16frm二级2016年notes part ii book1market risk measurement and management梦轩考资专业提供CFAFRM全程讲义

专业提供CFAFRM全 专业提供CFAFRM全 专业提供CFAFRM全 PartIIFRM®eAstheVPofAdvancedDesignationsatKa nSchweser,Iampleasedtohavetheopportunitytohelpyouprepareforthe2016FRM0Exam.GettinganearlystartonyourstudyprogramisimportantforyoutosufficientlyPrepare.,.Practice.,.Perform™onexamday.Proper nningwillallowyoutosetasideenoughtimetomasterthelearningobjectivesinthePartIIcurriculum.Nowthatyou'vereceivedyourSchweserNotes™,here'showtogetStep1:AccessYourOnline mandlogintoyouronlineaccountusingthebuttonlocatedinthetopnavigationbar.Afterloggingin,selecttheappropriatepartandproceedtothedashboardwhereyoucanaccessyouronlineproducts.Step2:CreateaStudyCreateastudynwiththeSchweserStudyCalendar(locatedontheSchweserdashboard).ThenviewtheCandidateResourceLibraryon-demandsforanintroductiontocoreconcepts.Step3:PrepareandReadyourOurclear,concisestudynoteswillhelpyoupreparefortheexam.Attheendofeachreading,youcananswertheConceptCheckerquestionsforbetterunderstandingofthecurriculum.AttendaWeeklyAttendourLiveOnlineWeeklyClassorreviewtheon-demandarchivesasoftenasyoulike.OurexpertfacultywillguideyouthroughtheFRMcurriculumwithastructuredapproachtohelpyoupreparefortheexam.(Seeourinstructionpackagestotheright.Visit mtoorder.)PracticewithSchweserPro™izeyourretentionofimportantconceptsandpracticeansweringexam-stylequestionsintheScheePrQBankandtakingseveralPracticeExams.UseSchweser'sQuickSheetforcontinuousreviewonthego.mtoStep4:FinalAfewweeksbeforetheexam,makeuseofourOnlineReviewWorkshopPackage.Reviewkeycurriculumconceptsineverytopic,performbyworkingthroughdemonstrationproblems,andpracticeyourexamtechniqueswithour8-hourliveOnlineReviewWorkshop.UseSchweser'sSecretSauce®forconvenientstudyonthego.Step5:AspartofourOnlineReviewWorkshopPackage,takeaSchweserMockExamtoensureyouarereadytoperformontheactualFRMExam.Putyourskillsandknowledgetothetestandgainconfidencebeforetheexam.

nWayforFRM®Instruction)PremiumPlus™)PremiumInstructionPackageLiveInstruction*:RemembertojoinourLiveOnlineWeeklyClass.Registeronlinetodayat Again,thankyoufortrusting nSchweserwithyourFRMExam

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*Dates,times,andinstructorssubjectto 专业提供CFAFRM全 专业提供CFAFRM全 PARTIIBooK1:MARKETRisKMEASUREMENTANDMANAGEMENTETOTHE2016SCHWESERNOTESREADINGASSIGNMENTSANDLEARNINGOBJECTIVES1:EstimatingMarketRiskMeasures:AnIntroductionand2:Non-parametric3:ParametricApproaches(II):Extreme

©vVl©02015K11 a6:MessagesfromtheAcademicLi fortheTradingBook7:SomeCorrelationBasics:Properties,Motivation,8:EmpiricalPropertiesofCorrelation:HowDoCorrelationsBehaveintheRealWorld?11:EmpiricalApproachestoRiskMetricsand12:TheScienceofTermStructure13:TheEvolutionofShortRatesandtheShapeoftheTerm14:TheArtofTermStructureModels:15:TheArtofTermStructureModels:Volatilityand16:OISDiscounting,CreditIssues,andFunding

, c.Page】FRM2016PARTIIBOOK1:MARKETRISKMEASUREMENTAND©2015Kan,Inc.,d.b.a.KanSchweser.Allrights PrintedintheUnitedStatesofAmerica.ISBN:978-1-4754-3807-PPN:3200-721Required er:GARP®doesnotendorse,promoter,eview,orwarranttheaccuracyoftheproductsservicesofferedbyKanSchweserofFRM®relatedinformation,nordoesitendorseanypassratesclaimedbytheprovider.Further,GARP®isnotresponsibleforanyfeesorcostspaidbytheusertoKanSchweser,norisGARP®responsibleforanyfeesorcostsofany orentityprovidinganyservicestoKanSchweser.FRM®,GARP®,andGlobalAssociationofRiskProfessionalsTMaretrademarksownedbytheGlobalAssociationofRiskProfessionals,Inc.Thesematerialsmaynotbecopiedwithoutwrittenpermissionfromtheauthor.Theunauthorizedduplicationofthesenotesisaviolationofglobalcopyrightlaws.Yourassistanceinpursuingpotentialviolatorsofthislawisgreatlyappreciated. er:TheSchweserNotesshouldbeusedinconjunctionwiththeoriginalreadingsassetforthbyGARP®.Theinformationcontainedinthesebooksisbasedontheoriginalreadingsandisbelievedtobeaccurate.However,theiraccuracycannotbeguaranteednorisanywarrantyconveyedastoyourultimateexamPage ©2015 n, 专业提供CFAFRM全 ETOTHE2016ThankyoufortrustingKa nSchwesertohelpyoureachyourgoals.WeareverypleasedtobeabletohelpyouprepareforthePartIIFRMexam.Inthisintroduction,Iwanttoex intheresourcesincludedwiththeSchweserNotes,suggesthowyoucanbestuseSchwesermaterialstopreparefortheexam,anddirectyoutowardothereducationalresourcesyouwillfindhelpfulasyoustudyfortheexam.BesidestheSchweserNotesthemselves,therearemanyeducationalresourcesavailable.JustloginusingtheindividualusernameandpasswordyoureceivedwhenyoupurchasedtheSchweserNotes.Sh"sTl1TheSchweserNotesconsistoffourvolumesthatincludecompletecoverageofallFRMassignedtopicsandlearningobjectives(LOs),ConceptCheckers(multiple-choicequestionsforeverytopic),andSelf-Testquestionstohelpyoumasterthematerialandcheckyourretentionofkeyconcepts.OnlinePracticeToretainwhatyoulearn,itisimportantthatyouquizyourselfoften.WeofferanonlineversionoftheSchweserProTMQBank,whichcontainshundredsofPartIIpracticequestionsandex nations.Quizzesareavailableforeachtopicoracrossmultipletopics.BuildyourownexamsbyspecifyingthetopicsandthenumberofSchweserofferstwofull4-hourpracticeexams.Theseexamsareimportanttoolsforgainingthespeedandskillsyouwillneedtopasstheexam.ThePracticeExamsbookcontainsanswerswithfullex nationsforself-gradingandevaluation.UseyourOnlineAccessto luswhenyouwillstartandwhatdaysoftheweekyoucanstudy.TheonlineSchweserStudyCalendarwillcreateastudy njustforyou,breakingthecurriculumintodailyandweeklytaskstokeepyouontrackandhelpyoumonitoryourstudyprogress.©2015Kan, Page】】

etothe2016ThePartIIFRMexamisaformidablechallenge(covering78assignedreadingsandalmost500learningobjectives),andyoumustdevoteconsiderabletimeandefforttobeproperlyprepared.Therearenoshortcuts!Youmustlearnthematerial,knowtheterminologyandtechniques,understandtheconcepts,andbeabletoanswer80multiplechoicequestionsquicklyand(atleast70o/o)correctly.Agoodestimateofthestudytimerequiredonaverageis250hours,butsomecandidateswillneedmoreorlesstime,dependingontheirindividualbackgroundsandexperience.Tohelpyoureallymasterthismaterialandbewell-preparedfortheFRMexam,weofferseveralothereducationalresources,including:OnlineWeeklyOurOnlineWeeklyClassisofferedeachweek,beginninginFebruaryfortheMayexamandAugustfortheNovemberexam.Thisonlineclassbringsthe alattentionofaclassroomintoyourhomeorofficewith30hoursofreal-timeinstruction,ledbyeitherDr.JohnPaulBroussard,CFA,FRM,orDr.GregFilbeck,CFA,FRM,CAIA.Theclassoffersin-depthcoverageofdifficultconcepts,instantfeedbackduringlectureandQ&Asessions,anddiscussionofsampleexamquestions.Archivedclassesareavailableforviewingatanytimethroughouttheseason.CandidatesenrolledintheOnlineWeeklyClassalsohavefullaccesstosupplementalon-demand instructionintheCandidateResourceLibraryandane-mailaddresslinktousetosendquestionstotheinstructoratanytime.Late-SeasonLate-seasonreviewandexampracticecanmakeallthedifference.OurReviewPackagehelpsyouevaluateyourexamreadinesswithproductsspecificallydesignedforlate-seasonstudying.ThisReviewPackageincludestheOnlineReviewWorkshop(8-hourliveandarchivedonlinereviewofessentialcurriculumtopics),theSchweserMockExam(one®4-hourexam),andSchweser'sSecret (concisesummaryoftheFRM ©2015 n, 专业提供CFAFRM全 etothe2016PartIIExamInpreparingfortheexam,payattentiontotheweightsassignedtoeachknowledgewithinthecurriculum.ThePartIIexamweightsareasfollows: 1MarketRiskMeasurementand202CreditRiskMeasurementand203OperationalandIntegratedRisk4RiskManagementandInvestment15%410%8HowtoTherearenoshortcutstostudyingforthisexam.Expect

totestyouinawaywillrevealhowwellyouknowthePartIIcurriculum.Youshouldbeginstudyingearlyandsticktoyourstudy n.YoushouldfirstreadtheSchweserNotesandcompletetheConceptCheckersforeachtopic.Attheendofeachbook,youshouldanswertheprovidedSelf-Testquestionstounderstandhowconceptshavebeentestedinthepast.YoushouldfinishtheoverallcurriculumatleasttwoweeksbeforetheFRMexam.ThiswillallowsufficienttimeforPracticeExamsandfurtherreviewofthosetopicsyouhavenotyetmastered.Iwouldliketotakethisopportunitytothankthecontentdevelopers,editors,andgraphicdesignerswhoworkedcountlesshourstocreatethe2016FRMSchweserNotes.IwouldespeciallyliketothankAdam

DerekBurkett,CFA,FRM,CAIA;Greive,CFA;CraigProchaska,CFA;KentWestlund,CFA;KurtSchuldes,CFA,CAIA;LauraGoetzinger;KatherineBourgeois;HeatherWalley;ChrisZobin;AlyssaBrunner;JuliePorter;AllieBottcher;BeccaDargatz;AlissaKnop;DavidGriswold;GenevieveKrets LindseyCasto;andJaredHeintzfortheircontributions.BestEricSmith,CFA,FRMFRMProductManagerKanSchweser©2015Kan, Page 专业提供CFAFRM全 READINGASSIGNMENTSANDLEARNINGOBJECTIVESREADING Sons,2005)."EstimatingMarketRiskMeasures:AnIntroductionandOverview,"Chapter3(page"Non-parametricApproaches,"Chapter"ParametricApproaches(II):ExtremeValue,"Chapter

(page(page 3rdEdition(NewYork:McGrawHill,2007)."BacktestingVaR,"Chapter"VaR ,"Chapter (page"MessagesfromtheAcademicLi tureonRiskMeasurementfortheTradingBook,"BaselCommitteeonBankingSupervision,WorkingPaperNo.19,Jan2011.(page51)GunterMeissner,CorrelationRiskModelingandManagement(NewYork:Wiley,"SomeCorrelationBasics:Properties,Motivation,Terminology,"Chapter (page"EmpiricalPropertiesofCorrelation:HowDoCorrelationsBehaveinRealWorld?,"Chapter (pageChapter (page"FinancialCorrelationModeling-Bottom-UpApproaches,"Chapter (pageBruceTuckman,Fixed Page

"EmpiricalApproachestoRiskMetricsandHedging,"Chapter6"TheScienceof

(page(page015K,I. 专业提供CFAFRM全 Book1"TheEvolutionofShortRatesandtheShapeoftheTermStructure,"Chapter8"TheArtofTermStructureModels:Drift,"Chapter

(page(page"TheArtofTermStructureModels:VolatilityandDistribution,"Chapter10(pagePrenticeHall,2014)."OISDiscounting,CreditIssues,andFundingCosts,"Chapter"Volatility ,"Chapter

))】BookReadingAssignmentsandLearningLEARNINGEstimatingMarketRiskMeasures:AnIntroductionandAftercompletingthisreading,youshouldbeableEstimateVaRusingahistoricalsimulation (pageEstimateVaRusingaparametricapproachforbothnormalandlognormalreturndistributions.(page4)EstimatetheexpectedshortfallgivenP/Lorreturndata.(pageDefinecoherentriskmeasures.(pageEstimateriskmeasuresbyestimatingles.(pageEvaluateestimatorsofriskmeasuresbyestimatingtheirstandarderrors.(pageInterpretQQplotstoidentifythecharacteristicsofadistribution.(pageAftercompletingthisreading,youshouldbeableApplythebootstraphistoricalsimulationapproachtoestimatecoherentriskmeasures.(page15)Describehistoricalsimulationusingnon-parametricdensityestimation.(pageCompareandcontrasttheage-weighted,thevolatility-weighted,thecorrelation-weightedandthefilteredhistoricalsimulationapproaches.(page17)Identifyadvantagesanddisadvantagesofnon-parametricestimation(pageAftercompletingthisreading,youshouldbeableExintheimportanceandchallengesofextremevaluesinriskmanagement.(page25)Describeextremevaluetheory(EVT)anditsuseinriskmanagement.(pageDescribethepeaks-over-threshold(POT)approach.(pageCompareandcontrastgeneralizedextremevalueandPOT.(pageEvaluatethetradeoffsinvolvedinsettingthethresholdlevelwhenapplyingtheGPdistribution.(page27)ExintheimportanceofmultivariateEVTforriskmanagement.(pageBacktestingAftercompletingthisreading,youshouldbeableDefinebacktestingandexceptionsandexintheimportanceofbacktestingVaRmodels.(page33)ExinthesignificantdifficultiesinbacktestingaVaRmodel.(pageVerifyamodelbasedonexceptionsorfailure (pageDefineandidentifytypeIandtypeIIerrors.(pageExintheneedtoconsiderconditionalcoverageinthebacktestingframework.(page37)DescribetheBaselrulesforbacktesting.(page ©2015 n, 专业提供CFAFRM全 Book1Aftercompletingthisreading,youshouldbeable intheprinciplesunderlyingVaRmap ,anddescribethemap (page41) inhowthe processcapturesgeneralandspecificrisks.(pageDifferentiateamongthethreemethodsofmap portfoliosoffixed securities.(page44)Summarizehowtomapafixed eportfoliointopositionsofstandardinstruments.(page44)Describehow ofrisk

supportstresstesting.(page inhowVaRcanbeusedasaperformance ark.(pageDescribethemethodofmap forwards,forwardrateagreements,interestrateswaps,andoptions.(page46) Aftercompletingthisreading,youshouldbeable inthefollowinglessonsonVaRimplementation:timehorizonoverwhichVaRisestimated,therecognitionoftimevaryingvolatilityinVaRriskfactors,andVaRbacktesting.(page51)Describeexogenousandendogenousliquidityriskandex inhowtheymightbeintegratedintoVaRmodels.(page52)CompareVaR,expectedshortfall,andotherrelevantriskmeasures.(pageCompareunifiedandcompartmentalizedriskmeasurement.(pageComparetheresultsofresearchon"top-down"and"bottom-up"riskaggregationmethods.(page54)Describetherelationshipbetweenleverage,marketvalueofasset,andVaRwithinanactivebalancesheetmanagementframework.(page55)Aftercompletingthisreading,youshouldbeableDescribefinancialcorrelationriskandtheareasinwhichitappearsinfinance.(page59) inhowcorrelationcontributedtotheglobalfinancialcrisisof2007to(pageDescribethestructure,uses,andpayoffsofacorrelationswap.(pageEstimatetheimpactofdifferentcorrelationsbetweenassetsinthetradingbookontheVaRcapitalcharge.(page66) intheroleofcorrelationriskinmarketriskandcreditrisk.(pageRelatecorrelationrisktosystemicandconcentrationrisk.(pageEmpiricalPropertiesofCorrelation:HowDoCorrelationsBehaveintheRealAftercompletingthisreading,youshouldbeableDescribehowequitycorrelationsandcorrelationvolatilitiesbehavethroughoutvariouseconomicstates.(page83)Calculateameanreversionrateusingstandardregressionandcalculatethecorrespondingautocorrelation.(page84)Identifythebest-fitdistributionforequity,bond,anddefaultcorrelations.(page87)©2015Kan, 】BookReadingAssignmentsandLearningStatisticalCorrelationModels-CanWeApplyThemtoAftercompletingthisreading,youshouldbeable Evaluatethelimitationsoffinancialmodelingwithrespecttothemodelitsel£calibrationofthemodel,andthemodel'soutput.(page93) AssessthePearsoncorrelationapproach,Spearman'srankcorrelation,Kendall'sT,andevaluatetheirlimitationsandusefulnessinfinance.(pageAftercompletingthisreading,youshouldbeable Exinthepurposeofcopulafunctionsandthetranslationofthecopulaequation.(page106)DescribetheGaussiancopulaandexinhowtouseittoderivethejointprobabilityofdefaultoftwoassets.(page107)SummarizetheprocessoffindingthedefaulttimeofanassetcorrelatedtoallassetsinaportfoliousingtheGaussiancopula.(pageAftercompletingthisreading,youshouldbeable ExinthedrawbackstousingaDVOI-neutralhedgeforabondposition.(pageDescribearegressionhedgeandexinhowitcanimproveastandardDVOI-neutralhedge.(page117)Calculatetheregressionhedgeadjustmentfactor,beta.(pageCalculatethefacevalueofanoffsettingpositionneededtocarryoutaregressionhedge.(page118)Calculatethefacevalueofmultipleoffsettingswappositionsneededtocarryouttwo-variableregressionhedge.(pageCompareandcontrastlevelandchangeregressions.(pageDescribeprincipalcomponent ysisandexinhowitisappliedtoconstructingahedgingportfolio.(page120)TheScienceofTermStructureAftercompletingthisreading,youshouldbeable Calculatetheexpecteddiscountedvalueofazero-couponsecurityusingabinomialtree.(page127)Constructandapplyanarbitrageargumenttopriceacalloptiononazero-couponsecurityusingreplicatingportfolios.(page127)Definerisk-neutralpricingandapplyittooptionpricing.(pageDistinguishbetweentrueandrisk-neutralprobabilities,andapplythisdifferencetointerestratedrift.(page130)Exinhowtheprinciplesofarbitragepricingofderivativeson

beextendedovermultipleperiods.(pageDefineoption-adjustedspread(OAS)andapplyittosecuritypricing.(pageDescribetherationalebehindtheuse CalculatethevalueofaconstantmaturityTreasuryswap,givenaninterestratetreeandtherisk-neutralprobabilities.(page134)Evaluatetheadvantagesanddisadvantagesofreducingthesizeofthetimestepsonthepricingofderivativesonfixed esecurities.(page137)EvaluatetheappropriatenessoftheBlack-Scholes-Mertonmodelwhenderivativeson esecurities.(pageDescribetheimpactofembeddedoptionsonthevalueof e(page ©2015 n, 专业提供CFAFRM全 Book1ReadingAssignmentsandLearningTheEvolutionofShortRatesandtheShapeoftheTermAftercompletingthisreading,youshouldbeable intheroleofinterestrateexpectationsindeterminingtheshapeofthetermstructure.(page144)Applyarisk-neutralinterestratetreetoassesstheeffectofvolatilityontheshapethetermstructure.(pageEstimatetheconvexityeffectusingJensen'sinequality.(pageCalculatethepriceandreturnofazerocouponbondincorporatingariskpremium.(page152)Aftercompletingthisreading,youshouldbeableConstructanddescribetheeffectivenessofashortterminterestratetreeassumingnormallydistributedrates,bothwithandwithoutdrift.(page159)Calculatetheshort-termratechangeandstandarddeviationoftheratechangeusingamodelwithnormallydistributedratesandnodrift.(page160)Describemethodsforaddressingthepossibilityofnegativeshort-termratesintermstructuremodels.(page161)Constructashort-termratetreeundertheHo-LeeModelwithtime-dependentdrift.(page163)Describeusesandbenefitsofthearbitrage- modelsandassesstheissueoffittingmodelstomarketprices.(page163)Describetheprocessofconstructingasimpleand biningtreeforashort-termrateundertheVasicekModelwithmeanreversion.(page164)CalculatetheVasicekModelratechange,standarddeviationoftheratechange,expectedrateinTyears,andhalflife.(page167)DescribetheeffectivenessoftheVasicekModel.(pageAftercompletingthisreading,youshouldbeableDescribetheshort-termrateprocessunderamodelwithtime-dependentvolatility.(page174)Calculatetheshort-termratechangeanddeterminethebehaviorofthedeviationoftheratechangeusingamodelwithtimedependentvolatility.(page174)Assesstheefficacyoftime-dependentvolatilitymodels.(pageDescribetheshort-termrateprocessundertheCox-Ingersoll-Ross(CIR)andlognormalmodels.(page176)Calculatetheshort-termratechangeanddescribethebasispointvolatilityusingCIRandlognormalmodels.(pageDescribelognormalmodelswithdeterministicdriftandmeanreversion.(pageAftercompletingthisreading,youshouldbeable inthemainconsiderationsinchoosingarisk- rateforderivativesvaluation.(page184)DescribetheOISrateandtheLIBOR-OISspread,andex intheiruses.(page185)EvaluatetheappropriatenessofOISrateas forthe rate.(pageDescribehowtousetheOISzerocurveindeterminingforwardLIBORratesvaluingswaps.(page©2015Kan, Page】BookReadingAssignmentsandLearningVolatilityAftercompletingthisreading,youshouldbeableDefinevolatilitysandvolatilityskew.(pageExintheimplicationsofput-callparityontheimpliedvolatilityofcallandputoptions.(page193)Comparetheshapeofthevolatilitys(orskew)totheshapeoftheimplieddistributionoftheunderlyingassetpriceandtothepricingofoptionsontheunderlyingasset.(page194)Describecharacteristicsofforeignexchangeratedistributionsandtheirimplicationsonoptionpricesandimpliedvolatility.(page195)Describethevolatilitysforequityoptionsandforeigncurrencyoptionsprovidepossibleexnationsforitsshape.(pageDescribealternativewaysofcharacterizingthevolatility .(pageDescribevolatilitytermstructuresandvolatilitysurfacesandhowtheymaybeusedtopriceoptions.(page197)Exintheimpactofthevolatilitys onthecalculationofthe"Greeks."(page197)Exintheimpactofasingleassetpricejumponavolatility .(page ©2015 n, 专业提供CFAFRM全 ThefollowingisareviewoftheMarketRiskMeasurementandManagementprinciplesdesignedtothelearningobjectivessetforthbyGARP®.Thistopicisalso

ESTITINMARKETRISKMEASURES:ANINTRODUCTIONANDOVERVIEW"

TopicInthistopic,thefocusisontheestimationofmarketriskmeasures,suchasvalueatrisk(VaR).VaRidentifiestheprobabilitythatlosseswillbegreaterthanapre-specifiedthresholdlevel.Fortheexam,bepreparedtoevaluateandcalculateVaRusinghistoricalsimulationandparametricmodels(bothnormalandlognormalreturndistributions).OnedrawbacktoVaRisthatitdoesnotestimatelossesinthetailofthereturnsdistribution.Expectedshortfall(ES)does,however,estimatethelossinthetail(i.e.,aftertheVaRthresholdhasbeenbreached)byaveraginglosslevelsatdifferentconfidencelevels.Coherentriskmeasures alriskaversionacrosstheentiredistributionandaremoregeneralthanexpectedshortfall. le(QQ)plotsareusedtovisuallyinspectifanempiricaldistributionmatchesatheoreticaldistribution.ESTIMATINGTobetterunderstandthematerialinthistopic,itishelpfultorecallthecomputationsofarithmeticandgeometricreturns.Notethattheconventionwhencomputingthesereturns(aswellasVaR)istoquotereturnlossesaspositivevalues.Forexample,ifaportfolioisexpectedtodecreaseinvalueby$1million,weusetheterminology"expectedlossis$1million"ratherthan"expectedprofitis-$1Profit/lossdata:Changeinvalueofasset/portfolio,interimpayments,Dt.

attheendofperiodtplusArithmeticreturndata:Assumptionisthatinterimpaymentsdonotearnareturn(i.e.,noreinvestment).Hence,thisapproachisnotappropriateforlonginvestmenthorizons.Geometricreturndata:Assumptionisthatinterimpaymentsarecontinuouslyreinvested.Notethatthisapproachensuresthatassetpricecanneverbenegative.©2015Kan, 】TopicCrossReferencetoGARPAssignedReading-Dowd,ChapterHISTORICALSIMULATIONLO1.1:EstimateVaRusingahistoricalsimulationEstimatingVaRwithahistoricalsimulationapproachisbyfarthesimplestandmoststraightforwardVaRmethod.Tomakethiscalculation,yousimplyorderreturnobservationsfromlargesttosmallest.TheobservationthatfollowsthethresholdlossleveldenotestheVaRlimit.Weareessentiallysearchingfortheobservationthatseparatesthetailfromthebodyofthedistribution.Moregenerally,theobservationthatdeterminesVaRfornobservationsatthe(1-a)confidencelevelwouldbe:(axn)+1.value(e.g.,95%)whereasthesignificancelevel,usuallydenotedasa,ismuchsmaller(e.g.,5%).ToillustratethisVaRmethod,assumeyouhavegathered1,000monthlyreturnsforasecurityandproducedthedistributionshowninFigure1.YoudecidethatyouwanttocomputethemonthlyVaRforthissecurityataconfidencelevelof950/o.Ata950/oconfidencelevel,thelowertaildisysthelowest5o/ooftheunderlyingdistribution'sreturns.Forthisdistribution,thevalueassociatedwitha950/oconfidencelevelisareturn-15.5o.Ifyouhave$1,000,000investedinthissecurity,theone-monthVaRis$155,000(-15.5x$1,000,000).Figure1:HistogramofMonthly v �

ofLossMonthlyPage ©2015 n, 专业提供CFAFRM全 TopicExample:IdentifyingtheVaRIdentifytheorderedobservationinasampleof1,000datapointsthatcorrespondstoVaRata950/oconfidencelevel.SinceVaRistobeestimatedat950/oconfidence,thismeansthat50/(i.e.,50)oftheorderedobservationswouldfallinthetailofthedistribution.Therefore,the51storderedlossobservationwouldseparatethe

oflargestlossesfromtheremaining950/oor'sNote:VtiRisthe lethatseparatesthetailfromtheboyInotherwords,shouldVtiRbethe50thobservation(i.e.,axn),the51stobservation[i.e.,axn)+1),orsomecombinationftheseobservations?Inusingthehistoricalsimulationmethodhasbeencalculatedasjust:axn),inthiscase,asthe50thobservation.Example:ComputingAlonghistoryofprofit/lossdatacloselyapproximatesastandardnormal(meanequalszero;standarddeviationequalsone).Estimatethehistoricalsimulation

VaRusingTheVaRlimitwillbeattheobservationthatseparatesthetaillosswithareaequalto50/fromtheremainderofthedistribution.Sincethedistributioniscloselyapproximatedbythestandardnormaldistribution,theVaRis1.65(50/ocriticalvaluefromthez-table).RecallthatsinceVaRisaone-tailedtest,theentiresignificancelevelof50/oisinthelefttailofthereturnsdistribution.Fromapractical ,thehistoricalsimulationapproachissensibleonlyifyouexpectfutureperformancetofollowthesamereturngeneratingprocessasinthepast.Furthermore,thisapproachisunabletoadjustforchangingeconomicconditionsorabruptshiftsinparametervalues.©2015Kan, 】TopicPARAMETRICESTIMATIONLO1.2:EstimateVaRusingaparametricapproachforbothnormalandlognormalIncontrasttothehistoricalsimulationmethod,theparametricapproach(e.g.,thedelta-normalapproach)explicitlyassumesadistributionfortheunderlyingobservations.ForthisLO,wewill yzetwocases:(1)VaRforreturnsthatfollowanormaldistribution,and(2)VaRforreturnsthatfollowalognormalIntuitively,theVaRforagivenconfidenceleveldenotesthepointthatseparatesthetaillossesfromtheremainingdistribution.TheVaRcutoffwillbeinthelefttailofthereturnsdistribution.Hence,thecalculatedvalueatriskisnegative,butistypicallyreportedasapositivevaluesincethenegativeamountisimplied(i.e.,itisthevaluethatisatrisk).Inequationform,theVaRatsignificancelevelais:VaR(ao/o)=-µp/L+<Yp/LX

µ

crdenotethemeanandstandarddeviationoftheprofit/lossdistribution denotesthecriticalvalue(i.e.,le)ofthestandardnormal.Inpractice,theparametersµandcrarenotlikelyknown,inwhichcasetheresearcherwillusethemeanandstandardExample:ComputingVaR(normalAssumethattheprofit/lossdistributionforXYZisnormallydistributedwithanannualmeanof$15millionandastandarddeviationof$10million.CalculatetheVaRatthe95°A>and99o/oconfidencelevelsusingaparametricapproach.Aner:VaR(5o/o)

-$15million+$10

x

=$1.5million.

toloseatmost$1.5millionoverthenextyearwith95o/oconfidence.Equivalently,XYZexpectstolosemorethan$1.5millionwitha5o/oprobability.VaR(lo/o)=-$15million+$10millionx2.33=$8.3million.NotethattheVaR(at99o/oconfidence)isgreaterthantheVaR(at95%confidence)asfollowsfr