Portfolio Margining Using PCA Latent Factors

Finance and Economics Discussion Series Federal Reserve Board, Washington, D.C. ISSN 1936-2854 (Print) ISSN 2767-3898 (Online) Portfolio Margining Using PCA Latent Factors Shengwu Du; Travis D. Nesmith 2025-016 Please cite this paper as: Du, Shengwu, and Travis D. Nesmith (2025). “Portfolio Margining Using PCA Latent Factors,”FinanceandEconomicsDiscussionSeries2025-016. Washington: BoardofGovernors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2025.016. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Portfolio Margining Using PCA Latent Factors ShengwuDu&TravisD.Nesmith∗ QuantitativeRiskAnalysis BoardofGovernorsoftheFederalReserveSystem February11,2025 Abstract Filteredhistoricalsimulation(FHS)—asimplemethodofcalculatingValue-at-Riskthat reactsquicklytochangesinmarketvolatility—isapopularmethodforcalculating marginatcentralcounterparties. However,FHSdoesnotaddresshowcorrelationcan varythroughtime. Typically,inmarginsystems,eachriskfactorisfilteredindividuallysothatthecomputationalburdenincreaseslinearlyasthenumberofriskfactors grows. Weproposeanalternativemethodthatfiltershistoricalreturnsusinglatent riskfactorsderivedfromprincipalcomponentanalysis. Wecomparethismethod’s performancewith“traditional”FHSfordifferentsimulatedandconstructedportfolios. Theproposedmethodperformsmuchbetterwhentherearelargechangesincorrelation. Italsoperformswellwhenthatisnotthecase,althoughsomecareneedsto betakenwithcertainconcentratedportfolios. Atthesametime,thecomputational requirementscanbereducedsignificantly. Backtestingcomparisonsareperformed usingdatafrom2020whenmarketswerestressedbytheCOVID-19crisis. Keywords: portfoliorisk;Value-at-Risk;margin;CCPs;principalcomponentanalysis (PCA);historicalsimulation;FHS JEL:g0,g2 1 Introduction Central counterparties (CCPs) have to calculate margin requirements for any portfolio presentedforclearing. AsCCPmarginrequirementsaregenerallydefinedasmeetinga ∗20thandCSts.,Washington,DC20551;E-mail:shengwu.du@frb.gov;travis.d.nesmith@frb.gov WewishtothankourcolleaguesinQuantitativeRiskAnalysisandFinancialStabilityAssessmentformany fruitfuldiscussions. WealsogreatlyappreciateYangHeppe’sexcellentresearchassistance. Wealsowish tothankElaineZhangforherexcellentdiscussionandtworeviewersfortheirhelpfulcomments,bothat WFEClear2024. TheviewspresentedaresolelyourownanddonotnecessarilyrepresentthoseoftheFederal ReserveBoardoritsstaff.Anyremainingerrorsareoursoleresponsibility. 1

percentileattheportfoliolevel,thesecalculationsresembleestimatingportfolioValue-at- Risk(VaR).1 VaRestimatesthesizeoflossthatcanbeexpectedataspecifiedprobability leveloveracertainperiod.2 CCPsoftenaddadditionalchargestotheirmargin,butthe core is a market risk measure, like VaR. CCPs have to make these margin calculations frequently—at least daily if not more often—implying that both accurate and efficient risk calculations are needed for the range of possible portfolios each CCP might clear. Furthermore,marginrequirementisa,ifnotthe,primaryriskmanagementtoolforCCPs, manyofwhichthemselvesareconsideredtobesystemicallyimportanttothefinancial system. Designing robust margin systems for CCPs is therefore both challenging and criticallyimportant. AnincreasingnumberofCCPsbasetheirmarginonformsoffiltered historicalsimulation(FHS). Suchmethodsareefficienttocalculate,butcanmissimportant riskcharacteristics. Thechallengeistodevelopalternativesthatmaintaintheefficiencyof FHSmethods,whileimprovingaccuracy. Toclarify,FHSisacommonmethodforcalculatingVaR;itisarefinementofhistorical simulation. Historical simulation calculates VaR nonparametrically as the percentile of thedatasample. Itassumesthatthehistoricaldatarepresentsasimulation,whereeach observation is an independent draw from a fixed underlying distribution. This method hastheadvantageofsimplicity,buttheimplicitassumptionthatthewholedatasample isproducedbyafixeddistributionisverystrong. Animmediateimplicationisthatthe volatilityoftheseriesshouldbeconstantinthesample. However,thevolatilityoffinancial returnsisnearlyalwaystime-varying. FHSaddressestime-varyingvolatilitydirectlybyscalinghistoricalvolatilitytoresemblethevolatilityofthemostrecentmarketdata; itthereforeestimatesconditionalVaR. Intuitively,itrescalesthehistoricaldatatomakeitmorelikeasimulationconductedunder currentlyobservedvolatility. Gurrola-PerezandMurphy(2015)showthatfilteringgreatly improves historical simulation. Many CCPs use FHS to calculate margin, usually with conditionalvolatilityestimatedsemi-parametricallybyanexponentially-weightedmoving average(EWMA).3 Nevertheless,thistraditionalFHSVaRmodelhasaseverelimitation. Itrescalesthe varianceinthesample,butdoesnotchangetheobservedcorrelations. Thisresultcanbe easilyseen. Let𝑋 and𝑌 betwovariablesattime𝑡,withvariancesdenoted𝜎 and𝜎 . 𝑡 𝑡 𝑋,𝑡 𝑌,𝑡 Forsimplicity,letusassumebothhaveameanofzero. Then,ifthecurrenttimeperiod is𝑇,inFHSthevariablesarenormalizedsothat𝑋̂ = 𝜎 𝑋,𝑇𝑋 andsimilarlyfor𝑌;buttheir 𝑡 𝜎 𝑡 𝑡 𝑋,𝑡 1TheinternationalstandardsarefoundinCPMI-IOSCO(2012). SomeCCPsexceedthestandardsand calculateexpectedshortfallratherthanVaR. 2TheliteratureonVaRisvast;twotextbooktreatmentsofmanyareJorion(2006)andAlexander(2009). 3EstimatingEWMArequiressettingonlyasingledecayparameter. 2

Figure1: Time-varyingcorrelationofS&P500stocksusingEWMAestimator correlationisnotchangedas, 𝜎 𝜎 𝜎 𝜎 𝔼[ 𝑋,𝑇𝑋 𝑌,𝑇𝑌] 𝑋,𝑇 𝑌,𝑇 𝔼[𝑋𝑌] 𝜌 = 𝔼[𝑋̂ 𝑡 𝑌 𝑡 ̂] = 𝜎 𝑋,𝑡 𝑡𝜎 𝑌,𝑡 𝑡 = 𝜎 𝑋,𝑡 𝜎 𝑌,𝑡 𝑡 𝑡 𝑋̂ 𝑡 ,𝑌 𝑡 ̂ √ 𝔼[(𝑋̂ 𝑡 )2] √ 𝔼[(𝑌 𝑡 ̂)2] √ 𝔼[( 𝜎 𝜎 𝑋 𝑋 , , 𝑇 𝑡 𝑋 𝑡 )2] √ 𝔼[( 𝜎 𝜎 𝑌 𝑌 , , 𝑇 𝑡 𝑌 𝑡 )2] 𝜎 𝜎 𝑋 𝑋 , , 𝑇 𝑡 √𝔼[(𝑋 𝑡 )2] 𝜎 𝜎 𝑌 𝑌 , , 𝑇 𝑡 √𝔼[(𝑌 𝑡 )2] = 𝜌 . 𝑋,𝑌 𝑡 𝑡 Thisresultisaspecialcaseofpopulationandsamplecorrelationbeinginvarianttoindependentlinearscaling. Theimplicitassumptionthatvolatilitycouldbetime-varying,butcorrelationconstant, isbothnon-intuitiveandlikelycounterfactual. Observedcorrelationsduringperiodsof highervolatilityarecertainlyhigher(Solniketal.,1996). However,theincreasecanbethe resultofselectionbias(Boyeretal.,1999,LonginandSolnik,2001,ForbesandRigobon, 2002). Nevertheless,time-varyingassetcorrelationisnearlyaswell-establishedastylized factastime-varyingvolatility,includinginthesesamepapers;seealsoGroenenandFranses (2000),Baur(2006),BramanteandGabbi(2007)andLietal.(2024)amongmanyothers. To illustrate,Figure1showsthatthecorrelationsbetweencomponentsoftheS&P500exhibit pronouncedvariationandpossiblyregimeswitches. Tozoominonalargeshock,when VIX—acommonmeasureofexpectedvolatilityinU.S.equitymarkets—spikesinMarch2020 duetoCOVID-19,thecorrelationamongindividualstockssignificantlyincreased. This phenomenaissocommonthatithasashorthand: “allcorrelationsgotoone.” Theproblem isthatinthepresenceoftime-varyingcorrelationFHSVaRlikelyunderestimatestailrisk, especiallyforportfoliosexposedtocorrelationrisk. Partlytoaddressthisissue,Aramonteetal.(2013)proposedaVaRmethodologybased onaDynamicFactorModel(DFM). Themethodcanhandlebothtime-varyingvolatilities andcorrelationsforalargesetoffinancialvariables. DFMrepresentseachindividualrisk factorbyalinearcombinationoflatentriskfactorsandtheresidual. Thelatentfactorsare derivedfromprincipalcomponentanalysis(PCA).4 Theymodeleachlatentriskfactorasa multivariategeneralizedautoregressiveconditionalheteroskedasticity(GARCH)process. 4TheliteratureonPCAisvast. Shlens(2014)providesatutorial;JolliffeandCadima(2016)andJolliffe (2022)providemorerecentoverviews. 3

Thereis,however,asecondissuewithFHSthatisspecificallyachallengeformargin systems: thepotentialnumberofriskfactors. Thehighdimensionalityofmarginsystems andtherequirementthatcalculationsworkforanyclearedportfoliomeansthatthereisa highpremiumplacedoncomputationalefficiency. FHScanbeapplieddirectlytoaportfolio, treatingitasoneriskfactor. Alternatively,eachpositioninaportfoliocanbedefinedasa riskfactor;eachfactoristhenfilteredseparatelyandbeforebeingcombinedtoproduce theportfolioVaR. ForCCPmargining,thefirstapproachwouldentailrecalculatingthe filtereverytimeeachclearedportfoliochangedeveryday. Thesecondmethodinvolves a univariate calculation for every cleared contract every day. With a large number of changing portfolios and a large space of contracts, either approach can impose a high computationalburden. Inpractice, CCPshavegenerallyfollowedthesecondapproach, suggestingthecomputationalburdenofthesecondapproachispreferable. ToaddressshortcomingsinCCPs’useofFHS,theVaRmodelneedstoaddresstimevarying correlation without losing the other properties that make FHS popular, in particular its ability to be applied to a variety of portfolios, while maintaining reasonable computationalefficiency. AlthoughAramonteetal.(2013)intendedfortheirmethodto becomputationallyefficient,theDFMapproachrequiresMonteCarlosimulationofthe returnsofthelatentriskfactorsbasedontheestimatedGARCHprocess. Suchsimulations maybetoocomputationallyburdensomeformarginsystems. Thispaperproposesanalternativethattriestoaddressbothtime-variationinvolatility andcorrelation,whileremainingcomputationallyfeasibleatscale. Themethodestimates portfolioVaRbyapplyingFHStothemajorlatentriskfactorsderivedusingPCA,rather thantothevolatilitiesofindividualriskfactors. Afterfilteringandscalingthehistorical returns of latent risk factors, we then build the historical return of each individual risk factors using the PCA loading matrix. Portfolio P&L history is calculated using those returnsofindividualriskfactors. PortfolioVaRandothertailriskmeasurethencanbe estimatedusingtheP&Lhistory. Implicitly,weareassumingthattheprincipalcomponents arerelativelystablethroughtime. Thisassumptionisarguablymuchmorereasonablethan assumingthatcorrelationsarestable. ThemethodisdenotedPCAFHS. Itissomewhat similartoorthogonalGARCHorEWMA(Alexander,2008,2009,McNeiletal.,2015). Toreducecomputationalburden,onlyafewprincipalcomponentsareused. Consequently,theresidualriskofeachindividualriskfactorisnotrescaled. Byusingonlythe firstfewprincipalcomponents,themethodfocusesontheprimarysystematicdriversof risk. Besides reducing computational burden, using the first few principal components means only those principal components are assumed to be stable, representing market structure,ratherthanassumingthatallprincipalcomponentsarestablewhensmallerones aremuchnoisier. Nevertheless,forcertainportfolioswithconcentratedidiosyncraticrisk, differentprincipalcomponentsmayneedtobecalculatedtoaccountforthespecificrisk exposure. UsingPCAtoreducethecomplexityassociatedwithahighnumberofriskfactorscan bedonesuccessfullyifalimitednumberofprincipalcomponentsaccountforthemostof thevarianceinthejointdistribution. Itthereforewouldnotbesurprisingforourmethod 4

to work in applications where factor modeling is the dominant paradigm, for example placeswheretheyieldcurveisthedominantriskfactorandathree-factormodelcaptures most of the risk. Such circumstances include rates, bonds, and futures. See Alexander (2008,ChapterII.2)forexamplesanddiscussionofapplyingPCAininterestratesensitive portfolios. Tomakeitharder,weexploretheperformanceofthismethodinequitymarkets, where factor models are used, but the factor structure is less strongly established and consistent. OurapplicationismotivatedbysuchworkasLalouxetal.(2000),PafkaandKondor (2002)andPlerouetal.(2002),amongothers,thathaveappliedRandomMatrixTheory (RMT) to study large dimensional financial time series systems, such as stock markets. Thoseresearcheshavefoundthattheeigenvaluesofthecorrelationmatrixofstockreturns are consistent with those calculated using random returns, with the exception of a few largeeigenvalues,implyingalargedegreeofrandomnessinthemeasuredcross-correlation coefficients. Inaddition,theyfindthatthese“deviatingeigenvectors”arestableintime. They analyze the components of the deviating eigenvectors and find that the largest eigenvalue corresponds to an influence common to all stocks. Moreover, their findings indicated that these large eigenvalues, which do not conform to random returns, had eigenvectors that were more stable over time. These findings support using the major latentfactorsderivedfromthePCAmethodtomodelthesystematicriskinequitymarkets atalargescale. Nevertheless,thisapplicationisarguablymorechallenging;ifthemethod performswellhere,weexpectitsperformancewouldbeconfirmedformarginingassets thataregenerallyrecognizedtohavestrongstablefactorstructure. ToevaluatethePCAFHSmethod,weperformtwosetsoftests. Inthefirst,wesimulate data that have a variety of sharp changes in volatility or correlation. We then conduct comparisonsofPCAFHScoverageat99thpercentileagainsttraditionalFHS. Weinclude othercomparisons,particularlyagainsttheDFMmethodofAramonteetal.(2013). Fora volatilitybreakthatdoesnotchangecorrelation,PCAFHSperformsaswellastraditional FHS, and both perform acceptably. This result may be surprising, as this simulation exactlyfollowsthenullassumptionunderlyingofFHS. Thesituationchangesdramatically whencorrelationchangesareevaluated. PCAFHScontinuestoperformwell,evenasthe performanceoftraditionalFHSsharplydegrades,somuchsothatthemodelisstrongly rejected. Interestingly, for simulations with changing correlations, the performance of PCAFHSconsistentlyapproachesresultsproducedbythefullparametricestimationof theDFMmethod,butwithmuchlowercomputationalcosts. Subsequently,wetestthemethodwithconstructedequityportfoliosover2019to2021, sothatthevolatileCovid-19periodisincluded. Foradiversifiedportfolio,thetestclearly acceptsthePCAFHSmethod,whiletraditionalFHSisontheborderline. Theresultsare verysimilarforalong/shortequityportfolio. Foraportfolioconcentratedinidiosyncratic positions,whichischosenexplicitlytoviolatethePCAFHSmethod’sfocusonsystematic risk,theinitialapplicationofPCAFHSisrejectedwhileFHSisaccepted. Interestingly,the performanceofbothmethodsissimilarinresponsetotheinitialCOVIDshock,withone largeexceedance,butFHSkeepsmarginelevatedandavoidssubsequentsmallbreaches 5

thatimpactthePCAFHSmethod’scoverage. Severalalternativestoimprovethemodel performanceinthissituationareexamined. Tomorethoroughlyexaminepotentiallimitationsoftheproposedmethod,weexplore someparticularlychallengingsimulatedportfoliosincludingsparseportfolios,whichare morelikelytodeviatefromthebroadermarketbehaviorandsomesimulatedlong/short portfolios, which have sharply different correlations than observe in the market. The performanceofthePCAFHSmethodcandegrade. Theseresultsreflectthechallengeof producingamarginsystemthatworksforanyportfolio,andillustrateshowCCPsneedto closelymonitormarginperformance,particularlyforsmallerandidiosyncraticportfolios. Intheend,wefindthatthePCAFHSprovidesanalternativeVaRmethodforcalculatingmargin,improvingperformanceinthefaceofcorrelationchangesandbreakswhile retainingmanyofthepracticalfeaturesthatmakestraditionalFHSappealing. The remainder of the paper is organized as follows. Section 2 describes a general frameworktostudytheimpactofcorrelationchangeonportfolioVaRestimation. Section3 detailstheestimationofthePCAFHSVaR. Section4providesseveralsimulationstostudy PCAFHSVaRperformance. Section5providestheempiricalanalysisonstockportfolio. PerformanceofthedifferentmethodsofcalculatingVaRandtheassociatedstatisticaltests are compared for both the simulations and the empirical analysis. Section 6 has some moresimulationslookingattheimpactofsparseportfoliosandportfoliodirectiononthe method’s performance. The last section contains concluding remarks and thoughts for futureresearch,followedbyanappendix. 2 Correlation in FHS VaR FHS-VaRdoesnotcapturechangingcorrelationsbetweentime-varyingvolatilities;only unconditional correlation among the filtered variables is captured (Pritsker, 2006). Sun and Zhang (2021) provides a theoretic framework to study how correlation impacts in FHS VaR estimation. They examined two approaches to estimate portfolio VaR using a FHStechnique. ThefirstapproachappliesFHStoindividualriskfactors,andthesecond approachestimatestheFHSVaRbasedontheportfolio’sP&Lhistory. Theyfoundthat applyingFHStoindividualriskfactorscouldunderestimateportfolioVaR,whenthereis largechangeincorrelation,asthisapproachonlyconsidersthesampleaveragecorrelation. Toillustratetheproblem,letusassumeasingleriskfactor𝑟 followsaGARCH(1,1) 𝑖 process: 𝑟 = 𝜎 𝑒 𝑖,𝑡 𝑖,𝑡 𝑖,𝑡 𝑒 ∼ 𝑁(0,1) (1) 𝑖,𝑡 𝜎2 = 𝜔 +𝛼𝑟2 +𝛽𝜎2 𝑖,𝑡 𝑖 𝑖 𝑖,𝑡−1 𝑖,𝑡−1 withparameters𝜔,𝛼,and𝛽. Aportfolio’sP&Lwithposition𝑤 onriskfactor𝑟 is 𝑖 𝑖 PR = ∑𝑤𝑟 . (2) 𝑡 𝑖 𝑖,𝑡 𝑖≤𝑁 6

TorunFHSonsingleriskfactor,set 𝜎 𝜎 𝑖,𝑇 𝑖,𝑡 𝑟̄ = 𝑒 = 𝜎 𝑒 (3) 𝑖,𝑡 𝑖,𝑡 𝑖,𝑇 𝑖,𝑡 𝜎 𝑖,𝑡 sothat𝑟̄ isthefilteredreturnattime𝑡 −𝑇. 𝑖,𝑡 Asanapproximation,weassumeportfoliodailyreturnsfollowanormaldistribution, sotheVaRattime𝑇canbeestimatedbyapplyingaconstantmultipliertothestandard deviationofportfolioreturn: 2 VaR2 = Factor⋅𝐸[PR2] = Factor⋅𝐸[(∑𝑤𝜎 𝑒 ) ]. (4) 𝑇,𝑝 𝑡 𝑖 𝑖,𝑇 𝑖,𝑡 𝑖≤𝑁 If we assume that all the risk factors have same volatility 𝜎 at time 𝑇, and 𝜌 is the averagecorrelationbetweenriskfactorsforthetimeperiodfrom1to𝑇,wecanrewritethe portfolioVaRat𝑇: VaR = Factor⋅𝜎 𝐸[(∑𝑤𝑒 )] ≈ Factor⋅𝜎 ⋅𝜌. (5) 𝑇,𝑝 𝑇 𝑖 𝑖,𝑡 𝑇 𝑖≤𝑁 Equation(5)showsthatbyapplyingtheFHStechniqueonsingleriskfactors,theestimated portfolioVaRisafunctionofriskfactors’volatilityattime𝑇andsampleaveragecorrelation forthetimeperiodfrom1to𝑇. Time-varyingcorrelationisnotconsidered. Time-varyingcorrelationhasbeenextensivelystudiedinfinancialliterature. Awellknown example is Engle (2002), which developed a Dynamic Conditional Correlations modeltoaddressthetimevaryingnatureofcorrelationbetweenfinancialassetreturns. Aramonteetal.(2013)builtonthisworkindevelopingtheirDFMmethod. Toillustratetheimpactoftime-varyingcorrelationontheportfolioVaRestimation, assumethateachindividualriskfactorinaportfoliocanbemodeledbyaone-factormodel: 𝑟 = 𝜌 𝐼 + 1−𝜌2 𝑒 (6) 𝑖,𝑡 𝑖,𝑡 𝑡 √ 𝑖,𝑡 𝑖,𝑡 where𝐼 isthesystematiccommonriskfactor,𝜌 isthetime-varyingcoefficientforindi- 𝑡 𝑖,𝑡 vidualriskfactor𝑖,and𝑒 istheidiosyncraticresidualrisk. Thenaportfolio’sP&Lwith 𝑖,𝑡 position𝑤 onriskfactor𝑖canbewrittenas 𝑖 PR = ∑𝑤𝑟 = ∑(𝑤𝜌 𝐼 +𝑤 1−𝜌2 𝑒 ). (7) 𝑡 𝑖 𝑖,𝑡 𝑖 𝑖,𝑡 𝑡 𝑖√ 𝑖,𝑡 𝑖,𝑡 𝑖<𝑁 𝑖<𝑁 Byassumingzerocorrelationbetweensystematicrisk𝐼 andidiosyncraticrisk𝑒 as 𝑡 𝑖,𝑡 wellasbetweenidiosyncraticrisk𝑒 and𝑒 ,wehave 𝑖,𝑡 𝑗,𝑡 2 2 PR2 = (∑𝑤𝜌 𝐼) +(∑𝑤 1−𝜌2 𝑒 ) = 𝐼2⋅ ∑ 𝑤𝑤𝜌 𝜌 +∑[𝑤(1−𝜌 )𝑒 ] 2 . (8) 𝑡 𝑖 𝑖,𝑡 𝑡 𝑖√ 𝑖,𝑡 𝑖,𝑡 𝑡 𝑖 𝑗 𝑖,𝑡 𝑗,𝑡 𝑖 𝑖,𝑡 𝑖,𝑡 𝑖<𝑁 𝑖<𝑁 𝑖,𝑗<𝑁 𝑖<𝑁 7

Wefurtherassumethatallindividualriskfactorshavethesameaveragecorrelation𝜌 𝑡 andthenumberofriskfactors,𝑁,islarge,thenwecandroptheresidualrisktermin(8)to produce PR2 ≈ 𝜌𝐼2 = 𝜌𝜎2 (9) 𝑡 𝑡 𝑡 𝑡 𝐼,𝑡 where𝜎 isthevolatilityofsystematicrisk. 𝐼,𝑡 LetusassumethatportfolioreturnsfollowanormalGARCH(1,1)process. Portfolio VaRthencanbewrittenas VaR2 = Factor⋅𝜎2 = Factor⋅(𝜔 +𝛼 𝑃𝑅2+𝛽𝜎2 ) (10) 𝑝,𝑇 𝑝,𝑇+1 𝑝 𝑝 𝑇 𝑝,𝑇 = Factor⋅(𝜔 +𝛼 𝜌 𝜎2 +𝛽𝜎2 ) 𝑝 𝑝 𝑇 𝐼,𝑇 𝑝,𝑇 Ifcorrelationchangeswithoutchangingvolatility,thechangeofportfolioVaRwillbe: VaR2 −VaR2 = Factor⋅𝛼 (𝜌 −𝜌 )𝜎2 (11) 𝑝,𝑇 𝑝,𝑇−1 𝑝 𝑇 𝑇−1 𝐼,𝑇 showinghowchangesincorrelationcanimpactVaR—linearlyinthiscase—eveninsimplifiedexamples. Missingtheimpactofcorrelationchangesseemslikeasignificantholein theFHS-VaRmodel. Inthenextsection,weproposeamethodtofillthishole. 3 FHS VaR Model Using Latent Factors from PCA As an alternative to traditional FHS, we propose filtering based on principal components. Effectively,werotatethedatatoconstructuncorrelatedriskfactors—theprincipal components—andthenhistoricallyfilterthoselatentriskfactors. Thevolatilityofeach selectedlatentfactorisnormalizedandthenrescaledtobeconsistentwiththecorresponding latent factor in the current data. In contrast to traditional FHS, when the historical returnsofindividualriskfactorsarerebuiltboththe volatility andcorrelation hasbeen normalizedtobemoreconsistentwithcurrentvolatilityandcorrelation. To calculate principal components, denote historical data with 𝑇 observations on 𝑁 correlated asset or risk factor returns by a 𝑇 × 𝑁 matrix 𝑋. PCA will produce up to 𝑁 uncorrelated returns, called the principal components of 𝑋, each component 𝑃 being a simplelinearcombinationoftheoriginalreturnsas: 𝑃 = ∑𝑋 𝑊𝑇 (12) 𝑘,𝑡 𝑖,𝑡 𝑖,𝑘 𝑖<𝑁 where𝑊isthematrixofeigenvectorsof𝑋′𝑋/𝑇. Theweightsintheselinearcombinations aredeterminedbytheeigenvectorsofthecorrelationmatrixof𝑋,andtheeigenvaluesof thismatrixarethevariancesoftheprincipalcomponents. Theprincipalcomponentsare orderedaccordingtothesizeofeigenvaluesothatthefirstprincipalcomponent,theone correspondingtothelargesteigenvalue(i.e.,theonewiththelargestvariance)explains mostofthevariation;inahighlycorrelatedsystemthelargesteigenvaluewillbemuch largerthantherestandonlythefirstfeweigenvalueswillbesignificantlydifferentfrom 8

zero. Thus,insuchsystems,onlyafewprincipalcomponentsarerequiredtorepresentthe originalvariablestoahighdegreeofaccuracy. Since𝑊isanorthogonalmatrix,wecanrewrite(12)as𝑋 = 𝑃𝑊,thatis 𝑥 = ∑𝑊𝑇𝑃 . (13) 𝑖,𝑡 𝑖,𝑘 𝑘,𝑡 𝑘<𝑁 Weusethefirst𝑟 principalcomponents,whicharethekeyriskfactorsforthesystem;itis importanttochooseonlyafewofthese,astheempiricalresultsshow(Plerouetal.(2002, pp. 9–13)and(Lalouxetal.,2000,pp. 14,15)). Thenwecanrewrite(13)as 𝑥 = 𝑤𝑇 𝑃 +𝑤𝑇 𝑃 +⋯+𝑤𝑇𝑃 +𝜀 (14) 𝑖,𝑡 𝑖,1 1,𝑡 𝑖,2 2,𝑡 𝑖,𝑟 𝑟,𝑡 𝑖,𝑡 where𝜀 istheerrortermpickinguptheapproximationfromusingonlythefirst𝑟ofthe 𝑖,𝑡 𝑁principalcomponents. Alexander(2001,2008)proposedanO-GARCH/O-EWMAmodeltobuildaprincipal componentcovariancematrixforalarge𝑁‐dimensionalmultivariateprocess,wherethe name reflects that the principals components are orthogonal. Following Alexander’s framework,wecanestimatethevolatilityforthoseprincipalcomponentsusingeithera GARCHorEWMAmodel. Toreduceestimationrequirements,weusetheEWMAmodel: 𝜎2 = (1−𝜆)𝑃2 +𝜆𝜎2 . (15) 𝑃 ,𝑇+1 𝑘,𝑇 𝑃 ,𝑇 𝑘 𝑘 ApplyingFHStoprincipalcomponentreturns: 𝜎 𝜎 𝑃 ,𝑇+1 𝑃 ,𝑡 𝑃̄ = 𝑘 𝑘 𝑒 = 𝜎 𝑒 . (16) 𝑘,𝑡 𝑃 ,𝑡 𝑃 ,𝑇+1 𝑃 ,𝑡 𝜎 𝑘 𝑘 𝑘 𝑃 ,𝑡 𝑘 ThesingleriskfactorreturnbasedonFHSusingthefirst𝑟principalcomponentscanbe writtenas: 𝑥̄ = 𝑤𝑇 𝜎 𝑒 +𝑤𝑇 𝜎 𝑒 +⋯+𝑤𝑇𝜎 𝑒 +𝜀 . (17) 𝑖,𝑡 𝑖,1 𝑃 1 ,𝑇+1 𝑃 1 ,𝑡 𝑖,2 𝑃 2 ,𝑇+1 𝑃 2 ,𝑡 𝑖,𝑟 𝑃 𝑟 ,𝑇+1 𝑃 𝑟 ,𝑡 𝑖,𝑡 Foraportfoliowithlargenumberofriskfactors,wecanignoretheresidualrisk𝜀. Thena 𝑖 diversifiedportfolio’sP&Lwithposition𝑢 onriskfactor𝑖canbewrittenas 𝑖 𝑃𝑅 = ∑𝑢𝑥̄ ≈ ∑ 𝑢𝑤𝑇𝜎 𝑒 . (18) 𝑡 𝑖 𝑖,𝑡 𝑖 𝑖,𝑗 𝑃 𝑗 ,𝑇+1 𝑃 𝑗 ,𝑡 𝑖≤𝑁 𝑖≤𝑁,𝑗≤𝑟 PortfolioVaRisthenestimatedfromtheP&Ltimeseriesdata. Tocapturethemost recentcorrelationimpact,weproposetousetheEWMAcorrelationorcovariancematrixto derivethePCfactors’loadingmatrix. Factors’volatility𝜎 canbeestimatedusingeither 𝑃 𝑇+1 𝑗, EWMAorGARCH(1,1). GivenPCfactorreturnisaweightedaveragereturnofindividual riskfactor,𝜎 willbedeterminedbythevolatilityandcorrelationofindividualrisk 𝑃 𝑇+1 𝑗, factors. To test the effectiveness of this method, we next compare its performance to other methodsinavarietyofsimulations. 9

4 Simulation Studies We run four different simulations to examine the model performance of PCA FHS VaR. First,westudythescenariowhencorrelationisconstantwhilevolatilityhasasharpregime switch. This simulation tests how PCA FHS VaR responds to a pure volatility change; this scenario corresponds precisely to the assumptions of traditional FHS. Second, we examine the case when volatility is constant, but correlation experiences a significant regimechange. Third,westudythemodel’sperformancewhencorrelationhasmultiple regimeswitches. Lastly,wetesttheFHSVaRmodelinascenariowhenbothvolatilityand correlationregimeschangetogether. Thisscenarioismoreconsistentwiththestructureof financialtimeseriesdata. Foreachsimulation,weestimatethe99%VaRfortheportfoliousingthreedifferent techniques: FHS, PCA FHS, and DFM. The FHS method estimates VaR based on the traditional FHS technique by filtering individual risk factors. PCA FHS is the approach described in previous section. The DFM method is based on the method proposed by Aramonteetal.(2013),alsodiscussedpreviously. Forthefirsttwosimulations, wealso considertheindexapproach,whichappliesthestandardFHStechniquetotheportfolioP&L historydatacalculatedfromthepercentagechangeofportfoliovalue. Forsimplicitybut alsotomakeitmorechallengingforthePCAmethodstobeeffective,weuseonlythetop threeprincipalcomponentstoestimatePCAFHSandDFMVaRforoursimulationstudies. The next section will show that the model performance could change using different numbersofprincipalcomponent,especiallyfornon-diversifiedportfolios. Theperformanceofthedifferentmethodswillbeevaluatedbylookingattherealized coverageratio,thenumberofobservedbreaches,andtwostandardstatisticaltests: the Kupiectest(Kupiec,1995)andtheConditionalCoverageIndependence(CCI)test(Christoffersen,1998).5 TheKupiectestisthestandardtestofwhetherthenumberofbreachesis consistentwiththetargetedcoveragequantile. Itistwo-sidedandcanrejectfortoomany ortoofewbreaches. TheCCItestevaluateswhetherbreachesareclusteredratherthan independentlydistributedaswouldbeexpectedifthemodelisaccurate. Eachsimulation has300observations. Inthefirsttwo,theregimechangeoccursafter50observationsso thatthereisroughlyayearworthofdailyobservationsafterthechange. Thisdurationis designedtoreflecthowtheKupiectestisoftenimplementedinpractice. Nevertheless,as weexpectbreachestooccurmorefrequentlyfollowingregimechanges,theCCItestmay moreaccuratelyreflecthowwellthemodelsadjust. Thefirstscenarioisabaselineanalysistocomparesmodelperformancewhenthere is no correlation change and only a jump in volatility. As said, this simulation exactly matchestheunderlyingassumptionsoftheFHSmethod. Wesimulatedtwosetsofdata fromstudentt-distributionwithdegreeoffreedom4(set𝐴andset𝐵). Thisdistribution isfat-tailedandempiricallyfitsthetypicalestimatedlog-returndistributionoffinancial assets. Bothsetsofdatahave100variableswithvarietyofvolatilitylevels. Dataset𝐴and 5Although Gurrola-Perez (2018) suggests that validation of FHS VaR models needs to go beyond just backtesting,comparisonsinthispaperprimarilywillsticktobacktestingresults. 10

Figure2: simulationstudy1–PortfolioVaRvsdailyP&Lwithvolatilityswitching Table3: PortfolioVaRperformancewithvolatilityswitching Model Breaches Cov. Ratio POFTest* CCItest* FHS 6 98.0% 2.348 5.570 PCAFHS 6 98.0% 2.348 5.570 DFM 5 98.4% 1.120 5.570 RealVaR 2 99.3% 0.380 0.128 IndexVaR 7 97.6% 3.916 6.319 *The𝜀=.05quantilesforthePOFtestandfortheCCItestare3841and5991respectively. set𝐵bothhave0.5averagecorrelation,butvolatilityinset𝐵istentimeshigherthanin set𝐴. Then,webuildatimeserieswithvolatilityregimeswitchingbyappendingset𝐵to set𝐴. Weexamineanequalweightedportfolioacrossall100riskfactors. Figure2presentsthetimeseriesofP&Lmovesandmarginestimatesforeachmethod forthefirstscenario. Visually,thejumpinvolatilityiseasytosee,asmovementsinthe first 50 observations are much more muted than the subsequent observations. Table 3 presentsthe99%VaRbacktestingresults. Althoughthissimulationisdesignedtomimic theunderlyingassumptionsofthestandardFHSmethod,PCAFHSproducesverysimilar results;bothmethodshave6breachesin300simulateddateswith98%coverageratios. This resultdemonstratesthatthePCAFHSadaptstochangingvolatilityinlinewiththestandard method. ThePCAresultsareachievedeventhoughthetop3principalcomponentsexplain only50%ofoverallvariance. TheDFMmethodperformsslightlybetterthanbothFHSand PCAFHSintermsofthecoverageratio. AllthethreemodelsareacceptedbytheKupiec testandtheCCItest. TheDFMmethodperformsslightlybetterontheKupiectestdue toitslowernumberofbreaches,butperformsthesameasthetwoFHSmethodsonthe CCI test. Surprisingly, the index VaR has more breaches and fails both the Kupiec and CCI tests. The result could highlight another limitation of traditional FHS that its VaR estimationreliesontheforecastedvolatilitythatisusedtoscalethehistoricaldata. Ifthe forecastedvolatilityisoverlysensitivetothemethodused,themodelperformancecould becomprised,especiallywhenthevolatilityofthedatasamplevariesfrequently. 11

Figure4: Simulationstudy2–PortfolioVaRvsdailyP&Lwithcorrelationswitching Second,weexamineascenariowithcorrelationswitching. Again,wesimulatedtwo setsofdatafromstudentt-distributionwithdegreeoffreedom4(set𝐴andset𝐵). Both sets of data have 100 variables with variety of volatility levels, and the volatilities are thesameinbothsetsofdata. Set𝐴haslowcorrelationbetweenthevariables,whileset 𝐵 has high correlation. Then we build a time series with correlation regime switching byappendingset𝐵toset𝐴. Again,weexaminedanequalweightedportfolioacrossall 100riskfactorsandestimatethe99%VaRfortheportfoliousingthesamefourdifferent techniques: FHS,PCAFHS,DFM,andanIndexmethod;asdiscussedpreviously. Allthe estimationmethodsuse500daysofhistoryuptotime𝑇tocalculatedailyVaR. RealVaR attime𝑇istheparametricVaRbasedonthepre-specifieddistributionusedtosimulatethe data. WealsocomparetheaveragecomputationaltimeforthetraditionalFHS,PCAFHS, andDFMmethods. Figure4presentstheVaRanddailyP&Linpercentagefor300timeperiodswherethe correlationregimeswitchesinthetime𝑡 = 50. Sincethereisnochangeinvolatilitybetween tworegimes,correlationswitchingisthedriverofchangesinportfolioVaR. Nevertheless, thereisavisualjumpintheapparentvolatilityoftheportfoliothatlooksverysimilarto thevolatilitydrivenjumpinFigure2. Thissimilarityillustratesthatidentifyingthedriver ofchangesinportfoliovolatilityisnotstraightforward,andhowrestrictingattentionsolely tochangesinvolatility—astraditionalPCAimplicitlydoes—canbemisleading. Table5on thefollowingpageshowsthenumberofbreaches,oneyearcoverageratio,andKupiectest, aswellasConditionalCoverageIndependence(CCI)Test. Averageruntimesfortraditional FHS,PCAFHS,andDFMmethodsarealsopresentedinthetable. As expected, FHS VaR seriously underestimates the tail risk when the correlation switches from low to high. There are over 23 VaR breaches during the studied period when correlation switched. The coverage ratio is only 90.8% for the FHS VaR model. BothKupiectestandCCItestrejectthemodel. TraditionalFHSVaR,aswediscussedin theprevioussection,effectivelyusestheaveragecorrelationofthedataset. Themodel responsestothecorrelationchangegraduallyandcanonlycapturethecorrelationchange whenthemajorityofsampledataarerollingintothehighcorrelationregime. Consequently, itunderperformsforaperiodafterthecorrelationjumps. 12

Table5: PortfolioVaRperformancewithcorrelationswitchingfromlowtohigh Model Breaches Cov. Ratio* POFTest* CCItest* RunTime FHS 23 90.8% 62.810 57.640 0.15 PCAFHS 6 97.6% 3.550 3.822 0.04 DFM 4 98.4% 0.760 4.871 20.00 RealVaR 2 99.2% 0.108 0.144 IndexVaR 1 99.6% 1.176 1.196 *The𝜀=.05quantilesforthePOFtestandfortheCCItestare3841and5991respectively. BothPCAFHSandDFMmethodssignificantlyimproveVaRestimationperformance. PCAFHShas6VaRbreachesduringthestudiedperiodwhileDFMonlyhas4. Thecoverage ratioforPCAFHSis97.6%andDFMachieves98.4%,bothclosetothe99%coverageratio objective. Kupiec POS test and CCI test accept both PCA FHS and DFM VaR models. Interestingly, variance explained by the top 3 principal components varies from only 33%priortothecorrelationchangeto94%afterwards. Itisworthpointingoutthatthe simulationiseffectivelyaregimechange—alwaysachallengeforVaRcalculations–sothe abilityofthesetwomodelstoadaptquicklyisimpressive. TheDFMmethoddoesperformslightlybetterhere. But,thereisacost. PCAFHS,asa nonparametricmethod,hastherelativeadvantageofcomputationefficiency. Theaverage runtimeforPCAFHSinthissimulationisonly 0.04secondscomparedto20.0seconds fortheDFMmodel.6 AmongtraditionalFHS,PCAFHS,andDFM,PCAFHSisthemost efficientapproach. Outofthefourapproaches,IndexVaRhasthebestmodelperformancewithcoverage ratioover99%andonlyoneVaRbreach. ThisresultisnotsurprisingbecauseIndexVaRis applyingFHStoportfolioP&Lhistory,andfactorsinbothsystematicandidiosyncratic risk. Nevertheless,thesecondsimulationstudyshowsthatFHSVaRmodelcancapture thetailriskwhenthereisasignificantchangeinthecorrelation. Tofurtherexamineperformance,ourthirdsimulationstudiesmodelperformancewhen therearemultiplemajorcorrelationchanges. Thecorrelationchangesfromlowtohigh,a coupleoftimes. WecomparethemodelperformanceforthreeVaRmodel: traditionalFHS, PCAFHS,andDFM. Figure6onthenextpagepresentsthe99%VaRsagainstportfoliodaily P&LandTable7onthefollowingpageshowsthenumberofbreaches,oneyearcoverage ratio,POFtest,andCCItest. Again,traditionalFHSunder-performsinthisscenario,asit doesnotadapttothecorrelationregimeswitches;itscoverageratioislessthan99%,and boththePOFandCCItestsrejectthemodel. ThePCAFHSandDFMmodelsperformwell inthisscenario. Theircoverageratioisoverthe99%requirement,andbothstatisticaltests acceptthosetwomodels. 6Runtimesarecalculatedusingadesktoppersonalcomputerwithoutanyattempttooptimizeperformance. Theruntimesprovideametricforcomparingrelativeefficiency,butanindustryapplicationwouldcertainly bemoreefficient. 13

Figure6: Simulationstudy3–PortfolioVaRvsdailyP&Lwithmultiplecorrelationregimes Table7: PortfolioVaRperformancewithmultiplecorrelationregimes Model Breaches Cov. Ratio POFTest* CCItest* FHS 7 97.6% 3.916 6.077 PCAFHS 3 99.0% 0.000 0.060 DFM 2 99.3% 0.380 0.408 *The𝜀=.05quantilesforthePOFtestandfortheCCItestare3841and5991respectively. Ourfourthsimulationstudiesthemodelperformancewhenbothvolatilityandcorrelationchange. Wemaketwochangestothesimulateddatasets. First,whencorrelation isswitchingfromlowtohigh,wealsoincreasethevolatilitylevelforalltheriskfactors. Second, we make the data switching back to low correlation and low volatility regime sothattherearetworegimeswitchesinthetimeseriesdata. Thisallowsustocompare FHSVaRwithPCAFHSVaRresultswhenbothcorrelationandvolatilityreturntonormal afteraspike. Figure 8 on the next page presents the 99% VaRs against portfolio daily P&L. First regimeswitchishappenedat datapoint 𝑡 = 50 andsecondregimeswitchat datepoint 𝑡 = 150. Thereisasignificantjumpinportfolio’sdailyPLvolatilitywhenbothcorrelation andvolatilityincrease. TraditionalFHSVaRfailstorespondtoregimechangeandunderestimatestheportfoliotailrisk,especiallyinthebeginningoftheperiodwhentheregime changed. Instarkcontrast,PCAFHScapturesthechangeandisveryresponsivetotherisk profiledynamic,drivenbybothcorrelationandvolatilitychanges. Whenbothcorrelation andvolatilityswitchbackfromhightolow,DFMVaRrespondsquicklytothechangein the portfolio risk profile. The VaR estimated using DFM method quickly decreases and revertstotheleveloflowvolatilityandcorrelationregimeperiod. PCAFHSandtraditional FHSVaRadjusttotheregimechangemoresmoothlyandstayelevatedforanextended period. Table9onthefollowingpageshowsthecoverageratio,numberofbreaches,Kupiec POF test, and CCI test. FHS has 8 VaR breaches to fail the 99% coverage ratio test. The 14

Figure 8: Simulation study 4–Portfolio VaR vs daily P&L with correlation & volatility changes Table9: PortfolioVaRperformancewithbothvolatilityandcorrelationchanges Model Breaches Cov. Ratio POFTest* CCItest* FHS 8 97.3% 5.778 6.218 PCAFHS 2 99.3% 0.381 0.408 DFM 2 99.3% 0.381 0.408 *The𝜀=.05quantilesforthePOFtestandfortheCCItestare3841and5991respectively. KupiectestsandCCItestbothrejectthemodelfor99%VaR. PCAFHS,ontheotherhand, onlyhas2breachesandmeetthe99%coverageratiotest. ItalsopassedbothKupiecPOF and CCI statistic test. Again,in this simulation, the variance explained by the principal components varies from 33% in low correlation regimes to 94% in the high correlation regime. ThemorecomputationallyintensiveDFMmethodhadidenticalresults. Therearedifferentmethodsforcalculatingprincipalcomponents. Totestthepotential sensitivitytodifferentestimationmethods,wererunthePCAFHSmodelonthisfourthset ofsimulateddatawithprincipalcomponentsderivedfromfourmethods: thefirstmethod usestheEWMAestimatecovariancematrix;thesecondmethodusesthesamplereturn dataset;thethirdmethodusescovariancematrixfilteredusingRMTtechniqueofPlerou etal.(2002);andthelastmethodisbasedonrobustly-estimationofthecovariancematrix usingthetechniqueproposedbyMaronnaandZamar(2002). ForeachPCAFHSestimation, weusethetopthreelargestPCsaslatentriskfactors. Table10onthenextpageshowstheperformanceof99%VaRsand95%VaRsforfour differentPCAFHSmethods. Wedidnotfindmeaningfuldifferencesinmodelperformance across the four methods. All of them have the same number of breaches and pass the tests. This lack of sensitivity to the PCA method used in VaRs estimation highlights thateigenvectorscorrespondingtolargeeigenvaluesareconsistentlyidentifiedandare relativelystableovertime,asPlerouetal.(2002)hasshown. Importantly,givenitsefficient estimation,usingEWMAtoestimatethecovariancematrixseemstoproduceresultsthat areconsistentwiththeothermethods. 15

Table10: PortfolioVaRperformancewithdifferentPCAFHSmethods 99%VaR 95%VaR Model Cov. Breaches POFtest*/ Cov. Breaches POFtest*/ Ratio CCItest* Ratio CCItest* PCAFHS1 99.3% 2 0.381 96.0% 12 0.021 0.408 1.855 PCAFHS2 99.3% 2 0.381 96.0% 12 0.021 0.408 1.855 PCAFHS3 99.3% 2 0.381 96.0% 12 0.021 0.408 1.855 PCAFHS4 99.3% 2 0.381 96.0% 12 0.021 0.408 1.855 *The𝜀=.05quantilesforthePOFtestandfortheCCItestare3841and5991respectively. 5 Empirical Study In this section, we study whether the PCA FHS VaR model can address the correlation riskforportfolioswithdifferentriskprofiles. Werunempiricalstudiesforthreedifferent portfolios: alongonlydiversifiedstockportfolioacrosstheS&P500index,a130-30(long 130short30)balancedportfolio,aswellasaportfolioconcentratedonnameswithhigh idiosyncraticrisk. Sinceweexcludethenon-systematicriskinourPCAFHSmodelsetting, studyingaportfoliowithconcentratedidiosyncraticriskprovidesasterntestofthemethod. In all three cases, we use daily returns on the S&P 500 stocks from the Center for ResearchinSecurityPrices,LLC(CRSP)anduseonlystocksthathavenon-missingreturns onalltradingdaysfrom2017to2022. Thetotalnumberofstocksintheuniverseis385. Sample portfolios have $100000 market values and portfolio weights remain constant throughoutthesampleperiod. Werundaily99%VaRcalculationsusingtraditionalFHS andPCAFHSmethods,thencomparetheirperformancesduringtheCOVID-19crisisalong fourdimensions: thenumberofVaRbreaches,thecoverageratio,KupiecPOFtest,and CCItest. WestudythePCAFHSVaRresultsusingthefourdifferentmethodsforderiving principalcomponents,thatwerediscussedinthesimulationstudysection. DiversifiedPortfolio Plerouetal.(2002)hasshownthatthetoplatentfactorderivedfromS&P500stockreturns canconsistentlyrepresentthesystematicriskinequitymarket. Givenawelldiversified portfolioismainlyexposedtosystematicmarketrisk,weexpectthatourPCAFHSVaR with only the top two latent factors will perform well for this portfolio. We could of courseincludemore,buttestingtheeffectivenessofthemethodwithonlytwomakesthe test more severe. Figure 11 on the following page presents the percentage of variance 16

Figure11: Percentageofvarianceexplainedbytop2PCsforS&P500 explained by the top two principal components of S&P 500 stock returns. The top two principalcomponentsalwaysexplainmorethan55%ofvarianceduringtheperiod. The varianceexplainedisoftenagoodbithigherand,whenmarketvolatilityescalatedduring theCOVID-19crisis,thepercentageexceeded90%. Table12comparestheperformancebetweenFHSVaRandPCAFHSVaRduringCOVID- 19 crisis period. For 99% VaR, the FHS model has 6 breaches over the one-year period coveringtheCOVID-19financialmarketturmoil. itsVaRcoverageratioisbelowthe99% requirement. AlthoughbothKupiecandCCItestssuggeststhatthemodelperformance isacceptable,thetestresultsareveryclosetothecriticalvaluetorejectthemodel. All four PCA FHS methods achieve a better VaR performance, even limiting the method to Table12: FHSandPCAFHSVaRperformancefordiversifiedportfolio 99%VaR 95%VaR Model Cov. Breaches POFtest*/ Cov. Breaches POFtest*/ Ratio CCItest* Ratio CCItest* FHS 98.0% 6 3.555 94.7% 16 0.951 5.957 2.530 PCAFHS1 99.0% 3 0.095 95.0% 15 0.496 0.164 1.651 PCAFHS2 98.7% 4 0.769 95.0% 15 0.496 0.887 1.651 PCAFHS3 98.7% 4 0.769 95.0% 15 0.496 0.887 1.651 PCAFHS4 98.7% 4 0.769 95.0% 15 0.496 0.887 1.651 *The𝜀=.05quantilesforthePOFtestandfortheCCItestare3841and5991respectively. 17

Figure13: DailyP&LvsVaRfordiversifiedportfolio onlyusethefirsttwolatentfactors. EWMAbasedPCAFHSVaRperformsbestwithonly threebreaches. Italsohasthecoverageratioof99%level. BoththeKupiectestandtheCCI testacceptthemodelwith95%confidence. VaRfromtheotherthreePCAFHSmethods have very similar performances. All of them pass the Kupiec test and CCI test with 4 VaRbreachesoverthetestingperiod. For95%VaR,theresultsshowthattheperformance betweenPCAFHSandtraditionalFHSareverysimilar,asFHShasonlyonemorebreach. BothmodelsareacceptedbyboththeKupiecandCCItests. Figure13presentsthehistoryofportfolio’sdailyP&LanddailyVaRresults. When market volatility elevated during the COVID-19 period, correlation among stocks also increasedsignificantly. ThesuperiorperformanceofthePCAFHSmethodthroughthis periodofmarketstressdemonstratesthebenefitofcapturingcorrelationchangesinVaR estimation,especiallyathigherquantiles. TotestwhetheraddingmorelatentfactorswillsignificantlyimproveVaRestimation, wererunourPCAFHSmodelwithadditionalPCAlatentriskfactors. TheaverageVaR duringthe2020COVID-19periodestimatedusingthefourPCAFHSmethodswithdifferent numbersofPCsispresentedinTable14onthenextpage. Again,allfourdifferentPCAFHS methodsproduceverysimilarVaRresults. AddingmorePCstoVaRestimationdoesnot changetheVaRresultsmeaningfullyforthisdiversifiedportfolio. Thisresultisexpected givenawell-diversifiedportfolioismainlyexposedtosystematicmarketrisk, whichis wellcapturedbythetopprincipalcomponent. ThelastcolumnofTable14presentstheaverageruntimesfortheEWMAapproach. Theincreaseintimeasthenumberofprincipalcomponentsincreasesreflectsthatthereisa trade-offbetweentheeffectivenessandtheefficiencyforthePCAmethods. Implementation of a PCA FHS margin system would have to consider where to set the trade-off for a particularmarkettypeofportfolios. Forthediversifiedequityportfolio,asmallnumber offactorsissufficienttocalculateVaReffectively,sothecalculationcanbechosentobe 18

Table14: Average99%VaRwithdifferentnumberofPCsfordiversifiedportfolio PCFactors EWMA SampleAvg. RMTadj. Robust RunTime 1PC −3113.38 −3047.98 −3092.39 −3089.37 2.0 2PC −3121.12 −3048.35 −3099.65 −3091.32 2.0 5PC −3058.82 −2960.70 −3005.94 −2999.13 3.0 10PC −3086.62 −2975.45 −3030.45 −3025.80 5.0 30PC −3144.34 −2995.30 −3051.37 −3044.12 12.0 *TheruntimeisaveragerunningtimeinsecondstoestimateVaRonce. extremelyefficient. 130-30portfolio This section evaluates the PCA FHS model performance for another popular portfolio: the 130-30 strategy portfolio. This strategy is often called a long/short equity strategy, referringtoaninvestingmethodologypopularlyusedbyinstitutionalinvestors. A130-30 designation implies using a ratio of 130% of starting capital allocated to long positions madepossiblebytakingin30%ofthestartingcapitalfromshortingstocks. Thisstrategy tendstohavelargerexposuretosystematicriskthanatraditionaldiversifiedlong-only portfolio. Figure15onthefollowingpageshowsthehistoryofthedailyVaRvsportfolioP&L. Table 16 on the next page compares VaR results between traditional FHS and PCA FHS methodsduringtheCOVID-19crisisperiod. AllourfourPCAFHSmethodsperformswell inmeasuringthetailriskforthistypeofportfolio. For99%VaR,PCAFHSmethodsonly havethreebreachesoverthe300-dayperiod,theyallmeetthe99%coverageratiotestand are accepted by the Kupiec test as well as CCI test. For 95% VaR, PCA FHS has 15 VaR breaches,twobreacheslessthanFHS. TheKupiectestandCCItestfavorthePCAFHS methodtoo. TraditionalFHSVaRhasmorebreachesandfallsbelowthecoverageratio requirement. TheKupiectestandCCItestresultsareclosetothecriticalvaluetorejectthe model. Thisresultisexpectedgiventhisportfolioisalsodiversifiedandmainlyexposedto systematicrisk. TofurthertestourPCAFHSVaRmodel,weperformthesameanalysisfordifferent long/shortcombinationportfolios. Table17onpage21presentsthecoverageratioresult usingEMWAbasedPCAFHSandtraditionalFHS. Forhedgeratioupto70%,PCAFHSVaR with only the top two principal components performed better than traditional FHS. Idiosyncraticriskexposureforthoseportfoliosisrelativelysmallcomparedtosystematic riskexposure. Forthehigherhedgeratioportfolio,idiosyncraticriskstartstooutweigh generalmarketriskduetotheextensiveoffsettingduetotheshortpositions. Interestingly, startingata170-70strategyportfolio,thetraditionalFHSmethodstartsoutperformingthe PCAFHSmethodandevenoverestimatesriskatnearlyfullyoffsetlevels. Theseresults suggestaneedtoincluderesidualriskintheVaRestimationinsuchcases; eithermore 19

Figure15: DailyP&LvsVaRfor130-30portfolio(PCAFHSwithtop2PCs) latentfactorsareneededtoachievebetterperformanceorlatentfactorsneedtobetailored forthespecificresidualriskremainingintheportfoliobycalculatingprincipalcomponents ontheactualpositions. Thetradeoffswillbeexaminedmorecloselyinthenextsection whichconsidersconcentratedportfolios. Table16: Average99%PCAFHSVaRfor130-30portfolio 99%VaR 95%VaR Model Cov. Breaches POFtest*/ Cov. Breaches POFtest*/ Ratio CCItest* Ratio CCItest* FHS 98.0% 6 3.555 94.3% 17 1.540 5.785 1.605 PCAFHS1 99.0% 3 0.069 95.0% 14 0.096 0.126 1.737 PCAFHS2 99.0% 3 0.095 95.0% 15 0.496 0.126 1.567 PCAFHS3 99.0% 3 0.095 95.0% 15 0.496 0.126 1.567 PCAFHS4 99.0% 3 0.095 95.0% 15 0.496 0.126 1.567 *The𝜀=.05quantilesforthePOFtestandfortheCCItestare3841and5991respectively. 20

Table17: VaRcoverageratioforlong/shortportfolios 99%VaR 95%VaR Long/Short coverage coverage hedgeratio PCA FHS PCA FHS 100/0 99.00% 98.40% 95.60% 94.00% 100/10 99.00% 98.40% 95.60% 94.00% 100/20 99.00% 98.40% 96.40% 94.00% 100/30 99.00% 98.00% 96.00% 94.00% 100/40 99.00% 98.40% 95.60% 94.00% 100/50 99.00% 97.60% 97.20% 94.00% 100/60 98.80% 98.40% 95.20% 94.40% 100/70 98.00% 99.60% 94.76% 94.94% 100/80 98.00% 99.60% 93.62% 95.88% 100/90 91.83% 99.91% 84.20% 96.06% 100/100 64.12% 100.00% 59.84% 97.26% ConcentratedPortfolio ThissectionexaminestheperformanceofPCAFHSVaRmodelforaportfoliowithconcentratedpositionsonstockswithlargemarketvolatility. Wetesttwodifferentapproaches toderivethePCAlatentriskfactors. ThefirstapproachestimatesPCAlatentriskfactors usingthedailyreturnof385stocksinourdatasample. Thesecondapproachonlyusesthe stockreturnintheconcentratedportfolio. Giventhataconcentratedportfolioismainly exposedtoidiosyncraticrisk,weexpectthatPCAFHSmethodusingonlythetopprincipal componentsfortheoverallmarketcouldunderestimatethetailriskforthisportfolio. More factorsmightbeneededtocapturetheidiosyncraticrisk. Figure 18 on the next page presents portfolio P&L against traditional FHS VaR and PCAFHSVaR. Table19onthefollowingpageshowstheaveragePCAFHSVaRresults during 2020 COVID-19 period with the top two PCs. As expected, PCA FHS produces consistentlysmallerestimatesofVaRthanFHSVaRhere. Thetoptwoprincipalcomponents cannot capture the idiosyncratic risk in the portfolio and consequently PCAFHS underestimatesthetailrisk. For99%VaR,EWMAPCAFHShas9VaRbreachesoverthe 300-dayperiodwhileotherPCAFHSmethodshave13breaches. AllPCAFHSmodelsfail tomeet99%thecoverageratiotestandarerejectedbytheKupiectestandCCItest. Similar resultsareobservedfor95%VaR. Ontheotherhand,thetraditionalFHSmodelperforms wellonthisconcentratedportfolio. Atthe99%level,traditionalFHSmethodonlyhastwo VaRbreachesoverthe300-dayperiod. Itmeetsthe99%coverageratiotestrequirement. At95%,FHSVaRhasonly12breacheswithcoverageratioover95%. TheKupiectestand CCItestacceptthetraditionalFHSmodelforboth99%and95%VaR. Thisresultindicates 21

Figure18: DailyP&LvsVaRforconcentratedportfolio(PCAFHSwithtop2PCs) thattraditionalFHSVaRmodelcancaptureidiosyncraticriskthatmaybemoredrivenby volatilityratherthancorrelation. Thispoorperformanceshowsthatthetoptwoprincipalcomponentsareinsufficient toaccuratelyestimateVaRfortheconcentratedportfolio. Itcanbeaddressedbyadding more PCs into the model estimation. Figure 20 on the next page shows portfolio P&L againsttraditionalFHSVaRaswellasPCAFHSVaRbasedon30 PCs. VaRestimations fromPCAFHSandtraditionalFHSareclosetoeachother. however,traditionalFHSmodel Table19: FHSandPCAFHSVaRperformanceforconcentratedportfolio(withTop2PCs) 99%VaR 95%VaR Model Cov. Breaches POFtest*/ Cov. Breaches POFtest*/ Ratio CCItest* Ratio CCItest* FHS 99.3% 2 0.108 96.0% 12 0.021 0.188 0.417 PCAFHS1 96.7% 9 9.661 90.0% 25 9.189 10.731 17.071 PCAFHS2 95.0% 13 21.395 87.0% 29 15.462 21.571 25.309 PCAFHS3 95.0% 13 21.395 87.7% 29 15.462 21.571 25.309 PCAFHS4 95.0% 13 21.395 87.0% 33 22.979 21.571 28.945 *The𝜀=.05quantilesforthePOFtestandfortheCCItestare3841and5991respectively. 22

Figure20: DailyP&LvsVaRforconcentratedportfolio(PCAFHSwithtop30PCs) stillhasbettermodelperformancewithfewerbreaches. AsseeninTable21,calculatingthe PCAFHSmodelwith30PCssignificantlyreducesthenumberofbreachesatthe99%VaR level. Clearly,thePCAFHSmodelneedsmorelatentfactorsthantwoforthisconcentrated portfolio. Toseehowperformanceimproveswithmorelatentfactors,Table22onthenext pagepresentstheaverageVaRnumbersusingPCAFHSmethodswithaincreasingnumber ofPCs. WhileaddingmoreprincipalcomponentscanimprovePCAFHSVaRperformance, italsorequiresmorecomputationandreducesthemethod’sefficiencybenefit. Asseenin Table21: FHSandPCAFHSVaRperformanceforconcentratedportfolio(with30PCs) 99%VaR 95%VaR Model Cov. Breaches POFtest*/ Cov. Breaches POFtest*/ Ratio CCItest* Ratio CCItest* FHS 99.3% 2 0.108 96.3% 11 0.197 0.187 0.415 PCAFHS1 99.0% 3 0.069 93.0% 21 4.577 0.126 9.795 PCAFHS2 98.0% 6 3.555 91.7% 25 10.327 3.571 17.945 PCAFHS3 98.0% 6 3.555 91.7% 25 10.327 3.571 17.945 PCAFHS4 98.0% 6 3.555 92.4% 23 6.527 3.571 13.294 *The𝜀=.05quantilesforthePOFtestandfortheCCItestare3841and5991respectively. 23

Table22: Average99%PCAFHSVaRforconcentratedportfolio PCFactors EWMA SampleAvg. RMTAdj. Robust RunTime* 1PC −3194.50 −2661.46 −2698.81 −2695.97 2.0 2PC −3284.58 −2878.23 −2795.35 −2814.58 2.0 5PC −3749.87 −3560.97 −3584.53 −3524.78 3.0 10PC −3935.98 −3720.82 −3837.27 −3760.70 5.0 30PC −4236.35 −3844.83 −3880.20 −3809.51 12.0 *TheruntimeisaveragerunningtimeinsecondstoestimateVaRonce. thelastcolumn,movingfromtwotothirtyfactorsincreasesthecomputationtimebyan orderofsix. Thisresultsuggeststhatforaportfoliowithlargeidiosyncraticrisk,itmaybe necessarytomoredirectlymodeltherisk. AnalternativeapproachtoenhancethemodelperformanceforPCAFHSistoconduct PCAonstockreturnsinthisconcentratedportfolioonly. Notably,theprincipalcomponents wehaveused to estimate VaR forthis concentrated portfolio arecalculated using all of S&P500stockreturns,althoughthisportfolioisconcentratedonlyon20stocks. Tailoring themethod,wecanestimatePCAFHSVaRusingthetopprincipalcomponentsderived fromthereturndataforstocksinthisportfolioonly. Ifthosetopprincipalcomponents capture a large portion of the volatility for this portfolio, our PCA FHS model should perform well for this concentrated portfolio. Table 23 on the following page presents the results of both upside and downside 99% VaR estimated using the first 2 principal componentsderivedwiththisnewapproach. WeonlytestthePCAFHSmethodbased onsamplecorrelationmatrixgivenourestimatedVaRnumbersarenotsensitivetothe methodusedtoconductPCA. Table23onthenextpageshowsthatPCAFHSmodelachievesthesameperformance asthetraditionalFHSmethodwhenestimatingtailriskforaconcentratedportfolio. The twoapproacheshavethesamenumberofVaRbreachesaswellasthecoverageratio. The Kupiec test and CCI test results suggest both models are acceptable. Figure 24 on the followingpagedisplaysthedailyhistoryof99%VaRvsportfolioP&Linpercentageofportfoliovalue. Ingeneral,whenmarketvolatilityismuted,VaRestimatedbyPCAFHSusing portfoliospecificlatentfactorsissimilartotheestimatefromtraditionalFHS. However, whenmarketvolatilityspikesasinMarch2020,thejumpinVaRisnoticeablyhigherwith PCAFHScomparedtotraditionalFHS. SincethePCAFHSmodelimplicitlyincludesthe sharpchangesincorrelationsobservedinthisperiod,thisresultisperhapsnotsurprising. 6 Limitations TheresultstodatehaveshownhowthePCAFHSmodelcanbeaneffectivemarginmethod inthefaceofchangingcorrelations. However,theresultsalsosuggestthatthemethod’s effectivenessdependsonhowexposedaportfolioistomoresystematicriskfactorsrather 24

Table23: FHSandPCAFHSVaRforconcentratedportfolio(PCAfromselectedriskfactors) 99%VaRUpside 99%VaRDownside Model Cov. Breaches POFtest*/ Cov. Breaches POFtest*/ Ratio CCItest* Ratio CCItest* FHS 99.0% 3 0.095 99.3% 2 0.108 0.731 0.731 PCAFHS 99.0% 3 0.095 99.3% 2 0.108 0.185 0.185 *The𝜀=.05quantilesforthePOFtestandfortheCCItestare3841and5991respectively. Figure24: DailyP&LvsVaRforconcentratedportfolio(PCAFHSwithtop2PCsusing selectedriskfactors) thanidiosyncraticrisks. Inonesense,suchresultshighlightthechallengefacedbymargin systems: effectivelyestimatingriskforanyfeasibleportfolioisadifficultproblem. Inthis section,wefurtherstudythelimitationsofthePCAFHSmodelfortwodifferentportfolio characteristics: sparsityandoffsets. Sparsity Thefirstfactorweexamineissparsity. Ifaportfoliohasrelativelyfewstocksitmaybe moreexposedtoidiosyncraticrisk. Tolookattheimpact,wesimulate100riskfactorsfrom amultivariatestudentt-distributionwithdegreeoffreedom4andanaveragecorrelation of0.5. WearenotchangingtheparameterssothattheoverallVaRisconstant. Wethen takesamplesoftheriskfactors,sotheresultingriskwilldependonthenumberofrisk factors. Westartwithalltheriskfactorsandthentakesmallerandsmallersamples. Smaller samples,ofcourse,canintroducemoresamplingvolatilityduetolessdiversification. The resultsareshowninTable25onthenextpage. Thetableshowstheportfoliovolatility,the actualparametricVaR,andVaRestimatedbyPCAFHSintwoways. Thefirstestimates principalcomponentsfromalltheriskfactors,whilethesecondusesonlytheriskfactors intheportfolio. Ineachcase,threeprincipalcomponentsareused. 25

Table25: 99%PCAFHSVaRforshrinkingportfolios PCAFHSVaR Numberof Parametric All Portfolio factors Volatility VaR factors factors 100 2.59 6.12 6.13 6.13 90 2.60 6.07 6.08 6.02 80 2.66 6.20 6.20 6.17 70 2.62 6.11 6.12 6.30 60 2.62 6.11 6.11 6.34 50 2.63 6.21 6.17 6.70 40 2.74 6.37 6.30 6.88 30 2.77 6.48 6.35 6.72 20 3.02 7.04 6.90 7.15 10 3.02 7.05 6.45 7.00 It is informative to compare the results here to those for long/short portfolios we studiedintheprevioussection. Whentheshortportfolioreached70%—sothattheresidual exposure was around 30%—the PCA FHS performance started to degrade compared to traditionalFHS. Similarly here, whentheportfoliocontains30%oftheriskfactors, the VaRestimatestartsfalling2%ormorebelowtheactualparametricvaluewhencalculated usingallriskfactors. Inthiscase,switchingtoPCAFHSestimationusingonlyselected riskfactorsisabetterapproachtocapturethetailriskoftheportfolio. Takentogether,theseresultssuggesttheperformanceofPCAFHSforportfoliowith moreunsystematicexposures,whetherduetoportfolioconcentrationorextensivehedging, would need to be monitored. Such a specific portfolio might bring less systematic risk, butcouldstillresultinlargelossestoaCCPespeciallywhentheexposureissizableand concentrated. Direction&Offsets Buildingonthelong/shortresults,hereweconductasimilarsimulationtotheoneusedto lookatsparsity,butweallowconstructingportfolioswithbothlongandshortpositionsand allowtheportfoliostoevenbeprimarilyshortindirection. Soforexample,a40/60portfolio herewouldinclude40longsimulatedriskfactorsand60short,allequallyweighted. Here weonlypresentresultsusingthetop3principalcomponentscalculatedusingalltherisk factors. The results are in Table 26 on the following page. Not surprisingly, as the hedging ratiogrows,theperformanceofthePCAFHSestimateworsens. Specifically,thePCAFHS VaRestimateswhen40to50%ofthepositionsareoffsetorhedgedfallwellshortofthe parametric estimates. This result is intuitive as hedged portfolios are mostly exposed 26

Table26: 99%PCAFHSVaRforlongandshortportfolios Long Short Parametric PCAFHS factors factors Volatility VaR VaR 90 10 2.58 4.82 4.84 80 20 2.70 3.81 3.83 70 30 2.63 2.36 2.28 60 40 2.62 1.29 1.18 50 50 2.74 0.72 0.23 40 60 2.87 1.28 1.10 30 70 3.05 2.35 2.27 20 80 3.52 3.80 3.81 10 90 3.57 4.82 4.85 toidiosyncraticrisk,andthetopprincipalcomponentsdonotmodelitaswell. Adding moreprincipalcomponentscanimprovemodelperformanceinthiscase. Forexample,for the50/50portfolio,using40principalcomponentscanremarkablyimproveperformance: thePCAFHSVaRestimateincreasesto0.73almostidenticaltotheparametricestimate. However, thissolutionreducestheefficiencyofthismodelastheruntimeincreasesto 16secondsfrom2seconds. 7 Conclusion TraditionalFHSVaR,whichfiltersvolatilityforeachindividualriskfactor,doesnotexplicitlyaddressthetime-varyingcorrelationobservedinfinancialreturns. Whenthereismajor correlationregimeswitchingamongindividualriskfactors,thetraditionalFHSmodelcannotrespondtothecorrelationchange;itwilleitherunderestimateoroverestimateportfolio tailrisk,dependingonportfoliocorrelationexposure. CCPswhousetraditionalFHSVaR to estimate margin requirements could therefore under-collateralize cleared portfolios. ThispaperproposesanewmethodtoestimateportfolioVaRbyapplyingfilteringtothe historicalreturnofPCAlatentriskfactors. Bothsimulationandempiricalstudiessupport thatthisPCAFHSmethodcaneffectivelycapturecorrelationdynamicsandaddressrisks drivenbythecorrelationchanges. Backtestinganalysisperformedusingdatafromthe 2020COVID-19financialcrisisshowsthatforportfolioswithlimitedidiosyncraticrisk exposure,PCAFHSVaRoutperformsFHSVaR. ThePCAFHSmethodalsoprovidesanefficientandtime-savingalternativeforestimatingportfolio’sVaR. TherequiredrunningtimeforthePCAFHSmethodismuchlessthan traditionaltheFHSmethodforlargeportfolios. CCPscouldpotentiallyusethePCAFHS methodasanefficientmarginmethodonitsown. Therearelimitationstothemethod. Portfoliosthataremoreexposedtoidiosyncratic risk—eitherbecausetheyarehighlyconcentratedorbecausetheyhedgethesystematic 27

risk—maynotbesufficientlymarginedbyanaiveapplicationoftheproposedmethod. Of course,traditionalFHSorothermarginmethodsmayalsostruggletoperformacrossall possibleportfoliotypes. But,moreworktomapthelimitationsoftheproposedPCAFHS methodlikelyareneededbeforeitisusedaprimarymarginmethod. Someofthelimitations may be addressed by including more principal components than we have used in our analysis. Doing so increases the effectiveness at the cost of decreasing efficiency. An implementation for a particular market would need to explore the trade off in order to satisfythesecompetingobjectives. Nevertheless,theefficiencyanddemonstratedability of the PCA FHS method to respond to correlation changes would seem to argue for its adoptionatleastasabackstoporacomparisonforothermarginmethods,particularlythe widelyadoptedtraditionalFHS,whichignorescorrelationdynamics. Furthermore,althoughthesimulationstudiesandempiricalexercisesinthispaperwere limitedtoequityportfolios,wesuspectthatthemethodwouldperformwellforotherasset classes. Inparticular,ratesarecommonlymodeledwithjustafewprincipalcomponents,so themethodwouldseemnaturalthere. AlsoAlexander(2001,2008)appliedtheO-EWMA modeltocommodities,suggestingtheapproachmayextendtofuturesmarkets. Inaddition, CME Group (2023, pp. 7) indicates that the new SPAN2 model for futures implements some form of correlationscaling as well. Because of the potential nonlinear exposures, the method likely would outperform straight FHS for equity options, although neither methodmightbesufficient. Furtherstudiesareneeded, however, toexaminePCAFHS modelperformanceforotherassetclassesandportfolioscontainingmultipleassetclasses. References Alexander,Carol.2001,MasteringRisk: Volume2-Applications.FinancialTimes/Prentice Hall. Alexander, Carol. 2008, Practical Financial Econometrics, Market Risk Analysis, volume II. Wiley, URL http://www.wiley.com/WileyCDA/WileyTitle/ productCd-0470998016.html. Alexander,Carol.2009,MarketRiskAnalysis: Value-at-RiskModels,MarketRiskAnalysis, volumeIV.Wiley,URLhttp://www.wiley.com/WileyCDA/WileyTitle/ productCd-0470998016.html. Aramonte,Sirio,MariusdelGiudiceRodriguez,andJasonWu.2013,“Dynamicfactor Value-at-Risk for large heteroskedastic portfolios.” J Bank Financ, 37(11): 4299–4309. doi:10.1016/j.jbankfin.2013.07.038. Baur,Dirk.2006,“MultivariateMarketAssociationandItsExtremes.”J.Int.Finan.Markets, 16(4): 355–369.doi:10.1016/j.intfin.2005.05.006. Boyer, Brian H., Michael S. Gibson, and Mico Loretan. 1999, “Pitfalls in Tests for Changes in Correlations.” Technical report, Federal Reserve Board, URL 28

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Appendix: Correlation change and Portfolio VaR Byassumingtheunderlyingassetreturnsfollowanormaldistribution,7 aportfolioVaR withtwoassets,withanotationemphasizingthedependenceoncorrelation,equals: VaR𝜌 = 𝐹 ∗𝜎 𝜌 = 𝐹 ∗ 𝑤 2𝜎 2+𝑤 2𝜎 2+2𝜌 𝑤 𝑤 𝜎 𝜎 port √ 1 1 2 2 12 1 2 1 2 where 𝐹 is a constant and ∀𝑖 ∈ {1,2} 𝑤 is the portfolio weight on asset 𝑖 and, 𝜎 is the 𝑖 𝑖 volatilityofasset𝑖. Thecorrelationbetweenasset𝑖andasset𝑗isdenoted𝜌 . Thenotation 𝑖𝑗 𝜎 𝜌 denotesportfoliovariance. ForbothVaR𝜌 and𝜎 𝜌 valuesforparticularcorrelations port port will be denoted by either a letter for an unspecified value or a number for a particular value,soVaR𝑎 willbetheValue-at-Riskwith𝜌 = 𝑎and𝜎0 willrepresenttheportfolio 12 port variancewhenthetwoassetsareuncorrelated,whichequals√𝑤 1 2𝜎 1 2+𝑤 2 2𝜎 2 2. Let 𝜌𝑎 and 𝜌𝑏 denote different correlations in different periods. Then, the squared 𝑖𝑗 𝑖𝑗 relativechangeinVaR,holdingallothervariablesfixedis 2 VaR𝑏 2 𝜎 p 𝑏 ort 𝑤 1 2𝜎 1 2+𝑤 2 2𝜎 2 2+2𝜌 1 𝑏 2 𝑤 1 𝑤 2 𝜎 1 𝜎 2 2(𝜌 1 𝑏 2 −𝜌 1 𝑎 2 )𝑤 1 𝑤 2 𝜎 1 𝜎 2 ( ) = ( ) = = 1+ . VaR𝑎 𝜎 p 𝑎 ort 𝑤 1 2𝜎 1 2+𝑤 2 2𝜎 2 2+2𝜌 1 𝑎 2 𝑤 1 𝑤 2 𝜎 1 𝜎 2 (𝜎𝑎 ) 2 port Thisequationshowsthatthesquaredrelativechangeisalinearfunctionofthechangein correlationfrom𝑎to𝑏. LetΔ(𝜌) = 𝜌𝑏 −𝜌𝑎 denotethechange. Then,wehave 12 12 2(𝜌 1 𝑏 2 −𝜌 1 𝑎 2 )𝑤 1 𝑤 2 𝜎 1 𝜎 2 Δ(𝜌)⋅2𝑤 1 𝑤 2 𝜎 1 𝜎 2 = 2 2 (𝜎𝑎 ) (𝜎𝑎 ) port port 2 2 Δ(𝜌)⋅(𝜎1 ) −(𝜎0 ) port port = 2 (𝜎𝑎 ) port 2 2 (VaR1 ) −(VaR0 ) = Δ(𝜌)⋅( ). (VaR𝑎 ) 2 ThisresultcanbeusedtoshowthatthechangeinVaRisequaltothechangeincorrelation timestherelativesensitivityofVaRtocorrelation,asmeasuredbyhowmuchVaRwould changegoingfromnocorrelationtoperfectcorrelation,as 2 2 (VaR1 ) −(VaR0 ) (VaR𝑏)2 = (VaR𝑎)2⋅[1+Δ(𝜌)⋅( )] (VaR𝑎 ) 2 2 2 = (VaR𝑎)2+Δ(𝜌)⋅((VaR1 ) −(VaR0 ) ). 7Theanalysiscouldbereadilyextendedtoellipticaldistributions,likeStudent-tdistributions,becausethey alsohaveananalyticalformulathatdependssimilarlyoncorrelation(Dobrevetal.,2017). 31

ThenthepercentagechangeinVaRforacorrelationchangeattime𝑡is 2 2 ΔVaR Δ𝜌 𝑡 (VaR1 ) −(VaR0 ) ≈ ⋅( ). VaR𝜌 𝑡 2 (VaR𝜌 𝑡)2 Forportfolioswithmorethantwoassets,ananalogousresultholdssothat 2 2 max (Δ𝜌) (VaR1 ) −(VaR0 ) ΔVaR 𝑖,𝑗 𝑡 ≤ ⋅( ). VaR𝜌 𝑡 2 (VaR𝜌 𝑡)2 where max (Δ𝜌) is the maximum change of correlation across all assets 𝑖 and 𝑗 in the 𝑖,𝑗 𝑡 portfolio. Undertheassumptionofnormalityormoregenerallyellipticity,itisstraightforwardtocalculatetheVaRunderperfectandzerocorrelationevenforlargeportfolios. Itis thereforeeasytocalculatecorrelationsensitivityfortheportfolioandboundthechangein VaRforachangeinthecorrelationmatrix. Notsurprisingly,holdingvolatilityconstant,for agivenchangeincorrelation,thechangeinVaRwillbelargerforportfolioswithhigher correlationsensitivity. 32