Chapter4

BondPriceVolatility

4-1Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallLearningObjectivesAfterreadingthischapter,youwillunderstandtheprice-yieldrelationshipofanoption-freebondthefactorsthataffectthepricevolatilityofabondwhenyieldschangetheprice-volatilitypropertiesofanoption-freebondhowtocalculatethepricevalueofabasispointhowtocalculateandinterprettheMacaulayduration,modifiedduration,anddollardurationofabondwhydurationisameasureofabond’spricesensitivitytoyieldchanges4-2Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallLearningObjectives(continued)Afterreadingthischapter,youwillunderstandthespreaddurationmeasureforfixed-rateandfloating-ratebondshowtocomputethedurationofaportfolioandcontributiontoportfoliodurationlimitationsofusingdurationasameasureofpricevolatilityhowpricechangeestimatedbydurationcanbeadjustedforabond’sconvexityhowtoapproximatethedurationandconvexityofabondthedurationofaninversefloaterhowtomeasureaportfolio’ssensitivitytoanonparallelshiftininterestrates(keyratedurationandyieldcurvereshapingduration)4-3Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallReviewofthePrice-YieldRelationshipforOption-FreeBondsAsillustratedinExhibit4-1(SeeOverhead4-5):Anincreaseintherequiredyielddecreasesthepresentvalueofitsexpectedcashflowsandthereforedecreasesthebond’sprice.Andecreaseintherequiredyieldincreasesthepresentvalueofitsexpectedcashflowsandthereforeincreasesthebond’sprice.AsshowninExhibit4-2(SeeOverhead4-6):Theprice-yieldrelationisnotlinear.Theshapeoftheprice-yieldrelationshipforanyoption-freebondisreferredtoasaconvexrelationship.4-4Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallExhibit4-1Price–YieldRelationshipforSixHypotheticalBondsRequiredYield(%)PriceatRequiredYield(coupon/maturityinyears)9%/59%/256%/56%/250%/50%/256.00112.7953138.5946100.0000100.000074.409422.81077.00108.3166123.455695.841788.272270.891917.90538.00104.0554110.741091.889178.517867.556414.07138.50102.0027105.148289.986474.258765.953712.47958.90100.3966100.996188.498371.110564.701711.33918.99100.0395100.098888.167670.431864.423611.09759.00100.0000100.000088.130970.357064.392811.07109.0199.960499.901388.094370.282464.362011.04459.1099.605399.019987.765469.616464.085510.80939.5098.045995.253986.321466.777362.87239.824210.0096.139190.872084.556563.488161.39138.720411.0092.462483.068581.155957.671258.54316.876712.0088.959976.357277.919752.714455.83955.42884-5Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallExhibit4-2

ShapeofPrice-YieldRelationshipforan

Option-FreeBondPriceMaximumPriceYield4-6Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallPriceVolatilityCharacteristics

ofOption-FreeBondsTherearefourpropertiesconcerningthepricevolatilityofanoption-freebond:Althoughthepricesofalloption-freebondsmoveintheoppositedirectionfromthechangeinyieldrequired,thepercentagepricechangeisnotthesameforallbonds.Forverysmallchangesintheyieldrequired,thepercentagepricechangeforagivenbondisroughlythesame,whethertheyieldrequiredincreasesordecreases.Forlargechangesintherequiredyield,thepercentagepricechangeisnotthesameforanincreaseintherequiredyieldasitisforadecreaseintherequiredyield.Foragivenlargechangeinbasispoints,thepercentagepriceincreaseisgreaterthanthepercentagepricedecrease.

Anexplanationforthesefourpropertiesofbondpricevolatilityliesintheconvexshapeoftheprice-yieldrelationship.4-7Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallPriceVolatilityCharacteristics

ofOption-FreeBonds(continued)CharacteristicsofaBondthatAffectitsPriceVolatilityTherearetwocharacteristicsofanoption-freebondthatdetermineitspricevolatility:couponandtermtomaturity.First,foragiventermtomaturityandinitialyield,thepricevolatilityofabondisgreater,thelowerthecouponrate.Thischaracteristiccanbeseenbycomparingthe9%,6%,andzero-couponbondswiththesamematurity.Second,foragivencouponrateandinitialyield,thelongerthetermtomaturity,thegreaterthepricevolatility.ThiscanbeseeninExhibit4-3(SeeOverhead4-9)bycomparingthefive-yearbondswiththe25-yearbondswiththesamecoupon.4-8Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallEXHIBIT4-3InstantaneousPercentagePriceChangeforSixHypotheticalBondsSixhypotheticalbonds,pricedinitiallytoyield9%:9%coupon,5yearstomaturity,price=100.00009%coupon,25yearstomaturity,price=100.0006%coupon,5yearstomaturity,price=88.13096%coupon,25yearstomaturity,price=70.35700%coupon,5yearstomaturity,price=64.39280%coupon,25yearstomaturity,price=11.0710Yield(%)Changeto:ChangeinBasisPointsPercentagePriceChange(coupon/maturityinyears)9%/59%/256%/56%/250%/50%/256.00-30012.8038.5913.4742.1315.56106.047.00-2008.3223.468.7525.4610.0961.738.00-1004.0610.744.2611.604.9127.108.50-502.0052.4212.728.90-100.401.000.421.070.482.428.99-10.040.100.040.110.050.249.011-0.04-0.10-0.04-0.11-0.05-0.249.1010-0.39-0.98-0.41-1.05-0.48-2.369.5050-1.95-4.75-2.05-5.09-2.36-11.2610.00100-3.86-9.13-4.06-9.76-4.66-21.2311.00200-7.54-16.93-7.91-18.03-9.08-37.8912.00300-11.04-23.64-11.59-25.08-13.28-50.964-9Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallPriceVolatilityCharacteristics

ofOption-FreeBonds

(continued)EffectsofYieldtoMaturityWecannotignorethefactthatcreditconsiderationscausedifferentbondstotradeatdifferentyields,eveniftheyhavethesamecouponandmaturity.Holdingotherfactorsconstant,thehighertheyieldtomaturityatwhichabondtrades,thelowerthepricevolatility.Toseethis,comparethe9%25-yearbondtradingatvariousyieldlevelsinExhibit4-4(SeeOverhead4-11).The1stcolumnofExhibit4-4showstheyieldlevelthebondistradingat,andthe2ndcolumngivestheinitialprice.The3rdcolumnofExhibit4-4indicatesthebond’spriceifyieldschangeby100basispoints.The4thand5thcolumnsofExhibit4-4showthedollarpricedeclineandthepercentagepricedecline.The4thand5thcolumnsofExhibit4-4alsoshow:highertheinitialyield,thelowerthepricevolatility.Animplicationofthisisthatforagivenchangeinyields,pricevolatilityisgreater(lower)whenyieldlevelsinthemarketarelow(high).4-10Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallEXHIBIT4-4PriceChangefora100-Basis-PointChangeinYieldfora9%25-YearBondTradingatDifferentYieldLevelsYieldLevel(%)InitialPriceNewPriceaPriceDeclinePercentDecline7$123.46$110.74$12.7210.308110.74100.0010.749.709100.0090.879.139.131090.8783.077.808.581183.0776.366.718.081276.3670.555.817.611370.5565.505.057.161465.5061.084.426.758110.74100.0010.749.70

a

Asaresultofa100-basis-pointincreaseinyield.4-11Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallMeasuresofBondPriceVolatilityMoneymanagers,arbitrageurs,andtradersneedtohaveawaytomeasureabond’spricevolatilitytoimplementhedgingandtradingstrategies.Threemeasuresthatarecommonlyemployed:pricevalueofabasispointyieldvalueofapricechangeduration4-12Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallCopyright?2010PearsonEducation,Inc.PublishingasPrenticeHall4-13MeasuresofBondPriceVolatility(continued)PriceValueofaBasisPointThepricevalueofabasispoint,alsoreferredtoasthedollarvalueofan01,isthechangeinthepriceofthebondiftherequiredyieldchangesby1basispoint.Notethatthismeasureofpricevolatilityindicatesdollarpricevolatility

asopposedtopercentagepricevolatility(pricechangeasapercentoftheinitialprice).Typically,thepricevalueofabasispointisexpressedastheabsolutevalueofthechangeinprice.Pricevolatilityisthesameforanincreaseoradecreaseof1basispointinrequiredyield.

Becausethismeasureofpricevolatilityisintermsofdollarpricechange,dividingthepricevalueofabasispointbytheinitialpricegivesthepercentagepricechangefora1-basis-pointchangeinyield.Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHall4-14MeasuresofBondPriceVolatility(continued)YieldValueofaPriceChangeAnothermeasureofthepricevolatilityofabondusedbyinvestorsisthechangeintheyieldforaspecifiedpricechange.Thisisestimatedbyfirstcalculatingthebond’syieldtomaturityifthebond’spriceisdecreasedby,say,Xdollars.ThenthedifferencebetweentheinitialyieldandthenewyieldistheyieldvalueofanXdollarpricechange.Pricevolatilityisthesameforanincreaseoradecreaseof1basispointinrequiredyield.Thesmallerthisvalue,thegreaterthedollarpricevolatility,becauseitwouldtakeasmallerchangeinyieldtoproduceapricechangeofXdollars.Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHall4-15MeasuresofBondPriceVolatility(continued)DurationTheMacaulaydurationisonemeasureoftheapproximatechangeinpriceforasmallchangeinyield:whereP

=priceofthebondC

=semiannualcouponinterest(indollars)y

=one-halftheyieldtomaturityorrequiredyieldn

=numberofsemiannualperiods(numberofyearstimes2)M

=maturityvalue(indollars)Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHall4-16MeasuresofBondPriceVolatility(continued)DurationInvestorsrefertotheratioofMacaulaydurationto1+yasthemodifiedduration.Theequationis:wherey

=one-halftheyieldtomaturityorrequiredyield.Themodifieddurationisrelatedtotheapproximatepercentagechangeinpriceforagivenchangeinyieldasgivenby:wheredP=changeinprice,dy=changeinyield,P

=priceofthebond.Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHall4-17MeasuresofBondPriceVolatility(continued)DurationBecauseforalloption-freebondsmodifieddurationispositive,themodifieddurationequation,(dP/dy

)(1/P),statesthatthereisaninverserelationshipbetweenmodifieddurationandtheapproximatepercentagechangeinpriceforagivenyieldchange.Thisistobeexpectedfromthefundamentalprinciplethatbondpricesmoveintheoppositedirectionofthechangeininterestrates.Exhibit4-5(seeOverhead4-18)andExhibit4-6(seeOverhead4-19)showthecomputationoftheMacaulaydurationandmodifieddurationoftwofive-yearcouponbonds.Thedurationscomputedintheseexhibitsareintermsofdurationperperiod.EXHIBIT4-5CalculationofMacaulayDurationandModifiedDurationfor5-Year9%BondSellingtoYield9%Couponrate:9.00%Term(years):5Initialyield:9.00%Period,tCashFlowPVof$1at4.5%PVofCFt×PVCF1$4.500.9569374.3062204.3062224.500.9157294.1207858.2415634.500.8762963.94333511.8300044.500.8385613.77352615.0941054.500.8024513.61103018.0551464.500.7678953.45553120.7331874.500.7348283.30672823.1470984.500.7031853.16433325.3146694.500.6729043.02807027.2526210$104.500.64392767.290443672.90442100.00000826.878994-18Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallEXHIBIT4-6CalculationofMacaulayDurationandModifiedDurationfor5-Year6%BondSellingtoYield9%Couponrate:6.00%Term(years):5Initialyield:9.00%Period,tCashFlowPVof$1at4.5%PVofCFt×PVCF1$3.000.9569372.8708132.8708123.000.9157292.7471905.4943733.000.8762962.6288907.8866643.000.8385612.51568410.0627353.000.8024512.40735312.0367663.000.7678952.30368713.8221273.000.7348282.20448515.4313983.000.7031852.10955516.8764493.000.6729042.01871318.1684110103.000.64392766.324551663.24551Total88.130923765.895204-19Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallCopyright?2010PearsonEducation,Inc.PublishingasPrenticeHall4-204-20MeasuresofBondPriceVolatility(continued)DurationIngeneral,ifthecashflowsoccurmtimesperyear,thedurationsareadjustedbydividingbym,

thatis,WecanderiveanalternativeformulathatdoesnothavetheextensivecalculationoftheMacaulaydurationandthemodifiedduration.Thisisdonebyrewritingthepriceofabondintermsofitspresentvalueofanannuityandthepresentvalueoftheparvalue,takingthefirstderivative,anddividingbyP:wherethepriceisexpressedasapercentageofparvalue.Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHall4-21MeasuresofBondPriceVolatility(continued)PropertiesofDurationThemodifieddurationandMacaulaydurationofacouponbondarelessthanthematurity.TheMacaulaydurationofazero-couponbondequalsitsmaturity;butazero-couponbond’smodifieddurationislessthanitsmaturity.LowercouponratesgenerallyhavegreaterMacaulayandmodifiedbonddurations.Thereisaconsistencybetweenthepropertiesofbondpricevolatilityandthepropertiesofmodifiedduration.Forexample,apropertyofmodifieddurationisthat,ceterisparibus,abondwithalongermaturitywillhaveagreatermodifiedduration.Generally,alowercouponrateimpliesagreatermodifieddurationandagreaterpricevolatility.Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHall4-22MeasuresofBondPriceVolatility(continued)ApproximatingthePercentagePriceChangeThebelowequationcanbeusedtoapproximatethepercentagepricechangeforagivenchangeinrequiredyield:wheredP=changeinprice,P

=priceofthebondand

dy=changeinyield.Supposethattheyieldonanybondchangesby100basispoints.Then,substituting100basispoints(0.01)fordyintotheaboveequation,weget:Thus,modifieddurationcanbeinterpretedastheapproximatepercentagechangeinpricefora100-basis-pointchangeinyield.Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHall4-23MeasuresofBondPriceVolatility(continued)ApproximatingtheDollarPriceChangeModifieddurationisaproxyforthepercentagechangeinprice.Investorsalsoliketoknowthedollarpricevolatilityofabond.Forsmallchangesintherequiredyield,thebelowequationdoesagoodjobinestimatingthechangeinpriceyield:dP

=(dollarduration)(dy)wheredP=changeinpriceanddy=changeinyield.Whentherearelargemovementsintherequiredyield,dollardurationormodifieddurationisnotadequatetoapproximatethepricereaction.Durationwilloverestimatethepricechangewhentherequiredyieldrises,therebyunderestimatingthenewprice.Whentherequiredyieldfalls,durationwillunderestimatethepricechangeandtherebyunderestimatethenewprice.Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHall4-24MeasuresofBondPriceVolatility(continued)SpreadDurationMarketparticipantscomputeameasurecalledspreadduration.Thismeasureisusedintwoways:forfixedbondsandfloating-ratebonds.

Aspreaddurationforafixed-ratesecurityisinterpretedastheapproximatechangeinthepriceofafixed-ratebondfora100-basis-pointchangeinthespreadbond.PortfolioDurationThusfarwehavelookedatthedurationofanindividualbond.Thedurationofaportfolioissimplytheweightedaveragedurationofthebondsintheportfolios.Portfoliomanagerslookattheirinterestrateexposuretoaparticularissueintermsofitscontributiontoportfolioduration.Thismeasureisfoundbymultiplyingtheweightoftheissueintheportfoliobythedurationoftheindividualissuegivenas:contributiontoportfolioduration=weightofissueinportfolio×

durationofissue.Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHall4-25MeasuresofBondPriceVolatility(continued)PortfolioDurationPortfoliomanagerslookatportfoliodurationforsectorsofthebondmarket.Theprocedureisthesameforcomputingthecontributiontoportfoliodurationofasectorasitisforcomputingthecontributiontoportfoliodurationofanindividualissue.Aspreaddurationforaportfoliooffixed-ratebondscanalsobecomputed.Inthiscase,theportfoliodurationisdividedintotwodurations.ThefirstisthedurationoftheportfoliowithrespecttochangesinthelevelofTreasuryrates.Thesecondisthespreadduration.TheaboveisillustratedinExhibit4-7(seeOverhead4-26)foraportfoliousingsixsectors.EXHIBIT4-7CalculationofDurationandContributiontoPortfolioDurationforaAssetAllocationtoSectorsoftheLehmanBrothersU.S.AggregateIndex:October26,2007SectorPortfolioWeightSectorDurationContributiontoPortfolioDurationTreasury0.0004.950.00Agency0.1213.440.42Mortgages0.4493.581.61CommercialMortgage-BackedSecurities0.1395.040.70Asset-BackedSecurities0.0173.160.05Credit0.2746.351.741.0004.52Total2.00026.529.044-26Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallCopyright?2010PearsonEducation,Inc.PublishingasPrenticeHall4-27MeasuresofBondPriceVolatility(continued)PortfolioDurationExhibit4-8(seeOverhead4-28)showsthesizeofeachsectorintheLehmanBrothersU.S.AggregateIndex.Youcanseetheimportanceofeachsector.ThedurationforeachsectorisshowninthethirdcolumnandusesthesamevaluesasinExhibit4-7.Thedurationfortheportfolioisshowninthelastrowofthethirdcolumn.ThecalculationofthespreaddurationfortherecommendedportfolioallocationandtheLehmanBrothersU.S.AggregateIndexareshowninExhibit4-9(seeOverhead4-29)andExhibit4-10(seeOverhead4-30),respectively.Whiletheportfolioandtheindexhavethesameduration,thespreaddurationfortherecommendedportfoliois4.60vs.3.49fortheindex.Thelargerspreaddurationfortherecommendedportfolioisexpectedgiventhegreaterallocationtonon-Treasurysectors.EXHIBIT4-8CalculationofDurationandContributiontotheLehmanBrothersAggregateIndexDuration:October26,2007SectorWeightinIndexSectorDurationContributiontoIndexDurationTreasury0.2304.951.14Agency0.1053.440.36Mortgages0.3813.581.36CommercialMortgage-BackedSecurities0.0565.040.28Asset-BackedSecurities0.0103.160.03Credit0.2196.351.391.0004.56Total2.00126.529.124-28Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallEXHIBIT4-9CalculationofSpreadDurationandContributiontoPortfolioSpreadDurationforanAssetAllocationtoSectorsoftheLehmanBrothersU.S.AggregateIndex:October26,2007SectorPortfolioWeightSectorSpreadDurationContributiontoPortfolioSpreadDurationTreasury0.0000.000.00Agency0.1213.530.43Mortgages0.4493.621.63CommercialMortgage-BackedSecurities0.1395.040.70Asset-BackedSecurities0.0173.160.05Credit0.2746.351.791.0004.60Total2.00021.889.204-29Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallEXHIBIT4-10CalculationofSpreadDurationandContributiontotheLehmanBrothersAggregateIndexSpreadDuration:October26,2007SectorWeightinIndexSectorSpreadDurationContributiontoIndexSpreadDurationTreasury0.2300.000.00Agency0.1053.530.37Mortgages0.3813.621.38CommercialMortgage-BackedSecurities0.0565.040.28Asset-BackedSecurities0.0103.160.03Credit0.2196.531.431.0003.49Total2.00121.886.984-30Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallExhibit4-11

MeasuresofBondPriceVolatilityandTheirRelationshipstoOneAnotherNotation:D=MacaulaydurationD*=modifieddurationPVBP=pricevalueofabasispointy=yieldtomaturityindecimalformY=yieldtomaturityinpercentageterms(Y=100×

y)P=priceofbondm=numberofcouponsperyear4-31Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallExhibit4-11

MeasuresofBondPriceVolatilityandTheirRelationshipstoOneAnother(continued)Relationships:4-32Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallConvexityBecauseallthedurationmeasuresareonlyapproximationsforsmallchangesinyield,theydonotcapturetheeffectoftheconvexityofabondonitspriceperformancewhenyieldschangebymorethanasmallamount.Thedurationmeasurecanbesupplementedwithanadditionalmeasuretocapturethecurvatureorconvexityofabond.InExhibit4-12(seeOverhead4-34),atangentlineisdrawntotheprice–yieldrelationshipatyieldy*.Thetangentshowstherateofchangeofpricewithrespecttoachangeininterestratesatthatpoint(yieldlevel).4-33Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallExhibit4-12

LineTangenttothePrice-YieldRelationshipPriceYieldy*p*ActualPriceTangentLineaty*(estimatedprice)4-34Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallConvexity(continued)Ifwedrawaverticallinefromanyyield(onthehorizontalaxis),asinExhibit4-13(seeOverhead4-36),thedistancebetweenthehorizontalaxisandthetangentlinerepresentsthepriceapproximatedbyusingdurationstartingwiththeinitialyieldy*.Theapproximationwillalwaysunderstatetheactualprice.Thisagreeswithwhatwedemonstratedearlierabouttherelationshipbetweenduration(andthetangentline)andtheapproximatepricechange.Whenyieldsdecrease,theestimatedpricechangewillbelessthantheactualpricechange,therebyunderestimatingtheactualprice.Ontheotherhand,whenyieldsincrease,theestimatedpricechangewillbegreaterthantheactualpricechange,resultinginanunderestimateoftheactualprice.4-35Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallExhibit4-13.PriceApproximationUsingDurationPriceYieldy3p*ActualPriceTangentLineaty*(estimatedprice)y*y1y2y4ErrorinEstimatingPriceBasedonlyonDurationErrorinEstimatingPriceBasedonlyonDuration4-36Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallCopyright?2010PearsonEducation,Inc.PublishingasPrenticeHall4-37Convexity

(continued)MeasuringConvexityDuration(modifiedordollar)attemptstoestimateaconvexrelationshipwithastraightline(thetangentline).Thedollarconvexitymeasure

ofthebond:Theapproximatechangeinpriceduetoconvexityis:Thepercentagechangeinthepriceofthebondduetoconvexityortheconvexitymeasureis:Thepercentagepricechangeduetoconvexityis:Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallConvexity

(continued)MeasuringConvexityExhibit4-14(seeOverhead4-39)andExhibit4-15(seeOverhead4-40)demonstratehowtocalculatethesecondderivative,annualizeddollarconvexitymeasure,andannualizedconvexitymeasureforthetwofive-yearcouponbonds.Theconvexitymeasureisintermsofperiodssquared.Ingeneral,ifthecashflowsoccurmtimesperyear,convexityisadjustedtoanannualfigureasfollows:4-38EXHIBIT4-14CalculationofConvexityMeasureandDollarConvexityMeasureforFive-Year9%BondSellingtoYield9%Couponrate:9.00%Term(years):5Initialyield:9.00%Price:100Period,tCashFlow1/(1.045)t+2t(t+1)CFt(t+1)CF(1.045)t+214.500.87629697.88624.500.8385612722.64134.500.8024515443.33244.500.7678959069.11054.500.73482813599.20164.500.703185189132.90174.500.672904252169.57184.500.643927324208.63294.500.616198405249.56010104.500.58966311,4956,778.18612,9807,781.0204-39Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallEXHIBIT4-15CalculationofConvexityMeasureandDollarConvexityMeasureforFive-Year6%BondSellingtoYield9%Couponrate:6.00%Term(years):5Initialyield:9.00%Price:88.1309Period,tCashFlow1/(1.045)t+2t(t+1)CFt(t+1)CF(1.045)t+213.000.87629665.25723.000.8385611815.09433.000.8024513628.88843.000.7678956046.07353.000.7348289066.13463.000.70318512688.60173.000.672904168113.04783.000.643927216139.08893.000.616198270166.37310103.000.58966311,3306,680.89112,3207,349.4464-40Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallConvexity

(continued)ApproximatingPercentagePriceChangeUsingDurationandConvexityMeasuresUsingdurationandconvexitymeasurestogethergivesabetterapproximationoftheactualpricechangeforalargemovementintherequiredyield.SomeNotesonConvexity

Threepointstoknowforabond’sconvexityandconvexitymeasure.Convexityreferstothegeneralshapeoftheprice-yieldrelationship,whiletheconvexitymeasurerelatestothequantificationofhowthepriceofthebondwillchangewheninterestrateschange.Theapproximationpercentagechangeinpriceduetoconvexityistheproductofthreenumbers:?,convexitymeasure,andsquareofthechangeinyield.Inpracticedifferentvendorscomputetheconvexitymeasuredifferentlybyscalingthemeasureindissimilarways.4-41Copyright?2010PearsonEducation,Inc.PublishingasPrenticeHallConvexity

(continued)ValueofConvexity

Uptothispoint,wehavefocusedonhowtakingconvexityintoaccountcanimprovetheapproximationofabond’spricechangeforagivenyieldchange.Theconvexityofabond,however,hasanotherimportantinvestmentimplication,whichisillustratedinExhibit4-16(seeOverhead4-43).Theexhibitshowstwobonds,AandB.Thetwobondshavethesamedurationandareofferingthesameyield;theyhavedifferentconv