信用风险优化中的期望短缺模型及基于非数值算法求解

200555:100026788(2005)0520063206,(,300072):,,;,.Fredrik,.,,.:,,.:;;;;:F830:AExpectedShortfallModelinginOptimizingCreditRiskPortfolioandItsSolutionBasedontheNon2NumericalAlgorithmsZHANYuan2rui,ZHANGJian2long(SchoolofManagementl,TianjinUniversity,Tianjin300072,China)Abstract:ExpectedShortfall(ES)isanewtoolforcreditriskmeasureandoptimization.Astheriskmeasuretool,ESrepresentsthetailinformationoflossandisfavorabletokeepawaytheextremefinanceriskwithverylittleprobability.Asthetoolforriskoptimization,itsimultaneouslyadjustsallpositionsintheportfolioinordertooptimizeES,andsimultaneouslygaincorrespondingVaR.FredrikbuildsalinearprogrammodelwithExpectedShortfallwhichcansimultaneouslyoptimizeExpectedShortfallandVaRofportfolio,butthismodelhasthedrawbackofdimensionobstacle.Toovercomethisdrawback,werevertittoanon2linearprogrammodelagainandsolveitbyaGeneticAlgorithmwithconstraintsandSimulatedAnnealingAlgorithm.Exampleillustrationshowsthattheoptimizedportfoliosstandarddeviation,VaRandExepectedShortfallaredecreasedobviouslyunderthenearlysameexpectedyieldthroughtwomethods,butSimulatedAnnealingAlgorithmhasbettereffectonoptimizingportfoliosES.Keywords:expectedshortfall;geneticalgorithmwithconstraints;simulatedannealingalgorithm;VaR;CreditMetrics:2004203224:(79970041):(1944-),,;(1979-),,.1(ExpectedShortfall,ES)VaR,(CVaR)VaR,VaR.VaR,,,.,,.VaR(),,()..,VaR,VaR;,Fredrik[1],©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.,,VaR,,.,Fredrik,.,,VaR,.2f(x,y):RnRmRxRnyRm,yp:RmR,(x,a):(x,)=f(x,y)p(y)dy,(1),.VaR(x,),,:(x,)=min{R:(x,)}.(2)ESf(x,y)f(x,y)(x,),(x):(x)=(1-)-1f(x,y)(x,)f(x,y)p(y)dy.(3)xXRn.Rockafellar[2]:(x)XRn(4).F(x,)=+(1-)-1yRm(f(x,y)-)+p(y)dy(4)minRF(x,)=minxX(x)VaR,F(x,)VaRES.Fredrik[1]:min(z,)=+vJj=1zj,(5)s.t.zjni=1((bi-yji)xi)-,j=1,,J,(6)zj0,j=1,,J,(7)lixiui,(8)ni=1qixi=ni=1qini=1bixi=ni=1bi,(9),v=((1-)J)-1,x=(x1,,xn)i,;b=(b1,,bn);y=(y1,,yn),;J;f(x,y)=(b-y)Tx;(8),li,ui;li0;(9),q=(q1,,qn).(x3,3,z3),x3P,(z3,3)ES,3VaR.Fredrik,.20,J=20000,40000,,.Fredrik,:4620055©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.min(x,)=+vJj=1(b-yj)xT-+,(10)lixiui,i=1,,n,(11)ni=1qixi=ni=1qini=1bixi=ni=1bi,(12)+:x+=x,x00,x0,n+1(n),.Fredrik,.:[1].x3,3(VaR),(x3,3)(ES).Fredrik[1],.3:,.311[3],.20,53671022,ES.:,,,.VaR,,20000,yij(i=1,,20;j=1,,20000)(:20,000),20000;0199,20000,VaR200[4].20000,,VaR30411,(4)ES65012.2020000yij,i=1,,20;j=1,,20000,=0199,J=2000,l=0,u=2,i=1,,J,:v=((1-)J)-1=01005,b=[696188,99172,99174,96182,95152,87143,92147,955170,489131,298149,98145,195112,54174,95163,286190,174185,93184,763130,97164,495147]T,beq=20i=1bi=5367102.312,,.:1)(12)(10),,:min(x,)=(x)+vJj=1((b-yj)3xT-(x))+,(13)lxiu,i=1,,19,j=1,,J,(14)(beq-x(1:19)3b(1:19))Pb(20)u,(15)(beq-x(1:29)3b(1:19))Pb(20)l.(16)x(1:19)x119,x(20)=(beq-x(1:19)3b(1:19))Pb(20),565©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.:x=x(1:19);2):fs=(beq-x(1:19)3b(1:19))Pb(20);fs1=(|fs-l|-(fs-l))P2;fs2=(|u-fs|-(u-fs))P2;pp=111(fs1+fs2).3)(x,),aimf:aimf=ppP(1+111^(-01013(x,))).4):f(x)=aimf;..313;19;30;130;,jpselect=fj(x)P30j=1fj(x);,pcross=016;,pmutation=0104.1,:,,(,ES),,.[5],:x3=[112174,017449,113891,112425,013793,011406,014504,115824,013596,119442,111388,015047,116621,012355,010595,010076,110120,014305,018431,119774]TES=290.1,VaR=147.4.1314,Metropolis[6],,.:(VeryFastSimulatedAnnealing)[7]312.:yi=T(k)sgn(u-015)[(1+1PT(k))|2u-1|-1],(17)k;T(k)=T0k1PN;,017110;0151101PN;u[0,1];sgn(x)=1,x00,x=0-1,x0yi0,x(i,k+1)=x(i,k)+yi(Bi-x(i,k)),(18)yi0,x(i,k+1)=x(i,k)+yi(x(i,k)-Ai),(19),x(i,k)[Ai,Bi],x(i,k)ki,i=1,,19;(18)(19)x(i,k+1)[Ai,Bi].k,E(k)=aimf(k);6620055©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.r(k)=exp(E(k)PT(k)),E(k)=E(k)-E(k+1).E(k)0,,x(i,k+1)=x(i,k+1);E(k)0,Metropolis:[0,1],r(k),,x(i,k+1)=x(i,k+1);,,x(i,k+1)=x(i,k).Metropolis,,.:1)x(i,1),i=1,,19,E(1);2)x(i,k),E(k);3)E(k)0,x(i,k+1)=x(i,k+1);E(k)0,Metropolis;4),,,,.:1000;=0191;T0=10000;1PN=110;Ai=0,Bi=2.:x3=[115784,110275,017109,117729,013385,010861,015529,014225,015097,116808,012191,113768,010563,013970,011917,011909,111800,115464,011458,119212];ES=22716;VaR=14419.,2.21,.1,,,,.4611%,4312%;5219%,5317%;3815%,5118%.,;,,.1:-0.95%-0.41%-0.43%50.7321.8222.82142.9777.0681.25VaR313.1VaR147.4VaR144.9ES471.8ES290.1ES227.64,,;,,VaR.Fredrik[1](82)765©1995-2005TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.MaoDing2xiang.Acombinationalevaluationmethodresultinginconsistencybetweensubjectiveandobjectiveevaluationintheleastsquaressense[J].ChineseJournalofManagementScience,2002,10(5):21-26.(inChinese)[9].[J].,2002,(10):10-12.XuQiang.Researchofcombinationevaluationmethods[J].JiangsuStatistics,2002,(10):10-12.(inChinese)[10],,.[J].(),2003,42(5):667-672.ChenGuohong,ChenYantai,LiMeijuan.Researchonthecombinationevaluationsystem[J].JournalofFudanUniversity(NaturalScience),2003,42(5):667-672.(inChinese)[11]ChenGuohong,LiMeijuan,ChenYantai.Theresearchonmeasurementofdriftabouttheevaluationconclusionofsinglemethod[A].Proceedingsof2003InternationalConferenceonManagementScience&Engineering[C].Harbin:HarbinInstituteo