固定收益证-券Interest-Rate-Mod课件

Chapter17

Interest-RateModels

1Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallLearningObjectivesAfterreadingthischapter,youwillunderstandwhataninterest-ratemodelishowaninterest-ratemodelisrepresentedmathematicallythecharacteristicsofaninterest-ratemodel:drift,volatility,andmeanreversionwhataone-factorinterest-ratemodelisthedifferencebetweenanarbitrage-freemodelandanequilibriummodelthedifferenttypesofarbitrage-freemodelsandwhytheyareusedinpracticethedifferencebetweenanormalmodelandalognormalmodeltheempiricalevidenceoninterestratechangesconsiderationsinselectinganinterestratemodelhowtocalculatehistoricalvolatility2Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallMathematicalDescriptionofOne-FactorInterest-RateModelsInterest-ratemodelsmustincorporatestatisticalpropertiesofinterest-ratemovements.ThesepropertiesaredriftvolatilitymeanreversionThecommonlyusedmathematicaltoolfordescribingthemovementofinterestratesthatcanincorporatethesepropertiesisstochasticdifferentialequations(SDEs).Arigoroustreatmentofinterest-ratemodelingrequiresanunderstandingofthisspecializedtopicinmathematics.3Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallMathematicalDescriptionofOne-FactorInterest-RateModels(continued)Themostcommoninterest-ratemodelusedtodescribethebehaviorofinterestratesassumesthatshort-terminterestratesfollowsomestatisticalprocessandthatotherinterestratesinthetermstructurearerelatedtoshort-termrates.Theshort-terminterestrate(i.e.,shortrate)istheonlyonethatisassumedtodrivetheratesofallothermaturities.Hence,thesemodelsarereferredtoasone-factormodelswherethe“onefactor”istheshortrate.Theotherratesarenotrandomlydeterminedoncetheshortrateisspecified.Usingarbitragearguments,therateforallothermaturitiesisdetermined.4Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallMathematicalDescriptionofOne-FactorInterest-RateModels(continued)Therearealsomulti-factormodelsthathavebeenproposedintheliterature.Themostcommonmulti-factormodelisatwo-factormodelwherealong-termrateisthesecondfactor.Inpractice,however,one-factormodelsareusedbecauseofthedifficultyofapplyingevenatwo-factormodelaswellasempiricalevidencethatsupportsone-factormodels.5Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallMathematicalDescriptionofOne-FactorInterest-RateModels(continued)Whilethevalueoftheshortrateatsomefuturetimeisuncertain,thepatternbywhichitchangesovertimecanbeassumed.Instatisticalterminology,thispatternorbehavioriscalledastochasticprocess.Thus,describingthedynamicsoftheshortratemeansspecifyingthestochasticprocessthatdescribesthemovementoftheshortrate.Itisassumedthattheshortrateisacontinuousrandomvariableandthereforethestochasticprocessusedisacontinuous-timestochasticprocess.6Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallMathematicalDescriptionofOne-FactorInterest-RateModels(continued)Therearedifferenttypesofcontinuous-timestochasticprocessesusedininterest-ratemodeling.Inallofthesemodelsbecausetimeisacontinuousvariable,theletterdisusedtodenotethe“changein”somevariable.Specifically,inthemodelswelet

r=theshortrateandthereforedrdenotesthechangeintheshortratet=timeandthusdtdenotesthechangeintimeorequivalentlythelengthofthetimeinterval(foraverysmallintervaloftime)z=arandomtermanddzdenotesarandomprocess7Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallMathematicalDescriptionofOne-FactorInterest-RateModels

(continued)ABasicContinuous-TimeStochasticProcessAbasiccontinuous-timestochasticprocessfordescribingthedynamicsoftheshortrate(r)isgivenby:dr=bdt+σdz

dr=thechangeintheshortrateb=expecteddirectionofratechangedt=thechangeintimeorequivalentlythelengthofthetimeinterval(foraverysmallintervaloftime)σ=standarddeviationofthechangesintheshortratez=arandomtermanddzdenotesarandomprocessTheexpecteddirectionofthechangeintheshortrate(b)iscalledthedrifttermandσiscalledthevolatilityterm.Thechangeintheshortrate(dr)overthetimeinterval(dt)dependsontheexpecteddirectionofthechangeintheshortrate(b)arandomprocess(dz)thatisaffectedbyvolatility8Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallMathematicalDescriptionofOne-Factor

Interest-RateModels(continued)ABasicContinuous-TimeStochasticProcessTherandomnatureofthechangeintheshortratecomesfromtherandomprocessdz.Theassumptionsarethattherandomtermzfollowsanormaldistributionwithameanofzeroandastandarddeviationofone(i.e.,isastandardizednormaldistribution)thechangeintheshortrateisproportionaltothevalueoftherandomterm,whichdependsonthestandarddeviationofthechangeintheshortratethechangeintheshortrateforanytwodifferentshortintervalsoftimeisindependent

Theexpectedvalueofthechangeintheshortrateisequaltob,thedriftterm.Inthespecialcasewherethedrifttermiszeroandthevarianceisone,itcanbeshownthatthevarianceofthechangeintheshortrateoversomeintervaloflengthTisequaltoTandthereforethestandarddeviationisthesquarerootofT.9Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallMathematicalDescriptionofOne-FactorInterest-RateModels

(continued)ItôProcessNoticethatintheequation,dr=bdt+σdz,thatneitherthedriftterm(b)northestandarddeviationofthechangeintheshortrate(σ)dependsoneithertheleveloftheshortrate(r)andtime(t).Thereareeconomicreasonsthatmightsuggestthattheexpecteddirectionoftheratechangewilldependonthelevelofthecurrentshortrate;thesameistrueforσ.Wecanchangethedynamicsofthedrifttermandthedynamicsofthevolatilitytermbyallowingthesetwoparameterstodependontheleveloftheshortrateand/ortime.Wecandenotethatthedrifttermdependsonboththeleveloftheshortrateandtimebyb(r,t);thesameistrueforσ,e.g.,σ(r,t).Thus,wecanwritedr=b(r,t)dt+σ(r,t)dz

Thecontinuous-timestochasticmodelgivenbytheaboveequationiscalledanItoprocess.10Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallMathematicalDescriptionofOne-FactorInterest-RateModels

(continued)SpecifyingtheDynamicsoftheDriftTermInspecifyingthedynamicsofthedriftterm,onecanspecifythatthedrifttermdependsontheleveloftheshortratebyassumingitfollowsameanreversionprocess.Bymeanreversionitismeantthatsomelong-runstablemeanvaluefortheshortrateisassumed.Wedenotethisvalueby.So,ifrisgreaterthan,thedirectionofchangeintheshortratewillmovedowninthedirectionofthelong-runstablevalueandviceversa.Themeanreversionprocessthatspecifiesthedynamicsofthedrifttermis:

b(r,t)=α(r–)whereαiscalledthespeedofadjustmentbecauseitindicatesthespeedatwhichtheshortratewillmoveorconvergetothelong-runstablemeanvalue.11Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallMathematicalDescriptionofOne-FactorInterest-RateModels

(continued)SpecifyingtheDynamicsoftheVolatilityTermTherehavebeenseveralformulationsofthedynamicsofthevolatilityterm.Ifvolatilityisnotassumedtodependontime,thenσ(r,t)=σ(r).Ingeneral,thedynamicsofthevolatilitytermcanbespecifiedasfollows:σrγdzwhereγisequaltotheconstantelasticityofvariance.Theaboveequationiscalledtheconstantelasticityofvariancemodel(CEVmodel).TheCEVmodelallowsustodistinguishbetweenthedifferentspecificationsofthedynamicsofthevolatilitytermforthevariousinterest-ratemodelssuggestedbyresearchers.12Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallMathematicalDescriptionofOne-FactorInterest-RateModels

(continued)SpecifyingtheDynamicsoftheVolatilityTermFortheVasicekinterestratemodel,welookatthecaseforγ=0.Substitutingzeroforγintotheequationσrγdz,wegetthefollowingtheCEVmodelidentifiedbyVasicekwhofirstproposedit:γ=0:σ(r,t)=σIntheVasicekspecificationoftheCEVmodel,volatilityisindependentoftheleveloftheshortrateasintheequationofdr=bdt+σdz(wherebisthedriftterm)andisreferredtoasthenormalmodel.Inthenormalmodel,itispossiblefornegativeinterestratestobegenerated.13Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallMathematicalDescriptionofOne-FactorInterest-RateModels

(continued)SpecifyingtheDynamicsoftheVolatilityTermFortheDothaninterestratemodel,welookatthecaseforγ=1.Substitutingoneforγintotheequationσrγdz,wegetthefollowingtheCEVmodelspecifiedbyDothanwhofirstproposedit:γ=1:σ(r,t)=σrIntheDothanspecificationoftheCEVmodel,volatilityisproportionaltotheshortrate.Thismodelisreferredtoastheproportionalvolatilitymodel.14Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallMathematicalDescriptionofOne-FactorInterest-RateModels

(continued)SpecifyingtheDynamicsoftheVolatilityTermFortheCox-Ingersoll-Ross(CIR)interestratemodel,welookatthecaseforγ=½

.Substitutingoneforγintotheequationσrγdz,wegetthefollowingtheCEVmodelproposedbyCIR:γ=½:σ(r,t)=TheCIRspecification,referredtoasthesquare-rootmodel,makesthevolatilityproportionaltothesquarerateoftheshortrate.Negativeinterestratesarenotpossibleinthissquare-rootmodel.15Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallArbitrage-FreeVersusEquilibriumModelsArbitrage-FreeModelsInarbitrage-freemodels,alsoreferredtoasno-arbitragemodels,theanalysisbeginswiththeobservedmarketpriceofasetoffinancialinstruments.Thefinancialinstrumentscanincludecashmarketinstrumentsandinterest-ratederivatives,andtheyarereferredtoasthebenchmarkinstrumentsorreferenceset.Theunderlyingassumptionisthatthebenchmarkinstrumentsarefairlypriced.Arandomprocessforthegenerationofthetermstructureisassumed.Basedontherandomprocessandtheassumedvaluefortheparameterthatrepresentsthedriftterm,acomputationalprocedureisusedtocalculatethetermstructureofinterestrates.Themodelisreferredtoasarbitrage-freebecauseitmatchestheobservedpricesofthebenchmarkinstruments.16Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallArbitrage-FreeVersusEquilibriumModels(continued)Arbitrage-FreeModelsThemostpopulararbitrage-freeinterest-ratemodelsusedforvaluationare:theHo-LeemodeltheHull-WhitemodeltheKalotay-Williams-FabozzimodeltheBlack-KarasinkimodeltheBlack-Derman-ToymodeltheHeath-Jarrow-Mortonmodel17Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallArbitrage-FreeVersusEquilibriumModels(continued)Arbitrage-FreeModelsThefirstarbitrage-freeinterest-ratemodelwasintroducedbyHoandLeein1986.IntheHo-Leemodel,thereisnomeanreversionandvolatilityisindependentoftheleveloftheshortrate.Thatis,itisanormalmodelwhereγ=0.IntheKalotay-Williams-Fabozzimodel,changesintheshort-ratearemodeledbymodelingthenaturallogarithmofr;noallowanceformeanreversionisconsideredinthemodel.TheHeath-Jarrow-Morton(HJM)modelisageneralcontinuoustime,multi-factormodel;ithasreceivedconsiderableattentionintheindustryaswellasinthefinanceliterature.18Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallArbitrage-FreeVersusEquilibriumModels

(continued)EquilibriumModelsAfaircharacterizationofarbitrage-freemodelsisthattheyallowonetointerpolatethetermstructureofinterestratesfromasetofobservedmarketpricesatonepointintimeassumingthatonecanrelyonthemarketpricesused.Equilibriummodels,however,aremodelsthatseektodescribethedynamicsofthetermstructureusingfundamentaleconomicvariablesthatareassumedtoaffecttheinterest-rateprocess.Inthemodelingprocess,restrictionsareimposedallowingforthederivationofclosed-formsolutionsforequilibriumpricesofbondsandinterestratederivatives.Inthesemodelsafunctionalformoftheinterest-ratevolatilityisassumedhowthedriftmovesupanddownovertimeisassumed19Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallArbitrage-FreeVersusEquilibriumModels(continued)EquilibriumModelsIncharacterizingthedifferencebetweenarbitrage-freeandequilibriummodels,onecanthinkofthedistinctionbeingwhetherthemodelisdesignedtobeconsistentwithanyinitialtermstructure,orwhethertheparameterizationimpliesaparticularfamilyoftermstructureofinterestrates.Arbitrage-freemodelshavethedeficiencythattheinitialtermstructureisaninputratherthanbeingexplainedbythemodel.Basically,equilibriummodelsandarbitragemodelsareseekingtododifferentthings.20Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallArbitrage-FreeVersusEquilibriumModels

(continued)EquilibriumModelsInpractice,therearetwoconcernswithimplementingandusingequilibriummodels.Manyeconomictheoriesstartwithanassumptionabouttheclassofutilityfunctionstodescribehowinvestorsmakechoices.Thesemodelsarenotcalibratedtothemarketsothatthepricesobtainedfromthemodelcanleadtoarbitrageopportunitiesinthecurrenttermstructure.21Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallEmpiricalEvidenceonInterest-RateChangesInareviewofinterest-ratemodels,onecanencounterthefollowingissues:thechoicebetweennormalmodels(i.e.,volatilityisindependentofthelevelofinterestrates)andlogarithmmodelsifinterestratesarehighlyunlikelytobenegative,theninterest-ratemodelsthatallowfornegativeratesmaybelesssuitableasadescriptionoftheinterest-rateprocess22Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallEmpiricalEvidenceonInterest-RateChanges(continued)VolatilityofRatesandtheLevelofInterestRatesThedependenceofvolatilityonthelevelofinterestrateshasbeenexaminedbyseveralresearchers.Theearlierresearchfocusedonshort-termratesandgaveinconclusivefindings.OrenCheyettefoundthatfordifferentperiodstherearedifferentdegreesofdependenceofvolatilityonthelevelofinterestrates.Wheninterestratesarebelow10%,therelationshiphasbeenfoundtobeweak.Theimplicationisthatinmodelinginterestrates,onecanassumethatinterest-ratevolatilityisindependentofthelevelofinterestratesinanenvironmentwhereratesarelessthandoubledigit.Thatis,inmodelingthedynamicsofthevolatilitytermthenormalmodelcanbeused.23Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallEmpiricalEvidenceonInterest-RateChanges(continued)NegativeInterestRatesWhileourfocusisonnominalinterestrates,weknowthatrealinterestrateshavebeenfoundtobenegativeononlyrareoccasions.Thereasonisthatifthenominalrateisnegative,investorswillsimplyholdcash.Itisfairtosaythatwhilenegativeinterestratesarenotimpossible,theyareunlikely.Thesignificanceofthisisthatonemightarguethataninterest-ratemodelshouldnotpermitnegativeinterestrates(ornegativeratesgreaterthanafewbasispoints).24Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallSelectinganInterest-RateModelTheeaseofapplicationisacriticalissueinselectinganinterest-ratemodel.Forconsistencyinvaluation,aportfoliomanagerwouldwantamodelthatcanbeusedtovalueallfinancialinstrumentsthatareincludedinaportfolio.Inpractice,writingefficientalgorithmstovalueallfinancialinstrumentsthatmaybeincludedinaportfolioforsomeinterest-ratemodelsthathavebeenproposedintheliteratureis“difficultorimpossible.”25Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallSelectinganInterest-RateModel(continued)Basedontheempiricalevidence,somehaveconcludedthatthenormalinterestratemodelisasuitablemodel.Whatisimportantfromapracticalperspectiveisnotjustwhetherthenormalmodeladmitsthepossibilityofnegativeinterestratesbutwhethernegativeinterestratesmayhaveasignificantimpactonthepricingoffinancialinstruments.Whilethejuryisstillout,theconsensusseemstobethatnegativeinterestratesdonohaveanimpactinmodelinginterestrates.26Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallEstimatingInterest-RateVolatilityUsingHistoricalDataOneoftheinputsintoaninterest-ratemodelisinterest-ratevolatility.Marketparticipantsestimateyieldvolatilityinoneoftwomethods.Thefirstmethodisbyestimatinghistoricalinterestvolatility.Thismethoduseshistoricalinterestratestocalculatethestandarddeviationofinterest-ratechangesandforobviousreasonsisreferredtoashistoricalvolatility.Thesecondmethodismorecomplicatedandinvolvesusingmodelsforvaluingoption-typederivativeinstrumentstoobtainanestimateofwhatthemarketexpectsinterest-ratevolatilityis.27Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallEstimatingInterest-RateVolatilityUsingHistoricalData(continued)Basically,inanyoptionpricingmodel,theonlyinputthatisnotobservedinthemodelisinterest-ratevolatility.Sincetheexpectedinterest-ratevolatilityobtainedisbeing“backedout”ofthemodel,itisreferredtoasimpliedvolatility.DatainExhibit17-1canbeusedtoexplainhowtocalculatethehistoricalvolatilityasmeasuredbythestandarddeviationbasedontheabsoluteratechangeandthepercentagechangeinrates.(SeetruncatedversionofExhibit17-1inOverhead17-28.)ThehistoricalinterestratesshowninExhibit17-1aretheweeklyreturnsforone-monthLIBORfrom7/30/2004to7/29/2005.TheobservationsarebasedonbidratesforEurodollardepositscollectedaround9:30A.M.Easterntime.28Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallExhibit17-1DataforCalculatingHistoricalVolatility:One-Monthfrom7/30/2004to7/29/2005Date1-MonthLIBOR(%)AbsoluteRateChange(bps)PercentageRateChange(%)7/30/20041.438/6/20041.4964.1108/13/20041.5121.3338/20/20041.5210.6608/27/20041.5421.307…..…..…..…..7/22/20053.3861.7917/29/20053.4461.760Average3.981.69WeeklyVariance6.841.78WeeklyStdDev.2.621.33AnnualizedStdDev.18.869.6229Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallEstimatingInterest-RateVolatilityUsingHistoricalData(continued)Wecancalculatethehistoricalvolatilitybymeasuringthestandarddeviationbasedontheabsoluteratechangeandthepercentagechangeinrates.Ifcomputingaweeklystandarddeviation,wemustannualizethestandarddeviation.Theformulaforannualizingaweeklystandarddeviationistomultiplytheweeklystandarddeviationby.Formonthlydata,wemultiplyby.Fordailydata,wemultiplyby.Thedifferenceinthecalculatedhistoricalannualvolatilitycouldbesignificantdependingonthenumberoftradingdaysassumedinayear.30Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallAllrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmitted,inanyformorbyanymeans,electronic,mechanical,photocopying,recording,orotherwise,withoutthepriorwrittenpermissionofthepublisher.PrintedintheUnitedStatesofAmerica.31Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHall

Chapter18

AnalysisofBondswithEmbeddedOptions32Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallLearningObjectivesAfterreadingthischapter,youwillunderstandthedrawbacksofthetraditionalyieldspreadanalysiswhatstaticspreadisandunderwhatconditionsitwoulddifferfromthetraditionalyieldspreadthedisadvantagesofacallablebondfromtheinvestor’sperspectivetheyieldtoworstandthepitfallsofthetraditionalapproachtovaluingcallablebondstheprice–yieldrelationshipforacallablebondnegativeconvexityandwhenacallablebondmayexhibitithowthevalueofabondwithanembeddedoptioncanbedecomposedthelatticemethodandhowitisusedtovalueabondwithanembeddedoption33Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallLearningObjectives(continued)Afterreadingthischapter,youwillunderstandhowabinomialinterest-ratetreeisconstructedtobeconsistentwiththepricesfortheon-the-runissuesofanissuerandagivenvolatilityassumptionwhatanoption-adjustedspreadisandhowitiscalculatedusingthebinomialmethodthelimitationsofusingmodifieddurationandstandardconvexityasameasureofthepricesensitivityofabondwithanembeddedoptionthedifferencebetweeneffectivedurationandmodifieddurationhoweffectivedurationandeffectiveconvexityarecalculatedusingthebinomialmethod34Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallDrawbacksofTraditionalYieldSpreadAnalysisTraditionalanalysisoftheyieldpremiumforanon-Treasurybondinvolvescalculatingthedifferencebetweentheyieldtomaturity(oryieldtocall)ofthebondinquestionandtheyieldtomaturityofacomparable-maturityTreasury.ThelatterisobtainedfromtheTreasuryyieldcurve.Forexample,considertwo8.8%coupon25-yearbonds:Theyieldspreadforthesetwobondsastraditionallycomputedis109basispoints(10.24%minus9.15%).Thedrawbacksofthisconvention,however,are(1)theyieldforbothbondsfailstotakeintoconsiderationthetermstructureofinterestrates,and(2)inthecaseofcallableand/orputablebonds,expectedinterestratevolatilitymayalterthecashflowofabond.

YieldtoMaturity(%)9.1510.24Price$96.613387.0798IssueTreasuryCorporate

35Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallStaticSpread:AnAlternativeto

YieldSpreadIntraditionalyieldspreadanalysis,aninvestorcomparestheyieldtomaturityofabondwiththeyieldtomaturityofasimilarmaturityon-the-runTreasurysecurity.Suchacomparisonmakeslittlesense,becausethecashflowcharacteristicsofthecorporatebondwillnotbethesameasthatofthebenchmarkTreasury.Theproperwaytocomparenon-TreasurybondsofthesamematuritybutwithdifferentcouponratesistocomparethemwithaportfolioofTreasurysecuritiesthathavethesamecashflow.Thecorporatebond’svalueisequaltothepresentvalueofallthecashflows.36Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallStaticSpread:AnAlternative

toYieldSpread(continued)Thecorporatebond’svalue,assumingthatthecashflowsareriskless,willequalthepresentvalueofthereplicatingportfolioofTreasurysecurities.Inturn,thesecashflowsarevaluedattheTreasuryspotrates.Exhibit18-1showshowtocalculatethepriceofarisk-free8.8%25-yearbondassumingtheTreasuryspotratecurveshownintheexhibit.(SeetruncatedversionofExhibit18-1inOverhead18-7.)Thepricewouldbe$96.6133.Thecorporatebond’spriceis$87.0798,lessthanthepackageofzero-couponTreasurysecurities,becauseinvestorsinfactrequireayieldpremiumfortheriskassociatedwithholdingacorporatebondratherthanarisklesspackageofTreasurysecurities.37Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallExhibit18-1CalculationofPriceofa25-Year8.8%CouponBondUsingTreasurySpotRatesPeriodCashFlowTreasurySpotRate(%)PresentValue14.47.000004.251224.47.049994.105534.47.099983.962844.47.124983.825154.47.139983.692264.47.166653.5622….….….….464.410.100000.4563474.410.300000.4154484.410.500000.3774494.410.600000.350350104.410.800007.5278Theoreticalprice96.613438Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallStaticSpread:AnAlternativetoYieldSpread

(continued)Thestaticspread,alsoreferredtoasthezero-volatilityspread,isameasureofthespreadthattheinvestorwouldrealizeovertheentireTreasuryspotratecurveifthebondisheldtomaturity.ItisnotaspreadoffonepointontheTreasuryyieldcurve,asisthetraditionalyieldspread.Thestaticspreadiscalculatedasthespreadthatwillmakethepresentvalueofthecashflowsfromthecorporatebond,whendiscountedattheTreasuryspotrateplusthespread,equaltothecorporatebond’sprice.Atrial-anderrorprocedureisrequiredtodeterminethestaticspread.Exhibit18-2illustratesthecalculationofthestaticspreadfora25-year8.8%couponcorporatebond.(SeetruncatedversionofExhibit18-2inOverhead18-9.)39Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHallExhibit18-2CalculationoftheStaticSpreadfora25-Year8.8%CouponCorporateBondPresentValueifSpreadUsedIs:PeriodCashFlowTreasurySpotRate(%)100BP110BP120BP14.47.000004.23084.22874.226724.47.049994.06614.06224.058334.47.099983.90593.90033.894744.47.124983.75213.74493.737754.47.139983.60433.59573.5871….….….….….….464.410.100000.36680.35880.3511474.410.300000.33230.32500.3179484.410.500000.30060.29390.2873494.410.600000.27780.27140.265250104.410.800005.94165.80305.6677Totalpresentvalue88.547487.8029