTRATESANSWERFORBONDA,WEUSETHEDURATIONANDCONVEXITYMEASURESASGIVENBELOWFIRST,WEUSETHEDURATIONMEASUREWEADD100BASISPOINTSANDGETAYIELDOF9WENOWHAVEC40,Y45,N4,ANDM1,000NOTEINPARTAWECOMPUTEDTHEACTUALBONDPRICEANDGOTP982062PRIORTOTHAT,THEPRICESOLDATPARP1,000SINCETHECOUPONRATEANDYIELDWERETHENEQUALTHEACTUALCHANGEINPRICEIS9820621,00017938ANDTHEACTUALPERCENTAGECHANGEINPRICEIS17938/1,0000017938WEWILLNOWESTIMATETHEPRICEBYFIRSTAPPROXIMATINGTHEDOLLARPRICECHANGEWITH100BASISPOINTSGIVINGDY001ANDAMODIFIEDDURATIONCOMPUTEDINPARTBOF1805159,WEHAVPMOIFEDDURATIONYTHISISSLIGHTLYMORENEGATIVETHANTHEACTUALPERCENTAGEDECREASEINPRICEOF17938THEDIFFERENCEIS1793818051591793818051590011359USINGTHE1805159JUSTGIVENBYTHEDURATIONMEASURE,THENEWPRICEFORBONDAIS101859,08194DPTHISISSLIGHTLYLESSTHANTHEACTUALPRICEOF982062THEDIFFERENCEIS9820629819480114NEXT,WEUSETHECONVEXITYMEASURETOSEEIFWECANACCOUNTFORTHEDIFFERENCEOF0011359WEHAVECONVEXITYMEASUREHALFYEARS23212110/1NNNDPCCCYYYYPFORBONDA,WEADD100BASISPOINTSANDGETAYIELDOF9WENOWHAVEC40,Y45,N4,ANDM1,000NOTEINPARTAWECOMPUTEDTHEACTUALBONDPRICEANDGOTP982062PRIORTOTHAT,THEPRICESOLDATPARP1,000SINCETHECOUPONRATEANDYIELDWERETHENEQUALEXPRESSINGNUMBERSINTERMSOFA100BONDQUOTE,WEHAVEC4,Y0045,N4,ANDP982062INSERTINGTHESENUMBERSINTOOURCONVEXITYMEASUREFORMULAGIVESCONVEXITYMEASUREHALFYEARS342562414104/051693250501982Y224CONVEXITYMASURINMPEODRATHECONVXITYMEASURIYSADDINGTHEDURATIONMEASUREANDTHECONVEXITYMEASURE,WEGET180515900211661783994RECALLTHEACTUALCHANGEINPRICEIS9820621,00017938ANDTHEACTUALPERCENTAGECHANGEINPRICEIS17938/1,0000017938ORAPPROXIMATELY17938USINGTHE1783994RESULTINGFROMBOTHTHEDURATIONANDCONVEXITYMEASURES,WECANESTIMATETHENEWPRICEFORBONDAWEHAVEPR10178394,09814,0982160DNEWICADDINGTHEDURATIONMEASUREANDTHECONVEXITYMEASURE,WEGET180515900211661783994RECALLTHEACTUALCHANGEINPRICEIS9820621,00017938ANDTHEACTUALPERCENTAGECHANGEINPRICEIS17938/1,0000017938ORAPPROXIMATELY17938USINGTHE1783994RESULTINGFROMBOTHTHEDURATIONANDCONVEXITYMEASURES,WECANESTIMATETHENEWPRICEFORBONDAWEHAVE30569103956DPMOIFEDDURATIONYTHISISSLIGHTLYMORENEGATIVETHANTHEACTUALPERCENTAGEDECREASEINPRICEOF3896978THEDIFFERENCEIS38969783956350059382USINGTHE395635JUSTGIVENBYTHEDURATIONMEASURE,THENEWPRICEFORBONDBIS1039561,0459382DPTHISISSLIGHTLYLESSTHANTHEACTUALPRICEOF1,000THISDIFFERENCEIS1,0009993820618WEUSETHECONVEXITYMEASURETOSEEIFWECANACCOUNTFORTHEDIFFERENCEOF00594WEHAVE2321210/1NNNDPCCYCONVEXITYMASURHLFYESYYPFORBONDB,100BASISPOINTSAREADDEDANDGETAYIELDOF9WENOWHAVEC45,Y45,N10,ANDM1,000NOTEINPARTA,WECOMPUTEDTHEACTUALBONDPRICEANDGOTP1,000SINCETHECOUPONRATEANDYIELDWERETHENEQUALPRIORTOTHAT,THEPRICESOLDATP1,04055EXPRESSINGNUMBERSINTERMSOFA100BONDQUOTE,WEHAVEC45,Y0045,N10ANDP100INSERTINGTHESENUMBERSINTOOURCONVEXITYMEASUREFORMULAGIVES310211224545045/00CONVEXITYMSURHLFYES7,8103781THECONVEXITYMEASUREINYEARS22781039456CONVEXITYMASUREINPIODERYANOTEDOLLARCONVEXITYMEASURECONVEXITYMEASUREYEARSTIMESP194525641001,9452564THEPERCENTAGEPRICECHANGEDUETOCONVEXITYIS212DPCONVEXITYMASUREDYINSERTINGINTHEVALUES,WEGET178030974632DPTHUS,WEHAVE0097463INCREASEINPRICEWHENWEADJUSTFORCONVEXITYMEASUREADDINGTHEDURATIONMEASUREANDCONVEXITYMEASURE,WEGET395636590097263EQUALS3859096RECALLTHEACTUALCHANGEINPRICEIS1,0001,040554055ANDTHEACTUALNEWPRICEIS10385961,04596140,51,0394DPFORBONDATHISISABOUTTHESAMEASTHEACTUALPRICEOF1,000THEDIFFERENCEIS1,0003941,0000394THUS,USINGTHECONVEXITYMEASUREALONGWITHTHEDURATIONMEASUREHASNARROWEDTHEESTIMATEDPRICEFROMADIFFERENCEOF0618TO0394DCOMMENTONTHEACCURACYOFYOURRESULTSINPARTSBANDC,ANDSTATEWHYONEAPPROXIMATIONISCLOSERTOTHEACTUALPRICETHANTHEOTHERANSWERFORBONDA,THEACTUALPRICEIS982062WHENWEUSETHEDURATIONMEASURE,WEGETABONDPRICEOF981948THATIS0114LESSTHANTHEACTUALPRICEWHENWEUSEDURATIONANDCONVEXMEASURESTOGETHER,WEGETABONDPRICEOF982160THISISSLIGHTLYMORETHANTHEACTUALPRICEOF982062THEDIFFERENCEIS9821609820620098THUS,USINGTHECONVEXITYMEASUREALONGWITHTHEDURATIONMEASUREHASNARROWEDTHEESTIMATEDPRICEFROMADIFFERENCEOF0114TO00981FORBONDB,THEACTUALPRICEIS1,000WHENWEUSETHEDURATIONMEASURE,WEGETABONDPRICEOF999382THATIS0618LESSTHANTHEACTUALPRICEWHENWEUSEDURATIONANDCONVEXMEASURESTOGETHER,WEGETABONDPRICEOF1,000394THISISSLIGHTLYMORETHANTHEACTUALPRICEOF1,000THEDIFFERENCEIS1,0003941,0000394THUS,USINGTHECONVEXITYMEASUREALONGWITHTHEDURATIONMEASUREHASNARROWEDTHEESTIMATEDPRICEFROMADIFFERENCEOF0618TO0394ASWESEE,USINGTHEDURATIONANDCONVEXITYMEASURESTOGETHERISMOREACCURATETHEREASONISTHATADDINGTHECONVEXITYMEASURETOOURESTIMATEENABLESUSTOINCLUDETHESECONDDERIVATIVETHATCORRECTSFORTHECONVEXITYOFTHEPRICEYIELDRELATIONSHIPMOREDETAILSAREOFFEREDBELOWDURATIONMODIFIEDORDOLLARATTEMPTSTOESTIMATEACONVEXRELATIONSHIPWITHASTRAIGHTLINETHETANGENTLINEWECANSPECIFYAMATHEMATICALRELATIONSHIPTHATPROVIDESABETTERAPPROXIMATIONTOTHEPRICECHANGEOFTHEBONDIFTHEREQUIREDYIELDCHANGESWEDOTHISBYUSINGTHEFIRSTTWOTERMSOFATAYLORSERIESTOAPPROXIMATETHEPRICECHANGEASFOLLOWS2211DPYDYERODIVIDINGBOTHSIDESOFTHISEQUATIONBYPTOGETTHEPERCENTAGEPRICECHANGEGIVESUS222RYYPDTHEFIRSTTERMONTHERIGHTHANDSIDEOFEQUATION1ISEQUATIONFORTHEDOLLARPRICECHANGEBASEDONDOLLARDURATIONANDISOURAPPROXIMATIONOFTHEPRICECHANGEBASEDONDURATIONINEQUATION2,THEFIRSTTERMONTHERIGHTHANDSIDEISTHEAPPROXIMATEPERCENTAGECHANGEINPRICEBASEDONMODIFIEDDURATIONTHESECONDTERMINEQUATIONS1AND2INCLUDESTHESECONDDERIVATIVEOFTHEPRICEFUNCTIONFORCOMPUTINGTHEVALUEOFABONDITISTHESECONDDERIVATIVETHATISUSEDASAPROXYMEASURETOCORRECTFORTHECONVEXITYOFTHEPRICEYIELDRELATIONSHIPMARKETPARTICIPANTSREFERTOTHESECONDDERIVATIVEOFBONDPRICEFUNCTIONASTHEDOLLARCONVEXITYMEASUREOFTHEBONDTHESECONDDERIVATIVEDIVIDEDBYPRICEISAMEASUREOFTHEPERCENTAGECHANGEINTHEPRICEOFTHEBONDDUETOCONVEXITYANDISREFERREDTOSIMPLYASTHECONVEXITYMEASUREEWITHOUTWORKINGTHROUGHCALCULATIONS,INDICATEWHETHERTHEDURATIONOFTHETWOBONDSWOULDBEHIGHERORLOWERIFTHEYIELDTOMATURITYIS10RATHERTHAN8ANSWERLIKETERMTOMATURITYANDCOUPONRATE,THEYIELDTOMATURITYISAFACTORTHATINFLUENCESPRICEVOLATILITYCETERISPARIBUS,THEHIGHERTHEYIELDLEVEL,THELOWERTHEPRICEVOLATILITYTHESAMEPROPERTYHOLDSFORMODIFIEDDURATIONTHUS,A10YIELDTOMATURITYWILLHAVEBOTHLESSVOLATILITYTHANAN8YIELDTOMATURITYANDALSOASMALLERDURATIONTHEREISCONSISTENCYBETWEENTHEPROPERTIESOFBONDPRICEVOLATILITYANDTHEPROPERTIESOFMODIFIEDDURATIONWHENALLOTHERFACTORSARECONSTANT,ABONDWITHALONGERMATURITYWILLHAVEGREATERPRICEVOLATILITYAPROPERTYOFMODIFIEDDURATIONISTHATWHENALLOTHERFACTORSARECONSTANT,ABONDWITHALONGERMATURITYWILLHAVEAGREATERMODIFIEDDURATIONALSO,ALLOTHERFACTORSBEINGCONSTANT,ABONDWITHALOWERCOUPONRATEWILLHAVEGREATERBONDPRICEVOLATILITYALSO,GENERALLY,ABONDWITHALOWERCOUPONRATEWILLHAVEAGREATERMODIFIEDDURATIONTHUS,BONDSWITHGREATERDURATIONSWILLGREATERPRICEVOLATILITIES4）SUPPOSEACLIENTOBSERVESTHEFOLLOWINGTWOBENCHMARKSPREADSFORTWOBONDSBONDISSUEURATEDA150BASISPOINTSBONDISSUEVRATEDBBB135BASISPOINTSYOURCLIENTISCONFUSEDBECAUSEHETHOUGHTTHELOWERRATEDBONDBONDVSHOULDOFFERAHIGHERBENCHMARKSPREADTHANTHEHIGHERRATEDBONDBONDUEXPLAINWHYTHEBENCHMARKSPREADMAYBELOWERFORBONDU5）THEBIDANDASKYIELDSFORATREASURYBILLWEREQUOTEDBYADEALERAS591AND589,RESPECTIVELYSHOULDNTTHEBIDYIELDBELESSTHANTHEASKYIELD,BECAUSETHEBIDYIELDINDICATESHOWMUCHTHEDEALERISWILLINGTOPAYANDTHEASKYIELDISWHATTHEDEALERISWILLINGTOSELLTHETREASURYBILLFORANSWERTHEHIGHERBIDMEANSALOWERPRICESOTHEDEALERISWILLINGTOPAYLESSTHANWOULDBEPAIDFORTHELOWERASKPRICEWEILLUSTRATETHISBELOWGIVENTHEYIELDONABANKDISCOUNTBASISYD,THEPRICEOFATREASURYBILLISFOUNDBYFIRSTSOLVINGTHEFORMULAFORTHEDOLLARDISCOUNTD,ASFOLLOWS360DTDYFTHEPRICEISTHENPRICEFDFORTHE100DAYTREASURYBILLWITHAFACEVALUEFOF100,000,IFTHEYIELDONABANKDISCOUNTBASISYDISQUOTEDAS591,DISEQUALTO10059,647363DTDYFTHEREFORE,PRICE100,0001,6416798,35833FORTHE100DAYTREASURYBILLWITHAFACEVALUEFOF100,000,IFTHEYIELDONABANKDISCOUNTBASISYDISQUOTEDAS589,DISEQUALTO100589,63136DTDYFTHEREFORE,PRICEISPFD100,0001,6361198,36389THUS,THEHIGHERBIDQUOTEOF591COMPAREDTOLOWERASKQUOTE589GIVESALOWERSELLINGPRICEOF98,35833COMPAREDTO98,36389THE002HIGHERYIELDTRANSLATESINTOASELLINGPRICETHATIS556LOWERINGENERAL,THEQUOTEDYIELDONABANKDISCOUNTBASISISNOTAMEANINGFULMEASUREOFTHERETURNFROMHOLDINGATREASURYBILL,FORTWOREASONSFIRST,THEMEASUREISBASEDONAFACEVALUEINVESTMENTRATHERTHANONTHEACTUALDOLLARAMOUNTINVESTEDSECOND,THEYIELDISANNUALIZEDACCORDINGTOA360DAYRATHERTHANA365DAYYEAR,MAKINGITDIFFICULTTOCOMPARETREASURYBILLYIELDSWITHTREASURYNOTESANDBONDS,WHICHPAYINTERESTONA365DAYBASISTHEUSEOF360DAYSFORAYEARISAMONEYMARKETCONVENTIONFORSOMEMONEYMARKETINSTRUMENTS,HOWEVERDESPITEITSSHORTCOMINGSASAMEASUREOFRETURN,THISISTHEMETHODTHATDEALERSHAVEADOPTEDTOQUOTETREASURYBILLSM