债券市场分析与策略第7版答案4Word文件下载.docx

债券市场分析与策略第7版答案4Word文件下载.docx

《债券市场分析与策略第7版答案4Word文件下载.docx》由会员分享，可在线阅读，更多相关《债券市场分析与策略第7版答案4Word文件下载.docx（40页珍藏版）》请在冰豆网上搜索。

PRICEVOLATILITYCHARACTERISTICSOFOPTION-FREEBONDS

Therearefourpropertiesconcerningthepricevolatilityofanoption-freebond.（i）Althoughthepricesofalloption-freebondsmoveintheoppositedirectionfromthechangeinyieldrequired,thepercentagepricechangeisnotthesameforallbonds.（ii）Forverysmallchangesintheyieldrequired,thepercentagepricechangeforagivenbondisroughlythesame,whethertheyieldrequiredincreasesordecreases.（iii）Forlargechangesintherequiredyield,thepercentagepricechangeisnotthesameforanincreaseintherequiredyieldasitisforadecreaseintherequiredyield.（iv）Foragivenlargechangeinbasispoints,thepercentagepriceincreaseisgreaterthanthepercentagepricedecrease.

Anexplanationforthesefourpropertiesofbondpricevolatilityliesintheconvexshapeoftheprice-yieldrelationship.

CharacteristicsofaBondthatAffectitsPriceVolatility

Therearetwocharacteristicsofanoption-freebondthatdetermineitspricevolatility:

couponandtermtomaturity.

First,foragiventermtomaturityandinitialyield,thepricevolatilityofabondisgreater,thelowerthecouponrate.Thischaracteristiccanbeseenbycomparingthe9%,6%,andzero-couponbondswiththesamematurity.Second,foragivencouponrateandinitialyield,thelongerthetermtomaturity,thegreaterthepricevolatility.

EffectsofYieldtoMaturity

Abondtradingatahigheryieldtomaturitywillhavelowerpricevolatility.Animplicationofthisisthatforagivenchangeinyields,pricevolatilityisgreaterwhenyieldlevelsinthemarketarelow,andpricevolatilityislowerwhenyieldlevelsarehigh.

MEASURESOFBONDPRICEVOLATILITY

Moneymanagers,arbitrageurs,andtradersneedtohaveawaytomeasureabond’spricevolatilitytoimplementhedgingandtradingstrategies.Threemeasuresthatarecommonlyemployedarepricevalueofabasispoint,yieldvalueofapricechange,andduration.

PriceValueofaBasisPoint

Thepricevalueofabasispoint,alsoreferredtoasthedollarvalueofan01,isthechangeinthepriceofthebondiftherequiredyieldchangesby1basispoint.Notethatthismeasureofpricevolatilityindicatesdollarpricevolatilityasopposedtopercentagepricevolatility（pricechangeasapercentoftheinitialprice）.Typically,thepricevalueofabasispointisexpressedastheabsolutevalueofthechangeinprice.Pricevolatilityisthesameforanincreaseoradecreaseof1basispointinrequiredyield.

Becausethismeasureofpricevolatilityisintermsofdollarpricechange,dividingthepricevalueofabasispointbytheinitialpricegivesthepercentagepricechangefora1-basis-pointchangeinyield.

YieldValueofaPriceChange

Anothermeasureofthepricevolatilityofabondusedbyinvestorsisthechangeintheyieldforaspecifiedpricechange.Thisisestimatedbyfirstcalculatingthebond’syieldtomaturityifthebond’spriceisdecreasedby,say,Xdollars.ThenthedifferencebetweentheinitialyieldandthenewyieldistheyieldvalueofanXdollarpricechange.Thesmallerthisvalue,thegreaterthedollarpricevolatility,becauseitwouldtakeasmallerchangeinyieldtoproduceapricechangeofXdollars.

Duration

TheMacaulaydurationisonemeasureoftheapproximatechangeinpriceforasmallchangeinyield.

Macaulayduration=

whereP=priceofthebond,C=semiannualcouponinterest（indollars）,y=one-halftheyieldtomaturityorrequiredyield,n=numberofsemiannualperiods（numberofyearstimes2）,andM=maturityvalue（indollars）.

InvestorsrefertotheratioofMacaulaydurationto1+yasthemodifiedduration.Theequationis:

modifiedduration=

.

Themodifieddurationisrelatedtotheapproximatepercentagechangeinpriceforagivenchangeinyieldasgivenby:

=modifiedduration.

Becauseforalloption-freebondsmodifieddurationispositive,theaboveequationstatesthatthereisaninverserelationshipbetweenmodifieddurationandtheapproximatepercentagechangeinpriceforagivenyieldchange.Thisistobeexpectedfromthefundamentalprinciplethatbondpricesmoveintheoppositedirectionofthechangeininterestrates.

Ingeneral,ifthecashflowsoccurmtimesperyear,thedurationsareadjustedbydividingbym,thatis,

durationinyears=

WecanderiveanalternativeformulathatdoesnothavetheextensivecalculationsoftheMacaulaydurationandthemodifiedduration.Thisisdonebyrewritingthepriceofabondintermsofitstwocomponents:

（i）thepresentvalueofanannuity,wheretheannuityisthesumofthecouponpayments,and（ii）thepresentvalueoftheparvalue.BytakingthefirstderivativeanddividingbyP,weobtainanotherformulaformodifieddurationgivenby:

wherethepriceisexpressedasapercentageofparvalue.

PropertiesofDuration

ThemodifieddurationandMacaulaydurationofacouponbondarelessthanthematurity.TheMacaulaydurationofazero-couponbondisequaltoitsmaturity;

azero-couponbond’smodifiedduration,however,islessthanitsmaturity.Also,lowercouponratesgenerallyhavegreaterMacaulayandmodifiedbonddurations.

Thereisaconsistencybetweenthepropertiesofbondpricevolatilityandthepropertiesofmodifiedduration.Forexample,apropertyofmodifieddurationisthat,ceterisparibus,bondswithlongerthematuritywillhavegreatermodifieddurations.Also,generallythelowerthecouponrate,thegreaterthemodifiedduration.Thus,greatermodifieddurationswillhavegreaterthepricevolatility.Aswenotedearlier,allotherfactorsconstant,thehighertheyieldlevel,thelowerthepricevolatility.Thesamepropertyholdsformodifiedduration.

ApproximatingthePercentagePriceChange

Thebelowequationcanbeusedtoapproximatethepercentagepricechangeforagivenchangeinrequiredyield:

=（modifiedduration）（dy）.

Wecanusethisequationtoprovideaninterpretationofmodifiedduration.Supposethattheyieldonanybondchangesby100basispoints.Then,substituting100basispoints（0.01）fordyintotheaboveequation,weget:

=（modifiedduration）（0.01）=（modifiedduration）（%）.

Thus,modifieddurationcanbeinterpretedastheapproximatepercentagechangeinpricefora100-basis-pointchangeinyield.

ApproximatingtheDollarPriceChange

Modifieddurationisaproxyforthepercentagechangeinprice.Investorsalsoliketoknowthedollarpricevolatilityofabond.Forsmallchangesintherequiredyield,thebelowequationdoesagoodjobinestimatingthechangeinprice:

dP=（dollarduration）（dy）.

Whentherearelargemovementsintherequiredyield,dollardurationormodifieddurationisnotadequatetoapproximatethepricereaction.Durationwilloverestimatethepricechangewhentherequiredyieldrises,therebyunderestimatingthenewprice.Whentherequiredyieldfalls,durationwillunderestimatethepricechangeandtherebyunderestimatethenewprice.

SpreadDuration

Marketparticipantscomputeameasurecalledspreadduration.Thismeasureisusedintwoways:

forfixedbondsandfloating-ratebonds.

Aspreaddurationforafixed-ratesecurityisinterpretedastheapproximatechangeinthepriceofafixed-ratebondfora100-basis-pointchangeinthespread.

PortfolioDuration

Thusfarwehavelookedatthedurationofanindividualbond.Thedurationofaportfolioissimplytheweightedaveragedurationofthebondsintheportfolios.

Portfoliomanagerslookattheirinterestrateexposuretoaparticularissueintermsofitscontributiontoportfolioduration.Thismeasureisfoundbymultiplyingtheweightoftheissueintheportfoliobythedurationoftheindividualissuegivenas:

contributiontoportfolioduration=weightofissueinportfolioxdurationofissue.

CONVEXITY

Becauseallthedurationmeasuresareonlyapproximationsforsmallchangesinyield,theydonotcapturetheeffectoftheconvexityofabondonitspriceperformancewhenyieldschangebymorethanasmallamount.Thedurationmeasurecanbesupplementedwithanadditionalmeasuretocapturethecurvatureorconvexityofabond.

MeasuringConvexity

Duration（modifiedordollar）attemptstoestimateaconvexrelationshipwithastraightline（thetangentline）.WecanusethefirsttwotermsofaTaylorseriestoapproximatethepricechange.Wegetthedollarconvexitymeasureofthebond:

dollarconvexitymeasure=

Theapproximatechangeinpriceduetoconvexityis:

dP=（dollarconvexitymeasure）（dy）2.

Thepercentagechangeinthepriceofthebondduetoconvexityortheconvexitymeasureis:

convexitymeasure=

Thepercentagepricechangeduetoconvexityis:

Ingeneral,ifthecashflowsoccurmtimesperyear,convexityisadjustedtoanannualfigureasfollows:

convexitymeasureinyear=

ApproximatingPercentagePriceChangeUsingDurationandConvexityMeasures

Usingdurationandconvexitymeasurestogethergivesabette