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1、?33?10?Vol.33,No.10201310Systems Engineering Theory&PracticeOct.,2013:1000-6788(2013)10-2533-06:F270.3;O221:A!#$%&()(*(+-,/.(01(2(34(5687:91,;2,?A8B2(1.CDEFGIHKJLMNOPIHQ,CD100083;2.RSGIHTMNOPIHQ,CD100084)UVWYXZ_YMarkowitzab-cdef,ghijklmnYopq,rYstuvwyxz|ef,cy(yIyy),YstuvwYxz|YYef,YuvwYxz|efYuvwYxz|ef
2、Y,hWYXYsYcYtTnyo;uvwyxz|;Interval quadratic programming for the portfolio selection withshort sales allowedXU Xiao-ning1,HE Feng1,CHEN Rong2,ZHANG Qing-zhi2(1.Dongling School of Economics and Management,University of Science and Technology Beijing,Beijing 100083,China;2.School of Economics and Manag
3、ement,Tsinghua University,Beijing 100084,China)AbstractBased on the Markowitz M-V theory,an interval quadratic programming model of portfolioselection is proposed with short sales allowed.Approaches have been put forward by applying approachesto compare any two intervals and introducing acceptabilit
4、y degree and possibility degree of interval number.Consequently it converted uncertain quadratic portfolio model into certain quadratic portfolio model todeal with it.In addition,it compared two solutions of the model.Keywords short-selling mechanism;portfolio selection;interval number;interval quad
5、ratic programming;satisfactory solution1y,“”,y,?,?,?,?,?!#%$?&?(,?)?*?+,?,-?.?/?01?2?3:4?5?)?*?76?8T79,;:?,A?B?C?D?E?F7G?H?I?J?E?K?L?M?/?0,N?O?P?Q?R?SA?B?T?U?/?0?V?W?X?L1ZYy?_?a?K,?Lb?c?d!,e?f?g23?L.P?h?i?j?H?k,c?l?b?c?d?!nmf?P?Q?o?p?q?r?s?t?u,v?w$%xyz,?N?D?_b?|?S$Markowitz48?,5?-=?*?y,S?_?=?_?,?S5?
6、b?|?5?q?F_?+?X?q?L?s?S5L?Yb|,m?E?/?0T5:2011-09-05:(70802007,71172011);(NCET-12-0772);(FRF-TP-09-022A):(1988),:;:(1975),:,E-mail:;(1976),:;(1981),.2534?33?DY,?p?,?igz?,SL?NS?,?/?0?)/?0T?)/?0S?/?0?L?!,W,?)/?0?_?,1v?w/?0!#E,!$?%&T(,F)b?|?)/?0S*?+?,-?tu,.*?)?S/E?0,y?1?2?3465?/?7?8?9?:?;.Ida5?A0?B?C?/?7?
7、D?E?F?G?H 3JIGiove6K?L?MNO?PQRSBCT0?U?V?A8?9,WXYZY_2?3?a?bcJIed?fg7KOhiYZ1jklY1mn89opaLM,?UVA0qsr(tBCT/7DuEFvIxwyz8K|9K?UV?A/?7D?uE?F,WXr(tT1jOHS70/7Du89b?cvILi10KNA0?AEFvIe?0?,?UVADu89?,HhiklY89vIx11Kn0V-UVAEF,EF07OH1S/7EFabcvI12K?UVA,NMINIMAXD_00vI?U?VADu890,HEFTT0?0JI?0?,?UVA0BCDu89,yBCT23(BCeG),
8、O7EF1S7EFabc,UVs0UV0?UVA02vIY,WX=0mbc23aN4vI2?m?n?B?C?TA=aL,aU,B=bL,bU,kST,H?D_S:A+B=aL+bL,aU+bU,A B=aL bL,aU bUIB?C?T?A=?,Hm(A)=(aL+aU)/2,w(A)=(aU aL)/2,?m(A),w(A)SBCTA0jvI0BC1314:(1)(AImwBIAmBmand AwBwAImwBIAImwBIand AI6=BI(2)(AImBIAmBmAI?aUVA,Markowitz#$?UVA,BCTri=rLi,rUi?%i?0hi&)(,Eri=Ri,BCTRi=R
9、Li,RUi?%i?*+),-0hi&0)(,VI=(Iij)nnShi&)./r00212,123n45SBCT,6S?712vIxi?UV4,6+Pni=1xi=1,?UVO+08UV9n?I#?0,UV4TS:,:0;T?0?U?V?A0?B?C?57DuEFS:minZ(x)=xTV x=nXi=1nXi=1Iijxixjs.t.nXi=1RIixiRI0nXi=1xi=1,i=1,2,n(QP)?1Ax=(x1,x2,xn)TBEF(QP)0,_?SEF(QP)0acvI+acC0DA?SEF(QP)0aE,FSD.?2x D,A?G?H?na?c x D,?Z(x)?UVAvBCD
10、uEF(QP)TnBC,bvc?nB?C?TvIB?C?Du?89?qr(t?D?u8?9?o,aEv,qsr(tDu1516G2,Du89(QP)1Svmsr(tDu89:(min(m(Z(x),w(Z(x)s.t.x Dm(Z(x)=12nXi=1nXi=1(Lij+Uij)xixj,w(Z(x)=12nXi=1nXi=1(UijLij)xixj(QP1)?1msr(tDu89(QP1)v3n+cKPDu89(QP)vcvImsr(tDu89,r(tT;T(0 1),?SklT,vZxyNUVUVklvDz|zv;vIqsr(tDu,msr(tDu1STv;sr(tDu89:(minZ(x
11、)=(1 )m(Z(x)+w(Z(x)s.t.x D(QP2)?2TDu89(QP2)v3ncmsr(tDu89(QP1)v+cvI,aEv,?UVAvBCDu?1S?Tv?Du89abcvI3.2ghijn?nst?317B?C?TA=aL,aU,B=bL,bU,FLen(A)=aU aL,Len(B)=bU bL,_?P(A B)=max(0,Len(A)+Len(B)max(0,aUbL)Len(A)+Len(B)SA BvI?4EF(QP)vc,?=P(Pni=1RIixiRI0)SxiPni=1RIixiRI0v,H 0,1IcSUVAhiBCU?V?h?iB?Cv?&,xyUVhi
12、&BCvnvI?3 P(Pni=1RIixiRI0)vBCTPni=1RIixiRI01S7:Pni=1RLixi+(1 )Pni=1RUixi(1 )RL0+RU0IvI,7v?UVAvDu89:minZ(x)=(1 )m(Z(x)+w(Z(x)s.t.nXi=1RLixi+(1 )nXi=1RUixi(1 )RL0+RU0nXi=1xi=1,i=1,2,n(QP3)3.3ghijn?nst?518BCTA=aL,aU,B=bL,bU,(A B)=m(B)m(A)w(B)+w(A)SBCTA BvI?6EF(QP)vcxi,?=(RI0Pni=1RIixi)SBCTPni=1RIixiRI0
13、v,H 0,1,vZxyNUVBvz.Z,c./CvZ;x.?4,BCTPni=1RIixiRI01S719:nXi=1RLixi RL0nXi=1(1 )RLi+(1+)RUixi(1+)RL0+(1 )RU0qsr(tDuG(3.1)v,r(tT;T(0 1)j,89(QP)vcKP7v?UVAv;sr(tD?uEF(QP4)v+c:minZ(x)=(1 )m(Z(x)+w(Z(x)2536I33s.t.nXi=1RLixi RL0nXi=1(1 )RLi+(1+)RUixi(1+)RL0+(1 )RU0nXi=1xi=1,i=1,2,n(QP4)4?+?p1,p2,p3,V,T:?hi&
14、v0212S:V=0.13,0.260.20,0.250.17,0.340.20,0.250.62,0.830.45,0.530.17,0.340.45,0.530.3217,0.5208.UV-v?vh?i&?)(S:RI1=0.035,0.063,RI2=0.08,0.19,RI3=0.0625,0.243Ivhi&)(S:RI0=0.032,0.044I4.1ghijn?nstTGv?UVAv7vDuEF(QP3),ovsr(tT;TjovUVA2,UVsv!vA2(?1)I1MMMMMMMMMMMMMMMMMMMdM00.40.60.8100000X(1.1285,0.1487,0.2
15、771)(1.1065,0.2233,0.3298)(1.0912,0.2751,0.3664)(1.0714,0.3425,0.4139)(1.0446,0.4336,0.4781)Z(x)0.08620.06010.04650.03240.017300.40.60.810.30.30.30.30.3X(1.1055,0.152,0.2575)(1.0865,0.232,0.3185)(1.0737,0.2859,0.3596)(1.0575,0.3541,0.4116)(1.0364,0.443,0.4794)Z(x)0.08650.0610.04720.03270.017400.40.6
16、0.810.50.50.50.50.5X(1.079,0.1582,0.2371)(1.0643,0.244,0.3082)(1.0547,0.2998,0.3546)(1.043,0.3682,0.4112)(1.0284,0.4539,0.4822)Z(x)0.08920.06210.04790.03310.017500.40.60.810.70.70.70.70.7X(1.0321,0.1773,0.2094)(1.0274,0.2722,0.2996)(1.0245,0.3298,0.3543)(1.0212,0.3962,0.4174)(1.0174,0.4737,0.491)Z(x
17、)0.09240.0640.04920.03380.017700.40.60.8111111X(1.1142,0.6937,0.8079)(1.0806,0.641,0.7216)(1.0675,0.6203,0.6878)(1.0561,0.6024,0.6585)(1.0461,0.5867,0.6328)Z(x)0.10270.06890.05190.03490.0178?1:v,UV!vsr(tT;TZ,UVvkl;x,ZIv,UVvZ,vklZ;x,Ir(tklTv)(S:Z=0.0173,0.1027IH,?UVAvBCDuEFvT)(p,BCp,r(tT;Tjv,bvsr(tT?
18、I4.2ghijn?nstTGv?UVAv7vDuEF(QP4),ovsr(tT;TjovUVA2(?2)II10JKMLMN,O:PMQMRMSMTMUMVMWMXMYMZMMMMM_MMaMbdcMeMf25372MMMMMMMMMMMMMMMMMdMMdM00.40.60.8100000X(1.079,0.1582,0.2371)(1.0643,0.244,0.3082)(1.0547,0.2998,0.3546)(1.043,0.3682,0.4112)(1.0284,0.4539,0.4822)Z(x)0.08920.06210.04790.03310.017500.40.60.81
19、0.30.30.30.30.3X(1.1,0.1531,0.2531)(1.0819,0.2342,0.3161)(1.0697,0.2886,0.3583)(1.0544,0.3569,0.4113)(1.0347,0.4452,0.4799)Z(x)0.08790.06120.04730.03260.017400.40.60.810.50.50.50.50.5X(1.1103,0.1512,0.2615)(1.0907,0.23,0.3207)(1.0773,0.2835,0.3609)(1.0604,0.3516,0.4119)(1.0381,0.441,0.4791)Z(x)0.087
20、20.06080.0470.03260.017400.40.60.810.70.70.70.70.7X(1.1186,0.1499,0.2685)(1.0979,0.2268,0.3247)(1.0836,0.2796,0.3632)(1.0653,0.3474,0.4127)(1.0409,0.4376,0.4786)Z(x)0.08670.06050.04680.03250.017300.40.60.8111111X(1.1285,0.1487,0.2771)(1.1065,0.2233,0.3298)(1.0912,0.2751,0.3664)(1.0714,0.3425,0.4139)
21、(1.0446,0.4336,0.4781)Z(x)0.08620.06010.04650.03240.0173?2:v,UV!vsr(tT;TZ,UVvkl;x,ZIv,UVvZ,vkl;x,Ir(tklTv)(S:Z=0.0173,0.0892IO9m23bvsr(tk?l?S:Z1=0.0173,0.1027,Z2=0.0173,0.0892.9mnBCsr(tTv1.0.0173 0.0892 0.1027Z1Z21MMMMMMMMMMM1G?=v?B?C?G?v?,?:Z1jZ2,Z1jZ2?,Z2Z1IBCT723v720,BC(2)4Z1jZ2v,m(Z2)=0.05325 m(
22、Z1)=0.06,BCZ2BCZ1IH,P(Z2?UVA2v,?1j?2vUVA2jkl?,UV?vklG-?vU?VAhi?&?a;?,=?UVv?UVA2vI5?z?jYvj?,UVAv,D?AvBUV?b?v?E?FG?H?E?F=a?v?3JI?Markowitzv?-2?E?F,?v?,?U?V?Av?B?C?D?u?E?F,=Nrt?TB?C?,?BCjBCvm23abcvIY,=vmbc23aN4vI?UVA2?v,?+?v?CvIH,O?LMBCTv?UV2538I33A89,EF?Habc,zv?vIH,?UVA89+v#?j?C,#v?,O?UVAvhi?&kl&?E?TB
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