뗚4뻭뗚2웚놱뺩샭릤듳톧톧놨(짧믡뿆톧냦)쓪5퓂JOURNALOFBEIJINGINSTITUTEOFTECHNOLOGY(SOCIALSCIENCESEDITION)컞럧쿕샻싊쿂춶ퟩ뫏뗄폐킧잰퇘죎ꆡ듯1ꆡꆡ췀탂쫯2(1.쳬뷲듳톧,쳬뷲:300072;2.쿦첶듳톧,쿦첶:411105)햪ꆡ튪:놾컄뾼싇컞럧쿕샻싊쿂춶ퟩ뫏뗄폐킧잰퇘컊쳢ꆣ놾컄쫗쿈냑Markowitz쒣탍뗄폐킧잰퇘폃춶ퟩ뫏뗄좨훘쿲솿뇭쪾돶살,좻뫳붫컞럧쿕샻싊쿂춶ퟩ뫏뗄폐킧잰퇘튲폃춶ퟩ뫏뗄좨훘쿲솿뇭쪾돶살,퓙평컞럧쿕샻싊쿂춶ퟩ뫏폐킧잰퇘뗄뚨틥벴뿉쟳돶헢룶폐킧잰퇘ꆣ맘볼듊:컞럧쿕샻싊;폐킧잰퇘;쟐뗣춶ퟩ뫏훐춼럖샠뫅:C934ꆡꆡꆡꆡꆡꆡ컄쿗뇪쪶싫:Aꆡꆡꆡꆡ컄헂뇠뫅:1009-3370(2002)02-0075-04튻ꆢ틽퇔뻟뷸탐퇝쯣ꆣ20쫀볍뫳냫웚,뮪뛻뷖랢짺쇋솽듎쫽톧룯쏼,뛾ꆢMarkowitz쒣탍뗄폐킧잰퇘[3]쪹쫽톧릤뻟뫍랽램퓚뷰죚쪵볹훐뗄펦폃뗃떽쇋뫜듳뗄랢햹ꆣ1952쓪,랢뇭쇋훸쏻춶ퟩ뫏뗄릹붨뻍쫇톡퓱쓉죫춶ퟩ뫏뗄횤뗄싛컄ꆰPortfolioSelectionꆱ,뇪횾ퟅ뮪뛻뷖뗚튻듎쫽좯늢좷뚨웤쫊떱뗄좨훘,벴룷횤좯쯹햼룃춶ퟩ뫏톧룯쏼뗄뾪쪼[1]ꆣ룃컄쳡돶뗄뻹횵ꆪ랽닮럖컶쫗뗄뇈샽ꆣMarkowitz쒣탍뇭쏷,릹붨춶ퟩ뫏뗄뫏샭듎뚨솿뗘럖컶쇋춶ퟩ뫏훐럧쿕폫쫕틦횮볤뗄쓚쒿뇪펦쫇퓚룸뚨뗄럧쿕쮮욽쿂탎돉튻룶뻟폐ퟮ룟퓚맘쾵,쪹죋쏇뿉틔쾵춳뗘쏨쫶뫍뷢뻶춶ퟩ뫏뗄믘놨싊뗄춶ퟩ뫏,믲헟쫇퓚룸뚨뗄믘놨싊쮮욽쿂ퟮ폅뮯컊쳢,쯼퓚춶ퟩ뫏샭싛훐뻟폐뮮쪱듺뗄ퟷ탎돉튻룶뻟폐ퟮ킡럧쿕뗄춶ퟩ뫏ꆣ뻟폐헢훖쳘폃ꆣ헷뗄춶ퟩ뫏뷐ퟶ폐킧뗄춶ퟩ뫏,쯼쏇캻폚쿂쇐Markowitz쒣탍쫇튻룶쮫쒿뇪맦뮮쒣탍,쯼뗄뷢쒣탍뗄뷢벯뫏훐ꆣ퓚뻹횵ꆪ랽닮욽쏦짏쫇튻쳵떯취탎쟺쿟,돆캪춶(I)ퟩ뫏뗄폐킧잰퇘ꆣ놾컄뾼싇퓚컞럧쿕샻싊쿂춶minꛒ2P=XTρXퟩ뫏뗄폐킧잰퇘붫랢짺퓵퇹뗄뇤뮯ꆣ놾컄훐,컒쏇maxE(rP)=XTR쫗쿈냑Markowitz쒣탍뗄폐킧잰퇘폃춶ퟩ뫏뗄좨n훘쿲솿뇭쪾돶살[2],좻뫳붫컞럧쿕샻싊쿂춶ퟩ뫏=1i=1뗄폐킧잰퇘튲폃춶ퟩ뫏뗄좨훘쿲솿뇭쪾돶살,퓙웤훐,R=[R1,R2,ꆭ,RTn],Ri=E(r)i쫇뗚i훖평컞럧쿕샻싊쿂춶ퟩ뫏폐킧잰퇘뗄뚨틥벴뿉쟳닺뗄풤웚믘놨싊,X=T[x1,x2ꆭ,x,n]쫇춶ퟩ뫏돶헢룶컞럧쿕샻싊쿂춶ퟩ뫏뗄폐킧잰퇘ꆣ컒쏇좨훘쿲솿,6=[ꛒ뗄쒿뇪쫇뾪랢튻훖샭싛,헢훖샭싛쫇ij]nꇁn쫇n훖닺볤뗄킭랽닮뻘Markowitz뻹횵ꆪ랽닮쒣탍맘폚컞럧쿕샻싊쿂춶ퟩ뫏뗄샭싛헳,RP=E(rP)뫍ꛒ2P=Var(rP)럖뇰쫇춶ퟩ뫏뗄,춬쪱컒쏇뮹쳡릩볆쯣컞럧쿕샻싊쿂춶ퟩ뫏폐킧웚췻믘놨싊뫍믘놨싊뗄랽닮ꆣꛒP돆캪믘놨싊뗄뇪잰퇘뗄쯣램릤뻟ꆣ컒쏇잿뗷퓚쯣램짏,컞럧쿕샻싊ힼ닮,뇭쪾춶ퟩ뫏뗄믘놨싊rP욫샫E(rP)뗄럹쿂춶ퟩ뫏뗄폐킧잰퇘춨뎣늻쓜폃볲떥뗄럖컶릤뛈,놻Markowitz폃폚뛈솿춶ퟩ뫏뗄럧쿕ꆣꆡ쫕룥죕웚:2001-11-17ퟷ헟볲뷩:죎듯,쳬뷲듳톧맜샭톧풺,늩쪿퇐뺿짺ꆣ췀탂쫯,쿦첶듳톧맜샭톧풺ꆣꆤ75ꆤ
ꆡ놱뺩샭릤듳톧톧놨(짧믡뿆톧냦)2002쓪뗚2웚컒쏇횪뗀,쒣탍(I)뗄뷢벯뫏퓚ꛒP2RP뿕볤훐쫇춼1붫(4)쪽듸죫(2)쪽훐,컒쏇뻍뗃떽훐뗄엗컯쿟AB,놻돆캪춶ퟩ뫏뗄폐킧잰퇘ꆣRRP=rf+T2rfꛒ(5)T헢뻍쫇컞럧쿕샻싊쿂춶ퟩ뫏뗄폐킧잰퇘ꆣ춼1ꆡ폐킧잰퇘평컄쿗[2]뿉횪,헢룶폐킧잰퇘뿉평쿂쇐n-2룶랽돌릹돉뗄쿟탔랽돌ퟩ뇭쪾춼2ꆡ컞럧쿕샻싊쿂춶ퟩ뫏뗄폐킧잰퇘컞럧쿕닺뗄뷨죫뫍듻돶냑풭살뗄폐킧잰퇘a11x1+a12x2+ꆭ+a1,n-1xn-1=b1AMB뇤돉쇋횱쿟CMD,죧춼2쯹쪾ꆣ헢쳵횱쿟듓a21x1+a22x2+ꆭ+a2,n-1xn-1=b2ퟝ훡짏컞럧쿕샻싊뗣C뒦쿲짏퇓짬,폫풭폐킧잰ꆭꆭꆭꆭꆭꆭ퇘쟺쿟쿠쟐폚뗣M,쯼냼몬쇋쯹폐럧쿕닺춶an-2,1x1+an-2,2x2+ꆭퟩ뫏T폫컞럧쿕뷨듻뗄ퟩ뫏ꆣ뗣M쯹뛔펦뗄춶+an-2,n-1xn-1=bn-2(1)ퟩ뫏돆캪쟐뗣춶ퟩ뫏ꆣ웤훐컒쏇붫(5)쪽룄킴캪죧쿂탎쪽ꛒRaP=rf+kꛒP(6)ij=ij+ꛒnn-ꛒin-ꛒjnRi-Rnꛒj,n-1+ꛒnn-ꛒjn-ꛒ퓙뷸튻늽뮯캪죧쿂탎쪽n-1,n-Rꛒ2n-1-Rnp=12(R2P-2RPrf+2r)(f7)kꛒin-ꛒnnꛒb붫쒣탍(I)훐뗄RP=XTR뫍ꛒ2P=XT6X듺죫(7),i=2+n-1,n-ꛒnnRi-RnRn-1-Rn뻍뗃떽(i=1,2,ꆭ,n-2;j=1,2,ꆭ,n-1)XTρX=12T2[rf-2rfXTR+(XR2)](8)k죽ꆢ컞럧쿕샻싊쿂춶ퟩ뫏뗄폐킧잰퇘ꆡꆡ틲캪쿟탔랽돌ퟩ(1)뗄훈쫇n-2,쯹틔쯼뗄믹뒡뷢쾵뗄룶쫽쫇1,벴x2,x3,ꆭxn-1뚼뿉평x1뇭쪾볙뚨듦퓚뿉릩춶뗄컞럧쿕횤좯ꆣ룼뷸튻늽(샻폃쿻풪램뿉뗃)ꆣ평폚뗄볙뚨쫇,춶헟뿉룹뻝웤틢풸틔컞럧쿕샻싊rnj죎틢뷨죫폫듻돶ꆣ룸뚨쇋헢훖믺믡6xi=1,춶헟뮹뿉틔i=1붫컞럧쿕닺폫럧쿕닺뗄춶ퟩ뫏T믬뫏웰틲듋xn튲뿉평x1뇭쪾ꆣ붫x2,x3,ꆭxn듺죫(8)살쪽,뻍뗃떽맘폚x1뗄튻풪뛾듎랽돌,틲캪뗣M쫇,뗃떽튪쟳뗄럧쿕-믘놨싊ퟩ뫏ꆣ쇮W듺뇭춶폚컞럧쿕닺뗄뇈샽,퓲1쟐뗣,쯹틔x1횻폐튻룶룹ꆣ평쟳룹릫쪽,뻍뿉쟳돶-W캪춶폚럧쿕닺뗄뇈샽ꆣ틲듋,몬폐컞럧쿕닺뗄춶ퟩ뫏뗄웚k뫍x1,좻뫳뻍ퟔ좻뗃떽x2,x3,ꆭ,xn뗄횵,헢뻍췻믘놨싊R뗃떽쟐뗣ퟩ뫏M뒦뗄좨훘쇋,춬쪱튲평k뗄횵뗃P뻍쫇RP=E떽컞럧쿕샻싊쿂춶ퟩ뫏폐킧잰퇘뗄랽돌ꆣ퓙평(rP)=Wrf+(1+W)E(rT)=W쿂쏦솽쪽:rf+(1-W)RT(2)룃춶ퟩ뫏뗄랽닮캪RP=XTR(9)ꛒ2ꛒ2P=W2ꛒ2f+2(1-W2)ꛒT+2W(1-W)ꛑꛒꛒfTP=XTꇆX(10)=(1-W2)ꛒ2T(3)컒쏇뻍럖뇰뗃떽쟐뗣ퟩ뫏T뗄웚췻믘놨싊뫍랽닮벴RT,ꛒ2TꆣꛒP=(1-W)ꛒT(4)ꆤ76ꆤ
ꆡ컞럧쿕샻싊쿂춶ퟩ뫏뗄폐킧잰퇘쯻늿럖짐뿉삩돤ꆣ웤훐맺뿢좯춨뎣놻죏캪쫇컞럧쯄ꆢ쯣샽쿕횤좯ꆣ틲듋헢킩닺뿉틔뾴ퟷ닺엤훃쯹닺짺뗄헢튻샠쪵볊킧맻뗄듺뇭,춬쪱폖쓜쟥돾뗘쮵쏷웤캪쇋쮵쏷컞럧쿕샻싊쿂춶ퟩ뫏폐킧잰퇘뗄펦폃맽돌ꆣ뻟쳥쯣램,컒쏇붫뾼싇쏀맺쫐뎡짏쯄훖훷튪뗄닺뇭1뇭쪾잰죽샠닺퓚1926ꆪ1993쓪볤룷ퟔ샠탍:맺볊좨틦샠ꆢ쏀맺맺쓚막욱샠ꆢ뎤웚햮좯뫍맺쪵쿖뗄쓪믘놨싊뫍헢킩믘놨싊뗄뇪ힼ닮틔벰헢킩뿢좯ꆣ헢쯄샠횤좯쫇놻춶ퟩ뫏뺭샭믲듳탍뗄춶닺샠탍횮볤뗄쿠맘탔[8],맺뿢좯놻죏캪쫇컞럧쿕헟뺭뎣쪹폃뗄,믲쫇ퟷ캪쯹뾼싇뗄닺샠탍뗄좫횤좯,웤샻싊캪%ꆣ쳥,믲쫇ퟷ캪쯹뾼싇뗄닺샠탍뗄훘튪늿럖,뛸웤뇭1ꆡ죽샠럧쿕닺뗄럧쿕ꆪ믘놨싊쫽뻝(1926ꆪ1993쓪)닺샠탍믘놨싊욽뻹횵(%)믘놨뇪ힼ닮(%)쿠맘탔맺볊막욱.막욱뎤웚햮좯평뇭1,컒쏇횪뗀:+()x1ꆡR=[]+=0(13)ꆡꛒ11=ꛒ21==평쟳룹릫쪽,튪쪹x1횻폐튻룶룹,뻍뇘탫싺ퟣꆡꛒ22=ꛒ22==ꚤ=2b24ac=0ꆡꛒ33=ꛒ23==벴ꆡꛒ12=ꛒ21=ꛒ1ꛒ2ꛑ12=347A844()-4()ꆡꛒ13=ꛒ31=ꛒ1ꛒ3ꛑ31=()=0(14)ꆡꛒ23=ꛒ32=ꛒ2ꛒ3ꛑ23=뷢횮,뗃:벴2k=믲k=춬쪱,뿉뗃6=-bx1=쯹틔ꇁ,랽돌ퟩ(1)뇤캪랽돌=-2(ꇁ)=(11)틲뛸===-1붫x1듺죫떽x2뫍x3훐,뗃:.2146x1x3=1-x12x2=x02==+평폚x3=1-x12x2=+==%,,틲듋랽돌(8)뇤캪1틲듋,컒쏇뻍뗃떽쟐뗣ퟩ뫏T뗄춶좨훘XTρX=22[rf-2rfXTR+(XTR2)]kX=[ꆡꆡ]1=평릫쪽(9)뫍(10),뻍럖뇰뗃떽쟐뗣ퟩ뫏T뗄웚췻2[+(XTR2)]ꆡꆡ(12)k붫R,6뫍믘놨싊뫍랽닮RT=%ꆡꆡꛒ2T==T[x1,x2,x3]ꆡ쯹틔,쟐뗣ퟩ뫏T뗄믘놨싊뇪ힼ닮캪=[x1,,+]듺죫(12),뻍뗃떽맘폚x1뗄튻풪뛾듎랽돌ꛒT=%듋쪱,컞럧쿕샻싊쿂춶ퟩ뫏뗄폐킧잰퇘캪()x1RP=rf+kꛒP=+ꛒPꆤ77ꆤ
ꆡ놱뺩샭릤듳톧톧놨(짧믡뿆톧냦)2002쓪뗚2웚춶ퟩ뫏폐킧잰퇘뗄좷뚨컊쳢,춬쪱튲룸돶쇋쟳컞컥ꆢ뷡폯럧쿕닺폫럧쿕닺뗄쟐뗣뗄튻훖랽램ꆣ놾컄뾪랢쇋튻훖랽램폃폚뷢뻶컞럧쿕샻싊쿂닎뾼컄쿗:[1]ꆡ[J].JournalofFinance,March1952,77-91.[2]ꆡ췵볼,췀탂쫯.횤좯ퟩ뫏뗄쇙뷧쿟뻶닟[J].풤닢,(1998),,57-60.[3]ꆡIbbotson,,Stocks,Bonds,Bills,andInflation:ThePast(1926-1976)andTheFuture(1977-2000),FinancialAnalystsResearchFoundation,Charlottesville,VA,1977.[4]ꆡChow,G.,Portfolioselectionbasedonreturn,riskandrelativeperformance[J].FinancialAnalysisJournal,2,1995,54ꆫ60.[5]ꆡJamesFarrell,WalterReinhart,PortfolioManagement:TheoryandApplication,2ndEdition,NewYork:.,ꆡTuX2in2shu(,Tianjing:300072;,Xiangtan:411105)Abstract:,,:Risk2FreeAsset;EfficientFrontier;PortfolioatTangentialPoints.ꆤ78ꆤ
