BIS 美元，杠杆和利率平价偏离度

BISWorkingPaprsNo592Thdollar,banklragandthdiationfromcordintrstparitybyStfanAdji,WnxinDu,CathrinKochandHyunSongShinMontaryandEconomicDpartmntNombr2016JELclassification:F3,G1,G2Kywords:xchangrats,banklrag,crosscurrncybasisBISWorkingPaprsarwrittnbymmbrsofthMontaryandEconomicDpartmntofthBankforIntrnationalSttlmnts,andfromtimtotimbyothrconomists,andarpublishdbythBank.Thpaprsaronsubjctsoftopicalintrstandartchnicalincharactr.ThiwsxprssdinthmarthosofthirauthorsandnotncssarilythiwsofthBIS.ThispublicationisaailablonthBISwbsit(www.bis.org).©BankforIntrnationalSttlmnts2016.Allrightsrsrd.Brifxcrptsmaybrproducdortranslatdproiddthsourcisstatd.ISSN10200959(print)ISSN16827678(onlin)Thdollar,banklragandthdiationfromcordintrstparity∗StfanAdjiBankforIntrnationalSttlmntsWnxinDuFdralRsrBoardCathérinKochBankforIntrnationalSttlmntsHyunSongShinBankforIntrnationalSttlmntsNombr11,2016AbstractWdocumntthtriangularrlationshipformdbythstrngthofthUSdollar,crossbordrbanklndingindollarsanddiationsfromcordintrstparity(CIP).AstrongrdollargoshandinhandwithbiggrdiationsfromCIPandcontractionsofcrossbordrbanklndingindollars.DiffrntialsnsitiityofCIPdiationstothstrngthofthdollarcanxplaincrosssctionalariationsinCIParbitragprofits.Undrpinningthtrianglisthrolofthdollarasproxyforthshadowpricofbanklrag.Kywords:xchangrats,banklrag,crosscurrncybasisJELClassifications:F3,G1,G2∗ThiwsinthispaprthsolrsponsibilityofthauthorsandshouldnotbintrprtdasrflctingthiwsofthBankforIntrnationalSttlmnts,thBoardofGornorsofthFdralRsrSystmoranyothrprsonsassociatdwiththFdralRsrSystm.WthankClaudioBorio,KristaHughs,RobrtMcCauly,AndrasSchrimpfandVladyslaSushkoforryhlpfulcommnts.WalsothankBatlBrgrforxcllntrsarchsupport.1IntroductionOnofthmostsignificantdlopmntsinglobalfinancialmarktsinrcntyarshasbnthbrakdownofcordintrstparity(CIP).CIPisprhapsthbststablishdprinciplinintrnationalfinanc,andstatsthatthintrstratsimplicitinforignxchangswapmarktscoincidwiththcorrspondingintrstratsincashmarkts.Othrwis,somoncouldmakarisklssprofitbyborrowingatthlowintrstratandlndingatthhighrintrstratwithcurrncyriskfullyhdgd.Howr,thprinciplbrokdownduringthhightofth20082009crisis.AftrthGratFinancialCrisis(GFC),CIPdiationshaprsistdandhabcommorsignificantrcntly,spciallysincmid2014.Whydosuchapparntriskfrarbitragopportunitisxist?Incomptitimarkts,marktparticipantsarprictakrs,andcantakonanyquantityofgoodsatthprailingmarktpric.Thtxtbookarbitragargumntisthatsomoncouldborrowatthlowintrstratandlndoutatthhighrintrstrat,hainghdgdcurrncyriskcompltly.ThfailurofCIPwouldthrbyopnupthpossibilityofunlimitdrisklssprofits.Howr,intxtbooks,thrarnobanks.Inpractic,though,sucharbitragtypicallyntailsborrowingandlndingthroughbanks,andthcomptitiassumptionisiolatddutobalancshtconstraintsthatplaclimitsonthsizofthxposursthatcanbtaknonbybanks.Enfornonbanks,thirabilitytoxploitarbitragopportunitisrlyonbankstoproidlrag.Hnc,ifdiationsfromCIPprsist,itisbcausbanksdonotorcannotxploitsuchopportunitis.Ourfocus,thrfor,isonthbankingsctor,andthabilityofbankstotakonlrag.Thkymssagofourpapristhatthaluofthdollarplaysthrolofabaromtrofrisktakingcapacityincapitalmarkts.Inparticular,itisthspotxchangratofthdollarwhichplaysacrucialrol.DiationsfromCIPturnonthstrngthofthdollar;whnthdollarstrngthns,thdiationfromCIPbcomslargr.TothxtntthatCIPdiationsturnonthconstraintsonbanklrag,ourrsultssuggstthatthstrngthofthdollarisakydtrminantofbanklrag.Thcrosscurrncybasisisthdiffrncbtwnthdollarintrstratinthcash1USdollarbroadindxandthcrosscurrncybasisFigur1Indx,January1997=100BasispointsThrdlinshowsthFdralRsrBoard’sUStradwightdbroaddollarindx,withhighralusindicatingastrongrUSdollar.ThblulinisthsimplaragofthfiyarcrosscurrncybasisswapspradsforAUD,CAD,CHF,DKK,EUR,GBP,JPY,NOK,NZDandSEKisàisthUSdollar.Sourcs:BoardofGornorsofthFdralRsrSystm;Bloombrg.CrosscurrncybasisofuroagainstthdollarFigur2TimplotofUSDEURcrosscurrncybasisScattrchartofUSDEURcrosscurrncybasisSourc:Bloombrg.marktandthimpliddollarintrstratfromthswapmarktwhnswappingforigncurrncyintodollars.ThcrosscurrncybasismasursdiationsfromthCIPcondition.Figur1plotsthbroaddollarindx(inrd),whichisthtradwightdUSdollarxchangratagainstitsmajortradingpartnrs.Whnthrdlingosup,thdollarstrngthns.Thblulintrackstharagcrosscurrncybasisforthtnmostliquidcurrncisisàisthdollar.Wsthatthcrosscurrncybasisisthmirrorimagofdollarstrngth.Whnthdollarstrngthns,thCIPdiationswidn.Thisisspciallysointhlast24months,rflctingthstrongrdollar.Wsthsampattrnforindiidualcurrncis.Figur2showsthcrosscurrncybasisforthuro,orthCIPdiationsbtwndollarintrstratsandimplidintrstratsofswappingurosintodollarsinthFXmarkts.Wsthatthsizofthcrosscurrncybasisforthuroisthmirrorimagofthstrngthofthdollar,likmountainsrflctdinalak.Thrighthandpanlshowsadailyfrquncyscattrchartofthbasisandthdollarxchangrat.Thbasiscomosrycloslywiththxchangrat,natthdailyfrquncy.Aswdmonstratblow,thrisalsoanasstpricingrlationshipundrpinningthsmpiricalobsrations.ThxposurtothdollarxchangratispricdinthcrosssctionofCIPdiationsinthsnsthatariationsinsuchdiationsacross2USdollarbroadindxandthcrosscurrncybasisFigur1Indx,January1997=100BasispointsThrdlinshowsthFdralRsrBoard’sUStradwightdbroaddollarindx,withhighralusindicatingastrongrUSdollar.ThblulinisthsimplaragofthfiyarcrosscurrncybasisswapspradsforAUD,CAD,CHF,DKK,EUR,GBP,JPY,NOK,NZDandSEKisàisthUSdollar.Sourcs:BoardofGornorsofthFdralRsrSystm;Bloombrg.CrosscurrncybasisofuroagainstthdollarFigur2TimplotofUSDEURcrosscurrncybasisScattrchartofUSDEURcrosscurrncybasisSourc:Bloombrg.currncisarxplaindbythsnsitiityofthdiationstofluctuationsinthbroaddollarindx.ThCIPdiationsofdiffrntcurrncishadiffringxposurstothdollarfactor.Intrstingly,wdocumntarrsalofrols.Thclassical“safhan”currncis,suchasthJapansynandthSwissfranc,hathhighstxposurtothdollarfactor,andhighyilding“carry”currncis,suchasthAustraliandollarandthNwZalanddollar,hathlowstxposurtothdollarfactor.CurrnciswithhighrxposurtothdollarfactorxhibitlargrCIPdiationsandthrbyoffrgratrpotntialarbitragprofitsforbanks.Ourrsultspointtothfinancialchannlofxchangrats,throughwhichfluctuationsinthstrngthofthdollarstinmotionchangsincapitalmarktintrmdiationspradsthatrspondatahighfrquncy.Thntxportschannlofxchangratchangsisstandardinopnconomymacromodls,butthfinancialchannlislssstandard,andmayopratinthoppositdirctiontothntxportschannl.Undrthntxportschannl,itiswhnthdomsticcurrncydprciatsthatralconomicactiitypicksup.Bycontrast,thfinancialchannlapparstoopratinthoppositdirction;itiswhnthdomsticcurrncyapprciats,financialconditionsinthatcountryloosn,andCIPdiationsnarrow.WhydothCIPdiationsnarrowwhnthdomsticcurrncystrngthnsagainstthdollar?Undrpinningthisrlationshipisthrolofbanklragandcrossbordr3USdollardnominatdcrossbordrbanklndings.thUSdollarindxFigur3Compltsampl120quartrrollingwindowrgrssion,allsctors21CorsthQ12010–Q32015priod.LndingrfrstothxtnsionofloansandholdingsofdbtscuritisbyBISrportingbanks,whilallborrowrs(thtotalofbanksandnonbanks)rsidoutsidofthUnitdStats.Thblacklinisafittdrgrssionlin.PositichangsinthforignxchangratindicatanapprciationofthUSD.Forprsntationalpurposs,outlirswithFXratchangsxcding5%inabsolutaluhabndroppdfromthpanl,butnotfromthrgrssions.2ExtnsionofthLHPstaticrgrssiontoarollingwindowframworkof20quartrsprstimationforcrossbordrlnding(loansanddbtscuritis)dnominatdinUSdollars.Thpositionofdotswithrspcttothrticalaxisgisthcofficintstimat,whilthhorizontaltimlinrfrstothlastpriodofthrollingwindow.Rddotsindicatangaticofficintstimatthatisstatisticallysignificantatth10%ll.Sourcs:BISLocationalBankingStatistics;BISffctixchangratindics;BIScalculations.Crosscurrncybasis,USdollarindxandcrossbordrlndingFigur4ThbarsshowprcntaggrowthratsoftotalcrossbordrlndingdnominatdinUSdollars.ThrdlinplotsquartrlychangsinthbroadUSdollarindxinprcntagpoints,whilthblulinplotsthfirstprincipalcomponntofquartrlychangsinth5yarcrosscurrncybasisforG10currncisxprssdinbasispoints.Sourcs:Bloombrg;BoardofGornorsofthFdralRsrSystm;BISLocationalBankingStatistics;BIScalculations.banklndingindollars.Indd,wwillshowthxistncofa“triangl”thatcohrntlytistogthr(i)thaluofthdollar;(ii)thcrosscurrncybasis;and(iii)crossbordrbanklnding.Inthistriangl,adprciationofthdollarisassociatdwithgratrborrowingindollarsbynonrsidnts.ThrlationshipbtwncrossbordrdollarlndingandthdollarisillustratdinFigur3.Thlfthandpanlplotsthquartrlygrowthratofglobaldollardnominatdcrossbordrbanklndingflowsagainstthbroaddollarindx.Itralsthatthrisangatirlationshipbtwnthtwoariabls.Furthrmor,thrlationshipisstronglystatisticallysignificant.Aswdmonstratinthmpiricalsctionofthispapr,thstatisticalsignificancofthaborlationshipisrobusttocontrollingforanumbrofadditionalfactors.ThrighthandpanlofFigur3xaminshowthaborlationshiphasoldortim.Morconcrtly,itdisplaysthcofficintsobtaindfromrollingwindowrgrssionsofthquartrlygrowthratofdollardnominatdcrossbordrbanklndingonchangsinthaluofthdollar.Itapparsthat,whilthngatirlationshipbtwnthabotwoariablshasbnstrongthroughoutthsamplpriod,itgradually4strngthndorthdcadladinguptothGFCandrachdapakin2008duringthacutphasofthcrisis.Thismpiricalrgularityhassraldrirs,bothonthdmandfordollarcrditonthpartofborrowrsaswllasonthsupplyofdollarcrditbylndrs.Thngatirlationshipbtwndollarcrdit,aproxyforbanklrag,andthmagnitudofCIPdiations,thpricofbalancshtcapacity,pointinfaorofsupplydrirs.Thmchanismwhrbyadollardprciationladstoanincrasinthsupplyofdollarcrdithasbndubbdth“risktakingchannl”byBrunoandShin(2015a,b).Whnthrispotntialforaluationmismatchsonborrowrs’balancshtsarisingfromxchangratchangs,awakrdollarflattrsthbalancshtofdollarborrowrs,whosliabilitisfallrlatitoassts.Fromthstandpointofcrditors,thstrongrcrditpositionofborrowrsrducstailrisksinthcrditportfolioandcratssparcapacityforadditionalcrditxtnsionnwithafixdxposurlimitthroughaaluatrisk(VaR)constraintorconomiccapitalconstraint.Finally,washowthatsimilarrlationshipbtwnthxchangrat,thcrosscurrncybasisandcrossbordrbanklndingcanalsobfoundforthurointhpostcrisissampl.Thisisthcasdspitthfactthatthtriangularrlationshipisabsntforothrmajorcurrncis,whichpointstothuniqurolofintrnationalfundingcurrncisinaffctinglragandrisktakingiafluctuationsinxchangrataluations.ThbulkofthxistinglitraturonCIPdiationsfocussonthGFCandthEuropandbtcrisis(s,forxampl:Baba,Packr,andNagano(2008);Baba,McCaulyandRamaswamy(2009);BabaandPackr(2009);Coffy,Hrung,andSarkar(2009);Goldbrg,Knndy,andMiu(2011);GriffolliandRanaldo(2011);McGuirandonPtr(2012);Bottazzi,Luqu,Pascoa,andSundarsan(2012);andIashina,Scharfstin,andStin(2015)).Morrcntly,sralnwpaprsstudyingCIPdiationsinthpostcrisispriodhamrgd.Du,TpprandVrdlhan(2016)formallystablishCIParbitragopportunitisthatcannotbxplaindawaybycrditriskortransactioncosts,andprsntidncthatbankbalancshtcostsandasymmtricmontarypolicyshocksarthmaindrirsofCIPdiations.Boriotal(2016)andSushkotal5(2016)constructmpiricalproxisfornthdgingdmandofdiffrntnationalbankingsystmsandshowthatthyarconsistntwiththcrosssctionalariationsinCIPdiations.Liao(2016)focussoncorporatissuancpattrnsandlinksstratgicfundingcostarbitragacrosscurrnciswithCIPdiations.Iida,KimuraandSudo(2016)alsoxaminthrolofthrcntdirgncinthmontarypolicisofadancdconomisandpostcrisisrgulatoryrformsindriingCIPdiations.ThkycontributionofourpapristhuniquprspctiforlinkingthstrngthofthdollartoCIPdiationsthroughthlnsofbanklragandrisktaking.Ourpapralsoshdslightonthlarglitraturonintrmdiaryandmarginbasdasstpricing(forxampl,BrnankandGrtlr(1989),HolmstromandTirol(1997),BrunnrmirandPdrsn(2009),GarlanuandPdrsn(2011),HandKrishnamurthy(2012,2013),BrunnrmirandSanniko(2014),AdrianandShin(2014)andAdrian,EtulaandMuir(2014)).Furthrmor,ourpaprisrlatdtothmodlforxchangratdtrminationinthprsncoffinancialfrictionsprsntdinGabaixandMaggiori(2015),andthrolofthdollarinbilatralxchangratsasshowninVrdlhan(Forthcoming).Thrstofthpaprisorganisdasfollows.Sction2dfinsthcrosscurrncybasisandproidsanoriwofthtriangularrlationshipbtwnthdollarxchangrat,thcrosscurrncybasisandcrossbordrbanklndingindollars.Sction3outlinsamodltoxplainthtriangularrlationshipiathshadowcostofbanklrag.Sction4prsntsmpiricalidnconthrlationshipbtwnthUSdollarandthcrosscurrncybasis,andbtwnthdollarandcrossbordrbanklndingindollars.Sction5discusssthmrgncofthuroasaglobalfundingcurrncyduringthpostcrisispriod.Sction6concluds.62AnoriwofthissusWdfinthnyarcrosscurrncybasisofcurrncyiisàisthUSdollar,dnotdxit,t+n,asthdiationfromthCIPconditionbtwncurrncyiandthdollar:(1+y$t,t+n)n=(1+yit,t+n+xit,t+n)nSitFit,t+n,(1)whry$t,t+nisthnyardollarintrstrat,yit,t+nisthnyarintrstratincurrncyi,SitisthdollarspotxchangratofcurrncyiandFit,t+nisthoutrightforwardratforcurrncyi.Bothforwardandspotxchangratsardfindintrmsofcurrncyunitsiprdollar.Equialntly,inlogs,thcurrncybasisisqualto:xit,t+n=y$t,t+n−(yit,t+n−ρit,t+n),(2)whrρit,t+n1n[log(Fit,t+n)−log(Sit,t+n)]isthmarktimplidforwardprmiumtohdgforigncurrncyiagainstthUSdollar.Thcrosscurrncybasismasursthdiffrncbtwnthdirctdollarintrstratinthcashmarkt,y$t,t+n,andthimpliddollarintrstratinthswapmarkt,yit,t+n−ρit,t+n.Consistntwithmarktconntions,wfocusonthcrosscurrncybasisdridfrombnchmarkintrbankratsinthrspcticurrncy.ThxistncofangaticrosscurrncybasisimplisCIParbitragopportunitisforborrowingdollarsinthdollarintrbankmarktandlndingdollarsiathforignintrbanksmarktincombinationwiththFXswapmarkts.AsshowninDu,TpprandVrdlhan(2016),tharbitragprofitsassociatdwiththCIPtradscannotbxplaindawaybytransactioncostsorcrditrisk.Apparntly,banksdonotarbitragawaythsopportunitisdutoconstraintsonthirbalancshtcapacity.Nonrgulatdntitis,suchashdgfunds,obtainlragfromdalrbanks,andthusthbalancshtconstraintsfacingthbankingmarktrmainsatthcntrofattntion.Thrfor,suchCIPdiationsgithshadowpricofbankbalancshtcapacitis.Wusthaggrgatdollarindxandthbilatralxchangratisàisthdollar7tomasurthstrngthofthdollar.Inparticular,wchoosawidlyusdaggrgatdollarindx,thFdralRsrBoard(FRB)broaddollarindx,whichisthtradwightdaragofthforignxchangaluofthdollaragainstthcurrncisofabroadgroupofmajorUStradingpartnrs.Anincrasinthbroaddollarindxindicatsadollarapprciation.1WobtaindataoncrossbordrbanklndingflowsfromthBISlocationalbankingstatistics(LBS).ThycapturoutstandingclaimsandliabilitisofbankslocatdinBISrportingcountris,includingintragrouppositionsbtwnofficsofthsambankinggroup(BIS(2015)).Thlocationalstatisticsarcompildfollowingprinciplsthatarconsistntwithbalancofpaymntsstatistics.ThLBSalsoproidinformationonthcurrncycompositionofbanks’balancshts,whichinturnallowstocalculatquartrlyflowsthattakxchangratfluctuationsintoaccount.Inaddition,thstimatdquartrlyflowsarstatisticallyadjustdforbraksinsris.Furthrmor,thLBSfaturbrakdownsbyborrowingcountrpartysctorandcountry.Inthcontxtofourmpiricalxrciss,thsadditionalbrakdownsallowustodistinguishamongborrowrsfromdiffrntsctorsandcountris.Figur4prsntsanoriwthtriangularrlationshipbtwnthdollar,thcrosscurrncybasisanddollardnominatdcrossbordrbanklnding.Thrdlinshowsquartrlychangsinthbroaddollarindx,thblulintracksthfirstprincipalcomponntofquartrlychangsinthfiyarcrosscurrncybasisforG10currncis,2andthbarsrflctgrowthratsindollardnominatdcrossbordrbanklnding.Changsinthbroaddollarindxar75%corrlatdwithchangsinthfirstprincipalcomponntofthcrosscurrncybasis,and56%corrlatdwiththgrowthratsofdollardnominatdcrossbordrlnding.Thrfor,astrongrdollarisassociatdwithgratrCIPdiations(orahighrpricforbalancshtcapacity)andwithlowrgrowthratsindollardnominatdcrossbordrbanklnding.1Inrsultsthataraailabluponrqust,wobtainrysimilarmpiricalfindingsbyusingothraltrnatidollarindics,suchasthBISnominalffctixchangratindxorasimplaragofalldollarxchangratsforoursamplcurrncis.2Wprformaprincipalcomponntanalysisonquartrlychangsinth5yarcrosscurrncybasisforthAUD,CAD,CHF,DKK,EUR,GBP,JPY,NOK,NZDandSEKandfindthatthfirstprincipalcomponntxplains53%oftotalariations.8USdollardnominatdcrossbordrbanklndings.thUSdollarindxFigur3Compltsampl120quartrrollingwindowrgrssion,allsctors21CorsthQ12002Q32015priod.LndingrfrstothxtnsionofloansandholdingsofdbtscuritisbyBISrportingbanks,whilallborrowrs(thtotalofbanksandnonbanks)rsidoutsidofthUnitdStats.Thblacklinisafittdrgrssionlin.PositichangsinthforignxchangratindicatanapprciationofthUSD.Forprsntationalpurposs,outlirswithFXratchangsxcding5%inabsolutaluhabndroppdfromthpanl,butnotfromthrgrssions.2ExtnsionofthLHPstaticrgrssiontoarollingwindowframworkof20quartrsprstimationforcrossbordrlnding(loansanddbtscuritis)dnominatdinUSdollars.Thpositonofdotswithrspcttothrticalaxisgisthcofficintstimat,whilthhorizontaltimlinrfrstothlastpriodofthrollingwindow.Rddotsindicatangaticofficintstimatthatisstatisticallysignificantatth10%ll.Sourcs:BISLocationalBankingStatistics;BISffctixchangratindics;BIScalculations.Crosscurrncybasis,USdollarindxandcrossbordrlndingFigur4ThbarsshowprcntaggrowthratsoftotalcrossbordrlndingdnominatdinUSdollars.ThrdlinplotsquartrlychangsinthbroadUSdollarindxinprcntagpoints,whilthblulinplotsthfirstprincipalcomponntofquartrlychangsinth5yarcrosscurrncybasisforG10currncisxprssdinbasispoints.Sourcs:Bloombrg;BoardofGornorsofthFdralRsrSystm;BISLocationalBankingStatistics;BIScalculations.3ModlWsktchamodlofabanklocatdoutsidthUnitdStats,butwhichhassignificantUSdollarbusinss.Thbank’sdollarbusinsshastwoparts.Thfirstislndingindollarstoborrowrs,suchasmrgingmarktcorporats,whichhartaindsomcurrncymismatchonthirbalancshts.Forconcrtnss,considrthdollarborrowrstobmrgingmarktproprtydloprswhoborrowdollarsinordrtofinancdomsticralstatdlopmnts.Thscondlmntofthbank’sdollarbusinssistoproiddollarfundinginthFXswapmarkt.ThbanklndsdollarsinthFXswapmarktinxchangfordomsticcurrncy.Atmaturity,thbankrcisdollarsinxchangforthdomsticcurrncy.ThbankisrisknutralandisaprictakrinthdollarloanmarktaswllasinthFXswapmarkt.ThrisknutralbankmaximissprofitssubjcttoVaRconstraint,tobdscribdblow.Wadoptthfollowingnotation.Dnotbya1thdollaramountlnttomrgingmarktcorporatsanddnotbya2thdollarfacaluofFXswapclaims.Thbank9dosnotholdanyothrasst.Thbalancshtidntityofthbankis:a1+a2=+d(3)whristhbank’sbookquityindollartrmsanddisthdollaraluofdbtfinancing.Assumforsimplicitythatthdollarfundingdcanbraisdatthrisklssdollarratandthatthrisklssdollarratiszro.Thprofitrofthbankisthnginby:r=a1r1+a2r2(4)whrr1isthgrossrturnondollarloanstocorporatsandr2isthgrossrturntothbankinthFXswap.ThbankmaximissxpctdprofitssubjcttoaVaRconstraint.Thbank’soptimisationproblmcanbwrittnas:maxa1,a2E(r)subjcttoVaR≤whrE(r)isthxpctdprofitofthbank.Wlimitattntiontothchoicofa1anda2only,asthbank’sdbtfundingdfollowsfromthbalancshtidntity(3).AssumthatthVaRisamultiplαofthstandarddationofportfoliorturnσrsothatVaR=ασr.Thconstraintis:ασr≤(5)Wtransformthconstraintbysquaringbothsidsanddiidingbyα2togi:σ2r≤(α)210Thn,writthLagranganas:L=E(r)−λ(σ2r−(α)2)=E(r)−λσ2r+λ(α)2(6)whrE(r)isthxpctdrturnofthbank’sasstportfolio,λisthLagrangmultiplirofthVaRconstraintandσ2risthariancofthbank’sprofit.ThthirdtrminthLagranganLin(6)dosnotdpndona1ora2,andsowilldropoutwhnthfirstordrconditionistakn.ThfirsttwotrmsofthLagranganisaquadraticxprssion.Wusthshorthand:µ1=E(r1),µ2=E(r2),anddnotthcoariancmatrixofrturnsas:Σ=σ21σ12σ12σ22Wcanintrprtµ2−1asthsizofthCIPdiationinthabsolutalu(orthngatiofthcrosscurrncybasis).ItisthxpctdpayofffromlndingdollarsinthFXswapmarkt.Thinstorfacsmarktomarktrisk,andsothrissomriskinthtrad.ThfirsttwotrmsofthLagrangancanbwrittnasthquadraticform:[a1a2]µ1µ2−λ[a1a2]σ21σ12σ12σ22a1a2.Thfirstordrconditionis:µ1µ2=2λΣa1a2,Solingfora1anda2,thoptimalportfoliois:11a1a2=12λΣ−1µ1µ2.(7)Manwhil,thariancofprofitisginbythquadraticform:σ2r=a′Σawhra′isthtransposofa.From(7)whaσ2r=a′Σa=14λ2µ′Σ−1µFinally,fromthVaRconstraint,whaσ2r=(α)2.Thus,14λ2µ′Σ−1µ=(α)2ThLagrangmultiplirofthtransformdconstraintis:λ=α2µ′Σ−1µ(8)Thx