分位数回归视角下的金融市场风险度量研究进展

20096()No.62009(94)JOURNALOFFUZHOUUNIVERSITY(PhilosophyandSocialSciences)SerialNo.94:2009-05-08:教育部人文社会科学青年基金资助项目(07JC790046);福建省自然科学基金资助项目(2008J0192);福建省社会科学规划项目(2008B037):唐勇,男,江苏淮安人,福州大学管理学院副教授,博士;寇贵明,男,福建泉州人,福州大学管理学院硕士研究生分位数回归视角下的金融市场风险度量研究进展唐勇寇贵明(,350108):基于分位数回归的金融市场风险度量方法是一个崭新的研究领域因具有准确描述尖峰厚尾等数据特征和无需对序列分布做特定假设的优点,分位数回归而倍受关注梳理基于分位数回归的四种风险度量方法,包括表达式参数估计相应的检验过程存在的问题,以及这些度量方法在国内的研究状况,可为提高我国金融业风险管理水平提供必要的理论指导和实践方法:分位数回归;风险度量;VaR;金融市场:F830:A:1002-3321(2009)06-0039-06,J.P.Morgan2090VaR(ValueatRisk)VaR,VaRtGEDLogistic,VaR,,,VaR,,,,,,,KoenkerBassett[1],(OLS)(meanregression),,:;;;KoenkerBassett,,,39,:Y,FY(y),FY(y)=P(Yy)=th,FY(y)=y=F-1Y()=inf{Y|FY(y)!}(OLS)min∀Rn#ni=1(yi-x∃i)2,(medianregression),med(y|x)=xT(0.5),(0.5)min∀Rnyi-x∃iKoenkerBassett,:min∀Rn#ni=1(yi-!(xi,))(1)(checkfunction)(x)=x,x0(-1)x,x0∀(0,1),()Koenker,,[2]OLS,;,OLSLAD(leastabsolutedeviation),,,,,,,,,,,,,(一)VaR与一致性风险度量方法J.P.Morgan2090VaR,(VaR),(maximumpossibleloss),VaRVaR:,VaR;,VaR;,VaR,Artzner,EberHeath(Coherentmeasureofrisk)[3],,,RockfellarUryasev(CVaR)[4],∀,VaR,CVaR,CarloAcerbiDirkTascheExpectedShortfall(ES)CVaR[5]AcerbiES,(SpectralRiskMeasure),,[6]WangVaRVaR(Tail-VaR),,VaRCVaRDRM(DistortionRiskMeasure)[7][8],Copula,40(二)基于分位数回归(quantileregression,QR)的风险度量方法,,,,ARCHQuantileCAViaRCAREEWQR1.ARCHQuantile风险度量方法KoenkerZhao,ARCHQuantile[9],:yt=∀0+∀1yt-1+%+∀pyt-p+#t#t=(∃0+∃1#t-1+%+∃q#t-q)ztth,:Q#(It-1)=X∃t∀()th()VaR:-VaRt()=!t+X∃t∀()(2)[10]:Xt=(1,#t-1,%,#t-q)∃,∀()=(∃0Qz(),∃1Qz(),%,∃qQz())∃,Qz()zt∀()Q.R.,(1):∀^()=argmin#(#t-X∃t∃):∃=(∃0,∃1,%,∃q)∃ARCH,,GARCH2.CAViaR风险度量方法EngleManganelliCAViaR(ConditionalAutoregressionValueatRisk)[11],VaR(VaR),VaRt(1)CAViaRCAViaR:VaRt=0+#pi=1iVaRt-i+l(p+1,%,p+q;%t-1):Adaptive:VaRt=VaRt-1+1{[1+exp(G[yt-1-VaRt-1])]-1-&}SymmetricAbsoluteValue:VaRt=0+1VaRt-1(&)+2yt-1AsymmetricSlope:VaRt=1+2VaRt-1(&)+3(yt-1)++4(yt-1)-IndirectGARCH(1,1):VaRt=(1+2VaR2t-1+3y2t-1)1/2,Kuester,IndirectAR-GARCHCAViaR[12]:VaRt(&)=∀1rt-1+(0+1(VaRt-1-∀1rt-2)2+2(rt-1-∀trt-2)2)1/2,i0VaR,,(2)CAViaREngleManganelliCAViaR,,105,[0,1][-1,0],QR(Quantileregressionsum),quasi-Newton,10,QR,QR(3)CAViaR,EngleManganelliHit:Hitt()=I(yt-VaRt)-&ytHitt()1-&,-&Hit=X∋+utut=-&,(1-&)1-&,&X=(Hitt-i,VaRt)Hits,H0&∋=0,:∋^OLS=(X∃X)-1X∃Hit~N(0,&(1-&)(X∃X)-1)(DynamicQuantileTest,DQ):DQ=∋^∃OLSX∃X∋^OLS&(1-&)~(2(k+1)413.CARE风险度量方法,Koenkerexpectile,expectilesTaylorexpectilesVaRES,ES[13]expectile&expectile&Efron,ALS(Asymmetricleastsquares)[14]TaylorES:ESt(&)=(1+(1-2)&)!t()-(1-2)&E(yt)CAViaRCARE(ConditionalAutoregressiveExpectiles),SymmetricAbsoluteValueCARE:!t()=0+1!t-1()+2yt-1ES:ESt(&)=∃0+∃1ESt-1(&)+∃2yt-1∃i=(1+(1-2)&)i,i=0,1,2CARECAViaR,QRALS,0.0001,CAViaR,HitDQ,,EfronTibshiranibootstrap[15]4.EWQR风险度量方法,TaylorVaRESEWQR(ExponentiallyWeightedQuantileRegression)[16]),,:min#Tt=1)T-t(y-x∃t)(&-I(ytx∃t)):)T-t):)∀[0.8,1],0.005QR,)TaylorVaRGJR-GARCHCAViaRES,,EWQRES:ES^T=-1&#Tt=1)T-t#Tt=1)T-t(yt-x∃t^)(&-I(ytx∃t^))TaylorYuJones[17],Qt(&)=x∃t,:min#Tt=1Kh1(x-xt)∋+(-((y-x∃t)(&-I(ytx∃t))Wh2(y-yt)dyWh2(y-yt)Kh1,Wh2,:min#Tt=1)T-t(&(yt-x∃t)+(x∃t-yt)∗((x∃t-yt)/h2)+h2+((x∃t-yt)/h2))∗+Taylor,,,,,,,,,:(),[18],,Bootstrap42,;[19]()CAViaR,CAViaRVaR[20],(VaR),)VaR∗VaR,[21];()VaR,VaR[22],,,[23]TARCH2-CAViaR[24]CAViaR,,,,,TARCH2-CAViaR,,,VQR,CAREEWQR,,,VaRES,(DRM),,:[1]Koenker,R.,G.W.Bassett,)RegressionQuantiles∗,Econometrica,vo.l46,no.1(Jan.1978),pp.3350.[2]Koenker,R.W.,QuantileRegression.Cambridge,UK:CambridgeUniversityPress,2005.[3]Artzner,P.,Delbaen,F.,Eber,J.-M.,Heath,D.,)CoherentMeasuresofRisk∗,MathematicalFinance,vo.l9,no.3(1999),pp.203-228.[4]P.Krokhma,lJ.Palmquist,S.Uryasev.,)PortfolioOptimizationwithConditionalValue-at-RiskObjectivbesandConstaints∗,JournalofRisk,vo.l42,no.2(2002),pp.124129.[5]Acerb,iC.,NordioC.,SirtoriC.,)ExpectedShortfallasaToolforFinancialRiskManagement∗,(2001).[6]Acerb,iC.,)SpectralMeasuresofRisk:aCoherentRepresentationofSubjectiveRiskAversion∗,JournalofBankingandFinance,vo.l26,no.7(2002),pp.1505-1518.[7]WangS.S.,)InsurancePricingandIncreasedLimitsRatemakingbyProportionalHazardsTransforms∗,Insurance:MathematicsandEconomics,no.17(1995),pp.43-54.[8]WangS.S.,)PremiumCalculationbyTransformingtheLayerPremiumDensity∗,Bulletin,no.26(1996),pp.71-92.[9]Koenker,R.andZhao,Q.,)ConditionalQuantileEstimationandInferenceforARCHmodels∗,EconometricTheory,no.12(1996),pp.793-813.[10]此处表示左尾分位数,+Var表示损伯[11]Engle,R.F.,S.Manganell,i)CAViaR:ConditionalAutoregressiveValueatRiskbyRegressionQuantiles∗,Journalof43BusinessandEconomicStatistics,no.22(2004),pp.367-381.[12]Kuester,K.,S.Mittnik,M.S.Paolella,)Value-at-RiskPrediction:AComparisonofAlternativeStrategies∗,JournalofFinancialEconometrics,no.4(2006),pp.53-89.[13]Taylor,J.W.,)EstimatingValueatRiskandExpectedShortfallUsingExpectiles∗,JournalofFinancialEconometrics,no.6(2008),pp.231-252.[14]Efron,B.,)RegressionPercentilesUsingAsymmetricSquaredErrorLoss∗,StatisticaSinica,no.1(1991),pp.93-125.[15]Efron,B.,J.Tibshiran,iAnIntroductiontotheBootstrap,NewYork:ChapmanandHal,l1993.[16]Taylor,J.W.,)EstimatingValueatRiskUsingExponentially