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1、?市场风险测量X与管理FRMPartIIPrOgram基础班讲师:CrystalGao竺8巾i城5瞄方中40暇（师Session NO.Content%TopicWeightingsinFRMPartIISession1MarketRiskMeasurementandManagement20Session2CreditRiskMeasurementandManagement20Session3OperationalRiskandResiliency201.iquidityandTreasuryRiskMeasurementandSession415ManagementSession5RiskM
2、anagementandInvestmentManagement15Session6CurrentIssuesinFinancialMarket10(三)FrameworkMarketRiskMeasurementandManagement/kVaRandotherRiskMeasuresParametricApproachesNo-parametricApproachesSemi-parametricApproachesExtremevalueBacktestingVaRVaRM叩Ping”ModelingDependence:CorrelationsAndCopulasSomeCorrel
3、ationBasicsEmpiricalPropertiesofCorrelationFinancialCorrelationModeling,EmpiricalApproachestoRiskMetricsandHedges”TermStructureModelsofInterestRatesTheScienceofTermStructureModelsTheEvolutionofShortRatesandtheShapeoftheTermStructureTheArtofTermStructureModels:DriftTheArtofTermStructureModels:Volatil
4、ityandDistributionVolatilitySmilesxRiskMeasurementfortheTradingBookB三弟圆三mI0Z4SamSeonXs-H。0PUeIozLIAlX52JEOPUeeSlpeldd4U-Eeed!.ProfitandLossxProfit/LossP/L=K+%Pt.1jArithmeticReturnData:Pt+Dt-Pt-Pt+Dtr=1tPPt-t-GeometricReturnData:F+DtRt=】n(+-)=ln(l+r)t-!.ProfitandLossThedifferencebetweenthetworeturnsi
5、snegligiblewhenbothreturnsaresmall,butthedifferencegrowsasthereturnsgetbiggerwhichistobeexpected,asthegeometricreturnisalogfunctionofthearithmeticretun.“SinceWeWoUldexpectreturnstobeIoWOVerShOrtPeriodSandhigherOVerIongerperiods,thedifferencebetweenthetwotypesOfreturnisnegligibleOVerShOrtPeriOdSbutPO
6、tentiallySUbStHntialOVerlongerones.Approach1:NormalVaRWeassumethatarithmeticreturnsarnnormallydistributedWithmeanandStandGrddeviationVVaR=-(-%)VaR=-(-z)Pt.10 050.4350.3 O0.260.1A=-IqeqOjdProfit (OS8 (一)圜Example:Assumethattheprofit/lossdistributionforXYZisnormallydistributedwithanannualmeanof$16milli
7、onandastandarddeviationof$11million.CalculatetheVaRatthe95%and99%confidencelevelsusingaparametricapproach.VaR(5%)=-$16million+Sllmillion1.65=$2.15millionVaR(l%)=-$16million+Sllmillion2.33=$9.63millionLognormalVaRVaR = (l-ef)Pt-06VaR = 1 e0.5O -1-08-0-04-02 O 0204 0 08Lost (y0rofit (-)4 3 0.0. U=石 go
8、ldAssumethatgeometricreturnsarenormallydistributedwithmeanandstandarddeviation.ThisassumptionimpliesthatthenaturallogarithmofPtisnormallydistributed,orthatRitselfisIognorrriallydistributed.NormallydistributedgeometricreturnsimplythattheVaRisIognormallydistributed.圜Example:Adiversifiedportfolioexhibi
9、tsanormallydistributedgeometricreturnwithmeanandstandarddeviationof11%and21%,respectively.Calculatethe5%and1%lognormalVaRassumingthebeginningperiodportfoliovalueis$100.1.ognormalVaR(5%)=100(1-e011-21165)=$21.061.ognormalVaR(l%)=100(1-e011-21233)=$31.57Weareinterestedinasking:Ifdatalooksrightwhenweus
10、eparametricapproach?Whatwedois/Plotourdataonahistogramandestimatetherelevantsummarystatistics./Considerwhatkindofdistributionmightfitourdata.Aplotofthequantilesoftheempiricaldistributionagainstthoseofsomespecifieddistribution.TheshapeOftheQQPlottellsUSaIOtabouthowtheffipi4l-4bwt+n-mpaFes4-he-speed-Q
11、Re7Inparticular,iftheQQPlotislinear,thenthespecifieddistributionfitsthedata,andwehaveidentifiedthedistributiontowhichourdatabelong.Sa=WBnb-s-三E32IoZltIAlX528。PUeeslpeoAbootstrappedestimatewilloftenbemoreaccuratethana,raw,sampleestimate,andbootstrapsarealsousefulforgaugingtheprecisionofourestimates.D
12、rawbacksofHSBasicHShasthepracticaldrawbackthatitonlyallowsustoestimateVaRsatdiscreteConfidenCeintervalsdeterminedbythesizeofourdataset.Forinstance,theVaRatthe95.1%confidencelevelisaproblembecausethereisAGGoFFex)RdmgIGSSGbservatioRtogowithit.Withnobservations,basicHSonlyallowsustoestimatetheVaRsassoc
13、iatedwith,atbest,ndifferentconfidencelevels.”Non-parametricDensityEstimationNon-ParametriCdensityestimationOfferSapotentialsolution.Drawinstraightlinesconnectingthemid-pointsatthetopofeachhistogrambar(Polygon).TreatingtheareaunderthelinesasapdfthenenablesustoestimateVaRsatanyconfidencelevel.(m) Orig
14、inal histogram(b) Surrogate density function0 1-101Loss (*)ro1lt (-)2ExpectedShortfallkTheConditionalVaR(expectedshortfall)TheexpectedvalueofthelosswhenitexceedsVaR.MeasurestheaverageofthelossconditionalonthefactthatitisgreaterthanVaR.CVaRindicatesthepotentiallossiftheportfolioishitbeyondVaR.Because
15、CVaRisanaverageofthetailloss,onecanshowthatitqualifiesasasubadditiveriskmeasure.042.ExpectedShortfall圜Example:Giventhefollowing30orderedpercentagereturnsofanasset:-16,-14,-10,-7,-7,-5,-4,-4,-4,-3,-L-L0,0,0,1,2,2,4,68,9,11,12,12,14,18,21,23.CalculatetheVaRandexpectedshortfallata90%confidencelevel:Sol
16、ution:VaR(90%)=7,ExpectedShortfall=13.33.VaRvsESVaRcurveandEScurve:plotsofVaRorESagainsttheconfidencelevel.3.VaRvsESkThelongerthewindow,thesparsertheVaRcurve.”TheVaRcurveisfairlyunsteady,asitdirectlyreflectstherandomnessofindividuallossobservations,buttheEScurveissmoother,becauseeachESisanaverageoft
17、aillosses.”Astheholdingperiodrises,thenumberofobservationsrapidlyfalls,andwesoonfindthatwedonthaveenoughdata.“Evenifwehadaverylongrunofdata,theolderobservationsmighthaveverylittlerelevanceforcurrentmarketconditions.4.A/DofNon-parametricMethodskAdvantagesIntuitiveandconceptuallysimple;Donotdependonpa
18、rametricassumptions;Accommodateanytypeofposition;Noneedforcovariancematrices,nocursesofdimensionality;Usedatathatare(often)readilyavailable;Arecapableofconsiderablerefinementandpotentialimprovementifwecombinethemwithparametricadd-onstomakethemsemiparametric.4.A/DofNon-parametricMethodsDisadvantagesV
19、erydependentonthehistoricaldataset;Subjecttoghosteffect;Ifourdataperiodwasunusuallyquiet,non-parametricmethodswilloftenproduceVaRorESestimatesthataretoolowfortheriskweactuallyfacing,ViCeVerSa;Havedifficulty(actslowly)handlingss(permanentriskchange)thattakeplaceduringoursampleperiod;4.A/DofNon-parame
20、tricMethodsHavedifficultyhandlingextremevalue/Ifourdatasetincorporatesextremelossesthatareunlikelytorecur,theselossescandominatenon-parametricriskestimateseventhoughwedon,texpectthemtorecur;/Makenoallowanceforplausibleeventsthatmightoccur,butdidnotactuallyoccur,inoursampleperiod.4.A/DofNon-parametri
21、cMethodsProblemsfromLongWindowTheIongerthewindow:/ThegreatertheproblemswithagedClata;/Thelongertheperiodoverwhichresultswillbedistortedbyunlikely-to-recurpastevents,andtheIongerWeWillhavetoWaitforghosteffectstodis叩PeaK/Themorethenewsincurrentmarketobservationsislikelytobedrownedoutbyolderobservation
22、s;/Thegreaterthepotentialfordata-cUeionprobIems.xAcoherentriskmeasureisaweightedaverageofthequantilesofourlossdistribution.0=I0(p)Po(p)=weighingfunctionspecifiedbytheuser.xExponentialWeightingFunction-(i-)/thedegreeofourrisk-aversion”EstimatingexponentialspectralriskmeasuresasaweightedaverageofVaRs(
23、=0.05)Confidencelevel()VaRWeight()()VaR10%-1.281600.000020%-0.841600.000030%-0.524400.000040%-0.25330.00010.000050%00.00090.000060%0.25330.00670.001770%0.52440.04960.026080%0.8416036630.308390%1.28162.70673.4689Riskmeasure=mean()timesVaR)0.4226kTheestimatedoeseventuallyconvergetothetruevalueasngetsl
24、arge.EstimatesofexponentialspectralcoherentriskmeasureasafunctionofthenumberoftailslicesNumberoftailslicesEstimateofexponentialspectralriskmeasure100.4227501.37391001,58535001.789610001819750001,846110,0001.849850,0001.8529100,0001.8533500f0001,8536IoEmIAlX528。PUeeSlpeldd4U-lusedEsl.Age-weightedHist
25、oricalSimulationOnereturnobservationwillaffecteachofthenextnobservationsivourP/Lseries.Butafternperiodshavepassed,theobservationwillfalloutofthedatasetusedtocalculatethecurrentHSP/Lseries,andwillthereafterhavenoeffectOnP/L.xThisweightingstructurehasanumberofproblems.Oneproblemisthatitishardtojustify
26、givingenchObSerVatiOninOUrSarnPIePeriQdthesameweight.Theequal-weightapproachcanalsomakeIriSkestimatesUnreSPoDSiVetomajorevents.Theequal-weightstructurealsopresumesthateachobservationinthesampleperiodisequallylikelyandindependentoftheothersovertime.However,thisiidassumptionisunrealistic.l.Age-weighte
27、dHistoricalSimulationItisalsohardtojustifywhyanobservationshouldhaveaweightthatsuddenlygoestozerowhenitreachesagec.Ghosteffects/wecanhaveaVaRthatisundulyhigh(orlow)becauseofasmallclusterofhighlossobservations,orevenjustasinglehighloss,andthemeasuredVaRwillcontinuetobehigh(orlow)untilndaysorsohavepas
28、sedandtheobservationhasfallenoutofthesampleperiod.l.Age-weightedHistoricalSimulationBoudoukhrRichardsonandWhitelaw(BRW:1998)Wistheprobabilityweightgiventoanobservation1dayold.Acloseto1indicatesaslowrateofdecay,andafarawayfrom1indicatesahighrateofdecay.3121in入T(l 一入 =3(1) + 入3(1) + 入IQ)*)=l.Age-weigh
29、tedHistoricalSimulationMajorattractionsItprovidesanicegeneralizationOftraditionalHS,becauseWeCanFegad甘adi灯CrelHSaaSpeGiaL(ZewithZeFOdeea力of入I.AlargelosseventwillreceiveahigherWeightthanUndertraditionalHS,andtheresultingnext-dayVaRwouldbehigherthanitwouldotherwisehavebeen.Helpstoreducedistortionscaus
30、edbyeventsthatareunlikelytorecur,andhelpstoreducegsetfes./Asanobservationages,itsProbabilityWeightgraduallyfallsanditsinfluencediminishesgraduallyovertime.Whenitfinallyfallsoutofthesampleperiod,itsweightwillfallfromn(l)tozero,insteadoffrom1/ntozero.l.Age-weightedHistoricalSimulationkMajorattractions
31、.Age-weightingallowsustoIetOUrSamPlePeric)ClgrowWitheachnewObSerVatiori,soweneverthrowPOtentiallyVaIUableinformationaway.Thiswouldimproveefficiencyandeliminateghosteffects,becausetherewouldnoIongerbeanyjumpsinoursampleresultingfromoldobservationsbeingthrownaway.HullandWhite(HW1998)Weadjustthehistori
32、c日IretIImStoreflecthowvolatilitytomorrowisbelievedtohavechangedfromitspastvalues._nrtjt,i/rti=actualreturnforassetiondayt/ti=volatilityforecastforassetiondayt/i=currentforecastofvolatilityforassetiMajorattractionsIttakesaccountofvolatilitychangesinanaturalanddirectway.Itproducesriskestimatesthatarea
33、ppropriatelySenSitiVetoCUITCntVoIutilityestimates.ItallowsustoobtainVaRandESestimatesthatcanexceedthamaximumIOSSinOUFhistoricaldataset./Inrecentperiodsofhighvolatility,historicalreturnsarescaledupwards,andtheHSP/LseriesusedintheHWprocedurewillhavevaluesthatexceedactualhistoricallosses.ProducesSuperi
34、orVaRestimatestotheBRWone.”Correlation-weightedhistoricalsimulationCorrelation-weightingisalittlemoreinvolvedthanvolatility-weighting.Toseetheprinciplesinvolved,supposeforthesakeofargumentthatwehavealreadymadeanyvolatility-basedadjustmentstoourHSreturnsalongHull-Whitelines,butalsowishtoadjustthosere
35、turnstoreflectchangesincorrelations.FilteredhistoricalSimuIation(FHS)CombineshistoricalsimulationmodelwithGARCHorAGARCHmodel.“Thestepsareasfollows:Firstly,usethehistoricalreturntofindanysurpriseandthusreproducevolatilitywithGARCHorAGARCHmodel.Secondly,thesevolatilityforecastsarethendividedintotherea
36、lizedreturnstoproduceasetofstandardizedreturns,whichisI.I.D.Thethirdstageinvolvesbootstrappingfromthesetofstandardizedreturns.Finally,eachofthesesimulatedreturnsgivesusapossibleend-of-tomorrowportfoliovalue,andacorrespondingpossibleloss,andwetaketheVaRtobethelosscorrespondingtoourchosenconfidencelev
37、el.kMajorattractionsCombinethenon-parametricattractionsofHSwithasophisticated(e.g.,GARCH)treatmentofvolatility,andsotakeaccountOfChangingmarketVOlatiIityConditionSItisfast,evenforlargeportfoliosASWiththeearlierHWapproach,FHSallowsUStogetVaRandESestimatesthatCanexceedthemaximumhistoricallossinOUrdata
38、set.ItmaintainstheColTeIationSlTUCtUreinourreturn.ItcanbemodifiedtotakeaccountofautocorrelationsinassetreturnsItcanbemodifiedtoproduceestimatesofVaRorESconfidenceintervals.ThereisevidencethatFHSworkswell.IoZa!X5HoPUBBBA三eEXUJ!.IntroductionThefitteddistributionwilltendtoaccommodatethemorecentralobser
39、vations,ratherthantheextremeobservations,whicharemuchsparser.ATheestimationoftherisksassociatedwithlowfrequencyeventswithlimiteddataisinevitablyproblematic.Extreme-valuetheory(EVT):Centraltendencystatisticsaregovernedbycentrallimittheorems,butcentrallimittheoremsdonotapplytoextremes.Instead,extremes
40、aregovernedbyextreme-valuetheorems.xSupposewehavearandomlossvariableX,andweassumetobeginwiththatXisindependentandidenticallydistributed(iid)fromsomeunknowndistribution.ConsiderasampleofsizendrawnfromF(x),andletthemaximumofthissamplebeMnIfnislarge,wecanregardMnasanextremevalue.kUnderrelativelygeneral
41、conditions,thecelebratedFisher-Tippetttheoremthentellsusthatasngetslarge,thedistributionofextremes(i.e.,Mnconvergestothefollowinggeneralizedextreme-value(GEV)distribution:Thisdistributionhasthreeparameters.XUexp-(l+-),0F(X)=J,/xxexp-expJ,=0”:thelocationparameterofthelimitingdistribution,whichisameas
42、ureofthecentraltendencyofMn./,thescaleparameterofthelimitingdistribution,whichisameasureofthedispersionofMn.r,thetailindex,givesanindicationoftheshape(orheaviness)ofthetailofthelimitingdistribution.When0:Frechetdistribution,heavytails,liket-dist,Paretodist.When=0:Gumbeldistribution,lighttails,likenormalorlognormaldist.When0:Weibulldistribution,verylighttails,notusefulformodellingfinancialreturns.2.suJPe=三Rq=dHowdowechoosebetweentheGumbelandtheFrechet?WechoosetheEVdistributiontowhichtheextremesfromtheparentdistributionwilltend.Wecould
