固定收益证-券Measuring-Yield课件

Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHall3-1

Chapter3

MeasuringYield

Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHall3-2LearningObjectivesAfterreadingthischapter,youwillunderstandhowtocalculatetheyieldonanyinvestmenthowtocalculatethecurrentyield,yieldtomaturity,yieldtocall,yieldtoput,andcashflowyieldhowtocalculatetheyieldforaportfoliohowtocalculatethediscountmarginforafloating-ratesecuritythethreepotentialsourcesofabond’sreturnCopyright©2010PearsonEducation,Inc.PublishingasPrenticeHall3-3LearningObjectives

(continued)Afterreadingthischapter,youwillunderstandhowtocalculatetheyieldonanyinvestmentwhatreinvestmentriskisthelimitationsofconventionalyieldmeasureshowtocalculatethetotalreturnforabondwhythetotalreturnissuperiortoconventionalyieldmeasureshowtousehorizonanalysistoassessthepotentialreturnCopyright©2010PearsonEducation,Inc.PublishingasPrenticeHall3-4ComputingtheYieldorInternalRateofReturnonanyInvestmentTheyieldonanyinvestmentistheinterestratethatwillmakethepresentvalueofthecashflowsfromtheinvestmentequaltotheprice(orcost)oftheinvestment.Mathematically,theyieldonanyinvestment,y,istheinterestratethatsatisfiestheequation.

whereP

=priceoftheinvestment,CF

=cashflowinyeart=1,2,3,…N,y=yieldcalculatedfromthisrelationship(andalsocalledtheinternalrateofreturn),

N

=numberofyears.Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHall3-5ComputingtheYieldorInternalRateofReturnonanyInvestment(continued)Absentafinancialcalculatororcomputersoftware,solvingfortheyield(y)requiresatrial-and-error(iterative)procedure.Theobjectiveistofindtheyieldthatwillmakethepresentvalueofthecashflowsequaltotheprice.Keepinmindthattheyieldcomputedistheyieldfortheperiod.Thatis,ifthecashflowsaresemiannual,theyieldisasemiannualyield.Ifthecashflowsaremonthly,theyieldisamonthlyyield.Tocomputethesimpleannualinterestrate,theyieldfortheperiodismultipliedbythenumberofperiodsintheyear.Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHall3-6ComputingtheYieldorInternalRateofReturnonanyInvestment(continued)SpecialCase:InvestmentwithOnlyOneFutureCashFlowWhenthecasewherethereisonlyonefuturecashflow,itisnotnecessarytogothroughthetime-consumingtrial-and-errorproceduretodeterminetheyield.Wecanusethebelowequation:Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHall3-7ComputingtheYieldorInternalRateofReturnonanyInvestment(continued)AnnualizingYieldsToobtainaneffectiveannualyieldassociatedwithaperiodicinterestrate,thefollowingformulaisused:effectiveannualyield=(1+periodicinterestrate)m

–1

wherem

isthefrequencyofpaymentsperyear.Toillustrate,ifinterestispaidquarterlyandtheperiodicinterestrateis0.08/4=0.02,thenwehave:effectiveannualyield=(1.02)4–1=1.0824–1=0.0824or8.24%Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHall3-8ComputingtheYieldorInternalRateofReturnonanyInvestment(Continued)AnnualizingYieldsWecanalsodeterminetheperiodicinterestratethatwillproduceagivenannualinterestratebysolvingtheeffectiveannualyieldequationfortheperiodicinterestrate.Solving,wefindthatperiodicinterestrate=(1+effectiveannualyield)1/m

–1Toillustrate,iftheperiodicquarterlyinterestratethatwouldproduceaneffectiveannualyieldof12%,thenwehave:periodicinterestrate=(1.12)1/4–1=1.0287–1=0.0287or2.87%Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHall3-9ComputingtheYieldorInternalRateofReturnonanyInvestment(continued)AnnualizingYieldsWecanalsodeterminetheperiodicinterestratethatwillproduceagivenannualinterestratebysolvingtheeffectiveannualyieldequationfortheperiodicinterestrate.Solving,wefindthatperiodicinterestrate=(1+effectiveannualyield)1/m

–1Toillustrate,iftheperiodicquarterlyinterestratethatwouldproduceaneffectiveannualyieldof12%,thenwehave:periodicinterestrate=(1.12)1/4–1=1.0287–1=0.0287or2.87%Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHall3-10ConventionalYieldMeasuresBondyieldmeasurescommonlyquotedbydealersandusedbyportfoliomanagersare:CurrentYieldYieldToMaturityYieldToCallYieldToPutYieldToWorstCashFlowYieldYield(InternalRateofReturn)foraPortfolioYieldSpreadMeasuresforFloating-RateSecuritiesCopyright©2010PearsonEducation,Inc.PublishingasPrenticeHall3-11ConventionalYieldMeasures(continued)1)CurrentYieldCurrentyieldrelatestheannualcouponinteresttothemarketprice.Theformulaforthecurrentyieldis:currentyield=annualdollarcouponinterest/priceThecurrentyieldcalculationtakesintoaccountonlythecouponinterestandnoothersourceofreturnthatwillaffectaninvestor’syield.Thetimevalueofmoneyisalsoignored.Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHall3-12ConventionalYieldMeasures(continued)2)YieldToMaturityTheyieldtomaturity

istheinterestratethatwillmakethepresentvalueofthecashflowsequaltotheprice(orinitialinvestment).Forasemiannualpaybond,theyieldtomaturityisfoundbyfirstcomputingtheperiodicinterestrate,y,whichsatisfiestherelationship:whereP

=priceofthebond,C

=semiannualcouponinterest(indollars),M

=maturityvalue(indollars),andn

=numberofperiods(numberofyearsmultipliedby2).Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHall3-13ConventionalYieldMeasures(Continued)2)YieldToMaturity(continued)Forasemiannualpaybond,doublingtheperiodicinterestrateordiscountrate(y)givestheyieldtomaturity.Theyieldtomaturitycomputedonthebasisofthismarketconventioniscalledthebond-equivalentyieldTheyield-to-maturitycalculationtakesintoaccount(i)thecurrentcouponincome,(ii)anycapitalgainorlossrealizedbyholdingthebondtomaturity,and(iii)thetimingofthecashflows.Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHall3-14ConventionalYieldMeasures(Continued)3)YieldToCallThecallpriceisthepriceatwhichthebondmaybecalled.Thereisacallschedulethatspecifiesacallpriceforeachcalldate.Theyieldtocallassumesthattheissuerwillcallthebondatsomeassumedcalldateandthecallpriceisspecifiedinthecallschedule.Theyieldtocallcanbeexpressedasfollows:whereM

*=callprice(indollars)andn*=numberofperiodsuntiltheassumedcalldate(numberofyearstimes2)Forasemiannualpaybond,doublingtheperiodicinterestrate(y)givestheyieldtocallonabond-equivalentbasis.Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHall3-15ConventionalYieldMeasures(Continued)4)YieldToPutIfanissueisputable,itmeansthatthebondholdercanforcetheissuertobuytheissueataspecifiedprice.Theputschedulespecifieswhentheissuecanbeputandtheputprice.

Whenanissueisputable,ayieldtoputiscalculated.Theyieldtoputistheinterestratethatmakesthepresentvalueofthecashflowstotheassumedputdateplustheputpriceonthatdateassetforthintheputscheduleequaltothebond’sprice.Theformulafortheyieldtoputisthesameasfortheyieldtocall,butM

*isnowdefinedastheputpriceandn*isthenumberofperiodsuntiltheassumedputdate.Theprocedureisthesameascalculatingtheyieldtomaturityandtheyieldtocall.Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHall3-16ConventionalYieldMeasures(Continued)5)YieldToWorstApracticeintheindustryisforaninvestortocalculatetheyieldtomaturity,theyieldtoeverypossiblecalldate,andtheyieldtoeverypossibleputdate.Theminimumofalloftheseyieldsiscalledtheyieldtoworst.6)CashFlowYield

Amortizingsecuritiesinvolvecashflowsthatincludeinterestplusprincipalrepaymentandthecashfloweachperiodconsistsofthreecomponents:(i)couponinterest,(ii)scheduledprincipalrepayment,and(iii)prepayments.Foramortizingsecurities,marketparticipantscalculateacashflowyield,whichistheinterestratethatwillmakethepresentvalueoftheprojectedcashflowsequaltothemarketprice.Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHall3-17ConventionalYieldMeasures(Continued)7)Yield(InternalRateofReturn)foraPortfolioTheyieldforaportfolioofbondsisnotsimplytheaverageorweightedaverageoftheyieldtomaturityoftheindividualbondissuesintheportfolio.Itiscomputedbydeterminingthecashflowsfortheportfolioanddeterminingtheinterestratethatwillmakethepresentvalueofthecashflowsequaltothemarketvalueoftheportfolio.Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHall3-18ConventionalYieldMeasures(Continued)8)YieldSpreadMeasuresforFloating-RateSecuritiesThecouponrateforafloating-ratesecuritychangesperiodicallybasedonthecouponresetformula.Thisformulaconsistsofthereferencerateandthequotedmargin.Sincethefuturevalueforthereferencerateisunknown,itisnotpossibletodeterminethecashflows.Thismeansthatayieldtomaturitycannotbecomputed.Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHall3-19ConventionalYieldMeasures(Continued)8)YieldSpreadMeasuresforFloating-RateSecuritiesInstead,thereareseveralconventionalmeasuresusedasmarginorspreadmeasurescitedbymarketparticipantsforfloaters.Theseincludespreadforlife(orsimplemargin),adjustedsimplemargin,adjustedtotalmargin,anddiscountmargin.Themostpopularofthesemeasuresisthediscountmargin,whichestimatestheaveragemarginoverthereferenceratethattheinvestorcanexpecttoearnoverthelifeofthesecurity.Exhibit3-1showsthecalculationofthediscountmarginforasix-yearfloating-ratesecurity.(SeeOverhead3-20.)Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHall3-20Exhibit3-1CalculationoftheDiscountMarginforaFloating-RateSecurity

Floating-ratesecurity

Maturity:sixyears

Couponrate:referencerate+80basispoints

ReseteverysixmonthsPresentValueofCashFlowatAssumedAnnualMargin(basispoints)PeriodReferenceRateCashFlowa

80848896100110%5.45.12335.12245.12145.11955.11852105.44.86094.85904.85724.85354.85163105.44.61184.60924.60664.60134.59874105.44.37554.37224.36894.36234.35905105.44.15144.14744.14354.13564.13176105.43.93873.93423.92973.92083.91637105.43.73693.73193.72703.71713.71228105.43.54543.54013.53473.52403.51869105.43.36383.35803.35233.34093.335210105.43.19143.18543.17943.16733.161311105.43.02793.02163.01533.00282.99651210105.456.072955.945455.818255.564755.4385PresentValue=100.000099.826999.654199.309899.1381aForperiods1–11:cashflow=100(referencerate+assumedmargin)(0.5);forperiod12:cashflow=100(referencerate+assumedmargin)(0.5)+100.Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHall3-21PotentialSourcesofaBond’sDollarReturnAninvestorwhopurchasesabondcanexpecttoreceiveadollarreturnfromoneormoreofthesesources:theperiodiccouponinterestpaymentsmadebytheissueranycapitalgain(orcapitalloss—negativedollarreturn)whenthebondmatures,iscalled,orissoldinterestincomegeneratedfromreinvestmentoftheperiodiccashflowsThecurrentyieldconsidersonlythecouponinterestpayments.Theyieldtomaturity,yieldtocall,andcashflowyieldalltakeintoaccountthethreecomponents.Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHall3-22PotentialSourcesofaBond’sDollarReturn

(continued)DeterminingtheInterest-On-InterestDollarReturnTheinterest-on-interestcomponentcanrepresentasubstantialportionofabond’spotentialreturn.Thecouponinterestplusinterestoninterestcanbefoundbyusingthefollowingequation:whereCisthecouponinterestr

isthesemiannualreinvestmentratenisthenumberofperiodsCopyright©2010PearsonEducation,Inc.PublishingasPrenticeHall3-23PotentialSourcesofaBond’sDollarReturn

(continued)DeterminingtheInterest-On-InterestDollarReturnThetotaldollaramountofcouponinterestisfoundbymultiplyingthesemiannualcouponinterestbythenumberofperiods:totalcouponinterest=nCwhereCisthecouponinterest,r

isthesemiannualreinvestmentrate,andnisthenumberofperiods.Theinterest-on-interestcomponentisthenthedifferencebetweenthecouponinterestplusinterestoninterestandthetotaldollarcouponinterest,asexpressedbytheformulaCopyright©2010PearsonEducation,Inc.PublishingasPrenticeHall3-24PotentialSourcesofaBond’sDollarReturn

(continued)DeterminingtheInterest-On-InterestDollarReturnExample.Assumethatthecouponinterest(C)is$50,thesemiannualreinvestmentrate(r)is4.5%,andthenumberofperiods(n)is40.Whatistheinterest-on-interest?Usingourequationforinterestoninterestandinsertinginourgivenvaluesweget:Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHall3-25PotentialSourcesofaBond’sDollarReturn

(continued)YieldToMaturityandReinvestmentRiskTheinvestorrealizestheyieldtomaturityonlyifthebondisheldtomaturityandthecouponpaymentscanbereinvestedatthecomputedyieldtomaturity.Reinvestmentriskistherisk

thatfuturereinvestmentrateswillbelessthantheyieldtomaturityatthetimethebondispurchased.Therearetwocharacteristicsofabondthatdeterminetheimportanceoftheinterest-on-interestcomponentandthereforethedegreeofreinvestmentrisk:maturityandcoupon.Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHall3-26PotentialSourcesofaBond’sDollarReturn

(continued)YieldToMaturityandReinvestmentRiskForagivenyieldtomaturityandagivencouponrate,thelongerthematurity,themoredependentthebond’stotaldollarreturnisontheinterest-on-interestcomponentinordertorealizetheyieldtomaturityatthetimeofpurchase.Foragivenmaturityandagivenyieldtomaturity,highercouponrateswillmakethebond’stotaldollarreturnmoredependentonthereinvestmentofthecouponpaymentsinordertoproducetheyieldtomaturityanticipatedatthetimeofpurchase.Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHall3-27PotentialSourcesofaBond’sDollarReturn

(continued)CashFlowYieldandReinvestmentRiskForamortizingsecurities,reinvestmentriskisevengreaterthanfornonamortizingsecurities.Thereasonisthattheinvestormustnowreinvesttheperiodicprincipalrepaymentsinadditiontotheperiodiccouponinterestpayments.Typically,fornonamortizingsecuritiestheborrowercanacceleratetheperiodicprincipalrepayment,inparticular,aborrowerwilltendtoprepaywheninterestratesdecline.Ifaborrowerprepayswheninterestratesdecline,theinvestorfacesgreaterreinvestmentriskbecauseheorshemustreinvesttheprepaidprincipalatalowerinterestrate.Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHall3-28TotalReturnTheyieldtomaturityisapromisedyieldbecauseatthetimeofpurchaseaninvestorispromisedayield,asmeasuredbytheyieldtomaturity,ifbothofthefollowingconditionsaresatisfied:thebondisheldtomaturityallcouponinterestpaymentsarereinvestedattheyieldtomaturityThetotalreturnisameasureofyieldthatincorporatesanexplicitassumptionaboutthereinvestmentrate.Theyield-to-callmeasureissubjecttothesameproblemsastheyieldtomaturitybecauseitassumesthatthe:bondwillbehelduntilthefirstcalldatecouponinterestpaymentswillbereinvestedattheyieldtocallCopyright©2010PearsonEducation,Inc.PublishingasPrenticeHall3-29TotalReturn

(continued)ComputingtheTotalReturnforaBondTheideaunderlyingtotalreturnissimple.Theobjectiveisfirsttocomputethetotalfuturedollarsthatwillresultfrominvestinginabondassumingaparticularreinvestmentrate.Thetotalreturnisthencomputedastheinterestratethatwillmaketheinitialinvestmentinthebondgrowtothecomputedtotalfuturedollars.Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHall3-30ApplicationsoftheTotalReturn

HorizonAnalysisHorizonanalysisreferstousingtotalreturntoassessperformanceoversomeinvestmenthorizon.Horizonreturnreferstowhenatotalreturniscalculatedoveraninvestmenthorizon.

Anoften-citedobjectiontothetotalreturnmeasureisthatitrequirestheportfoliomanagertoformulateassumptionsaboutreinvestmentratesandfutureyieldsaswellastothinkintermsofaninvestmenthorizon.Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHall3-31CalculatingYieldChangesTheabsoluteyieldchange(orabsoluteratechange)ismeasuredinbasispointsandistheabsolutevalueofthedifferencebetweenthetwoyieldsasgivenbyabsoluteyieldchange=│initialyield–newyield│×100.Thepercentagechangeiscomputedasthenaturallogarithmoftheratioofthechangeinyieldasshownby

percentagechangeyield=100×

ln

(newyield/initialyield)wherelninthenaturallogarithm.Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHall3-32Allrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmitted,inanyformorbyanymeans,electronic,mechanical,photocopying,recording,orotherwise,withoutthepriorwrittenpermissionofthepublisher.PrintedintheUnitedStatesofAmerica.Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHall4-33

Chapter4

BondPriceVolatility

Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHall4-34LearningObjectivesAfterreadingthischapter,youwillunderstandtheprice-yieldrelationshipofanoption-freebondthefactorsthataffectthepricevolatilityofabondwhenyieldschangetheprice-volatilitypropertiesofanoption-freebondhowtocalculatethepricevalueofabasispointhowtocalculateandinterprettheMacaulayduration,modifiedduration,anddollardurationofabondwhydurationisameasureofabond’spricesensitivitytoyieldchangesCopyright©2010PearsonEducation,Inc.PublishingasPrenticeHall4-35LearningObjectives(continued)Afterreadingthischapter,youwillunderstandthespreaddurationmeasureforfixed-rateandfloating-ratebondshowtocomputethedurationofaportfolioandcontributiontoportfoliodurationlimitationsofusingdurationasameasureofpricevolatilityhowpricechangeestimatedbydurationcanbeadjustedforabond’sconvexityhowtoapproximatethedurationandconvexityofabondthedurationofaninversefloaterhowtomeasureaportfolio’ssensitivitytoanonparallelshiftininterestrates(keyratedurationandyieldcurvereshapingduration)Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHall4-36Exhibit4-2

ShapeofPrice-YieldRelationshipforan

Option-FreeBondPriceMaximumPriceYieldCopyright©2010PearsonEducation,Inc.PublishingasPrenticeHall4-37PriceVolatilityCharacteristics

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

Anexplanationforthesefourpropertiesofbondpricevolatilityliesintheconvexshapeoftheprice-yieldrelationship.Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHall4-38PriceVolatilityCharacteristics

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.Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHall4-39EXHIBIT4-3InstantaneousPercentagePriceChangeforSixHypotheticalBondsSixhypotheticalbonds,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%/256.00-30012.8038.5913.4742.1315.56106.047.00-2008.3223.468.7525.4610.0961.738.00-1004.0610.744.2611.604.9127.108.50-502.005.152.115.552.4212.728.90-100.401.000.421.070.482.428.99-10.040.100.040.110.050.249.011-0.04-0.10-0.04-0.11-0.05-0.249.1010-0.39-0.98-0.41-1.05-0.48-2.369.5050-1.95-4.75-2.05-5.09-2.36-11.2610.00100-3.86-9.13-4.06-9.76-4.66-21.2311.00200-7.54-16.93-7.91-18.03-9.08-37.8912.00300-11.04-23.64-11.59-25.08-13.28-50.96Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHall4-40EXHIBIT4-4PriceChangefora100-Basis-PointChangeinYieldfora9%25-YearBondTradingatDifferentYieldLevelsYieldLevel(%)InitialPriceNewPriceaPriceDeclinePercentDecline7$123.46$110.74$12.7210.308110.74100.0010.749.709100.0090.879.139.131090.8783.077.808.581183.0776.366.718.081276.3670.555.817.611370.5565.505.057.161465.5061.084.426.75

a

Asaresultofa100-basis-pointincreaseinyield.Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHall4-41MeasuresofBondPriceVolatilityMoneymanagers,arbitrageurs,andtradersneedtohaveawaytomeasureabond’spricevolatilitytoimplementhedgingandtradingstrategies.Threemeasuresthatarecommonlyemployed:pricevalueofabasispointyieldvalueofapricechangedurationCopyright©2010PearsonEducation,Inc.PublishingasPrenticeHall4-42Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHall4-42MeasuresofBondPriceVolatility(continued)PriceValueofaBasisPointThepricevalueofabasispoint,alsoreferredtoasthedollarvalueofan01,isthechangeinthepriceofthebondiftherequiredyieldchangesby1basispoint.Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHall4-43Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHall4-43MeasuresofBondPriceVolatility(continued)YieldValueofaPriceChangeAnothermeasureofthepricevolatilityofabondusedbyinvestorsisthechangeintheyieldforaspecifiedpricechange.Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHall4-44Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHall4-44MeasuresofBondPriceVolatility(continued)DurationTheMacaulaydurationisonemeasureoftheapproximatechangeinpriceforasmallchangeinyield:whereP

=priceofthebondC

=semiannualcouponinterest(indollars)y

=one-halftheyieldtomaturityorrequiredyieldn

=numberofsemiannualperiods(numberofyearstimes2)M

=maturityvalue(indollars)Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHall4-45Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHall4-45MeasuresofBondPriceVolatility(continued)DurationInvestorsrefertotheratioofMacaulaydurationto1+yasthemodifiedduration.Theequationis:wherey

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

=priceofthebond.Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHall4-46EXHIBIT4-5CalculationofMacaulayDurationandModifiedDurationfor5-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$104.500.64392767.290443672.90442100.00000826.87899Copyright©2010PearsonEducation,Inc.PublishingasPrenticeHall4-47EXHIBIT4-6CalculationofMacaulayDurationandModifiedDurationfor5-