x65x3c Markowitzx49x4ax66x67x68x69x63x6ax4fx6bx44x6bx6cx52x53x4dx4e
x57x58x7fx59x20 x47x20 Markowitz x5ex13x7cx1x32x72x71x25x26x25x3excx30x20x32x33x29x68x39x35x2c
x20x1x52x29x41x2dx5cx20x72x71x25x26x20x25x6fx4x67x34x22x60x11x77x76x43x5cx20x58x59x33x10x46x39x35x18x48x2e
x48x35x43x16x7fx18x23x14x20x27x28x29x41x2dx5cx20x72x71x25x26x25x3ex5cx4fx25 x48x2ex48x35x43x16x7fx18x3x39x58
x53x35x2ax2bx5x5cx4fx13x70x23x14x20x33x16x27x28x39x35x7fx46x29 x47x20xex5fx49x60x18x2bxex5fx49x60x9xaxb
x69x52x53x72x71x25x26x25x3ex5cx4fx20x34x35x29
x5ax2ex51x5bx5cx5dx20 x37x3ax11x32x29x5cx33xex41x4f (x58x53x62x3bx20x32x33x55x13x25 x1x32x25 x44x1x32
x33)x29x5cx33x28x77x33x3ex2cx4dx32x5x36x6cx5cx13x20x74x2cx36x6c)x29
x66x5ex20
? 4.1 x72x71x25x26x20x2bx32x41x34x72x71x25x26x25x3excx30x29
? 4.2x9xax72x71x20x72x71x25x26x25x3excx30x29
? 4.3x44x1x32x4x5x62x43x20x3bx56xbx69x5cx20x5ex13x7cx1x32x72x71x25x26x25x3excx30x20x20x29
? 4.4 x4cx70x3dx30x72x71x20x5ex13x7cx1x32x72x71x25x26x25x3excx30x20x20x29
? 4.5x22x60x11x77x76x43x5cx18x48x2ex48x35x43x16x7fx18x29
? 4.6 x72x71x25x26x25x3ex5cx4fx25 x48x2ex48x35x43x16x7fx18x34x58x53x35x2ax2bx5x5cx4fx20x33x16x3ax29
? 4.7x2bxex5fx49x60x20x5ex13x7cx1x32x72x71x25x26x25x3excx30x29
? x1fx41x20 x48x2ex48x35x43x16x7fx18x20Sharpe x72x42
? x1fx42x20 x32x33x47xdx64x442x29
x3bx31x3ax3ex49xdx4ex0x1xfx2cx21x29x48x6fx7cx661952x46x54Markowitzx54x2cx68
x14x25x7x8x69x63x2x1ax4ex3dx3ex57x58x4fx54x14x77x78x17x2cx49x76x22x44x15x45x75x54x70x77
x2x2ax5dx2cx68x20x21x27x54x69x7dx6fx38x2cx7dx2ex1x2cx68x20x21x69x6ax54x3bx6cx78 Markowitz
x1fx79x19x48x4fx1ex31x49x7x8x2cx68x14x25x18x48x51x43x73x47x48x3ex2bx20x21x27x22x23x25x4ex48
x10x18x58x47x48x20x21x27x69x7dx22x23x25x16x78x26x7ex62x67x20x8x21x22xdx3ex2bx20x21x64x20
x21x69x7d(expected return–variance of return)xfx55xex48x51x6cx0x1x4fx1ex31x2x15
x16x25x70x55xex27x4ex4fx4fx1ex54x78
4.1 x3ax3ex6cx79x27x47x48 Markowitz x2cx68x14x25x7x8x5fx60x21x44x58x54x68x4f
x55x56x78 Markowitzx65x63x54x2cx68x42x43x2cx73x51x51x2ax64x2cx68x48x51x6cx62x51x36x51x47
x48x2cx68x54x50x27x4ax4bx48x58x2x18x38x39x2cx68x54x20x21x27x8x5fx27x20x8x5fx60x78xfx2
x1x61x66x66x69x1ex48x3ax3ex2ex40x79x39x47x48x2cx68x4ax4bx54x55x56x8x5fx48x51x6cx61x62x1c
x1dx62x4ax55xex19x57x41x7dx48x27x65x63Markowitzx20x8x54x5fx60x78x7ax13x48x13x23x24x1c
x1dx62x4ax55xex48x18x38x39x2cx68x54x20x21x27x8x5fx27x65x63x48x51x6cx5dx8Markowtizx5f
71
x60x54x51x3axbx3ax54x66x49x1dx9x4ax66x25x2bx2cx68x42x43x2cx54x1cx1dx62x4ax55xex41x7dx78
x26x27x13x4bx21x70x54x36x37x3x4cx48x74x7ax25x7dx53x3ax3ex10 Markowitz x69x63x51x21x22
x23x54x69x1ex78
x61x62x51x17xbx4x50x27x78x2bx18x23x48x17xbx2x3cx62x54x48xfx50x27x2x2ex3cx62x54x78
x61x62x42x43x2cx51nx64x40x6fx2cx68x48x17x50x27x4ax4bx2nx2bx28x29x2ax6cx1;x2;:::;xn.
x25nx64x2cx68x54x4fx1ex14x25x23x3bnx75x24x6c = ( 1; 2;:::; n)x27x60x27x48x58x4f
x1ex14x25x54x50x27x4ax6x1
y = ￠x = 1x1 + 2x2 +￠￠￠+ nxn: (1)
x7fx61x62x1cx1dx62x4ax55xex41x7dx48x24x5ex66x49x11x2bx1cx1dx62x4ax65x68p (xaxbx10x2x62
x7ax49x28x29x2ax6cx69x25x44x54x65x68)x48x53x68x25nx64x2cx68x54x17xbx4ax4bx1p(x1);p(x2),
:::, p(xn). x58x2cx68x14x25 x54x17xbx4ax6xex2
p(y) = p( ￠x) = 1p(x1)+ 2p(x2)+￠￠￠+ np(xn):
x26x7ex23x68x19x8x2cx68x14x25x54(x5)x20x21x27 ry x1
ry = yp(y)
= 1x1 + 2x2 +￠￠￠+ nxnp(
1x1 + 2x2 +￠￠￠+ nxn)
= 1x1 + 2x2 +￠￠￠+ nxn
1p(x1)+ 2p(x2)+￠￠￠+ np(xn)
= w1r1 +w2r2 +￠￠￠+wnrn; (2)
x17x2c
ri = xip(x
i)
; i = 1;2;:::;n;
x10x3ex2x74x40x6fx2cx68x54x20x21x27x57
wi = ip(xi)
1p(x1)+ 2p(x2)+￠￠￠+ np(xn)
; i = 1;2;:::;n:
x1fx79x19x33x61x2x8x2cx68x14x25x54x17xbx4ax6x48x46wi x2x14x25x2cx65ix64x2cx68x54x17
xbx4ax6x49x17x2cx21xfx54x77x45(x2ex1fx10x51x23x25x2x50x54)x78x16x17x2
w1 +w2 +￠￠￠+wn = 1: (3)
x39x25x70x42x76x29x27x3cx48x31x32x1cx1dx62x4ax55xex41x7dx48x70x2bx2cx68x14x25x54x20
x21x27x57x40x6fx2cx68x20x21x27x75x76x54x3x4cx2xfx68x3cx54x48x4x2cx68x14x25x54x20x21x27
x9x66x40x6fx2cx68x20x21x27x54xdx74x7ax14x25xf (x4cx68x4x9x661x54x1cx1dx14x25)x48x51x6c
72
xdx74x7ax14x25xfx54x4cx68 wi x67x9x66x65 i x64x2cx68x54x17xbx4ax6x49x2cx68x14x25x54x17
xbx4ax6x2cx21xfx54x77x45x78x7ax58x48 Markowitzx69x63x51x2ex47x75x1cx1dx62x4ax55xexc
x2cx41x7dx78x2bx71x44x48x49Markowitz 1952x46x54x57x4fx64x67[9]x2cx48x70x73x65x63x5b
x2x18x38x39x2cx68x4x2cx68x14x25x54x20x21x27x8x5fx54x78x39x68x4fx6ex69x5cx54x76x77x27x3cx48
x3ax3ex7x8x5dx25x18x54x61xdxbx20x4ex70x57x68x3cx78x10x7ex48x3ax3ex20x8x56x5cx62x7ax2a
Markowitz x3dx22x5x6x79 x20 r1, r2;:::;rn x1d n xcx50x51x73x52x79x39x3a
x3bx3cx6bx15x5ax10x1d n x67xdxex79x5x6x21x0x21x7fx5bx10 R1 x7fx79x11x12x10x1dx77 n
x67xdxex79x3dx22x10x5x6x21x2a
R1 =
(
r = w1r1 +w2r2 +￠￠￠+wnrnjwi 2R; i = 1;:::;n;
nX
i=1
wi = 1:
)
x25x18x3ax3ex7ex7fx2ex20x2cx68x54x17xbx4ax4bx24x58x62x4ax65x68x78x7bx7ex48x25x2bx20
x21x27x69x25R1 x5ax2ex13x51xdx17xbx4ax6x11x54x50x62xbx21x7fx1fxfx54x79x7ax78x3ax3e
x24x13x54x31x63x2x2ax31x32Markowitzx5fx60x54x51x3axbx3ax66x49x48x33x34x49x24x17x54x38
x3dx56x48x70x62x66x49x11x2bx1cx1dx62x4ax65x68x48x53x68x25x70x20x21x27x69x25x1cx7ax7fx51x3a
x3exbx4bx21x70x54x79x7ax78
x1ax49x3ax3ex75x0x1x2cx68x4x2cx68x14x25x54x69x6ax3bx6cx78 Markowitz x17x46x54
x70x2bx45x75x76x22x2x2ax69x6ax3bx20x21x27x54x69x7dx6fx38x2cx7dx27x23x34x78x31x32 Vij =
Cov[ri;rj]x2ri x4rj x75x76x54x32x69x7dx48x33x34ry x54x38x2cx7d y x7x8x9xax56x5c
x6ex26x2a
2y = E
2
4
? nX
i=1
wiri ?
nX
i=1
wiE[ri]
!23
5
=
nX
i;j=1
wiwjE[(ri ?E[ri])(rj ?E[rj])]
=
nX
i;j=1
Vijwiwj: (4)
x58Markowitzx54x2cx68x14x25x7x8x69x63x54x8x5fx77x67x2(2)–(4).
Markowitzx47x48x54x5fx60x2x31x1fx3cx62wi,x53x68x2cx68x14x25x49x20x21x27x70x62
x18x48x69x6a (x20x21x27x54x69x7dx6fx38x2cx7d)x15x16x78x1x61x53x60x54x77x77x28x7cx48x3ax3e
x53x53x3bx56x5cx20x21x60x27x2a
w = (w1;w2;:::;wn)T; e = (1;1;:::;1)T;
? = (?1;?2;:::;?n)T; ?i = E[ri]; i = 1;2;:::;n;
73
V = (Vij)i;j=1;2;:::;n = (Cov[ri;rj])i;j=1;2;:::;n;
x51x2ex40xc w x1x3dx22x48 ?w = wT? x1x3dx22x10x5x6x48 w = (wTVw)1=2 x1x3d
x22x10x28x29x78x25x26x48 Markowitz x54x5fx60 (x10x7fxcx1x3dx29x24x56x57x38x39x3dx22x35
x36x22x23)x1
8
<
:
min 2w = wTVw = Pni;j=1 Vijwiwj
s:t: wTe = w1 +w2 +￠￠￠+wn = 1
?w = wT? = w1?1 +w2?2 +￠￠￠+wn?n = ˉ?
(5)
x25x18minx60x27x10x13x4bx54x2cx68x14x25x54x20x21x27x69x7d 2w x39x15x16x6x48 s:t: (sub-
ject tox54x13x1e)x60x27x65x18x38x3dx78x25x70x5fx60x54x1e ˉwxcx1x10x7x20x21 ?w x54x3c
x40x28x29x3dx22x78
x3bx68x4fx54x11x1x27x70x48x25x2x2bx8x6dx33x34x5fx60x48x4x10x2x49x78x2bx1cx1dx9
x26x65x18x38x3dx56x54x8x6dx65x68x54x39x15x16x6x54x5fx60x78x2ex1nx2bx28x29x2ax6cx54x32
x69x7dx20x21 V, x10x70x62x2x43x50x62x54x48x4x10x66x15x1f n x75x24x6c w, x10xcx7ax51
2w = wTVw > 0. x25x26x48x3ax3ex4bx2fx54x15x16x5cx65x68x2nx2bx2ax6cx54x43x50x62x8
x6dx65x68x78x78x2bx65x18x38x3dx7fx53nx75x2ax6cx11x2ax10x49x70x2bx67x69x44x78x47x48x19x43
x50x62x8x6dx65x68x54xcx5ax1dx48x23x24x5dx8x48x10x66x15x1fx35x62x54 ˉ?,x25x2bx5fx60x54x1e
x2x70x62x66x49x54x78
x31x32x35x61 w = 0 x19x48x44x4bx54x2ex9x26x2cx54x67x4bx2ex9x49x7cx52x41x7dx48x33
x34V x67x11xcx1x2x55x36x10x78x3bx62x20x21x70x62x2x43x13x20x21x48x5ax67x2x70x48x10x21
x10x7x54x4fx5cx26x70x62x2ex9x66x7d(x51x6cx70x62x4ex66x7d)x48x24x5ex10x54x19x20x21x66x49x78
xfx43x50x62x54x20x21x23x25x2x10x13x20x21x48x4x17x10x7x54x4fx5cx26x23x25x9x66x7dx78x3a
x3ex24x13x57x7fx4dx61x62V x2x3bx62x54x78x51x3x5fx60x3ax3ex32x19x24x13x12x65x63x78x25x18
x3ax3ex79x20x70x56x3bx62x20x21x54x70x2bx1dx60x2ax3bx62x20x21x54x74x14x4fx5cx26x5bx4ex66x7dx48
x16x17x2x3bx62x20x21x54x74x10x3ax1cx44x54x38x46x5bx4ex66x7dx78x25x18x74x14x4fx5cx26x2x5d
x45x6ex74x10x3ax1cx38x46x54x4fx5cx26x78x7ax75x48x31x32x70x2bx20x21x51x25x26x54x1dx60x48x10
x70x62x2x3bx62x20x21x78x25x70x1dx60x2x21x22 Sylvester x62x69x78x71x72x44x48x10x2x26x3b
x62x20x21x54x74x14x20x21x70x62x5ax2x3bx62x20x21x25x70x1dx60x1fx8x54x78
4.2 x1ax49x3ax3ex39x3x1fx44x27x55x44x25x70x5fx60x54x39x1ex78x70x75x54x8x6dx3bx62
x65x68x54xex44x2x6fx5bx24x44x54xdx5x1cx57x71x75x54x8x6dx3bx62x65x68x54xex44x2x6fx5b
x24x44x54xdx5x66x39x4bx48x58x78x2bx65x18x38x3dx60x27x2ax6cx65x49x78x2bx66x36x4bx54x66x69
x44x48xdx5x66x39x4bx11x25x2bx66x69x14x8x70x2bx14x5bx78x17x15x6fx77x67x10x7x5fx60x54x1ex78
Markowitzx15x1bx10x66n = 3x54x74x40x67x2x3cx3bx25x26x54x3x1fx69x55x27x65x63x39x1e
x54x78
x10x66n = 2,x5fx60x77x1x28x6dx78x49?1 6= ?2 x18x48x26x78x2bx65x18x9x26x67x23x1e
x8w1 x4w2 x27x78x2bx71x44x48x26
w1 +w2 = 1; w1?1 +w2?2 = ?w;
74
x7dx4x23x68
w1 = ?w ??2?
1 ??2
; w2 = ?w ??1?
2 ??1
:
2w = V11
?
w ??2
?1 ??2
?2
?2V12(?w ??1)(?w ??2)(?
1 ??2)2
+V22
?
w ??1
?1 ??2
?2
:
x5dx44x26x33x0x1ex69x23x68
(?1 ??2)2 2w = (V11(?w ??2)?V12(?w ??1))
2 +(V11V22 ?V 2
12)(?w ??1)
2
V11 :
x3ax3ex61x62V x2x3bx62x20x21x48x46V11 > 0 (x10x79x41x42x65x70x64x2cx68x20x21x27x54x69
x7dx2ex1x7d), V11V22 ?V 212 > 0 (x10x79x41x42x78x64x2cx68x20x21x27x54x7bx3x4cx68x2ex9
x66§1). x7bx7ex48x44x26x1fx14xfx4ex66x7dx57x39x58x10x70x62x23x60x27x1a(?w ?b)2 +c
x54x40x26x48x51x6ca > 0, c > 0. x25x26x70x27x48x13x12d = (?1 ??2)2,x3ax3ex67x68x19
w x57?w x9xax31x56x40x26x54x3x4cx26x2a
d 2w ?a(?w ?b)2 = c; a;c;d > 0:
x25x49( w;?w)x36x4bx44x2x70x38x3ex39x1cx54xex44x48x51x6cx26x66 w > 0,x10x73x51x6f
x5bx24x1fx54x44x3fx78
x3ax3ex58x23x10x44x29 n = 2 x54x74x40x54xcx5ax74x75x2ex8x21x37x1fx54x6bx12x78x44
V11 = 21, V22 = 22, V12 = 1 2r12,x17x2c 1 x1x65x70x64x2cx68x54x20x21x27x54x38x2c
x7dx48 2 x1x65x8x64x2cx68x54x20x21x27x54x38x2cx7dx48x58 r12 x1x78x64x20x21x27x54x7bx3
x4cx68 (x32x69x7dx57x78x2bx38x2cx7dx16x2x75x77)x78 r12 x23x25x47?1 x4 +1 x75x76x54x15
x1fx6x78x31x32r12 = 1,x4r1 x4r2 x7ex7fx3bx7bx3(x25x18V x2ex3bx62)x48x33x34
2w = ( 1w1 + 2w2)2 = ( 1(?w ??2)? 2(?w ??1))
2
(?1 ??2)2 :
x10x78x14x6fx69x48x51x61xd?1 > ?2,xex68
w = 1(?w ??2)? 2(?w ??1)?
1 ??2
:
x25x18x1x7cx2c ?w = ?1 x18x48 w = 1 x24x58 ?w = ?2 x18x48 w = 2, x3bx50x49
x73x51x70x64x47x55x78x25x70x69x6ax70x68x48x49r12 = 1x18x48x44x29x54x3ex39x1cx5dx5cx1x49
( w;?w)x36x4bx44x54x7fx38x77( 1;?1)x4( 2;?2)x54x18x1cx48x2ex1fx75x35x12 w < 0
x54x32x33x78x8x18x1cx66?w x6ax66x66x77(0;( 1?2? 2?1)=( 1? 2)). x17x2cx13x70x2b
xax38x67x2x49x25x64x74x75x54x33x69x6ax2cx68x20x21x27x78x31x32 r12 = ?1, x4 r1 x4 r2
x7ex7fx50x7bx3(x25x18V x5ax2ex3bx62)x48x33x34
2w = ( 1w1 ? 2w2)2 = ( 1(?w ??2)+ 2(?w ??1))
2
(?1 ??2)2 :
75
x10x78x14x6fx69x48x51x61xd ?1 > ?2, x33x34x1x7cx2c ?w = ?1 x18x48 w = 1 x24x58
?w = ?2 x18x48 w = 2,x3bx50x49x54x78x64x47x55x5bx51x23x25x78x17x2cx7ex1fx77( 1;?1)
x54x18x1cx1
w = 1(?w ??2)+ 2(?w ??1)?
1 ??2
:
x58x7ex1fx77( 2;?2)x54x18x1cx1
w = 2(?w ??1)+ 1(?w ??2)?
2 ??1
:
x25x78x38x18x1cx66x66?w x6ax54x10x70x77(0;( 1?2 + 2?1)=( 1 + 2)),x17x2cx13x70x2b
xax38x2x49x25x64x74x75x56x54x33x69x6ax3cx27x78x25x78x64x74x40x54x65x63x53x3ax3exfx75x16
x4dx48x8x2ex9xax1cx1dx62x4ax55xex54x60x64x2cx68x54x20x21x27x27x48x4x7ex1fx10x3ex54x14
x25x23x24x68x19x78x64x2ex10x54x33x69x6ax3cx27x78x25x70x68x48x15x79x54x70x14x28x29x2ax6cx2e
x70x62x2x70xbx2cx68x54x20x21x27x48x10x3ex51x23x25x57x1cx1dx62x4ax55xex5dx23x78x4dx70x69
x4bx48x25x78x64x7ex7fx3bx50x7bx3x74x40x5bx53V x2ex3bx62x78x75x39V x3bx62x54x6ax7bx75x70
x5ax3bx2x1x61x1bxdx25x78x64x74x75x78
x58x51x70x64xcx5ax74x40x2r12 = 0,x4r1 x57r2 x2ex7bx3x78x25x18x48
2w = ( 1w1)2 +( 2w2)2 =
2
2(?w ??1)
2 + 2
1(?w ??2)
2
(?1 ??2)2 :
x10x49 ( w;?w) x36x4bx44x2x7ex1fx77 ( 1;?1) x4 ( 2;?2) x54x3ex39x1cx78x3ax3ex23x24
x39x8 w x54x19x15x16x6x54x77 (x4x3ex39x1cx54x15x77) x1( 21 22=( 21 + 22);(?1 22 +
?2 21)=( 21 + 22)). x25x18x23x24x1fx79x19x48x73x75 1 x4 2 x5bx2ex1x7dx48x3ax3ex51
21 22
21 + 22 < minf
2
1;
2
2g:
x25x67x2x70x48x10x66x15x1fx78x64x20x21x27x2ex7bx3x54x2cx68x48x5x25x49x8x70x64x69x6ax77
x6ax27x78x31x5bx75x16x3x58x20x21x27x49x78x31x75x76x54x2cx68x14x25x27x78x25x18x36x51x20x19
x54x17x62x2cx76x74x40x23x24x72x76x78x10x3ex21x10x7x54x32x69x7dx20x21 V x5bx2x3bx62x54x48
x58x10x7x54xex44x5bx2x3ex39x1cx78
76
-
w
6
?w
PPP
PPP
PPP
PPP
PPP
PPP
PP
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
s
( 1;?1)
s( 2;?2)
r12 = ?1 r12 = 0 r12 = 1
x661x7dx0x67xdxex79xfx10
4.3 x1ax49x3ax3ex27x65x63x70x71x74x40x78x1x61x53x5fx60x66x66x65x63x48x3ax3ex2ex2f
V x55x36x48x4x10x66x15x1fw 6= 0,x51wTVw > 0. x3bx62x20x21x70x62x2x43x13x54x48x4
x17x10x7x54x4fx5cx26x2ex1x7dx48x24x2ax17x19x20x21 V ?1 x66x49x78x49x25x26x54x61xdx56x48
x6cx79x2bx35x61x25nx64x2cx68x2cx51x33x69x6ax2cx68x48x35xex10x54x20x21x27x53x57x17x62x2c
x68x54x20x21x27x5bx2ex7bx3x48x4dx53V x8x1ax7fx1x7dx54x4fx57x5cx78x17x6dx2x2ex66x49x43
x7dx14x25w,x53x68x17x10x7x54x14x25x69x6a w x1x7dx78x25x79x41x42x2ex66x49x11xbx2cx68
x54x14x25x9x10x66x70x64x33x69x6ax2cx68x57x6fx31x70x48x25xbx69x6ax2cx68x54x15x1fx14x25x1c
x7ax2x69x6ax2cx68x78x16x17x2x25x26x53x2bx35x5dx78x64x20x21x27x7ex7fx7bx7ax54x2cx68x32x49
x70x14x27x47x48x78x15x13x48x58x2bx35x61x78x64x2cx68x54x20x21x27x7ex7fx7bx3x54x74x40x48x45
x31x48x2ex3bx2dx51Cov2[r1;r2] = Var[r1]Var[r2],x4x2ex3bx2dV11V22 ?V 212 = 0,x7b
x1V x2ex1x3bx62x20x21x17x74x14x4fx5cx26x5bx7x4ex66x7dx78x2ex1fV x3bx62x25x70x38x3dx17
x71x58x2x1x61x68x4fx65x63x44x54x69x66x48x51x2ex79x41x42x1ax71x2cx2ex23x25x8x1aV x43x3b
x62x54x74x75x48x5ax2ex79x41x42x17V x43x3bx62x18x5fx60x67x36x51x1e1x78x10x18x48x3ax3ex58x75
x39x21x51x2cx68x54x72x73x5x6x21 ?i x57x74x75x76x4dx78x35xe?w x73x25xfx9x66x25x6ex63
x54x3ex2bx20x21x27x78x25x2x77x77x75x16x65x63x54x74x40x78
x31xbx4bx21x29x48 Markowitzx5fx60(5)x2x70x2bx8x6dx33x34x5fx60x78x25x70x5fx60
x3bx77x7ex54Lagrangex16x14x55x67x25x39x68x17x1ex78 (5)x54Lagrangex65x68x1
L(w;?1;?2) = wTVw??1(wT?? ˉ?)??2(wTe?1):
1V x35x36x37x38x39x3ax3bx41x3cx3dx37x4dx3ex3fx40x41x39x42x43x44x37x45x23x46x47x42x43x44x37x45x4dx3ex13x44x48x37x4dx3e
x49x2ex4ax4bx4cx4dx2ex4ax4bx37x4ex4fx50x71x26x51x70x23x52x44x48x37x4dx3ex53x37V x54x36x37x23
77
x17x1ew = ˉwx9xax54xcx75x38x3dx1
@L(ˉw;?1;?2)
@w = 2Vw??1???2e = 0; (6)
@L(ˉw;?1;?2)
@?1 = ˉ?? ˉw
T? = 0; (7)
@L(ˉw;?1;?2)
@?2 = 1? ˉw
Te = 0; (8)
x25x18 (6) x2x2b n x75x24x6cx9x26x78x26x66x25x2x2bx8x6dx26x33x34x5fx60x48x58x23x24x5d
x8x48x25x70x69x6ax14x5ax2 ˉwx1(5)x54x1ex24x58?1;?2 x1(5)x54Lagrangex16x14x54
x1dx33x38x3d(x46x3cx43x6f2)x78
x26x7ex23x1ex68
ˉw = 12V ?1(?1?+?2e) = 12V ?1(? e)
?
1
?2
?
:
x25x18x3ax3ex5dx2cx76x54x60x54x26x44x41x20x21x7bx16x54x40x26x78x4dx70x69x4bx48x7fx51
(? e)T ˉw =
ˉ?
1
?
;
x39x58
(? e)T ˉw = 12(? e)TV ?1(? e)
?
1
?2
?
=
ˉ?
1
?
:
x12
A = (? e)TV ?1(? e):
x31x32x25x2bx8x7bx20x21x54x19A?1 x66x49x48x33x34x26x7ex23x68
1
2
?
1
?2
?
= A?1
ˉ?
1
?
;
x24x5e
ˉw = 12V ?1(? e)
?
1
?2
?
= V ?1(? e)A?1
ˉ?
1
?
: (9)
x40x56x54x2x58x75x5dx8 A?1 x54x66x49x1dx78x10x7ex3ax3ex73x47x5dx8 A x2x3bx62x20x21x78
x2bx71x44x48x10x66x15x1fx2ex10x18x1x7dx54x71x68y1;y2,x51
(y1 y2)A
y
1
y2
?
= (y1 y2)(? e)TV ?1(? e)
y
1
y2
?
= (y1?+y2e)TV ?1(y1?+y2e) > 0;
78
x25x2x7bx1V ?1 x2x3bx62x20x21x78x26x7ex1fx68Ax2x3bx62x54x78
x15x13x48x3ax3ex5dx44x29x65x63x31x32x1ex41x62x69x40x26x78
x36x3f 4.1 x61x4x1cx2x50x51xdxexfx10x5x6x7x2(5)x7fx6bx57x58nx67xdxe
x79x8x9x11x12x13?i xaxbx25x5ex6bx11x12x13xdx50x51xcxdV x68x62x6bxfx10x79x8x9x11
x12x13 ˉ?xex62x6bx5dx5e(5)x73x3x7ax4a ˉwx1d
ˉw = V ?1(? e)A?1
ˉ?
1
?
: (10)
x7ex7f
A = (? e)TV ?1(? e): (11)
x12
A =
a b
b c
?
:
x33x34x10x66x14x25x54x3ex2bx20x21x27 ˉ? x4x15x16x69x7d ˉ 2 x75x76x7x8x9xax56x5cx60x54
x26x2a
ˉ 2 = ˉwTV ˉw = (ˉ? 1)A?1(? e)TV ?1VV ?1(? e)A?1
ˉ?
1
?
= (ˉ? 1)A?1
ˉ?
1
?
= (ˉ? 1) 1(ac?b2)
c ?b
?b a
? ˉ?
1
?
= a?2bˉ?+cˉ?
2
(ac?b2) :
x25x70x68x49( w;?w)x36x4bx44x48x24x16x69x6ax14x25 ˉwx54x20x21 ˉ?x57x69x6a ˉ x75x76x37
x8x61x70x38x3ex39x1cx54x1fx44x3f(x31xe2)x48x10xcx1x14x25x54x2cx11x78x17x2cx3ex39x1cx54
x15x77x10x7x12x13x20x40x56x57x3dx22wG. x75x16x18x38x26x44x26x6bx12x68x19x17x10x7x54x14
x25x20x21 ?G = b=c,x14x25x69x6ax1 G = 1=pc,x58
wG = V ?1(? e)A?1
?
G
1
?
= 1(ac?b2)V ?1(? e)
c ?b
?b a
? b=c
1
?
= V
?1e
c :
x3ex39x1cx54x44x44x32xcx1x24x14x2cx11x48x51x3axbx3ax44x54x59x70x77x21x10x7x54x14x25xc
x1x24x14x3dx22x78x10x3ex5bx4ex60x70x64x20x21x35x62x18x69x6ax15x16x54x14x25x24x58x69x6ax35
x62x18x20x21x15x4ex54x14x25x78x5dx20x21x4x69x6ax3cx41x14x25x78x2bxax38x48x33x34x51x3ax14
79
x25x10x7x21x22Paretox15x6dx14x25x48x4x10x10x3ex27x70x48x2ex23x25x60x19x78x69x4bx5bx77
x10x28x54x14x25x78x58x49x10x3ex75x76x25x7bx77x77x48xex23x5fx1ax69x5x6x14x43x69x28x29x78x3e
x39x1cx54x56x44x32xcx1x2cx14x2cx11x48x49x33x3axbx3ax44x54x59x70x77x5ax5bx4ex60x70x64x20
x21x35x62x18x69x6ax15x16x54x14x25x48xfx10x3ex54x20x21x49x69x6ax35x62x18x51x2ex2x15x4ex78x10
x66x4fx1ex31x27x70x48x17x7ax62x73x3x21x51x3axbx3ax78x25x10x62x54x44xex29x33x45x75x78x62
x53x16x5ex62x36x57x54x4fx1ex7x28x49x51x3axbx3ax44x7x47x62x54x4fx1ex14x25x78
-
w
6
?w
sx7fx5ex20x55x1x32x25x26
x70x6fx4x67 )
x25x6fx4x67PPi
G = 1pc
?G = bcs
s
x662x7d nx67xdxex79xfx10
x49 Markowitz x20x8x62x54x69x63x18x48x75x6bx12x51x79x7ax54x2cx68x14x25x54x51x3a
xbx3ax48x33x73x25x16x44x5fx17x78x2fx5ex19x611980x46x4ex48x6bx12x44x54x64x2cx68x54x14x25
x51x3axbx3ax48x5ax58x7bx17xexfx78x7bx7ex48x6bx12x51x3axbx3ax54x5fx60x4bx57x2 Sharpe
x20x8x1ex6fx1ex37x62x4ax55x56x54x70x2bx30x292x78xfx2x28x42x6bx12x26x7bx54x5fx3fx48xaxb
x75x6bx12x3x54x64x2cx68x54x14x25x51x3axbx3ax5bx23x49x3x33x18x10x7ex41x78
4.4 x24x44x54x65x63x2cx26x66x75x39V x3bx62x48x53x68nx64x2cx68x2cx2ex23x25x45x46
x33x69x6ax2cx68x78x10x66x51x33x69x6ax2cx68x74x40x48x3ax3ex23x24x61xdx35x44x29nx64x2cx68
x19x48x4dx19x58x51x65 0 x64x2cx68x1x33x69x6ax2cx68x48x51x6cx10x54x33x69x6ax3cx27x1x7fx28
x29x2ax6crf. x49x25x64x74x75x56x48x14x25x53x62x7ax1x9xaw0 +w1 +￠￠￠+wn = 1x54
2x55x56 ri ?rf = fli(rm ?rf) +"i, i = 1;2;:::;n, x57x3f "i x58 rm x35x59x55x13x5a "i x5bx3fx5cx35
x59x55x13x5dx5e Cov[ri;rj] = flifljVar[rm]; x5fx52x60x61x62x63x64x37xfx10x65x5fn2 x37x56x65x66x67x0nx37x56
x65x66x23
80
w0;w1;:::;wn. x3ax3ex1cx44w = (w1;:::;wn)T. x25x18x48x14x25x54x3ex2bx20x21x27x1
?w = w0rf +w1?1 +w2?2 +￠￠￠+wn?n;
x39x58
?w ?rf = w1(?1 ?rf)+w2(?2 ?rf)+￠￠￠+wn(?n ?rf) = wT(??rf);
x58x14x25x54x69x7dxex74x7ax1cx1 2w = wTVw:x12?0i = ?i ?rf, ?0w = ?w ?rf. x33
x34x1ax49x54x72x6x64x69x7dx14x25x7x8x5fx60x2ax1
‰ min 2
w = w
TVw = Pn
i;j=1 Vijwiwj
s:t: ?0w = wT?0 = w1?01 +w2?02 +￠￠￠+wn?0n = ˉ?0 (12)
x5ax67x2x70x48x40x26x44x57x6ax27x54x5fx60x54x48x3fx4cx49x66x5fx61x70x2bx65x18x38x3dx78 (12)
x54Lagrangex65x68x1
L(w;?) = wTVw??(wT?0 ? ˉ?0):
x17x1ew = ˉwx9xax54xcx75x38x3dx1
@L(ˉw;?)
@w = 2V ˉw???
0 = 0; (13)
@L(ˉw;?)
@? = ˉ?
0 ? ˉwT?0 = 0; (14)
x26x7ex23x1ex68
ˉw =
ˉ?0
(?0)TV ?1?0
?
V ?1?0: (15)
x58x7bx7x54x24x16x69x7dx1
ˉ 2 = ˉwTV ˉw
=
ˉ?0
(?0)TV ?1?0
?2
(?0)TV ?1VV ?1?0
= (ˉ?
0)2
(?0)TV ?1?0:
x25x67x2x70x48 ˉ x57 ˉ?0 = ˉ??rf x75x76x49( w;?w)x36x4bx44x54x3ex39x1cx3x4cx49x25x64
x74x40x5dx5cx1x78x38x18x1cx2a
ˉ = §ˉ?
0
((?0)TV ?1?0)1=2:
81
x26x66ˉ xcx2cx1x3bx48x25x78x38x18x1cx5ax5bx73x51x1fx17x54x44x38x7ax1cx48x51x7bx66x66?w
x6ax44x54rf x77 (x31xe 3)x78x17x2cx44x44x38x7ax1cx2x14x25x54x51x3axbx3ax57x56x44x38x7a
x1cx2x14x25x54x33x3axbx3ax78x39x57x58x79x7ax44x27x3cx48x33x69x6ax3cx27rf x57x5x1fx15x16
x69x6ax14x25x54x3ex2bx20x21x27 ?G x7bx77x48x7x8x75x16x78x7bx7ex48x10x3ex75x76x54x6ex6fx7
x8x31xe3x2cx21x27x78xfx39x69x63x44x27x70x48xdx15x36x51x69x26x70rf > ?Gx70x62x2ex23
x25x78x26(15)x48x31x32x14x25x20x21x27 ˉ?x9xa
ˉ?0 = ?
0TV ?1?0
eTV ?1?0 ;
x33x34 ˉwx9xa ˉwTe = 1x48x4x25x18x14x25x2cx33x69x6ax2cx68x54x77x45x1x7dx78x7bx7ex48x10
xcx2cx10x7nx64x69x6ax2cx68x14x25x54xbx3ax44x54x70x77x48x4x7bx7x54x3ex39x1cxcx2cx57
x7ax1cx7bx73x66x8x14x25x10x7x54x77x78x17eTV ?1?0 > 0x18x48x8x73x77( m;?m)x49x44
x44x38x7ax1cx44(x31xe3)x57x25x18rf < ?G. x17eTV ?1?0 < 0x18x48x8x73x77x49x56x44
x38x7ax1cx44x57x25x18rf > ?G. x17eTV ?1?0 = 0x18x48x71x72x44x70x68x25x26x54 ˉw x2e
x66x49x57x25x18rf = ?G,x58x78x38x7ax1cx2ax1x3ex39x1cx54x7dx5dx1cx78
-
w
6
?w
sx7fx5ex20x55x1x32x25x26?G =
b
c
G = 1pc
rf s##
##
##
##
##
##
##
##
#
s
( m;?m)
cc
cc
cc
cc
c
s
s
x663x7dx18x19x1axdxex14nx67xdxex79xfx10
x31x44x21x29x48x1cx33x69x6ax2cx68x54x2cx68x14x25x51x3axbx3ax2x70x38x7ax1cx78x25x38x18
x1cx7fxcx1x1dx38x68x69x74(CapitalMarketLine, CML)x78x7ax1cx54x6bx27x1(?0TV ?1?0)1=2.
x25x79x41x42x51x3axbx3ax44x54x59x70x14x25 ˉw x54x3ex2bx20x21x27 ?ˉw x4x38x2cx7d ˉw x75
x76x51x31x56x3x4cx2a ?
ˉw ?rf
ˉw = (?
0TV ?1?0)1=2:
82
x1ex14x54x77x6xcx1x14x25x54Sharpex50,x10x3bx27x73x6cx28x29x14x6x48x4x7bx6bx6cx69
x6ax58x23x25x1cx27x54x20x21x78x1cx33x69x6ax2cx68x54x2cx68x14x25x51x3axbx3ax54xcx77x67x49
x66x17x44x54Sharpex77x2x7fx68(?0TV ?1?0)1=2,x10x7ex7fx26x74x69x6ax2cx68x54x3ex2bx20
x21x27 ?x4x10x3ex76x54x32x69x7dx20x21 V x21x44x62x78x70x71x74x75x56x48x51x3axbx3ax7ax1c
x57x40x56x54x69x6ax2cx68x14x25x54x51x3axbx3ax7bx73x66x70x77( m;?m). x7bx7ex48x49x25x38
x7ax1cx44x54x59x70x77x21x10x7x54x3ex2bx20x21 ?ˉw x5bx51x56x5cx40x26x2a
?ˉw ?rf = flˉw(?m ?rf); (16)
x17x2c flˉw = ˉw= m. x25x70x9x26x5ax51x1ex6fx1ex37x62x4ax55x56x54x40x26x48xfx10x51x2e
x2x1ex6fx1ex37x62x4ax55x56x48x58x73x2x70x48x10x7x74x64x51x3bfl x54x2cx68x14x25x5x66x49
x51x10x26x20x21x54x51x3axbx3ax44x54x14x25x78x31x32x5d (16) x69x1ex1 ?ˉw x57 flˉw x75x76
x54x3x4cx48x33x34x10x54xex44x5ax2x70x38x18x1cx78x25x38x18x1cx7ex7fxcx1x38x39x68x69x74
(Security Market Line, SML)x78
4.5 x3ax3ex19x57x1fx68Markowitzx72x6x64x69x7dx2cx68x14x25x7x8x5fx60(5)x54
x1ex78x26x66x25x70x5fx60x54x1ex1cx23x30x31x1x1cx1dx69x6ax14x54x39x1ex48x7bx7ex48x1ex54x69
x25x51x70x2bxcx5ax54x1dx60x78x37x1fx1ex70x48x67x2x21x68x19x54x24x16x69x6ax14x25x9xaxdx19
x0x6ax69xfx48x4x24x16x69x6ax14x25x54x74x7ax14x25x1cx7ax2x24x16x69x6ax14x25x78x25x18x21x22
x73x78x3dx22x2x25x26x62x7ax54x2ax31x32x24x6cyx57x24x6cx1, x2 :::;xnx76x9xax56x5cx3
x4cx2a y = ?1x1 +?2x2 +￠￠￠+?nxn, ?1 +?2 +￠￠￠+?n = 1,x33x34yxcx1x1,
x2;:::;xn x54x74x7ax14x25x78x31x32x25x18x21x51?i > 0,x33x34y xcx1x1, x2;:::;xn
x54x5bx3dx22x78x5dx25x70xdx19x0x6ax69xfx45x20x60x54x48x17x7ex1ex54x60x29x67x2x56x5cx45x75
x54x1fx37x3bx27x39x36x3fx2a
x36x3f 4.2 x20xfx10wpx71wq x49xfx67x4x1cx2x50x51xfx10x5x6x7x2(5) (x15
(12))x79x6dx6ex8x9x11x12x13 ˉ?x1d?p x71?q x79x4ax6bx1x2?p 6= ?q. x5ex4x6bx7x62
x62x94.1 (x15x5dx18x19x1axdxex5ex7b) x7fx79x5fx60xaxbx0x5dx5e ˉw x67x61x63x19x1axf
x10x79x62x49x63x4ex5fx60x1dx60x61x72x65?,x4x5 ˉw = (1??)wp +?wq. x57x58wp x71
wq x24x67x73x64xfx10x6bx14?x610x711xcx45x6bx5dx5e ˉw = (1??)wp + ?wq x10x67
x73x64xfx10x0
xdx27 x12 ˉ? = (1??)?p + ??q. x33x34x3cx3bx1ex54x60x54x26(10) (x6f(15)),
x75x16x2cx68x48 ˉw = (1??)wp + ?wq x67x2x5fx60(5) (x6f(12))x54x1ex78x66x2x5fx60
x67x28x6dx1ex30x31x1x26?x3cx62 ˉ?x4x26 ˉ?x3cx62?. x5ex66x62x69x54x13x70x32x33x23x26
?x490x41x75x76x68x19 ˉ?x49?p x4?q x75x76x48x39x58x17wp x4wq x5bx10x7x51x3a
xbx3ax44x54x14x25x18x48 ˉwx5ax53x10x7x51x3axbx3ax44x54x14x25x78 2
x59x40 x20xfx10 w1, w2;:::;wk x49xfx67x4x1cx2x50x51xfx10x5x6x7x2 (5)
(x15(12))x79x6dx6ex8x9x11x12x13ˉ?x1d?1, ?2;:::;?kx79x4ax6bx1x2?ixcx45xaxbx25
x26x0x5ex4x6bx7x62x62x94.1 (x15x5dx18x19x1axdxex5ex7b)x7fx79x5fx60xaxbx0x5dx5ew1,
w2;:::;wkx79x7ex65xfx10 ˉw = ?1w1+?2w2+￠￠￠+?kwk, ?1+?2+￠￠￠+?k = 1x10
83
x67x61x63x19x1axfx10x0x57x58w1, w2;:::;wk x24x67x73x64xfx10x6bx14?1;?2 :::;?k >
0,x5dx5ew1, w2;:::;wk x79x66xfx10 ˉw = ?1w1 + ?2w2 +￠￠￠+ ?kwk x10x67x73x64
xfx10x0
x2cx68x23x24x3bx7fx33x54x68x4fx30x55x55x27x4ex4fx78
x62x694.2x54x6ex7ax49x66x2ax31x32x4fx1ex31x54x2cx68x4fx1ex44xex67x2x75x16x5ex62x6f
x7dx54xax43x4x69x6ax6bx11x25x43x49x72x6x64x69x7dx5fx60x54x15x6dx1ex2cx7x47x70x77x48x33
x34x62x47x48x7fx1fx2cx68x54x14x25x57x47x48x2cx68x54x78x64x14x25x54x14x25x2x70x26x54x78x25
x78x64x14x25x49x1ax71x2cx68x42x43x2cx23x25x67x2x78x64x3x1ax1bx28x54x63x10x40x3d(mutual
fund)x48x7bx7ex48x25x67x2x70x48x4fx1ex31x2excx47x48x7fx1fx2cx68x31x1fx14x25x48x73x47x47
x48x31x1fx1cx6ex25x78x64x40x3dx54x14x25x4x23x78x25x5ax2x25x70x62x69x1x37x34xcx1xdx8
x40x3dx33x5bx62x69xfx54x6ax7bx78 1981 x46x7x59x5bx57x58x5ax5bx68x31 J. Tobin (1918–)
x491958x46x5fx60x54[14]x2cx3bx2x3cx3bx70x64x33x69x6ax1ex37x57x70x64xcx5ax54x69x6ax1e
x37x54x14x25x27x63x2cx72xdx54x8x40x3dx33x5bx62x69x41x7dx48x39x58x68x19x70xbx5cx76x57x58
x69x4bx54x31x63x78x1x7ex48x7dx3ex51x18x0x1 Tobin x5ax2x2cx68x14x25x33x14x69x63x54x7b
x7cx7dx78
x8x40x3dx33x5bx62x694.2x58x40x3ax3ex54x2x2ax26x78x2bx24x16x69x6ax14x25x54x14x25x67
x23x36x41nx64x2cx68x54x1ex2bx14x25xbx3ax78x16x17x2x31x32x25x78x2bx14x25x5ax3cx41x2x78
x64x2cx68x48x33x34x51x61x25x78x64x2cx68x24x13x48x1x36x41x10x26x54x14x25xbx3ax48x17x62x2c
x68x67x2ax68xdx2ex40xfx78x5dx25x70x55x55x60x54x41x68x4fx40x26x48x3ax3ex68x19x31x56x62x69x78
x16x17x2x3ax3ex26x7ex13x6dx1fx68x1ex6fx1ex37x62x4ax55x56x78
x36x3f 4.3 x20px71q x67x0x67xdxex6bx1x2x15x5ax79x8x9x11x12x13?p 6= ?q.
x5dx5ex6fx70xdxe i xax5x3b p x71 q x47x6xax79xfx10x1bx7x79x62x49x63x4ex5fx60x1dx7d
x60x61x72x65fi 2R,x4x5
?i = (1?fi)?p +fi?q; (17)
Cov[ri;rp] = (1?fi)Var[rp]+fiCov[rp;rq]; (18)
Cov[ri;rq] = (1?fi)Cov[rp;rq]+fiVar[rq]: (19)
xdx27 x63x4ex76 x2ex40x61xd p = 1, q = 2, i = 3. x31x32x65 3 x64x2cx68x2e
x17x2axbx78x64x2cx68x54x14x25x21x36x41x54x14x25xbx3ax48x33x34x10x66x15x1f ˉ? 2R,x69x6a
(6)–(8)x10x66n = 3x51w3 = 0x54x1ex48x4x66x49x71x68 ˉw1; ˉw2,x4?1;?2,x9xa
ˉw1 + ˉw2 = 1
?1 ˉw1 +?2 ˉw2 = ˉ?
V11 ˉw1 +V12 ˉw2 ? 12?1?1 ? 12?2 = 0
84
V21 ˉw1 +V22 ˉw2 ? 12?1?2 ? 12?2 = 0
V31 ˉw1 +V32 ˉw2 ? 12?1?3 ? 12?2 = 0
xfx2x26x66?1 6= ?2, xbx47x2bx69x6ax10 ˉw1; ˉw2, x4?1;?2 x51x1dx70x1ex48x25x70x68xb
x47x2bx69x6ax54x1ex70x62x2x65x24x2bx69x6ax54x1ex78x10x18x48x26x66 ˉ? x23x47x15x1fx71x68x48
x3ax3ex5dx65x8x2bx69x6ax12x5fx48x1cx7ax23x24x70x48x31x32 ˉw1; ˉw2,x4?1;?2 x9xax65x70x3
x60x3x47x2bx69x6ax48x33x34x10x70x62x9xax65x24x2bx69x6ax78x25x4cx17x65x24x2bx69x6ax2x65
x70x3x60x3x47x2bx69x6ax54x1cx1dx14x25x18x78x51x23x25x48x4x66x49fi1;fi2;fi3,x53x68
fi1 +fi2V11 +fi3V12 = V31
fi1 +fi2V21 +fi3V22 = V32
fi2?1 +fi3?2 = ?3
fi2 +fi3 = 1
fi1 = 0
x12 fi3 = fi, xe fi2 = (1 ? fi). x1fx79x19 p;q;i x4 Vij x54x62x7ax48x7dx4x23x68
(17)–(19).
x62x49x76 x62x69x38x3dx71x72x44x5dx8x48x44x4bx54x69x6ax14x2cx48x31x32 ˉw1; ˉw2, x4
?1;?2 x9xax65x70x3x60x3x47x2bx69x6ax48x33x34x10x70x62x9xax65x24x2bx69x6ax78x47x48x19
x69x6a(6)–(8)x54x1ex70x62x2Markowitzx5fx60x54x1ex48x25x70x68x26xbx78x64x2cx68x54
x14x25x36x41x54x14x25xbx3ax70x62x5ax2x60x64x2cx68x54x14x25x36x41x54x14x25xbx3ax78x25x67
x2x70x65x60x64x2cx68x2ex17x2axbx78x64x2cx68x54x14x25xbx3ax78 2
x59x40(x1dx38x1dx35x36x28x72xe) x20xdxepx71qxaxbx62x94.3x79x7x20x6bx1
x2Cov[rp;rq] = 0. x5dx5ex6fx70xdxeixax5x3bpx71qx47x6xax79xfx10x1bx7x79x62
x49x63x4ex5fx60x1dx7ex11x12x13ri xaxbx21x7fx11x7axfx6x7x6x8x62x63x7dx9x12x7d
E[ri]?E[rp] = flpi (E[rq]?E[rp]); flpi = Cov[ri;rq]=Var[rq]: (20)
xexfx67x3xdxepx1dx11x41x42xfx10x12 mx4x6bx1x10q x11x12x, (20)x3bx1dx13fl x1d
x38x1dx35x36x28x72xe
E[ri]?E[rx] = fli(E[rm]?E[rx]); fli = Cov[ri;rm]=Var[rm]: (21)
x3xdxexx67x18x19x1axdxex4x6b (20)x3bx1dx14x15x79x1dx38x1dx35x36x28x72xe
E[ri]?rf = fli(E[rm]?rf); fli = Cov[ri;rm]=Var[rm]: (22)
x7ex7frf x67x18x19x1ax5fx13x0
85
x25x70x62x69x58x17x76x63x39x4dx70x2bx3ax3bx20x8x1ex6fx1ex37x62x4ax55x56x78x25x70x3a
x3bx2x25x26x27x3cx54x2ax61x62x42x43x2cx51x40x41x2cx68x78x33x34x26x25xbx2cx68x54x14x25x21
x40x41x54x14x25xbx3ax4ex60x61x25x70x42x43x54x5x1fxdx69x6ax64x20x21xfx70x75x78x26x8x40x3d
x33x5bx62x694.2,x25x70x42x43x54x5x1fxdx69x6ax64x20x21xfx70x75x7ex7fx23x24x26x78x2bxbx3a
x14x25x27x44x62x78x58x15x1fx2cx68x6fx2cx68x14x25x2ex17x2ax42x43x5x1fxdx69x6ax64x20x21xfx70
x75x54x1dx33xcx75x38x3dx1x25x70x2cx68x54x20x21x10x25x78x64x51x3ax14x25x9xaxdx77x7ax1e
x6fx1ex37x62x4ax55x56xf(x62x694.3)x78
x6ax27x54x1ex6fx1ex37x62x4ax55x56x2x10x68x69x3dx22x58x1x54x78x21x22x42x43x14x25x2
x5dx42x43x44x21x51x4fx1ex31x54x2cx68x4fx1ex54x5x4x21x40x41x54x14x25x78x31x32x10x2x51x3a
x14x25x48x33x34x44x29x62x69x76x63x2cx54mx67x24x3bx66x25x70x42x43x14x25x78xfx2x42x43x14
x25x2x35x2x51x3ax54x7ex10x7ex48 Sharpex0x1x48x31x32x61xdx21x51x4fx1ex31x5bx2xdx69x1d
x4fx1ex31xfx48x51x6cx62x3ex54x4fx1ex44xex5bx2x15x16xdx72x6x64x69x7dxfx54x6axex27x4ex4f
x54x48x33x34x59x2bx4fx1ex31x54x2cx68x7x8x5bx40x41x70x2bx51x3ax14x25x78x58x78x2bx51x3ax14
x25x54x2cx68x25x49x70x14x48x70x62x5ax40x41x70x2bx51x3ax14x25x78x25x2x7bx1x10x5ex28x67x40
x41x25x78x2bx51x3ax14x25x54x26x14x25(x7ax601)x78x26x7ex5ax23x24x1fx68x51x2ax2bx4fx1ex31x54
x21x51x2cx68x25x49x70x14x21x40x41x54x2cx68x14x25x5ax2x51x3ax54x57x16x17x2x42x43x14x25x2
x51x3ax54x78 Sharpex49x62x541964x46x57x4fx63x64[12]x2cx5dx25x70x61xdxcx1xdx1cx1d
x34x72x73x3dx22x59(homogeneity of investor expectations)xfx61xdx78x31x32x25x2bx61
xdx41x7dx48x44x29x1ex6fx1ex37x62x4ax55x56x67x41x7dx78
x2exfx31x7ex48x31x32x42x43x2cx36x51x33x69x6ax2cx68x48x33x34x7x8x57x42x43x14x25x2e
x7bx3x54x51x3ax14x25x48x3ax3ex67x51x7bx7x54xdx7dflx1ex6fx1ex37x62x4ax55x56xfx41x7dx78x31
x32x42x43x2cx51x33x69x6ax2cx68x48x33x34x13x61x62x42x43x14x25x2cx2ex45x6ex33x69x6ax2cx68x48
x67x23x68x19x42x43x14x25x21x4ex60x54x77x67x2x1ex6fx42x43x1cx57x69x6ax2cx68x51x3axbx3ax54
x73x77( m;?m). x49x25x64x74x40x56x48x59x70x51x3ax14x25x33x43x67x2x33x69x6ax2cx68x57x42
x43x14x25x54x14x25x48x4x59x70x4fx1ex31x21x7x47x54x51x3ax14x25x57x42x43x14x25x54x7dx3fx4c
x4cx49x66x33x69x6ax2cx68x32x33x78x58x25x18x48x7ex7fx54x1ex6fx1ex37x62x4ax55x56x41x7dx78x42
x43x14x25x2cx2ex45x6ex33x69x6ax2cx68x25x70x61x62x79x41x42x33x69x6ax2cx68x2x4fx1ex31x75x76
x54x70x64xcx1cx4fx1x54x25x65x78x7bx7ex48x21x51x33x69x6ax2cx68x54x5x4x1x7dx48x6fx31x70x48
x42x43x2cx54x21x51x33x69x6ax2cx68xcx2cx31x7bx78
x25x26x3ax3ex67x3bx4dx70x64x69x55x1fx68x61x1ex6fx1ex37x62x4ax55x56x78x25x70x69x55x40
x6fx44x2Sharpex9x6ax27x54x5ex55x78xfx2Sharpex6ax27x54x2cx68x2x39x3x1fxex44x44
x54x47x48x27x68x19x54(x4fx43x6f1)x78x49x25x70x76x1fx2cx48x35x61Markowitzx69x63x47x75
x54x69x63x61xdx24x19x48x15x45x75x54x61xdx2x42x43x14x25x54x51x3ax1dx78x1x53x42x43x14x25
x51x3ax48x7fx47x75x61xdx21x51x4fx1ex31x54x2cx68x4fx1ex14x25x51x3ax78x66x2x25x18x7fx2x6e
x42x21x51x4fx1ex31x10x50x27x54x3cx55x2x70x2ax54x78x3ax3ex24x13x58x23x24x39x72x73x62x4ax54
x3ax3bx27x4ex70x57x63x2cx1ex6fx1ex37x62x4ax55x56x78x25x53x49x24x13x27x27x28x51x3x63x29x78
x26x7ex7fx54x1ex6fx1ex37x62x4ax55x56(22)x23x68
ri ?rf = fli(rm ?rf)+"i; (23)
86
x17x2c"i x2x28x29xdx41x1dxfx48x9xaE["i] = 0. x2exfx31x7ex48x58x23x26
fli = Cov[ri;rm]=Var[rm]
x1fx68Cov[rm;"i] = 0. x7bx7ex48
Var[ri] = fl2Var[rm]+Var["]: (24)
x25x2bx9x26x57x7fx11x1ex4dx1x2ax38x39x5x6x10x28x29x5dx27x54x2ex2fx27x32x18x2fx27x4x68
x69x34x41x10x30x31x19x1ax32x32x18x2fx27x4x23x68x69x2cx33x10x22x30x31x19x1ax78x10x16x25x2b
x68x4fx60x54x26x48x3ax3ex7x8x10x25x78x2bx53x1ex51x3cx73x54x69x1ex78
x25x70x1ex6fx1ex37x62x4ax55x56(CAPM)x54x76x1fx57xbx70x51x2cx54x76x1fxfx2ex70
x26x78x3x66x10x57x6ax27x54x76x1fx54x3x4cx3ax3ex53x49x56x4bx65x63x78x5ex66x10x54x57x58x79
x7ax48x3bx31x3ax3ex24xbx19x57x70x1fx48x1ex6fx1ex37x62x4ax55x56x2x23x34x2cx68x54x66x4ax20
x21x57x42x43x54x66x4ax20x21x75x76x54x1cx1dx3x4cx54x78x66x2x67x66x49x25x70x55x56x25x35x49
x71x72x2cx11x7dx52x54x5fx60x78 1976x46x48 Famax49x62x54x70x6fx13x14xfx4ex54x52x2e[4]
x2cx5dx8x48x7dx52x1ex6fx1ex37x62x4ax55x56x9x4ax66x7dx52x42x43x14x25x54x72x6x64x69x7dx51
x3ax1dx48x58x25x26x54x7dx52x2x2ex23x4fx54x78 Rollx491977x46x5fx60x54[12]x2cx10x1ex6f
x1ex37x62x4ax55x56x54x7dx52x21x2x1dx6x7x71x3bx78x62x3ex0x1x42x43x14x25x2x35x51x3ax2
x33x55x7dx52x54x48x7bx1x10x47x75x1bx2ax21x51x2bx1fx54x1ex37x70x75x48x17x2cx2fx5ex45x46x50
x4ex1x42x43x66x16x54x1ex37x48x58x7dx52x5dxdx54x42x43x14x25x54x51x3ax1dx51x2ex25x70x68x42
x43x14x25x2x35x51x3ax78x3bx7bx1x51x25x26x54x63x77x48x42x53Rossx54APTx41x1CAPM
x54x43x4ex5x8x1ax78 APTx54x70x2bx28x73x49x66x10x51x2ex44x58x42x43x14x25x54x51x3ax1dx5f
x60x78x7ax58x48 Rollx49x10x70x66x63x64x2cx71x72x44x19x5dx8x3ax3ex25x18x54x62x694.3x54x76
x63x78x25x67x2x70x48 CAPMx54x41x7dx51x2ex47x75x42x43x14x25x2x51x3ax54x48x58x2x73x75
x25x60x19x70x2bx51x3axbx3ax44x54x14x25x48x10(x57x33x69x6ax2cx68x70x14)x67x25x14xdx42x43
x14x25xfx54x2ex3bx48x53x68x7bx7x54 CAPM x41x7dx78x25x67xdx1ex61x42x43x14x25x2x35x51
x3ax54x7dx52x5fx60x78
x22x23x31x7ex48 CAPMx58x49x4x0x1x7dx3ex21x7x3bx78x7dx3ex0x1x48 xdx60x0x1exf
x7x3b CAPM x58x2x51x3ax54x48x17x2cx42x43x14x25x7ex7fx67x47x74x64x42x43x5dx68 (Dow
Jonesx5dx68x48 S&Px5dx68x9x9)x78x26x7ex68x19x54flx23x25x36x51x67x4bx54CAPMx54
x79x7ax48xfx2x5x25x7ax7bx2cx68x62x4ax54x7cx6ax58x17x69x6ax70x75x78x51x70x64x28x6dx54x4f
x1exex0x67x2x16x5eCAPMx10x2cx68x54x4x4ax57x2cx68x54x71x72x42x43x4ax54x7dx3fx27x44
x62x2cx68x54x70x3cx78x7ex19x48x49x69x63x44x48 CAPMx21x2x41x1x11x64x40x2cx48x23x3bx27
x65x63x74x64x5fx60x78
4.6 x25x70x51x2cx48x3ax3ex53x4ex70x57x65x63 Markowitz x2cx68x14x25x7x8x69x63
x57x1cx1dx62x4ax55xex75x76x54x3x4cx78x49xbx4bx54x65x63x2cx48x3ax3ex5x61x62x42x43x2cx51
n x64x40x6fx2cx68x48x2fx5ex1x39x8x2cx68x14x25x54x15x6dx7x8x48x58x2ax62x25 n x64x2cx68
x54x20x21x27x32x69x7dx20x21x2x3bx62x54x78xfx2x25x26x54x2ax10x2x2excx75x54x78x2bx71x44x48
x3ax3ex1cx7ax23x24x49x65x8x51x54x40x6fxex4cx2cx27x65x63Markowitzx69x63x78
87
x1bx19x65x8x51x2cx54x40x6fxex4cx48x3ax3ex54x65x63x1cx7ax23x24x39xdx50x62xbx21 Hilbert
x72x76Mxfx8x5fx78x49x25x2bx72x76x2cx51x70x2bx7fx0x3bx5x6dx62x4ax65x68 p:M ! R.
xdx20x21x69R1xfx1cx7ax62x7ax1
R1 = fr 2Mjp(r) = 1:g
xfx47x75x61x62x10x2x74x7ax69x48x4x10x66x15x1fr1;r2 2 R1 x4x15x1fx71x68fi 2R,x5
x51(1?fi)r1 +fir2 2 R1:x7ax58x48x3ax3ex2ex47x75x61x62R1 x2x51x2ax75x54x48x4x10x23
x25x2x26x33x2ax2bxdx40x6fx2cx68xfx21x36x41x54x20x21x69x78x2ex1fx26px2x7fx0x65x68x48x25
x2bx74x7ax69x70x62x2x67x69x78x10x66x25x26x54x20x21x69x48x3ax3ex1cx7ax23x24x20x8x10x54x14
x25xbx3ax54x25x22x78x1fx79x19xbx4bx20x8x54x8x40x3dx33x5bx62x694.2, xdMarkowitzx14
x25xbx3ax66x49x38x3dxfx23x24x7ex1fx78x2bxdxbx3ax20x21x27xf rp x4 rq x27x20x8x2ax59x2b
x20x21x27 r 2 R1 x5bx2ex4dx17x2ax26 rp x4 rq x36x41x54x14x25xbx3ax78x25x6cx79x75x39x2a
x10x66x15x1fr 2 R1, r 6= rp, r 6= rq,x66x49fir 2R,x9xa
E[r] = (1?fir)E[rp]+firE[rq]
Var[r] > Var[(1?fir)rp +firrq]
x2ex1fx25x58x51x2ex2rp x4rq x36x41x10x66R1 x54x38x46x54x7fx1fx27x70x54x14x25xbx3ax54
x1dx33x38x3dx78x10x58x47x75rp x4rq x5bx49x10x3ex36x41x54x14x25xbx3ax44x54x38x3dx2a
E[r] = (1?fir)E[rp]+firE[rq]
Var
? r
1?fir ?
firrq
1?fir
?
> Var[rp]
Var
? r
fir ?
(1?fir)rp
fir
?
> Var[rq]
x15x13x78x2bx2ex9x26x57xbx4bx54x70x2bx2ex9x26x75x76x17x71x51x2ex2x7ex7fx7cx7dx54x78x3a
x3ex71x72x44x73x47x75x17x2cx54x78x2bx2ex9x26x78xfx49x56x4bx54x62x694.4x54x38x3d4)x2cx48
x3ax3ex58x2x5bx5dx10x3ex5cx44x78
x36x3f 4.4 x20x1ex62x43x12x44x45Mx1dx50x51x73x52x79x39x3ax3bx3cx47x7bxax79
Hilbert x44x45. p:M ! Rx1dMx7cx79x5fx60x68x69x6ax62x63x64x65x6bx6cx6dx6ex6fx70
x 2 M x71x6fx70 ? > 0, x73 p(?x) = ?p(x). x5ex4x7x62 p(§1M) 6= 0, x77x35
1M 2Mx67x18x19x1axdxe1 (x57x581 2M)x15x18x19x1axdxex79x7dx7exfx10(x57x58
1 =2M)x0x62x59
R1 = fr 2Mjp(r) = 1:g
x1x7x62x15x67Mx7fx79x7ex65x5bx0x5dx5ex21x7fx5cxcx1x2x26x63x7d
1) x60x61x3x7ax79x22x23 m 2 M, x2 E[rm] = E[m]=E[m2] 6= E[r1M] =
E[1M]=E[m],x4x5x6dx6ex6fx70x 2M,x73p(x) = E[mx];
88
2) (x23fl-x6x7x6x8x62x63x7dx9x6b zero-fl-CAPM)x60x61ru 2 R1, E[ru] 6= 0,
x4x5x6dx6ex6fx70r 2 R1,x73
E[r]?E[rv] = Cov[r;ru]Var[r
u]
(E[ru]?E[rv]);
x7ex7frv 2 R1,xaxbE[rv] 6= 0, Cov[ru;rv] = 0, E[rv] 6= E[ru].
3) x60x61 rp;rq 2 R1, E[rp] 6= E[rq], x4x5x6dx6ex6fx70 r 2 R1, x60x61x72x65
fi 2R,xaxb
E[r] = (1?fir)E[rp]+fir?q;
Cov[r;rp] = (1?fir)Var[rp]+firCov[rp;rq];
Cov[r;rq] = (1?fir)Cov[rp;rq]+firVar[rq]:
4)x60x61rp;rq 2 R1, E[rp] 6= E[rq],x4x5x6dx6ex6fx70r 2 R1,x60x61fir 2R,
xaxb
E[r] = (1?fir)E[rp]+firE[rq]
Var[r] > Var[(1?fir)rp +firrq]
Var
? r
1?fir ?
firrq
1?fir
?
> Var[rp]
Var
? r
fir ?
(1?fir)rp
fir
?
> Var[rq]
xdx27 1) ) 4) ( Hansen–Richardx68x4cx49x4ax62x9) xd1)x41x7dx78x261)
x23x1bx48 rm x57r1M x2ex63x1cx48x4x25x78x2bx24x6cx0x41Mx2cx54x70x2bx8x75x14x72x76
M2. x26Hilbertx72x76x54x3bx66x33x1ex62x69x48 Mx23x33x1ex1x25x2bx8x75x14x72x76x57
x10x54x3bx66x40x72x76x54x18x4x48x4
M = M2 +M?2 :
x25x67x2x70x48x10x66x15x1fx 2M,x66x49x71x68fixm x4fix1 x24x58"x 2M?2 ,x53x68
x = fixmrm +fix1r1M +"x; E[rm"x] = E["x] = 0:
xcx3fx2x48x10x66x15x1fr 2 R1,x66x49x71x68fim;fir 2R,x53x68
r = fimrm +firr1M +"r; E[rm"r] = E["r] = 0: (25)
89
x25x2x7bx1 r;rm;r1M 2 R1; x4 E[mr] = E[mrm] = E[mr1M] = 1, x10x18x48
E[m"r] = E[m]E[rm"r] = 0, x21x24x10x44x26x54x65x70x2bx9x26x78x17x16x44mx13x47
x68x4fx3ex2bx48x67x23x68x19fim +fir = 1,x4fim = 1?fir:x25x26x48x26(25)x23x68x48
E[r] = (1?fir)E[rm]+firE[r1M]+E["r]
= (1?fir)E[rm]+firE[r1M];
Var[r] = Var[(1?fir)rm +firr1M]+2Cov[(1?fir)rm +firr1M;"r]+Var["r]
= Var[(1?fir)rm +firr1M]+Var["r]
x44)x2cx54x9x26x4x65x70x2bx2ex9x26x10x66rp = rm;rq = r1M x41x7dx78 4)x2cx54x13
x78x2bx2ex9x26x54x2cx68x7ex7fx72xdx78
4) ) 3) xd4)x41x7dx78x25x17x71x2x70x68x15x1fr 2 R1 x5bx2ex25x17x2ax26rp
x4rq x21x36x41x54x14x25xbx3ax78x7bx7ex48 3)x54x41x7dx23x57x62x694.3 x70x26x2cx68x78xf
x2x3ax3ex25x18x49x8x4dx70x2bx18x38x2cx68x78x2bx71x44x48xdr 2 R1 x9xa
E[r] = (1?fir)E[rp]+firE[rq]:
x12
rfi = (1?fir)rp +firrq:
x33x34x10x66x15x1fx71x68– 2R,x3ax3ex5bx51
E[rfi +–(r?rfi)] = (1?–)E[rfi]+fiE[r] = E[r] = E[rfi]:
x7bx7ex48x264)x41x7dx48x24x58R1x2x74x7ax69x48 rfi+–(r?rfi) = (1?–)rfi+–r 2 R1,
x23x68
Var[rfi +–(r?rfi)] > Var[rfi]:
x5dx1ex14x3fx6fx48x7fx68
Var[rfi]+2–Cov[rfi;r?rfi]+–2Var[r?rfi] > Var[rfi]:
x31x32x47– > 0,x33x34x26x44x26x48x51x12– ! 0,x23x68
Cov[rfi;r?rfi] >?–2Var[r?rfi] ! 0:
xfx2x47– < 0,x26x10x26x54x76x69x48x7fx23x68x19
Cov[rfi;r?rfi] 6?–2Var[r?rfi] ! 0:
x15x13x54x31x63x2Cov[rfi;r?rfi] = 0,x4
Cov[rfi;r] = (1?fir)Cov[rp;r]+firCov[rq;r] = Var[rfi]: (26)
90
x3bx10x26x54x69x55x10x26x23x2cx68
Cov
?
rp; r?firrq1?fi
r
?rp
?
= 0;
Cov
?
rq; r?(1?fir)rpfi
r
?rq
?
= 0:
x58x25x78x2bx9x26x43)x2cx54x13x78x2bx9x26x78x5ex66x9x26(26)x48x73x2x25x78x2bx9x26
x75x4x78
3) ) 2) xd3)x41x7dx78x31x32Var[rp]Var[rq] = Cov2[rp;rq], x33x34rp x4
rq x7ex7fx7bx3x78x3bx31 4.2 x51x2cx21x5dx8x54x33x26x48x33x69x6ax3cx27x53x2 rp x57 rq
x54x14x25x78x4x66x49x71x68 , x53x68 rf = (1 ? )rp + rq. x26x66 E[rp] 6= E[rq],
x78x31x5ex5fx51x70x2bx2ex9x66 rf. x25x18x48x2ex40xd E[rp] 6= rf, xex23x47 ru = rp,
rv = rf = (1? )rp + rq:
x31x32 Var[rp]Var[rq] > Cov2[rp;rq]. x33x34 Var[rp] x4 Var[rq] x2cx5ex5fx51
x70x2bx2ex9x66Cov[rp;rq]. x2ex40xdVar[rp]?Cov[rp;rq] 6= 0. x25x18x12ru = rp,
rv = (1? )rp+ rq,x17x2c 2Rx9xaCov[ru;rv] = Cov[rp;(1? )rp+ rq] = 0;
x4
= Var[rp]Var[r
p]?Cov[rp;rq]
:
x26x66 6= 0, E[rp] 6= E[rq],x10x26x51E[ru] 6= E[rv]:
x10x25x78x64x74x40x48x3ax3ex5bx23x5drp = ru, rq = rv ? (1? )ru x4ex13)x2cx54
xbx78x2bx9x26x48x58x68x19
E[r] = (1?fir)ru +fir
r
v
?
(1? )ru
?
=
1?fir ? fir(1? )
?
ru + firrv
Cov[r;ru] = (1?fir)Var[ru]+firCov
?
ru; rv ? (1? )ru
?
=
1?fir ? fir(1? )
?
Var[ru]
x26x7ex7dx4x23x1fx682)x2cx75x39x54x9x26x78
2) ) 1) x4fx62x692.3x54x2cx68x78 2
x25x26x48x3ax3ex67x7ex7fx2cx68x61Markowitzx2cx68x14x25x7x8x69x63x3x1ex6fx1ex37
x62x4ax55x56x4x28x29x13x1ax7bx14x69x63x75x76x54x9x4ax1dx78x25x70x31x32x2x5fx7dx22x5ex54x78
x49x39x20x21x27x8x5fx54x57x4fx54Markowtizx69x63x2cx48x22x23x39x60x4bx44x3cx27x48x1cx1d
91
x62x4ax55xexdx15x7x8x19x57x7ax50x49x77x13x78x17x71x2ex7ax78x2bx71x44x48x8x69x63x51x36x51
x1fx1bx51x2ex10x20x21x27x54x2ex64x33x69x6ax2cx68x75x72x54x6dx3cx1ax44x66x49x78x7ax58x48x31
x32x75x39x51Markowitzx79x7ax56x54x14x25xbx3ax54x66x49x48x33x34x10x67x70x62x2x6ex1cx1d
x62x4ax55xex48x24x2ax2ex13x23x25x8x1ax72xdx54x6dx3cx1ax44x78x67x9x3fx70x48 Markowitz
x54x14x25xbx3ax44x62x61x70x64x1dx70x54x1cx1dx62x4ax55xex78x7ax1fx27x5ax70x26x78
x25x18x6x68x1fx79x54x2 Markowitz x54x14x25xbx3ax73x47x75x3bx78x2bxbx3ax14x25
x27x3cx62x78x58x26x25x78x2bxbx3ax14x25x67x25x7ex7fx3cx62x1cx1dx62x4ax55xex21x47x75x54x28
x29x13x1ax7bx14x78x2ex1x57x25x78x64x69x63x9x4ax54x1ex6fx1ex37x62x4ax55x56x5ax10x26x31x7ex78
x26x7ex8x5fx48x49x0x44x3ax3exbx4bx19x57x1fx79x19x54x2bx71x48x23x24x73x62x48x25xbx69x63
x54x68x4fx17x71x5bx2xdx36x4bx3x1fxfx48x4x10x3ex5bx73x44x58x70x2bx8x75x72x76x78x25x2bx72
x76x2x26x28x29x13x1ax7bx14mx4x33x69x6ax2cx68x54x55x74x14x251M x21x0x41x54x78x1ex6f
x1ex37x62x4ax55x56x21x44x58x54ru x4rv x24x58Markowitzx2cx68x14x25xbx3ax21x44x58x54
rp x4rq,x5bx2x25x2bx8x75x72x76x2cx54x78x2bx2ex63x1cx54x24x6cx78x7bx7ex48x1bx2ax17x2cx54
x70x10x48x67x25x39x68x17x62x78x10x78x58x50x62xbx21 Hilbertx72x76x2cx54x17x62x24x6cx5bx11
x7ax13x19x25x2bx8x75x72x76x44x48x24x68x19x7bx7x54xdx3ex2bx62x4ax6ex26xfx78x58x10x2cx68x42
x43x54x62x4ax5fx60x49x68x4fx44x15x13x5bx11x30x31x1x31x1fx3cx62x25x2bx8x75x72x76x2cx54x78
x2bxdx40x24x6cxfx78x25x67x2Black–Scholesx3exbx62x4ax69x63x8x1ax24xbx54x3dx3ex57x58
x4fx54x7bx21x21x49x78
4.7 x49x44x4bx54x65x63x2cx48x3ax3ex7cx52x61x62x2cx68x66x16x2x3bx2dx3cx72x54x78x31
x32x2ex3bx2dx3cx72x48x4x2ex3bx2dwi x47x50x6x48x33x34x5fx60x54x39x1ex2ex23x25x51(10)x6f
(15) x33x26x54x68x74x54x60x54x26x78x16x17x2x5fx60x54x39x1ex5ax2ex13x30x31x1x1cx1dx69x6a
x14x54x39x1ex78x17x71Markowitzx54x57x4fx63x64[8]x5cx2x7cx52x65x63x2ex3bx2dx3cx72x54
x74x40x48x51x6cx1x7ex20x8x5dx5ex74(critical line)x56x5ex78x25x18x68x19x54x51x3axbx3ax49
x2ex1cx33x69x6ax2cx68x54x74x40x53x2x40x41x18x1cx42x57x3ex39x1cx42x54x43x38x78x26x66x3ax56
x6ax42x5ex11x58x2ex3bx2dx3cx72x48x7bx7ex65x63x2ex3bx2dx3cx72x54x2cx68x14x25x7x8x5fx60x21
x51x1ax71x79x7ax78x3ax3ex49x25x15x13x70x16x51x10x7ex49x8x70x28x75x54x65x63x78
nx64x2cx68x54x2ex3bx2dx3cx72x54Markowitzx72x6x64x69x7dx14x25x7x8x5fx60x1
min 2w = wTVw = Pni;j=1 Vijwiwj
s:t: wTe = w1 +w2 +￠￠￠+wn = 1
?w = wT? = w1?1 +w2?2 +￠￠￠+wn?n = ˉ?
wi > 0; i = 1;2;:::;n:
(27)
x10x66x21x51x23x25x54x14x25x20x21 ˉ?x68x19x54x5fx60(27)x54x1ex54x7fx1fx67x2x25n
x64x2cx68x54(x2ex3cx72)x3cx40x28x29x3dx22x7fx1fx78 nx64x2cx68x54x51x3ax14x25x70x62x2x24
x16x69x6ax14x25x48xfx7ax75x2ex7ax78x57x3bx2dx3cx72x54x74x40x51x21x2ex10x54x2xaxbx54x24
x16x69x6ax14x25x54x20x21x73x23x25x49x48x76[mini ?i;maxi ?i]x2cx27x47x48x58x51x3axbx3a
x14x25x54x20x21x21x2x73x25x49x48x76[?nmin w;maxi ?i]x2cx27x47x48x25x18?nmin w x2x10
92
x7x14x25x54x15x16x69x6amin w x54x15x4ex20x21x48x10x2x5fx60(27)x2cx12x5fx65x8x2bx9
x26x65x18x24x13x21x68x19x54x1ex14x25x21x10x7x54x15x4ex20x21x78
x5fx60(27)x2x70x2bx1cx1cx1dx9x26x4x2ex9x26x65x18x54x8x6dx26x33x34x5fx60x78x10
x5ax23x3b Lagrange x16x14x55 (x6fx31x70 Kuhn-Tucker x38x3d) x27x39x1ex48x4 ˉw =
(ˉw1; ˉw2;:::; ˉwn)T x4?1, ?2, ” = (”1;:::;”n)T x2x17x1ex4Lagrangex16x14x54x1d
x75x38x3dx1x10x3ex9xa
8
>>>>
<
>>>
>:
2V ˉw??1???2e?” = 0;
ˉw1 + ˉw2 +￠￠￠+ ˉwn = 1;
ˉw1?1 + ˉw2?2 +￠￠￠+ ˉwn?n = ˉ?;
”i > 0; ˉwi > 0;”i ˉwi = 0;
i = 1;2;:::;n:
(28)
x35x61x2ex3bx2dx3cx72x57x3bx2dx3cx72x54x78x64x74x40x24x19x48x58x51x70x32x33x2cx68x3b
x2dx3cx72x48x4dx70x32x33x2cx68x2ex3bx2dx3cx72x54x2cx76x74x40x78x2ex40xdnx64x2cx68x54xb
k0 x64x2cx68x2x3bx2dx3cx72x54x78x33x34x7bx7x5fx60x54x1ex4Lagrangex16x14x1cx7ax23x24
x60x54x1x44x29x40x26x48xfx2x17x2cx54”i x75x17x41x9xax56x5cx38x3dx2a
”i = 0; i = 1;2;:::;k0; ”i > 0; ˉwi > 0; ”i ˉwi = 0; i = k0 +1;:::;n:
x1x61x60x54x69x66x48x3ax3ex5dnx2cx2cx68x21x14x41x54x69x25x60x27x1Sn. Sk0 x60x27x17xb
k0 x64x2cx68x54x69x25x78 SJ xex60x27Sn x54x70x2bx14x69x78
x26x66(28)x54x59x70x2bx1e ˉwx5x51x70xbx33x6c ˉwi > 0,x39x58x17x7bx7x54”i x1
x7dx48x66x2x3ax3ex7dx4x23x24x68x19x56x5cx4x60x2a
x59x23 4.1 x20 ˉw = (ˉw1; ˉw2;:::; ˉwn)T x1dx7x2(27)x6dx6ex11x12x1d ˉ?x79x7a
xcx4ax6bx2x61x7ex62x4ex5fx60(28)x79x36x60x7fx6bxaxb
ˉwi
‰ > 0; i 2 S
J = fj1;j2;:::;jlg‰ Sn;
= 0 i =2 SJ: (29)
x5dx5e ˉw x10x67x6d SJ x7fx79xdxex61x62x63x44x6bx14x6d Sn n SJ xax61x62x63x44x79x61
x63x19x1axfx10x0xexfx67x6b ˉw0 = (ˉwj1; ˉwj2;:::; ˉwjl)T x67 SJ x7cx79x61x62x63x44x79
Markowitzx7x2(5)x79x4ax0
x3ax3ex2ex40xcx9xax38x3d(29)x54SJ x1x14x25 ˉw x54x64x65x79x78x26x25x70x62x7a
x4(6)–(8)x54x1cx1dx1dx48x3ax3ex7dx4x23x68x56x5cx4x60x78
x59x23 4.2 x20 ˉwm = (ˉwm1 ; ˉwm2 ;:::; ˉwmn )T x1dx7x2 (5) x6dx6ex11x12x1d ˉ?m
x79x4ax6b m = 1;2, x2x15x5ax24x48 SJ x1dx46x66x5bx0x5dx5ex6dx6ex6fx70 fi 2 (0;1),
ˉwfi = (1?fi)ˉw1 + fiˉw2 x67x7x2(27)x6dx6ex11x12x1d ˉ?fi = (1?fi)ˉ?1 + fiˉ?2 x79
x4ax6bx1x2 ˉwfi x10x48SJ x1dx46x66x5bx0
93
x25x26x70x27x48x49 Sn x2cx5ex5fx51x78x64x20x21x2ex10x54x2cx68x18x48x5fx60 (27) x23
x25x51x1ex54x20x21 ˉ?x23x33x1x78x72,x70x72x40x41[mini ?i;maxi ?i]x2cx54x70x2bx48x76x48
x49x25x2bx48x76x44x48x10x66x59x70x2b ˉ?x48x5bx66x49x51x10x26x5dx38x69SJ x54x5fx60(27)x54
x1ex48x24x2ax10x3ex5bx2x10x70x2bx5dx38x69SJ x44x54x3bx2dx3cx72x54x5fx60(5)x54x1ex78x25
x2bSJ x5ex5fx45x6ex78x64x20x21x2ex10x54x2cx68x48x35xeSJ x44x54x3bx2dx3cx72x5fx60x54x1e
x2ex23x25x10 ˉ?x49x70x2bx48x76x44x54x6x5bx66x49x78x4dx70x72xex40x2ex41x48x76x48xfx10x7
x54x1ex5ax2x11x2bx5dx38x69x54x3bx2dx3cx72x5fx60x54x1ex78x7bx1 Sn x73x51x51x2ax2bx14x69
SJ,x21x24[mini ?i;maxi ?i]x70x62x2x51x2ax2bx25x26x54x48x76x4x51x2ax2bx77x54x51x69x78
x4x66x49 ˉ?l 2 [mini ?i;maxi ?i], l = 1;:::;p;x9xa
min
i
?i = ˉ?1 < ˉ?2 < ￠￠￠ < ˉ?p = max
i
?i;
x53x68x49x48x76(ˉ?l; ˉ?l+1)x2cx48x10x7x25x70x48x76x2cx54 ˉ?x54x1ex23x24x51x10x26x54x5dx38
x69SJ. x5ax67x2x70x48x10x66x25x70x48x76x2cx54 ˉ?x27x70x48x21x10x7x54x1ex5ax70x62x2SJ
x44x54x3bx2dx3cx72x54x5fx60x54 (27) x54x1ex78x5ex66x10x66x48x76x54x14x77 ˉ?l x4 ˉ?l+1 x27
x70x48x10x3ex10x7x54xbx3ax14x25x11Markowitzxcx1x67x68x3dx22(corner portfolio)x48
x37x51xfxcx5ax54x79x7ax78
x3ax3ex39xbx4bx54x65x63x2cx19x57x1bx2ax48x49x3bx2dx3cx72x54x74x40x56x48x31x32Sn x54
x14x25xbx3ax43x72x48x33x34x10x6fx2x70x2bx69x7dx77(x21x51?i x5bx7bx9)x48x6fx2x3ex39x1c
x54x1fx44x32(Sn x54x14x25x2cx2ex45x46x33x69x6ax1ex37)x48x6fx2x14x77x49x62x6ax44x54x78x38
x6bx27x7bx7ax54x7ax1c(Sn x54x14x25x2cx45x6ex33x69x6ax1ex37)x78x25x26x70x27x48x49x44x29x24
SJ x1x5dx38x69x54x1ex21x10x7x54 ˉ?x54x48x76x2cx48x17x2ex3bx2dx3cx72x54x14x25xbx3axcx62
x57SJ x44x3bx2dx3cx72x54x14x25xbx3ax45x25x48x4x10x6fx31x2x70x42x3ex39x1cx48x6fx31x2x70
x42x18x1cx78x58x24 ˉ?x1x36x2ax6cx48x5fx60(27)x54x6 x1x7bx2ax6cx54x65x68 (ˉ?)x54x1e
x2bxex44x53x2x25x64x3ex39x1cx42x6fx18x1cx42x43x38x58x41x54[mini ?i;maxi ?i]x44x54x7f
x0x26x65x68x54xex44x78x25x18x7fx0x1dx4x26x1dx2exfx18x38x2cx68x78x22x23x3cx3bx5dx38x69
x27x10x48x76[mini ?i;maxi ?i]x4ex4fx44x29x33x7x54x69x55x23x25x2ex1dx70x48xfx40x41x54
x65x68 (ˉ?)x2x36x51x48x3fx54x78x18x44x21x29x48x3ax3ex67x68x19x56x5cx62x69x2a
x36x3f 4.5 x20Sn x1dx7axcx22x44xdxex5bx0x5dx5ex15x79xax61x62x63x44x5fx60x21
x79xfx10x1bx7x22x44x6bx1x2x15x15x67x7axcx6cxbx6d(x47x73?i x24x25x26)x6bx15x67x73x52
x6ex6fx70x75x6ex15x4dx75x6ex79x71x15x0x7x2(27)x79x1c
(ˉ?) = min
w2Rn+;wTe=1;wT?=ˉ?
?wTVw￠1=2 ;
x7fx6ex11x12 ˉ?x79x64x65 x61(ˉ?; 1)x72x1cx7cx79x66x73x67x74x73x52x6ex6fx70x75x6ex15x4d
x75x6ex71x15x14xax79x5fx60x66x64x65x66x73x75x76x14x3 ˉ? > ?min w x4x6bx7ex77x64x65x66
x73(x6cx73x64x1bx7)x61( w; ˉ?)-x72x1cx7cx7bxax7axcx73x52x45x78x45x7cx79x5fx60x79x64x65
x79x66x73x0
94
x62x694.5x2x10x66x14x25xbx3ax58x1x54x48x10x66x51x3axbx3ax27x70x48x31x63x72xdx78
xfx51x3axbx3ax30x31x1x70x77x18x51x2ex25x1fx68x21x51?i x7bx9x48x58x51x23x25x2x15x16x69
x6ax14x25x54x20x21x15x4ex78x25x26x48x3ax3ex10x2ex3bx2dx3cx72x54x72x6x64x69x7dx14x25x7x8
x5fx60x54x1ex5ax67x51x70x2bx4ex2ax54x61x1ex78x47x75x0x14x1fx79x54x2x48x26x66(28)x2ex13
x2x1cx1dx69x6ax14x48x2ex3bx2dx3cx72x54x8x40x3dx33x5bx62x69x67x2ex13x41x7dx78x7bx7ex48x44x29
x5ex7ex54x1ex6fx1ex37x62x4ax55x56x5ax2ex13x41x7dx78x44x70x51x2cx21x20x8x54x60x64x69x63x54
x9x4ax1dx5ax2ex13x41x7dx78x25x16x17x2x70x68x48x31x32x3ax3ex64x2bx7ex1fx2ex3bx2dx3cx72x74
x40x54x14x25xbx3ax27x10x2cx68x62x4ax48x33x34x25x64x62x4ax55xex53x2ex13x2x1cx1dx62x4ax55
xex78x25x70x5fx60x6x68x7dx3ex4ex70x57x64x65x78x44x29x54x70xbx2bx71xdx15x51x50x1x70xb
xcxdx31x21x1fx79x19x78x2bx71x44x48x3ax3ex57x7fx23x24x3cx19x51x3x2cx56x6ax42x54CAPM
x54xcxdx78x2ex1fx48x3ax3ex49x13x4bx54x78x51x2cx7fx23x24x3cx19x48x49x70x71x57x58x72x73xe
x4cx2cx54CAPM,x4dx68x8x4fx1ex31x70x62x4dx7x8x2ex3cx72x54x4fx1exex0x78x66x2x49x70
x71x72x73xex4cx56x48 CAPMx49x3bx2dx3cx72x57x2ex3bx2dx3cx72x78x64x74x40x75x76x54x48x3f
x7fx53x2ax68x2ex68x74x78
x2fx6a Markowitz x72x71x25x26x25x3ex5cx4fx6x14x43x36x37x2ax11x11x12x58x59x33x20x65x1x5x6e
x58x62x37x58x1cx5cx4fx29x5dx3bx2exbx1x11x20x7x64x2fx60x2cx52x1ax1cx5xex4xdx28x29x19x27x5 Huang–
Linzenberger [6] x34 Ingersoll [8] x60x25x52x1ax9x20x3dx1fx2bx2c Markowitz x5cx4fx29 Jarrow
[9]x20x30x1x27x25x2bx16x5x28x51x4ax3bx56x5ex11x71xcx4CAPM,x5ax2ex2cx5ex13x7cx1x32x25x6fx20xexf
x4dx2bx2c Markowitz x5cx4fx29x3bx67x5x5dx2ex27x7bx48x33x28x7x64x2fx4ex3bx43x2cx2bx2c Markowitz x5c
x4fx29 Markowitz x5cx4fx6x32x33xfx23x2bx3dx5fx5bx5x2bx3fx35x29x43x7dx25x1x0x7bx29xbx53x19x1ax2cx4a
Constantinides–Malliaris [3]x20x4x5x29x43x1x7dx5x3cx69x7dx1cx50x37x38x3ex1dx29 x3cx47x8x2cx52x4
x5x5ax4fx37x38x1cx1dx29x2ex6bx7x64x4axex37x3ax11x32x25 xex41x4fx34x28x77x33x20x3bx18x64x44x60x48x3x6x25
x27x7x64x2fx7x5dxex67x5x79x4axex3bx18x32x33x7bx57x20x64x44x29xbx18x64x44x4ex48x6x6ex48x3bx2ex32x33x7b
x57(x20x23x15x1x52)x1x11x20x7x64x2fx7x5dxex29x3bx18x3ex2cx20x23x44x2bx16x53x1cx19x20x5bxdx12x19x1ax20
x10x1fx42x13x29
xfx11x2exex20x41x25x7x64x2fx6x2bx2c Markowitz x5cx4fx12x60x47x3cx43xex5fx49x60x20x29 x35x7e
x70x4fx6Markowitzx20x17x50x3ax4fx2e[10]x7x5x79x47x6xex26x4dx24x29x20x5x6b[11]x7x5x60x47x3c
x43x2bxex5fx49x60x20x29x6xex5fx49x60x20xbx69x5cx5 x48x3x40x4ax37x3ax1x37x25x2cx20x4dx60x17x5cx4fx5xc
x30x37x38x3ex3fx29x16x12x5x52x26x4dx20x11x12x5cx4fx2ex2fx4ex2ax9x29 x56x6x2bxex5fx49x60x20xbx69x5cx5x32x33
xfx25x1x5fx5bx29x19x1ax6x20x26x3bx1x41x2bx2cx20x3bx18x2bxex5fx49x60xbx69x20xex44x5 x2fx45x19x1ax20x63
x36[17]x29xbx53x41x2ex2cx20x47x3bxax3ex1dx20x4fx72x25x6fx4x67x20x1x52x29
x62x27x19x1ax6ex58x2exex3fx5x48x2ex48x35x43x16x7fx18x47Sharpe [14], Lintner [9]x34Mossin
[12] x77x35x6ax2ax2ex2cx20x29xex1ax7ex61xfx60x47x15x28x18x22x60x11x77x76x43x5cx52x5cx28x56x4dx2cx20x29x22
x60x11x77x76x43x5cx1cx19x4ex37x2cx53 Tobin [15] x34 Lintner [9]. x31x4x5fx26x7x64x2fx7x2cx4ax20
x48x2ex48x35x43x16x7fx18x20x72x42x4dx45 Sharpe [14] x20x72x42x29xbx3bx72x42x5 x62x27x19x1ax6x1fx4x7
x41x2bx2cx20x5x5dx17xex2dx5x6x32x33xfx47x2bx35x5dx36x4ax7fx20x29x6Constantinides–Malliaris [3]
x34 Huang–Linzenberger [6] x7x28x40x4ax41x42x10x62x73x4x67x25x26x13x20xexfx4dx42x62x4dx2cx3bx56
x20Black [1]x20x10x5ffl x48x2ex48x35x43x16x7fx18x13x29x19x1ax20x72x42x4dx45x19x1ax20x63x36[16],x6x7ex53
x19x1ax79x72x42x47x2ax3bx56x20x25x27x10x3cx20x0x1x72x71x2bx12x62x4x4dx20x25x6fx4x67x13x20x74x2cx36x6cx29x6
[17] x7x5x19x1ax79x52x2bxex5fx49x60x20xbx69x4ex4fx47x24x16x20xcx30x29x7dx53 Markowitz x25x6fx4x67
95
x2cx6x5 CAPMx34x37x3ax43x16x7ax32x20x2cx6x23x14x20x27x28x6Cochrane [2]x7x25x3bx18x24x16x20x4e
x4fx5 x3cx18x19x1axbx53x2ex2cx20x25x3dx9x2bx16x29 x43x5c4.4x7x201) ) 4)x33x37Hansen–Richard
x77x20x43x5cx5x28x471987x2x24x29x20Hansen–Richard [5]x7x20xex44x29
x27x53x48x2ex48x35x43x16x7fx18x20x6ex7excx30x5x6Huang–Linzenberger [6]x7x25x5x6x3bx6b
x7bx7cx4ex4fx29
x44x30x23x41x45x23
1. xdyp = p1x1 + p2x2 +￠￠￠+ pnxn, p = 1;2x1x78x2bx2cx68x14x25x54x50x27x4a
x4bx78x76x5dx8x2cx68x14x25y = y1 + y2 x54x20x21x27 ry x57y1x3 y2 x54x20x21x27
ry1 x4ry2 x75x76x54x3x4cx78xcx3fx2x5dx8x48x10x3ex2ex1nx64x2cx68x54x20x21x27
x54x14x25x54x7bx25x3x4cx78
2. xdpx4qx1x78x64x2cx68x48x6cx10x3ex54x20x21x27 rpx4rq x9xaE[rp] = E[rq].
x33x34x25x78x64x2cx68x54x14x25xbx3ax2x37x34x7ex2cx68x48x2cx68px2x25x78x64x2cx68
x54xbx3ax14x25x54x1dx75x38x3dx1Cov[rp;rq] = Var[rp].
3. x31x32nx64x2cx68x54x3ex2bx20x21x27x5bx7bx9x48xfx20x21x27x69x7dx23x25x2ex10x78x10
x18x48x10x3ex7fx5bx2x25x2ex7bx3x54x78x76x39x8x17x51x3ax14x25x48x51x1ex8x17x15x16
x69x7dx54x60x54x26x78
4. xdx2cx68 p x4x2cx68 q x51x10x26x54x20x21 (x3ex2bx20x21x27) x4x69x6a (x20x21x27
x38x2cx7d)x78x76x65x63x25x78x64x2cx68x54x14x25x48x51x5dx8x17xbx3ax14x25x57x32x69x7d
Cov[rp;rq]x75x76x54x3x4cx78
5. x10x66x15x79x54x2cx68x69x48x51x3axbx3ax2x35x70x62x66x49x7ex76x65x63x78x64x51x10x26
x54x69x6ax3x2ex10x54x20x21x54x2cx68x54x14x25x78x17x10x3ex54x7bx3x4cx68x1 1 x18x48
x10x3ex54x51x3axbx3ax66x49x2x7e
6. x31x32x49x2cx68x69x2cx51x78x64x20x21x2ex10x54x33x69x6ax2cx68x48x33x34x10x3ex10x8x2c
x68x69x54x51x3axbx3ax51x37x34x13x14x7e
7. xdx2cx68x14x25 wp = (w1p;w2p;:::;wnp)T, wq = (w1q;w2p;:::;wnq)T. x33x34
x10x3ex54x20x21x27x54x32x69x7d Cov[rp;rq] x8x32x26x3b n x64x2cx68x54x32x69x7d
x20x21x27x60x27x7ex76x26x7ex2cx68x48x31x32 wp x2x24x16x69x6ax14x25x48x51x6cx2ex2
x5x1fx24x16x69x6ax14x25x48x33x34x70x62x66x49x4dx70x2bx24x16x69x6ax14x25wq,x53x68
Cov[rp;rq] = 0.
8. x37x34x2x1ex6fx1ex37x62x4ax55x56x2cx54x42x43x14x25x7ex32x26x27x63x2cx42x43x14x25x2
x51x3ax14x25x48x51x6cx10x2ex45x6ex33x69x6ax2cx68x7e
9. x37x34x2x1ex6fx42x43x1cx7ex37x34x2x2cx68x42x43x1cx7ex46x3bx2cx68x54x20x21x27x27x1e
x8x10x3ex54x60x54x26x48x51x5dx8x10x3ex54x2ex10x6ex7ax58x7bx25x3x4cx78
96
10. x37x34x2x2cx68x20x21x54x4cx12x69x6ax4x43x4cx12x69x6ax7ex25x2ex25x5dx4cx12x69x6axc
x1xdx42x43x69x6axfx7e
11. x32x26x3cx7dx1ex6fx1ex37x62x4ax55x56x57Markowitzx2cx68x14x25x7x8x69x63x75x76
x54x3x4cx7e
12. x6fx51x1fx8x54x1ex6fx1ex37x62x4ax55x56x57x44x70x51x2cx54x1ex6fx1ex37x62x4ax55x56x51
x37x34x2ex10x7ex10x3ex75x76x51x37x34x3x4cx7e
13. x32x26x3cx7dMarkowitzx2cx68x14x25x7x8x69x63x3x1ex6fx1ex37x62x4ax55x56x4x1c
x1dx62x4ax55xex75x76x54x9x4ax3x4cx7ex32x26x0x4ax25x18x44x58x54x68x4fx6fx60x44x2
x70x64xdx36x4bx3x1fxfx7e
x2fx30x31x32
[1] Blackx48 F., 1972, Capital market equilibrium with restricted borrowing,
Journal of Business, 45: 444–454.
[2] Cochrane, J. H., 2001, Asset Pricing, Princeton University Press, Prince-
ton, N. J..
[3] Constantinides, G. M. & A. G. Malliaris, 1995, Portfolio theory, in R.
A. Jarrow, V. Maksimovic & W. T. Ziemba (eds), Finance, Elsevier,
Amsterdam, 1–30.
[4] Fama, E. F., 1976, Foundations of Finance, Basic Books, New York, NY.
[5] Hansen, L. P. & S. F. Richard, 1987, The role of conditioning informa-
tion in deducing testable restrictions implied by dynamic asset pricing
models, Econometrica, 55: 587–614.
[6] Huang, Chi-fu & R. H. Litzenberger, 1988, Foundations for Financial
Economics, Prentice–Hall, Englewood Cliffs, N. J..
[7] Ingersoll, J. E. Jr., 1987, Theory of Financial Decision Making, Rowman
& Littlefield Publishers, Maryland.
[8] Jarrow, R. A., 1988, Finance Theory, Prentice-Hall, Englewood Cliffs, N.
J..
[9] Lintner, J., 1965, The valuation of risk assets and the selection of risky in-
vestments in stock portfolios and capital budgets, Review of Economics
and Statistics, 47: 13–37.
97
[10] Markowitz, H., 1952, Portfolio selection, Journal of Finance, 7: 77–91.
[11] Markowitz, H., 1959, 1991 Second ed., Portfolio Selection: Efficient
Diversification of Investment, Basil Blackwell, Cambridge.
[12] Mossin, J., 1966, Equilibrium in a capital asset market, Econometrica,
34: 768–783.
[13] Roll, R., 1977, A critique of the asset pricig theory’s tests: Part 1, J.
Financ. Econ., 4: 129–176.
[14] Sharpe, W., 1964, Capital asset prices: a theory of market equilibrium
under conditions of risk, Journal of Finance, 19: 425–442.
[15] Tobin, J., 1958, Liquidity preference as bihavior toward risk, Rev. Econ.
Studies, 25: 65–86.
[16] x63x33x2cx3x7bx40x48 2002,x2cx68x14x25x7x8x54x51x3ax14x69x48 x65x7x3bx68x4fx4fx57x6b
55: 176–186.
[17] x63x33x2cx3x7bx40x48 2003, x2ex3bx2dx3cx72x2cx68x14x25x7x8x54x51x3ax14x69x48 x65x7
x3bx68x4fx4fx57x6b56:.
x5cx5d1x1cx1dx38x1dx35x36x28x72xex10Sharpex38x7a
Sharpex54x2cx68x40x66x25x26x54x5ex55x2ax10x66x15x1fx42x43x2cx54x2cx68(x6fx2cx68x14
x25) i, x10x57x42x43x14x25mx54x14x25x21x40x41x54x69x6ax64x20x21x3ex39x1cxcx62x57x1ex6f
x42x43x1cx7bx73x66x42x43x14x25x21x10x7x54x77( m;?m)x44x78x26ix4mx21x36x41x54x69
x6ax64x20x21x39x1cx23x3bx46x68x40x26x27x60x54x2a‰
= (fi) = ((1?fi)2Var[rm]+2fi(1?fi)Cov[ri;rm]+fi2Var[ri])1=2
? = ?(fi) = (1?fi)E[rm]+fiE[ri]
x25x18rm x4ri x33x3fx2x42x43x14x25mx57x2cx68(x14x25) ix54x20x21x27x78x1ex6fx42x43
x1cxex23x60x54x1??rf = ,x17x2crf x2x33x69x6ax3cx27x78x26x66x42x43x14x25x10x7
x54x77( m;?m) = (Var1=2[rm];E[rm])x49x1ex6fx42x43x1cx44x48x3ax3ex51
= E[rm]?rfVar1=2[r
m]
:
x4dx70x69x4bx48x3ax3ex7fx49x44x29x69x6ax64x20x21x39x1cx44x51 m = (0)x4?m = ?(0),
x58x26x66x1ex6fx42x43x1cx57x44x29x69x6ax64x42x43x39x1cx7bx73x66( m;?m)x48x46x8x39x1cx49
x7bx7x77x54x6bx27x5ax9x66 ,x4
= ?
0(0)
0(0) =
(E[rm]?E[ri])Var1=2[rm]
Var[rm]?Cov[rm;ri] :
98
x26x25x78x2b x54x60x54x26x48x7dx4x23x68x1ex6fx1ex37x62x4ax55x56x2a
E[ri]?rf = Cov[rm;ri]Var[r
m]
(E[rm]?rf):
x5cx5d2x1cx42x43x1dx1ex1fx202
1. x5bx79x20 x5bx49x42x41x7cx49x42x1c nx75x24x6cx72x76Rn x2cx54x69x25K xcx1x5b
x79x2x5d
8x;y 2 K;8? 2 [0;1]; (1??)x+?y 2 K:
x17x3x1fx79x7ax2x69x25 K x2cx54x15x1fx78x77x54x7fx38x1cx42x5bx49 K x2cx78x25x2bx25x22
x5ax25x76x77x19x70x71x54(x33x2ax75)x24x6cx72x76x44x78
xd f:K ! R, x17x2c K ‰ Rn x2x70x2bx26x69x78x31x32x10x66x15x1f x;y 2 K,
? 2 (0;1),x51
f((1??)x+?y) 6 (1??)f(x)+?f(y);
x33x34f xcx1x26x69K x44x54x5bx49x42x78 u:K !Rxcx1K x44x54x7cx49x42x48x2x5d
f = ?ux2x26x65x68x78x31x32x44x58x54x2ex9x26x5bx2x67x4bx2ex9x26x48x33x34x7bx7x54x25
x22x5bx23x0x44xdx67x4bxfx78x4x67x4bx26x65x68x6fx67x4bx7dx65x68x9x9x78
x10x66x6dx2ax6cx54x26x65x68f:(a;b) !R,x44x29x2ex9x26x54x6ex7ax23x39x24x56x54
xex3cx8x78
-
6
0 x
y
r
r
r
r
x
rf(x)
y
r
f(y) r
(1??)x+?y
r
rf((1??)x+?y)
rr(1??)f(x)+?f(y)
x66x64x65x79x66x73
99
x31x32b > y > z > x > a,x33x34x66x49? 2 (0;1),x53x68z = (1??)x+?y,
? = (z ?x)=(y ?x). x66x2
f(z) 6 y ?zy ?xf(x)+ z ?xy ?xf(y):
x26x7ex23x68
f(x)?f(z)
x?z 6
f(y)?f(z)
y ?z :
x31x32f x23x1fx48x33x34x33x3fx12z ! x, z ! y,x23x68
8x < y; f0(x) 6 f(y)?f(x)y ?x 6 f0(y):
x25x2bx38x3dx5ax2x23x1fx54f x1x26x65x68x54x1dx33x38x3d(x2cx68x75x3bx2cx6x62x69)x78x25
x26x48x23x1fx54x26x65x68f x23x62x7ax1x17x1fx68f0 x49x54x54x65x68x78x8x7bx23x1fx54x26x65
x68f x23x62x7ax1x17x8x7bx1fx68f00 > 0x54x65x68x78x10x26x48x23x1fx54x7dx65x68ux23x62
x7ax1x17x1fx68u0 x49x3fx54x65x68x78x8x7bx23x1fx54x7dx65x68ux23x62x7ax1x17x8x7bx1fx68
u00 6 0x54x65x68x78
x25xbx31x32x5bx23x76x77x19x2ex2ax6c x = (x1;x2;:::;xn)T x74x40x78x25x18x70x7b
x1fx68
f0(x) =
@f
@x1(x);
@f
@x2(x);:::;
@f
@xn(x)
?T
;
x1 f x49 x 2 K x73x54x7ex3bx24x6cx78 f0 x49 x x44x6dx50x2x5dx10x66x15x1f x0 2 K,
(f0(x0)?f0(x))T(x0?x) > 0. x57x7ex10x18x48xdx;y 2 K, h = y?x,x33x34x10x66
K x44x54x26x65x68x58x51
f(y)?f(x) = f(x+h)?f(x) > (f0(x))Th = (f0(x))T(y ?x):
x58x8x7bx1fx68
f00(x) =
0
BB
BB
BB
BB
@
@2f
@x21(x)
@2f
@x1@x2(x) ￠￠￠
@2f
@x1@xn(x)
@2f
@x1@x2(x)
@2f
@x22(x) ￠￠￠
@2f
@x2@xn(x)
￠￠￠ ￠￠￠ ￠￠￠ ￠￠￠
@2f
@xn@x1(x)
@2f
@xn@x2(x) ￠￠￠
@2f
@x2n(x)
1
CC
CC
CC
CC
A
x1f x49x 2 K x73x54Hessex20x21x78 f00 x49xx44x43x50x2x5dx8x20x21x43x50x62x78x7d
x65x68x54x74x40x72x76x78
Markowtitz x5fx60x2cx54xax38x65x68x54 Hesse x20x21x67x2x2cx68x20x21x27x54x32
x69x7dx20x21 V,x10x2x43x50x62x54x48x7bx7ex48x10x2x26x65x68x78x17x65x18x69x25x2x78x2bx1c
x1dx9x26x44x62x54x26x69x78
100
2. x31x35x54x27x44x79x22x10x55x2fx59x41x74x59x56x7fx3dx3fx40x1c
x3ax3ex49xbx4bx19x57x2cx68x48x10x66Hilbertx72x76(x45x46x51x2ax75x54x2ex55x72x76)
X x54x59x2bx67x14x72x76Y,x5bx23x24x51x56x5cx3bx66x33x1ex48
X = Y 'Y ?;
x4x10x66x15x1fx 2 X,x66x49x1dx70x54yx 2 Y x4y?x 2 Y ?,x53x68x = yx +y?x .
x26x7ex23x1fx68x2a x0x1x24x18x19x25x2dz x24x34xdx10x59x2x1cx14x4ex3x4x23Y x55
x2fx10x25x2dx32x56x18x36x23z x55x2fx32 x5x3z x18x36x4Y x10x6x7x39x2bx71x44x48x25x2
x7bx1 z x57 Y ? x2cx54x21x51x38x46x5bx3bx66x48x21x24x10x54x33x1ex26x2cx2ex23x25x51x43x7d
x54Y ? x41x33x78
nx2bx2ax6cx1;x2;:::;xn x54x5x6dx1cx1dx69x6a
a1xn +a2xn +￠￠￠+anxn = 0
x23x24x3cx2ex39x70x2bnx75x24x6c(x1;x2;:::;xn)x53x68x10x57nx75x24x6c(a1;a2;:::;an)
x3bx66x78x7bx7ex48xbx4bx54x31x63x23x3bx66x5x6dx1cx1dx69x6ax14x69x63x78x26x7ex23x68x19x2a x0
x1 k x19x66x5cx74x59x56x7fx8x30x10x56x7fx3dx10x25x18x36x56x4x17 k + 1 x19x66x5cx74x59
x56x7fx10x25x32 x5x3x34x17k +1x19x56x7fx18x36x4x2ckx19x56x7fx10x74x59x3dx22x39
x2cx68x25x75x2axdkx2bx1cx1dx5x6dx69x6ax1x2a
a11x1 +a12x2 +￠￠￠+a1nxn = 0
a21x1 +a22x2 +￠￠￠+a2nxn = 0
￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠ ￠￠￠ ￠￠￠
ak1x1 +ak2x2 +￠￠￠+aknxn = 0
x10x23x3cx2ex39 n x75x24x6c x = (x1;:::;xn), x53x68x10x57 k x2b n x75x24x6c ai =
(ai1;:::;aik), i = 1;:::;k;x5bx3bx66x78x12x25k x2bx24x6cx0x41x54x24x6cx72x76x1Y,
x33x34x9xax69x6ax54xx54x7fx1fx67x40x41Y ?. x31x32x5dx65k + 1x2bx1cx1dx5x6dx69x6a
x5ax3cx2ex70x2bnx75x24x6cak+1 = (ak+1;1;:::;ak+1;n)x57xx3bx66x54x60x54x26x48x33
x34x4x60x54x38x3dx2x70ak+1 x57Y ? x2cx54x21x51x38x46x5bx3bx66x78x7bx7ex48x10x70x62x2
Y x2cx54x38x46x48x4x10x23x60x27x41xbkx2bx24x6cx54x1cx1dx14x25x78
x44x29x31x32x2cx54x69x6ax54x5x6dx1dx38x3dx23x24x12x5fx48x4x0x1 k x19x74x59x56x7f
x8x30x10x56x7fx3dx10x25x18x36x56x4x17k + 1x19x74x59x56x7fx10x25x32 x5x3x34x17k + 1
x19x56x7fx18x36x4x2ckx19x56x7fx10x74x59x3dx22x39 x2bx71x44x48xdkx2bx1cx1dx69x6ax1x2a
a11x1 +a12x2 +￠￠￠+a1nxn = b1
a21x1 +a22x2 +￠￠￠+a2nxn = b2
￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠ ￠￠￠ ￠￠￠
ak1x1 +ak2x2 +￠￠￠+aknxn = bn
101
x2ex40xdx10x51x1ex1x0 = (x01;:::;x0n),x33x34x12y = x?x0 = (y1;:::;yn),x6ax69
x6ax2ax1kx2bx1cx1dx5x6dx69x6ax2a
a11y1 +a12y2 +￠￠￠+a1nyn = 0
a21y1 +a22y2 +￠￠￠+a2nyn = 0
￠￠￠￠￠￠￠￠￠￠￠￠￠￠￠ ￠￠￠ ￠￠￠
ak1y1 +ak2y2 +￠￠￠+aknyn = 0
x7bx7ex48x43x5x6dx5fx60x23x30x31x1x5x6dx5fx60x27x65x63x78
3. Markowitzx22x23x9x10x43xax1c
x47x48Markowitzx5fx60x2a
8<
:
min V(w) = 2w = wTVw = Pni;j=1 Vijwiwj
s:t: wTe = w1 +w2 +￠￠￠+wn = 1
?w = wT? = w1?1 +w2?2 +￠￠￠+wn?n = ˉ?
(1)
x12
H = fw 2RnjwTe = 1;wT? = ˉ?:g
x1x5fx60(1)x54x65x18x69x78x31x32 ˉw x2x5fx60(1)x54x1ex48x33x34x10x66w 2 H,x5bx51
V(w) > V(ˉw). x26x66Hx2x70x2bx26x69x48x46x10x66x15x1ft 2 (0;1), (1?t)ˉw+tw 2
H,x39x58x3ax3ex51
V(ˉw +t(w? ˉw))?V(ˉw) > 0: (2)
x12h = w? ˉw. xex26wx4 ˉwx5bx2H x54x38x46x48 hx9xax5x6dx1cx1dx69x6ax14
h1 +h2 +￠￠￠+hn = 0
?1h1 +?2h2 +￠￠￠+?nhn = 0 (3)
x10x18x48x59x2bx9xax44x29x38x3dx54hx5ax70x62x10x7x70x2bw = ˉw + h 2 H. x26(2)
x23x68
limt!0+ V(ˉw +th)?V(ˉw)t > 0;
x44x26x1ex14x2x65x68V x3ax69x24hx54x69x24x1fx68x48x10x9x66
@V(ˉw)
@w1 h1 +
@V(ˉw)
@w2 h2 +￠￠￠+
@V(ˉw)
@wn hn:
x44x26x70x68x48x10x10x66x15x1fx9xa (3) x54 h x43x50x78xfx2x31x32 h x9xa (3), x33x34
?hx5ax9xa(3). x7bx7ex48x71x72x44x10x66x15x1fx9xa(3)x54hxcx62x51
@V(ˉw)
@w1 h1 +
@V(ˉw)
@w2 h2 +￠￠￠+
@V(ˉw)
@wn hn = 0: (4)
102
x3cx3bx44x51x2cx54x31x32x48x3ax3ex23x1bx48x25x2bx10x66hx54x69x6ax70x62x2xbx78x2bx69x6a
x54x1cx1dx14x25x48x4x66x49x71x68?1;?2,x53x68
@V(ˉw)
@w1 = ?1?1 +?2
@V(ˉw)
@w2 = ?1?2 +?2
￠￠￠￠￠￠ ￠￠￠ ￠￠￠￠￠￠￠￠￠
@V(ˉw)
@wn = ?1?n +?2
x4
V11 ˉw1 +V12 ˉw2 +￠￠￠+V1n ˉwn ? 12?1?1 ? 12?2 = 0
V21 ˉw1 +V22 ˉw2 +￠￠￠+V2n ˉwn ? 12?1?2 ? 12?2 = 0
￠￠￠￠￠￠￠￠￠￠￠￠ ￠￠￠ ￠￠￠
Vn1 ˉw1 +Vn2 ˉw2 +￠￠￠+Vnn ˉwn ? 12?1?n ? 12?2 = 0
x6f
2V ˉw??1???2e = 0: (5)
x7ax75x48x31x32x66x49?1;?2,x53x68 ˉw 2 H x9xa(5)x48x33x34x10x10x66x15x1fx9xa
(3)x54hxcx62x51(4)x41x7dx78x26x66V x2x26x65x68x48x46x10x66x15x1fx9xa(3)x54h
xcx62x51
V(ˉw +h)?V(ˉw) > limt!0+ V(ˉw +th)?V(ˉw)t = 0;
x4x10x66x15x1fx9xa(1)x54x65x18x38x3dx54wx51V(w) > V(ˉw):x25x67x2x70x48x5fx60
(1)x54x65x18x38x3dx4(5)x78x31x49x70x14x48x5ax2x5fx60(1)x54x1ex54x1dx33x38x3dx78
4. x57xbxcx52x54x10Markowitzx22x23x1c
x1ax49x47x48x2ex3bx2dx3cx72x54Markowitzx5fx60x2a
8
>><
>>:
min V(w) = 2w = wTVw = Pni;j=1 Vijwiwj
s:t: wTe = w1 +w2 +￠￠￠+wn = 1
?w = wT? = w1?1 +w2?2 +￠￠￠+wn?n = ˉ?
w1;w2 :::;wn > 0
(6)
x25x18x4cx17 ˉ? 2 [mini ?;maxi ?]x18x48x5fx60x78x51x1ex78
x1cx7ax12H = fw 2RnjwTe = 1;wT? = ˉ?:g. x3ax3ex2ex40x58x61x62x2a
x60x61 ?w 2 H,x4x5 ?w1; ?w2;:::; ?wn > 0.
103
x25x2bx38x3dx17 ˉ? 6= mini ?x6fmaxi ?x18x5x2x41x7dx54x78x25x2x7bx1x10x66x15x1fx65i
x64x2cx68x48x10x5x25x57x20x21x15x4ex54x2cx68x6fx20x21x15x16x54x2cx68x70x14x4dx41x70x2bx2e
x3bx2dx3cx72x54x14x25x48x53x68x14x25x20x21x68x28x9x66 ˉ?. x7ax13x48x5dx25xbx14x25x13x32x49
x70x14x4dx41x70x2bx2ex3bx2dx3cx72x54x59x2bx41x33x54x77x45x5bx1x3bx54x14x25x48x8x14x25x21
x10x7x54 ?wx67x4dx9xax44x29x38x3dx78x5ex66 ˉ? = mini ?x6fmaxi ?x54x74x40x7bx10x27
x70x77x77x28x6dx48x23x3bx56x4bx54x72xdx69x55x4ex4fx65x63x78
x2exfx3cx8x48x74x33x6cx43x50x54x24x6c ˉw 2 H x2x5fx60(27)x54x1ex54x1dx33xcx75
x38x3dx1
8w 2 H; maxfV(w)?V(ˉw);?w1;:::;?wng> 0: (7)
x7bx7ex48x69x25
A = fx = (x0;x1;:::;xn) 2Rn+1 j 9w 2 H;x53x68
V(w)?V(ˉw) 6 x0;?wi 6 xi;i = 1;:::;n:g
x57x69x25
B = fy = (y0;y1;:::;yn) 2Rn+1jyj < 0;j = 0;1;:::;n:g
x2ex7bx66x78 B x74x7ax2x26x69x78x58x10x66x15x1fx1;x2 2 A,x66x49w1;w2 2 H,x53x68
V(wp)?V(ˉw) 6 xp0; ?wpi 6 xpi; p = 1;2; i = 1;:::;n:
x7bx7ex48x10x66x15x1f? 2 (0;1),
V((1??)w1 +?w2)?V(ˉw) 6 (1??)(V(w1)?V(ˉw))+?(V(w2)?V(ˉw))
6 (1??)x1 +?x2;
?[(1??)w1i +?w2i] 6 (1??)x1i +?x2i; i = 1;:::;n;
x25x18x3bx19x26x65x68V x54x1dx60x48x4x10x9xa
V((1??)w1 +?w2) 6 (1??)V(w1)+?V(w2):
x25x70x68(1??)x1 + ?x2 2 A,x24x2aAx5ax2x26x69x78x25x26x48x3ax3ex67x23x7x3bx26
x69x33x5bx62x69x2ax66x49”0;”1;:::;”n x2ex7fx1x7dx48x53x68
8x 2 A;8y 2 B; ”0x0 +”1x1 +￠￠￠+”nxn > ”0y0 +”1y1 +￠￠￠+”nyn:
x25x70x2ex9x26x54x1fx14x23x15x79x38x5dx66x7dx48x7bx58,
8x 2 A; ”0x0 +”1x1 +￠￠￠+”nxn > 0:
104
x10x18x48x26x66x0;x1;:::;xn x5bx23x24x15x79x4ex48x46xcx2cx51”0;”1;:::;”n > 0. x4d
x70x69x4bx48xbx4bx54x61x62x58x23x1fx8”0 > 0. x35xex48x49x44x26x2cx47xi = wi,x23x68
x19x10x66x15x1fw 2 H,x51
”1w1 +”2w2 +￠￠￠+”nwn = 0:
x16x5ex61x62x48 H x2cx51x74x2bx33x6cx5bx1x3bx54x24x6c ?w. x25x53x68x44x26x4cx17x21x51
”i = 0x18x78x51x23x25x78x57x10x3ex2ex7fx1x7dx5dx23x78x25x26x48xcx2cx51”0 > 0. x2ex40
x47”0 = 1. x16x5ex69x25Ax54x62x7ax48x26x7ex1fx68
8w 2 H; V(w)?”1w1 ?”2w2 ?￠￠￠”nwn > V(ˉw):
x5d ˉwx4ex19x1ex14x48x58x25x68x19
?”1 ˉw1 ?”2 ˉw2 ?￠￠￠?”n ˉwn > 0:
x26x66x21x51x54”i > 0, ˉwi > 0,x25x4cx17
”i ˉwi = 0; i = 1;:::;n;
x18x78x51x23x25x78x7bx7ex48x15x13x51x2a ˉw = (ˉw1;:::; ˉwn) 2 H x4x22x23(27)x10x25x10
x26x27x28x29x2ax2bx54x1cx63x64”1;:::;”n > 0, xdx40
8w 2 H; V(w)?”1w1 ?￠￠￠?”nwn > V(ˉw):
ˉwi > 0; ”i ˉwi = 0; i = 1;:::;n:
xbx4bx2cx68x54x2x38x3dx54xcx75x1dx78x58x38x3dx54x1dx33x1dx3x15x2x74x7ax54x78
x3ax3ex13x5d V(w;”) = V(w) ? ”1w1 ? ￠￠￠ ? ”nwn x27x4ex43x44x70x51x2cx54
V(w), x51x4ex4fx7ex7fx70x26x54x65x63x48x31x32x67x68x19x2a ˉw = (ˉw1;:::; ˉwn) 2 H x4
x22x23(27)x10x25x10x26x27x28x29x2ax2bx54x1cx63x64”1;:::;”n > 0, ?1;?2 2R, xdx40
V11 ˉw1 +V12 ˉw2 +￠￠￠+V1n ˉwn ? 12?1?1 ? 12?2 = ”1
V21 ˉw1 +V22 ˉw2 +￠￠￠+V2n ˉwn ? 12?1?2 ? 12?2 = ”2
￠￠￠￠￠￠￠￠￠￠￠￠ ￠￠￠ ￠￠￠
Vn1 ˉw1 +Vn2 ˉw2 +￠￠￠+Vnn ˉwn ? 12?1?n ? 12?2 = ”n
x33x34
2V ˉw??1???2e?” = 0:
xexf
ˉwi > 0; ”i ˉwi = 0; i = 1;:::;n:
105
