IMF-宏观审慎压力测试模型综述（英）-2023.8

MacroprudentialStress-TestModels:ASurvey

DavidAikman,DanielBeale,AdamBrinley-Codd,GiovanniCovi,Anne-CarolineHüserandCaterinaLepore

WP/YY/173

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©2023InternationalMonetaryFund

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IMFWorkingPaper

MonetaryandCapitalMarketsDepartment

ATCr0prUdentiTIMtreSS-SeStA0deIS:昌MUrVey

PreparedbyDavidAikman,DanielBeale,AdamBrinley-Codd,GiovanniCovi,Anne-CarolineHüserand

CaterinaLepore*

AuthorizedfordistributionbyHirokoOura

August2023

IMFWorkingPapersdescriberesearchinprogressbytheauthor(s)andarepublishedtoelicit

commentsandtoencouragedebate.TheviewsexpressedinIMFWorkingPapersarethoseofthe

author(s)anddonotnecessarilyrepresenttheviewsoftheIMF,itsExecutiveBoard,orIMFmanagement.

ABSTRACT:Inthispaper,wesurveytherapidlydevelopingliteratureonmacroprudentialstress-testing

models.Thescopeofthesurveyincludesmodelsofcontagionbetweenbanks,modelsofcontagionwithinthewiderfinancialsystemincludingnon-bankfinancialinstitutionssuchasinvestmentfunds,andmodelsthat

emphasisethetwo-wayinteractionbetweenthefinancialsectorandtherealeconomy.Ouraimistwo-fold:first,toprovideareferenceguideofthestate-of-the-artforthosedevelopingsuchmodels;second,todistilinsights

fromthisendeavourforpolicy-makersusingthesemodels.Inourview,themodellingfrontierfacesthreemain

challenges:(a)ourunderstandingofthepotentialforamplificationinsectorsofthenon-bankfinancialsystem

duringperiodsofstress,(b)multi-sectoralmodelsofthenon-bankfinancialsystemtoanalysethebehaviouroftheoveralldemandandsupplyofliquidityunderstressand(c)stresstestingmodelsthatincorporate

comprehensivetwo-wayinteractionsbetweenthefinancialsystemandtherealeconomy.Emerginglessonsforpolicy-makersarethat,foragiven-sizedshockhittingthesystem,itseventualimpactwilldependon(a)the

sizeoffinancialinstitutions'capitalandliquiditybuffers,(b)theliquidationstrategiesfinancialinstitutionsadoptwhentheyneedtoraisecash,and(c)thetopologyofthefinancialnetwork.

JELClassificationNumbers:

G21,G22,G23,G32

Keywords:

Stresstesting;system-widemodels;contagion;systemicrisk;market-basedfinance;real-financiallinkages;macro-prudentialpolicy.

Author’sE-MailAddress:

CLepore@

2

Contents

1Introduction

3

2Abriefprimeroncontagionchannels

4

3Contagioninthebankingsector

5

3.1Directcontagionviathesolvencychannel

6

3.2Directcontagionviathefunding-liquiditychannel

8

3.3Indirectcontagionviathemarket-liquiditychannel

9

3.4Contagioninvolvingcollateral

11

3.5Interactionofdirectandindirectcontagion

12

4Contagioninthenon-bankfinancialsystem

14

4.1Overviewofthenon-bankfinancialsystem

14

4.2Contagionmechanismsbetweennon-bankfinancialinstitutions

15

4.2.1Directcontagionviathesolvencychannel

15

4.2.2Directcontagionviathefunding-liquiditychannel

16

4.2.3Indirectcontagionviathemarket-liquiditychannel

16

4.2.4Contagioninvolvingcollateral

17

4.3Sector-specificapproaches

18

4.3.1Fundstresstesting

18

4.3.2CCPstresstesting

23

4.4Multi-sectormodelsofthefinancialsystem

23

5Feedbacksbetweenthefinancialsectorandtherealeconomy

25

5.1Theory

25

5.2Empiricalevidence

26

5.2.1Thebankresiliencechannel

26

5.2.2Theborrowerresiliencechannel

26

5.3Incorporatingmacrofinancialfeedbacksintomacroprudentialstresstestingmodels

26

5.3.1DSGEmodels

27

5.3.2Semi-structuralmodels

30

5.3.3Networkmodels

31

5.3.4Otherapproaches

31

6Outputsfromsystem-widemodelling

34

6.1BalanceSheet-BasedIndicators

34

6.1.1Networkmeasures

34

6.1.2Model-basedmeasures

34

6.2PrudentialTools

35

7Concludinglessonsfromthissurvey

38

7.1Lessonsforpolicymakersandforthedesignofsupervisorystresstests

38

7.2Lessonsforresearchersdevelopingmacroprudentialstresstestingmodels

39

References

41

3

“Perhapsatsomefuturedate,asfinancialinstitutionsandriskmanagementsystemsevolve,aggregatestresstestswillbefoundtobeawaytotapintothisnewdatasourcethatwouldprovideforward-lookinginformation aboutaggregateriskexposuresthatwouldbeofusetofinancialfirms,centralbanks,andotherfinancialregulators.”

CommitteeontheGlobalFinancialSystem,April2000.

1Introduction

Thedevelopmentofframeworkstoconductsystematicstresstestsofthebankingsystemhasbeenoneofthemostimportantinnovationsinfinancialregulationinthepost-GlobalFinancialCrisisera.Theseframeworkshaveinformedthecalibrationofbankcapitalrequirementsinmanyjurisdictions;theyhavealsoprovidedregulatorswithvaluableforward-lookinginformationontheresilienceoftheirbankingsystemstoshocksnotpreviouslyexperienced,includingtheimpactofadisorderlyBrexitandtheCovid-19pandemic.

Alongsidethis,therehasbeensignificantresearcheffortinrecentyearstoexpandourunderstandingofhowfinancialsystemsbehaveunderstressviathedevelopmentofmacroprudentialstresstestingmodels.Whilethetypicalsupervisorystresstestcentersaroundanassessmentofthedirect,first-roundimpactofagivenstressscenarioonindividualbanks’profitabilityandcapitalusingacombinationofbank-reportedestimatesanddesktopanalysisbytheregulator,theresearcheffortbycontrasthasfocusedonmodellingfeedbackloopswithinthefinancialsystemthatcanamplifytheimpactofanyexternalshock.Thisisacomplementaryeffort,whichovertimeitishopedwillprovideregulatorswithrichertoolsforidentifyingvulnerabilitiesandevaluatingpoliciesdesignedtomitigatesystemicrisk.

Inthispaper,wesurveytherapidlydevelopingliteratureonmacroprudentialstresstestingmodels.Ouraimistwo-fold:first,toprovideareferenceguideofthestate-of-the-artforthosedevelopingsuchmodels;second,todistilinsightsfromthisendeavourforpolicymakersthatmayinformsupervisorystresstests.Relativetootherrelatedsurveys(see

Aymannsetal.

(2018),

Andersonetal.

(2018)and

Greenlawetal.

(2012)),ourmain

contributionistotakestockofprogressindevelopingmodelsthatextendbeyondthebankingsystemtocapturecertainsectorsofthebroaderfinancialsystem.Inthissenseoursurveyanswersdirectlytorecentcallstobetterunderstandthenon-bankfinancialsystem,thefinancialsystemasawholeandreal-financiallinkages(

Cunliffe,

2020;

GieseandHaldane,

2020

).

Forthepurposesofthissurvey,wedefineamacroprudentialstresstestingmodelasonethatpermitsanexaminationoftheresilienceofthefinancialsystem–orcomponentsofit–understress,takingintoaccountplausiblebehaviouralresponsesofinstitutionswithinthesystemandtheknock-onconsequencesofthoseactionsforothersinstitutionsinthesystem.Feedbackandamplificationchannelsarethereforefrontandcenter.Thescopeofthesurveyincludesmodelsofcontagionbetweenbanks,modelsofcontagionwithinthewiderfinancialsystemincludingnon-bankfinancialinstitutionssuchasinvestmentfunds,andmodelsthatemphasisethetwo-wayinteractionbetweenthefinancialsectorandtherealeconomy.Wefocustotheextentpossibleonquantitativemodelsgroundedingranularbalancesheetdatathatallowuserstoexaminethemagnitudeofdifferentchannels.Thatsaid,thestateoftheliteratureissuchthatmuchoftheworkavailableiscastinstylised,conceptualmodelsandsectionsofthesurveyreviewinsightsfromsuchmodels.

Overall,theliteraturewesurveyisatdifferentstagesofmaturitydependingontheareaofthefinancialsys-temstudied.Theliteratureoncontagiondynamicsinthebankingsystemisrelativelywellestablished.Thisisparticularlysoformodelsfocusedonsolvencycontagionoperatingviainterbankexposuresandmodelsfocusedontheinterplaybetweenassetfiresalesandleveragerequirements.Incontrast,theliteratureoncontagionchannelsoperatingviafundingliquidity,includingliquidityhoardingeffectsanddynamicsoperatingviacollat-eralisedfundingmarkets,remainsinitsinfancy.Arecentstrandofthemacroprudentialstresstestingliteratureattemptstomodelfeedbackandamplificationchannelsinthebroaderfinancialsystem,withaparticularem-phasisonthepotentialforanamplificationloopoperatingviafiresalesandredemptionsintheinvestmentfundsector.Despitethisrecentattention,thepotentialforcontagioninthenon-bankfinancialsectorremainslesswellunderstood.Finally,whilethereisalong-establishedliteratureexaminingtheimplicationsofembeddingfinancialfrictionsandcrudebankingsystemsinmacroeconomicdynamicgeneralequilibriummodels,veryfewpaperstodatehaveattemptedtoincorporatesuchreal-financialsectorlinkagesinamacroprudentialstresstestingsetting.

Giventhestateoftheliterature,itisperhapstooearlytoexpecttofindrobust,widelyagreeduponresults.Thatsaid,therearesomenoteworthyfindingsthatarecommoninmanyofthepaperswesurvey.First,estimatesofsolvencycontagionlossesinthebankingsystemtendtobesmallwhenmodelsarecalibratedtocurrentbalancesheetsandthecurrentconfigurationofinterbankexposures.Intuitively,thisreflectsthepost-GlobalFinancialCrisisbuildupinequitycapitalinthebankingsystemandthereductioninthescaleofinterbankexposuresoverthesameperiod.Anothercommonfinding,albeitanimplicitone,istheimportance

4

ofusablebuffersofcapitalandliquidityformitigatingthescaleofcontagionlosses.Forinstance,thereisalargedispersioninthemagnitudeoflossestimatesinthemodelsoffiresaleswesurvey.Thisdispersioncanbetracedtodifferentialassumptionsaboutcapitalbufferusabilityandpriceimpactestimation,withthelargestlossestimatesbeinginmodelsthatassumebankshavefixedleverageratiotargetswithnobuffer.Third,anemergingresultsfromtheliteratureonfundstresstestingisthattheseverityoftheoutcomesdependonassumptionsabouttheliquidationstrategyoffundsafteraredemptionshock.

Section

2

providesanoverviewofthedifferentcontagionchannelsthatcanoperateinthefinancialsystem-includingbanksandnon-banks-andbetweenthefinancialsystemandtherealeconomy.Havingaclassificationofthechannelsupfrontwillprovideanoverarchingstructuretothesurveyandhelpthereadernavigatethefollowingsections.

Section

3

reviewsthemodellingofcontagionchannelswithinthebankingsector.Wedonotaimtoprovideacompleteoverviewoftheliteratureinthisarea,andreferreaderstosurveysby(

GlassermanandYoung,

2016;

H¨user,

2015

)forsuchdetail.Ratherouraimistofixideasabouthowparticularfeedbackandamplificationmechanismsoperateinarelativelywellunderstoodsetting.

Section

4

beginswithabriefoverviewofthemaintypesofnon-bankfinancialinstitutions(NBFI),settingouttherolestheyplayandthetraditionalstructureoftheirbalancesheets.Wethenexaminewhichofthecontagionchannelsdescribedearlierapplytotheseentities;wereviewthenascentliteraturethathasattemptedtomodelthesecontagionchannels,andwepointthereadertoaccountsofhistoricalexamplesofsystemicstressinthissector.Followingthis,wezoominonrecentadvancesinmodelsoffundandcentralclearingcounterparty(CCP)stresstesting.Andwefinishthesectionbycoveringthehandfulofpioneeringmodelsofsystem-widestresstesting,modelsthatincludemultiplesectorsandtheirinteractions.

Section

5

coversmodelsofreal-financiallinkages.Thissectiontakesstockofmodelstoassesscontagionduetoreal-financiallinkages.Whilethelinkagesbetweentherealeconomyandthefinancialsectorhavebeenthe

subjectofagreatdealofstudywithinthemacroeconomicsliterature,1

veryfewmodelsattempttoincorporatetheselinkagesintomacro-prudentialstress-testingmodels.Withthatinmind,thatsectionwillfirstprovideabriefandselectiveoverviewofthe(verybroad)literaturecoveringtherelevantlinkagesbetweenthefinancialsectorandrealeconomy.Thenitwilldiscussmodelswhichincorporatetheselinkages,organizingtheliteratureintothedifferentmodellingapproachesusedanddiscussingtheirrelativemerits.

Section

6

discussesindicatorsthatcancaptureandsummarisethesourcesofriskforbanksandotherfinancialinstitutions.Overall,thissectionaimstoshedlightuponthedifferenttypesofoutputsthatamacroprudentialstresstestingmethodologymaydeliver.

Intheconcludingsectionofthepaper,wedrawoutsometake-awaylessonsfromthissurvey.Wefirstdiscusslessonsforfinancialstabilitypolicymakers,wheretheemphasisisonemerginginsightsfromthisliteraturethatcaninformthedesignofsupervisorystresstests.Wethendiscusslessonsforresearchersinvolvedindevelopingmacroprudentialstresstestingmodels,whereouremphasisisprioritisingaddressinggapsinthisliteraturethatimpedetheutilityofthesemodelsforinformingpolicy.

2Abriefprimeroncontagionchannels

Inthissectionweprovideanoverviewofthedifferentcontagionchannelsthatcanoperateinthefinancialsystem-includingbanksandnon-banks-andbetweenthefinancialsystemandtherealeconomy.Havingaclassificationofthechannelsupfrontwillprovideanoverarchingstructuretothesurveyandhelpthereadernavigatethefollowingsections.

Theacademicliteraturehasidentifiedtwokeytypesoflinksalongwhichcontagioncanpropagate:directandindirectlinks.Examplesforthosedirectlinksareloanexposuresintheinterbankmarket,leveragedinvestmentfundsdebt-likeliabiltiesheldbyotherfinancialinstitutionsandbanks’loanstonon-financialfirms.Indirectlinksincludeforexampleoverlappingportfoliosorcorrelatedassets.

Directlinksconnectborrowersandlenders,andhencethetriggerforcontagioncanbecausedbythedistressofeithertheborrowerorthelender.Iftheborrowerisindistress(forexamplebecauseitdefaults),thisimpliesitisunabletorepayitsliabilitiestoitscounterparties.Sincetheseliabilitiesareotheragents’assets,theseagentsmaynowgetintrouble,therebyaffectingtheircounterparties.Thisishowadefaultcascadestarts.Thiscontagionchannelworksviacounterpartyrisk,wheretheborrowercannotpaybackthelender.Itisthereforecalledinthissurveydirectcontagionviathesolvencychannel.Ifthelenderisindistress(forexamplebecauseofaliquidityshock),itmaydecidetoincreasetheircostoflendingorpulltheirfundingaltogether.Thisisturnwillcausealiquidityshockfortheborrowerwhichmayalsousesimilardefensiveactionswithhisowncounterparties.Thiscontagionchannelworksviafundingriskandisthereforecalledinthissurveydirectcontagionviathefunding-liquiditychannel.

Indirectlinksconnectagentsholdingthesameorsimilarassetsviachangesinassetprices.Thiscontagionchannelworksviamarket-liquidityriskandisthereforecalledinthissurveyindirectcontagionviathemarket-

1Forarecentsurvey,see

ClaessensandKose

(2018)

.

5

liquiditychannel.Thiscontagionchannelcanoperateoverdifferenttimescales.Onlongtimescales,acrossyearsordecades,itconcernsagents-suchasinsurancecompanies-withlonginvestmenthorizonswhocanadjusttolongtermtrendsinassetprices.Onashortertimescale,suchasdays,weeksandmonths,contagioncanbecausedbyfastsalesofassetsatdistressedprices,oftencalledfiresales.Assetsalescanbedrivenbyinvestors’redemptions,suchasinthecaseofinvestmentfunds,orviolationofcapitaladequacyconstraints,suchasinthecaseofbanks,forexample.

Inthefollowingsections

3,

4

and

5

weprovidemoredetailsonthesechannelsforeach‘system’undercon-sideration,beginningwiththe‘narrowest’systemi.e.,bankingsystem,thenthewiderfinancialsectorincludingnon-banks,andfinallyturningtointeractionsbothtoandfromtherealeconomy.Whendiscussingcontagioninthebankingsectorandbeyondbanks,insections

3

and

4

respectively,wereserveaseparatediscussionforcontagionchannelsinvolvingcollateral.Thisisbecausetheuseofcollateralfortradingandlending,whilemitigatingcounterpartyrisk,cangeneratebothdirect(viathefunding-liquiditychannel)andindirect(viafiresales)contagionchannels.

Figure1:Contagionchannels.

3Contagioninthebankingsector

Inthissection,wefirstreviewthestateoftheartinmodelsthatcapturecontagionchannelsforthebankingsectorinisolation.Wethenreviewpapersthatattempttomodeldifferentcontagionchannels’interaction.Ourfocusisonmodelsthatcanbecalibratedwithgranulardatafrombanks’regulatoryreturns,andhencebeusedtoprovidequantitativeanalysistoinformfinancialstabilitypolicy.Onealternativeapproachtomodellingcontagioninbankingsystemsistousereconstructedinterbanknetworkdata;anotheristousefinancialmarketdatatoinferthestrengthofbilateralnetworkconnectionsbetweenbanks,Wereferthereaderto

Gandyand

Veraart

(2017)forareviewoftheliteratureonreconstructedinterbanknetworks,andto

DieboldandYılmaz

(2014),

Engleetal.

(2015)and

TobiasandBrunnermeier

(2016)forareviewoftheliteratureonusingfinancial

marketinformationtoestimatenetworks.

TherehasbeenanotableincreaseintheavailabilityofgranularbankingsectordatasincetheGlobalFinancialCrisisof2008-2009,andthishasenabledmodellerstomakesignificantprogressinquantifyingtheimportanceofthesecontagionchannels.Inthissurvey,wefocusonreviewingsuchprogressintheacademicliteratureoncontagionmodelsinthebankingsector.Areviewofthemacro-prudentialstresstestingframeworksofregulatoryandmonetaryauthoritiesisoutofthescope.Wereferthereadersto

Andersonetal.

(2018)

.

Overall,itisimportanttobemindfulofthefactthatthesemodelsarehighlysensitivetodifferentmodellingassumptionsandthatsomeparameters(e.g.themarketliquidityofspecificassetclasses)arehardtocalibrate.Hence,pointestimatesfromthesemodelsshouldbeinterpretedwithcaution.Amoreprofitableapproachforusingsuchmodelsinapolicysettingistheframeworktheyprovideforunderstandingwhatmighthappen.Aretherereasonablecalibrationsofkeyparameterswhereseverecontagionoccurs?Whatwouldweneedtobelieve

6

aboutshocks,balancesheetpositionsormarketliquidityforthistobethecase?Whichpolicyinterventionsaremostsuccessfulatsteeringthesystemawayfromsuchdireoutcomes?

3.1Directcontagionviathesolvencychannel

Theseminalpaperthatintroducedamodellingframeworkforanalysingdirectcontagionviathesolvencychannelintheinterbankmarketis

EisenbergandNoe

(2001)

.Thepaperconsideredanetworkoffirmswithinterfirmdebtclaimsandanalysedthevectorofclearingpaymentsthatoccuruponthedefaultofonefirm.Inthismodel,whenabankisnotabletorepayitsdebtinfullitdefaults.Thisinturncantriggeracascadeoffurtherdefaults,whenthecreditorsthatfailtoreceivesomepaymentsfromtheircounterpartiesarenotabletopaytheirowncreditors.Theauthorsshowthataclearingpaymentvectoralwaysexistsand,undermildconditions,isunique.Thismodelhasbeenhighlyinfluentialandhassetthebasisforanextensiveliteratureapplyingitsclearingmechanismstothestudyofdifferentaspectsofaninterconnectedfinancialsystem.

The

EisenbergandNoe

(2001)modelis,however,basedonsomestrongsimplifyingassumptions

.Inpar-ticular,itassumesthatwhenabankdefaults,thefullfacevalueofitsremainingassetsisdistributedtoitscreditorspro-rata.Inreality,bankdefaultsgeneratesignificantlegalcostsandtherearesubstantialdelaysinpayingbackcreditors

.2

Furthermore,claimsonabank’sassetshavedifferentseniorityinpractice.

Elsinger

etal.

(2009)hasconsideredtheimplicationsofaccountingforcrossholdingsofequityaswellasadetailed

senioritystructureofdebtintheinterbanknetwork.Inthismodelthevalueofequityanddebtofthebanksaredeterminedendogenously.Whilein

EisenbergandNoe

(2001)equityvaluesareconvexanddebtvaluesare

concaveintheexogenousincome,thisisnotthecaseanymore.

RogersandVeraart

(2013)haveextendedthisframeworktoallowfornon-zerobankruptcycosts

.Intheirmodelwhenabankdefaultsitdoesnotrealizethefullvalueofitsassetsbutonlyafraction.Introducingdefaultcostsgenerateincentivesforrescueconsortia,thatisthereisabenefitforsolventbankstorescueinsolventones.However,theauthorsnotethatgiventhepracticalchallengesinimplementingarescueconsortium,alenderoflastresortwouldberequiredasanappropriatecoordinationmechanisminthisframework.Atthesametime,asalsodiscussedin

Elliottetal.

(2014)whichaccountsfordiscountinuouslosseswhenbanks’valuefallsbelow

thefailurethreshold,non-zerobankruptcycostsalsocreateincentivesforabanktoincreaseitsfailurecostsandmakeitsfailuremorelikely,inordertoincreaseitsnegotiatingpower.

Elliottetal.

(2014)adoptsthemodeltostudyhowcascadesoffailuresdependonthenetworkstructure,topic

highlydebatedintheliterature.3

Inparticular,theyareinterestedinstudyinghowcascadesoffailuresdependonthenetworkintegration(i.e.dependenceoncounterpartiesintermsoflevelofexposures)anddiversification(i.e.thenumberoffirms’cross-held).Increasingintegrationleadstoincreasedexposures,hencecanincreasethelikelihoodofacascadeonceaninitialfailureoccurs.However,itcanalsodecreasethelikelihoodofobservingafirstfailure.Diversificationincreasesinterdependenciesinthenetwork,allowingcontagiontocascade,butit

alsomakesfirmslesssensitivetootherfirms’failures.4

Asreportedin

H¨user

(2015),whilethereisagreement

ontheexistenceofthistrade-off,theliteraturehasnotprovidedaconclusiveanswerontheultimateeffectofconnectivityanditsdesirablelevel,answerwhichdependsonotherfactorsaswell.Forexample,anadditionalfactortoconsiderispotential‘errorsinthestructureofthecontractnetwork’asdefinedby

Battistonetal.

(2016),thatisinformationregardinghowmanycontractsabankhasandwithwhichcounterpartiesmaybe

incorrect.Higherdiversificationcanresultinmarketparticipantsandregulatorsknowinglesspreciselytheprobabilityofsystemicdefault.Anotherfactortoconsideristhesizeoftheinitialshock.

Acemogluetal.

(2015)findsthatamorediversifiednetworkofinterbankliabilitiesleadstoalessfragilefinancialsystem,in

thepresenceofrelativelysmallshocks.However,whenshocksbecomeslargermoreinterconnectednetworkstructurescanfacilitatecontagionandcreateamorefragilesystem.Thisisbecauseunderlargeshocksexcessliquidityofthebankingsystemcanbecomeinsufficienttoabsorblosses,andinalessdiversifiednetworklossesaresharedwiththecreditorsofthedistressedbanksonly,protectingtherestofthebanks.Thisfindingissimilartotheoneby

Georg

(2013),whenbankscanoptimizetheirbalancesheetandasaresulttheinterbanknetwork

structure,andisconsistentwiththe‘robust-yet-fragile’propertyasexplainedby

Haldane

(2009).

Glasserman

andYoung

(2015)adoptsafull-fledgeddistributionofshocksandanalyzetheprobabilityofdefaultcascades

thatareattributabletonetworkconnections.Theyshowthat,whilesolvencycontagionlosseshaveonlylimited

2Asanexampleofsuchcosts,

Denisonetal.

(2019

)estimatethetotalvaluedestructionassociatedwithLehmanBrothers’Chapter11Bankruptcy