财务风险管理CH15.pptx
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财务风险管理CH15.pptx
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,Chapter15,MarketRiskVaR:
Model-BuildingApproach,1,TheModel-BuildingApproach,ThemainalternativetohistoricalsimulationistomakeassumptionsabouttheprobabilitydistributionsofthereturnsonthemarketvariablesThisisknownasthemodelbuildingapproach(orsometimesthevariance-covarianceapproach),2,MicrosoftExample(page323-324),Wehaveapositionworth$10millioninMicrosoftsharesThevolatilityofMicrosoftis2%perday(about32%peryear)WeuseN=10andX=99,3,MicrosoftExamplecontinued,Thestandarddeviationofthechangeintheportfolioin1dayis$200,000Thestandarddeviationofthechangein10daysis,4,MicrosoftExamplecontinued,Weassumethattheexpectedchangeinthevalueoftheportfolioiszero(ThisisOKforshorttimeperiods)WeassumethatthechangeinthevalueoftheportfolioisnormallydistributedSinceN(2.33)=0.01,theVaRis,5,AT&TExample,Considerapositionof$5millioninAT&TThedailyvolatilityofAT&Tis1%(approx16%peryear)TheSDper10daysisTheVaRis,6,Portfolio(page325),NowconsideraportfolioconsistingofbothMicrosoftandAT&TSupposethatthecorrelationbetweenthereturnsis0.3,7,S.D.ofPortfolio,AstandardresultinstatisticsstatesthatInthiscasesX=200,000andsY=50,000andr=0.3.Thestandarddeviationofthechangeintheportfoliovalueinonedayistherefore220,227,8,VaRforPortfolio,The10-day99%VaRfortheportfolioisThebenefitsofdiversificationare(1,473,621+368,405)1,622,657=$219,369WhatistheincrementaleffectoftheAT&TholdingonVaR?
9,TheLinearModel,WeassumeThedailychangeinthevalueofaportfolioislinearlyrelatedtothedailyreturnsfrommarketvariablesThereturnsfromthemarketvariablesarenormallydistributed,10,MarkowitzResultforVarianceofReturnonPortfolio,11,CorrespondingResultforVarianceofPortfolioValue,12,siisthedailyvolatilityoftheithasset(i.e.,SDofdailyreturns)sPistheSDofthechangeintheportfoliovalueperdayai=wiPisamountinvestedinithasset,CovarianceMatrix(vari=covii)(page328),13,AlternativeExpressionsforsP2page328,14,FourIndexExampleUsingLast500DaysofDatatoEstimateCovariances,15,VolatilitiesandCorrelationsIncreasedinSept2008,16,Correlations,Volatilities(%perday),AlternativesforHandlingInterestRates,Durationapproach:
LinearrelationbetweenDPandDybutassumesparallelshifts)Cashflowmapping:
Variablesarezero-couponbondpriceswithabout10differentmaturitiesPrincipalcomponentsanalysis:
2or3independentshiftswiththeirownvolatilities,17,HandlingInterestRates:
CashFlowMapping(page330-333),Wechooseasmarketvariableszero-couponbondpriceswithstandardmaturities(1mm,3mm,6mm,1yr,2yr,5yr,7yr,10yr,30yr)Supposethatthe5yrrateis6%andthe7yrrateis7%andwewillreceiveacashflowof$10,000in6.5years.Thevolatilitiesperdayofthe5yrand7yrbondsare0.50%and0.58%respectively,18,Examplecontinued,Weinterpolatebetweenthe5yrrateof6%andthe7yrrateof7%togeta6.5yrrateof6.75%ThePVofthe$10,000cashflowis,19,Examplecontinued,Weinterpolatebetweenthe0.5%volatilityforthe5yrbondpriceandthe0.58%volatilityforthe7yrbondpricetoget0.56%asthevolatilityforthe6.5yrbondWeallocateaofthePVtothe5yrbondand(1-a)ofthePVtothe7yrbond,20,Examplecontinued,Supposethatthecorrelationbetweenmovementinthe5yrand7yrbondpricesis0.6TomatchvariancesThisgivesa=0.074,21,Examplecontinued,Thevalueof6,540receivedin6.5yearsin5yearsandbyin7years.Thiscashflowmappingpreservesvalueandvariance,22,UsingaPCAtoCalculateVaR(page333to334),Supposewecalculatewheref1isthefirstfactorandf2isthesecondfactorIftheSDofthefactorscoresare17.55and4.77theSDofDPis,23,WhenLinearModelCanbeUsed,PortfolioofstocksPortfolioofbondsForwardcontractonforeigncurrencyInterest-rateswap,24,TheLinearModelandOptions,Consideraportfolioofoptionsdependentonasinglestockprice,S.Defineand,25,LinearModelandOptionscontinued,AsanapproximationSimilarlywhentherearemanyunderlyingmarketvariableswherediisthedeltaoftheportfoliowithrespecttotheithasset,26,Example,ConsideraninvestmentinoptionsonMicrosoftandAT&T.Supposethestockpricesare120and30respectivelyandthedeltasoftheportfoliowithrespecttothetwostockpricesare1,000and20,000respectivelyAsanapproximationwhereDx1andDx2arethepercentagechangesinthetwostockprices,27,ButtheDistributionoftheDailyReturnonanOptionisnotNormal,Thelinearmodelfailstocaptureskewnessintheprobabilitydistributionoftheportfoliovalue.,28,ImpactofGamma(SeeFigure15.1,page337),29,PositiveGamma,NegativeGamma,TranslationofAssetPriceChangetoPriceChangeforLongCall(Figure15.2,page337),30,TranslationofAssetPriceChangetoPriceChangeforShortCall(Figure15.3,page338),31,QuadraticModel(page338-340),ForaportfoliodependentonasingleassetpriceitisapproximatelytruethatsothatMomentsare,32,QuadraticModelcontinued,33,Whenthereareasmallnumberofunderlyingmarketvariablemomentscanbecalculatedanalyticallyfromthedelta/gammaapproximationTheCornishFisherexpansioncanthenbeusedtoconvertmomentstofractilesHoweverwhenthenumberofmarketvariablesbecomeslargethisisnolongerfeasible,MonteCarloSimulation(page340-341),TocalculateVaRusingMCsimulationweValueportfoliotodaySampleoncefromthemultivariatedistributionsoftheDxiUsetheDxitodeterminemarketvariablesatendofonedayRevaluetheportfolioattheendofday,34,MonteCarloSimulationcontinued,CalculateDPRepeatmanytimestobuildupaprobabilitydistributionforDPVaRistheappropriatefractileofthedistributiontimessquarerootofNForexample,with1,000trialthe1percentileisthe10thworstcase.,35,SpeedingupCalculationswiththePartialSimulationApproach,Usetheapproximatedelta/gammarelationshipbetweenDPandtheDxitocalculatethechangeinvalueoftheportfolioThisisalsoawayofspeedingupcomputationsinthehistoricalsimulationapproach,36,AlternativetoNormalDistributionAssumptioninMonteCarlo,InaMonteCarlosimulationwecanassumenon-normaldistributionsforthexi(e.g.,amultivariatet-distribution)CanalsouseaGaussianorothercopulamodelinconjunctionwithempiricaldistributions,37,ModelBuildingvsHistoricalSimulation,Modelbuildingapproachcanbeusedforinvestmentportfolioswheretherearenoderivatives,butitdoesnotusuallyworkwhenforportfolioswhereTherearederivativesPositionsareclosetodeltaneutral,38,
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