投资学第六讲

Lecture7

BondValuationCompoundedinterestrateAssumeannualinterestrateisr,ifAnnuallycompounded,SemianuallycompoundedContinuouslycompoundedBondpricePB= PriceofthebondCt= interestorcouponpaymentsF

=FaceValue(ParValue)T = numberofperiodstomaturityr = discountratet=1+20=

PB401(1+.03)t10001(1+.03)20Ct =40(SA)P =1000T =20periodsr =3%(SA)PB=$1,148.77

SolvingforPrice:e.g.10yearbond,couponrate8%,currentinterestrate6%,parvalue$1,000SolvingforpriceE.g.10yearbond,couponrate8%,parvalue$1,000,whatisthebondpriceiftheinterestrateis10%or8%?E.g.30yearbond,couponrate8%,parvalue$1,000,whatisthebondpriceiftheinterestrateis10%,8%,or6%Factorsaffecttheprice:discountrate,timetomaturityYieldtoMaturityInterestratethatmakesthepresentvalueofthebond’spaymentsequaltoitspriceSolvethebondformulaforrYieldtoMaturityExample10yrMaturity CouponRate=7%Price=$950Solveforr=semiannualrater=3.8635%PricesandYieldshaveaninverserelationshipWhenyieldsgetveryhighthevalueofthebondwillbeverylowWhenyieldsapproachzero,thevalueofthebondapproachesthesumofthecashflowsBondPricesandYieldsPriceYieldPricesandYieldInverserelationshipbetweenpriceandyieldAnincreaseinabond’syieldtomaturityresultsinasmallerpricedeclinethanthegainassociatedwithadecreaseinyieldLong-termbondstendtobemorepricesensitivethanshort-termbondsBondPricingRelationshipsAsmaturityincreases,pricesensitivityincreasesatadecreasingratePricesensitivityisinverselyrelatedtoabond’scouponratePricesensitivityisinverselyrelatedtotheyieldtomaturityatwhichthebondissellingBondPricingRelationships(cont’d)AmeasureoftheeffectivematurityofabondTheweightedaverageofthetimesuntileachpaymentisreceived,withtheweightsproportionaltothepresentvalueofthepaymentDurationistheslopeoftheprice-yieldcurveexpressedasafractionofthebondpriceDurationcanbeusedtomeasurethesensitivityofbondpricechangestoyieldchangesDurationDuration:CalculationY:discountrateCFt:cashflowonyeartDurationcalculationexampleTwoyearbond,couponrate6%,yieldtomaturity8%,parvalue$1,000,computethedurationComputethebondpricefirst60/(1+8%)+1060/(1+8)2=964.33Computetheduration[1*60/(1+8%)+2*1060/(1+8)2]/964.33=1.94RulesforDurationRule1Thedurationofazero-couponbondequalsitstimetomaturityRule2Holdingmaturityconstant,abond’sdurationishigherwhenthecouponrateislowerRule3Holdingthecouponrateconstant,abond’sdurationgenerallyincreaseswithitstimetomaturityRule4Holdingotherfactorsconstant,thedurationofacouponbondishigherwhenthebond’syieldtomaturityislowerPricechangeisproportionaltodurationP/P=-Dx[(1+y)/(1+y)D*=modifieddurationD*=D/(1+y)P/P=-D*xyDuration/PriceRelationshipDuration/pricerelationship-example30yearbond,couponrate8%,yieldtomaturity9%,currentprice$897.26,modifieddurationis11.37years,ifyieldtomaturityincreasesto9.1%,whatisthepricechanges?P=-D*x(y)*P=-11.37*897.26*(9.1%-9%)=-9.36Pricedecreases$9.36ConvexityTheactualrelationshipbetweenbondpriceandyieldsisnotlinear,durationruleisagoodapproximationforsmallchangesinbondyieldWhenyieldchangeisbig,needtoconsiderconvexityConvexitymeasuresthecurvatureoftheprice-yieldcurve,itisthesecondderivative(therateofchangeofheslope)oftheprice-yieldcurvedividedbythebondpriceYieldPriceDurationPricingErrorfromconvexityDurationandConvexityCorrectionforConvexityCorrectionforConvexity:Convexity-example30yearbond,parvalue$1,000,couponrate8%,yieldtomaturity8%,supposethebondpaycouponannually.ifyieldincreasefrom8%to10%,computethepricechangea)usingdurationonly;b)usingdurationandconvexityConvexity-exampleModifiedduration=12.16/(1+8%)=11.26PricechangebasedondurationonlyP=-D*x(y)*P=-11.26*0.02*1000=-225.2Pricedecreases$225.2Convexity-exampleComputeconvexityPricechangebasedondurationandconvexityPricedecreases$182.7,bigdifferencebecausetheyieldchangeislargeRulesforConvexityRule1Holdingmaturityconstant,abond’sconvexityishigherwhenthecouponrateislowerRule2Holdingthecouponrateconstant,abond’sconvexitygenerallyincreaseswithitstimetomaturityRule3Holdingotherfactorsconstant,theconvexityofacouponbondishigherwhenthebond’syieldtomaturityislowerImmunizationofinterestrateriskNetworthimmunizationDurationofassets=DurationofliabilitiesNetPresentValueofasset>=NetPresentValueofLiabilityAssumeinterestcanbereinvestedatcurrentinterestrateImmunizationofinterestraterisk-exampleSupposeafirmhasasinglepaymentof$1,931in10years,r=10%,ifinterestrateincrease,thevalueofthefirm’sassetwilldrop,thefirmmaynothaveenoughfundtopaytheobligation,howcanthefirmimmunizefrominterestraterisk?Example(Cont’)PVoftheliability(1931*(1+10%)-10)=745Durationofsinglepaymentliabilityequaltotimetomaturity,10yearsToimmunize,thefirmsearchsecuritiesthathavedurationof10yearsandnetpresentvaluenolessthan745A20yearbond,parvalue$1,000,couponrate7%,hasadurationof10yearsandNPVof745,thefirmpurchasethisbondtoimmunizeinterestrateriskofitsliabilityEffectofInterestRatechangesonterminalvaluesinyear10Ratesstayat10%Ratesfallto4%Ratesriseto16%Accumulatedvalueofinterestpaymentsreceivedandreinvested70*1.10970*1.04970*1.169………70*170*170*1Total=1115Total=842Total=1492Marketvalueinthe10thyear8161243565Grandtotal193120852057TotalLiability1931193119311931NetSurplus0154126conclusionWheninterestratefall,thevalueofyourliabilityincrease,however,thevalueofyourassetalsoincrease

Wheninterestrateincrease,thevalueofyourassetfall,however,thevalueofyourliabilityalsofallThatishowimmunizationworks

Casestudy—theorangecountyOrangecountymanagedaninvestmentpoolintowhichseveralmunicipalitiesmadeshort-terminvestments.Atotal7.5billionwasinvestedinthispoolsandthismoneywasusedtopurchasesecurities.Usingthesesecuriteisascollateral,thepoolborrowed12.5billionfromWallStreetbrokerages,andthesefundswereusedtopurchaseadditionalsecurities.The20billiontotalwasinvestedprimarilyinlongtermfixedincomesecuritiestoobtainahigheryieldthanshort-termsecurities.Furthermore,asinterestrateshortlydeclinedasdidin1991-1994,angreaterreturnwasobtained.Casestudy—theorangecountyHowever,thingsfellapartin1994wheninterestrateraisedsharply.Assumetheinitialdurationoftheinvestedportfoliowas10years,theaveragecouponinterestontheportfoliowas8.5%offacevalue,thecostofWallStreetmoneywas7%,andshort-terminterestwerefalling0.5%peryearforfouryearsbeforeitraisedto2%inthefifthyear.CaseStudy—theorangecountyNowassume:Portfolioisrebalancedannuallytomaintaindurationof10yearsMoneyborrowfromWallStreetismaintainedat12.5billionOrangecountymakesinterestrateondepositatrateprevailedatthebeginningoftheyearCaseStudy—theorangecountyYear1Year2,V=23.68,r=20.86CaseStudy—theorangecountyYear3,V=25.81,r=19.02%Year4,V=28.14,r=17.51%CaseStudy—theorangecountyInyear5,interestrateincreasedsharplyInyear6,portfoliovaluedecreasedto:RelationshipbetweenyieldtomaturityandmaturityiscalledtermstructureofinterestrateInformationonexpectedfutureshorttermratescanbeimpliedfromyieldcurveTheyieldcurveisagraphthatdisplaystherelationshipbetweenyieldandmaturityThreemajortheoriesareproposedtoexplaintheobservedyieldcurveTermStructure

ofInterestRatesYieldsMaturityUpwardSlopingDownwardSlopingFlat

YieldCurvesYieldCurveTheupwardyieldcurveismorecommonthandownward-slopingtermstructures,thismayreflectmarketanticipatedtheupwardtrendinthegenerallevelofinterestrates,orreflectstheliquiditypremiumsinholdinglong-termbondsThedownwardslopingyieldcurveisconsistentwithamarketexpectationofadeclineininterestrateExpectationsLiquidityPreferenceMarketSegmentationTheoriesofTermStructureExpectationsTheoryThetermstructureisdeterminedbythemarket’sexpectationsregardingfutureinterestrates(thefutureratereferstothefutureexpectedyieldstomaturityon1-yearbond)Foragivenperiod,themarketexpectstogetthesamerateofreturnonallbonds,regardlessoftheirtermtomaturity.(e.g.oneyearandtwoyearbondallhaverateofreturnof10%forthecurrentyear)marketexpectationtheoryNow,oneyearinterestrateis3%,themarketexpectedfutureinterestrateis7%oneyearfromnow,and10%twoyearsfromnow,computethetermstructuralofinterestrate(assumingarithmeticmeanyieldtomaturity)MarketExpectationTheoryYieldtomaturityfortwo-yearbond(3%+7%)/2=5%Yieldtomaturityforthree-yearbond(3%+7%+10%)/3=6.67%3%5%6.67%1-year23fn=one-yearforwardrateforperiodnyn=yieldforasecuritywithamaturityofnForwardRatesfrom

marketexpectationtheoryExampleofForwardRate3yr=7% 2yr=6% computetheforwardrateinyear3(1+7%)3=(1+6%)2(1+f)f=9.02%Note:thisisexpectedone-yearrateofreturnforallbondsinyear3AnarbitrageexampleSupposetherearetwozerocouponbonds:Bond1:maturein1year,currentpriceis95Bond2:maturein2years,currentpriceis96Itisimpossiblebecauseforwardrateisnegative.Bond2isovervalued.Short95sharesofbond2andpurchase96sharesofbond1,initialinvestmentiszero.Profitattheendofyear2is100,get100arbitrageprofit.Extractzero-couponyieldfromcoupon-bearingbondsInpractice,zero-couponyieldcurvecannotalwaysbeobserveddirectly.Sometimeswecanonlyobservepriceofcoupon-bearingbonds.Wehavetoestimatetermstructureofinterestrate.Asimplewayistotreateachcouponpaymentasazero-couponbond.Example:supposeyouobservetwo1yearbonds,bohtpayinterestsemiannually:Bond1:couponrateis8%,currentpriceis986.1Bond2:couponrateis10%,currentpriceis1004.78.Extractzero-couponyieldfromcoupon-bearingbondsDefined1andd2asthediscountfactorfor6monthandoneyearFindingzero-couponyieldcurve—bootstrapapproachAcommonmethodtofindzero-couponyieldcurvefromcouponbearingbondsisbootstrapapproach.Example:

principleTime-tomaturityAnnualcouponBondprice1000.25097.51000.5094.910010901001.5896100212101.61002.751099.8BootstrapmethodSincethefirstthreebondspaynocoupons,theyieldcorrespondingtothematuritiesofthesebondscanbeeasilycalculated.Thecontinuoustimethree-monthyieldis:Andyoucancomputesix-monthrateis10.47%,oneyearrateis10.54%BootstrapApproachTheforthbondlasts1.5years,thepaymentsareasfollows:6months:4dollarOne-year4dollar1.5year104Denotethe1.5-yieldasR,itfollowsthat:Solvingthis,weget1.5-yearyieldis10.68%.Usingthesamemethod,youget2-yieldrateof10.81%BootstrapmethodPointscorrespondingtootherintermediatematuritiesareobtainedbylinearinterpolation.Thesixthbondprovidescashflowsasfollows:3month:5dollar9month:5dollar1.25years:5dollar1.75years:5dollar2.25years:5dollar2.75years:105dollarUsinglinearinterpolation,thediscountratefor9-monthis(10.47%+10.54%)/2=10.505%BootstrapmethodYoucanalsofind1.25yearrateis10.61%;1.75yearrateis10.745%.Supposethatthe2.75yearrateisR,then,the2.25yearrateisThus,the2.75-yearyieldisgivenby:Solvingthis,R=10.87%,nowyoucandrawazero-couponyieldcurveLong-termbondsaremoreriskyInvestorswilldemandapremiumfortheriskassociatedwithlong-termbondsYieldcurvehasanupwardbiasbuiltintothelong-termratesbecauseoftheriskpremiumForwardratescontainaliquiditypremiumandarenotequaltoexpectedfutureshort-termratesLiquidityPremiumTheoryLiquidityPremiumNow,oneyearinterestrateis3%,themarketexpectedfutureinterestrateis7%oneyearfromnow,and10%twoyearsfromnow,inadditionmarketrequire2%liquiditypremiumforholding2-yearbondand3%liquiditypremiumforholdingthree-yearbonds,computethetermstructuralofinterestrate(assumingarithmeticmeanyieldtomatu