This paper presents preliminary fi ndings and is being distributed to economists and other interested readers solely to stimulate discussion and elicit comments. The views expressed in this paper are those of the authors and are not necessarily refl ective of views at the Federal Reserve Bank of New York or the Federal Reserve System. Any errors or omissions are the responsibility of the authors.

Federal Reserve Bank of New YorkStaff Reports

Staff Report No. 596February 2013

Darrell Duffi eDavid Skeie

James Vickery

A Sampling-Window Approach to Transactions-Based Libor Fixing

REPORTS

FRBNY

Staff

Duffie: Stanford University (e-mail: duffie@stanford.edu). Skeie, Vickery: Federal Reserve Bank of New York (e-mail: david.skeie@ny.frb.org, james.vickery@ny.frb.org). The authors thank David Hou and Ali Palida for outstanding research assistance, as well as Spence Hilton, Antoine Martin, Jamie McAndrews, and Simon Potter for comments. The views expressed in this paper are those of the authors and do not necessarily reflect the position of the Federal Reserve Bank of New York or the Federal Reserve System.

Abstract

We examine the properties of a method for fixing Libor rates that is based on transactions data and multi-day sampling windows. The use of a sampling window may mitigate problems caused by thin transaction volumes in unsecured wholesale term funding markets. Using two partial data sets of loan transactions, we estimate how the use of different sampling windows could affect the statistical properties of Libor fixings at various maturities. Our methodology, which is based on a multiplicative estimate of sampling noise that avoids the need for interest rate data, uses only the timing and sizes of transactions. Limitations of this sampling-window approach are also discussed.

Key words: shadow banking, financial intermediation

A Sampling-Window Approach to Transactions-Based Libor FixingDarrell Duffie, David Skeie, and James VickeryFederal Reserve Bank of New York Staff Reports, no. 596February 2013JEL classification: G01, G10, G18, G28

1

1.IntroductionThisnoteconsidersanapproachtoconstructingLiborfixingsusingtransactionsdataandmulti‐daysamplingwindows.1Forinstance,onecouldfixthe3‐monthLiborrateonagivendateastheaverageoftheactualinterestratesonall3‐monthloansintherelevanthistoricalsamplewhosetransactionsdatesarewithinthetrailing10businessdays.This“10‐daysamplingwindow”ismerelyforpurposesofillustratingtheconcept.Wewillexaminetheinfluenceofthesamplingwindowonsamplingnoiseandconsideradditionaltechniquesfor“fattening”thesampleandweightingthedatasoastoreducesamplingnoiseandmitigatebiases.Wealsoconsiderthepotentialrangeofapplicationsofthisapproach,andsomeofitsdisadvantages.LiborprovidesanestimateoftheinterestrateatwhichmajorbanksactiveinLondonmayborrowfromotherbanksonanunsecuredbasis.TheBritishBankersAssociation(BBA)currentlyreportsLiboronadailybasisfor10currenciesand15maturitiesbetweenovernightandoneyear.2Thesedailyinterestrate“fixings”areconstructedbasedonbanksubmissions.Eachofapanelofbanksself‐reportsitsownestimatedhypotheticalborrowingratesateachtenor.Notably,Liborisnotcurrentlycomputeddirectlyfromactualloantransactionrates.PublishedLiborratesarereferencedinthesettlementofmanyformsoffinancialcontracts,includingcorporatebondsandloans,mortgages,aswellasinterest‐ratefutures,swapsandoptions.AttentionhasrecentlyfocusedonthepotentialtoaddressshortcomingsofthesurveyapproachtoLiborwithafixingmethodthatissomehowbaseddirectlyonactualloantransactionsdata.Whileadvocatingfortheretentionofasubmission‐basedapproach,theWheatleyReviewofLibor(H.M.Treasury,2012)recommendsthatLiborsubmissionsshouldbe“explicitlyandtransparentlysupportedbytransactiondata.”ItalsooutlinesguidelinesforhowthisprincipleshouldbeimplementedinpracticebyLiborpanelbanks.3Thejudgmentandexpertiseofsubmittingbanksstillplaysaroleunderthisapproach.AnalternativewouldbetocomputeLibordirectlyasanaverageofindividualtransactionrates.Oneconcernoversuchanapproach,however,istherelativesparsenessofdailyinterbankunsecuredloantransactionsatcertainmaturities,

1Liborstandsfor“LondonInterbankOfferedRate”.2ThenumberofcurrenciesandmaturitiesisplannedtobereducedinthefutureinlinewiththerecommendationsoftheWheatleyReviewofLibor(H.M.Treasury,2012).Seesection2.3Theseguidelines(section4.8oftheWheatleyReview)layoutahierarchyoftransactiontypesthatbanksshouldusewhendeterminingtheirsubmissions.Highestpriorityisgiventotransactionsintheunsecuredinterbankdepositmarket,particularlythoseundertakenbythecontributingbank.Intheabsenceofrelevanttransactiondatatheguidelinessuggestthatexpertjudgmentshouldbeusedtodeterminethebank’ssubmission.Theyalsostatethat“submissionsmayalsoincludeadjustmentsinconsiderationofothervariables,toensurethesubmissionisrepresentativeofandconsistentwiththemarketforinter‐bankdeposits”,suchasplacinglessweightonnon‐representativetransactions.

2

particularlyduringperiodsoffinancialstress.Afixingthatisbasedonrelativelyfewtransactionscouldhaveexcessivesamplingnoiseandcouldalsocreateaheightenedincentiveforsomemarketparticipantstotransactwiththepurposeofinfluencingthedailyfixing.(Inastock‐marketcontext,Carhart,Kaniel,Musto,andReed(2002)discussevidenceoftransactionsdesignedto“paintthetape.”)TheWheatleyReport(H.M.Treasury,2012)indicatesthattherearetoofewtransactionstosupportLiborinmanyofthecurrency‐maturitypairsforwhichLiboriscurrentlyreported.4Weshow,however,thatatleastforsomeofthemoreactiveU.S.dollarmaturities,theuseofasampling‐windowapproachwouldsignificantlyreducethenoisinessoftransactions‐basedaverageinterestrates.Thisapproachwouldalsoimproverobustnesstomisreportingincentives.Theapproachcouldbeexploitedeitherasthebasisforanewfixingrateforasubsetofcurrenciesandmaturities,orasasourceofadditionalinformationinjudgingthevalidityofotherfixingmethods.Weillustratethetransaction‐windowapproachusingtwopartialdatasetsmeasuringunsecuredwholesalelendingactivity.Thefirstisahistoricaldatasetofbrokeredinterbankloans.ThesecondisasetofputativeunsecuredloansinferredfromFedwirepaymentsusingastatisticalalgorithmdevelopedbystaffoftheResearchGroupoftheFederalReserveBankofNewYorkthatextendstheworkofFurfine(1999).(SeeKuo,Skeie,VickeryandYoule,2013foradetaileddescriptionofthisalgorithm.)Wenotethatwhilethesedatasetsareusefulforillustratingourapproach,neithercouldbeusedinpracticeasthebasisforconstructingatransaction‐basedindexofbankfundingcosts.Inparticular,weemphasizethattheKuoetal.statisticalalgorithmidentifiesterminterbankloanswitherror.Historically,algorithmsbasedontheworkofFurfinehavebeenusedasamethodofidentifyingovernightortermfederalfundstransactions.TheResearchGroupoftheFederalReserveBankofNewYorkhasrecentlyconcludedthattheoutputofitsalgorithmbasedontheworkofFurfine5maynotbeareliablemethodofidentifyingfederalfundstransactions.6ThispaperthereforereferstothetransactionsthatareidentifiedusingtheResearchGroup’salgorithmasovernightortermloansmadeorintermediatedbybanks.Useoftheterm“overnightortermloansmadeorintermediatedbybanks”inthispapertodescribetheoutputoftheResearchGroup’salgorithmisnotintendedtobeandshouldnotbeunderstoodtobeasubstituteforortorefertofederalfundstransactions.

4Forthisreason,andbecauseoftheirlowusage,theWheatleyReviewrecommendsdiscontinuingLiborfortenorsof4,5,7,8,10and11months,anddiscontinuingLiborentirelyforfivecurrencies.ReportingofLiboristocontinuefortheUSDollar,Euro,JapaneseYen,UKPoundandSwissFranc.5Itshouldbenotedthatforitscalculationoftheeffectivefederalfundsrate,theFederalReserveBankofNewYorkreliesondifferentsourcesofdata,notonthealgorithmoutput.6Theoutputofthealgorithmmayincludetransactionsthatarenotfedfundstradesandmaydiscardtransactionsthatarefedfundstrades.Someevidencesuggeststhatthesetypesoferrorsinidentifyingfedfundstradesbysomebanksmaybelarge.

3

Giventhelimitationsofexistingdatasets,atransaction‐basedindexwouldrequireconstructingacentralizedandauditablerepositoryofrelevantinterbanktransactions.OnepossiblemethodologyforreportingthenecessarydataistheTradeReportingandComplianceEngine(TRACE),developedbyFINRAforthereportingofindividualtradesincertaintypesoffixed‐incomesecurities.Insection4ofthispaperwealsohighlightanumberofconceptuallimitationsofthesamplingwindowapproach,andconsiderpotentialsolutions.Oneimportantissueisthatafixingbasedonalaggedmovingwindowwillreflectstaleinformationduringperiodswhenmarketconditionschangerapidly,forexampleaftermonetarypolicyannouncements,orattheonsetofafinancialcrisis.Forapplicationsthatallowhindsight,suchasex‐postcorroborationofothermethodsforfixingLibor,atwo‐sidedsamplingwindowcouldbeused,incorporatingtransactiondatafromboththedaysbeforeandafterthefixingdate.Thiscouldmitigatethestaleness.Atwo‐sidedsamplingwindowisofcourseinfeasibleifthefixingneedstobepubliclyreleasedinrealtime.Asecondpotentialconcernisthattheavailablesampleofunderlyingwholesaleloantransactionsmaybesmallevenwithamulti‐daysamplingwindow,particularlyduringperiodsofmarketstress.Onewaytomitigatethisproblemcouldbetoconsiderawiderrangeofunsecuredfundinginstrumentswhenconstructingthetransaction‐basedindex.2.WiderSamplingWindowsSupposethereisasourceofactualtransactionsdataonlargeunsecuredloanstobanksinthedesiredborrowerclass.Incasethevolumeofinterbankloantransactionsisviewedasinsufficient,onemaywishtoconsiderawiderrangeofsourcesofunsecured“wholesale”fundingtomajorbanks,perhapsincludingcertificatesofdeposit,commercialpaperandsoon.EvenforaglobalcurrencysuchastheU.S.dollar,thereareextremelyfewlargeunsecuredloantransactionsatmanyofthematuritiesatwhichLiboriscurrentlyfixed.Evenasampling‐windowapproachwouldnotbereliableinsuchcases.Alternativesforthese“sparselypopulated”maturitiesincludeinterpolation,improvingthecurrentsurvey‐basedapproach,oracessationofLiborfixings,asrecommendedbytheWheatleyReviewofLibor.Fortunately,thematuritiesatwhichtherearefewtransactionssuitablefordeterminingareferenceratearealsolessimportantforapplications.Forexample,therearerelativelyfewderivatives,bonds,andotherinstrumentsthatreference9‐monthLibor.ThemostcommonlyreferencedLiborratesinmajorcurrenciesarethosewithmaturitiesofonemonth,threemonthsand,toalesserextent,sixmonths,asindicatedbyasurveyappearingintheWheatleyReviewofLibor.Wefocusoncurrenciesandmaturitiesforwhichtheaggregate‐sampletransactionsfrequencyispotentiallysufficienttoconsiderforafixing,orforvalidationofafixing.

4

EvenforrelativelyactiveU.S.dollarloanmaturitiessuchas1monthand3months,wewillshowthatasubstantialproportionalreductioninthesamplingnoiseassociatedwithatransactions‐basedfixingcanbeachievedwiththeuseofarollingsamplingwindow.Thisisnotsurprising,buttheempiricalmagnitudeoftheeffectisnotable.Moreover,ourmethodologydoesnotrelyonaccesstotheinterest‐ratedatathemselves,butratheronthetimesandsizesoftransactions.Ourapproachisinthespiritofstatisticalfiltersthatattempttoextractlonger‐frequencymovementsintime‐seriesdata(suchastheHodrick‐PrescottfilterortheKalmanfilter).Thisapproachcouldbeemployedinatleasttwoways:1)toprovideareplacementtothecurrentquote‐basedapproachfordeterminingtheLiborfixing;2)incorroborationofaquote‐basedorpoll‐basedLiborfixing,forexampleaspartoftheprocessofstrengtheningoversightofLibor.Inthefirstapplication,itwouldbenecessarytouseaone‐sidedlaggingwindow,sincethefixingwouldneedtobeannouncedinrealtime.Forex‐postvalidationpurposes,however,itwouldbepossibletouseatwo‐sidedsamplingwindowtoconstructthefixing,incorporatingbothpastandfuturedata.7Ournumericalexamplesbelowfocusonaone‐sidedwindow.Fromastatisticalfilteringpointofview,atwo‐sidedsamplingwindowwouldloweraveragethedegreeofsamplingerror.Oursimpleillustrativeexampleisafixingofthe3‐monthrateonagivendateastheaverageoftheratesonall3‐monthloansintherelevanthistoricalsamplewhosetransactionsdateiswithinthetrailing10businessdays.Onemayalsowishtouseasamplingwindowbasedonmaturity.Weelaborateandgeneralizeasfollows.SupposeonewantstocreateanestimateR(t,m)ofa“representative”m‐monthmaturityloanrateondayt.LetS(t,m;w,d)bethesubsetofallloansintheentirerelevanthistoricalsampleavailableonthefixingdatetwhosetransactiondateiswithinthetrailingwdaysandwhosematurityiswithinddaysofm.OnecouldfixR(t,m)asthevolume‐weightedaverageoftheloanratesinthisfixingsampleS(t,m;w,d).Forexample,foralag(w)of10daysandamaturitywindow(d)of5days,thefixingsampleforthethree‐monthborrowingrate(thatis,m=3months)onagivenday(t),sayMarch15,2013,wouldincludealltransactionsintherelevantpoolwithloanoriginationdatesbetweenMarch1,2013andMarch15,2013,inclusive(thatis,laggingbynomorethan10businessdays),withloanmaturitiesofthreemonthsplusorminus5businessdays.Inchoosingthelaggingtransaction‐datewindowwandthecenteredmaturitywindowd,onecantradeoffthebenefitofincreasedsamplesizeagainstthecostofbiasesassociatedwithincreasinglystaleoroff‐maturitydata.Inthelastsection,weexplorethebenefitsandcostsofreducingtheweightsappliedtothetransactionsaccordingtothetimelag,inordertomitigatestalenessbias.Inpractice,therelevanttermloanmaturitiesappeartobetightlyconcentratedaroundthestandardmaturitiesof1month,3month,and6months.Itmaybe7WethankSimonPotterforalertingustothispoint.

5

arguedthatitisrelativelypointlesstouseanon‐trivialmaturitywindow.Ontheotherhand,fatteningupthesamplebyincludingsimilar‐maturityloantransactionswouldlowersamplingnoisesomewhatandseemsunlikelytocreateimportantbiases.Theuseofamaturitywindowalsolowersthepotentialincentiveforloanmarketparticipantswhosetransactionsaresampledtocustomizetheirmaturitydatessoastoavoidenteringthefixingsample.3.EmpiricalMethodsandResultsInthissectionwepresentaproportionalsampling‐noisemeasureandourempiricalevidenceregardinghowvariationinthesamplingwindowandotherdatafiltersaffectsthe“thinness”ofthedataunderlyingapotentialtransaction‐basedLiborindex.3.1DatasourcesWedonothaveaccesstoacomprehensivetransaction‐leveldatabaseofunsecuredwholesaleloans.Intheabsenceofsuchdata,weillustrateourapproachusingtwopartialdatasources:

1. Adatasetofbrokeredinterbanktransactionsfromtheperiod2000‐04.2. Statisticallyinferredtransactionsbasedoninterbanktransfersoffederal

reservespassingoverFedwireFundsService(“Fedwire”),alarge‐valuepaymentsystemoperatedbytheFederalReserve,fromtheperiod2007‐12.

ThefirstofthesedatasourceswaspreviouslyusedbyBartolini,HiltonandMcAndrews(2010)andobtainedfromBGCBrokers,oneofthefourlargestU.S.interbankbrokeragefirms.Thesedatarepresentoneoftheonlydirecttransaction‐levelresearchdatasetsforUS‐dollar‐denominatedinterbankloansavailableforresearch.However,thisdatasethasanumberoflimitations.First,thedataareavailableonlyforahistoricaltimeperiodfromJanuary1,2000untilSeptember27,2004.Thissamplepre‐datesthe2007‐08financialcrisisandthepost‐crisisperiod.Second,thedatacoveronlybrokeredloans,whichrepresentonlyasubsetoftheinterbankmarket,andrepresentonlytradesnegotiatedthroughasinglebroker.Theidentitiesoftradecounterpartiesarenotprovided.Finally,thedatacoveronlyinterbankloans,andthusdonotincludeotherunsecuredfundinginstruments(suchaswholesaletimedeposits)thatmaybeusefulforconstructingatransaction‐basedLiborfixing.TheseconddatasourceisasetoftermloansmadeorintermediatedbybanksinferredfrompaymentspassingoverFedwireusingastatisticalalgorithmdevelopedinKuo,Skeie,Vickery,andYoule(2012)(KSVY).TheKSVYalgorithmisageneralizationofFurfine(1999),whoappliedthemethodtoidentifypotentialovernightloans,nottermloans.TheideabehindtheKSVYalgorithmisthatmostwholesaleinterbankloansaresettledoveralarge‐valuepaymentsystem.Inthe

6

caseofUS‐dollarloans,thisislikelytobeeitherFedwireorClearingHouseInterbankPaymentsSystem(“CHIPS”).TheKSVYalgorithmsearchesfortransactionpairsconsistingofa“send”leg(frompartyAtopartyB)foralargeround‐lotamount,anda“return”leg(fromBtoA)onasubsequentdateforaslightlylargeramount,suchthattheimpliedannualizedinterestrateisawholenumberofbasispointsandsuchthatthetransactionpairmeetscertainothercharacteristics.Forthepurposesofthispaper,thealgorithmisusedtoidentifyputativeinterbanktransactionsforwhichboththesendingandreturnlegpassoverFedwirebetweenJanuary1,2007andMay1,2012.ThemostimportantdisadvantageoftheKSVYinferencesisthatthesetofidentifiedtransactionpairsareinferences,notdirectobservationsoftermloans.Itisdifficulttoverifyatthispointhowwellorpoorlythesepairscorrespondtoactualunsecuredtransactions.KSVYdohoweverpresentsometestssuggestingthattheresultsofthealgorithmareinformative.Forexample,KSVYshowthatpriortotheonsetofthefinancialcrisis,thedistributionofimpliedinterestratesoftheseputativeloansisclusteredtightlyaroundtheLiborfixingrate,implyingthattheresultsarenotstatisticalnoise.Aswediscussedintheintroductiontothispaper,itisimportanttoemphasizethatthismethodissubjecttobothType‐IandType‐IIclassificationerrors(failurestodiscoveractualloans,andinferredloansthatarenotactualloans).OneparticularconcernisthattheproximatecounterpartiesidentifiedintheFedwiredatamaybeactingonlyascorrespondents,ratherthanbeingtheultimateborrowerandlenderoffunds.Thisisespeciallyrelevantifauserofthedatawantstorestricttheirsampletoaparticularsubsetofborrowers.Notably,recentresearchbyArmantierandCopeland(2012)concludesthattherelatedovernightFurfinealgorithmperformspoorlyinidentifyingovernightfederalfundsloansconductedbytwolargebanks.8(Note:FederalfundsloansareasubcategoryofinterbankloanswhicharenotsubjecttoU.S.reserverequirements.)Giventheissuesdescribedabove,weemphasizethatneitherofthedatasourcesweconsidercouldreliablybeusedinpracticeasthebasisforcomputingatransaction‐basedreplacementforLibor.Inpractice,suchafixingwouldpresumablyrequirethecreationofarecordlogofactualwholesaleloans(whetherrestrictedtointerbankloans,orencompassingawidersetofunsecuredinstruments),whichcouldbeaggregatedorauditedbyregulatorsorotheroutsideparties.Inthemeantime,however,intheabsenceofasuitabledatabaseofactualterminterbankloans,ananalysisofthesetwodatasetsprovidesatleastaroughideaoftheeffectofthesizeofthesamplewindowandotherfiltersontherobustnessofthesampling‐windowapproach.Giventhelimitationsofthedatasources,wedonot

8Inpartbecauseoftheseconcernswedonotmakeuseofmeasuredinterestratesinthispaper,foreitherdatasource.Instead,werestrictouruseofthesedatasourcestotransactiontimes,maturities,andsizes.

7

presentsampling‐windowestimatesoftheinterbankrateitself,insteadwefocusonhowasamplingwindowapproachwouldaffecttherelativesamplingnoiseassociatedwithatransaction‐basedinterbankindex.3.2ResultsBearinginmindtheimportantcaveatsdescribedabove,weusethesetwodatasourcestocomputeestimatesoftherelativesamplingnoiseassociatedwithanillustrativeUS‐dollarindexrate,forvariousdatafiltersandmaturities.Figure1andTable1illustratetheeffectofchangingthesamplingwindowfortheimpliedsample‐volatilitymultiplierV(t),aproportionalsamplingnoisemeasurethatisbasedonthenumberandrelativesizesofloansinthefixingsampleS(t,m;w,d).Specifically,V(t)isthesquarerootofthesumofthesquareddollar‐sizeweightsoftheloansinS(t,m;w;d).Forexample,ifthefixingsampleS(t,m;w,d)includestwoloans,ofamounts$40millionand$60million,thentherelativesizeweightsare0.4and0.6.Thesumofthesquaredweightsis0.16+0.36=0.52,soV(t)is0.72.Ifoneweretoassumethat,conditionalon“fundamental”loan‐marketinformation,theindividualloanratesinagivenday’sfixingsampleareuncorrelatedandhavethesamestandarddeviationD(t),thenthefixingR(t,m)hasaconditionalstandarddeviationofD(t)V(t).Undertheseconditions,intheaboveexampleofafixingsamplewithtwoloansofamounts$40millionand$60million,thesamplevolatilitymultiplierof0.72meansthattheassociatedsize‐weightedaverageinterestratehasastandarddeviationthatis72%ofthatforafixingratebasedonasingleloantransaction.Thesestatisticalassumptionsdonotapplyinpracticeandwedonotrelyonthem,butthesample‐volatilitymultiplierV(t)neverthelessgivesusagoodideaoftherelativeeffectofthelengthofthesamplingwindowontherobustnessofthesample.ArelativelyhighsamplingvolatilitymultiplierV(t)meansthattherearerelativelyfewloansdominatingthesample,andthereforelittleopportunityfor“diversification”ofthesamplingnoise.Atitsmaximum,forthecaseofasinglesampledloan,V(t)=1.Asthenumberofloansbecomeslargeandthefractionofanyoneloansizerelativetothetotalquantityofloansbecomessmall,V(t)approacheszero,bythelawoflargenumbers.WeemphasizethatV(t)saysnothingaboutthelevelsorvolatilitiesofinterestratesintheinferred‐loansample.Rather,V(t)isdeterminedentirelybythenumberandrelativesizesoftheloansinthefixingsamplefordatet.Withinterest‐ratedatafromactualtransactions,onecouldalsodirectlystudythesamplestandarddeviationsoftheratesinthefixingsamples,andtheeffectofthesamplingwindowonbiasesandrelativenoise.GiventhepotentialformisclassificationusingtheKSVYalgorithm,weavoidusingtheinferredloaninterestrates.

8

Forthe3‐monthmaturity,Figure1belowplotsthetimeseriesofV(t)basedona10‐daysamplingwindowfromthetwotransaction‐leveldatasources.9

Figure1:Time‐seriesplotofV(t)

ThedailysamplevolatilitymultiplierV(t)for3‐monthmaturityloans.Thesampleisbasedonaminimumtransactionsizeof$25manda10‐daysamplingwindow.Thesampleperiodis2000‐2004forthebrokereddata,and2007‐2012fortheFedwireinferences.A.Brokeredinterbankdata

B.Fedwireinferences

9ThebrokereddatasampleusedtoconstructFigure1aswellassubsequentfiguresandtablesincludesbothEurodollarandtermFederalfundsinferredtransactions(asdiscussedinBartolinietal.,2010,thedatasetincludesaflagwhichindicatesthetransactiontype;weretainbothcategories).Similarly,fortheFedwireinferences,wepresentresultsbasedontheentiredatasetofinterbankloaninferences,ratherthanattemptingtorestrictthesampletoaparticularloantype.

0

.2

.4

.6

.8

1

SVM

01jan2000 01jul2001 01jan2003 01jul2004Date

0

.2

.4

.6

.8

1

SVM

01jan2007 01jan2008 01jan2009 01jan2010 01jan2011 01jan2012Date

9

Figure1showssubstantialvariationovertimeinthedailysample‐volatilitymultiplierV(t),forbothdatasources.Thesample‐volatilitymultipliermeasuredfromthebrokereddataisconsistentlyhigherthanthatforFedwire‐inferreddata.Thisisnotsurprising,giventhatthebrokereddatacaptureonlyasmallsegmentofthemarket(thosebrokeredinterbankloansintermediatedbyasinglebroker).ThedifferenceinV(t)betweenthetwodatasourcescouldalsopartiallyreflectfalse“matches”intheFedwireinferences,differencesinthesampleperiod,andotherfactors.Table1showsthemedianacrosstheperiodofthesamplevolatilitymultiplierV(t),forvariousmaturitiesandsamplingwindowslags,normalizedbythemedianofV(t)for3‐monthmaturityloansandasamplewindowlagof10days.Wevariedthesamplingwindowfromtwodaysto20days,andconsideredmaturitiesof1,3,and6months.(Thenormalizingcellassociatedwitha10‐daysamplingwindowand3‐monthmaturitythusalwaysshowsavalueof1.)Thetablealsoreportssummarystatisticsfromthetwodatasources.Table1:RelativevaluesofV(t)fordifferentmaturitiesandsamplingwindows

MedianvaluesofthesamplevolatilitymultiplierV(t),forvariouscombinationsoflagwindowandmaturity,normalizedbythemedianvalueofV(t)foralagwindowof10daysandamaturityof3months.Thesampleperiodis2000‐2004forthebrokereddata,and2007‐2012fortheFedwireinferences.Brokeredinterbankloans      Maturity 

      1 month  3 months  6 months 

Lag window (days)  2  1.04  1.57  2.22 

5  0.81  1.31  1.66 

10  0.61  1.00  1.36 

15  0.51  0.85  1.17 

20  0.46  0.77  1.05 

Fedwireinferences      Maturity 

      1 month  3 months  6 months 

Lag window (days)  2  1.16  1.68  2.50 

5  0.88  1.33  2.13 

10  0.67  1.00  1.63 

15  0.56  0.84  1.37 

20  0.50  0.76  1.23 

10

Table1showsthatinbothdatasources,thesamplingnoiseasmeasuredbyV(t)issignificantlygreateratlongermaturitiesandforshortersamplingwindows.Forbothdatasources,V(t)istwotothreetimeslargerforsix‐monthloansthanforone‐monthloans.Thisisnaturalinpartfromthefactthatlonger‐termloansrolloverlessoftenthanshorter‐termloans.(Thatis,theratiooftheflowofloanstothestockofloansislowerinsteadystateforlonger‐maturityloans.)Inanycase,ourpreliminaryresultssuggestcautionoverwhetheritwouldbepossibletoconstructarobustLiborfixingfromunderlyingloantransactionsforlonger‐termloanssuchassixmonths.Table2presentssummarystatisticsofthedatausedtoconstructthesampling‐windowLiborindex.Forbothdatasources,theaverageacrossthesampleperiodofthenumberofinferred3‐monthloantransactionswithina10‐daysamplingwindowislow,8and25transactionsrespectivelyforthebrokereddataandFedwireinferences.Again,careshouldbetakenininterpretingthesestatisticsgiventhatneitherdatasourceiscomprehensive.Table2:Summarystatistics(10daywindow,3monthmaturity)SummarystatisticsfortheestimatedsamplevolatilitymultiplierV(t),aswellasthenumberoftransactionswithinthe10daysamplingwindow,andtheaveragetransactionsize.Sampleperiodis2000‐04forthebrokereddata,and2007‐12fortheFedwireinferences.p10,p25etc.referstopercentilesoftherelevantdistribution.Brokereddata

   Mean  p10  p25  p50  p75  p90  StDev 

SVM  0.48  0.29  0.35  0.45  0.56  0.71  0.17 

# of Transactions in Window  8.13  2  4  7  11  16  5.65 

Transaction Size ($mm)  78.69  25  40  50  100  150  59.47 

Fedwireinferences

   Mean  p10  p25  p50  p75  p90  StDev 

SVM  0.30  0.21  0.24  0.28  0.34  0.41  0.10 

# of Transactions in Window  25.45  13  18  24  31  41  10.59 

Transaction Size ($mm)  110.81  25  38  54  110  246  213.74 

3.3AlternativespecificationsWehaveexperimentedwithvariousotherdatafilters.Intheappendix,wepresenttwovariations.Thefirstconsidersaminimumtransactionsizeof$100million,ratherthan$25million.Applyingthishighersizecutoffinevitablyreducesthenumberofeligibletransactionsatanypointintime,andthusraisesV(t).Onebearsinmind,however,thatthe“root‐mean‐squared”definitionofV(t)impliesthataloan

11

ofsize$100millionhasarelativeimpactonV(t)thatis16timesthatofa$25millionloan,whenbothsizesarepresentinafixingsample.Secondly,wehaveexperimentedwithanapproachinwhichmoreweightisgiventotransactionsclosertodatet.Seesection4belowforadiscussion.Inunreportedcalculations,wealsoexperimentedwithexpandingthewidthofthematuritywindow(byfivedaysineachdirection).Wefoundthatthishasonlyasmalleffectonthenumberofeligibletransactions.4.SomeDisadvantagesofThisApproach,andTheirMitigationInthissectionwediscusssomeimportantpotentialdisadvantagesofafixingthatisbasedonasampling‐windowapproach:(i)theeffectofusinglaggeddataonthetimelinessoftheresultingLiborfixing,(ii)theriskofalackofunderlyingtransactionsdata,evenwithinasamplingwindow,and(iii)possiblecalendar‐dateeffects.Wealsoconsidersomemitigantsoftheseproblems.Afirstdisadvantageofthesampling‐windowapproachisthatthefixingannouncedonagivendaywouldbebasedinpartonlaggeddatathatmaynolongerberepresentativeofmarketconditions.Thatis,thefixingratecouldbesomewhatstaleduringperiodsofrapidchangesinmarketconditions,forexamplearoundthetimesofsignificantcentral‐bankmonetarypolicyannouncements,orattheonsetofafinancialcrisisorotherperiodinwhichbankfundingcostsareshiftingrapidly,suchasAugust9,2007andtheperiodfollowingit.Theinformationthatmarketparticipantsandregulatorslearnfromtheresulting“Libor”reportcouldthereforebestale.Thereisnosingle“true”interbankborrowingrate,andnosamplingmethodisperfect.Onemaywishtocomparethebiasandsamplingnoiseofthesampling‐windowtransactions‐basedapproachthatwehavedescribedwiththoseofotherfeasiblemethods,includingthecurrentmethodforfixingLibor.Forapplicationsinvolvingbondorswapcontracts,thestalenessintroducedbyasamplingwindowmeasuredindaysisrelativelyunimportant.Afterall,aninvestorholdingapositioninswapsorfloating‐ratenotesisconcernedwiththelevelof3‐monthloanratesthatisgenerallylikelytoprevailseveralyearsintothefuture,andisprobablynotsointerestedinvariationin3‐monthloanrateswithinasmalltimewindowthatbeginsinseveralyears.Apartfromitsroleinfinancialcontracting,Liborisalsousefulforassessingcurrentmarketconditions.However,evenduringtherecentfinancialcrisis,Kuo,SkeieandVickery(2012)showthatmovementsinLiboroverallcommovequitecloselywithanumberofotherpubliclyavailableindices(suchassecondary‐marketCDratesandEurodollaryieldsreportedintheFederalReserveH.15report).Thesealternative

12

indices,whichwouldbemoresensitivetoshort‐termmarketshocks,wouldremainavailabletopolicymakersandmarketparticipants.Wealsonotethatintermsofrevealinginformationtomarketparticipants,asampling‐windowfixingapproachallowstherecoveryofmostofthe“fresh”marketinformationthatispresentintheunderlyingdata.Giventhatthedifferencebetweenthefixingrateondaytandthatonthepreviousdayt‐1iscausedbydroppingobservationsfromdatet‐w(foralagwindowofw)andaddingobservationsfromthelatestdatet,observerscanapproximatelyinvertthemoving‐averageproceduresoastoestimatetheimpliedaveragerateoftransactionsthatoccurredonthelatestavailabledate.Ofcourse,itwouldalsobepossibletosimplyreleasetheaveragetransactionrateforeachday,asdiscussedfurtherbelow.Onecouldreducethebiasassociatedwithstalenessbyweightingthedatawithinthefixingsamplebasedonthetimelag,usingweightsthatdecaywiththelag,sayexponentially.Inordertoillustratetheimpactonsamplingnoiseofde‐weightingstaledata,weexploredtheeffectofanexponentialdecayintransactionweightsthatgivesobservationswitha10‐daylagonly50%oftheweightappliedtoobservationsonthecurrentday.(Thiscorrespondstoaweightfactorof0.933raisedtothepowerofthenumberofdayslagging.)Thisdegreeofde‐weightingofstaletransactionscausesarelativelysmalldegradationinsamplingnoise.10Forexample,for3‐monthinferredtransactionsobtainedfromFedwiredatafor2007‐2012,wesawinTable2thatthemeansamplevolatilitymultiplieris0.30.Withaweightdecayfactorof0.933perdayoflag(50%de‐weightingof10‐dayoldobservations),thesamedataareassociatedwithameansamplevolatilitymultiplierof0.31,about3%higher.Theestimatedeffectsonsamplingnoiseofde‐weightingstaledataaresimilarlymutedinallofthecasesthatwehaveexamined,asdemonstratedinadditionalchartsandtablesfoundintheappendices.Itistobecautionedthattheseresultsarepreliminaryandonlyforillustrativepurposes.Inadditiontopublishingthesampling‐window‐basedfixingrate,onecouldalsopublishsomepropertiesoftheunderlyingdata,suchasthedailyaveragerate,thedailynumberoftransactions,orthesample‐volatilitymeasure.Whilefinancialcontractswouldpresumablybetiedtothefixingrate,otherpublishedinformationbasedonthesamplecouldprovideadditionalusefulinformationandcould

10Inordertogainsomeintuitionforthelimitedimpactofdecayingweightsonthesamplevolatilitymultiplier,considerarelativelyadversecaseinwhichthetransactionsareconcentratedatthefirstandlastdateofa10‐daysamplewindow.Twoequallysizedtransactionsateachendofthe10‐daysamplingwindow,withoutdecay,wouldhaveasamplevolatilitymultiplierofV(t)=(0.52+0.52)0.5=0.707.Withweightsdecayingproportionatelybyafactorof0.933perday,or50%over10days,wewouldhaveV(t)=[(0.5/k)2+(0.5×0.5/k)2]0.5,wherek=0.5+0.25=0.75,implyingV(t)=0.74.So,indeed,eveninthisrelativelyextremesituation,theelevationofthesamplevolatilitymultiplierV(t)duetodecayisonlyabout5%.

13

potentiallybeusedincontracting,forexampleinordertoallowfinancialcontractstobetiedtomarketliquidityortothequalityofthefixingsample.Aseconddisadvantageofasampling‐windowapproachisthatitisnotguaranteedtoproducereliableresultsunderallmarketconditions.Iftherearetoofewtransactionsatagivenmaturitytoprovideevenareasonableestimateofmajor‐bankborrowingrates,marketparticipantswillneverthelessrequireareferencerateonwhichtobasethesettlementofderivativesandfloating‐rateloancontracts.FortheU.S.dollarmarket,ourresultsbasedonalimiteddatasetsuggestsomehopeforthefeasibilityoftransaction‐basedfixing,usingsamplingwindows,for1‐monthand3‐monthmaturities.Inanycase,onemaywishtointroducerobustnesssafeguardsinthedefinitionofthefixingsampleS(t,m;w,d),suchasexpandingthefixingsamplewheneverthereisinsufficientdataforareliablefixing.Forinstance,onecouldtakethesamplewindowtobeafixednumberofdaysortheminimumnumberofdaysnecessarytoincludeagivenvolumeoftransactions,whicheverisgreater.11AsanalternativetofixingLiborbasedonunsecuredborrowingrates,ithasbeensuggestedthatLibormightbereplacedwithabenchmarkratebasedonsecuredlendingtransactions.Prominentamongthesuggestedsecuredinterestratesis“GCFrepo,”whosemarketisdescribedbyFlemingandGarbade(2003).12Thisapproachwouldintroduceseveralpotentialcomplications,however.First,forGCFrepo,thereremainrobustnessconcernsoverwhetherthereisasufficientvolumeofGCFrepotransactionsattherelevantmaturities.Second,GCFreporatesareonlyindirectlyconnectedtobanks’unsecuredcostoffunds,whichreducestheusefulnessofGCFrepoasthebasisforanindexrateforfinancialcontracting.Forcommercialbanksandbankholdingcompanies,unsecuredborrowingisgenerallyamuchlargersourceofoverallfundingthansecuredborrowing.Unsecuredborrowingisalsotraditionallytheprimarysourceoffundingonthemargin.(Forsecuritiesdealers,securedborrowingisalargersourceoffundingandamoretypicalmarginalsourceoffunding,relativetobanks.)Further,Libor‐basedswapsareheavilyusedforrisk‐managementandpricediscoveryfortheunsecureddebtofnon‐financialcorporations.BasingLiboronasecuredborrowingratewouldreduceitsusefulnesshereaswell.Third,usingasecuredfinancingratesuchasGCFreporaisesthe

11ArelatedconcernisthataLiborfixingbasedonasamplingwindowapproachcouldbecomedistortedaroundkeycalendardates,suchastheendofaquarterorcalendaryear.Counterpartiesmayforexamplelengthenorshortenthematurityofotherwisestandardcontractstoinfluencewhethertheycoverparticularfinancialstatementdates,forwindow‐dressingpurposesorforotherreasons.Thiscouldaffectthesetofcontractswhosematuritiesliewithinagivenrange(d)aroundastandardmaturitysuchasonemonthorthreemonths.Inourexamples,wesetthisdaterangetobeconstant,butitmaybenecessarytoadjustdinsuchsituations.12TheDTCCpublishesanaverageovernightGCFreporateforthreetypesofcollateral:Treasuries,agencyMBS,andagencydebt.TradinginfutureslinkedtotheseindicesbeganinJuly2012.Seehttps://globalderivatives.nyx.com/nyse‐liffe‐us/dtcc‐gcf‐repo‐index‐futures/settlement‐procedures

14

questionofhowtotreatlegacyLibor‐basedfinancialcontracts,ofwhichthereareenormousquantities.AcounterpartyreceivingLiboronalegacycontractwouldnotwillinglyreceiveinsteadtheGCFreporate,whichistypicallymuchlower.Replacing“legacyLibor”withanapproximationofunsecuredratesthatareestimatedfromsecuredfinancingrateswouldlikelyleadtoasubstantialamountofcontractualdispute.Thisalsoraisesthepossibilityoftwoparallelmarkets,atleastduringatransitionperiod,with“legacy”and“new”benchmarksbasedonunsecuredandsecured(repo)rates,respectively.Theassociatedtransitionwouldbeawkwardandlengthy,andinvolvesplittingliquidityacrossthetwomarketswithanattendantlossinmarketefficiency.Inanycase,asampling‐windowapproachcouldalsobeusedfortermreporates,providedtherearesufficientdata.TheWheatleyReport(H.M.Treasury,2012)reviewsotheralternativeapproachesandbenchmarks,suchastheovernightindexswaprate(OIS),andprovidesadescriptionoftheiradvantagesanddisadvantages.

15

ReferencesArmantier,OlivierandAdamCopeland(2012).“AssessingtheQualityof“Furfine‐based”Algorithms,”FederalReserveBankofNewYorkStaffReports,No.575October.Bartolini,Leonardo,SpenceHiltonandJamesJ.McAndrews(2010).“SettlementDelaysintheMoneyMarket”,JournalofBankingandFinance34,934‐945.Carhart,Mark,RonKaniel,DavidMusto,andAdamReed(2002)“LeaningfortheTape:EvidenceofGamingBehaviorinEquityMutualFunds,”JournalofFinance57,661‐693.Fleming,MichaelandKennethGarbade(2003).“TheRepurchaseAgreementRefined:GCFRepo,”FederalReserveBankofNewYorkCurrentIssuesinEconomicsandFinance,9(June).Furfine,Craig(1999)“TheMicrostructureoftheFederalFundsMarket,”FinancialMarkets,Institutions&Instruments8,24‐44.Kuo,Dennis,DavidSkeie,JamesVickery(2012),“AComparisonofLibortoOtherMeasuresofBankBorrowingCosts,”WorkingPaper,FederalReserveBankofNewYork.Kuo,Dennis,DavidSkeie,JamesVickery,andThomasYoule(2012),“IdentifyingTermInterbankLoansfromFedwirePaymentsData,”WorkingPaper,FederalReserveBankofNewYork.H.M.Treasury(2012).“TheWheatleyReviewofLibor:FinalReport,”H.M.Treasury,London,September,2012.

16

Appendix:OtherDataFiltersAppendixA1.Minimumtransactionsizeof$100m(ratherthan$25m)ThestatisticsshownherearecomputedforthesamedataasthoseunderlyingFigure1andTable1,withtheexceptionthatthetransactionssizeshaveaminimumof$100m,ratherthanaminimumof$25m.FigureA1.Time‐seriesplotofV(t)i.Brokereddata

ii.Fedwireinferences

0

.2

.4

.6

.8

1

SVM

01jan2000 01jul2001 01jan2003 01jul2004Date

0

.2

.4

.6

.8

1

SVM

01jan2007 01jan2008 01jan2009 01jan2010 01jan2011 01jan2012Date

17

TableA1.StatisticsforV(t)MedianacrossthesampleperiodofthesamplevolatilitymultiplierV(t)forthematurityandsamplingwindowlengthshown,normalizedbythemedianofV(t)forasamplingwindowof10daysandmaturityof3months.i.Brokereddata      Maturity 

      1 month  3 months  6 months 

Lag window (days)  2  1.00  1.41  1.41 

5  0.79  1.41  1.41 

10  0.54  1.00  1.41 

15  0.44  0.78  1.41 

20  0.38  0.70  1.41 

ii.Fedwireinferences      Maturity 

      1 month  3 months  6 months 

Lag window (days)  2  1.22  1.73  2.42 

5  0.92  1.42  2.42 

10  0.66  1.00  1.71 

15  0.54  0.83  1.54 

20  0.48  0.74  1.43 

18

AppendixA2.UsingexponentialdecayThestatisticsshowninFigureA2andTableA2arecalculatedusingthesamesamplesasthoseofFigure1andTable1,exceptthatweincorporateexponentialdecayoverthesamplingwindow.FigureA2.Time‐seriesplotofV(t)(i)Brokereddata

(ii)Fedwireinferences

0

.2

.4

.6

.8

1

SVM

01jan2000 01jul2001 01jan2003 01jul2004Date

0

.2

.4

.6

.8

1

SVM

01jan2007 01jan2008 01jan2009 01jan2010 01jan2011 01jan2012Date

19

TableA2.StatisticsforV(t)MedianfortheperiodofthesamplevolatilitymultiplierV(t)fortheindicatedmaturityandsamplingwindowlengthshown,normalizedbythemedianforasamplingwindowof10daysandmaturityof3months.(i)Brokereddata      Maturity 

      1 month  3 months  6 months 

Lag window (days)  2  1.03  1.55  2.19 

5  0.80  1.28  1.63 

10  0.61  1.00  1.35 

15  0.52  0.86  1.18 

20  0.47  0.78  1.09 

(ii)Fedwireinferences      Maturity 

      1 month  3 months  6 months 

Lag window (days)  2  1.15  1.67  2.48 

5  0.88  1.31  2.10 

10  0.67  1.00  1.63 

15  0.57  0.86  1.38 

20  0.53  0.79  1.27