奇异方差-协方差矩阵的种风险资产投资组合的两基金分离

-n1123(1.510006)2.5106313.510275)--n-029;F224A.1952MarkowitzH[1]MarkowitzH-Σ-nnΣ||0Σ≈[2]MarkowitzHΣWW,abWnαβ1αβ+=aWWbWαβ=+αβ1αβ+=aWWbWαβ=+W-,aWbb,aWWnΣ-[2]Σn------------------------------------(70471018)(200267).1978-1963-1356037362702039329261Email:[e-mail].n1,2,,nξξξ'12(,,,)nnnRrrrR∈='(1,1,1)n,IR=∈Wwu',,)nww12(,=nnn2σn',)n12(,ξξξ=ξ-Σ')Ecov(,)([()(EEξξξ−ξξξ−==iξiup2pσrn0fnσpσ2nσ2pσ1uσ(,)uσEσ−2σu(,)uσEσ−12,,,nηηη302,,,nkkk01122nnkkkkηηηη=++++η12,,,nηηη412,,,nηηη12,,,riiiηηη12,,,riiiηηη(1,2,)iniη=12,,,riiiηηη)11(niiiPaaη===∑1niiiaη=∑a=.nun2nσ12'''min..,1nwnnWWstWRuWIσ=Σ==[2]An||0Σ=F12,,(riiirn)ξξξrFn1F21FF=2FF=12,,iiiξξξξ∃01rtttiiikkξξ==+∑t=1rttikk==−∑ifr=1rtiik==∑(1tr=+F=1ri=∑(1−,)n2FF=''1rr)fu−=12(,,rrRrr12rξξ'(1,1,,rI=()nfur()rrrfrHRIr=−nw,,)r10rk+0*tw0=)**1(,0nrrWw−+=',0)rξ12,,rξξ[2]rn1,,rnξ+12,,rξξξ01,,ttkktrkR∈,)ntr(1(1r+,,)n0(1),+A[2]A1.t10,1rtiik=−=∑1,,n1Fnr)2)tik0tk(1,tr=+212(2)2'min.(rnrWRstWRrσ∈=Σ+n',),rrrrrRR×∈∈Σ,,ξ-1)rR∈1(rrfrRrH−−Σ−**(,=*=W1Ww'1()rrfRIr−Σ−**12,)w∈)'*1'*01(1,2(1)niittitrnttrftr=+=+=−==−∑∑≠'**110(1)rfrrIWrwk++−=(2,,trn=+'*01(1)nttrtrkwIWr=+=−∑f1'*1****110(1)nfririittiitrrrkI=++−=−=−∑(1,2,,ir)='*),0,0)rfr****'1210(1(,,)(nrrrnrI−+++−=='1r+n.A1||n0Σ=F12,,(riiirn)ξξξrn1F22F1FF=FF=1A[3-4]u21nnur≥frnrAuC≥'1''10rrrrrrAIRRI−−=Σ=Σ'10rrI−rrCI=Σ-ΣnA1-21rr1r+1r+22()2αβ1αβ+=WWabWαβ=+αβ1αβ+=aWbWWαβ+=1212-n1r+1r+rnu3'110'110(1)(1,2,,(1)fririirrfrrrkI−=−=−=)1'12()((,,,)nfrrfrrurRrwwH−−Σ−==)rWwIHkr10r+1rik+(1,2,,ir=)110(1,2,,)friirrkkirk++==(,Kk=',)rk12k10f+rrkq=(3)''(1(1rrWIWqI−−=−'1121(,,,,)rr=))WrKWw4'1()rnfrrrfHurIRIr−=+Σ−(4)10erW+=eW1'1()()rrrferrrrfRIrWWIRIr−−Σ−==Σ−'1erWI=nurf=(4)W1rKqθ+−=WWθθ==GθGr'1'(1)(1)rrrWIWqIW+−−=−KWW''''1''''''''11(1)(1)(1)00(1)(1)(1)(1)00()(1)rrerrrrrererrrerrrrWIWKIWKKWWIWWIqqIWqIWKKWWIWIWWIIWqqWIWIWWθ+++−−−−−==+=+−−−−−=+−=+−=+−W''(1)rrWIWI+−=e1r()WW11r+θ+W1er+W1rθ+θ+WWaabb1111(1)(1)earrebrrWaWaWWbWbWθθ++++=+−=+−(1a)a−b(5)(6)b≠eWθ+1xy+=1xxbab−=xbabα−=−=+abaαβ=+1=(nWanbn'wI)(1ntr=+∑'wI)(1ntr=+∑11(1)eabrbarbθ++−−=−−=−WWaWxWxy11rryW+=+(5)(6)abxbaxabab−−=+−−aab−+−−axabβ−=−WWabWαβ1αβ+=WWuuabuaub≠(4)αβ=+b3W2-,aWbn1r+αβαβ+WWaWbαβ=+Wwuuu''1212(,,,),,,,)aaabbbanb==abnnu≠A[2]7101(1,2,,)aaaiittinaattrftrwwkirkwWr=+=−==−∑8101(1,2,,)bbbiittinbbttrftrwwkirkwWr=+=−==−∑1'12()((,,,)anfrrfaaaarrurRrwwH−−Σ−==)Ww1'12()((,,,)bnfrrfbbbbrrurRr−−Σ−==)(7)α×(8)β+×91'01()()()(1,2,()(1())nabababiiiitttitrnababtttrftrαβαβαβαβαβ=+=++=+−+=+=−+∑∑,)rnuuuababnnαβ=+1αβ+=1'121'1122()((,,,)(())()(,,,abnfrrfababababrrabnnfrrfarabababnnurRrαβαβαβαβαβ−−−Σ−==+−Σ−==+++))b=+bn10910Ww'121122(,,)(,,,)ababanabnαβαβαβαβ==+=+++'12(,,,)nww=111'01(1,2,,(1)nabiittitrnabttrftr=+=+=−==−∑∑)nuAWuuababnnαβ=+-n1Σ21r+WuWnau,abWau≠nbbuαβ1αβ+=WWaWbαβ+=αβ1αβ+=1rrAC≥abuuαβ+2abuufrαβ+≥abαβ=+2auuubαβ=+(4)WWaWbαβ=+auuubαβ=+2W2-uu,aWbnab1r+auu≠bαβ1αβ+=1rrAC≥abuuαβ+2abuufrαβ+≥WWabWαβ+=uuaubαβ=+3[2]1Σn-MarkowitzHrr4()W-W,aWbnαβ1αβ+=WWaWbαβ=+αβ1αβ+=WWabWαβ=+3W-u,aWbbnaubauu≠Wαβ1αβ+=WWaWbαβ=+αβ1αβ+=auubrfαβ+≥WWabWαβ=+.-nWW,aWbb,aW([5])1[2]2rr1+r-43-nrr[1]MarkowitzH.Portfolioselection[J].JournalofFinance,1952,(4):77-91.[2]-[J]n2005221107-113.[3].[M].2000.[4].[M].2001.[5][J]20061.TheEfficientFrontierFeatureOfRiskyAssetsWithSingularVariance-covarianceMatrixYaoHai-xiang1,Maqing-hua1YIJian-xin2,LiZhong-fei31.FacultyofInformationScienceandTechnology,GuangdongUniversityofForeignStudies,GuangZhou510006,China(2.SchoolofMathematicalSciences,SouthChinaNormalUniversity,GuangdongGuangZhou,516031,China)(3.LingnanCollege,ZhongshanUniversity,Guangdong,Guangzhou,510275,China)Abstract:Underthearbitrage-freemarkethypothesisthepaperstudiestwo-fundseparationtheoremofnkindsriskyassetsportfolioinamean-variancemodel,inthecasetheirvariance-covariancematrixissingular.Keywords:Portfolio;Efficientfrontier;Variance-covariancematrix;Singular;two-fundseparationtheorem