Ebook paper new10july...AN INTEGRATED APPROACH TO COST-RISK ANALYSIS IN PUBLIC DEBT MANAGEMENT 2 An Integrated Approach to Cost-Risk Analysis in Public Debt Management Massimo Bernaschi

MASSIMO BERNASCHI,IAC - CNR

ROBERTO MOREA, SOGEI

LUCIO SARNO, Cambridge Judge Business School,

University of Cambridge

FABRIZIO TESSERI, MEF - Department of Treasury

FEDERICA VERANI, MEF - Department of Treasury

DAVIDE VERGNI, IAC - CNR

An IntegratedApproach toCost-Risk Analysisin Public DebtManagement

ANINTEGRATEDAPPROACHTOCOST-RISKANALYSISINPUBLICDEBTMANAGEMENT

1

IndexAbstract.........................................................................................................................................2

Introduction...................................................................................................................................3

MathematicalModel......................................................................................................................4

Interestratescenarios....................................................................................................................4

Modellingthedynamicsofinterestrates..............................................................................................5Anchoringtheyieldcurvesusingexogenousinformationonshorttermrates.....................................9Thedebtportfoliocashflowandthebalanceconstraint....................................................................11TheESA2010costfunction..................................................................................................................12AnalyticalstudyontheESA2010evolution.........................................................................................13

TheCost/RiskAnalysis..................................................................................................................14

Classicalmeasuresofcostandrisk......................................................................................................15Evolutionmeasuresofcostandrisk.....................................................................................................16

CVAmodelling..............................................................................................................................17

SoftwareArchitecture..................................................................................................................20

Conclusions..................................................................................................................................22

Appendix......................................................................................................................................22

Generationofinterestratecurves.......................................................................................................22

References...................................................................................................................................27


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AnIntegratedApproachtoCost-RiskAnalysisinPublicDebtManagement

MassimoBernaschi1,2RobertoMorea3LucioSarno2,4FabrizioTesseri2FedericaVerani2DavideVergni1,2

AbstractPublicdebtmanagementrequiresaccurateanalysisofthecostandtheriskassociatedwithfeasiblechoicesofpublicdebtportfolios.TheItalianTreasuryemploysanin-house,custom-builtsetofmodelsandsoftwaretoolsaimedatselectingportfoliosofsecuritiesthatsatisfythegovernment’sborrowingneeds while fulfilling all the relevant regulatory constraints, thereby delivering suitable tradeoffsbetweencostandriskonanefficientfrontier.Inthispaper,wepresentadescriptionoftheapproachfollowedbytheItalianTreasury,includingthestudyofthedynamicsofthecashflowgeneratedbythepublicdebtandthedevelopmentofstochasticmodelsfortheevolutionofthemainriskfactors(i.e.,interest ratesand inflation). Then,wedescribehowseveraldynamicmetricsof costand risksareintegratedwithinascenariogenerator inamodularsoftwarepackagecalledSAPE(SistemaAnalisiPortafoglidiEmissione–SystemfortheAnalysisofPublicDebtIssuances).Thissystemsupportsthepublic debtmanager by providing accurate quantitative estimates of the expected effects of theirchoicestakingintoaccountnotonlyshockstointerestratecurvesbutalsoexogenousforecastsaboutthefuturebehavioroftheriskfactors.Thesoftwarecanalsobeusedasanaccountingtoolfortheoutstanding securities of the Italian public debt, and includes various satellite modules for theevaluationofotherrelevantmetrics,suchastheCreditValueAdjustmentforderivativeinstruments.

The latest version of the paper has been presented during the September 2019 “Public DebtManagement Network Conference” organized in Paris by the PDM - Public Debt Management -Network,aninitiativefosteredbytheOECD,theItalianTreasuryandtheWorldBanktodisseminatePublicDebtmanagementtechniques;itwillbepublishedintotheproceedingsoftheconference.

1IstitutoperleApplicazionidelCalcolo“M.Picone"(IAC)-CNR2DipartimentodelTesoro-Ministerodell’EconomiaedelleFinanze3Sogei4CambridgeJudgeBusinessSchool,UniversityofCambridge


3

Introduction

Publicdebtmanagementistheprocessofdesigningandexecutingastrategyforthemanagementofgovernment debtwith theultimate goal to satisfy the financing needs of the government. In thisprocessitisnecessarytoconsideranumberofrelevantissues,relatedtocost,risk,maturitystructure,andchoiceofinstrumentsfromabroaduniverseofassets.Debtmanagersgenerallyrelyonacost-riskanalysisoftheirstockofliabilitiestomakechoicesofdebtinstrumentsandportfoliocomposition.

Thiskindofanalysisrequiresabroadsetoftools.Akeycomponentofthetoolkitrequiredisaformalframeworkthatallowstoproducestochasticsimulationsofmultipletermstructuresofinterestratesand macroeconomic variables. From these simulations, which constitute a set of possible futurescenarios forthe interestrates’evolution,debtmanagerscandeterminetherelationshipbetweencostandrisktheyfaceunderdifferentfeasibleportfolios.Theprocessis,inpractice,mademuchmorecomplexbyanumberofexogenousfactors(i.e.,factorsnotunderthecontrolofdebtmanagers)thatinclude,forexample,theinfluenceofcentralbanksonthepathofinterestrates,politicalrisk,globalrisksofvariousform,andregulatoryconstraints.Surprisingly,theliteratureinthisareaisscant,withfewpapersattemptingtodescribethemodellingstrategiesandchallenges faced in theprocessofevaluation of the cost and risk trade-off associated with different financing strategies. Stochasticsimulationmodellingisusedinpublicdebtdecisionmakingbymanygovernmentsaroundtheworld.JusttociteafewwementionCanada(seeBolder2002,2003,2011),theSwedishNationalDebtOffice(seeBergstrom,HolmlundandLindberg2002), theNationalBankofDenmark (2005), theUKDebtManagementOffice(seePickandAnthony2006),theFrenchTreasury(seeRenneandSagne2008),theTurkishTreasury(seeBalibekandMemis2012)andtheU.S.Treasury(seeBeltonetal.2018).Foran overview of stochastic debt strategy simulationmodelling inOECD countries, see Risbjerg andHolmlund(2005)whileageneraldiscussionondebtstrategiesinEuropecanbefoundinBalling,GnanandHoller (2013). There are alsomore academicworks on public debtmanagement that can beconsideredasfoodforthoughtfordebtmanagers,seeforexampleDottoriandManna(2016).Finally,concerning practical applications, mathematical methods on cash flow forecasting and cost-riskanalysis for public debt has been endorsed by the World Bank, which has developed a toolkitspecificallydesignedbyitsdebtandriskmanagementexperts(seePanzer2015).

The Italian Treasury employs an in-house, custom-built set models and software tools aimed atselectingportfoliosofsecuritiesthatsatisfythegovernment’sborrowingneedswhilefulfillingalltherelevantregulatoryconstraints.Inthispaper,wepresentadescriptionoftheapproachfollowedbytheItalianTreasury,includingthestudyofthedynamicsofthecashflowgeneratedbythepublicdebtandthedevelopmentofstochasticmodelsfortheevolutionofthemainriskfactors(i.e.,interestratesandinflation).Then,wedescribehowseveraldynamicmetricsofcostandrisksareintegratedwithina scenario generator in a modular software package called SAPE (Sistema Analisi Portafogli diEmissione–SystemfortheAnalysisofPublicDebtIssuances).Thissystemsupportsthepublicdebtmanagerbyprovidingaccuratequantitativeestimatesoftheexpectedeffectsoftheirchoicestakingintoaccountnotonlyshocksto interestratecurvesbutalsoexogenousforecastsaboutthefuturebehavioroftheriskfactors.ThesoftwarecanalsobeusedasanaccountingtoolfortheoutstandingsecuritiesoftheItalianpublicdebt,andincludesvarioussatellitemodulesfortheevaluationofotherrelevantmetrics,suchastheCreditValueAdjustmentforderivativeinstruments.Weprovidebothadescriptionofthegeneralframeworkandseveralillustrationsofthewaythissetoftoolsisutilizedinpractice.


4

MathematicalModelThestudyoftheevolutionofthepublicdebtandtheassessmentofthecostandriskofaportfolioofgovernmentsecuritieswithrespecttoasetofscenariosofthefutureinterestrates,requireasetupthat represents in aquantitativewayall theelements that contribute to the cash flowsof all therelevant securities in the portfolio. The Italian Treasury Department issues six different types ofdomesticsecuritiesincludingthreewithafloatingrateandasmallnumberofforeignsecuritiesforatotal amount of 2,004billion euro plus a derivative portfoliowith a notional of 99 billion euro atDecember2019.Eachtypeofsecuritycanbeissuedwithdifferentmaturitiesandthetotalnumberofdifferentsecuritiesisabouttwenty(seeTable1).

Securities Amount(mln€) % Maturities(months) Coupon IndexedBOT 113,929 5.68 6,12 ZeroCoupon CCTeu 125,586 6.26 60,84 FloatingCoupon Euribor6MCTZ 51,139 2.55 24 ZeroCoupon BTP 1,440,016 71.83 36,60,…,360,600 FixedCoupon BTPs€i 151,707 7.57 60,120,180,360 FloatingCoupon HIPCBTPItalia 77,558 3.87 48,72,96 FloatingCoupon FOIForeignDebt 44,567 2.22 36,…,600 Fixed/Floating Euribor,CPI

Table1ItalianGovernmentdebt:breakdownbyinstrumentswiththeirfeatures.

Thekeyelementsneededforareliableandeffectivedebtmanagementprocessarethestochasticmodelsfortheevolutionofinterestratesandthemathematicalrepresentationofthecashflowsasafunctionofthecompletedebtportfolio.

InterestratescenariosAkeyinputtoimplementthecost-riskanalysisoftheItalianpublicdebtissuanceportfoliosaretheout-of-samplescenariosforthetermstructureofinterestratesrelevanttothesetofsecuritiesissuedbytheItalianDepartmentofTreasuryandtothesetofderivativetransactions.AclassofmodelsofthetermstructureofinterestrateshasbeendevelopedfortheItaliansovereignbondtermstructure,the Italianbreak-even inflation (BEI) termstructure,and for theswapcurvesofbothEuroandUSdollar.Themaingoalofthesemodelsistoproducejointstochasticout-of-samplescenariosforthesedifferentyieldcurvesonthebasisofcalibrationsthatrespect theirobserved in-sampleproperties,whileensuringinternalconsistencyofthegeneratedscenarios.


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Startingfrom2010,thedevelopmentofthefirstreleaseofthemodelwasrelatedtothesovereign(nominal)andbreak-even-inflationrates,withthepurposeofpricingandforecastingnominal(BOT,CTZandBTP)andinflation-linked(BTP€i)securitiesandthenderivingendogenouslytherealcurve.Asimilar approach has been adopted by the Federal Reserve for the estimation of the real curvereferringtoinflation-linkedsecurities(TIPS).Inthisregard,seetheworkofGürkaynaketal.(2008).ThenthemodelwasextendedtothetermstructureofEURswapratesinordertopriceandforecastthenewsovereign(CCTeu)securitiesindexedtotheEuriborratesandalsoallowthemanagementofplainvanilladerivativecontracts,suchasinterestratesswapwithfloatinglegsindexedtotheEuriborrates. In2012,themodelwasfurtherextendedincludingthespecificationofthetermstructureofUSDswaprates,inordertomanagecost-riskanalysisonbondsdenominatedinforeigncurrency,priceandforecastderivativecontracts,suchascrosscurrencyswap,andimprovetheaccuracyofEURswapyieldcurvescenarios.Thesovereign,break-eveninflation(BEI)andEURswapcurvesareestimatedand simulated jointly, whereas the USD swap curve is modeled individually and so is consideredexogenoustotheothers.

ThelastrefinementofthemodelwasthenrelatedtothedevelopmentofamodulelinkingthetermstructureofItalianbreak-eveninflationratestothescenariosgeneratedfortheEurobreak-evenrates,inordertopriceandforecastthesecuritiesdesignedforretailinvestorsandindexedtoItalianinflation,namelyBTPItalia.

Allthesespecificationsareintegratedinamodulededicatedtothegenerationofstochasticscenarios,whichinterfacesthemainmodulecalculatingcostandriskoftheportfoliosofissuances.Thescenariogenerator also incorporates the possibility of using different stochastic models for generatingmedium/long-term scenarios for interest rates and inflation rates, allowing the evaluation of theexpectedperformance, in termsof cost-risk analysis, of different strategies related topublic debtissuancepolicies.

Modellingthedynamicsofinterestrates

TheapproachadoptedattheDepartmentofTreasuryformodellingthedifferenttermstructuresofinterestratesrequirestheexecutionofmultiplesteps.ThefirststeprelatestotheseminalworkofNelsonandSiegel(1987),extendedbySvensson(1994).TheNelson-Siegel-Svensson(NSS)approachuses an exponential components framework to characterize the entire yield curve within a four-dimensional parameters framework. This setup essentially decomposes the yield curve into fourfactors:thelevel,theslopeand,twocurvaturefactorsoftheyieldcurve.

Asingleyieldcurvecanberepresentedas:

! " = %& + %(1 − exp −

".&

".&

+ %/1 − exp −

".&

".&

− −exp −"

.&

+ %01 − exp −

".(

".(

− −exp −"

.((1)

where! " isthezero-couponrateformaturity"and%&,%(,%/,%0,.&e.(areestimatedparameters,whichhaveaneconomicinterpretationthatimpliesthefollowingrestrictions:

%& > 0, %& + %& > 0, .& > 0, .( > 0.


6

Thisclassofspecificationsisbroadlyadoptedbymanycentralbanksandmanyfinancialinstitutionsfortheestimationofthetermstructureofinterestrates.

Thesecondcurvaturefactor,%0,allowsmoreflexibilityandisusefulinthepresenceofmultiplelocaloptima and to better fit term structure data for long termmaturities (> 10 years), which is veryrelevantforthecaseoftheItaliansovereignbonds.Indeed,theItalianTreasuryissuesbondsupto50-yearmaturityand theallowance fora secondcurvature factorhasbeen found tobecrucial tocapturetheverylongendofthetermstructureinourNSSestimation.

TheNSSspecificationhastwohumps,andtheirpositionsareidentifiedby.&and.(.Theseparameterscanbeeitherestimatedorfixed.Settingfixedvaluesfor.&and.(yieldslowervolatilityinestimationandbettercontroloftheevolutionprocessof%parameters.

Themainstepscarriedoutinthedevelopmentoftheentiresetofmodelsarethefollowing:

• unrestricted in-sample estimation,usingdailydata,of theNSSparameters foreachcurve:sovereign (Bot, Ctz, Cct, Btp), BEI (Zero Coupon Inflation Swap based on euro area HICPconsumerprice index),Euroswap(Euribor+Eurirsrates)andUSDswap(Libor+USDswaprates);

• identificationofaveragevaluesof.&and.(forallcurvesandnewmodelestimationwiththecalibratedvaluesof.&and.(;

• time aggregation from daily tomonthly frequency (applying NSS fit to the end ofmonthestimates);

• calibration of parameters and specification of the dynamics for all%& -%0 of all the termstructuresthroughavectorautoregression(VAR)approachforthejointevolutionofthebetasofthecurves;

• out-of-samplestochasticsimulationofthemodel.

Thein-sampleestimationoftheyieldcurvesallowstoderiveendogenouslytherealratescurveandtheswapspreads(withrespecttosovereignrates).Indetail,ifwedefinethebreak-eveninflation(BEI)curveastheratesvectorthatmakesequivalent(forinvestors)toholdnominalorinflation-linkedsecurities,weget:

BEI: ; = !:

<=> ; − !:?@AB ; (2)

whichimplies

!:?@AB ; = !:

<=> ; − BEI: ; (3)AsimilarapproachiscommonlyadoptedonUSdatafortheestimationoftherealcurvereferredtoUSinflation-linkedsecurities(TIPS).SeeGürkaynaketal.(2008).

Similarly,theswapspreadcurveisobtainedendogenouslyasthedifferencebetweensovereign(nominal)andswaprates:

SPREAD: ; =!:<=> ; –!:

KLAM ; (4)

NOP; = 0.25,0.5, … , 30years.Thefitofthein-sampleestimatesis,ingeneral,excellentandweobserveverysmallpricingerrorsforallfourcurves,asshowedinthefollowinggraphs(seeFigure1,Figure2andFigure3).


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Figure1Nelson-Siegel-Svenssondailyestimation(sovereignandbreak-even-inflationrates)

Figure2Nelson-Siegel-Svenssondailyestimation(EuroandUSDswaprates)


8

Figure3Nelson-Siegel-Svenssondailyestimation(BTP€irealyields)

Afterin-sampleestimationweperformajointcalibrationofnominal,EuroBEIandeuroswapcurvesthataimsatfindingoutanevolutionprocessforparameters%&-%0thatmayproduceout-of-sampleforecasts(scenarios).As inDieboldandLi (2006),weadoptamultivariatespecificationbasedonaVARmodel:

S: = T + US:V& + W:(5)

whereS:isa12-elementvectorforthe%parameters(fourforeachofthethreecurves),Tisavectorofintercepts,Uisa12x12coefficientmatrixdescribingtheautoregressiveschemeofparameters.TheinnovationsW: are assumed to be normally distributed in the estimation, which is carried out bymaximumlikelihood.

The VAR model, after estimation and general-to-specific model reduction designed to eliminateredundantcoefficients1,issimulatedout-of-sampletoproducejointforecastsforthethreesetsof%s.Thismethodensuresconsistencyfortheout-of-samplescenariosofthetermstructuresofsovereign(nominal),BEIandEURswaprates.TheUSDswapcurveisestimatedandcalibratedseparately,anditslevel%&entersinthelevelequationoftheswapcurveineuro.

1Thegeneral-to-specificprocedureisperformediteratively,eliminatingineveryiterationtheleastsignificantcoefficientintermsofitsp-value.


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The original multivariate specification ensured fulfillment of the statistical properties of yields,observed in-sample. These properties relate to no arbitrage conditions through the variance andcovarianceoftheratesfromthetermstructuremodels.

However,aftertheglobalfinancialcrisis,someoftheno-arbitrageconditionsareinconsistentwiththedata.Therefore,someoftherecentsimulationsallowforthisfeatureinthescenariogeneration.Stochasticout-of-samplesimulationsofthemodelareperformedusingabootstrap2methodappliedtotheVARresiduals.Atleast1000differentscenarioshavetobeboostrappedinordertohavestableresultsonaverage.Theprocedureisusedtogeneratenotonlypointforecasts,butalsomultiplesetsofscenarios(distributions)consistentwiththein-samplestatisticalpropertiesoftheyieldcurves.Theparametersareregularly(re-)calibratedsoastoincludethemostrecentobservationsinthehistoricalestimationsampleandcheckingforstructuralstability,particularlyaftertheshocksobservedinrecentyearstotheItaliansovereignratesandtotheEuropeanswapcurve.

RegardingthesecuritiesindexedtoEuropeanandItalianinflation(BTP€iandBTPItalia)thescenariosproducedfortheBEIcurveonEuroAreainflationareusedtogenerateconsistentprojectionsforboththe HICP (Euro Area) and FOI (Italy) consumer price indices.3In particular, the Italian break-eveninflationratesareobtainedthroughthesimulationofasetoflinearequationsinwhichtheItalianBEIratesareregressedontheEuroAreaBEIrates.Further,consideringthestrongrelationshipbetweenprice indices and BEI behavior, we adopt a linear specification also for HICP and FOI, which areregressedontheirlaggedvaluesandontheBEIvaluesat5-yearmaturity.

Anchoringtheyieldcurvesusingexogenousinformationonshorttermrates

Whenusing themethodsdescribedabove, it isoften thecase that there isa specificprioron thebehaviorofinterestratesthatcanbeusedincalibration.Thepriorcouldbeaspecificbeliefoftheissuerorthemarketexpectation(survey)ofoneormoreinterestratesoveraparticularhorizon.Thisistypicallyaprioraboutfutureshort-termrates.Inthiscase,itisdesirableto"anchor"therelevantrates and generate simulations that condition on such prior. In recent years a new module for“anchoring”yieldcurveshasbeendeveloped,startingfromtheworkofAltavillaetal. (2013),whoproposedamethodologyfortilting(orrotating)theyieldcurveusingexogenouspriorsonshorttermrates(e.g.surveydata).

The anchoring technique exploits the informational advantageof survey expectations about shortyieldsinordertoimprovetheaccuracyofyieldcurveforecastsgivenbyabasemodel.Theanchoringmethodology is independent from the base model, and the basic idea is to tilt the yield curves

2Theinnovationsgeneratedduringthestochasticsimulationwiththebootstrapmethodareextractedrandomly(withrepetition)fromthesetofresidualsofthestochasticequationsoftheβparametersobservedin-sample.Theuseofthistechnique,comparedtotheextractionfromanormaldistribution(MonteCarlo),maybemoreappropriateincaseswheretheresidualsofthedynamicequationsarenotnormallydistributed.3TreasuryBondslinkedtoEuro-zoneinflationaresecuritiesthatprovideinvestorswithprotectionagainstpriceincreases.Boththeprincipaltoberedeemedatmaturityandtheircoupons,paidsemannually,areadjustedforinflationintheEuro-zone,asmeasuredbytheHarmonisedIndexofConsumerPrices(HICP),excludingtobacco.MonthlydatafortheEurostatindexmaybefoundattheStatisticalOfficeoftheEuropeanCommunitieswebsite(Eurostat).BtpItaliaisagovernmentsecuritythatprovideinvestorswiththeprotectionagainstanincreaseinthelevelofpricesinItaly:boththecoupons,whicharepaidsemi-annually,andtheprincipal,therevaluationofwhichisalsopaidsemi-annually,areindexedtotheItalianinflation,asmeasuredbyISTAT–theItalianNationalBureauofStatistics-throughtheFOInationalindex,“Prezzialconsumoperlefamigliedioperaieimpiegati”,withtheexclusionoftobaccoproducts.


10

generated by the base model, incorporating the exogenous priors on short rates, without re-estimatingtheparametersorchangingtheoriginalspecification.

Themethod isbasedontheprojectionof theout-of-sampledensity forecasts ftonto thespaceofdensitiesthathaveconditionalmeanequaltotheexogenouspriorandthatareclosesttoft,according

toaKullback-Leibermeasureofdivergence.Basically,ifthevectorofexpectedyieldsh-periodsaheadhasmeanX:YZandvariance-covariancematrixΣ:YZ ,andattime\wehavesurveydataonthefirst

(i.e.,shortest)Pmaturitiesrelatedtoperiod\ + ℎ,X:YZ,&:?�

,thetilted(anchored)vectoriscalculated,

startingfromtheaveragescenarioofthemodel,asfollows:

X:YZ∗ =

X:YZ,&:?`

X:YZ,?Y&:> − Σ:YZ,(& Σ:YZ,&&V&(X:YZ,&:? − X:YZ,&:?

` )(6)

wherethevariance-covariancematrixΣ:YZ,canberewrittenas:

Σ:YZ =

Σ:YZ,&&?b?

Σ :YZ,&(?b(>V?)

Σ :YZ,(&(>V?)b?

Σ :YZ,(((>V?)b(>V?)

Ifweconsider,forexample,theanchoringtoanexogenousvalueoftheratewiththeshortestmaturity(r=1,e.g.,the3-monthsrate),theprojectionofthesingletermstructureproducedbythedynamicsofthefactorsoftheNelson-Siegel-Svenssoncurveisobtainedbysettingthe3-monthsrateequaltotheexogenousvalue(prior)andmodifyingthedatafortheothermaturitiesaccordingtoequation(6).

RecentlytheanchoringmodulehasbeenimplementedintothescenariogeneratorofSAPEandsothistechnique canbe applied to any single scenario of the yield curves generatedwith the stochasticsimulationsofthebaselinetermstructuremodel.

Thefirstreleaseoftheanchoringmoduleperformedthetiltof thecurvesonlyonthefirstout-of-sampleobservation,inlinewiththeresultsofAltavillaetal.(2013).Usuallytheexogenousprioristhesovereignrateofthezero-couponinstrumentwiththeshortestmaturity(BOT3months).Moreover,consideringthatthetermstructuresofsovereign,BEIandEURswapratesaresimulatedjointly,wedevelopedamethod for selecting thepriors alsoonBEI andEUR swap rates, so as to ensure theconsistencywith theoneseton the3-monthsnominal rate. Indetail,weset theother twopriorstakingtheaverageofthe3-montheuriborratesandofthe2-yearBEIratescalculatedonthefirstout-of-sampleobservationofthefivebaselinescenarioswhichareclosesttothe3-monthsnominalrateprior,withtheconstraintthattheselectedscenariosmusthavethesamedirection(sign)intermsofrotation.

The second versionof the anchoringmodulehas been recently released, thatmakespossible therotation(tilt)ofthecurvesnotonlyforthefirstout-of-sampleobservation,butalsoforthefollowingones,forexampleupto12monthsonwards.Thisextensionallowstosimulatethemodel,forexample,grantingtheconsistencywithdifferentscenariohypothesesrelatedtotheexpectationsintermsofissuancepolicyandmonetarypolicychoices.


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ThedebtportfoliocashflowandthebalanceconstraintBydebtportfoliowemeanthecollectionofdomesticsecuritiesissuedbytheItalianTreasurythatarestill on the market, that is securities that have not yet reached their maturity. A quantitativedescriptionofthedebtportfoliobenefitsfromestablishingprecisenotationandthedefinitions:

• The indexc represents the type of security, wherec ∈ 1. . ef andef is the number ofdifferent securities issued by the Italian Treasury. Securities of the same kind but withdifferentmaturitiesareconsidereddifferent;

• The index\represents themonthsboth in thebalanceperiodand in theplanningperiod,

where\ ∈ gK. . gh ,gKisthestartingmonthandghisthefinalmonth;

• ij \ indicatesthenominalvalueofsecuritiesoftypecissuedattime\;• Tj \ indicatesthematurity,measuredinyears,ofsecuritiesoftypecissuedattime\;• kj \ indicatesthepriceofsecuritiesoftypecissuedattime\;• elm : indicatesthenumberofcouponsthatwillbepaidforbondsoftypecissuedattime\;

• nj \ indicatesthefixedcouponpercentageforbondsoftypecissuedattime\orthefixedrealcouponpercentageforinflationindexedbondsoftypecissuedattime\;

• nj o; \ indicates the floating couponpercentagepaid for the couponoof bondsof typecissuedattime\;

• Tj(o; \)indicatesthetimetomaturity,measuredinyears,ofthecouponoofbondsoftypecissuedattime\;

• qj \ indicatesthespreadforthecouponsofbondscissuedattime\;• Pr g; \ indicatesthezerocouponratefordomesticissuancesformaturitygobservedattime

\;• PK g; \ indicatestheeuroswaprateformaturitygobservedattime\;• stuv \ andwxt \ aretheEuropeanandItalianInflationIndexes,respectively;

• tj(g; \)indicatestheindexcoefficientattimegofbondsoftypecissuedattime\.

Todescribethecashflowsfortheportfolio,weusetheindicatorfunctiony q; \ ,whosevalueis1ifq = \,0otherwise.Theunitarycashflowatmonthqforbondcissuedattime\iscomposedbythreeterms:

• Thenetincomeatissuancetime\:

ej q; \ = y q; \kj \

100;

• ThecouponpaymentattimeTj(o; \)witho ∈ 1. . elm :

uj q; \ = y q; \ + Tj(o; \) tj(Tj(o; \); \)nj o; \z{m |

B}& ;

• ThereimbursementattimeTj

~j q; \ = y q; \ + Tj tj Tj; \ .

Usingtheabovedefinedfunctions,itispossibletorepresentthecashflowforthewholeportfolioofdomesticbondsatanytimeqasfollows:


12

wr q = ij \ ej q; \ − uj q; \ − ~j q; \

K

:}KV>m

z�

j}&

.

Besidesthecashflowofthedomesticportfolio,itisnecessarytoincludethePrimaryBudgetSurplus(PBS), the cash flowof the securities issued in foreign currency andderivative transactions in thebalanceequation.AstothePrimaryBudgetSurplus,anyforecastisdifficultduetoissueslikepoliticaldecisionsandchangesinthestateoftheeconomy.However,weassumethatthePBSisgivenasaninputparametereverymonthsodefiningasequencePBS(s).Astothecashflowinforeignbondsandderivatives,weconsideritasanothergivenquantityw@ q ;howeverthatquantitymaybeadaptedaccordingtothesimulatedinterestratesscenariosforthosecomponentsinw@ q whichareindexedtoinflationortosomeinterestrate(e.g.,aninterestrateswap).Thecashflowofbonds’issuancesandpaymentsgoes throughaBankof Italyaccount,ownedby theTreasury, calledTreasuryCashAccount.Byregulation,thereisapositivelowerbound,% = 10billioneuro,ontheamountofmoneythataccountmusthold.WeindicatebyguU q theamountofmoneyintheTreasuryCashAccountattimeq.Thenthebalanceconstraint,i.e.theconstraintthatmustberespectedinordertobalancetheincomeandexpenditureofthepublicdebt,canbewrittenas

guU q = guU q + 1 + wr q + w@ q + vSá q ≥ %.

Thisisthefundamentalconstraintthatguaranteesthepaymentofcouponsandthereimbursementof bonds atmaturity. For the historical issuances, the constraint has necessarily been respected,whereasinthesimulationtimewindowtheissuedquantitiesij \ havetobeadaptedtofulfillthebalanceconstraint.

TheESA2010costfunction

Thedefinitionof a suitableobjective function is anelementofpivotal importance inanydecisionmakingprocess.Inpublicdebtmanagement,thereisanaturalcosttoconsiderthatisthecostofthedebtservice,i.e.theinterestexpenses.Actually,thereareanumberofpossiblechoicesforquantifyingthe interest expenses. However, since it falls within the regulatory parameters of the Europeancommunity,wefollowtheEuropeanSystemofNationalandRegionalAccounts(ESA2010),thatisthelatestinternationallycompatibleEUaccountingframeworkforasystematicanddetaileddescriptionofaneconomy.

TheESA2010criteriarefertoagiventimeperiod,say[\&, \(],and,looselyspeaking,considerforeachbond its total cost (original issue discount, coupons and uplift for securities inflation indexed) asdistributedoveritsexistenceperiod,namelyfromissuancetomaturity,takingtheportionofthecostthathasanintersectionwiththeconsideredtimeperiodoftheESA2010.Tobemoreprecise,eachcostassociatedtoabondhasanaturalreferenceperiod(e.g.,sixmonthsforacouponorthecompletelifetimeofthebondfortheoriginalissuediscount)thatcanbeindicatedas[\K, \@],andimpactsontheESA2010proportionallytotheratiobetween(N),thenumberofdaysofintersectionbetweentheESA2010periodandthereferenceperiodofthecost,and(D),thenumberofdaysofthereferenceperiodofthecost.Inthefollowing,wepresenttheimpactontheESA2010ofthevarioustypesofcostsforaunitaryquantityofasecurityoftypecissuedattime\:

• ESA2010costintheperiod \&, \( fortheoriginalissuediscount

ãj=år [\&, \(]; \ = 1 −

kj \

100

[\&, \(] ∩ [\, \ + Tj \ ]

[\, \ + Tj \ ];


13

• ESA2010costintheperiod \&, \( forthecouponsofthesecurityoftypecissuedattime\

ãjl=éM [\&, \(]; \ = tj(Tj(o; \); \)nj o; \

z{m |

B}&

[\&, \(] ∩ [\ + Tj o − 1; \ , \ + Tj(o; \)]

[\, \ + Tj \ ];

• ESA2010costintheperiod \&, \( fortheuplift

ãjéMBåh: [\&, \(]; \ = tj min \ + Tj \ , \( ; \ − tj max \, \& ; \ .

Finally, the ESA 2010 cost in the period \&, \( for thewhole portfolio of domestic bonds can berepresentedas

ã [\&, \(] = ij \ ãj=år [\&, \(]; \ + ãj

l=éM=< [\&, \(]; \ + ãjéMBåh: [\&, \(]; \

:ëV&

:}:íV>m

z�

j}&

.

AnalyticalstudyontheESA2010evolutionTheaboveequationisaverygeneralrepresentationoftheinterestexpenditure,sinceitaccountsforanykindofsecurity.Hereafter,welimitourinvestigationoftheESA2010dynamicstotwotypesofsecurities,zero-couponbondsandbondswithfixedcoupons,andassumethat:

1. eachsecuritywithcouponsisissuedatparvalue,i.e.,kj \ =100;2. eachsecurityisissuedthefirstdayofamonthandthetimeframefortheevaluationofthe

ESA2010costisalwaysanintegernumberofmonths;3. theamountofeachsecurity issuedeverymonthdoesnotdependontimebutonlyonthe

typeofsecurity.Ifìistheinitialdebtattime\ = 0,wesupposethatìisdistributed,accordingtoafractionîj,amongthedifferentsecuritiescandtheamountforeachsecurityisuniformlydistributedacrossthematurity,Tj,ofthesecuritysothat:

ij =ìîjTj

.

The thirdassumption is themost importantone. Itmeans that the totaldebtand its compositionremainconstant:onlymaturingsecuritiesrequirenewissuances.Couponsarepaid,forinstance,usingtheprimarybudgetsurplus.AsimplifiedversionoftheESA2010interestexpenditureisusefultostarttheanalysisofthemonthlyinterestexpenditure.Theassumptionthatthegovernmentissueseitherzero-couponbondsorparvaluebondsmakesitpossibletoderivetheESA2010interestexpenditureasafunctionofpastinterestrates.Inparticular,theESA2010requirementofspreadingtheoriginalissuecostalongthewhole lifeofasecurity impliesthatthecostofazero-couponbond inagivenmonth,ï,duringthelifeofthebondcissuedattime\,isequalto

ãj=år ï; \ =

Pr Tj \ ; \

12

andasimilarrelationholdsfortheESA2010costsofacoupon


14

ãjl=éM ï; \ =

Pr Tj \ ; \

12

since, for fixedcouponbonds issuedatparvalue,nj \ = Pr Tj \ ; \ /2,andthatcosthas tobedistributedalongsixmonths.Wecannowdefineó g asthetotalESA2010costinasinglemonth,g,as

ó ï = ij \Pr Tj \ ; \

12

òV&

:}òV>m

z�

j}&

= ij ï − TPj ï − T

12

>m

>}&

z�

j}&

,

where,tosimplifythenotation,weusethechangeofvariablesT = ï − \andPj \ = Pr Tj \ ; \ without ambiguity, since in this paragraph we limit our analysis to a fixed domestic bond;Pj \ indicates the spot rate for issuancesof thedomestic bondchavingmaturityTj issuedat time\.Assumption3 simplifies theaboveexpressionsinceij \ is fixed,dependingonlyon theportfoliocomposition

ó ï =ì

12

îjTj

Pj ï − T

>m

>}&

z�

j}&

.

IfwedefinetheaveragevalueofratePj \ intheperiodbetween ï −Tj,ï − 1 as

~j ï =1

TjPj ï − T

>m

>}&

themonthlyESA2010becomes

ó ï =ì

12îj~j ï

z�

j}&

.

WearenowreadytowritedowntheESA2010simplifiedexpression:

ã [\&, \(] = ó ï

:ë

ò}:í

=ì

12îj ~j ï

:ë

ò}:í

z�

j}&

=ì

12îj~j

z�

j}&

\( − \&

where~j istheaverageofthespotrateassociatedtothesecuritycintheperiod \&, \( .Theaboveexpression shows that, using reasonable approximations, the ESA 2010 cost has a very intuitiveexpression as a linear function of timewith a coefficient that is the composition of the portfoliopercentagesmultipliedbytheaveragerates.TheactualESA2010dynamicsthatwecomputewithoursoftwaretoolshowsthatthelinearapproximationcanbeconsideredagoodproxyandthegrowthrateof theESA2010, i.e., the functionó ï , canbe consideredas a simplebut informative costfunction.

TheCost/RiskAnalysisBothmathematicalmodels,theoneusedforthegenerationofpossiblescenariosoffutureinterestratesandtheotherdescribingthecashflowsofthepublicdebtandtheESA2010costfunction,canbeusedseparatelyandincombinationinseveralways.Firstofall,itispossibletocomputefinancialindicatorsusefulforthecompilationofaccuratebalancereports,sincethemodelconsiderstherealoutstandingdebt (includingalso issuances in foreigncurrenciesandderivativecontracts). It isalso


15

possible toestimate future interest expendituresdue to changeseither in the compositionof theportfolioofpublicdebtsecuritiesortothedynamicsofinterestrates,e.g.,iftheyremainlowforthenextfiveyearsortheyundergoshocks(what-ifanalysis).However,themost interestinganalysis isprobablytheonebasedonMonteCarlosimulationsinwhichtheperformanceofdifferentpossiblefundingportfoliosiscomparedwithrespecttoalargenumberofpossiblescenariosofinterestratesevolution. This is the so called cost-risk analysis that is a fundamental task that the public debtmanagermustbeabletoperforminaveryreliableway.Intheprevioussubsection,wedescribed,inanalytic form, the dynamics of interest-expenditure for a portfolio under certain assumptions.Hereafter,weextendanddeepentheanalysisconsideringtheactualevolutionofinterestexpenditurewithout imposing any constraint and we determine the probability distribution of the ESA2010interestexpenditurewithrespecttoastochasticdistributionofinterest-ratepaths.

ClassicalmeasuresofcostandriskClassically,thechoiceofthedebtportfolioreliesonatrade-offbetweencostandriskmeasuresthatmustbestatisticallydefinedandbasedonthegovernmentobjectivefunction.Themostcommonlyusedcostandriskmeasuresarethemeanandthestandarddeviationoftheobjectivefunction(theESA2010inourcase),respectively,butotherfunctionshavebeendeveloped(seeFig.4forapictorialrepresentationofcostandriskmeasures)whicharemoresuitableforthemanagementofpublicdebt,liketheCostatRisk.

Figure4.SchematicrepresentationoftheESA2010ProbabilityDistributionandassociatedcostandriskmeasures.

TheCostatRisk (CaR) isakeyreference indicator indebtstrategiessimulationanalysisbecause itdependsonboththeexpectedvalueandriskcharacteristicsoftheprobabilitydensityfunctionoftheinterestexpenditure(Jorion2006).TheCaRisthemaximumvalueoftheinterestexpenditurethatcanbeexpectedwithaprobabilitypoveragiventimehorizon.Informula

CaRM = minb

î|Prob ã \&, \( ; v, á < î ≥ k ,

whereweextendthenotationintroducedintheESA2010costfunctionSubsectionindicatingwithã [\&, \(]; v, á theESA2010costintheperiod[\&, \(]foraportfoliovofpublicdebtsecurities,andfor a scenarioá of the interest rates’ evolution. The quantityã [\&, \(]; v, á can be considered astochastic variable since, for any portfoliov it depends on the scenarioá . The CaR indicates theinterestexpenditurethattheGovernmentcanexpectnottoexceedwithaconfidencelevelp.Itoffersasimplemetricsoftheinterestcostthatthegovernmentcouldincurunderunfavorablecircumstances.Thehighertheprobabilityp,thegreateristheconsiderationofunfavorablerealizationsoftheinterest


16

expenditureandthusthegreateristheattentiontoriskinthechoiceofthedebtportfolio.TwoothermetricsofriskareoftenconsideredinrelationtotheCaR:theRelativeCostatRiskandtheExpectedShortfall.TheRelativeCaRisgivenbythedifferencebetweentheCaRandtheexpectedvalueoftheinterestexpenditureoveragivenhorizon.Itcanbeseenasameasureofthewidthoftheinterestexpenditureprobabilitydensityfunction.Itisdefinedas

RelCaRM = RelCaRM − ã \&, \( ; v, á ú,where, hereafter,with ∙ ú we indicate the average valueof thequantity ∙ with respect to therandomvariableá.

TheExpectedShortfall(ES)istheexpectedvalueoftheinterestexpenditureincasetheCaRmeasureisexceeded.Itisdefinedas

ESM = ã [\&, \(]; v, á û|ü†°¢£§.

ESindicatestheexpectedinterestexpenditureconditionalontherealizationofthe1–ppercentageof worst possible scenarios. It provides an estimate of the expected interest expenditure underextremelyunfavorable circumstances.As theESdependson the thicknessof theupper tailof theprobability distribution of interest expenditure, it complements the CaRmeasure in case of non-normalityofthedistributionofinterestexpenditure.Asamatteroffact,itisworthnotingthatifthedistributionofinterestexpenditurewerenormal,theabove-mentionedstatisticalindicatorswouldbefullydeterminedbythemeanandthestandarddeviationofthedistribution.

Evolutionmeasuresofcostandrisk

Beforeshowinganexampleofcost-riskanalysis,itisworthnotingthereisanimportantlimitintheuseofthestatisticalmeasuresjustdescribed:theyrefertoafixedtimeinterval.Theconsequenceisthatthevalueofallthemeasures,and,therefore,thecost/riskanalysis,dependsonthechosentimeinterval.However,byusingtheresultspresentedintheAnalyticalstudyontheESA2010evolutionsubsection,inwhichweshowedthat,undercertainassumptions,thereisalineardependenceoftheESA2010costonthetimeinterval,wemayintroducenewquantitiesthatproviderobustmeasuresofcostandriskthatareonlylooselydependentonthetimewindow.Asafirstexample,intheleftpanelofFigure5weshowtheESA2010growthdynamicsoftheactualportfoliooftheItaliangovernmentbondsonDecember12,2019accordingtotwohundredpossiblescenariosofinterestrates.

Figure5.ESA2010growthpath(ontheleft)andESA2010distributionafterthreeyearsofsimulation(ontheright).

At any time of the simulation period (5 years in the present case), it is possible to compute thedistributionoftheESA2010interestexpenditure(rightpanel).Asamatteroffact,asshowedinFigure


17

5 also the growth of the ESA2010 cost computed exactly, without any approximation, can beconsidered,withagooddegreeofapproximation,aslinear.InawaysimilartowhatpresentedintheAnalyticalstudyontheESA2010evolutionsubsection,weindicatewithó ï; v, á theESA2010costin monthï for a portfoliov of public debt securities and for a scenarioá of the interest rates’evolution.ToobtaintheESA2010costinthetimewindow[\&, \(]itissufficienttocomputethesum

of the functionó ï; v, á in theperiod[\&, \(]:ã [\&, \(]; v, á = ó ï; v, á:ëò}:í

.Therefore, for

thepurposeofcomputingareasonableapproximationofthegrowthrateoftheESA2010cost,itispossible to use the functionó ï; v, á . However, since the actual growth of the ESA2010 is notperfectly linear,differentgrowthratemetricsmaybedefinedforspecificneedsofthepublicdebtmanager.ItispossibletoconsidertheinitialgrowthrateR:í v, á = ó \&; v, á (ifoneisinterested

intheearlyevolutionofthedebt),thebestgrowthrateR• v, á =òV:í ∙¶ [:í,ò];ß,ú

|ë®©|í

òV:í ë|ë®©|í

(ifoneis

interestedinthetypicalgrowth),orthefinalgrowthrateR:ë v, á = ó \(; v, á (ifoneisinterested

inthelongtermgrowth).

SincewithMonteCarlosimulationsitispossibletoevaluatetheperformanceoftheportfoliovovera large number of scenarios, we may define for each of the above mentioned growth rates aprobabilitydistributionandthecorrespondingmeasuresofcostsandrisks.Theideaisthatinsteadofcomputingthecostsandrisksmeasures(suchastheCaR)overtheESA2010distributionatafixedtime, they are computed on the distribution of the chosen growth rate. Therefore, assuming, forinstance,thatadebtmanagerisinterestedinthetypicalgrowthrateoftheESA2010cost,itispossibletogenerateanefficientfrontier,associatedtodifferentdebtportfolios,byusingtheaverageofthe

bestfittothegrowthrate Rh v, á úanditsstandarddeviation™ R• v, á withrespecttoalarge

enoughsetofscenarios(seeFigure6).

Figure6.EfficientFrontier.

CVAmodellingThis Section describes the development of themathematicalmodel and the scenarios generator,whichareimplementedinasatellitemoduleofthesystemdedicatedtothecalculationoftheCreditValuationAdjustment(CVA)forcollateralizedderivativetransactions.


18

PriortointroducingthedetailsoftheCVA,itisimportanttounderstanditspurposewithinthecontextofthesocalledCreditSupportAnnex(CSA).ACSAisanagreementthatregulatesthecreditsupport(collateral)tomitigatethecreditriskofOTCderivativetransactions.Itdefinestheconditionsunderwhich,atgivendates,collateralisposted,ortransferred,betweenthecounterpartiesthatenterintoanOTCderivativecontract,accordingtothefollowingelements:

§ frequency:timeintervalbetweentwocontiguouscollateralpostings;§ minimumtransferamount:thesmallestamountofcollateralexchanged;belowthisamount

nocollateralisposted;§ threshold:referencemark-to-marketabovewhichthecollateralshouldbeposted.

Inordertomeetspecificneeds,ItalianTreasurycustomizedthecanonicalCSAframeworkaddingapotentiallyfourthelementtolimitthemaximumamountofcollateraltobepostedateachdate.Thisleads to a non-standard framework with increasing cash flows, which gradually should ensure acoverageofthecreditriskwithalimitedburdenforItalianTreasury.

Posting collateral against a derivative position usually cancels the credit risk of that position.ComparedtoastandardCSA,anon-standardframeworkreducesthecreditriskofacertainamount.ThemetricstomeasurethisamountisthengivenbytheCreditValuationAdjustment.

Givenaliabilityportfolio,theCVAisthedifferencebetweenitsvalueinabsenceofriskandthevaluecalculatedconsideringacounterparty’sdefault.Inotherwords,itisthecreditadjustmentappliedtoderivativestransactiontoaccountforcounterparty’sdefault.

Let´KM=:betheexpectedlossesthatcanoccurinthetimeinterval[0,T]:

´KM=: = o¨≠ ⋅ ãã ⋅ ìv

whereo¨≠is the lossgivendefault,ããis theexpectedexposureandìvis thedefaultprobability.Assuming the recovery rate is equal to(1 − o¨≠) and constant at default time, the CVAmay becomputedwiththefollowingexpression:

uØU = o¨≠ ⋅ ã∞ ì \ ⋅ ãã \ = "]≠ìv(0, \)

±

≤

TheCVAcalculationbeginswiththegenerationofasetofnscenarios.Ascenarioisasetofmforward-looking,monthlytermstructuresofinterestrates,eachonerepresentingthespotcurveati-thmonthinthefuture.Thenumberofmonthsusuallyfitsthetime-to-maturityofthelongestswapinportfolio.

Thescenariosaregeneratedthroughthespecificationofastandardone-factorHull&Whitemodel.With this model, an array of instantaneous forward rates (short rates), one per each month ofsimulation,isderivedfromthefollowingformula:

≠P \ = y \ − ≥P \ ≠\ + ™≠¥(\)

where:

§ P(\)istheshortrate§ ≠¥(\)isaWienerprocess™isthevolatilityoftheshortrate§ ≥isthemeanreversionrate§ yisadriftfunction


19

Fromthesimulationoftheshortrate’spath,afullyieldcurveisgeneratedateachsimulationdatethroughanaffinefunction.Thistermstructurerepresentsthesinglescenarioandiscomposedbyonehundred maturities, where each point is the forward rate calculated every six-months, from thematurityofsixmonthsuptofiftyyears.

LatestfinanceguidelinessuggestdiscountingthefuturevalueofaderivativeusingthefactorsfromanOvernightIndexedSwapcurve.Thisrequirestogeneratetwocorrelatedsetsofscenarioshavingthesamecardinality:thefirstontheswapIBORcurve,todeterminetheforwardrates,thesecondontheswapOIScurve,tocomputethediscountfactors.

Thecalculationofmark-to-marketsisiteratedforeveryswapinportfolio,foreachofthengeneratedscenarios,devisingan;×Tarrayofmark-to-markets,wheremisthenumberofmonthsbetweenthesimulationdateandtheexpiryoftheswapwiththelongestmaturity.

Generatingtherawmark-to-marketsratherthanthefinalEEallowstoperformfurtheranalysisandpost-processing,suchastheapplicationofthenon-standardCSAframework.Inthiscaseithasbeenadoptedatwo-stepprocedurewhereinthefirststepeverymark-to-marketisrecalculatedaccordingtothenon-standardCSAconstraintsdescribedabove,whileinthesecondthepositivefilteredmark-to-markets are separated from the negative ones. Finally, both subsets of mark-to-markets aregrouped into twom-sizedarrays,whichactually contain theExpectedPositive (EPE)andNegative(ENE)Exposuresforeachmonthofsimulation,asshowninfigure7.

WkW≤≤ ⋯ WkW≤<⋮ ⋱ ⋮

WkW>≤ ⋯ WkW><⇒

ãvã≤⋮

ãvã>

T\T≤≤ ⋯ T\T≤<⋮ ⋱ ⋮

T\T>≤ ⋯ T\T><

T\T≤≤ ⋯ T\T≤<

⋮ ⋱ ⋮T\T>≤ ⋯ T\T><

W;W≤≤ ⋯ W;W≤<⋮ ⋱ ⋮

W;W>≤ ⋯ W;W><⇒ãeã≤⋮

ãeã>

Figure7.EPEandENEcalculation.

Bothsetsofmark-to-marketarestoredinam-sizedarray.

For each simulationmonth the Expected Exposure value is discounted by the spotOIS curve andmultipliedby thecorrespondingdefaultprobability (DP)derived fromthe termstructureofCreditDefaultSwapusingastandardmoralhazardmodel.ThefinalvalueofCVAcomesfromthesumofallvaluesinthedatasetmultipliedbytheLossGivenDefault.

Since the computational cost of this procedure is incompatible with response time requested bycounterparties,there istheneedtooptimize itbyprocessingmultiplescenariosatonce.Thusthisapplicationfollowsamulti-threadedapproachwhichensuresaspeedupthatgrowslinearlywiththenumberofthreads.

mtm mtmCSA1ststep

2ndstep


20

Inordertomanageaheterogeneoussetofinstrumentsthelogicbehindcalculationshasbeenmovedintoapricinglibrary,towhichtheapplicationislinked.Currentlythislibraryhastheabilitytopricethefollowingtypesofinstruments:

§ interestrateswaps§ crosscurrencyswaps§ swaptions

Eachinstrumentismodelledthroughaclasswhichenclosescomplexdatastructuresandalgorithmsneededtoperformthecalculationofmark-to-market.Themajoradvantagesofthissolutionare:

§ reusability:thepricinglibrarycanbeused,andeffectivelyitis,byotherapplications§ extensibility:becauseofitsstructure,functionalitiescanbeeasilyaddedtothelibrary

Forexample,overtimethelibraryhasbeenenrichedtomanagesomefeaturesoftheswap,suchasamortizingswaps,differentpaymentfrequenciesbetweenthetwolegsofaswapandbreakclauses.The last functionality models the possibility that a swap may be terminated before its naturalexpiration(earlyterminationoption(ETO),leadingtoaseconddatasetofrawmark-to-markets.Inthisdataset, the swap’s cashflows after the early termination date are simply discarded and thus thecontributionoftheswaptothemark-to-marketoftheportfolioafterthatdateisnull.

SoftwareArchitectureTheSistemaAnalisiPortafoglidiEmissione(SAPE)softwareisaquitesophisticatedtool[1],madeofasetofmodulesthatinteracteachotherthroughspecificApplicationProgrammingInterfaces(API)andconventions.

The core of the system is the Computational Engine (CE). The CE determines debt cash flowsconsidering the outstanding government bonds and the future (simulated) interest rates. The CEindicates what will be the future issuances of any kind of security based on a given portfoliocomposition and an exogenous estimate of the budget surplus. In the CE there is full integrationamongthedomesticdebt,theforeigncurrencydebtandthederivativesinstruments,thatresultsinacompleteanddetaileddescriptionoftheportfolioofgovernmentsecurities.TheCEcomputesallthequantitiesofinterestforthepublicdebtmanager(e.g.,thedebtcostaccordingtotheESA2010criteria,thedurationoftheportfolio,etc…)andprovidesinputtotheCost/Riskanalysismodule.TheCEcanbeusedalsoasanaccountingtooltodouble-checktheoutputofothercommercialsoftwareproductsin use at the Italian Treasury for themanagement of public debt. The CEmodule is written in Clanguageforefficiencyreasons.

AcrucialcomponentoftheSAPEistheInterestRatesgenerationmodulethatsimulatestheevolutionofinterestratesaccordingtothedifferentstochasticmodelsdescribedearlierinthispaper,including,amongothers,theVARmodelandtheHull-Whiteone-factormodel.AllthescenariogeneratorsarewritteninClanguage.

The Cost/Risk module analyses and shows the results ofMonte Carlo simulations comparing theperformance of different portfolios under several measures of cost, including both the ESA2010accrualcostandthecash-flowcostsinagiventimeinterval.TheCost/Riskmodulemakespossibletotraceanefficientfrontierofthepublicdebt,bychoosingsuitableportfoliosandcostandriskmeasures.ThatmoduleisdevelopedpartinClanguageandpartinthePerlscriptinglanguage.Oneofthereasons


21

forusingPerlisthatatthetimewestartedthedevelopmentoftheSAPE,Perlofferedabettersupportformanaging(i.e.,readingandwriting)Excelsheets,thatareoneofthemainoutputformatsrequiredbytheItalianTreasurystaff.

Finally, there is the CVA module, designed to analyze collateralized derivative transactions asdescribedinthepreviousSection.

TheSAPEsoftwarerequiresinputdatacomingfrommultiplesources:

i) theGEDI(GEstioneDebitoItaliano)systemthatprovidesalltheinformationaboutnewissuances,derivatives,non-domesticsecuritiesandinterestratesreal-timeandhistoricaldata.ThosedataarestoredinarelationaldatabaseandretrievedusingaWebservice;

ii) the SAPEDB database of historical issuances that provides information about theoutstandingdomesticbondsand thehistoricalvaluesof somemacro indexes (euribor,HICP,FOI);

iii) theInterestratesgenerationmodule.

Finally, there is an outputmodule that stores all the relevant files of each simulation in a SQlitedatabase.Thedatabasemakessearchandretrievaloperationsveryeasyandefficient.This,inturn,facilitates,forinstance,theset-upofsimulationsthatarelikeotherscarriedoutinthepast.Inaddition,thismodulealsodealswith thecreationofgraphsandreports topresent theresultsofnumericalsimulationsinawaythatiseasytointerpret.

SAPEoffersacomprehensiveinteractiveGUIbasedonPerl/Tkthatincludesacontext-sensitivehelp.However,itispossibletorunmanyofthesoftwaremodulesalsoinbackgroundtoproduceresultsinbatchmode.Inparticular,theinterestratesgenerationmodulecanproducethousandsofscenariosthat can be used for other purposes or simply to carry out a detailed statistical analysis of theirbehavior.Allthesoftwarecomponentsarehighlyportable,andtheymayrunonMacOSX,LinuxandWindows,thelatterbeingtheplatforminuseatItalianTreasurypremises.

Figure8summarizestheSAPEorganizationandtherelationsamongthemodules.Inparticular,SAPEDB,GEDIandtheInterestrategeneratoraredatasources,whereastheCEandtheCVAcarryouttherequiredcomputationswhoseresultsareanalyzedandsavedintheremainingtwomodules.

Figure8TheorganizationoftheSAPEsoftware


22

ConclusionsThe models used by the Italian Treasury for public debt management and the software thatimplementsthesemodels,describedinthispaper,aretheresultofalonglastingefforttosupporttheItalianTreasuryintheirdecisionsrelatingtoissuanceportfolios.ThemathematicalmodelofthepublicdebtsecuritiesactuallyissuedbytheItaliangovernmentallowsforfinedetailstobeincorporatedinthecalculations,inordertofaithfullyreproducethecashflowsoftheentirepublicdebtportfolio.Thismeans that the model can be used both as balance computational tool and as an expenditureforecastingtoolinaunifiedframeworkofanalysis.

Atthecoreoftheforecastingmodules,thereistheinterestratesscenariogenerator.Extensiveworkhasbeencarriedouttoassureitsreliabilityandguaranteerobustnessandconsistencyofsubsequentanalysisbothintermsofcost/riskfrontierandCVAcomputations.Thecost/riskanalysisreliesonadynamiccomputationofvariouscostandriskmeasures inordertoreducethedependenceof theresultsonaspecificcomputationaltimewindowasmuchaspossible.

Whiletheworkdonetodevelopthetoolsdescribedinthispaperhasbenefittedfrommuchacademicliteratureontermstructuremodellingandcreditriskmeasurement,itshouldbeclearthatnoexistingmodelintheliteraturecanbeusedtoaddressthecomplexissuesthatsurroundthedebtmanagementoperations of the Italian Treasury. This means that several of the research hurdles that wereencounteredhadnosolutioninexistingpublishedresearch.Ingeneral,forexample,agovernmentissuing different types of debt instruments (e.g. nominal, inflation-linked, foreign currencydenominated) needs tools that allow the generation of scenarios for multiple term structures ofinterest rates for these instruments simultaneously. Apart from the obvious complexity ofendogenouslygeneratingmultipletermstructures,thereisafurtherneedtosatisfyregulatoryorself-imposed constraints in the determination of feasible issuance portfolios, which makes standardcost/risk optimization tools inadequate. This need formulti-term-structure, constrainedmodellingcontrasts with the common approach in published research that tends to analyze a single termstructureofinterestratesinisolation.WehopethatthedescriptionofthismodelandofthegeneralframeworkwithinwhichtheItalianTreasuryoperatesintermsofdebtmanagementwillspurfurtherresearch among scholars interested in this area which can, in turn, help finding more effectivesolutions.Indeed,sincenewcircumstancesandrequirementsmanifestthemselvesatregularintervalsin theworldofpublic debtmanagement, researchanddevelopment remainsongoing in termsofmodels,metrics, and considerationofnew typesof financial instruments that the ItalianTreasuryexpectstoproposetothemarket.

Appendix

GenerationofinterestratecurvesInthisappendixwepresentsomeresultsaboutthescenariosgeneratormodeloftermstructuresofinterest rates and inflation implemented in SAPE.As statedabove, theobjectiveof the stochasticmodel istheproductionofscenariosthatcanbeusedforconductingcost-riskanalysesonfeasibleportfoliosofissuancesofsovereignbonds.Inthisregardtheparametersofthevectorautoregressive(VAR)modelarecheckedregularlyforstabilityandre-calibratedbyusingmostrecentmarketdata.Clearlythemodelsconsideredherearenottobeconsideredasstructuralmodelsbased(solely)ondeep parameters, rather as reduced-form characterizations of the unknown underlying processgeneratinginterestratesandinflationdata.Hence,instabilityoftheparametersistobeexpectedandre-calibrationisrequiredfromtimetotime.


23

Hereafter,wedescribeanexampleofscenariosgenerationforthecost-riskanalysescarriedoutbytheItalianDepartmentoftheTreasury.ItisworthnotingthatthedatadonotnecessarilycorrespondtoadistributionofscenariosactuallyusedbytheItalianTreasury.

Intheexamplepresentedbelow,thestochasticscenariosoftheyieldcurvesaregeneratedusingthecalibrationofthemodelbasedonthehistoricalsampleofdatafromJanuary2005toOctober2019.Thetermstructuresconsideredareobtainedconsidering:12maturities(from3monthsto50years)forthesovereigncurve,10maturities(from3monthsto30years)fortheEuropeanswapcurve,15maturities(from2yearsto30years)fortheEuropeanbreak-eveninflation,and11maturities(from3monthsto10years)fortheUSDswapcurve.Thiscalibrationshowsexcellentin-samplebehavior,withverysmallpricingerrorsacrossthedailyhistoricalsample.Asanillustrativeexample,inFigure9andinFigure10isshownthefitoftheestimatedyieldcurvesfortheobservationsofOctober2006andJune2016(endofmonth):

Figure9ActualvsFittedin-sampleyieldcurves


24

Figure10ActualvsFittedin-sampleyieldcurves

ThetimeseriesoftheβfactorsoftheNelson-Siegel-Svenssonspecifications,estimatedinsample,arethen projected out-of-sample through the generation of stochastic scenarios, obtained bybootstrappingtheresidualmatrixoftheVARsystem.Thesescenariosarethenusedforthefollowingcost-risk analyseson theportfoliosof issuances, in combinationwith the liquidity and refinancingconstraints of the Italian public debt structure.Overall, the procedureoutlined abovedefines thestochasticcomponentoftheentireprocessofbuildingtheefficientcost-risk frontieronwhichtheTreasuryassessesitsissuingpolicyforfutureperiods.

Fordescriptiveandinterpretativepurposes,itisinterestingtoanalyzetheout-of-samplebehaviourof the yield curves, through a generation of 5,000 scenarios simulated from November 2019 toDecember2025.Ifweconsiderasynthesisofthefourdifferentyieldcurvessimulatedbythemodel,Figure11showstheaverageout-of-samplebehaviorofthefourtermstructures.


25

Figure11Averageout-of-samplescenarios(Sovereign,BEI,EuroswapandUSDswaptimeseries)

It is also interesting to observe the behaviour of the curves in case of anchoring the short termsovereignrate(3monthsBOT)toanexogenousprior.Theanchoringmoduledescribedinthemaintext is frequentlyusedbythe ItalianTreasuryfor“what-if”analyses,grantingtheconsistencywithdifferenthypothesesrelatedtoexpectationsintermsofissuanceandmonetarypolicychoices.

Forexample,assumingapriorof the3-monthrateequal to0.1% in the firstmonthof theout-of-samplesimulation(November2019),theaveragerotation(tilt)ofthesovereignyieldcurveturnsouttobeasfollows(seeFigure12):


26

Figure12AnchoringtheSovereignyieldcurve.

Consistently, all the 5,000 simulated scenarios are tilted applying themonthly increments of thebaselinesimulationtothefirst,anchored,out-of-sampleobservation.


27

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