金融资产的VaR和CVaR风险的优良估计

:1003-207(2006)05-0001-06VaRCVaR,(,430074):VaRCVaR,,-,,,,,,:;;;;:F830:A:2005-12-01;:2006-09-03:(0385008):(1965-),(),,,,:.1,,VaR(Val2ueatRisk),VaR,,,VaR,,,,[1-2],,,VaR,,VaR[3-5],[6-7](Condition2alValueatRisk,CVaR),VaR,VaR[6-8]CVaR,VaR,VaR[9]CVaRESVaRCVaRVaRCVaR,VaRCVaRVaR,[2]:,[10],VaR;,[11]VaR,VaR;MonteCarlo,[12],[13]MCMC;[14]CVaR,[6-7],,,145200610ChineseJournalofManagementScienceVol.14,No.5Oct.,2006,CVaR;GARCH[15][16][17],,,(),,,VaRCVaR,,,,,VaRCVaR(UMVUE)(,BLUE)(,BLIE),2VaRCVaR,3VaRCVaR,42VaRCVaR[1,6-7]VaRCVaR,[1,6-7]-VaRCVaRR,RN(,2)[1,6,18],(011,1%5%),VaRCVaRVaR=g1(,)=-+-1(1-)(1)CVaR=g2(,)=-+c()(2)c()=[-1(1-)]/0,()N(0,1),()N(0,1),-1(1-)0(011)N(0,1)1-,3VaRCVaR:VaRCVaRg1,g2,-VaRCVaR,RX1,X2,,Xn,VaRCVaRUMVUEBLUEBLIE,311VaRCVaRUMVUE-,g(,)=-+l(l0)(VaRCVaR)UMVUE,1[19]X=(X1,X2,,Xn)dP(x)=c()expkj=1jTj(x)d(x),,x=(x1,x2,,xn),T(X)=(T1(X),T2(X),,Tk(X))2[20]S(X1,X2,,Xn){P},g(),g()UMVUE,S(X1,X2,,Xn)1X1,X2,,XnN(,2)(-,0)ni1i1d1,g=-+lUMVUEg3=-ŠX+lcnni=1(Xi-ŠX)2,cn=(n-12)/[2(n2)],ŠX=1nni=1Xip(x)dx=12nexp-122n1(xi-)2dx1dxn=c(1,2)exp[1T1(x)+2T2(x)]d(x)1=2,2=-122,T1(x)=n1xi,T2(x)=ni=1x2i,c(1,2)=12nexp-n222,d(x)=dx1dxn22006={(1,2)-1,-20},,1,(T1(X),T2(X))g3=-ŠX+lcnni=1(Xi-ŠX)2g=-+l,2([20]),E(g3|(T1,T2))=g3(g3(T1,T2))gUMVUE11-,,(1)(2)VaRCVaRUMVUE:VaR3=-ŠX+-1(1-)cnni=1(Xi-ŠX)2(3)CVaR3=-ŠX+c()cnni=1(Xi-ŠX)2(4),312VaRCVaRBLUE-,g(VaRCVaR)BLUEVaRCVaR?l0,E(ni=1aiXi)=ni=1ai-+l=g,PR,0,g,gg()(,BLUE,)Gauss2Markov,BLUE,-Gauss2Markov,VaRCVaRBLUEGauss2Markov(X,Y,2In)(EX=Y,Var(X)=2In),Var(X)=2InVar(X)=2G,G,Gauss2Markov[20],G0,nB,G=B2ŽX=B-1X,“Y=B-1Y,EŽX=B-1EX=B-1YB=“YB;Var(ŽX)=B-1Var(X)B-1=2In(ŽX,“Y,2In)Gauss2Markov(LSE)‰LS=(“Y“Y)-1“YŽX=(YG-1Y)-1YG-1X(5)3[20](X,Y,2In),Y,„LSBLUE4[20](X,Y,2In),Y,PRp,„LSBLUEGauss2MarkovHX1,X2,,Xni1i1d1,X(1)X(2)X(r)r(1rn),Y(i)=X(i)-,i=1,2,,r,Y(1)Y(2)Y(r)N(0,1)ni1i1d1rEY(i)=ai,i=1,2,,rCov(Y(i),Y(j))=vij,1i,jr:ai,vijn,r,,X(i)=+Y(i)=+ai+i,i=(Y(i)-ai),i=1,2,,rX=(X(1),X(2),,X(r)),a=(a1,a2,,ar):EX=(Ir,a)Var(X)=2Vr=2(vij)rrIr1rGauss2Markov,H5X1,X2,,XnN(,2)i1i1d1,rBLUE‰‰=IrV-1rIraV-1rIraV-1rIraV-1ra-1IraV-1rX¦L1XL2X(6)35:VaRCVaRVar‰„=2L1VrLrL1VrL2L2VrLrL2VrL2¦2ABBC(7)HH(5)3452-,X(1)X(2)X(r)Rni1i1d1r,X=X(1)X(2)X(r)),g=-+lrBLUE‰g=-‰+l„=-L1X+lL2X22,,VaRCVaRBLUEVaR3=-L1X+-1(1-)L2X(8)CVaR3=-L1X+c()L2X(9)nr,(4)-(8)L1,L2,,[21]n,Matlab313VaRCVaRBLIEg=-+l(VaRCVaR)BLIE3X1,X2,,XnN(,2)i1i1d1,‰„BLUE,,g(,)=-+l^g=-‰+l+B1+C„,H[20]21715,[20]21715:,3BLIE2BLUE3-,,(1)(2)VaRCVaRBLIE:VaR3=-‰+-1(l-)+B1+C„(10)CVaR3=-‰+c()+B1+C„(11)32(6)4,300,20041011022004111103Rt=logPt-logPt-1,200,ttmatlab6154111Jarque2bera-417951e-0040101360136233128764185151,300,,Jarque2Bera,JB0105519915,0101912103,,q2q,q2q,,,,,412VaRCVaR-5%1%,2[6-7]VaRCVaR(,,,);VaRCVaR(42006,r200);VaRCVaR2300VaRCVaRVaRFVaRVaRCVaRFCVaRCVaR5%UMVUE010229BLUE010284BLIE0102840102430102160102860103580103580103110102551%UMVUE010322BLUE010404BLIE010404010345010261010369010465010465010377010310,UMVUE,F,BLUEBLIE,,r200,BLUEBLIE,r(,),:(20),[21]L1,L2,,5VaRCVaR,,,,,,VaRCVaRVaRCVaR(UMVUE)(,BLUE)(,BLIE),,,,,,,,,,(20),,:[1].[M].:,2001:71-83,195-327[2]JorionP,.VaR:-[M].:,2000:96-105,301-322[3]ArtznerP.,DelbaenF.,EberJ.M.,HeathD1.Coher2entmeasureofrisk[J]1MathematicalFinance,1999,9(3):203-228[4],,.VaR[J].,1999,51(6):15-18[5]MauserH.,RosenD.1BeyondVaR:frommeasuringrisktomanagingrisk[J]1ALGOResearchQuarterly,1999,1(2):5-20[6]RockafellerT.,UryasevS.1Optimizationofconditionalvalue2at2risk[J]1JournalofRisk,2000,2(3):21-42[7]RockfellerT.,UrvasevS.1Conditionalvalue2at2riskforgenerallossdistribution[J].JournalofBanking&Fi2nance,2002,26(7):1443-1471.[8]PflungG.Ch..SomeRemarksontheValue2at2RiskandtheConditionalValue2at2Risk[M].In:UryasevS,Ed.ProbabilisticConstrainedOptimization:MethodologyandApplications.Boston:KluwerAcademicPublishers,2000[9]AcerbiC.,andTascheD..Onthecoherenceofexpectedshortfall[J].JournalofBanking&Finance,2002,26:1487-1503[10]J.S.Butler&BarrySchachter.Improvingvalue2at2riskestimatesbycombiningkernelestimation[J].Proceed2ings,FederalReserveBankofChicago,1996,May:363-380[11],.VaR[J].,2002,16(4):33-37[12]BoyleP.,BroadieM.,GlassermanP..MonteCarlomethodforsecuritypricing[J].JournalofEconomicDy255:VaRCVaRnamicsandControl,1997,21:12671321[13],,.MCMCVaR[J].,2000,3(2):54-61[14]SteliosD.Bekiros,DimitrisA.Georgoutsos.EstimationofValue2at2Riskbyextremevalueandconventionalmeth2ods:acomparativeevaluationoftheirpredictiveperfor2mance[J].InternationalFinancialMarkets,InterestandMoney,2005,15:209-228.[15],.VaRES[J].,2004,22(1):84-90[16]EvisKelleziandManfredGilli.ExtremeValueTheoryforTail2RelatedRiskMeasures[Z].[17]O.Scaillet.Nonparametricestimationandsensitivityanal2ysisofexpectedshortfall[J].MathematicalFinance,2004,14:115-129[18],.CVaR[J].,2004,24A(4):442-448[19],,,.[M].:,1985:84-85[20],,.[M].:,1998:95-154[21].[M].:,1987:247-313OptimalEstimationofValue2at2RiskandConditionalValue2at2RiskLIUXiao2mao,DUHong2jun(MathematicsDepartment,HuazhongUniversityofScienceandTechnology,Wuhan430074,China)Abstract:Inthispaper,statisticalmethodisusedtoimprovetheestimationofvalue2at2risk(VaR)an