(27)-CFA固定收益证券

CFA固定收益证券Chapter3.10SpotRatesandBondPricingII:Bootsrapping第三章第十节：即期利率与债券定价II:使用自助法计算即期利率BootstrappingAfterconstructingtheyieldsofon-the-runissuesusinglinearinterpolation,let’slookathowtoconverttheyieldsintothetheoreticalspotratecurveusingBootstrapping.9/13/2023Page1Assumetheparvalueis$100,allTreasurysecuritiesaretradingatparvaluebecauseYTM=couponrate.Throughouttheanalysisandillustrationtocome,itisimportanttorememberthatthebasicprincipleisthatthevalueoftheTreasuryCouponsecurityshouldbeequaltothevalueofthepackageofzero-couponinstrumentsthatduplicatesthecouponbond’scashflow.BootstrappingFirst,let’sstartwiththe6-monthT-bill.AT-billisazero-couponinstrument,soitsYTMof5.25%isequaltothe6-monthspotrate.Similarly,forthe1-yearT-bill,theYTMof5.5%istheone-yearspotrate.Giventhesetwospotrates,wecancomputethe1.5-yearspotrate.Thepriceofatheoretical1.5-yearcouponTreasuryshouldequalthepresentvalueofthreecashflowsfroman1.5-yearcouponTreasury,wheretheyieldusedfordiscountingisthespotratecorrespondingtothecashflow.Sincethecouponratefora1.5-yearcouponTreasuryis5.75%,wegetthecashflowsforthisTreasurysecurityare9/13/2023Page2BootstrappingBecausethe6-monthspotrateandone-yearspotrateare5.25%and5.50%,respectively,weknowthatz1=0.02625andz2=0.0275,thepresentvalueofthe1.5-yearcouponTreasurysecurityisBecausethepriceofthe1.5-yearcouponTreasurysecurityis$100,thefollowingrelationshipmusthold:Wecansolveforthetheoreticalsemi-annualperiodic1.5-yearspotrateasfollows:9/13/2023Page3Doublethissemi-annualrate,weobtainthetheoretical1.5-yearspotrate,s1.5=5.76%BootstrappingGiventhe1.5-yearspotrate,wecanobtainthetheoretical2-yearspotratefollowingthesameapproach.Thecouponrateofa2-yearcouponTreasuryis6.0%,sothecashflowsareThepriceofthe2-yearcouponTreasurysecurityis$100,thefollowingrelationshipmusthold:Solveforz4,anddoubleittoget2-yearspotrate,s2=6.02%.Wecanfollowthisapproachsequentiallytoderivethetheoreticalspotratesfortheremainingperiods.9/13/2023Page4TheoreticalTreasurySpotRates9/13/2023Page5yearyieldTheoreticalTreasurySpotRatesExample1:UsethetheoreticalTreasuryspotrateswederivedtopricea3-year,6%semi-annualcouponTreasurynotewithaparvalueof$1000.Solution:9/13/2023Page6ParBondYieldCurveAparbondyieldcurve,orparcurve,reflectsthecouponratethatahypotheticalbondateachmaturitywouldneedtohavetobepricedatpar.Itcanbederivedfromspotratecurve.Considera3-yearannual-paybondandspotratesforone,two,andthreeyearsares1=1%,s2=%ands3=3%,respectively.Thefollowingequationcanbeusedtocalculatethecouponratenecessaryforthebondtobetradingatpar:Substitutethevalueofspotratesintothee