金融风险Measuring market risk VaR approach.ppt

3 Measuringmarketrisk VaRapproach 3 1Introduction3 2UnderstandingVaR3 3Riskmetrics3 4Historicsimulation3 5Mote Carlosimulation3 6BISstandardizedmodel 3 1Introduction J P MorganG30BISInvolvingprobabilitycomponentalongwithlossseveritycomponentinriskmeasurement dealingwiththelimitationofsensitiveapproachesVaRRevolutionExtensiontoCreditandOperationalriskandthereforeintegratedriskmanagement AdvantagesofVaRapproach oversensitivityapproach completemeasureofrisk Measuringriskusingthesameunit dollarAggregateviewofaportfolioriskaccountingforleverageandcorrelationeffectsintegratednature notonlyderivativesbutalsoallotherfinancialinstruments andcanbebroadenedfrommarketrisktoothertypesoffinancialriskone numberindicator 3 2UnderstandingVaR QuestionsleadingustoVaRmeasureDefiningValueatRiskKeyelementsofcalculatingVaRWhatdoesPDFcurvetell Approachestoprobabilitydistribution typesofVaR WorkingoutVaRthroughariskfactor QuestionsleadingustoVaRmeasure Asaportfoliomanager youmaybeaskedbyyourbossfollowingquestions Q1 Givenamarketchangeorshock howmuchcouldyourportfoliosuffer Q2 Ifitturnsouttobeabaddaytomorrow whatistheworstlossofyourinvestment Thefirstisasensitivityquestion andyoucangiveaclearanswerafteryoudoasensitivitymeasureasweshowedinpreviouslectures Thesecondisnotaclearquestion Beforeyoutrytoanswerityouhavetoaskback Whatdoyoumeanpreciselyby abadday or Howbadthedayyousupposeittobe SensitivityQfollowedbyprobabilityQ Q1leadingtoVaRmeasure ExampleofsensitivityQ Givena25bpsyieldrise howmuchcouldyourinvestmentportfolio P 1m MD 2ys suffer A dP D dR 1 R P MD dR P 2 0 0025 1 000 000 5 000 Youmaybefurtherasked Howlikelycoulditbethecase Ormoreprecisely giventhenormalmarketcondition howlikelywouldyourportfoliosuffernotmorethanthatamountofmoneyoveratargetholdingperiod AVaRquestion Morefrequently thequestionisputinanotherway Whatistheworstlossyourportfoliocouldsufferoveratargetholdingperiodwithprobabilityofagivenlevel say1 undernormalmarketcondition AstandardVaRquestion Define badday andcompleteQ2 Q2leadingtoVaRmeasure Oneeasywaytodefinea badday istodefinethelosssizeorseverity Butthiswillmakethequestionmakenosense Itisreasonable andmeaningful todefinea badday insuchawaythatthedayissobad orthelossissoseverethatsuchaday orloss occursundernormalmarketconditiononlyonceoutofevery20 tobedefined tradingdays putanotherway thechance probability oftheoccurrence loss isonly5 tobedefined Ifitturnsouttobeabaddaytomorrow whatwillbetheworstlossofyourinvestment giventhatdayhappensonceoutofevery20tradingdays AstandardVaRquestion DefiningValueatRisk Definition TheworstlossoveragiventargetholdingperiodatagivenconfidencelevelundernormalmarketconditionTwopre specifiedvariables HoldingPeriod 1day 1weekormore ConfidenceLevel 95 99 orevenhigher VaRisananswerto Undernormalmarketcondition whatistheworstlosscouldmyinvestmentportfoliosufferwithaprobabilityof5 withinonetradingday Iam95 99 surethattheinvestmentportfoliowillsuffernotmorethan 5 000 20 000 lossover1dayundernormalmarketcondition KeyelementsofcalculatingVaR HoldingperiodConfidencelevelProbabilitydistributionofreturnProbabilitydensityfunction PDF inthecaseofcontinuousrandomvariableWorkingouttheprobabilitydistributionofreturnisthemostdifficultpartoftheVaRwork Holdingperiod DEARandMore than one dayVaR Holdingperiodisthetargettimehorizonduringwhichyouholdyourinvestmentposition alsotargetmeasuringperiod One dayVaR orDailyVaRisusuallycalculated especiallyinthecaseofRiskmetricsmodel ItisusuallytermedDailyEarningatRisk DEAR More than one dayVaRcanbederivedfromDEARfromfollowingformula Undertheassumptionthatmarketvalidityisconstantovertime N DayVaR DEAR NBISrequires10 dayVaR Confidencelevel ProbabilityofthelossDefiningthe badday frequencyofthebaddayLevelofconfidenceoveryourlossforecast riskestimate Confidenceinterval statisticallytermed 95 Riskmetricsvs 99 BISrequirementThehighertheCL thebiggertheVaRnumber WhatcouldhappenifCLissettoeither100 or0 Answersarerightstatementsbutnonsense WhatdoesPDFcurvetell aEb E EE ProbabilityDistributionofarandomvariableXX randomreturn orvalue L G ofinvestmentE expectedreturn theprobabilityweightedaveragevalueofallpossibleoutcomesoftherandomvariable SD variance measureofdispersion expectedsurprisesVarianceistheprobabilityweightedsumofthesquareddeviationsofalltheoutcomesfromtheirexpectedvalue orsimply theexpectedsquareddeviationoftherateofreturnfromitsexpectation PDF1 F a P X a PDF2 N a P X a X X Area P X a Area P a X b Area P X b 1 F b 68 26 WhatdoesPDFcurvetell aEb a E Eb E ProbabilityDistributionofarandomvariableXPDFtellstheprobabilitythattherandomvariable X doesnotexceedthespecifiedcriticalvaluea F a P X a whichcanbeillustratedbythesizeofthearealefttothecriticalvalue Thesizeoftheareaindicatestheprobabilitythatthevariablefallwithinthetwocriticalvalues ThewholeareaunderneaththePDFcurveis1 PDF1 F a P X a PDF2 N a P X a X X Area P X a Area P a X b Area P X b 1 F b 68 26 Normaldistribution Anormaldistributioncanbecompletelydescribedbytwofeatures expectedvalue E andstandarddeviation NDissymmetricandbell shaped smalltailoneitherside whichononehandmeansprobabilityofextremevalueoneithersideissmall ontheotherhandmeanschangesoftherandomvariablearecenteredarounditsmeanvalue 68 26 ofchangesfallwithin1SDoneitherside 95 44 within2SDs 99 74 within3SDs90 within1 65SDs 95 within1 96SDs 98 within2 33SDsNormaldistributionassumptioninfinanceandfattailproblem a E Eb E PDF N a P X a X 68 26 ConfidencelevelandcriticalvalueofaPDF SupposethePDFofyourinvestmentportfolioisgivenasfigureinnextslide Supposeyourcurrentpositionis 100 000 theoreticallyreflectingtheexpectedvalueofrandomlyfluctuatingvalueofyourportfolio SupposetheSD 5 000 1 65 8 250Thecriticalvalue1 65 awayfromEintheleftsideis 8 250inlossor 91 750invalue whichindicates With95 confidencelevel youbelieveyourlosswillnotexceed 8 250 G LofPortfolio ValueofPortfolio 8 2500 8 250 91 750 100 000 108 250 E 1 65 EE 1 65 90 5 5 PDFofyourinvestmentportfolio Approachestoprobabilitydistribution typesofVaR HistoricalsimulationVariance covarianceapproachMonte Carlosimulation Howaboutwhentheprobabilitydistributionofyourportfolioisnotdirectlyknown Dueto 1 Integratedcontributionfromdifferentriskfactors likeinterestrisk FXrisk etc 2 Aggregationofmultipletypesofassets likepositionsinbonds equity commodities 3 Moreimportantly shorthistoryanddynamicadjustmentofyourportfolioThen weturnto 1 decomposingtheriskintoriskfactors 2 workingoutthePDFofriskfactors 3 andlinkingtheriskfactorchangetoyourportfoliovaluechangeusingsensitivity WorkingoutVaRthroughariskfactor Marketrisk Estimatedpotentiallossunderadversecircumstances valuepricePotential VolatilityVaR ofthe sensitivityof adversemoveofriskpositionthepositioninyieldfactor VaR position PricevolatilityofthepositionPricesensitivity unit or 1sensitivity i e 1position ssensitivitytoariskfactor likeinterestrate stockprice changePricevolatility unit or 1volatilitye g 1position svolatility Marketrisk Estimatedpotentiallossunderadversecircumstances valuepricePotential VolatilityVaR ofthe sensitivityof adversemoveofriskpositionthepositioninyieldfactor VaR position Pricevolatilityofanasset valuepriceProbability VolatilityVaR ofthe sensitivityof Distributionofriskpositionthepositionofriskfactorfactor givenaunitchangeinriskfactorVaR 100 000 0 005 1P 1bp 16 5bps 95 CL 9 250 95 CL VaR P S DR valueofyourposition Unitsensitivityofyourpositiongivenaunitchangeofriskfactor BadChanges Volatility PDF ofriskfactor 1position svolatility Definethe badchange ofriskfactor Equivalenttodefininga badday Itiseasytodefineoridentifythe baddirection ofriskfactorchange e g riseininterestrate yield isadversechangedirectiontoyourlongpositioninbond Howtodefineamorespecific badchange ismoretroublesome youneedtospecify howbadthat changewillbe Oneeasywaytodefinea badchange istodefinethesizeorseverityofthechange Butthisdoesnotgeneratetoomuchsenseforriskmanagementsincetheprobabilityisnotspecified Everyday undernormalmarketcondition youexperienceadversechangeswhilethemarketfluctuates someadversechangesfrequentlyoccurbutaretolerabletoyouduetotheirrelativelysmallsize othersareseriousorevendisastroustoyou butdonothappenveryoften Asariskmanager yourfocusisonthelatter Itisreasonable andmeaningful todefineabadchangeinsuchawaythatabadchangeisanadversechangewhichhasatmost 5 chancetooccurundernormalmarketcondition putanotherway itoccursonceoutofevery 20 timesofexperimentorobservation Thisdefinitioncombinestogethertheseverityandprobabilityofthe badchange 3 3Riskmeticsmodel IntroductionDennisWeatherstone sOrderBasicmethodologyandprocessCalculatingVaRforanaiveinvestmentportfolioofanFIFixed incomesecuritiesFXEquitiesPortfolioaggregationCriticismsagainstRiskmetricsBISregulationonVaR basedinternalmodelsinlargebanks IntroductiontoRiskmetricsmodel InternalmodelofJ P Morgan 1994 NormaldistributionisassumedformarketchangeOne dayholdingperiod95 confidencelevelBenchmarkofmarketriskmanagement DennisWeatherstone sOrder Atcloseofbusinesseachdaytellmewhatthemarketrisksareacrossallbusinessesandlocations Inanutshell thechairmanofJ P Morganwantsasingledollarnumberat4 15pmNewYorktimethattellshimJ P Morgan smarketriskexposurethenextday especiallyifthatdayturnsouttobea bad day TherequiresingledollarnumberisDailyEarningsatRisk DEAR ordailyVaR Anontrivialjob ForeignFixedExchangeEmergencyincomeSTRIT CommoditiesDerivativesEquitiesMarketsProprietaryTotalNumberofactivelocations1412511871114Numberofindependentrisk takingunits3021816141119120ThousandsofTransactionsPerday 5 5520BillionsofdollarsindailyTradingvolume 10 301150 Basicmethodologyandprocess CalculatingDEARfiguresforeachofthebusinesslines riskfactors StandaloneriskFixed incomesecuritiesFXEquity PortfolioAggregation PortfolioRiskDifferenttradingpositionsaggregatedDifferentriskfactorsaggregatedCorrelationeffectconsidered CalculatingVaRforanaiveinvestmentportfolioofanFI SupposeanFIhasfollowinginvestmentpositions 1 a 1millionmarketvaluepositionin7 yearzero couponbonds 2 Swf1 6millioninspotSwissfrancs FXrateisWsf1 60 atthedailyclose 3 1milliontradingpositioninstocksthatreflectaU S stockmarketindex WhatistheDEAR VaR 95 confidencelevel Fixed incomesecurities Suppose 1 TheFIhasa 1millionmarketvaluepositionin7 yearzero couponbonds 2 Today syieldonthesebondsis7 243 ThenS MD D 1 R 7 1 7 243 6 527 3 Thedailychangeofyieldisnormallydistributedanditsvolatility is10bpsDEAR P S DR 95 CL 1million 6 527 1 65 10bpc 95 CL 1million 1 077 95 CL 10 770 95 CL Pricevolatility MD Potentialadversechangeinyield 6 527 0 00165 1 077 DEAR Marketvalueofposition Pricevolatility 1 000 000 01077 10 770 FromDEARtomore than one dayVaR Tocalculatethepotentiallossformorethanoneday N dayVAR DEAR NExample Forafive dayperiod VAR 10 770 5 24 082 FX InthecaseofForeignExchange DEARiscomputedinthesamefashionweemployedforinterestraterisk DEAR P S DR 95 CLDEAR Dollarvalueofposition Pricevolatility Suppose 1 theFIhadaSwf1 6millioninspotSwissfrancs FXrateisWsf1 60 atthedailyclose ThismeansP 1million 2 ChangesoftheFXratearenormallydistributedandthehistoricalvolatility ofdailychangesinthespotFXrateis56 5bps DEAR P S DR 95 CL 1million 1 1 65 56 5bps 95 CL 1million 93 2bps 95 CL 9 320 95 CL Equities Accordingtomodernportfoliotheory therearetwotypesofrisktoanequitypositioninanindividualstock Totalrisk Systematicrisk UnsystematicriskU riskcanbelargelydiversifiedawayinaverywell diversifiedportfolio S riskreflectsthecomovementofthatstockwiththemarketportfolio forwhichthestockmarketindexcanbeaproxy ThesensitivityofastocktothemarketportfolioisgivenbyCAPMmodel E ri rf i E rM rf i iM M2 Forawell diversifiedstockportfolio DEAR P S DR 95 CL wherethemarketreturnvolatilityistakenas1 65sM P DIndex 95 CLDEAR Dollarvalueofposition Pricevolatility Iftheportfolioreplicatesthestockmarketindex 1 Inlesswelldiversifiedportfolio theeffectofU riskshouldbeconsidered IfCAPMmodeldoesnotofferagoodexplanationofassetpricingcomparedto say multi indexAPTmodel adegreeoferrorshouldbebuiltintoDEARcalculation Suppose 1 TheFIholdsa 1milliontradingpositioninstocksthatreflectaU S stockmarketindex Then 1 2 Thedailyreturn changeofvalue onthestockmarketindexisnormallydistributedanditsvolatilityis2 DEAR P DR 95 CL 1million 1 1 65 2 95 CL 1million 3 3 95 CL 33 000 95 CL PortfolioAggregation individualDEARsoftheFI BondDEAR 10 770 95 CLFXDEAR 9 320 95 CLSUM 53 090EquityDEAR 33 000 95 CLSimplysummingupindividualDEARsdoesnotcomplywithmodernportfoliomanagementtheoryforcorrelationanddiversificationeffectisnottakenintoaccount InordertoaggregatetheDEARsfromindividualexposureswerequirethecorrelationmatrix Three assetcase DEARportfolio DEARa2 DEARb2 DEARc2 2rab DEARa DEARb 2rac DEARa DEARc 2rbc DEARb DEARc 1 2 correlationmatrix r Seven YearZeroSwf 1U SStockIndexSeven yearzero 2 4Swf 1 1U Sstockindex AggregatingindividualDEARintoportfolioDEAR DEARportfolio DEARz 2 DEARswf 2 DEARu s 21 2 2Xrz SwfXDEARzXDEARswf 2Xrz U S XDEARzXDEARU S 2XrU S SwfXDEARU SXDearswf DEARportfolio 10 77 2 9 32 2 33 2 2 2 10 77 9 32 2 4 10 77 33 2 1 9 32 33 39 969 53 090Portfolioeffect 53 090 39 969 13 121 PortfolioDEARSpreadsheet InterestRateRiskNotionalAmounts U S millionsequivalents FXRiskTotal112345710interestSpotFXPortfolioTotalmonthyearYearsYearsyearsyearsyearsyearsDEARFXDEARAustraliaAUDBelgiumBEFCanadaCADDenmarkDKKFrance19 301148FFR48Germany 1930 1127DEM27ItalyLIRJapanYENNetherlandsNLGSpainESBSwedenSEKSwitzerlandGBPUnitedKingdomGBPUnitedStates101076USD76Total1010151151Portfolioeffect 62 62 RISKDATAPRINTCLOSETotalDEAR 000S 8989 CriticismsandshortcomingsofRiskmetrics Assumptionofasymmetricnormaldistributionforallassetreturns Forsomeassets suchasoptionsandshort termsecurities bonds thisishighlyquestionable Limitationofnormalmarketconditionandsupplementarystresstestingorscenarioanalysis BISregulationonVaR basedinternalmodelsinlargebanks IncalculatingDEAR adversechangeinratesdefinedas99thpercentile ratherthan95thunderRiskMetrics Minimumholdingperiodis10days meansthatRiskMetrics dailyDEARmultipliedby 10 Capitalchargewillbethehigherof Previousday sVAR orDEAR 10 AverageDailyVARoverprevious60daystimesamultiplicationfactor 3 Subjecttoback testing 3 4HistoricorBackSimulation AdvantagesBasicideaProcessofHistoricSimulationWeaknesses Advantages SimplicityDoesnotrequirenormaldistributionofreturns whichisacriticalassumptionforRiskMetrics Doesnotneedcorrelationsorstandarddeviationsofindividualassetreturns Basicidea Revalueportfoliobasedonactualprices returns ontheassetsthatexistedyesterday thedaybefore etc usuallyprevious500days Thencalculate5 worst case 25thlowestvalueof500days outcomes Only5 oftheoutcomeswerelower ProcessofHistoricSimulation Converttoday sFXpositionsintodollarequivalentsattoday sFXrates MeasuresensitivityofeachpositionCalculateitsdelta MeasureriskActualpercentagechangesinFXratesforeachofpast500days Rankdaysbyriskfromworsttobest Examples Example1 TextbookPage244 246Example2 Page257 QuestionsandProblemsNO 16 Weaknesses Basicassumption therecentpastdistributionofexchangeratesisanaccuratereflectionofthelikelydistributionofFXratechangesinthefuture thatexchangeratechangeshavea stationary distribution Disadvantage 500observationsisnotverymanyfromstatisticalstandpoint Increasingnumberofobservationsbygoingbackfurtherintimeisnotdesirable Couldweightrecentobservationsmoreheavilyandgofurtherback 3 5MonteCarloSimulation Toovercomeproblemoflimitednumberofobservations synthesizeadditionalobservations Perhaps10 000realandsyntheticobservations Employhistoriccovariancematrixandrandomnumbergeneratortosynthesizeobservations Objectiveistoreplicatethedistributionofobservedoutcomeswithsyntheticdata 3 6RegulatoryModels BIS includingFederalReserve approach MarketriskmaybecalculatedusingstandardizedBISmodel Specificriskcharge Generalmarketriskcharge Offsets Subjecttoregulatorypermission largebanksmaybeallowedtousetheirinternalmodelsasthebasisfordeterminingcapitalrequirements BISStandardizedModel Specificriskcharge Riskweights absolutedollarvaluesoflongandshortpositionsGeneralmarketriskcharge reflectmodifieddurations expectedinterestrateshocksforeachmaturityVerticaloffsets AdjustforbasisriskHorizontaloffsetswithin betweentimezones Terms ValueatRisk Va