固定收益證-券Analysis-of-Bonds-with-Embedded-Options課件

Chapter18

AnalysisofBondswithEmbeddedOptions18-1Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallLearningObjectivesAfterreadingthischapter,youwillunderstandthedrawbacksofthetraditionalyieldspreadanalysiswhatstaticspreadisandunderwhatconditionsitwoulddifferfromthetraditionalyieldspreadthedisadvantagesofacallablebondfromtheinvestor’sperspectivetheyieldtoworstandthepitfallsofthetraditionalapproachtovaluingcallablebondstheprice–yieldrelationshipforacallablebondnegativeconvexityandwhenacallablebondmayexhibitithowthevalueofabondwithanembeddedoptioncanbedecomposedthelatticemethodandhowitisusedtovalueabondwithanembeddedoption18-2Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallLearningObjectives(continued)Afterreadingthischapter,youwillunderstandhowabinomialinterest-ratetreeisconstructedtobeconsistentwiththepricesfortheon-the-runissuesofanissuerandagivenvolatilityassumptionwhatanoption-adjustedspreadisandhowitiscalculatedusingthebinomialmethodthelimitationsofusingmodifieddurationandstandardconvexityasameasureofthepricesensitivityofabondwithanembeddedoptionthedifferencebetweeneffectivedurationandmodifieddurationhoweffectivedurationandeffectiveconvexityarecalculatedusingthebinomialmethod18-3Copyright?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

18-4Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallStaticSpread:AnAlternativeto

YieldSpreadIntraditionalyieldspreadanalysis,aninvestorcomparestheyieldtomaturityofabondwiththeyieldtomaturityofasimilarmaturityon-the-runTreasurysecurity.Suchacomparisonmakeslittlesense,becausethecashflowcharacteristicsofthecorporatebondwillnotbethesameasthatofthebenchmarkTreasury.Theproperwaytocomparenon-TreasurybondsofthesamematuritybutwithdifferentcouponratesistocomparethemwithaportfolioofTreasurysecuritiesthathavethesamecashflow.Thecorporatebond’svalueisequaltothepresentvalueofallthecashflows.18-5Copyright?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.18-6Copyright?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….….….….464.410.100000.4563474.410.300000.4154484.410.500000.3774494.410.600000.350350104.410.800007.5278Theoreticalprice96.613418-7Copyright?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.)18-8Copyright?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….….….….….….464.410.100000.36680.35880.3511474.410.300000.33230.32500.3179484.410.500000.30060.29390.2873494.410.600000.27780.27140.265250104.410.800005.94165.80305.6677Totalpresentvalue88.547487.802987.079618-9Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallStaticSpread:AnAlternativetoYieldSpread(continued)Exhibit18-3showsthestaticspreadandthetraditionalyieldspreadforbondswithvariousmaturitiesandprices,assumingtheTreasuryspotratesshowninExhibit18-1.(SeetruncatedversionofExhibit18-3inOverhead18-11.)Noticethattheshorterthematurityofthebond,thelessthestaticspreadwilldifferfromthetraditionalyieldspread.Themagnitudeofthedifferencebetweenthetraditionalyieldspreadandthestaticspreadalsodependsontheshapeoftheyieldcurve.Thesteepertheyieldcurve,themorethedifferenceforagivencouponandmaturity.AnotherreasonforthesmalldifferencesinExhibit18-3isthatthecorporatebondmakesabulletpaymentatmaturity.Thedifferencebetweenthetraditionalyieldspreadandthestaticspreadwillbeconsiderablygreaterforsinkingfundbondsandmortgage-backedsecuritiesinasteepyieldcurveenvironment.18-10Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallExhibit18-3ComparisonofTraditionalYieldSpreadandStaticSpreadforVariousBondsaSpread(basispoints)BondPriceYieldtoMaturity(%)TraditionalStaticDifference25-year8.8%CouponBondTreasury96.61339.15———A88.547310.06911009B87.803110.1510011010C87.079810.2410912011….….….….….….5-year8.8%CouponBondTreasury105.95557.36———J101.79198.35991001K101.38678.451091101L100.98368.551191201aAssumesTreasuryspotratecurvegiveninExhibit18-1.18-11Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallCallableBondsandTheirInvestmentCharacteristicsThepresenceofacalloptionresultsintwodisadvantagestothebondholder:callablebondsexposebondholderstoreinvestmentriskpriceappreciationpotentialforacallablebondinadeclininginterest-rateenvironmentislimitedThisphenomenonforacallablebondisreferredtoaspricecompression.Iftheinvestorreceivessufficientpotentialcompensationintheformofahigherpotentialyield,aninvestorwouldbewillingtoacceptcallrisk.18-12Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallCallableBondsandTheirInvestmentCharacteristics(continued)TraditionalValuationMethodologyforCallableBondsWhenabondiscallable,thepracticehasbeentocalculateayieldtoworst,whichisthesmallestoftheyieldtomaturityandtheyieldtocallforallpossiblecalldates.Theyieldtocall(liketheyieldtomaturity)assumesthatallcashflowscanbereinvestedatthecomputedyield—inthiscasetheyieldtocall—untiltheassumedcalldate.Moreover,theyieldtocallassumesthattheinvestorwillholdthebondtotheassumedcalldatetheissuerwillcallthebondonthatdate.Often,theseunderlyingassumptionsabouttheyieldtocallareunrealisticbecausetheydonottakeintoaccounthowaninvestorwillreinvesttheproceedsiftheissueiscalled.18-13Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallCallableBondsandTheirInvestmentCharacteristics

(continued)Price-YieldRelationshipforaCallableBondTheprice–yieldrelationshipforanoption-freebondisconvex.Exhibit18-4(seeOverhead18-15)showstheprice–yieldrelationshipforbothanoncallablebondandthesamebondifitiscallable.Theconvexcurvea–a'istheprice–yieldrelationshipforthenoncallable

(option-free)bond.Theunusualshapedcurvedenotedbya–bistheprice–yieldrelationshipforthecallablebond.Thereasonfortheshapeoftheprice–yieldrelationshipforthecallablebondisasfollows.Whentheprevailingmarketyieldforcomparablebondsishigherthanthecouponinterest,itisunlikelythattheissuerwillcallthebond.Ifacallablebondisunlikelytobecalled,itwillhavethesameconvexprice–yieldrelationshipasanoncallablebondwhenyieldsaregreaterthany*.18-14Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallExhibit18-4

Price-YieldRelationshipforaNoncallableandCallableBondPriceYieldy*bNoncallableBonda’-aa’aCallableBonda-b18-15Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallCallableBondsandTheirInvestmentCharacteristics(continued)Price-YieldRelationshipforaCallableBondAsyieldsinthemarketdecline,thelikelihoodthatyieldswilldeclinefurthersothattheissuerwillbenefitfromcallingthebondincreases.Theexactyieldlevelatwhichinvestorsbegintoviewtheissuelikelytobecalledmaynotbeknown,butwedoknowthatthereissomelevel,sayy*.Atyieldlevelsbelowy*,theprice-yieldrelationshipforthecallablebonddepartsfromtheprice-yieldrelationshipforthenoncallablebond.Forarangeofyieldsbelowy*,thereispricecompression–thatis,thereislimitedpriceappreciationasyieldsdecline.Theportionofthecallablebondprice-yieldrelationshipbelowy*issaidtobenegativelyconvex.18-16Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallCallableBondsandTheirInvestmentCharacteristics(continued)Price-YieldRelationshipforaCallableBondNegativeconvexitymeansthatthepriceappreciationwillbelessthanthepricedepreciationforalargechangeinyieldofagivennumberofbasispoints.Forabondthatisoption-freeanddisplayspositiveconvexity,thepriceappreciationwillbegreaterthanthepricedepreciationforalargechangeinyield.ThepricechangesresultingfrombondsexhibitingpositiveconvexityandnegativeconvexityareshowninExhibit18-5(seeOverhead18-18).Itisimportanttounderstandthatabondcanstilltradeaboveitscallpriceevenifitishighlylikelytobecalled.18-17Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallExhibit18-5PriceVolatilityImplicationsofPositiveandNegativeConvexity

AbsoluteValueofPercentagePriceChangeChangeinInterestRatesPositiveConvexityNegativeConvexity-100basispoints X% LessthanY%+100basispoints LessthanX% Y%

18-18Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallComponentsofaBondwithanEmbeddedOptionTodevelopaframeworkforanalyzingabondwithanembeddedoption,itisnecessarytodecomposeabondintoitscomponentparts.Acallablebondisabondinwhichthebondholderhassoldtheissueracalloptionthatallowstheissuertorepurchasethecontractualcashflowsofthebondfromthetimethebondisfirstcallableuntilthematuritydate.Theownerofacallablebondisenteringintotwoseparatetransactions:buysanoncallablebondfromtheissuerforwhichshepayssomepricesellstheissueracalloptionforwhichshereceivestheoptionpriceAcallablebondisequaltothepriceofthetwocomponentsparts;thatis,callablebondprice=noncallablebondprice–

calloptionpriceThecalloptionpriceissubtractedfromthepriceofthenoncallablebondbecausewhenthebondholdersellsacalloption,shereceivestheoptionprice.Graphically,thiscanbeseeninExhibit18-6(seeOverhead18-20).Thedifferencebetweenthepriceofthenoncallablebondandthecallablebondatanygivenyieldisthepriceoftheembeddedcalloption.18-19Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallExhibit18-6

DecompositionofaPriceofaCallableBondPriceYieldy**bNoncallableBonda’-aa’aCallableBonda-by*PNCBPCBNote:Aty**yieldlevel:PNCB=noncallablebondprice PCB=callablebondprice PNCB-PCB=calloptionprice18-20Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallComponentsofaBondwithanEmbeddedOption

(continued)Thelogicappliedtocallablebondscanbesimilarlyappliedtoputablebonds.Inthecaseofaputablebond,thebondholderhastherighttosellthebondtotheissueratadesignatedpriceandtime.Aputablebondcanbebrokenintotwoseparatetransactions.Theinvestorbuysanoncallablebond.Theinvestorbuysanoptionfromtheissuerthatallowstheinvestortosellthebondtotheissuer.Thepriceofaputablebondisthenputablebondprice=non-putablebondprice+putoptionprice18-21Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallValuationModelThebondvaluationprocessrequiresthatweusethetheoreticalspotratetodiscountcashflows.Thisisequivalenttodiscountingataseriesofforwardrates.Foranembeddedoptionthevaluationprocessalsorequiresthatwetakeintoconsiderationhowinterest-ratevolatilityaffectsthevalueofabondthroughitseffectsontheembeddedoptions.Dependingonthestructureofthesecuritytobeanalyzed,threemodelscanbeusedtoaccountforthevaluationeffectofembeddedoptions.Thefirstmodelisforabondthatisnotamortgage-backedsecurityorasset-backedsecurityandwhichcanbeexercisedatmorethanonetimeoveritslife.Thesecondcaseisabondwithanembeddedoptionwheretheoptioncanbeexercisedonlyonce.Thethirdmodelisforamortgage-backedsecurityorcertaintypesofasset-backedsecurities.18-22Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallValuationModel(continued)ValuationofOption-FreeBondsThepriceofanoption-freebondisthepresentvalueofthecashflowsdiscountedatthespotrates.Toillustratethis,wecanusethefollowinghypotheticalyieldcurve:Wecansimplifytheillustrationbyassumingannual-paybonds.Usingthebootstrappingmethodology,thespotratesandtheone-yearforwardratescanbeobtained.

MaturityYears123YieldtoMaturity(%)3.504.004.50MarketValue100100100

Years123SpotRate(%)3.5004.0104.541One–YearForwardRate3.5004.5235.580

18-23Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallValuationModel

(continued)ValuationofOption-FreeBondsEXAMPLE.Consideranoption-freebondwiththreeyearsremainingtomaturityandacouponrateof5.25%.Thepriceofthisbondcanbecalculatedinoneoftwoways,bothproducingthesameresult.Thecouponpaymentscanbediscountedatthezero-couponrates:Thesecondwayistodiscountbytheone-yearforwardrates:18-24Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallValuationModel(continued)IntroducingInterest-RateVolatilityWhenweallowforembeddedoptions,considerationmustbegiventointerest-ratevolatility.Thiscanbedonebyintroducinganinterest-ratetree,alsoreferredtoasaninterest-ratelattice.Thistreeisnothingmorethanagraphicaldepictionoftheone-periodforwardratesovertimebasedonsomeassumedinterest-ratemodelandinterest-ratevolatility.18-25Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHall18-26ValuationModel(continued)Interest-RateModelAsexplainedinthepreviouschapter,aninterest-ratemodelisaprobabilisticdescriptionofhowinterestratescanchangeoverthelifeofafinancialinstrumentbeingevaluated.Aninterest-ratemodeldoesthisbymakinganassumptionabouttherelationshipbetweenthelevelofshort-terminterestratesandinterest-ratevolatility(e.g.,standarddeviationofinterestrates).Theinterest-ratemodelscommonlyusedarearbitrage-freemodelsbasedonhowshort-terminterestratescanevolve(i.e.,change)overtime.Theinterest-ratemodelsbasedsolelyonmovementsintheshort-terminterestratearereferredtoasone-factormodels.Morecomplexmodelswouldconsiderhowmorethanoneinterestratechangesovertime.Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHall18-27ValuationModel(continued)Interest-RateLatticeExhibit18-7(seeOverhead18-28)showsanexampleofthemostbasictypeofinterest-ratelatticeortree,abinomialinterest-ratetree.Thecorrespondingmodelisreferredtoasthebinomialmodel.Inthismodel,itisassumedthatinterestratescanrealizeoneoftwopossibleratesinthenextperiod.Inthevaluationmodelwepresentinthischapter,wewillusethebinomialmodel.Valuationmodelsthatassumethatinterestratescantakeonthreepossibleratesinthenextperiodarecalledtrinomialmodels.Morecomplexmodelsexistthatassumethatmorethanthreepossibleratesinthenextperiodcanberealized.Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallExhibit18-7

Three-YearBinomialInterest-RateTreer0Nr1HNHr1LNLr2HHNHHr3HHHNHHHr3HHLNHHLr2LLNLLr3LLLNLLLr2HLNHLr3HLLNHLLToday 1Year 2Years 3Years

18-28Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHall18-29ValuationModel

(continued)Interest-RateLatticeReturningtothebinomialinterest-ratetreeinExhibit18-7(aswasseeninOverhead18-28),eachnode(boldbluecircle:)representsatimeperiodthatisequaltooneyearfromthenodetoitsleft.EachnodeislabeledwithanN,representingnode,andasubscriptthatindicatesthepaththatone-yearforwardratestooktogettothatnode.HrepresentsthehigherofthetwoforwardratesandLthelowerofthetwoforwardratesfromtheprecedingyear.Forexample,nodeNHHmeansthattogettothatnodethefollowingpathforone-yearratesoccurred:Theone-yearraterealizedisthehigherofthetworatesinthefirstyearandthenthehigheroftheone-yearratesinthesecondyear.Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHall18-30ValuationModel(continued)Interest-RateLatticeExhibit18-7(aswasseeninOverhead18-28)showsthenotationforthebinomialinterest-ratetreeinthethirdyear.Wecansimplifythenotationbylettingrtbethelowerone-yearforwardratetyearsfromnowbecausealltheotherforwardratestyearsfromnowdependonthatrate.Exhibit18-8(seeOverhead18-31)showstheinterest-ratetreeusingthissimplifiednotation.Beforewegoontoshowhowtousethisbinomialinterest-ratetreetovaluebonds,wefirstneedtofocusonwhatthevolatilityparameter(

)intheexpressione2

representshowtofindthevalueofthebondateachnodeCopyright?2010PearsonEducation,Inc.PublishingasPrenticeHallExhibit18-8Three-YearBinomialInterestRateTreewithOne-YearForwardRatesr0Nr1e2NHr1NLr2e4NHHr3e6NHHr3e4NHHr2NLLr3NLLLr2e2NHLr3e2NHLLToday 1Year 2Years 3Years

Lower1-yrforwardrate r1 r2 r318-31Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallValuationModel(continued)VolatilityandtheStandardDeviationInthebinomialmodel,itcanbeshownthatthestandarddeviationoftheone-yearforwardrateisequaltor0.Thestandarddeviationisastatisticalmeasureofvolatility.Fornowitisimportanttoseethattheprocessthatweassumedgeneratesthebinomialinterest-ratetree(orequivalently,theforwardrates)impliesthatvolatilityismeasuredrelativetothecurrentlevelofrates.EXAMPLE.Ifis10%andtheone-yearrate(r0)is4%,whatisthestandarddeviationoftheone-yearforwardrate?Whatisifr0=12%?r0=4%×10%=0.4%or40basispointsr0=12%×10%=1.2%or120basispoints18-32Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallValuationModel(continued)DeterminingtheValueataNodeInthebinomialmodel,wefindthevalueofthebondatanodeisasillustratedinExhibit18-9(seeOverhead18-34).Calculatethebond’svalueatthetwonodestotherightofthenodewherewewanttoobtainthebond’svalue.Thecashflowatanodewillbeeitherthebond’svalueiftheshortrateisthehigherrateplusthecouponpaymentthebond’svalueiftheshortrateisthelowerrateplusthecouponpayment.thevalueisthepresentvalueoftheexpectedcashflowsTogetthebond’svalueatanodewefollowthefundamentalprocessforvaluation:theappropriatediscountratetouseistheone-yearforwardrateatthenode.18-33Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallExhibit18-9CalculatingaValueataNodeOne-YearRateatNodeWhereBond’sValueIsSoughtBond’sValueinHigher-RateStateOneYearForwardBond’sValueinLower-RateStateOneYearForwardCashFlowinHigher-RateStateCashFlowinLower-RateStateVr*VL+CVH+C18-34Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallValuationModel(continued)ConstructingtheBinomialInterest-RateTreeToconstructthebinomialinterest-ratetree,weusecurrenton-the-runyieldsandassumeavolatility,σ.Therootrateforthetree,r0,issimplythecurrentone-yearrate.Inthefirstyeartherearetwopossibleone-yearrates,thehigherrateandthelowerrate.Whatwewanttofindisthetwoforwardratesthatwillbeconsistentwiththevolatilityassumption,theprocessthatisassumedtogeneratetheforwardrates,andtheobservedmarketvalueofthebond.Thereisnosimpleformulaforthis.Itmustbefoundbyaniterativeprocess(i.e.,trialanderror).ThestepsaredescribedinOverheads18-36,18-37,and18-38andillustratedinExhibits18-10and18-11(seeOverheads18-39and18-40).18-35Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHall18-36ValuationModel(continued)ConstructingtheBinomialInterest-RateTreeStep1:Selectavalueforr1.Recallthatr1isthelowerone-yearforwardrateoneyearfromnow.Inthisfirsttrialwearbitrarily

selectedavalueof4.5%forr1.Step2:Determinethecorrespondingvalueforthehigherone-yearforwardrate.Thisrateisrelatedtothelowerone-yearforwardrateasfollows:r1(e2).ThisvalueisreportedatnodeNH.Step3:Computethebond’svalueoneyearfromnow.Thisvalueisdeterminedasfollows:3a.Thebond’svaluetwoyearsfromnowmustbedetermined.3b.Calculatethepresentvalueofthebond’svaluefoundin3ausingthehigherrate.ThisvalueisVH.3c.Calculatethepresentvalueofthebond’svaluefoundin3ausingthelowerrate.ThisvalueisVL.3d.AddthecoupontoVHandVLtogetthecashflowatNHandNL,respectively.3e.Calculatethepresentvalueofthetwovaluesusingtheone-yearforwardrateusingr*,sowecancompute:and.Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallValuationModel(continued)ConstructingtheBinomialInterest-RateTreeStep4:Calculatetheaveragepresentvalueofthetwocashflowsinstep3.Thisisthevalueatanodeis.Step5:Comparethevalueinstep4withthebond’smarketvalue.Ifthetwovaluesarethesame,ther1usedinthistrialistheoneweseek.Thisistheone-yearforwardratethatwouldbeusedinthebinomialinterest-ratetreeforthelowerrate,andthecorrespondingratewouldbeforthehigherrate.If,instead,thevaluefoundinstep4isnotequaltothemarketvalueofthebond,thismeansthatthevaluer1inthistrialisnottheone-periodforwardratethatisconsistentwith(1)thevolatilityassumptionof10%,(2)theprocessassumedtogeneratetheone-yearforwardrate,and(3)theobservedmarketvalueofthebond.Inthiscasethefivestepsarerepeatedwithadifferentvalueforr1.[Note.Ifwegetavaluelessthan$100,thenthevalueforr1istoolargeandthefivestepsmustberepeated,tryingalowervalueforr1.]Inthisexample,whenr1is4.5%wegetavalueof$99.567instep4,whichislessthantheobservedmarketvalueof$100.Therefore,4.5%istoolargeandthefivestepsmustberepeated,tryingalowervalueforr1.18-37Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallValuationModel(continued)ConstructingtheBinomialInterest-RateTreeAfterwecomputer1,wearestillnotdone.Supposethatwewantto“grow”thistreeforonemoreyear—thatis,wewanttodeterminer2.Nowwewillusethethree-yearon-the-runissuetogetr2.Thesamefivestepsareusedinaniterativeprocesstofindtheone-yearforwardratetwoyearsfromnow.Butnowourobjectiveisasf