CH11 The Term Structure of Interest Rates.docx
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CH11 The Term Structure of Interest Rates.docx
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CH11TheTermStructureofInterestRates
Chapter11TheTermStructureofInterestRates
THEYIELDCURVEANDTHETERMSTRUCTURE
Thegraphicaldepictionoftherelationshipbetweentheyieldonbondsofthesamecreditqualitybutofdifferentmaturitiesiscalledtheyieldcurve.ThesecurvesaretypicallyconstructedfromU.S.Treasurybonds,sincesuchsecuritieshaveeffectivelynodefaultorliquidityrisks.
UsingtheYieldCurvetoPriceaBond
Atfirstglancebondpricingusingayieldcurveshouldbearelativelysimplematter—justtaketheTreasuryyieldforthecomparablematurityandaddariskpremium.Butsecuritieswiththesamematuritycanhavedifferentyields,dependinguponthecouponrateandcashflowpattern.Becauseofthedifferentcashflowpatterns,thesameinterestratecannotbeusedtodiscountallcashflows.
ConstructingtheTheoreticalSpotRateCurve
Abetterwaytothinkaboutthesebondsisaspackagesofzero-couponbonds,whereintheinterestpaidforeachperiodisthepriceatmaturity.Determiningthevalueofeachzero-couponbondcanbederivedfromatheoreticalspotratecurveofTreasurybonds(thecurveistheoreticalbecausethereareveryfewzero-couponTreasuries).ThisprocessiscalledbootstrappingandemploysthebasicprinciplethatthevalueoftheTreasurycouponsecurityshouldbeequaltothevalueofthepackageofzero-couponsecurities.
UsingSpotRatestoValueaBond
Thetheorysuggeststhattheoreticalpriceofabondisequaltothepresentvalueofthecashflows.Whateconomicforcewillassurethatactualpricewillnotdepartsignificantlyfromitstheoreticalprice?
Sinceanycouponbondcanbeviewedasapackageofzero-couponinstrument,therateonazerocouponbondiscalledthespotrate.Thisisshownbyaprocessreferredtoasstrippingasecurity.Eachsecuritycreatedhasamaturityvalueandmaturitydateequaltothedatewhenthecouponpaymentofmaturityvalueisdue.Inthismanner,eachsecurityistreatedasazerocouponbond.ItistheprocessofcouponstrippingandsyntheticallycreatingaTreasurybondbybuyingthezerocouponbondsthatwilldrivethepriceofT-bondtoitsvalueasdeterminedbythespotrates.
FORWARDRATES
Aforwardrateistherateonaloanatsomefutureperiod,e.g.theyieldonone-yearbondsayearfromnow.Theimplicitforwardratecanbecalculatedfromtheknowledgeofcurrentspotratesandfindinganindifferencerate.Forexample,assumethespotrateofone-yearbondis8%andfortwo-yearbondsitis10%.Supposenowthataninvestorwouldconsiderbuyingeitheratwo-yearbond(long-term)ortwoone-yearbonds(short-term).Theyieldfortheperiodmustbethesame.Toreachthatindifferencepointtheinvestormustobtain12%nextyearonhisone-yearbondtomatchthetwo-yearoffering.Hencetheimplicitforwardratebecomes12%.Ofcourse,duetocompoundingandsemi-annualpaymentperiods,theactualsituationismorecomplex.Thegeneralformulais:
f=(1+z2)2/(1+z1)–1
wheref=theforwardrateforthesecondoneperiodbond
z1=thebondequivalentyieldofthetheoreticalspotrateforoneperiod
z2=thebondequivalentyieldofthetheoreticalspotrateforasecondperiod
Afutureinterestratecalculatedfromeitherthespotratesortheyieldcurveiscalledaforwardrateoranimpliedforwardrate.
RelationshipbetweenSpotRatesandShort-TermForwardRates
Ingeneral,therelationshipbetweenaT-periodspotrate,andimpliedforwardratesisasfollows:
Zt=[(1+z1)(1+f1)(1+f2)(1+f3)...(1+ft-1)]1/t–1
HistoricalShapesObservedfortheTreasuryYieldCurve
Whentheyieldcurveisupwardsloped,itiscalledpositivelyslopedyieldcurve.Theconventioninthemarketistorefertoapositivelyslopedyieldcurvewhosematurityspreadasmeasurebythe6-monthand30-yearyieldsasanormalyieldcurvewhenthespreadis300basispointsorless.Whenthespreadismorethan300bp,theyieldcurveissaidtobeasteepyieldcurve.Themarketscanalsoseeaninvertedyieldcurve,andaflatyieldcurve.
DETERMINANTSOFTHESHAPEOFTHETERMSTRUCTURE
Twomajortheoriesevolvedtoaccountfortheseobservedshapesoftheyieldcurve:
theexpectationtheoryandthemarketsegmentationtheory.Severalformsoftheexpectationtheoryincludethepureexpectationtheory,theliquiditytheory,andthepreferredhabitattheory.Thepureexpectationtheorypositsthatnosystematicfactorsotherthantheexpectedfutureshort-termrateaffectsforwardrates.Theothertwotheoriesassertthatotherfactorsareinvolved.Thelasttwoformsoftheexpectationtheoryarereferredtoasbiasexpectationstheory.
ThePureExpectationsTheory
Accordingtothepureexpectationtheory,theforwardratesrepresentexpectedfuturerates.Theentiretermstructureatanypointintimereflectsthemarket’scurrentexpectationsofthefamilyoffutureshort-termrates.Inotherwords,thelong-termratesrepresenttheaverageoffutureandpresentshort-termrates.Arisingtermstructureindicatesthatthemarketexpectsshort-termratestorise.Ifinterestratesareexpectedtoincrease,investorswilllendlong-termonlyiftheyarecoaxedtodosobyhigheryields.Hencethecurvewilltiltupward.
Risksassociatedwithbondinvestment:
Thetheoryneglectstherisksinherentininvestinginbonds.Tworiskscauseuncertaintyaboutthereturnoversomeinvestmenthorizon:
(1)theuncertaintyaboutthepriceofthebondattheendoftheinvestmenthorizon,calledthepricerisk,and
(2)theuncertaintyabouttherateatwhichtheproceedsfromabondthatmaturespriortothematuritydatecanbereinvesteduntilthematuritydate,calledthereinvestmentrisk.
Interpretationsofthepureexpectationtheory:
Thebroadestinterpretationofthistheoryreliesupontheassumptionthatholdingperiodreturnmustbeidenticalforanyinvestingstrategythatspanstheinvestor'schosenholdingperiod.Inotherwords,allcombinationsofbondsthatcompriseafive-yeartermshouldhavethesametotalyield.Amodificationofthisideaisconsideredtobemoreaccurate.Thelocalexpectationshypothesissaysthatlong-termbondswillhavethesamereturnswheninvestorshaveshortinvestmenthorizons.Thefinalinterpretation,calledthereturn-to-maturityexpectationstheory,holdsthatinvestorswillequatetheyieldonapackageofbondswithazero-couponbondwhichhasthesamematurity.Onlythesecondversionisthoughttoholdmuchpromiseindescribingactualmarketrates.
TheLiquidityTheory
Thistheorysuggeststhatinvestorsdonotlikeuncertainty,whichtendstoincreaselinearlyovertime.Theyprefertoholdliquidassetsandmustbecoaxedoutofsuchholdingsandintolong-termassetsbyhigherrates.ThisisanoutgrowthofKeynesiantheoryandimpliesthatthecurvemusthaveanupwardslope.Attheveryleastitassumesthatinvestorsdemandaliquiditypremiumovertheaverageofexpectedfuturerates.Accordingtotheliquiditytheoryofthetermstructure,theimplicitforwardrateswillnotbeanunbiasedestimateofthemarket’sexpectationsofthefutureinterestratesbecausetheyembodyaliquiditypremium.
ThePreferredHabitatTheory
Thepreferredhabitattheoryacceptstheviewthatthetermstructurereflectstheexpectationsofthefuturepathofinterestratesaswellasaliquidityriskpremium.Butitrejectstheviewthattheriskpremiummustriseuniformlywithmaturity.Investorsorborrowersareassumedtoprefercertainmaturitysectorsandtheyarereluctanttoshiftoutofthosesectorsunlessinducedbyapremiumtodoso.Theclassicexamplesofinvestorswithstrongmaturitypreferencesarebanks(whichbythenatureoftheirliabilitiesstructuresprefershort-terminvestments)andinsurancecompanies(whichforthesamereasonprefertostaywithlong-termsecurities).
MarketSegmentationTheory
Themarketsegmentationtheoryrepresentsastrongerversionofthepreferredhabitattheory,whereinshiftsarenotmadeoutofmaturitysectors.Hencethecurvecanbehumped,negativeorpositiveinslope,dependinguponactivitiesinvariousfinancialmarketsectors.IncreasedgovernmentborrowingintheformofT-notes,wouldraiseratesamongmedium-termsecurities.Theactualbehaviorofthetermstructureseemstofollowcharacteristicsofpureexpectationsinthatarbitragebetweenratesofvariousmaturitiesoccursaccordingtothemathematicalunderpinningsthattheexpectationshypothesisprovides.Segmentationrepresentsavalidalternative,butitlackstheabilitytoexplainmanyobservedshapesandmovementsoftheyieldcurveovertime.Somecombinationofthetwotheories,asrepresentedbyeitheraliquiditypremiumorpreferredhabitatapproach,isprobablythemostcorrectviewofinterestratemovementsovertime.
ANSWERSTOQUESTIONSFORCHAPTER11
(Questionsareinboldprintfollowedbyanswers.)
1.
a.Whatisayieldcurve?
b.Historically,whyhastheTreasuryyieldcurvebeentheonethatismostcloselywatchedbymarketparticipants?
a.Ayieldcurveisageographicalrepresentationofthetermstructure(therelationshipbetweenyieldtomaturityandthetermtomaturity)ofsecuritieshavingthesamecreditrating.
b.Treasurysecuritiesarerisk-freewithrespecttocreditandliquidity,andtheiryieldsserveasbenchmarksforothersecurities.
2.Whatismeantbyaspotrate?
Thespotrateistheyieldonazero-couponTreasurysecuritywiththesamematurityasthesecurityunderconsideration.Sometimesreferredtoasthetheoreticalspotrate,sinceitmustbederived,astherearefewzero-couponTreasuriesoutstanding.
3.Explainwhyitisinappropriatetouseoneyieldtodiscountallthecashflowsofafinancialasset.
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