Fixed_Income_Attribution_Model(债券投资组合归因分析方法和模型)

FixedIncomeAttributionModel

Thispaperproposesaflexiblefixedincomeattributionmodelthatcanbeadaptedtoalargerangeofinvestmentprocesses.TheStatProreturnbreakdownmodelcanfitaverylargerangeoffixedincomeinvestmentprocesses.Pricingtechniquesareusedtounderstandtheimpactbond-specificfactorshaveonsecurityreturns.Thisallowsustogiveanaccurateanddetailedsecurityperformancedecomposition.Bondspecificsnaturallyleadtopartic-ularmanagementmethodsandthereforerequireadaptedex-postperformanceattributionanalysis.

MathieuCubilié

isaperformancemeasurementconsultantatStatProPlc,whereheisproductmanagerforStatProFixedIncome,thecompanybondperformancemeasurementandattributionsystem.MathieuholdsapostgraduatedegreeinMarketFinanceandEconomicsfromToulouseUniversityandbeganhiscareeratSinopia-Asset-Management,HSBC-groupquantitativemanagementsubsidiary,inParis,whereheexperiencedbondportfoliomanagement.lendsitselftotheuseofderivativeproducts.Unfortunately,futurescontractsarereplicatingbond

Oncethevariousperformancecontributionsarecom-maturitybehaviorsratherthanbondindexreturns.Asaputed,theycanbeconsideredaselementarybuildingresult,bondmanagerspracticebenchmarksampling.blocksthatcanberearrangedaccordingtotheportfolioThistechniqueconsistsoftryingtotracklargeindices

withalimitednumberofassets.managementdecisionprocess.INTRODUCTION

Thesinglecurrencyreturnattributionmodelisflexible

enoughtopermitconcurrentelementaryreturndecom-positions.Itisalsoabletoattributeatdifferentlevelsofdetailcorrespondingtoseveralavailableattributioneffectaggregationcapacities.Tofinish,wewillseehowthissinglecurrencymodelcaneasilybeusedinamulti-currencyenvironment.FIXEDINCOMEMARKETS

Bondandequitymarketsareclearlydifferent;equityindicesareintra-dayquoting,whileveryfewbondindicesare.Asaconsequence,bondmanagersrequiremarketindicatorsduringthedaytoruntheirinvest-ments.Actually,they’reusingasampleofliquidbondsrepresentingthevariousmarketliquidmaturities.ThissetofbondsisknownastheYieldCurve.TheYieldCurvesarecontinuouslyquotedandthereinformationisaccessable.Theyrepresentdifferentmarkets:mainly

localcurrencyTreasurymarketsandalsocreditmar-THEPRICEEFFECTkets.

Amajordifferencewithequitymarketsisthatbond

Indexreplicationmanagementisadifficultexercise;marketsaretraded“over-the-counter”whileequitybondindicescontainseveralthousandsofsecurities,marketsareorganizedmarkets.Pricesareproprietytomanywithverylargeminimaltradesize.Thismeansbrokersandindexproviders.Thisleadstoasituationmanagerswouldrequirehugeamountsofinvestmentswherepricedifferencebetweenasecurityheldinatocloselyreplicateanindex.Thissituationdefinitelyportfolioandintheindexcanbeverysignificant.ThisWinter2005/2006

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Inamulti-currencyuniverse,aftermakingyieldmarketandcurrencyallocationdecisions,managershavetodefinewhichbondinparticulartheywanttoinvestin.Theyusuallyhavedurationobjectivesagainstthebenchmarkreflectingtheirmarketexpectation.Marketanticipationsaremainlyyieldcurvemovementsandcreditspreadvariation.Bondswillthereforebeselect-edfortheircharacteristics:theirduration,thespreadtheyoffer,andallotherspecificities.

Toachievethisgoal,bondmanagementisruledmorebyriskcontrolthantheequitymarket.Inthebondarea,riskisfirstreflectedbytheDuration.MANGEMENTPROCESSES

Despitethefactthattherearemanyinvestmentprocess-es,bondmanagersgenerallyfacethesameconsiderations.

Fixed_Income_Attribution_Model(债券投资组合归因分析方法和模型)

phenomenon,knownas“pricedifferenceeffect,”canrepresentalargepartoftheexcessreturn.Thisphe-nomenoncanexistinanorganizedmarketwherebackofficesystemsarenotfedbyindexdata.Thisismuchlessimportanceinequitymarkets,asvolatilityremainshighcomparedtothepricedifferenceeffect.Forbonds,thepricedifferenceeffectcanbeevenmoreimportantthantheexcessreturn.Insuchcase,aperformanceattributionexercisemaybemisleading.

Thereturnformedbytwoconsecutiveintermediarypricesisconsideredasecurityreturncontribution.Withthisapproach,itispossibletogoasfarasrequiredinthereturnbreakdownandtoadheretotherealmanage-mentprocess.

Itisnotsufficienttoworkatthesecuritylevelonly,asmostofthemanagementdecisionsarenotmadeinde-pendentlyfromoneanother,whichmakestheirimpactdifficulttomeasure.Forinstance,selectingalong-termmaturitybondontheyieldcurvealsomeans,inthegeneralcaseselectingahighdurationbond.Ifthisbondisacorporateone,thisalsomeansitissupposedtodeliverahigheryieldthanthereferencemarket.Ofcourse,thisalsoexposestheinvestmenttospreadvari-ationsthatwillimpactthesecurityreturn.

Whateverthereferenceaccountreturnsource(backorfrontoffice),afirstrequirementistocleantheinitialportfolioreturnfromthepriceeffect.Thismeansthatthefirststepistorepriceportfoliosandbenchmarks,usingthesamepricebase.Forsimplicityreasons,itiscommonlyacceptedthattheportfolioisrepricedusingthebenchmarkpricesforcommonsecurities.

Noncommonsecuritieswillkeepthebackofficeprices.Whatabouttransactions?PricingtechniquescanonlyThisisextendedtocommonexchangeratesforsecuri-explainpricereturnsandthereforeignorestheimpacttiesinmulti-currencyportfoliosusingbenchmarksrates.oftransactionsmadebythemanager.Forthisreason,

wealsothinkanaccurateattributionmodelmustbetransactionbasedandmustclearlyidentifytheir

impact.

RETURNCONTRIBUTIONS

Forsimplicityreasonswewillconsiderplainvanillabondsinthischapter.

Let’sfocusontheportfolioreturn,andlet’srunafirst

Inthefollowing,wewillconcentrateonunderstandingdecompositionofeachinstrumentpricereturn.Inaddi-theportfoliorecomputedreturn.tion,wewillmeasurethetradeimpactbycomputing

thereturnusingtheModifiedDietzandtheall-inprice

rManagement=rreference+rPricereturnandconsidertheirdifference.SECURITYPRICINGTECHNIQUES

InordertounderstandtheportfolioorbenchmarkrModified

return,itmakessensefirsttoanalyzetheperformanceofeachinstrument.

Dietz

=

MVe MVs ∑CFt

t=1

n

MVe+∑CFt×

t=1

n

s t(SeeAppendixA)s e

AsModifiedDietzreturncapturestradereturnbydefi-Eachsecurityisassociatedwithapricingfunction.

nition,wecanderive

Suchafunctionestablishesalinkbetweenthesecurity

rTrade=rModifiedDietz rPricepriceanditsyieldtomaturity.Shockingtheyieldto

maturitywithelementaryyieldvariationsrelatedto

marketconditionsallowsustogenerateasetofinter-Fromnowon,let’sconsiderthepricereturn,.

Price

mediarypricesbetweenthebeginningandtheendofperiodprice.

Therearethreemajorsourcesofreturncommonly

identifiedwhencloselylookingatabond.

r

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rPrice=rCarry+rYieldCurve+rCredit1

Afirstcomponentofabondreturnisthecarryreturnreflectingtheimpactoftime.Then,theYieldcurvereturnrevealshowthereference(Treasury)marketreactedovertheanalysisperiod.Thisreturniscreatedbyglobalyieldcurvemovements.Atlast,thecreditreturnisthepartofthereturn,whichisspecifictothebondandnottothereferenceyieldmarket.Thiscorre-spondstochangesinthesecurityspreadwiththerefer-enceyieldcurve.Ifwemaketheassumptionabondpricedependsontimeandonitsyieldtomaturity,thenfromapricingpointofviewwecangeneratethefol-lowingprices.

2

rCarry=

PCarryPt 1

1

rYield

Curve

=

PYield

Curve

PCarry 1

1

rCredit=

PtPYieldCurve

Thisleadstothesplitoftheaccount(orbenchmark)returnintermsofmainsourcesofreturn.DETAILEDRETURNCONTRIBUTIONS

spondingtodatet-1andtotheyieldtomaturityatthisEachofthethreemainsourcesofreturncanfurtherbedate.decomposedandgiveamoredetailedvisiononhow

returnswereconstructed.Thiscanstillbeachieved

PtCarry=P(t,yt 1),istheintermediarypricetakingthankstotheuseofpricingfunctions.intoaccountthetimepassingby.Itissimplyobtained

byswitchingthedateattheendofperiodoneinthefor-mula.

Thecarryreturncanbesplitusingtwopossiblebreak-,istheintermediarydowns:

pricecapturingtheYieldCurvemovement.Itcanbeobtainedbyconsideringthesecurityatdatetandby Couponandconvergencereturnaddingthecorrespondingyieldcurvechangetothestartofperiodyieldtomaturity.Inthisbreakdown,thereturnduetothebondyieldto

maturitycanbedecomposedintotwodistinctmecha-PtCredit=P(t,yt 1+YC[t 1,t]+δ[t 1,t]),istheendofperi-nismsthecouponreturncomponentandthesimplepas-odprice.Itcanalternativelybecomputedfromthesageoftime,denotedasrolldowneffect,reflectingtheyieldcurvechangeandthespreadchangeovertheperi-bondconvergenceeffect.od.

Pt 1=P(t 1,yt 1),isthestartofperiodpricecorre-

Pt

YieldCurve

=P(t,yt 1+YC[t 1,t])

Whereδ[t 1,t]allowsPt

Credit

=Ptandwhere

rCarry=rCoupon+rConvergence

Thecouponreturnissimplythecouponratemultipliedbythetime.

yt=YCt+δt

(SeeAppendixBforsecurityspreadcalculationmethod)

Fromthisvariousintermediaryprices,wecanwrite:rCoupon=C.dt

Theconvergenceeffectreflectsthefactthatbondsareusuallynottradedatpar,butthemarketpricemustcon-vergetowardpar.Theconvergenceeffectisobtainedbythedifferencebetweentheyieldtomaturityandthecouponrate,multipliedbythetime.

rPrice=

Pt

1Pt 1

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rRollDown=(yt C).dt

Treasuryandspecificreturn

anticipatethevariousyieldcurvemovements,known

asShift,Twist,andButterflymovementsandthereforemaketheirinvestmentchoices.

TheShiftmovementreflectstheparallelmovementoftheyieldcurve.TheTwistmovementcorrespondstothesteepeningorflatteningshiftsvariationoftheyieldcurveandtheButterflymovementisthecurvaturechangeoftheyieldcurve.ThereareindeedvariousmethodstocomputetheShift,Twist,andButterflymovements,andthisarticlewon’tdescribehowthesevariousmovementsaremeasured.

AnadditionalmovementisthenaturalagingprocessofthesecuritiesallalongtheYieldCurve.Itcorrespondstothefactthatinastandardconfiguration,yieldsaredecreasingwithmaturities.Shift,TwistandButterflymovementsarethereforecalculatedforconstantsecuri-tymaturities(startofperiodmaturities)whiletheroll-downcapturesthechangeofmaturitiesduetothetimepassingby.

Aconcurrentpossibledecompositioninsistsmoreontheoriginofthecarryeffect.PartofitcomesfromthelocalTreasurymarketrepresentedbythelocalgovern-mentyieldcurve.Anotherpartcomesfromspecificcarryeffectduetothepurchaseofbonds(governmen-talorothers)notlocatedonthereferenceyieldcurve.Thesebondspresentaspreadwiththereferenceyieldcurve.(SeeAppendixA)

rCarry=rTreasury+rSpecific

With,

rTreasury=yt.dtrSpecific=

δt.dt

ThisbreakdownrequiressystematicallyrecoveringtheAsamathematicalapproach,werecommendtheuseofcorrespondingmaturitytreasuryyieldtomaturity.theNelson-Siegelformulation(1987).Theygiveapar-simoniousfittoasetofmarketforwardrates.Thefit-tingfunctiontheyproposedependsonfourparameters:

ascaleparameterandthreeotherparametersusedto

Theyieldcurvereturncanalsobebrokendownamongdescribetheshapeandtheasymptoticalbehaviorofthevariouscomponentsreflectingyield-curvemanage-curve.AproperrearrangementoftheBetasgivesthement.Shift,theTwistandtheButterflyfigures.Oncecomputedforeachmaturity,thethreeyieldcurve

shiftscaneasilybeusedtoshockthesecurityyieldto

Mostmanagersseetheirinvestmentsasbetstheyhave-maturityinthepricingfunctions.Thisgivesyieldcurvedoingagainsttheyieldcurvevariations.Theyusuallyshiftreturnscorrespondingtoyieldcurvemanagement

rYieldCurve=rShift+rTwist+rButterfly+rRoll down

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PtShift=P(t,yt 1+S[ot 1,t])

121

Let’sconsiderS[0PtTwist=P(t,yt 1+S[o+St 1,t],S[t 1,t],S[t 1,t]arerespectively[t 1,t])t 1,t]yieldchangesduetoShift,Twist,andButterflymove-Butterfly12

=P(t,yt 1+S[ot 1,t]+S[t 1,t]+S[t 1,t])mentsovertheinterval[t-1,t]foragivensecurity.Pt

S[RD

t 1,t]istheyieldcurvechangecorrespondingtoroll-Roll down012RDP=P(t,y+S+S+S+Stt 1downmovement.[t 1,t][t 1,t][t 1,t][t 1,t])

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Fixed_Income_Attribution_Model(债券投资组合归因分析方法和模型)

Thisgives:rShift=

PShiftPCarry

rButterfly=

1

PButterflyPTwist

1

rRoll down=

Creditshiftsreturn

PRoll down

1

PButterfly

rTwist=

PTwist

1PShift

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Fixed_Income_Attribution_Model(债券投资组合归因分析方法和模型)

45CreditreturncanbeseenthesamewayasyieldcurveCreditYCSp[t 1,t]=S[3t 1,t]+S[t 1,t]+S[t 1,t]

returnbyconsideringtheshiftmovementsofthecredit

yieldcurvescomparedtothereferenceyieldcurve.Thisgives:Thismeansthefollowingbreakdowncanbeachieved:

rCredit=rCreditYieldCurve+rBondSelection

Thecredityieldcurvespreadisthedifferencebetweenthecredityieldcurveandthereferenceyieldcurve:

12

PtCreditShift=P(t,yt 1+S[0+S+St 1,t][t 1,t][t 1,t]+S

S[RD+S[t3 1,t])t 1,t]

PtCreditTwist=P(t,yt 1+S[0+S[1t 1,t]+S[2+t 1,t]t 1,t]S[RD+S[t3 1,t]+S[t4 1,t])t 1,t]

CreditYCSpt=CreditYCt YCt

Asanexample,let’sconsideracredityieldcurve

formedbyAAratingsecurities.TheCredityieldcurvePtCreditButterfly=P(t,yt 1+S[0+S[1t 1,t]+S[2+t 1,t]t 1,t]returnrepresentstheimpactofthiscredityieldcurve

RD345

)variationcomparedtothereferenceAAAyieldcurve.S[t 1,t]+S[t 1,t]+S[t 1,t]+S[t 1,t]

Thebondselectionreturnisthereforethepartofthe

creditreturnduetothespecificspreadbehavioroftheTheendofperiodpriceincludesthebondspecificsecurity.Thisspecificspreadismeasuredwiththespread.Credityieldcurvesothat:

yt=YCt+CreditYCSpt+σt

3

1

PtSelection=P(t,yt 1+S[o+S[+S[2t 1,1]+S[RD+t 1,1]t 1,1]t 1,1]

CreditYieldCurve=CreditShift+

rCreditTwist+rCreditButterfly

Let’sassumewealsouseNelson-Siegelmathematicalmodelingforthecredityieldcurve.Itisthereforepos-sibletoderivethevariouscreditshiftreturnsfromtheWhereσ[t 1,t]allowsPtSelection=Ptcredityieldcurvereturn.

rr

S[4t 1,1]+S[5t 1,1]+σ[

t1,1])

Thecreditshiftreturnthereforerepresentstheparallel

variationofthecreditspread(betweenAAcreditcurve

andAAAreferenceyieldcurve).Thecredittwistreturnrepresentsthereturnduetoslopevariationofthecred-itspreadandcreditbutterflyreturnisthereturndueto

http://www.77cn.com.cningpricingfunctions,thiscouldbewrittenasfollowed:

345

S,S,SLet’sconsider[t 1,t][t 1,t][t 1,t]arerespectivelycreditspreadchangesduetoshift,twist,andbutterflymovementsovertheinterval[t-1,t]andσ[t 1,1]thebond

specificspreadvariationoverthesameperiod.Wewill

callselectionreturnthereturngeneratedbythespecif-icspreadvariationofthebond.Thisreturnreflectspart

ofthereturnduetothenatureofthesecurityindepend-entlyfromthereferencemarketandthecreditmarket.(SeeappendixBforbondspreadcalculationmethod)Thecredityieldcurvecanbewrittenas:Winter2005/2006

rCreditShift

P

=tButterfly 1Pt

CreditShift

Apossiblealternativebreakdownistoconsiderthe

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Fixed_Income_Attribution_Model(债券投资组合归因分析方法和模型)

swapyieldcurve.ThisyieldcurveisformedoftheWhereσ[t 1,t]allowsPtSelection=Ptratesofferedbybankstoinitiateswapcontracts.

rCredit=rSwapSpread+rSelection

Let’sconsiderSwapYCSp[t 1,t],σ[t 1,t]arerespective-lythevariationoftheswapyieldcurvespreadsandthevariationofthebondspecificspreadovertheperiodwe

study.TheswapyieldcurvespreadcorrespondstothedifferencebetweentheSwapyieldcurveandtherefer-enceyieldcurve:SwapYCSpt=SwapYCt YCt.

Theswapyieldcurvereturnreflectsthereturngenerat-edbythespreadvariationbetweentheswapyieldcurve

andthereferenceyieldcurve.Theselectionreturn Sectorandselectionreturnreflectsthereturnsofferedbythespecificbondspread

existingbetweenthebondyieldtomaturityandtheAnotherwaytodecomposetheCreditreturnistocon-swapyieldcurve.sidermanagershaveallocatedtheirinvestmentsaccord-ingtosectorchoicesandinsideeachsectoraccordingtoWhere,yt=YCt+SwapYCSpt+σtabondselectionprocessrelatedtoeachissuecharacter-istics.Thisrequiresusingsectoryieldcurves4andcon-SwapSpread1

sideringthesectorspreadvariationandthespecificPt=P(t,yt 1+S[ot 1,t]+S[t 1,t]+

bondspreadvariation.With:2

S[t 1,t]+SwapYCSp[t 1,t])

SectorYCSpt=SectorYCt YCtrCredit=rSector+rSelection

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12

PtSelection=P(t,yt 1+S[ot 1,t]+S[t 1,t]+S[t 1,t]+SwapYCSp[t 1,t]+σ[t 1,t])

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Fixed_Income_Attribution_Model(债券投资组合归因分析方法和模型)

12

PtSector=P(t,yt 1+S[0t 1,t]+S[t 1,t]+S[t 1,t]+

Singlecurrencyattributionmodel

SectorYCSp[t 1,t])

12PtSelection=P(t,yt 1+S[0+S+St 1,t][t 1,t][t 1,t]+

Thesinglecurrencyversionisbasedonreturncontribu-tiondifferences.

SectorYCSp[t 1,t]+σ[t 1,t])

Whereσ[t 1,t]allowsPt

Selection

r=P B

=Pt

5

+r∑ω(r

r=∑(wr wr)+∑(w

B

B,Carry

r=∑ωP(rP,Carry+rP,Curve+rP,Credit)

B,Curve

+rB,Credit)

PP,Carry

wBrB,Carry)+∑(wPrP,Curve

BB,CurvePP,Credit

r

wBrB,Credit)

Anexampleofmoredetailedexcess-returndecomposi-tionwouldgive:6

rP,Coupon+rP,Convergence

Belowisasummaryofthevariousmainsourcesof

r=∑ωP +rP,Shift+rP,Twist+rP,Butterfly+rP,Roll down returnbreakdownsthatcanbedone.

+r +r,,PSectorPSelection

FIXEDINCOMEATTRIBUTIONMODELSNowthatwehaveidentifiedthevarioussourcesof

returnforabond,let’sformulateasinglecurrencyfixedincomeattributionmodel.

rB,Coupon+rB,Convergence ω+r+r+r+r∑B B,ShiftBTwist,,, down BButterflyBRoll

+r +r B,SectorB,Selection

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ωBr+∑(ωr,Butterfly ω+B,Twist)PPBrB,Butterfly)ωr∑(+∑(ω∑(ωr

PP,Roll down

PP,Sector

∑(ω+∑(ω

r=∑ωPrP,Coupon ωBrB,Coupon+

()

PP,ConvergencePP,Shift

r

ωBrB,Confergen

ce

r

ωBrB,Shift)+∑(ωPrP,Twist ωBrB,Roll

down

)

Calculatingthedurationandtheyieldcurvepositioning

effectsrequiresthecreationofsyntheticornotionalportfolioswewillcomparetothebenchmark.Let’sdefinetheAccountDashavingthesamedurationastheportfolio,butsamepositioningontheyieldcurve(weightsallalongtheyieldcurve)asthebenchmark.OnanysegmenttheAccountDYieldCurvereturncanbemeasuredasfollowed:

)

r

ωBrB,Sector)+ ωBrB,Selection

PP,Selection

)

ThisYieldCurvereturnisobtainedbyusingthebench-marksegmentyieldcurvereturnandbyadjustingitby theratiomadeofthetotalaccountandbenchmarkmod-ifieddurations.Carryeffecthasbeendecomposedinthecouponcarryeffectandconvergencecarryeffect.YieldcurveeffecthasbeendecomposedintermsofShift,Twist,Butterfly,andRoll-downeffects.Thecrediteffectisinterpretedasasectorchoiceeffectandabondselec-tioneffect.

So,foranysegmenti,wecansplittheyieldcurveeffectintoitsdurationandyieldcurvepositioningcompo-nent:

Yieldcurveeffecti=Durationeffecti+Yieldcurvepositioningeffecti

AsdescribedinFigure5,otherexcess-returndecompo-Tounderstandthesesourcesofreturn,weneedtointro-sitionscanbemadeaccordingtotheportfoliomanage-ducesyntheticportfolios.mentprocess.Let’snowintroduceaslightlydifferentwaytosplittheyieldcurveeffect.Thesetwoeffectsarecomputedasfollowed:

DurationandYieldCurvepositioningeffects

iiii

Durationeffecti=wBrD,Curve wBrB,Curve

Thegoalofthissectionistomakeapparenthowthe

durationmanagementbetandhowtheimpactoftheThiseffectmeasurestheimpactofhavingadifferentassetallocationbetallalongtheyieldcurve,canbeyieldcurvereturnduetoadifferentmodifieddurationmeasured.betweenthebenchmarkandtheaccount.

iiiii

Let’sconsiderthefollowingexcess-returndecomposi-Yieldcurvepositioning=wPrP,Yieldcurve wBrD,Yieldcurvetion:

Thisreturngivestheyieldcurvepositioningmanage- r=Carryeffect+Curveeffect+Crediteffectmentimpact.Thiscorrespondstothereturnrelatedto

differentmaturityallocationbetweentheportfolioand

Dependingonthemanagementprocess,manyman-thebenchmark.agerssplittheYieldCurveeffectintermsofduration

betsandYieldCurvepositioningbets.TheiranalysisatAsasummary,hereisatemplateshowingthevarioussecuritylevelmakeslittlesense;thisiswhywewillexcess-returnbreakdownsthatcanbedone.driveourapproachatsegmentlevel.

Forinstanceatypicalwayofdecomposingmainfixed

Yieldcurvepositioningeffect+Crediteffectincomeattributioneffectscouldbetofocusonthe

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Treasurycarryeffectandonthespecificcarryeffect.Thiscanbeusefulwhenpartoftheinvestmentsweremadeoncorporatebonds.Also,theyieldcurveeffectisseeningeneralasthecombinationofadurationbetandofayieldcurveallocationstrategy.Toconclude,ifweconsiderthesamemanagerinvestedpartoftheaccountincorporatebonds,itmaybeofinteresttomonitorthesectorchoiceeffectaswellasthespecificbondselec-tioneffectinsideeachsector.Thisisofcoursemean-ingfulwhenthecorporatepartoftheportfoliohasbeenmanagedbysectors.WORKINGEXAMPLE

Theannualmanagementfeeforthisaccountis1.75%.

Themanagementfeesimpactiscomputedbyusingthe

Theaccountyieldcurvereturncontributioncanbeannualmanagementfees:obtainedonanymaturitybandasfollows:

r

i

P,YieldCurve

= MD×w× YC

iPiP

i

[t;t+1]

Inthesameway,thebenchmarkyieldcurvereturncon-tributioncanbeobtainedonanymaturitybandasfol-lows:

iiiirB,YieldCurve= MDB×wB×

YC[t;t+1]

Thedurationaccountyieldcurvereturncontributioncanbeobtainedoneachmaturitybandbydoing:

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Effectsareobtainedbydifferentiatingaccountandmodelonlyrequirestheadditionofamarketallocation

effectreflectingtheyieldcurvemarketchoiceandcom-benchmarkcontributions.

putedasastandardassetallocationeffect.Currencyallocationeffectalsoneedstobecalculated.Wecanimaginemoredetailedversionsofthistemplateusingthevariousdecompositionpossibilitieswepoint-edoutinthereturncontributionssection.Apartofthisattribution“tree”whichhasnotbeenmentionedinthisarticleistheresidualeffect,whichistheportionofthelocalcurrencyexcessreturnwhichisnotexplainedbytheattribution.Thetradeimpactistheeffectoftransac-Here,theYieldCurveeffecthasbeenreplacedbythetionsmadeatdifferentpricesthanthevaluationprices.Durationeffectandtheyieldcurvepositioningeffect.

TheDurationeffectforanymaturitybandiscalculatedSUMMARYasthedifferencebetweenthedurationaccountyield

curvereturnandthebenchmarkyieldcurvereturnforFixedincomeattributionfirstrequiresusingthesamethissegment.Theyieldcurvepositioningeffectiscal-priceandexchangeratereferenceintheportfolioandinculatedasthedifferencebetweentheaccountyieldthebenchmarksoastofocusonlyontheexcessreturncurvereturnandthedurationaccountyieldcurvereturnpart,whichcanbeattributabletomanagementdeci-sions.Basedonthis,aflexibleattributionframeworkisforagivensegment.

neededtodecomposetheexcessreturninawayreflect-ingthemanagementprocess.Aflexiblecalculationtool Multi-currencyattributionmodel

iscertainlythekeydriverforflexiblereporting.With

Inamulti-currencyenvironment,sameeffectscanalsosuchadescribedattributionframework,wecanimag-bemeasuredonsinglecurrencyassetclasses.The

ineafirstreportinglineshowingdetailedattribution

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forbondmanagersandanotherreportinglinesumma-APPENDIXArizingmainattributioneffectsforclientshavinglessknowledgeinthisarea.ModifiedDietzreturn:NOTATIONS

y,

X[t 1,1]

Xt,MD,YC,C,δ,

,

t,dt,S,

CreditYCSpt,SwapYCSpt,SectorYCSpt,

σ,

P,B,

rP,X,

i,j,

YC,

w,

eCurrency,

Yieldtomaturity

VariationofquantityXbetweent-1andtQuantityXatdatetModifieddurationReferenceyieldcurveCouponrate

Givenabond,δrepresentsthespreadthatmustbeaddedtotheyieldcurveratetoobtainthebondyTime

ElementarytimevariationDesignates

ayieldmarket

shift

IstheCreditYieldcurvespreadatdatet

IstheSwapYieldcurve

spreadatdatetIstheSwapYieldcurve

spreadatdatet

Designatesthesecurityspreadwiththeswapyieldcurveorwiththecredityieldcurve,orwiththesectoryieldcurve.IstheportfolioreturnIsthebenchmarkreturnIsthereturncontributionoffactorXtoanassetclass

oftheportfolioAssetclassindexSecurityindex

Yieldcurvechangeforoverthereferenceperiod,alsostatedas

Assetclassweight

Partoftheexcessreturnduetocurrencyallocationeffect

MVeMVsCFtest

istheendofperiodmarketvalueisthestartofperiodmarketvalue

arecashflowsoccurringduringthereferenceperiod

istheendofperiod

designatesthestartofperiod

reflectsthemomentatwhichatransactionoccurs

ModifiedDietzreturnstakecashflowsimpactintoaccountandgivesanapproximationoftheinternalrateofreturn.Theshorter

thereferenceperiod,thebettertheapproximation.APPENDIXB

δt0,thesecurityspreadwiththereferenceyieldcurve

t

t00thebondcouponsforeachfollowingmaturityandwhereYCtisthetmaturityreferenceyieldcurverate.(SeeG.R.A.P.document[2003])APPENDIXC

Let’sassumethepriceofabonddependsontimeandonitsyieldtomaturity.

p=f(t,y)

Let’snowdifferentiatethisexpression:

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Fixed_Income_Attribution_Model(债券投资组合归因分析方法和模型)

threedifferentdecompositionswehaveproposed.

Providedthedifferentbreakdownmethodswepro-videdinthereturncontributionchapter,thereaderwillunderstandwejustfocusonafewpotentialexamples.

6

MDisthebondsensitivitytoyieldcurvevariation.Weobtain:

perf=y.dt MD.dy

Byassumingtheyieldtomaturitycanbesplitamongtheyieldcurveyieldandthesecurityspread,weget:

perf=y.dt MD.dYC MD.dSpread

Whichcanbewrittenas:

perf=rCarry+rYieldCurve+rSpread

REFERENCES:

G.R.A.P.(GroupedeRecherchéenAttributiondePerformance),researchgroupinperformanceattribu-tion,“TraitementdesProduitsdeTaux,”Fall2003.Nelson,CharlesR.andAndrewF.Siegel,“ParsimoniousModelingofYieldCurve,”JournalofBusiness,October1987,Vol.60,Issue4,pp.473-489.ENDNOTES

1

Thesecurityreturncanbeseenasanadditionofvari-oussub-returnsbyperformingaTaylordevelopmentasshowninAppendixC.

2Returnandreturncontributionareregardedequiva-lentterminologies.

3

Thisalsomeansthat:CreditYCSpt+σt=δt4

Orsectoroptionadjustedspreads.

5

Theselectionreturnwillofcoursebedifferentinthe

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TheJournalofPerformanceMeasurement