Value-at-risk

Value-at-risk:TechniquestoAccountforLeptokurtosisandAsymmetricBehaviorinReturnsDistributionsLindsayA.LechnerandTimothyC.Ovaert**UniversityofNotreDame,NotreDame,IN,USA*Correspondingauthor:146Multi.Res.Bldg.,NotreDame,IN,[e-mail]sinthefinancialmarketshaveshowngreatinstabilityandhighvolatility.Inordertocapturetheamountofriskafinancialfirmtakesoninasingletradingday,riskmanagersuseatechnologyknownasvalue-at-risk(VaR).TherearemanymethodologiesavailabletocalculateVaR,andeachhasitslimitations.Manypastmethodshaveincludedanormalityassumption,whichcanoftenproducemisleadingfiguresasmostfinancialreturnsarecharacterizedbyskewness(asymmetry)andleptokurtosis(fat-tails).Design/methodology/approachThispapercomparestheStudent-t,autoregressiveconditionalheteroskedastic(ARCH)familyofmodels,andextremevaluetheory(EVT)asameansofcapturingthefat-tailednatureofareturnsdistribution.FindingsRecentresearchhasutilizedthethirdandfourthmomentstoestimatetheshapeindexparameterofthetail.Otherapproaches,suchasextremevaluetheory,focusontheextremevaluestocalculatethetailendsofadistribution.ByhighlightingbenefitsandlimitationsoftheStudent-t,autoregressiveconditionalheteroskedastic(ARCH)familyofmodels,andtheextremevaluetheory,onecanseethatthereisnooneparticularmodelthatisbestforcomputingVaR(althoughallofthemodelshaveproventocapturethefat-tailednaturebetterthananormaldistribution).Originality/valueThispaperdetailsthebasicadvantages,disadvantages,andmathematicsofcurrentparametricmethodologiesusedtoassessvalue-at-risk(VaR),sinceaccurateVaRmeasuresreduceafirm’scapitalrequirementandreassurecreditorsandinvestorsofthefirm’srisklevel.1Value-at-risk:TechniquestoAccountforLeptokurtosisandAsymmetricBehaviorinReturnsDistributionsAbstractThelastfewyearsinthefinancialmarketshaveshowngreatinstabilityandhighvolatility.Inordertocapturetheamountofriskafinancialfirmtakesoninasingletradingday,riskmanagersuseatechnologyknownasvalue-at-risk(VaR).Inrecentyears,VaRanalysishasbeentheprominentriskmeasurebecauseitallowsthemanagertotaketheriskofanentireportfolioandreduceittoasinglenumberthatcanbecomparedtootherportfoliosaswellasothertradingdays.TherearemanymethodologiesavailabletocalculateVaR,andeachhasitslimitations.Manypastmethodshaveincludedanormalityassumption,whichcanoftenproducemisleadingfiguresasmostfinancialreturnsarecharacterizedbyskewness(asymmetry)andleptokurtosis(fat-tails).ThispapercomparestheStudent-t,autoregressiveconditionalheteroskedastic(ARCH)familyofmodels,andextremevaluetheory(EVT)asameansofcapturingthefat-tailednatureofareturnsdistribution.Thepaperconcludesthatonecannotutilizeasinglemodelasthebestapproach,duetothetradeoffsbetweenaccuracyandcomputationaltime,theconfidenceinterval,andthetypeofasset.1.IntroductionHistoryhasshownthatbillionsofdollarscanbelostinashorttimeduetofailureincontrollingfinancialrisks.Withsuchextremeevents,thedevelopmentofreliablemethodstomonitorfinancialriskhasbecomeincreasinglyimportant(Bormetti2006,GilliandKellezi2006).Riskmeasuresarecriticalforcharacterizingfinancialinvestmentdecisions(Hwangand2Pederson2004);aseachinstitutionmeasurestheamountofriskittakesonoveraperiodofaday,week,month,oryear.AsimposedbytheBaselCommitteeonBankingSupervision,afinancialinstitutionisobligatedtomeetcapitalrequirementstocoverpotentiallossesduetosourcesofriskduringnormaloperations:namely,creditrisk,operationalrisk,andmarketrisk(Bormetti2006).AsnotedbyHopper(1996),afinancialfirmmaywanttounderstandpotentiallossestoitsportfoliosinordertobetterallocateitsfundsandplanforpaymentstoinvestors.Inrecentyears,value-at-risk(VaR)hasbecomethemostpopularamongriskmanagersasthebestandsimplestmethodtopredictlossesofanasset,portfolio,orevenanentirefirm.TheVaRofaportfoliomeasuresthemaximumlosssufferedduringaspecifictimehorizonwithinagivenconfidencelevel(i.e.99%),conditionedsuchthatthecompositionremainsunchanged.Forexample,ifVaRis$2M,thentheaveragelossoftheportfoliowillnotexceed$2Movertheone-dayhorizonon99outof100tradingdayswitha99%confidencelevel.VaRhasbecomeastandardcomponentforriskmanagementbecauseofitsconceptualsimplicityandaccuracyofestimatingriskatareasonablecomputationalcost.Ofcourse,thisutilitydoesnotalwaysimplyreliability.TherearemanydifferentmethodstocalculatedVaR,rangingfromasimplehistoricalsimulationtoacomplexsemi-parametricapproach.MostVaRestimatesareevaluatedfromhistorically-estimatedprobabilitydensityfunctions(PDFs),whichlimitthepredictivepoweroffutureriskmeasures(seeFigure1forreturnseriesPDF).ThemajorsourceoferrorinpredictingfutureVaRsfromhistoricaldataistheactualshapeofthePDF,whichcandiffersignificantlyfromtheoneusedinthepast.Forthisreason,economistsarenowfocusingtheireffortsoncreatingarobustparametricVaRmodelthat3notonlyaccountsforleptokurtosisandskewnessinthereturnsdistribution,butiscomputedinareasonableamountoftime.ThispaperprovidesanoverviewofVaRanddescribessomeofthemostrecentcomputationalapproaches.Section2introducesmethodsofcomputingVaR,detectingnon-normality(usingQ-Qplots,sampleMEF,andmomentanalysis),andc