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x43x16x52x28x18x48x2ex48x35x43x16x7fx18(CAPM)x47x33x16x20x29
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x66x5ex20
? 2.1x70x2bx7dx43x3ax20x70x47x40x3cx4dx43x16x52x28x29
? 2.2 x4cx2bx7dx43x3ax20x70x47x40x3cx4dx43x16x52x28x29
? 2.3x22x55x72x71x5cx5dx20x60x2ex7fx18x29x37x3ax43x16x52x28x34x58x53x35x2ax2bx5x29
? 2.4 x58x53x35x2ax2bx5x20x5cx59x4ex4fx29x70x3dx30x72x71x39x35x7fx78x25x26x29
? 2.5 x2bx32x3ex5dx11x34x3ex6x2bx32x5x60x14x29
? 2.6x48x2ex48x35x43x16x7fx18x18x37x3ax43x16x52x28x20x33x16x3ax29
? x1fx4x20 x32x33x47xdx64x441x29
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x54x25x65 (x2cx68)x48xfx2x5bx25x3bx5bx27x31x12x78x3dx3ex1ex37x66x16x5bx2x1x61x24x5bx36
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x23x25x25x26x56x78
x10x3dx3ex1ex37x62x4ax54x70x64x30x46x54x55x55x2x2ax4x6bx17x50x27x4ax6x54x74x64x23
x25x1dx48x7ax13x6bx12x17x36x72x6x48x51x13x24x25x70x36x72x6x27x2ex1x10x54x4x6x78x6fx31
x3bx68x4fx11x1x27x70x48x5dx3dx3ex1ex37x54x50x27x4ax6x3cx2ex70x2bx28x29x2ax6cx48x33x34x25
32
x2bx28x29x2ax6cx54x68x4fx3ex2bx67x23x17x2ex8x3dx3ex1ex37x54x17xbx4ax6x78x37x1fx7x3bx18x48
x73x75x7ex1fx70x62x54x69x55x27x4x6bx50x27x4ax6x54x25x27x33x6cx48x7ax13x12x8x17x68x4fx3e
x2bx48x67x0x1x5fx60x19x57x68x19x1ex44x78x25x26x54x55x55x31x10x25x27x63x25x23x4fx4ex6fx50
x70x26x6x31x78x1ax21x37x1bx48x25x27x63x14x17x66x10x70xbxdx4bx5fx60x54xcxdx48x58x10xdx4b
x5fx60x54x62x4ax67x2x25x26x1ex44x54x78x49x3dx3ex57x58x4fx54x5x3excxdx2cx48x7dx3ex5ax4d
xfx36x7ax1ex25x26x27x0x4ax3dx3ex1ex37x62x4ax5fx60x78x58x3dx3ex57x58x4fx5fx3fx2cx2ex8x54
x15x45x75x54x70x57x2x35x62x61x25x26x54x55x55x48x58x20x8x3bx33x6dx3cx61xdx27x62x4ax57x49
x3exbx62x4ax5fx60x2cx48x7ex1ex54x33x6dx3cx61xdx53x1x50x27x4ax6x25x70x28x29x2ax6cx40x4b
xcx5ax54(x2ex10x66x68x76x23x25x1dx54)x25x27x33x6cx48x10x3ex75x76x54x3x4cx67xcx1x1ex37
x62x4ax40x6fx62x69x78
x49x44x70x51x2cx48x3ax3ex5ax19x57x3bx60x61x54x68x4fx55x56x27x47x29x32x26x3cx3bx68x4f
x27x1ex44x25x26x54x5fx60x78xfx2x49x33x18x48x3ax3ex1x61x66x66x26x31x38x11x48x36x51x0x4e
x77x77x46x3cx54x68x4fx11x1x78x10x66x7ax7bxdx33x5bx4fx1x67x33x5bx37x8xfx54x33x6dx3cx61xd
x49x36x51x68x3cx54x68x4fxex4cx54x74x75x56x48x60x54x44x5ax51xfx2ex55x24x54x1ex69x78x49x25
x70x51x2cx48x3ax3ex67x75x21x51xdx68x4fx40x26x5cx48x53x17x2cx54x60x54x2ex13x51x15x1fx6ex24x78
x2ex1fx49x25x70x51x54x65x63x2cx48x3ax3ex1ax6cx5dx75x39x21x71x54x7ex1ex54x33x6dx3cx61xdx32
x49x70x17x48x58x79x65x63x70x64xex6dx33x6fx54x33x6dx3cx61xdx78x10x67x2xdx1cx1dx62x4ax55
xexfx78x25x70x62x4ax55xex54x2ex3bxfx4ex48x10x71x72x44x23x3bx27x25x46Black–Scholesx3e
xbx62x4ax69x63x24xbx54x3dx3ex57x58x4fx54x3x15x21x51x74x75x31x32x78
2.1 x49x3ax3ex17xbx54x65x63x2cx48x33x6dx3cx61xdx54x70xbx10x75x53x2ex1xdx6ex69xf
x27x0x1x78x3ax3ex54x6cx75x5fx60x2x32x26x5dx10x68x4fx55x56x5cx78x1x61x66x66x69x1ex48x3a
x3ex79x65x63x36x51x2ex3cx62x1dx54x55x56x48x13x65x63x51x2ex3cx62x1dx54x55x56x78x65x63x36x51
x2ex3cx62x1dx54x55x56x48x5ex2x65x63x70x2bx36x51x28x29x2ax5cx54x69x68x42x43x48x6fx31x70x5e
x65x63x70x2bx34x35x5fx60x48x17x31x32x2x70xax61x7ax54x78x7ax58x48x4x53x31x7ex48x5ax19x57x25
x49x7dx2dx2ex76x5fx78
x49x3ax3ex5fx38x54x70xbx76x29x2cx48x3ax3ex5bx73x65x63x8x3ex55x56x48x4x39x18x76x54x3a
x3bx27x3cx48x61x62x3ax3ex54x55x56x2cx73x51xdx17xbxfx57xdx50x27xfx78x2bx26x36x45x1x69x66x14
x4fx48x24x13x3ax3ex5dx0x49x37x0x7ax78x21x22x38x39x68x69x2x26Kx64(x37x38)x38x39x58x17x74
x64x14x25x21x14x41x78x25xbx2cx68x11x61x62x1x50x27x4ax4bx19x1bx54x70x64x45x46x78x21x22x2cx68
x62x4ax5fx60x48x67x2x75x16x5ex2cx68x50x27x54x4ax4bx24x58x70xbx61xdx27x3cx62x10x17xbx54
x4ax4bx78x25K x64x2cx68x23x24x3bx4ex60x10x3ex54x50x27x4ax4bx54x3bx71x68x1;:::;xK > 0
x27x60x27x78x17x5x1fxex23x3cx2ex70x2bK x75x5cx24x6cx = (x1;:::;xK)T,x25x18T x60
x27x3ax6fx78x25x26x56x36x7ax19x57x45x6ex2dx2ex57x58x4fx44x54x61xdx2ax4ax4bx23x24x3bx3bx71
x68x27x60x27x48x58x2ex2x31x12x7fx36x2fx2cx33x26x48x15x1fx4ax4bx5bx51x70x2bx15x16x50x51x6b
x6cx6dx6ex57x42x43x44x73x51x70x64x4ex59x5bx38x11x54x4ax4bx48x58x2ex2x31x12x7fx66x16x2cx19
x19x66x49x2dx2ex64x4ax4bx57x4ax4bx2x3bx71x68x48x58x2ex47x48x51x43x3bx4ax4bx54x2cx68(x70x5b
x2ex6x54x2cx68x17x71x5ax2x66x72x4fx10x54)x57x31x7ex9x9x78x2ex1fx24x56x54x65x63x10x4ax4b
x2x35x1x3bx51x33xcx3fx75x39x78x16x17x2x70x2bx2cx68x14x25x54x4ax6x67x2ex70x62x2x3bx54x78
33
x58x49x24x13x54x69x63xex4cx2cx48x2cx68x57x2cx68x14x25x19x57x36x51x6fx60x48x3fx78
x1ax49x27x47x48x2cx68x4fx1ex78x2cx68x4fx1ex4fx1x23x26x70x2b K x75x4fx24x6c =
( 1;:::; K)x27x60x27x48x25x18 1 x60x27x70x1x651x64x2cx68x54x68x6cx48 2 x60x27x70x1
x652x64x2cx68x54x68x6cx48x31x7ex9x9x48x24x5e K x60x27x70x1x65K x64x2cx68x54x68x6cx78
x25x18x1x61x68x4fx44x73x69x69x66x14x4fx48x61xdx25xb i x5bx23x24x2x15x79x71x68x48x58x2e
x2x5ex71x72x36x2fx2cx48x2cx68x54x68x6cx5bx73x25x2x1ex68x57x10x18x48x17 ix2x50x68x18x48x25
x18x54xdx70x1xfx17x71x2xdx3cx8xfx78x16x17x2 x54x7bx7ax24x6c? = (? 1;:::;? K)
x2x60x27x57 x3bx28x7bx7ax54x4fx1ex2fx30x78x25x26x54 K x75x24x6cx57x7fxcx1x3bx1cx1dx3c
x3dx22(portfolio)x6fx31x3bx1cx1dx3cx3ex3f (strategy)x78x7bx1x59x64x2cx68x49x50x27x54x4a
x4bx2x19x1bx54x48x21x24x70x2bx4fx1ex14x25x6fx31x70x66x4fx1exex0 x54x50x27x4ax6x67x11
x25x69x1ex61x62 (x25x17x71x5ax2x70x64xdx1cx1dx62x4ax55xexf) x1x4fx24x6c x57x5cx24x6c
xx54x16x2x2a
￠x = 1x1 + 2x2 +￠￠￠+ KxK: (1)
x26x66 i x23x24x47x50x6x48x25x2bx68x6x5ax2x23x3bx23x50x54x71x68x78x25x26x48x21x22x2cx68
x42x43x67x2x14x25x6fxex0 x54x7fx1fx78x31x32x42x43x44x10x4fx1exex0x36x51x15x1fx2ax10x48
x33x34 x23x47x15x1f K x75x24x6cx78x66x2x21x22x2cx68x42x43x48x3bx68x4fx54x11x1x27x70x48
x5ax67x2x70x2bx71x68x74x44x54K x75x24x6cx72x76x78x3ax3ex5dx10x44x2eM.
x5ex66x36x28x22x23x2x25x26x27x20x54x2ax19x1bx70x2bx4fx1ex14x25x50x27x54x4ax6x48x75
x3cx62x8x14x25x17xbx54x4ax6x78x26x66x49x3ax3exaxbx54x2cx68x42x43x55x56x2cx36x51x15x1f
x57x58x4fx10x75(x45x31x36x51x4fx1ex64x34x35x31x48x36x51x36x37x48x36x51x34x35x48x36x51x66x16
x9x9)x78x7bx7ex48x3ax3ex2ex23x25x49x25x18x7ex1fx74x64x5cx76x57x58x3x4cx27x1x2cx68x62x4ax78
x3ax3ex25x56x54x73x2xdx7bx10x62x4axfx48x4x31x1fx16x5ex40x6fx2cx68x54x4ax4bx48x27x1x2cx68
x14x25x62x4ax78
x1ax49x3ax3ex27x5dx7ax7bxdx33x5bx4fx1x67x33x5bx37x8xfx54x55x24x54x33x6dx3cx61xdx46
x3cx5cx78x3ax3ex54x5fx60xex4cx19x57x2ax1x70x2bK x75x24x6cx72x76x78x58x62x4ax5fx60x33x43
x2x75x10x25x2bKx75x24x6cx72x76x2cx54x59x70x2bx24x6cx49x8x70x2bx71x68x27x2ex1x10x54x17
xbx4ax6x78x33x6dx3cx61xdxex2x75x10x25x64x62x4ax62x8x40x41x55xex78x25x26x54x55xex15
x17x75x39x54x2ex73x20x71x48x23x33x1x25x26x24x2bxex6dx2a
1. x40x41x28x29x18xdx10x3dx22x32x42x2cx43x44x24x18xdx10x36x28x39x25x2x7bx1x31x32
x51x78x2bx14x25 x4 ? x54x50x27x4ax6 (x46x1fx79x48x3ax3ex25x18x51x70x2bx79x44x38x3dx2
x14x25x54x50x27x4ax6x2x26(1)x62x7ax54x78x25x2bx62x7ax5ax2x70x2bx45x75x61xd)x2x70x26
x54x48x58x10x3ex17xbx54x4ax6xex2 x77 ? x75x66x45x48x33x34x4fx1ex31x67x23x7ex1fx70x4e
x70x2b ,x3cx8x70x2b? x49x17xbx46x70x47x5bx48x58x17x50x27x66x7x18x62x51x33x66x7x54xe
xf(x25x18x36x7ax7fx10x66x16x2ex8x70xbx61xdx48x4x17xbxdx3cx72xfx70x2bx14x25x23x19x50
x27x13x66x7)x78x25x67x53x62x5bx68x61x6dx3cx78x39x58x48x77x61x33x6dx3cx61xdx78x26x66x10x7
x59x2bx14x25x54x50x27x4ax6x48x73x10x7x17x1dx70x54x17xbx4ax6x48x62x4ax5fx60x67x2ax1x31
x1fx27x3cx62x70x2bx36x28x49x42x48x10x2x70x2bx71x2ax6cx54x71x6x65x68x2a p:R!R,x17x2c
Rx60x27x71x68x7fx1fx78 px54x36x2ax6cx2x14x25x54x50x27x4ax6x48x7bx2ax6cx2x14x25x54x17
34
xbx4ax6x78x46x1fx79x48 px54x36x2ax6cx2ex2M x2cx54x38x46x48x4px2ex2x2cx68x14x25(K
x75x24x6c)x54x65x68x48x58x2x2cx68x14x25x50x27x4ax6(x71x68)x54x65x68x78
2. x3dx22x10x4ax4bx4cx10x42x2cx28x29x43x44x4dx4ex44x3dx22x10x42x2cx28x29x10x4fxdx4c
x42x39x25x2x7bx1x31x32x25x78x2bx6cx2ex7bx9x54x3fx48x31x10x14x25x54xdx6x5fx4axfx57xdx7d
x50x4axfx2ex10x78x66x2x7dx3ex67x23x3cx3bx17x2cx54x7dx4ax48x5x3bx77x31xdx6x5fxfx70x1x48
xdx7dx50xfx3cx8x33x26x54xex0x48x5bx68x6dx3cx78
3. x3dx22x10x51x28x23x52x28x43x44x18x53x39x49x3ax3ex65x63x54x42x43x55x56x2cx48x2ex44
x58x66x16x21x3x56x42x45x75x72x54x42x43x1fx10x78x7bx7ex48x42x43x44x54x2cx68x14x25x54x70x4a
x57x3cx4ax7x8x70x2ax78x35xex7dx3ex67x23x7ex1fx70x3cx7dx4ax27x5bx68x6dx3cx78
4. x3dx22x10x42x2cx28x29x43x44x4dx4ex51x3dx22x30x27x10x42x2cx28x29x52x41x39x25x70x33
x62x54x69x26x57x652x77x72xdx78x35xex7dx3ex23x24x3cx3bx17x2cx54x7dx4ax48x5x3bx77x31xdx70
x53x14xfx70x1x14x25x48 xdx33x54xfx3cx8x14x25x41x33x54xex0x48x27x5bx68x6dx3cx78
5. x40x41x29x25(x28x29x54x55)x10x3dx22x32x42x2cx56x29x25x39x25x70x33x62x31x32x2ex41
x7dx48x67x4dx8x1ax17xbx2ex13x5bx70x1x70x2bx14x25x48x50x27x58x46x70x47x5bx78x57xbx4bx2e
x10x54x2x48xbx4bx60x2bxex6dx54x61xdx2ex41x7dx48x21x5bx68x54x6dx3cx2x49x10x70x2bx18x3e
x68x19x54x78x58x25x70xex6dx54x61xdx2ex41x7dx48x6dx3cx23x25x2x5ax18x3ex5bx68x54x78x2ex40
xcx13x70x64x6dx3cx2xdx5ax18x6dx3cxfx78
x25x24x2bx2ex10xex6dx54x33x6dx3cx61xdx79x7ax3cx14x27x19x57xfx7bx7cx48xfx2x31x32
x3bx68x4fx40x26x27x60x54x48x33x67x21x1x28x68x3cx73x78
x2cx57x58x36x59x10x2cx2dx20x2ex2fx36x28x10x54x19x5bx5cx2a
1. (x5dx36x28x5ex5f)x60x61x62x63x64x65p:R!R.
2. (x55x66x5cx36x28x5ex5f) px67x68x69x6ax64x65x6bx6cx6dx6ex6fx70x68x72x65? > 0x71
x72x65y,x73p(?y) = ?p(y):
3. (x66x5cx36x28x5ex5f) p x67x69x6ax64x65x6bx6cx6dx6ex6fx70x72x65 ? x71x72x65 y, x73
p(?y) = ?p(y):
4. (x74x59x36x28x5ex5f) px67x75x76x64x65x6bx6cx6dx6ex6fx70x72x65?;?x71x6fx70x72x65
y;z,x73p(?y + ?z) = ?p(y) + ?p(z):x77x78x79x62x63x64x65x7ax62x73x77x78x79
x7bx7cx7d p(y) = ay,x7ex7fax67x72x65x0
5. (x55x74x59x36x28x5ex5f) p x67x68x75x76x64x65x6bx6c p x67x75x76x64x65x6bx1x2x3
y > 0x4x6b p(y) > 0:x77x78x79x62x63x64x65x7ax62x73x77x78x79x7bx7cx7d p(y) = ay,
x7ex7fa > 0.
x49x70x71x54x2cx68x42x43x69x63xex4cx56x48x25x24x2bxex6dx54x33x6dx3cx61xdxdx15x5b
x7x8x9xax78xfx2x3ax3ex5axfx75x16x60x19x2ex9xax25xbx61xdx54x42x43x1ax71x78x45x31x48
35
x26x66x47x48x66x16x35x3bx75x72x54x74x75x48x10x70x64x2cx68x23x25x51x2ex64x4ax4bx78x25x67x2e
x9xax25x18x54xdx23x62x4ax55xexfx78 xdx6x5fx4axfx57xdx7dx50x4axfx2ex10x3x70x3cx4ax2ex10
(p(y) 6= ?p(?y))x6fxdx70x53x14x4axfx57xdx33x54x4axfx2ex10x48x67x2ex9xaxdx3bx5x6dx62
x4ax55xexfx3 xdx5x6dx62x4ax55xexfx6fxdx1cx1dx62x4ax55xexfx78x66x49xdx5ax18x6dx3cxfx18x48
x67x2ex9xaxdx3bx1cx1dx62x4ax55xexfx78x31x7ex9x9x78x24xbx49x69x63x44x65x63x3dx3ex5fx60
x18x48x5dx42x43x69x55x5cx48x5x0x1x45x6ex24x2bxex6dx54x7ex1ex54x33x6dx3cx61xdx5bx7x8
x9xax78xfx2x1ax49xcxdx4bx27x4bx22x1x48x3ax3ex67x23x5dxex6dx33x68x21x0x2fxbx78x3b
x31x3ax3ex49x44x70x51x2cx5dx8x48 Markowitzx2cx68x14x25x7x8x69x63x3x1ex6fx1ex37x62x4a
x55x56(CAPM)x4Rossx54APTx69x63x17x71x5bx73x47x75x1cx1dx62x4ax55xex48x2fx5e
x3ax3ex24x13x23x24x2cx68x48x49xfx70x71x54x50x27x4ax6x2ex3cx62x54x74x40x56x48xbx78x31x17
x61x62x3bx5x6dx62x4ax55xex41x7dx18x5bx57x1cx1dx62x4ax55xex9x4ax78x25x67x79x41x42x25xb
x69x63x51x2ex75x39xdx5ax18x6dx3cxfx2ex66x49x48x39x58x5ax24x3bx66x66x49xdx5ax18x6dx3cxfx54
x11xbx4fx1x3dx3ex4fx75x72x54xcxdx78x25xbx5bx2x13x3fx78x3ax3ex1ax18x58x2x79x1bx19x3a
x3ex54x28x6dx55x56x44x27x48x33x67x2x2cx68x14x25x50x27x4ax6x2x3cx62x71x68x54x74x40x78
x1ax49x3ax3ex79x61x62(x23x62x4ax55xex4)x3bx5x6dx62x4ax55xex41x7dx78x49x25x70x38
x3dx56x48x3ax3ex23x24x62x7ax70x2bx2cx68x14x25x54x20x21x27x78xd = ( 1;:::; K) x1x42
x43 M x2cx54x70x2bx14x25x48x17x50x27x4ax6 p( ￠ x) 6= 0. x33x34 r = ￠ x=p( ￠ x)
x67xcx1 x54x5x6x21x78xcx3fx2x48x2cx68xi (xbx4bx19x57x20x19x48x3ax3ex24x13x5x3bx65
i x64x2cx68x54x50x27x4ax6x27x60x27x8x2cx68x36x50) x54x20x21x27x1 ri = xi=p(xi). x17
x6ex7ax2x14x25x54x50x27x4ax6x57x17xbx4ax6x75x77x78x46x1fx79x48x25x18x21x70x54x20x21x27
x57x7ax10x44x53x3bx54x20x21x27x51x21x2ex10x78x7ax10x44x53x3bx54x20x21x27x57x7fx2x14x25x54
x50x27x4ax6x4x17xbx4ax6x75x7dx57x17xbx4ax6x75x77x48x10x57x44x29x62x7ax7bx7d?1. xf
x2x49x69x63x65x63x44x48x53x3bx3ax3exaxbx54x62x7ax51x2dx2ex69x66x75x73x78x7bx7ex48x3ax3e
x24x13x67x7ex3bxaxbx54x62x7ax78x26x31x49x7x3bx6fx51x7a(x24x58x2dx2ex7ex3bx10x26x62x7ax54
x52x2e)x2cx51x3x20x21x27x54x31x32x18x48xcx2cx1fx79x19x17x2cx54x7dx3fx78x58x51x75x1fx79x54
x2x48x26x66x3ax3ex7x62x68x69x6ax62x63x8x9xaxbx6bx7axcxdxexfx10x79x11x12x13x14x15
x79x16x65xfx10x79x11x12x13x67x7ax78x79x0x25x70x77x2x4ex25xdx7fx4axfx54x2ax78x2bx4fx1ex31
x17x70x10x26x54x2cx68x48x33x34x62x3ex54x4fx1ex20x21x27x2x70x26x54x48x58x57x62x3ex54x4fx1e
x6cx33x3x78
x17xbx4ax6x1x3bx54x14x25x54x20x21x27x6fx50x5ax23x3cx2ex11x2bx14x25x54x50x27x4ax6x78
x17xcx39x2x10x21x10x7x54x17xbx4ax6x70x62x11. x25x2x7bx1x26x3bx5x6dx62x4ax55xex48
p(r ) = p
￠x
p( ￠x)
?
= p( ￠x)p( ￠x) = 1:
x25x26x70x27x48x3ax3ex23x24x62x7a(x18x19x1a)x11x12x13x67x3x1bx63x1cx1d1x79x1ex1fx63x1cx0
x25x70x62x7ax57x3ax3ex36x18x69x1ex54x20x21x27xdx15x51x77x2ax5bx48x17x71x3bx2x26x66x31x44
x62x7ax20x21x27x48x68x4fx44x78x51x25x26x54x69x66x78x31x32x15x7ax10x27x62x7ax20x21x27x48x10
x67x36x51x25x26x54x28x73x78x4x53x1cx1dx62x4ax55xex41x7dx48x5ax4dx44x58x70x66x73x69x14x27
x2ex7ex69x66x54p(1). x24x13x48x3ax3ex67x7ex3bx25x2bx20x21x27x62x7ax78
36
x56x4bx54x28x6dx62x69x2x70xax61x7ax54x48xfx2x10x37x51x45x75x79x7ax78
x36x3f 2.1 x61x75x76x62x63x8x9x7x20x21x6bx6fx70 (x22x23x18x19x1a) xdxexfx10
x79x11x12x13x24x25x26x0
xdx27 xdx62x4ax65x68px9xar = 1=p(1). x65ix64x2cx68x54x20x21x27x1ri. x33
x34
ri = xip(x
i)
= xix
ip(1)
= 1=p(1) = r:
x4ri x57rx7bx9x78x10x66x70x71x54x2cx68x14x25 = ( 1;:::; K),x17x20x21x27 r x1
r = 1x1 +￠￠￠+ KxKp(
1x1 +￠￠￠+ KxK)
= 1x1 +￠￠￠+ KxK(
1x1 +￠￠￠+ KxK)p(1)
= r;
x17x2cx75x39 x54x50x27x4ax6x2ex1x7dx78 2
x25x70x62x69x54x76x69x79x7ax43x7fx28x6dx48xfx17x31x63x71x43x7fx45x75x78x7fx6 r =
1=p(1) xcx1x20x21x78x2ex1fx3ax3ex44x4bx54x31x63x2x49x70x2bx7ex7fx36x51x2ex3cx62x1dx54
x55x56x2cx68x19x54x78x31x32x55x56x2cx51x2ex3cx62x1dx48x4x66x49x11xbx17x50x27x4ax4bx51x2e
x64x23x25x1dx54x2cx68x48x33x34x3ax3ex5dx17x2cx54x33x2ex3cx62x1d(x4xdx33x69x6axf)x54x2cx68
x33x5bx6fx27x4dx41x70x2bx14x55x56x48x1cx25x26x1cx1dx62x4ax55xex68x19x72xdx54x31x63x2ax15
x1fx33x69x6ax2cx68x14x25x54x20x21x27x2x70x26x54x78x25x18x7bx7x54 r = 1=p(1) x53x21x3c
x73x1excx1x2cx28x29x20x21x78x49x44x70x51x2cx3ax3ex65x63x3exbx62x4ax6ex26x18x48x65x70x64
x76x77x54x16x5ex67x49x66x7ex45x33x18x54x33x69x6ax3cx27x61x62x11)x78x62x63x44x48x25x70x31x63
x3cx71x5ax2Black–Scholesx3exbx62x4ax6ex26x2cx68x2cx54x3x1dx78
2.2 x1ax49x3ax3ex27x65x63x70x2bx50x27x1cx2ex3cx62x1dx54x2cx68x42x43x55x56x78x1cx61
xdx55x56x2cx51K x64x2cx68x78x25xbx2cx68x54x50x27x4ax4bx23x25x5bx2x2ex3cx62x54x78x2cx68
x54x50x27x2ex3cx62x4ax4bx3bx25x27x63x2cx54x2ax2bx2cx2dx27x47x29x78x25x18x17x7ax5ax79x41x42
x11x64x61x62x48x7bx1x35x61x3bx28x29x2ax6cx27x47x29x2ex3cx62x1dx24x19x48x3ax3ex5ax23x25x3b
x17x62x68x4fx25x22x27x47x29x2ex3cx62x1dx78xfx2x25x27x63x2x5fx3fx15x41x1ax54x68x4fx4fx4ex78
x2ex11x1x1bx48x3dx3ex57x58x4fxcxdx2cx54x2ex3cx62x1dx48x58xfx5fx2ex3bx28x29x2ax6cx27x23
x34x78x3ax3ex25x18x21x70x54x28x29x2ax6cx67x2x1bx9x25x27x63x2cx62x7ax54x28x29x2ax6cx48x10
x23x25x47x15x1fx71x68x6x48x58x10x47x11x2bx71x68x48x76x54x6x54x25x27x23x3bx8x28x29x2ax6c
x54x33x6cx65x68x27x23x34x78x49x70xbx21x28x6dx54x55x56x2cx48x7fx61x62x50x27x73x25x51x51x2a
x64x70x71x78x45x31x48x49x44x70x51x54x28x6dx55x56x2cx48x50x27x73x51x78x64x23x25x78x25x18x21x44
x58x54x28x29x2ax6cx5ax73x23x25x47x78x2bx2ex10x54x6x78x49x25x64x74x75x56x48x68x4fx44x4dx28
x6dxfx2ex78xfx49xaxbx54x65x63x2cx3ax3ex2ex47x75x25x70x28x5cx61x62x78
37
x57x24xbx70x26x48x25K x64x2cx68x54x50x27x4ax4bx1cx60x27x1x1;:::;xK 0,xf
x25 K x2bx50x27x4ax4bx5bx2 (x3b) x28x29x2ax6cx48x58x70x71x2ex2x77x7ex3bx71x68x78x44x49
x 0 x60x27 x x47x3bx68x54x25x27x11. x2ex1fx24x13x3ax3exfx5fx3bx10x78x44x49 x > 0
xexdx49x4ex66x7dx54x25x27x1x3bx54x43x50x28x29x2ax6cx78x31x32xi x2x3bx71x68x48x6fx31x21
x3cx73x1ex70x48x2x62x7fx3bx28x29x2ax6cx48x33x34x25x67x79x41x42x17x50x27x4ax4bx2x3cx62x54x78
x7bx7ex48x8xi x67x4ex60x70x64x33x69x6ax2cx68x78x7ax58x48x49x3ax3ex56x4bx54x65x63x2cx48x2e
x61x62x42x43x2cx66x49x33x69x6ax2cx68x48x6fx31x66x49x33x69x6ax2cx68x14x25x78x25x26x54x65x63
x67x21x37x51x70x71x1dx78 K x64x2cx68x54x5x1fx3bx10x3ex54x50x27x28x29x4ax6x27x60x27x48xe
x23x3cx2ex70x2bK x75x28x29x5cx24x6cx = (x1;:::;xK)T. x57x24xbx54x2ex10x49x66x48x25
x2bx24x6cx54x33x6cx5bx2x28x29x2ax6cx78x5ex66x2cx68x14x25x6fx4fx1exex0x48x5ax1cx7ax2x70
x2bK x75x4fx24x6c = ( 1;:::; K),x17x79x7ax5ax57x24xbx70x26x78x17x33x6cx23x24x2x3c
x62x54x6cx48x5ax23x24x2x28x29x2ax6cx78x25x75x3cx65x63x54x5fx60x2x37x34x27x44x62x78x2ex1f
x3ax3ex49x25x18x62x7ax2cx68x42x43M x18x48x73x47x48x3cx62x1dx54x14x25x48x4x73x47x48 ix2
x77x7ex71x68x54x74x40x78x25x26x54 x54x7fx1fx1cx7ax2x70x2bx71x68x74x44x54K x75x24x6cx72
x76x78x10x57x44x51x2cx62x7ax54M x6fx60x44x2x70x26x54x78xfx70x2bx14x25 x54x50x27x4ax6
x53x2x62x7fx4fx24x6c x57x28x29x5cx24x6cxx54x16x2x2a
￠x = 1x1 + 2x2 +￠￠￠+ KxK: (2)
x25x2bx68x6xex2x70x2bx28x29x2ax6cx48x10x7ax7bx14x25x54x50x27x4ax6x54x2ex3cx62x1dx78x42
x43M x2cx54x14x25x54x50x27x28x29x4ax6x21x40x41x54x28x29x2ax6cx7fx1fM,x2ex40xcx10x3e
x1x5dx2fx30x10x40x36x31x6x48x4Mx62x7ax1
M = fyj9 2 M; y = ￠x:g
xdx50x62xbx21 (contingent claim)xfx25x2bx1ex79x41x42x17x50x27x4ax6x2x2ex3cx62x54x78
x58xdx23x66x16x54xfx2x5dx25x70x50x62xbx21x23x24x57x42x43M x2cx54x11x2bx14x25x7bx10x7x78
x49x21xcxdx54x5fx60x54x70x62xex4cx56x48x31x32x21x44x58x54x50x62xbx21x5bx2x23x66x16x54x48
x25x64x42x43x67x2x21x22xdx7ex7fx42x43xfx78x25x18x48x31x32x13x51x1cx1dx62x4ax55xex48x33x34
x59x70x50x62xbx21x67x5bx23x7ex1fx40x6fx2cx68x54x62x4ax27x62x4ax78x25x64x62x4ax54x5ex7ex3b
x2x3ax3ex49x44x70x51x2cx20x19x1fx54x65x8x64x76x77x78
x1ax49x3ax3ex27x68x3cxaxbx74x40x56x54x33x6dx3cx61xdx62x4ax55xex78x25x18x40x6fx44
x23x6ax32x2ex30x1ex16x33x78x1dx70x2ex10x54x2xaxbx54x62x4ax65x68x53x2ex13x2x4ex60x3cx62
x54x14x25x50x27x4ax6x54x71x68x54x65x68x48x58x2xdx23x66x16x54x50x62xbx21xfx54x50x27x28x29
x4ax6x7fx1fMx44x54x65x68x78x37x1fx1ex70x48x10x23x24x60x54x1
x34x57x58x36x59x10x2cx2dx20x2ex2fx36x28x10x54x19x5bx5cx2a
1. (x5dx36x28x5ex5f)x60x61x62x63x64x65p:M!R.
2. (x55x66x5cx36x28x5ex5f) px67x68x69x6ax64x65x6bx6cx6dx6ex6fx70x68x72x65? > 0,x73
p(?y) = ?p(y):
38
3. (x66x5cx36x28x5ex5f) p x67x69x6ax64x65x6bx6cx6dx6ex6fx70x72x65 ?, x73 p(?y) =
?p(y):
4. (x74x59x36x28x5ex5f) px67x75x76x64x65x6bx6cx6dx6ex6fx70x72x65?;?x71x6fx70y;z 2
M,x73p(?y +?z) = ?p(y)+?p(z):
5. (x55x74x59x36x28x5ex5f) p x67x68x75x76x64x65x6bx6c p x67x75x76x64x65x6bx1x2x3
x > 0x4x6b p(x) > 0:x77x35x > 0x36x37xx1dx22x38x39x3ax3bx3cx6bx1x2x15x1d
x68x79x3dx13x3ex6ex23x0
x79x7ax25x18x57xbx4bx54x48x3fxdx15x73x51x70x73x48x4x62x4ax65x68px54x62x7ax74x39
x71x68x74Rx2ax1x23x66x16x54x50x62xbx21x7fx1fM,xfx2x17x68x4fx44x54x7dx3fx58x2x75
x1fx79x54x78x17x2cx15x45x75x54x2Mx51x24x6cx72x76x54x31x4dx78x2bx71x44x48x15x16x10x54x62
x7ax48x2exfx2cx68x48x31x32y;z 2M,x33x34x10x66x15x1fy;z 2Mx4x15x1fx71x68?;?,
x5bx51?y +?z 2M. x7bx7ex48x13x4bx54x60x38x62x4ax55xex5bx51x79x7ax78
x56x4bx3ax3ex67x75x16x5ex25x26x54x69x63xex4cx27x65x63x3dx3ex1ex37x62x4ax5fx60x48x6f
x31x21x3cx73x1ex70x48x65x63x50x62xbx21x54x62x4ax5fx60x78x3ax3ex54x65x63x73x47x75x1cx1dx62
x4ax55xex78x5ex66x3bx1cx1dx62x4ax55xe(x10x4ex60xdx7ex1ex54x33x6dx3cx61xdxf)x48x49x3exb
x62x4ax5fx60x24xbx48x1ax18x36x51x3bx78x25x18x3fx66x20x40x26x31x48x49x3dx3ex57x58x4fx54x64
x67x2cx48 xdx33x6dx3cx61xdxfx7ex7fx67x5dx3bx1cx1dx62x4ax55xex78xfx2x5ax51x7bx17x2ex54x64
x67x2cx48 xdx33x6dx3cx61xdxfx3bx27x5dx1cx1dx62x4ax55xex48x17x2cx2fx5ex45x46Modigliani–
Miller x4 Merton x54x57x4fx64x67x78x7bx7ex48x49x3ex26x64x67x18x48x70x62x75x12x7bx7cx17
x2cx54xdx33x6dx3cx61xdxfxdx40x5dx37x34x78x3bx31x3ax3exbx4bx20x19x1fx54x48x49Cochrane
[2]x2cx5dx1cx1dx62x4ax55xexcx1xdx6dx70x4ax4bx55xexfx48x3bx1cx1dx62x4ax55xexcx1xdx33
x6dx3cxfx78x25x23x25x4dx59x11xfx77x78xfx2xdx6dx70x4ax4bx55xexfx25x70x7bx11x75x16x0x14
x6fx1ex78x6fx51x7ax53x7cx52x7ex3bx25x18x54xdx24x2bxex6dxfx54x70x55x78
2.3 x1ax49x3ax3ex53x5dx1cx1dx62x4ax55xex2ex1xdx6ex69xfx27x2ax7dx50x62xbx21x62
x4ax5fx60x54x69x63xex4cx78x1x7ex48x3ax3ex58x47x75x5dx44x4bx19x57x2ax7dx54xex4cx4ex70x57
x46x3cx5cx78x6cx79x3ax3ex23x1fx79x19x54x2x48x49x25xbx62x4ax55xex54x62x7ax2cx48 xdx23x66x16
x54x50x62xbx21xfx7fx1fMxdx15x77x2cx68x42x43 (x2cx68x14x25x7fx1f) M x21x45x75x48x7b
x1x49x15x13x54x62x7ax2cx48x3ax3ex73x3cx19M,x58M x19x57x34x2ex78x25x26x48x3ax3ex23x24
x18x38x39x28x29x2ax6cx54x69x25 M x8x5fx27x65x63x5fx60x48x2excx13x20x58 M. x45x75x54
x2Mx75x51x24x6cx72x76x54x31x4dx48x4x31x32y;z 2 M,x33x34x10x66x15x1fx71x68?;?,
?y +?z 2M.
x49x25x26x54x5ex7ex56x48x3ax3ex70x7axcxdxex41x42x7fx79x1ex62x43x12x44x45x6bx67x46x39
x3ax3bx3cx47x7bxax79x7axc(x72)x48x3cx44x45x0 xdx40x6fx2cx68xfxex2x5dx25x2bx50x62xbx21
x72x76x2cx54x70x2bx69x25x78x58xdx23x66x16x54x50x62xbx21xfxex2x5dx10x2x11xbx40x6fx2cx68
x54x1cx1dx14x25(x25x18x54x14x25x7x8x2combination,xfx2x5dx10x69x1ex1portfolio,
x5ax36x51x6e)x78x25x26x2ax7dx54xex4cx77xbx4bx75x77x78x6cx79x2xdx40x6fx2cx68xfx54x2bx68
39
x2ex70x62x2x51x2ax64x48x10x5ax23x24x51x33x2ax64x78x2fx5ex25x33x2ax64xdx40x6fx2cx68xfx7bx25
x76x58x23x24x2x1cx1dx33x3x54x48x4x67x5ax2ex2x17x62x2cx68x54x14x25x78x49x25x64x74x75x56x48
x50x62xbx21x72x76x53x2x70x2bx33x2ax75x24x6cx72x76x78x33x2ax75x24x6cx72x76x54x25x22x79x7a
x2ex49x1bx9x1cx1dx4ex68x52x6ax2cx51x4ax48xfx2x3bx51x2ax75x24x6cx72x76x54x69x63x12x69x1e
x10x5ax2ex4dx51x2ex5fxexfx78 xdx40x6fx2cx68xfx54x2bx68x33x2ax49x71x72x42x43x2cx2x51x23x25
x54x78x45x31x49x2cx68x42x43x2cx5dx18x76x7bx46x3x28x29x7bx46x5bx47x48x4ex12x48x2ex10x18x76x3
x2ex10x43x25x5fx4fx54x10x70x6ex66x54x6ax73x5bx23x0x1x2x2ex10x54x2cx68x48x33x34x0x1x42
x43x44x51x33x2ax64x2cx68x77x0x1x73x51x51x2ax64x2cx68x75x21x25x69x70xbx78x24x13x3ax3ex70
x71x51x2ex61x62xdx40x6fx2cx68xfx73x51x51x2ax64x78x33x26x56x17x71x5ax2ex4dx49x3ax3ex2ex5fx68
x4fx44x54xexfx78
x51x61x50x62xbx21x72x76x24x13x48x3ax3ex67x23x10x50x62xbx21x65x63x62x4ax55xex78x31
x32x1cx1dx62x4ax55xex41x7dx48x58xdx40x6fx2cx68xfx7fx5bx23x24x62x4ax48x51x6cx4ex25x1cx1dx62
x4ax55xex48x33x34x21x51x23x66x16x54x50x62xbx21x5ax5bx23x24x1dx70x62x4ax78x31x32x23x66x16
x54x50x62xbx21x54x7fx1fx67x9x66x1ex2bx50x62xbx21x72x76x48x33x34x21x51x50x62xbx21x5bx23
x24x1dx70x62x4ax78x25x18x48x67xcx8x2cx68x42x43x2x49x4ax68x69x48x35xex67xcx1x57x49x4a
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x5dxdx23x66x16x54x50x62xbx21x72x76xfx2cx54x62x4ax65x68x4bx1dx19x1ex2bx50x62xbx21x72x76x2c
x12x78x49x68x4fx44x48x10x23x60x54x1x25x26x70x2bx5fx60x2ax70x2bx16x24x6cx72x76x2cx54x1cx1d
x65x68x32x26x4bx1dx19x70x2bx45x6ex10x54x4ex24x6cx72x76x2cx12x78x10x66x25x2bx5fx60x48x3ax3e
x73x75xdx55x70x56x48x70x2bx6dx2ax6cx54x1cx1dx65x68x32x26x4bx1dx1x8x2ax6cx54x1cx1dx65x68x48
x67x23x69x1ex48x2ex1x70x2bx68x4fx5fx60x10x51x33x2ax2ex2bx1ex48x51x6cx25xbx1ex23x24xfx28
x79x78x25x10x70x2bx3dx3ex5fx60x27x70x2ex2x70x2bx28x1cx6cx78x7bx7ex48x1cx1dx62x4ax55xex10
x66x2ex7ex7fx42x43x54x50x62xbx21x5fx60x27x70x2x2ex63x54x78x25x18x48x3bx1cx1dx62x4ax55xe
x67x25x49x8x21x25x69x54x1ex1cx78xfx2x10x66x7ex7fx42x43x27x70x48x1cx1dx62x4ax55xex19x57
xax63x78x25x67x2x1x37x34x3ax3ex23x24x79x1bx2ax49x1cx1dx62x4ax55xex54x61x62x56x4ex4fx2d
x2ex45x75x54x65x63x78x24x44x25xbx3fx3cx27x36x4cx33x10x48x17x68x4fx10x75x5ax24x1x28x6dx78x7a
x58x48x10x3cx71x67x2x3dx3ex57x58x4fx68x4fxex4cx2cx54x7bx21x78x3ax3ex49x24x13x53x4bx27x4b
x1fx4dx19x25x70x77x78
x2ex1fx3fx7fx70x1bx27x48x44x4bx54xex4cx4ex40x7fx7ex70x71x78x25x26x70x71x54xex4cx2c
xfxfx68x8x70xbx51x3dx3ex79x7ax54x31x32x78x66x2x3ax3ex7fx47x75x20x54x61xdx78x25x2bx4f
x79x7x20x67x7dx1ex62x43x12x39x3ax3bx3cx79x50x51x73x52x0x70x2bx28x29x2ax6cx54x69x7dx23x34
x54x2x8x28x29x2ax6cx23x25x51x54x2ax5cx54x43x3bx78x21x22xdx50x62xbx21x28x29x2ax6cxfx67x2
x3ax3ex49x17xbx3cx70x2bx2cx68x6fx2cx68x14x25x54x50x27x2ex3cx62x4ax6x78x31x32x3ax3ex5dx10
x2ex1x70x2bx28x29x2ax6cx27x3cx48x33x34x61x62x10x54x69x7dx51x2ax2x25x69x54x78x35xex53x79
x41x42x50x27x2ex3cx62x4ax6x54x28x29x2ax5cx2cx33x51x10x78x49x25x26x61x62x56x48x3ax3ex23x24
x49x50x62xbx21x24x6cx72x76x48x0x4ex20x54x68x4fx31x4dx78x4x10x66x15x1fy;z 2M,x62x7a
x10x3ex54x75x53x1y;zx54x8x7bx20E[yz],x25x18E[￠]x60x27x28x29x2ax6cx54x68x4fx3ex2bx78
x33x34x10x67x40x41x70x2bx10x2x72x76x78x3ax3ex13x4ex70x57x54x61x62x2x2ax25x2bx10x2x72x76
x2x7ex74x54x48x4x10x2x70x2bHilbertx54x27(x51x3x54x68x4fx1bx4ax4fx43x6f)x78
40
x1x70x68x10x66x69x7dx51x2ax54x28x29x2ax6cx25x26x56x2x25x69x54x48x3ax3ex79x27x1bx64
x70x56x70x2bx28x29x2ax6cx54x32x69x7dx4x69x7dx54x62x7ax78xdy;zx1x78x2bx28x29x2ax6cx78xe
Cov[y;z] = E[(y ?E[y])(z ?E[z])] = E[yz]?E[y]E[z]
xcx1yx4z x75x76x54x55x56x57x78x17y = z x18x48
Var[y] = Cov[y;y] = E[(y ?E[y])2] = E[y2]?E2[y]
xcx1 y x54x56x57x78x10x66x15x79x54x28x29x2ax6cx27x70x48x10x54x68x4fx3ex2bx4x69x7dx5bx2e
x70x62x66x49x78x2ex1fx33x64x28x29x2ax6cx5bx2xdx58x71xfx54x78x3ax3ex1ax49x61x62x50x62xbx21
x54x69x7dx51x2ax48x2x5dx10x54x68x4fx3ex2bx4x69x7dx5bx66x49x48x51x6cx5bx47x51x2ax6x78
x78x2bx28x29x2ax6cx54x69x7dx4x32x69x7dx75x76x51x70x2bx45x75x54Cauchyx2ex9x26x2a
Cov2[y;z] 6 Var[y]￠Var[z];
x17x2cx9x49x17x6cx4cx17y = az + bx18x41x7dx48x25x18ax4bx1x15x79x7fx68x78x7bx7ex48
x17y x4z x54x69x7dx5bx51x2ax18x48x10x3ex75x76x54x32x69x7dx5ax51x2ax48x39x58x13x26x32x69
x7dx54x62x7ax48x67x23x68x19yx4zx54x8x7bx20E[yz]x5ax51x2ax78x25x26x49x26x69x7dx51x2a
x54x28x29x2ax6cx21x4dx41x54x24x6cx72x76x2cx48x67x23x10x17x2cx54x15x1fx78x2bx38x46 y;z x3b
E[yz]x27x62x7ax10x3ex54x10x2x78x75x16x52x2cx48x25x26x62x7ax54x10x2x2x4ex25x10x2x54x75
x39x54x78x10x2x54x28x73x2x0x1x61x70x64x68x4fx31x4dx78x21x3cx73x1ex70x48x10x2x53x24x6c
x72x76x2cx51x61x3ax3bx54x25x22x48x16x17x2x51x61x78x2bx24x6cxdx3bx66xfx54x25x22x78x37x1fx1e
x70x48x10x66x44x4bx62x7ax54x50x62xbx21 Hilbertx72x76Mx27x70x48x21x22y;z 2 Mx3b
x66x48x2x5dE[yz] = 0.
xdx50x62xbx21x72x76x2 Hilbert x72x76xfx25x70x61xdx49x50x62xbx21x72x76x2x26x51
x2ax2bx40x6fx2cx68x36x41x18x70x62x41x7dx48x7bx1x25x18x21x40x41x54x24x6cx72x76x70x62x2x51
x2ax75x54x48x58x51x2ax75x54 Hilbert x72x76x67x2x21x22xdx2ex55x72x76xfx48x10x70x62x2x7e
x74x54x78xfx2x3ax3ex51x18x5ax75x65x63x33x2ax75x54x50x62xbx21x72x76x78x16x17x2x49x65x63
Ross x54 APT x69x63x18x67x2x31x7ex78x15x13x3ax3ex58x47x75x61x62x48x1cx1dx62x4ax65x68
p:M !Rx2x7fx0x54x78x25x49Mx2x51x2ax75x72x76x18x2x2ex41x5fx60x54x48x7bx1x15
x1fx51x2ax75x72x76x44x54x1cx1dx65x68x70x62x2x7fx0x54x78xfx2x17 M x2x33x2ax75x72x76
x18x48x25x2bx61x62x2xcx75x54x78x62x63x44x49x65x63Rossx54x6dx3cx62x4ax69x63x18x48x10x7b
x17x66x21x22xdx7dx5dx33x6dx3cx61xdxfx78x3ax3ex53x49x65x47x51x24x4ex59x27x28x25x70x77x78
x18x44x21x29x48x2ex1x3ax3ex54x69x63x54x8x5fx77x54xdx6ex69xfx2x25x26x3x38x2a
x37x38x2ex2fx2a
1. x1ex62x43x12x44x45Mx67x7ax56x50x51x73x52x79x39x3ax3bx3cx7bxax79x48x3cx44x45x0
2. x57x58x6dx6ex6fx70 y;z 2 M, x62x59 E[yz] x1dx15x5ax79x5bx5cx6bx5dx5e M x67
Hilbertx44x45x0
41
3. x62x63x64x65p:M!Rx1dx75x76x5fx60x64x65x0
x49x25x26x54x61xdx56x48x3ax3ex7dx4x23x68x56x4bx54x40x6fx62x69x2a
x36x3f 2.2 (x2ax2bx61x1bx62x63x63x64x36x3f) x49x44x29x40x6fx61xdx56x48x66x49x1d
x70x54m 2M,x53x68x10x66x15x1fy 2M,x51p(y) = E[my]:
x25x18x54mx2x50x62xbx21x72x76Mx2cx54x38x46x48x4x10x6fx50x5ax23x24x3cx2ex70
x64x2cx68x14x25x78x10xcx1Mx2cx54x2ax2bx61x74x62x63(Stochastic Discount Factor,
SDF)x78
x2ex1x70x38x68x4fx62x69x27x3cx48x28x29x13x1ax7bx14x54x66x49x2xfx36x65x54x48x7bx1x10
x2x21x22xdRieszx60x27x62x69xfx54x18x38x76x63x78x4x53x2Rieszx60x27x62x69x5axfx28x69
x1ex78x7fx25x18x54x74x40x27x3cx48x3ax3ex23x24x25x26x27x28x29x62x69x54x2cx68x2axdR0 ‰M
x1x21x51x17xbx4ax6x1x7dx54x50x62xbx21x7fx1f(x24x13x3ax3ex53xcx10x1xdx66x4ax20x21x14
x72x76xf)x48x4x10x66x15x1fy 2 R0, x51p(y) = 0. x33x34x26x66x62x4ax65x68px2x2bx7f
x0x1cx1dx65x68x48 R0 x4dx41Mx54x70x2bx67x1cx1dx14x72x76x78x25x58x67x14x72x76x70x62x23
x24x60x27x1x57x11x2bx24x6cx68x18(x6fxdx3bx66xfx48x4x57x8x24x6cx54x10x2x1x7d)x54x24x6c
x7fx1fx78x5dx25x2bxdx68x18x24x6cxfx2ex24x17x54x3dx3bx44x54x2ax5cx48x67x23x24x68x19x1dx70x54
x28x29x13x1ax7bx14m.
x7ax58x48x2ex1x70x38x3dx3ex57x58x4fx54x62x69x48x25x38x62x69x2x79x41x22x3dx54x78x10
x70x68x48x49x70x2bx25x69x54x3dx3ex57x58x4fx54x1ex37x62x4ax69x63x54xex4cx2cx48x15x1fx62x4a
x55xex48x73x75x10x2x1cx1dx62x4ax55xex48x33x34x10x67x70x62x10x7x70x2bx28x29x13x1ax7bx14x78
x25x2bxex4cxaxbx36x51x15x1fx57x58x4fx10x75x78xfx2x15x1fx57x58x4fx65x63x68x19x54x62x4a
x55xex48x73x75x10x1cx7ax2(x7fx0)x1cx1dx54x48x33x34x10x70x62x51x7bx7x54x28x29x13x1ax7b
x14x78x45x31x48x6x4fx54xdxdx4bx69x63xfx0x1x48x70x43x6ex36x54xdx4bx67x2x10x46xdx31x27
x70x50x27x4ax6x54x68x4fx3ex2bx1x7dx54xdx4bx78x7bx7ex48x5dx70x43xdx4bx54x50x27x4ax6x54
x68x4fx3ex2bx2ex1xdx4bx54x17xbx4ax6x48x33x34x17x7bx7x54x28x29x13x1ax7bx14x67x2x7fx681
(x31x321 2 M)x78x7fx31x48 Black–Scholesx3exbx62x4ax69x63x2cx48x15x13x54x3exbx62x4a
x2x3exbx54x50x27x4ax6x54x10x11x64xdx9x4ax25x27x9x51x3bxfx54x68x4fx3ex2bx78x25x64xdx9
x4ax25x27x9x51x3bxfx53x6ax73x54x17xbx4ax6x68x28x2x17(x13x1a)x50x27x4ax6x54x68x4fx3e
x2bx78x26x7ex5ax23x62x8x11x64x10xdx68x76x25x27x51x3bxfx54x28x29x13x1ax7bx14x78x17x62x31x59
x2bx4fx1ex31x10x42x43x44x54x74x64x40x6fx2cx68x5bx51x16x5ex36x57x6dx6ex54x7cx6ax24x58x7x28
x54x62x4ax78x25x64x62x4ax17x71x5ax2x49x8x61x70x2bx37x51x8x4fx1ex31xcx39x54x28x29x13x1a
x7bx14x78x4fx1x3dx3ex4fx2cx54x2dx2ex65x63x5ax2ex1fx2x49x8x57x57x4fx65x63x2ex10x54x28x29
x13x1ax7bx14x78x31x7ex9x9x78
x25x26x70x27x48x3ax3ex67x2ax7dx61x70x2bx65x63x3dx3ex1ex37x62x4ax5fx60x54x68x4fxex4cx78
x49x25x2bxex4cx2cx48x50x62xbx21x25x2bx25x22x57x69x7dx51x2ax54x28x29x2ax6cx9x10x14x27x78
x58x50x62xbx21x7fx23x45x46x2cx68x3x2cx68x14x25x3x3exbx3x1ax36x2cx68x3x3dx3ex1ex37x9x9
x25x22x78x58x51x49x3ax3ex54x51x7ax23x25x2ex3bx48xfx2x49x3dx3ex71x2x54x52x2ex2cx57x7fx53
42
x3bx54xdx5x69(position)xfx5ax23x5cx49x17x2cx78x58x28x29x13x1ax7bx14x66x49x62x69x58x40x3a
x3ex48x2fx5exdx62x4ax55xexfx5ax9x10x66x11x2bx50x62xbx21x6fx69x7dx51x2ax54x28x29x2ax6cx78
xfx2x25x18x54x21x22xdx9x10xfx3 xdx45x46xfx5bx2x5dx10x3ex49x73x74x44x5bx2x70x2bx69x7d
x51x2ax54x28x29x2ax6cx48x58x51x2ex2x5dx10x3ex49x75x76x44x5ax2x9x10x54x78x4x53x49x68x4f
x2cx48x24x6cx57x28x29x2ax6cx5ax44x2ex49x10x29x44x10x7ax48x58x73x2x49xaxbx25x26x54xex4c
x2cx48x70x2bx28x29x2ax6cx67x23x3cx41x70x2bx24x6cx72x76x2cx54x24x6cx48x53x68x51x3x24x6cx72
x76x54x31x32x5bx25x33x19x25x64x74x75x56x54x28x29x2ax6cx44x27x78
x1ax4ex68x4fx59x0x1x70x2bx68x4fx25x22x2x26x62x7ax10x54x6ex69x1fx4cx27x44x62x54x48
x58x57x10x21x37x34x53x79x33x3x78x4ex68x4fx59D. Hilbert (1861–1945) (x3ax3ex19x57xc
x19x1fx24x62x4x53x54x77x44x24x6cx72x76x48x2ex1fx33x17x7ax2x13x7dx25x26x4x53x54) x67x70
x1fx48x49x62x54x3x1fx6ex69x1fx4cx2cx48x5dxdx77x3x1cx3x4bxfx21x41xdx6ax14x3x6bx6cx3x6dx6e
x6fxfx48x51x2ex13x14x6ex69x1fx4cx6fx50x78xfx2x25x9x3fx23x25x73x10x68x4fx59x51x3ax48x7bx1
x62x3ex73x3x21x62x3ex21xcxdx54x70xbx25x22x54x7bx25x75x76x54x36x37x3x4cx78x58x10x70x2b
x77x7ex7dx27x70x48x25x26x70x17x53x24x13x48x62x6cx79x4dx55x19x54x2xdx6ax14x3x6bx6cx3x6dx6e
x6fxfx57xdx77x3x1cx3x4bxfx54x10x29x52x7ax2ex10x78x1ax4ex4ex4fx1fx4cx54x7bx7dx31x3ex49xf
x4ex6ax3bx44x38x11x61x68x4fx59x54x25x64x70x7ax10x48x53x68x62x3ex49x20x8x20x7bx11x44x2ex4e
x47x48x77x7ex7dx4dxdx2bx64x36x7axfx78x31x32x57x7fx5fx36x54x2x48x25xbx20x7bx11x6fx31x12x7d
x2ex1bx21xfx48x6fx31x53x7dx28x79x61x551x78
x5ex66x13x4bx25xbx3dx3ex53x1ex49x10x29x44x21x2x25x2ex7bx10x48xfx2x49xaxbx54xe
x4cx2cx48x10x3ex7fx41x61x19x2ex44x7bx10x54x28x29x2ax6cx48x51x6cx23x24x10x10x3ex4ex4fx24x6c
x5x12x78x17x7ax48x3ax3ex5ax23x24x49x68x4fx44x2fx5cx53x77x31x2cx68x57x3exbx51x21x48x3fx78xf
x2x73x75x1fx79x70x2bx25x22x54x19x2ex4x10x29x44x54x2ax10x24x58x21x65x63x54x5fx60x54x74x60x48
x25x64x48x3fx51x2ex2x70x62x75x0x1x54x78
2.4 x51x61x28x29x13x1ax7bx14x24x13x48x3ax3ex7dx4x23x24x68x19x70xbx1ex37x62x4ax54
x40x6fx1dx60x78x2bx71x44x48x26x32x69x7dx54x62x7ax48x3ax3ex51
Cov[y;m] = E[my]?E[m]E[y]:
x7bx7ex48x26x40x6fx62x4ax6ex26x48x23x68
p(y) = E[m]E[y]+Cov[y;m]: (3)
x1x61x70x68x25x2bx60x27x26x54x57x58x79x7ax48x3ax3ex79x27x0x1xdx33x69x6ax2cx68xf (x6fxdx33
x69x6axbx21xf)x78x49x44x4bx62x7ax50x62xbx21x72x76Mx18x3ax3ex36x51x22x23x20x58x33x69x6a
x2cx68x78x21x22x33x69x6ax2cx68x2x5dx17x50x27x4ax6x2x3cx62x7fx68(x6fx31x21x3cx73x1ex70x48
x9x66x7fx68x54x25x27x11)x54x2cx68(x6fx31x21x70x71x1ex70x48 xdx2cx68x14x25xfx3 xdx50x62
xbx21xfx9x9)x78x70x2bx42x43x2cx2ex70x62x51x33x69x6ax2cx68x78x45x31x48xaxbx54x2cx56x6a
1x3bx14x1cx18x20x19x5x47x4x18x2xdx37x51x79x20x10x21x13x4fx13x20 x28x37x4fx25 x44x16x4fx25 x5cx62x4fx29x2b
x61x36x63x2ex24x25x7ax61x62x28x1ax20x3bx25x67x19x47x7x66x5x4fx10x4fxdx13x65x4dx29
43
x42x2cx48x67xfxfx60x19x70x64x33x69x6ax2cx68(x56x19x7ex7fx23x7ex3bx5fx3ex56x71x68x2ex1x33
x69x6ax2cx68)x78x25x26x48x44x4bx62x7ax54x50x62xbx21x42x43Mx2cx5ax2ex70x62x75x51x33x69
x6ax2cx68x78x67x9x3fx70x48x7fx681x23x25x2ex2Mx54x38x46x78
x49x7ex7fx74x75x56x48x33x69x6ax2cx68x54x2ex3bx2x1x1ex37x62x4ax5fx60x2ax7dx70x2bx7b
x10x6bx6cx6dx6ex78x67x5ex5cx76x57x58x54x12x6bx2cx48x3ax3ex7fx7fx75x5dx50x51x6bx6cx6dx6ex13
x12x1x11x70x46x54xdx2ex2ax4ax4bxfx27x6bx12x48x78x25x53x57x58x54x2ax5cx23x24x7bx25x77x77x78
x31x32x33x69x6ax2cx681 2M,x33x34x10x54x4ax4bx67x2p(1) = E[m]:x3ax3ex49x24x13x5
x61x62E[m] 6= 0. x35xex4dx40x41xdx17xbx2ex6x5bx48x50x27x1ax62x6x5bxfx54xdx6dx3cxb
x21xfx78x25x26x48x33x69x6ax2cx68x54x20x21x27x7x8x21=p(1) = 1=E[m] = rf. x24x13x3a
x3ex67x3brf x27x60x27x33x69x6ax3cx27x48x51x6cx75x1fx79x19x10x2x70x2bx7bx10x66x11x2bx62x4a
x65x68x27x70x54x7fx68x78x10x57x42x43x2cx2x35x66x49x33x69x6ax2cx68x36x51x3x4cx78x40x6fx62
x4ax6ex26(3)x7bx58x7fx23x1ex41
p(y) = E[y]r
f
+Cov[y;m]: (4)
x25x2bx60x27x26x67x5dx70x2bx2cx68x6fx70x2bx50x62xbx21x54x17xbx4ax6x33x1ex1x78x32x33x78
xbx70x32x49 E[y]=rf x2x10x54x26x27x28x29x48x4x10x54x50x27x3ex2bx4ax6x10x33x69x6ax3c
x27x54x13x1ax57x13x70x32x33 Cov[y;m] xex2x10x54x28x29x28x29x78x10x2x26x66x50x27x4a
x6x23x25x51x54x28x29x8x30x21x0x14x54x48x23x24x3bx27x1ex4dx1x37x34x6ax73x54x17xbx4ax6
x4dx57x69x68x54x17xbx4ax6x51x21x2ex10x78x25x70x4ax6x2x50x27x4ax6y x57x28x29x13x1ax7b
x14 m x54x32x69x7dx78x25x70x32x69x7dx17x7ax5ax23x25x1x7dx78x26x7ex3ax3ex5fx1ax70x4ex72
xfx51x79x5ex54x50x62xbx21x48x10x3ex67x2x33x69x6ax4ax6x54x50x62xbx21x78x13x47x48x19x20
x21x27x54x62x7ax48x3ax3ex58x23x5fx1ax25x72x33x69x6ax4ax6x54x50x62xbx21x54x3ex2bx20x21x27
E[ry] = E[y=p(y)] = rf. x25x70x31x32x77x62x692.1x54x79x7ax21x0x77xbx78
xfx2x31x32x42x43x2cx2ex66x49x33x69x6ax2cx68x48x33x34x3ax3ex8x32x26x27x73x69x33x69
x6ax3cx27x58x17x51x3x3x4cx75x7ex25x2bx5fx60x7ex39x3dx3ex4fx54x3ax3bx27x47x48x48xdx15xf
xfx1ex44x78x7ax58x48x31x32x39x3ax3ex54x68x4fxex4cx27x47x48x5fx60x48x3ax3ex4dx5fx1ax48x26x7e
x23x68x19x70x2bxfx51x79x7ax54x25x22x78
x2bx71x44x48x7fx681x79x7ax2ex2Mx54x38x46x48xfx2x26x66Mx2cx54x38x46x5bx2x69
x7dx51x2a(x10x58x2x6ex68x4fx3ex2bx51x2a)x54x28x29x2ax6cx48x3ax3ex23x24x10x15x1fy 2M
x39x10x54x68x4fx3ex2bE[y]. x25x2bx68x4fx3ex2bx48x491 2 Mx18x23x3cx2e 1x57y x54x10
x2x78x58x491 =2 Mx18x48x10x7fx23x24x3cx2e Mx44x54x70x2bx7fx0x1cx1dx65x68x78x66x2
x3ax3ex13x26Rieszx60x27x62x69x23x1bx48x66x49x1dx70x54x38x461M 2 M,x53x68x10x66x15
x1fy 2M,x51E[y] = E[1My]:
x25x2b1M xcx1x2cx28x29x38x39x10x15x72x73x3dx22(mimic portfolio)x16x78x10x54x6e
x7ax2x17x42x43x26x40x41x40x6fx2cx68x36x41x18x48x25x2x2bx55x74x33x69x6ax2cx68x20x25x54x2cx68
x14x25x78x17x7ax48x10x6fx50x51x2ex70x62x2x33x69x6ax2cx68x48x35x43Mx2cx66x49x33x69x6ax2c
x68x78x17x33x69x6ax2cx681 =2Mx18x48x25x2bx33x69x6ax2cx68x54x55x74x14x251Mx49x2dx2ex1e
44
x69x5bx23x24x14x33x69x6ax2cx68x54x2ex3bx78xfx10x54x20x21x27x2ex13x21=E[m],x58x2x2ex3c
x62x541M=E[m]. x6x68x1fx79x54x2x48 1Mx51x31x56x1dx60x2a 0 < E[1M] < 1:x25x2x7b
x11Mx2x2bx43x7fx68x28x29x2ax6cx48x21x24x10x54x69x7dVar[1M] = E[12M]?E2[1M] >
0:xfE[12M] = E[1M]. x7bx7ex48x44x26x4cx170 < E[1M] < 1x18x78x51x23x25x78x16x17
x2x33x69x6ax2cx68x55x74x14x25x54x3ex2bx20x21x27x75x67x4bx16x66x33x69x6ax3cx27x78x25x2x43
x7fx51x79x5ex54x2bx78x10x5ex5fx49x69x63x44x58x40x3ax3ex48x49x70x2bx36x51x33x69x6ax2cx68x54
x42x43x44x48x2exfx66x49x70x2bx33x69x6ax3cx27x48x51x6cx58x51x70x2bx3ex2bx20x21x27x2ex9x66
x33x69x6ax3cx27x54x33x69x6ax2cx68x54x43x4ex5x78x10x66x2cx56x6ax42x27x70x48x25xdx15x2x70
x2bx6x68x56x71x2cx33x14x54xcxdx20x60x78
x51x61x33x69x6ax2cx68x55x74x14x25x24x13x48x3ax3ex58x23x1fx79x19x48x50x62xbx21x54x69
x6ax4ax6x5ax10x7x11x2bxdx55x74x14x25xfx78x2bx71x44x48
Cov[y;m] = E[my]?E[m]E[y] = E[(m?E[m]1M)y]:
x4x50x62xbx21x54x69x6ax4ax6x5ax2x70x2bx50x62xbx21x72x76x44x54x7fx0x1cx1dx65x68x48x17
x10x7x54xdx55x74x14x25xfx2Cm = m?E[m]1M. x7bx7ex48x21x22x33x69x6ax4ax6x54x50
x62xbx21x48x5ax67x2x57Cmx3bx66x54x50x62xbx21x78x1fx79x19x25x70x77x48x3ax3ex67x4dx5fx1a
x25x64x50x62xbx21x51x2ex59x4fx78x49xdx6x4fxdx4bx63xfx54x74x40x56x48x0x1x50x62xbx21x54
x17xbx4ax6x67x2x10x54x68x4fx3ex2bx78x25x18x48 p(y) = E[y], m = 1 (x6fx31x21x70x71x54
x74x40x48 m = 1M, xf E[m] = 1)x78x66x2x15x1fx50x62xbx21x5bx2x33x69x6ax4ax6x54x78
x49mx2x33x69x6ax2cx68(mx2x7fx68x48xfx2ex70x62x11)x6fx31mx571M x63x1c(x4
m = fi1M, fix1x71x68)x54x74x40x56x48x1cx7ax31x7ex78x49mx2x571M x2ex63x1cx54x69
x6ax50x62xbx21x54x74x40x56x48x3ax3ex23x24x1fx79x19x48x31x32E[m] 6= 0,x33x34x70x62x66x49
m0 2 M, x53x68 Cov[m;m0] = E[Cmm0] = 0, E[mm0] 6= 0, E[m0] 6= 0. x2bx71
x44x48x25x18mx571Mx54x1cx1dx14x25x25x0x41x70x2bx8x75x72x76x78 Cm = m?E[m]1M
x2x25x2bx8x75x72x76x2cx54x70x2bx57 m x2ex63x1cx54x24x6cx78x7bx7ex48x49x25x2bx8x75x72
x76x2cx57 Cm x3bx66x54x43x7dx24x6c m0 (x25x26x54x24x6cx17x7ax66x49) x70x62x57 m x2ex3b
x66 (x46x26x31x3bx67x4bx54x1cx1dx4ex68x43x55x2cx68x25x77)x78x4x66x49 m0 2 M, x9xa
E[Cmm0] = E[mm0]?E[m]E[m0] = 0, E[mm0] 6= 0. x25x26x54x24x6cx17x7ax58x75
x9xaE[m0] 6= 0:
x44x4bx54x15x13x70x42x63x2cx7bx17x45x75x78x10x53x2x3ax3ex49x56x4bx2cx68xdx7d-fl x1e
x6fx1ex37x62x4ax55x56(zero-beta CAPM)xfx18x54x3x1dx78x56x4bx3ax3ex58x4dx49x8x25x42
x63x2cx54x21x67x4bx54x76x29x78x25x2Black [1]x491972x46x2cx68x25x38x62x69x54x70x2bx3
x1dx4x60x78x18x19x1ax49x1x1bx48x3ax3ex1cx7ax23x24x49x70xbx52x4exex44x3cx19x72xdx4x60
x54x1x2x2cx68x78x58x3ax3ex3bx21x71x54x68x4fx76x77x27x3cx25x70x5fx60x48x67x5fx1ax25x2bx31
x32x4cx4cx2x70x2bx28x6dx54xdx36x4bx3x1fxfx5fx60x78
2.5 x70x14x20x21x27x48x3ax3ex13x49xaxbx25x2bx68x4fxex4cx2cx5dx10x4ex70x57x68x3cx78
xd M x1x50x62xbx21x72x76x48 m x1x17x28x29x13x1ax7bx14x78x33x34x10x66x15x1fx17xbx4a
x6x43x7dx54x50x62xbx21 y 2 M (p(y) = E[my] 6= 0), x62x7ax17x5x6x21x1 ry =
45
y=E[my]. x26x66Mx2Hilbertx72x76x48x46x3ax3ex5ax51ry 2M,x4ry x5ax2x70x2b
x50x62xbx21x48x17xcx77x49x66p(ry) = p(y)=p(y) = E[my]=E[my] = 1,x4x15x1fx20
x21x27 ry x54x17xbx4ax6x5x1 1. x7ax75x48x15x1fx17xbx4ax6x1 1 x54x50x62xbx21x48x5a
x70x62x2x11x2bx50x62xbx21 (x45x31x36x50)x54x20x21x27x78x25x26x48x3ax3ex62x7a
R1 = fr 2Mjp(r) = E[mr] = 1:g;
x51xcx10x1x5x6 (x21) x75x76x33x78x66x36x4bx2x60x75x72x76x2cx54x36x4bx25x22x54x76x77x78
x60x75x72x76x2cx54x36x4bx2x8x75x14x72x76(x10x7ex1fxax38x6ax77)x54x70x2bx36x77x78x66x36
x4bxex2x77x7fx72x76x54x75x68x5fx70x75x54(x67)x14x72x76x54x36x77x78x49x25x18x48x7bx7x54
x14x72x76x2
R0 = fr 2Mjp(r) = E[mr] = 0:g;
x4x17xbx4ax6x1x7dx54x50x62xbx21x7fx1fx78x15x47x70x2bx20x21x27 ry, x3ax3ex7dx4x23x24
x3cx19x48x10x66x59x2b rz 2 R1, x5bx51 rz ? ry 2 R0. x7ax75x5ax70x26x78x25x67x2x70x48
R1 = R0 + ry;x4x10x2R0 x54x36x77x78x26x66x51x25x26x54x3x4cx48x3ax3ex5axcR0 x2
x75x13x5x6(x21) x63x54x27x48x7bx1x17x2cx54x38x46x5bx2x7bx10x66x11x2bx20x21x27x54x7dx4a
x40x41x54x78x25x18x4x24x13x48x3ax3ex53x57x7fx5dxdx27xfx79x37x0x78
x47x75x1fx79x54x2x48x20x21x66x36x4bR1x2ex2x1cx1dx72x76x48x4x31x32y;z 2 R1,x3a
x3ex68x2ex19?y+?z 2 R1:xfx2x31x32?+? = 1,x33x34x3ax3ex67x51?y+?z 2 R1:
x24x6cx72x76x2cx51x25x26x1dx60x54x14x69xcx1x73x78x79x78x66x36x4bx2x24x6cx72x76x2cxdx15
x4ex54xfx74x7ax3bx14x69x48x4x3bx45x6ex10x54x74x7ax69x70x62x2x7fx72x76x78
x20x21x66x36x4bR1 x10x3dx3ex57x58x4fx69x63xcx3fx45x75x78x2bx71x44x48x5x3ex54x3dx3e
x4fxcxdx5bx2ex39x2cx68x6fx2cx68x14x25x54x4ax4bx8x5fx27xcxdx54x48x58x5bx39x2cx68x6fx2c
x68x14x25x54x20x21x27x8x5fx27xcxdx78 Markowitzx2cx68x14x25x7x8x69x63x3x1ex6fx1ex37
x62x4ax55x56x9x5bx2x31x7ex78
2.6 x49x25x70x51x54x15x13x70x51x48x3ax3ex27x5dx8x1cx1dx62x4ax55xex57x1ex6fx1ex37
x62x4ax55x56x54x9x4ax1dx78x1x7ex48x3ax3ex79x27x70x70x37x34x2x1ex6fx1ex37x62x4ax55x56x78x57
x62x4ax55xex2ex10x54x2x48x1ex6fx1ex37x62x4ax55x56x2x3bx2cx68x14x25x54x3ex2bx20x21x27x27
x60x27x54x70x64x62x4ax6ex26x78x25x2bx6ex26x23x24x25x26x7ex1fx20x21x66x36x4bR1 x27x60x54x2a
x66x49ru 2 R1 x4x71x68fi 6= 0,x53x68x10x66x15x1fr 2 R1,x51
E[r]?fi = Cov[r;ru]Var[r
u]
(E[ru]?fi):
x3ax3ex79x27x1ex5ax70x73x2ax5bx48x24x61x1ex1cx1dx62x4ax55xex57x1ex6fx1ex37x62x4ax55x56x75
x76x54x3x4cx78x25x2bx2ax5bx67x2x15x28x6dx54x75x68x11x54x28x29x50x62xbx21x72x76x78
x61xdMx54x75x68x11,x33x34x49Mx2cx66x49x70x2bx43x7dx28x29x2ax6cz,x53x68
x15x1fx 2Mx5bx23x60x27x1x = ?xz,x17x2c?x 2Rx2x71x68x78x25x18x48 Mx2cx54x43
46
x7dx62x4ax65x68px31x32x2x1cx1dx54x48x70x62x23x60x27x1p(x) = E[mx] = ?m?xE[z2],
x17x2c m = ?mz; x6c ?m 6= 0. x47x48 ?x 6= 0 x54x74x40x78xe x x54x20x21x27 rx =
x=p(x) = ?xz=?m?xE[z2] = z=?mE[z2]: x58 m x54x20x21x27 rm = m=p(m) =
?mz=?2mE[z2] = z=?mE[z2]:x4Mx54xdx20x21x66x36x4bxf R1x67x73x51x70x2bx38x46x78
x26x7ex23x76x8x7bx7x54xdx1ex6fx1ex37x62x4ax55x56xfx41x7dx48x4
E[rx]?fi = flrxrm(E[rm]?fi); flrxrm = Cov[rx;rm]=Var[rm] = 1; (5)
x25x18fix23x47x15x79x43x7dx6x78
x7ax75x48xdx3bx5x6dx62x4ax55xex41x7dx48x4x7fx0x62x4ax65x68px10x66x15x1fx 2M
x4x15x1fx3bx68 ? > 0, x51 p(?x) = ?p(x): xfx2x2ex70x62x51 p(?x) = ?p(x). x10
x18x48 px58x9xap(§1M) 6= 0:x25x18x48x31x32x66x49ru 2 R1 x4fi 6= 0,x53x68x10x66
x15x1fr 2 R1,x51
E[r]?fi = Cov[r;ru]Var[r
u]
(E[ru]?fi);
x33x34x10x66r = 1M=p(1M)x4r0 = ?1M=p(?1M),x51
E[1M]
p(1M) ?fi =
Cov[1M;ru]
p(1M)Var[ru](E[ru]?fi);
E[?1M]
p(?1M) ?fi =
Cov[?1M;ru]
p(?1M)Var[ru](E[ru]?fi):
x39x58x48
E[1M]?fip(1M) = Cov[1M;ru]Var[r
u]
(E[ru]?fi);
?E[1M]?fip(?1M) = ?Cov[1M;ru]Var[r
u]
(E[ru]?fi):
x26x7ex67x1fx68x48 p(1M) = ?p(?1M): x13x0x44 p x2x7fx0x3bx5x6dx65x68x24x58 M
x21x75x54x48x7dx4x23x68px2Mx44x54x1cx1dx65x68x78
x25x26x48x3ax3ex67x49 1 x75x28x29x50x62xbx21x72x76x44x48x49x3bx5x6dx62x4ax55xex41
x7dx4p(§1M) 6= 0x54x61x62x56x48x2cx68x61x1cx1dx62x4ax55xex57x1ex6fx1ex37x62x4ax55x56
x54x9x4ax1dx78
1 x75x54x74x75x2x36x65x54x78xfx2x39x2cx3ax3ex23x24x3cx19x70x2bx1cx1dx62x4ax65x68
x32x26x57x70x2bx3ex2bx20x21x75x76x54x9x26x9x4ax78x17x2cx24x56x70xbx38x3dx2x45x75x54x2a 1.
x49x62x4ax65x68x54x1cx1dx60x27x2cx48x28x29x13x1ax7bx14 m x43x7dx48x51x6c E[m] 6= 0, x39
x58x23x39x8x17x7bx7x54x20x21x27 rm = m=E[m]. x58x1x1fx8x25x26x54x31x32x48x10x62
x4ax65x68x54x7bx7x61x62x23x7ex1fxdpx2x3bx5x6dx62x4ax65x68x48x6cp(§1M) 6= 0xfx27x23
x34x78 2. x44x58x54x7fx68fi 6= 0. x25x2bx7fx68x17Mx1x2ex75x18x48x10x10x7x57ru x2ex7b
47
x3x54x20x21x27 rv x54x68x4fx3ex2bx48x4fi = E[rv] 6= 0, Cov[ru;rv] = 0. x35x61x25x78
x77x24x19x48x17x62x76x1fx5bx23x3fx67x76x5cx58x68x78
x70x71x54x74x40x17x7ax2ex7x8x10 M x54x75x68x51x15x1fx2ax10x78xfx2x31x32x3ax3e
x39x24x6cx72x76x54x76x77x27x3cx48x25x38x9x4ax62x69x17x71x73x44x58x50x62xbx21x54x68x4fx3e
x2bx4x50x62xbx21x54x17xbx4ax6x78x64x6cx78x50x62xbx21x54x68x4fx3ex2bx23x3cx2ex50x62xb
x21x57 1M x54x10x2x48x58x50x62xbx21x54x17xbx4ax6x2x50x62xbx21x57x28x29x13x1ax7bx14
x54x10x2x78x13x47x48x19x10x2x54x3x1fx79x7ax71x72x44x2x70x2bx24x6cx19x4dx70x2bx24x6cx54
x7ax13x54x3dx3bx48x25x67x2x70x48x2ex23x50x62xbx21x72x76x54x75x68x2x2ex5fx78x15x13x73x44
x58xdx33x69x6ax2cx68x54x55x74x14x25xf 1M x57xdx28x29x13x1ax7bx14xf m x21x0x41x54x8
x75x72x76x48x17x62x50x62xbx21x5bx73x2x11x7ax13x19x25x2bx8x75x72x76x2cx27x47x48x78x56x4b
x2cx68x54x10x70x71x74x40x54x9x4ax1dx62x69x6fx60x44x73x2x70x38x8x75x72x76x2cx54x62x69x78
x7bx7ex48x1x7cx2cx72x76x2ex5dx5cx1 1 x75x48x3ax3ex58x47x75 m x57 1M x2ex63x1cx78x25
x70x38x3dx23x26 m x54x3ex2bx20x21 E[rm] = E[m=p(m)] = E[m]=E[m2] x57 1M
x54x3ex2bx20x21 E[r1M] = E[1M=p(1M)] = E[1M]=E[m] x2ex7bx9x27x1fx8x57x4
E[m2]E[1M] 6= E2[m]:x25x2bx38x3dx491M x1x33x69x6ax2cx681x18x48x79x41x42mx2e
x2x33x69x6ax2cx68x78
x36x3f 2.3 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:
x5dx5ex21x7fx0xcx1x2x26x63x7d
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];
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]); (6)
x7ex7frv 2 R1,xaxbE[rv] 6= 0, Cov[ru;rv] = 0, E[rv] 6= E[ru].
xexfx67x6bx57x58x41x42x7fx60x61x18x19x1axdxex6bx6c1 2 M,x5dx5ex10x73(x6x7
x6x8x62x63x7dx9x6b CAPM)
E[r]?rf = Cov[r;ru]Var[r
u]
(E[ru]?rf); (7)
x7ex7frf = 1=p(1) = 1=E[m]x1dx18x19x1ax5fx13x0
48
x13x14x6bx31)x152)xaxbx4x6bx47x60u = am + b1M,x7ex7fa 6= 0x71bx1d
x6fx16x72x65x6bx1x2x6fx70x73x7cx17x7bx7cx79ux79x11x12x13ru 2 R1 x24xaxb2),x7ex7f
x18x7ex67ru = rm = m=p(m)x4x6b 2)xaxbx0
xdx27 x61x62x38x3d1)x41x7dx78x33x34x10x66x15x1fx51p(x) 6= 0x54x2cx68xx27x70x48
x17x3ex2bx20x21x27 E[r]x9xa
E[r] = E[x]p(x) = E[x]E[mx]:
x12m0 2 M x9xa Cov[m;m0] = 0. x17x61x62p(1M) = E[m] 6= 0 x18x48x25x26x54
m0x2x66x49x54x48x51x6cx58x23x75x39p(m0) = E[mm0] 6= 0; E[m0] = 1:x2bx71x44x48x12
m0 = cm+d1M. xex26x38x3d
E[m0] = cE[m]+dE[1M] = 1
E[mm0] = cE[m2]+dE[m] = E[m]
x7dx4x23x68
c = E[m](E[1M]?1)E[1
M]E[m2]?E2[m]
; d = E[m
2]?E2[m]
E[1M]E[m2]?E2[m]:
x25x18x33x61x2ex1x7dx54x38x3dx26E[rm] 6= E[r1M]x68x19x78
x10x18x48x10x66rm = m=p(m) = m=E[m2], r0m = m0=p(m0) = m0=E[mm0]
x24x58r = x=E[mx] 2 R1;x3ax3ex51
E[r]?E[rm0] = E[x]E[mx] ? E[m
0]
E[mm0]
=
? E[x]
E[mx] ?
1
E[m]
?
?
? E[m0]
E[mm0] ?
1
E[m]
?
= E[x]E[m]?E[mx]E[mx]E[m] ? E[m
0]E[m]?E[mm0]
E[mm0]E[m]
x26x32x69x7dx54x62x7ax24x58x68x4fx3ex2bx5x12x54x1cx1dx1dx48x51x1fx79x19Cov[m;m0] = 0,
x7dx4x23x68
E[r]?E[rm0] = ? Cov[x;m]E[mx]E[m] = ?E[m
2]
E[m] ￠Cov
? x
E[mx];
m
E[m2]
?
= ?E[m
2]
E[m] Cov[r;rm]; (8)
xcx3fx2x48x17x = mx18x48x26x69x7dx54x62x7ax48x3ax3ex51
E[rm]?E[rm0] = ?E[m
2]
E[m] Var[rm] (9)
49
x77x77(8)x4(9)x48x3ax3ex7dx4x68x19
E[rx]?E[rm0] = Cov[rm;rx]Var[r
m]
(E[rm]?E[rm0]); (10)
x4x10x66m = ux51x38x3d2)x41x7dx78
x7ex19x48x3ax3ex58x23x1fx79x19x48 E[rm] = E[m]=E[m2] 6= 0;E[rm0] = 1=E[m] 6=
0; E[rm]?E[rm0] = ?Var[m]=E[m]E[m2] 6= 0.
x7ax75x48x61x62x38x3d2)x41x7dx78x33x34x3ax3ex6cx79xdx55x5dx3bx20x21x27x60x27x54x9
x26(6)x3ax67x1x3bx62x4ax65x68x4x2cx68x54x68x4fx3ex2bx54x9x26x78x1x7ex48x3ax3ex1fx79x19
x17E[rv] 6= 0x18x48x3ax3ex26x38x3d2)x23x1fx68x62x4ax65x68px2x5x6dx65x68x48x4x10x2e
xfx2Mx44x54x3bx5x6dx65x68x48x51x6cx10x66x15x1fx 2 M, x9xap(x) = ?p(?x).
x25x70x76x1fx23x7ex7fx31x10x6fx51x6fx7cx18x10x70x75x72x76x54x2cx68x70x26x4ex4fx78
x25x26x70x27x48x3ax3ex2ex40x61xd r = x=p(x), ru = u=p(u), rv = v=p(v),
E[x] = E[u] = E[v] = 1. x35xex48x3ax3ex73x47x3b ?x = x=E[x], ?u = u=E[u] x4
?v = v=E[v]x27x47x4ex, ux4v,x4ex4fx10x26x54x65x63x78x66x2x9x26(6)x67x2ax1
1
p(x) ?
1
p(v) =
E[xu]?1
p(x)p(u)
E[u2]?1
p2(u)
1
p(u) ?
1
p(v)
?
:
x7bx7ex48
p(v)?p(x) = E[ux]?1E[u2]?1(p(v)?p(u)):
x3ax3ex75x7x47x54 m = a0u + b01M x2x64x2bx10x9xa p(x) = E[mx]. x16x17x2
p(v) = E[a0uv + b0v1M], p(u) = E[a0u2 + b0u1M]. x7bx7ex48 a0;b0 x23x39x56x5cx69
x6ax1ex68(x1fx79x2a E[u] = E[v] = 1, E[uv] = E[u]E[v] = 1)x2a
p(v) = a0 +b0; p(u) = a0E[u2]+b0:
x25x2bx69x6ax14x54x4fx5cx26x1 1 ? E[u2] = E2[u] ? E[u2] = Var[u] 6= 0. x7b
x7ex48x69x6ax14x51x1ex78x10x18x48x7bx1 E[ru] 6= E[rv] x4 E[u] = E[v] = 1 x23x1fx68
p(u) 6= p(v),x21x24x25x26x1ex68x54a0 6= 0. x58x5dx25x78x2bx9x26x4ex1x44x26x48x3ax3ex7d
x4x68x19
a0 +b0 ?p(x) = E[ux]?1E[u2]?1(a0 +b0 ?a0E[u2]?b0):
x7bx7ex48 (x1fx79x2a E[x] = 1)
p(x) = a0E[ux]+b0 = E[(a0u+b0)x] = E[mx]:
x1fx79x19px2x5x6dx65x68x48x25x2bx9x26x10x21x51x9xap(x) 6= 0x54x 2 Mx41x7dx78
x13x26x65x68x54x7fx0x1dx48x10x5ax10x21x51x 2Mx41x7dx78x13x0x44x48 p(u) = E[mu] 6=
50
p(v) = E[mv]x4E[u] = E[v] = 1x23x1fx8mx571M x70x62x2ex63x1cx48x58x1fx8
E[rm] 6= E[1M],x7bx7ex48x38x3d1)x41x7dx78
x7ex19x48x2exfx52x2cx48x31x32 m 2 M x9xax38x3d 1), x33x34x10x66x15x1fx7fx68
a 6= 0x4b, u = am+bx5bx51x38x3d2)x41x7d(x7ax60)x78x7ax75x48x31x32ux9xax38x3d
2)x48x33x34x7fxcx7ax51x9xax38x3d1)x54x1dx70x54m = a0u + b0, a0 6= 0. x25x67x2cx68
x61x62x69x2cx15x13x54x63x73x78 2
x25x26x48x3ax3ex67x1fx8x61x1cx1dx62x4ax55xex54x9x4ax40x26x2ax1ex6fx1ex37x62x4ax55
x56(CAPM)x78x31x21x37x1bx48x1ex6fx1ex37x62x4ax55x56x2Sharpe ([9])x9x7dx491960
x46x4ex20x8x54x78x25x70x55x56x58x40x7dx3ex48x59x70x64x3dx3ex1ex37(x50x62xbx21) x54x3ex2b
x66x4ax20x21 E[r]?rf x5bx57x11x64xdx42x43x14x25xfx54x3ex2bx66x4ax20x21 E[ru]?rf x75
x76x51x1cx1dx3x4cx78x25x67x53x7dx3ex10x3dx3ex42x43x51x70x2bx1ex1fx54x76x22x78x7bx7ex48x17x10
x5fx4cx18x48x49x3dx3ex4fx7bx5bx4x3x5bx5bx0x14x61xfx4ex54x3cx30x48x51x6cx11xcx1xdflx3
x4xf (x7bx1x7dx3ex57x7fx3b fl x27x60x27x25x70x1cx1dx3x4cx2cx54x4cx68)x78x13x27x48 Black
[1]x58x45x10x36x51x33x69x6ax2cx68x54x42x43x17x4ex61x1ex6fx1ex37x62x4ax55x56x48x58x20x8x21
x70x71x54xdx7d-fl x1ex6fx1ex37x62x4ax55x56xfx78
x67x4bx1ex70x48x16x17x39x57x58x79x7ax44x27x70x48x25x70CAPMx51x43x2Sharpex9
x20x8x54CAPMx48x7bx1x25x18x51x36x51x70x8xdx42x43x14x25xf ux2x37x34x78x24x13x3ax3e
x53x39x17x62x3ax3bx27x65x63CAPMx48x51x5dx8xdx42x43x14x25xf ux54x4ex70x57x6ex7ax78x49
x24x13x54x4ex70x57x54xex4cx56x54x65x63x2cx48x3ax3ex23x24x5dx8x48 u x2x42x43x2cx21x51x69
x6ax2cx68x21x40x41x54x14x25x78x2ex1fx49x3ax3exaxbx54x65x63x2cx48x35x61xcx75x54x68x4fxe
x4cx4x1cx1dx62x4ax55xex24x19x48x36x51x15x1fx17x62x61xdx78x16x17x2x36x51x3x66x42x43x44
x54x2cx68x6cx54x15x1fx61xdx78x7bx7ex48x49xaxbx54x55x56x2cx2ex23x25x51xdx42x43x14x25xfx54
x25x22x78x5ex66x62x692.3x54x31x32xex14x17x66Ross [8]x4Dybvig and Ingersoll [4]x78
x49x31x18x6fx25x24xbx48x3ax3ex58x23x1fx79x19x56x5cx68x4fx4x60x2a
x59x23 x20xx71yx67x0xcx39x3ax3bx3cx0 fl = Cov[x;y]=Var[x]:x5dx5e
Var[y ?flx] = min
2R
Var[y ? x];
x6cflxx4x47x73xx79x16x65 xx7fx14yxcx51x79x50x51x62x63x0
xdx27 x2bx71x44x48
Var[y ? x] = Var[y]?2 Cov[x;y]+ 2Var[x]:
x1fx14x10 x39x1fx48x26x24x6x54xcx75x38x3d (x6fx3cx3bx8x6dx65x68x54x1dx60)x48x7dx4x23
x68x17 = fl = Cov[x;y]=Var[x]x18x48 Var[y ? x]x54x19x24x16x78 2
x26x25x38x4x60x23x1bx48x44x29x54x1ex6fx1ex37x62x4ax55x56x58x51x25x26x54x12x6bx79x7ax2a
flryruru = Cov[ry;ru]ru=Var[ru]x53x21x51 ru x2cx57ry x75x7dx54x69x7dx54x19x15x16x78
51
x25x7bx17x66x70x48x31x32x3ax3ex10x2cx68(x6fx21x70x71x54x50x62xbx21) yx54x20x21x27 ry x3
x66rux2ex1bx30x33x14x48x4x3bx15x16x8x16x55x48x39 x4fi,x53Var(ry? ru?fi)x54x69
x7dx15x16x48x33x34 x54x15x6dx4x6bx67x2flryrm,x58fix54x15x6dx4x6bx1rf(1?flryrm).
x25x70x68x48 flryrm x54x60x54x26x51x17xcx5ax6ex7ax48x10x79x41x42x10x2cx68x20x21x27x4xdx42
x43x20x21x27xfx54x71x72x68x5ex7x3b CAPMx48x57x7ex7fx54x1bx30x33x14x49x5x12x44x2x70
x26x54x78
x2fx6a x62x27x19x1ax6x10x11x41x26x13x7x41xdx5x70x47x40x3cx4dx1cx51x47x6 Modigliani–Miller
1958x2x20x58x1cx4fx2ex7x42x7dx2ex2cx20x29x6x31x4x51x7cx20x2ex4ax7x5x70x47x40x3cx4dx1cx19x2cx4aDuffie
[3], Ingersoll [6] x20x29x43x69x7dx5 x35x7x15x47x40x53x6x34x77x37x12x3bxax47x40x53x6 (x1ax4x16x13x9
x62x5x59x4dx16x13x37x62) x18x12x22xax47x40x53x6 (x1ax4x16x13x37x16x5x59x4dx16x13x9x16)x29xbx17x3bx4dx5
x32x33xfx6x3cx20x3dx26x2bx74x2cx20x3ex3fx29x19x1ax2cx4ax24x16 Jarrow [7] x7x52x70x47x40x3cx4dx20x29x43
x69x7dx5 x35x7x42x7dx2ex2cx10x43x16x7ax32x13x20xexfx5x23x44x1cx3fx28x34x3cx4d E[m] 6= 0 x6ex58x15x58x77
x12x3bxax47x40x53x6x38x64x6ax5x45x44x47x3bx18x3fx7bx4ex4fx29x16x12x5x19x1ax15x70x47x40x3cx4dx77xax10x51x14
x61x4cx13x5xbx3cx50x2bx16x61x4cx20x70x47x40x3cx4dx20x10x46x2ax20x42x7dx29
x42x7dx3x22x55x7fx18x20x69x7dx20x4dx4ex4fx43x16xcx30x2cx2ax6 Ross 1978 x2x24x29x20 [8] x7x29
x6x7ex53x5 Ross x21x2cx72x7ex20x70x47x40x3cx4dx33x16x53x43x16x7ax32x47x62x37x3ax7ax32x29xbx3bx43x5cx26x4d
x46x33x37x48x35x43x16x60x2ex43x5cx29x19x1ax23x6x26x11x4dx4ex4fxbx3bx6x32x33xfx4ex3dx41xfx20x43x5cx29 x3c
x47x52x53x11x12x58x59x33x20x9x58x77x4ex4fx5x19x1ax7ex61xfx46x8x2cx38x3ex3fx20x37x3ax43x16x52x28x29xbx3b
x45x2fx4ex4dx45Ross [8]x5x28x18x48x2ex48x35x43x16x7fx18x20x5cx28x28x47Dybvig–Ingsoll [4] x2ex2cx20x29
x42x7dx2ex2cx28x1ax23x14x20x33x16x3ax28x2cx53Cochrane [2]x29x2bx3fx19x1ax20x4ex4fx25x35x45x25x20x55x6ax29
Cochranex20x71xcx7x5x37x3ax43x16x52x28x46x3cx43x37x10x47xax2ax5 x56x19x1ax28x41x10x62x7fx4cx43x16x52
x28x13x2cx24x4dx4ex4fx5 x72x42x47x6ax55x20x5x23x44x45x44x47x3bx18x2bx74x2cx20x0x50x56x4fx29
x53x55x20x11x12x58x59x33x7x64x2fx60x24x25x3bx7fxex37x3ax43x16x52x28x18CAPMx20x33x16x3ax29 x3c
x47 Duffie [3] x28x15x48x35x43x16x60x2ex43x5cx2ax6x1cx2cxex6ax5x23x21x2cx3ex28x35x4dx2c CAPMx29x1e
x7cx7ax28x20x121x6bx29 Ingersoll [6]x20x122x6bx4ex47x13x50x3bx1fx20x64x65x29 Huang–Lizenberger
[5] x20x25x27x64x65x6x12 8 x6bx5xbx47x2bx37x47x2fx3x43x20x47x63x37x20x10x5ex11x43x16x13x29x5dx36x34x75x20
x47Jarrow [7]x7x4dx1ex24x25x24x16x4ex4fx29
x44x30x23x41x45x23
1. x76x12x8x70xbx3dx3ex1ex37x62x4ax5fx60x54x71x72x45x14x78x10x3ex25x35x3bx3ax3ex54x40
x6fx55x56x27x4x6x7e
2. x31x32x49x1ax71x2cx1cx1dx62x4ax55xex41x7dx48x33x34x21x51x22x4fx54x3cx27x5bx7x8x2
x70x26x54x78xfx2x71x72x74x75x51x43x31x7ex78x32x26x1ex4dx25x64x1ax44x7e
3. x32x26x3cx3bx24x6cx72x76x20x37x48x27x60x54x3x69x1ex33x6dx3cx61xdx54x74x64xex6dx48
x51x1ex4dx17x57x58x79x7ax78
4. x37x34x2x7ex7fx42x43x7ex10x7ax7bx61x70x64x32x26x54x1ax71x3dx3ex42x43x54x5dxdx7ex10
x54x40x6fx1dx60x2x37x34x7e
52
5. x10x26x2xdx2ex3cx62x70x71x56x54x45x46x62x4axfx48x70x64xdx4bx1ax1b(x45x31x48 xdx1c
x3ax14x29xf)x54xdx62x4axfx57x3dx3ex1ex37x62x4ax51x1fx16x6fx2ex10x7e
6. x37x34x2x50x62xbx21x54Hilbertx72x76x4x1cx1dx62x4ax55xex54x28x29x13x1ax7bx14x7e
7. x49x44x70x51x2cx65x63x54x28x6dx54x3exbx62x4ax55x56x2cx48x37x34x2x8x55x56x2cx54x50
x62xbx21 Hilbert x72x76x4x28x29x13x1ax7bx14x7ex31x32x50x27x51 S x64x70x71x48x33
x34x50x62xbx21 Hilbertx72x76x4x28x29x13x1ax7bx14x7fx2x37x34x7e
8. x37x34x2x50x62xbx21x54x18x76x4ax6x4x69x6ax4ax6x7ex10x3ex2x7bx10x37x34x58x1x54x7e
9. x32x26x62x7ax33x69x6ax2cx68x4x33x69x6ax3cx27x7ex32x26x69x1ex33x69x6ax2cx68x54x55x74
x14x25x7e
10. xdMx2x45x6ex33x69x6ax2cx68x54x50x62xbx21x72x76x48x41 2M. M1 ‰Mx1
Mx54x14x72x76x48xf1 =2 M1. x25x18x48x39Mx8x5fx27x3cx5fx60x48 1x571M1
x75x76x51x37x34x3x4cx7ex25x64x3x4cx51x37x34x57x58x79x7ax7e
11. x5dx44x60x2cx54x33x69x6ax2cx681x67x1x28x29x13x1ax7bx14m,x65x63x10x26x54x5fx60x78
12. x32x26x27x69x1ex1cx1dx62x4ax55xex4x1ex6fx1ex37x62x4ax55x56x54x9x4ax1dx7e
13. x5dx1ex6fx1ex37x62x4ax55x56x2cx54r 2 R1 x67x1r 2 R0, x4x5dx55x56x2cx54xdx20
x21xfx67x1xdx66x4ax20x21xfx48x33x34x55x56x8x32x34x27x76x29x7ex10x57x1cx1dx62x4ax55
xex7fx2x37x34x3x4cx7e
14. x2cx68x48x31x32m 2 Mx9xaCAPM (x62x692.3x54x38x3d2)x48x33x34x10x66x15
x1fx7fx68a 6= 0x4b, u = am+b1M x5ax9xaCAPMx78
15. x2cx68x48x10x66x78x2bx69x7dx51x2ax68x4fx3ex2bx2ex1x7dx54x28x29x2ax6cxx4y,x31x32
E[x] = Cov[x;y]Var[y] E[y]; E[y] = Cov[x;y]Var[x] E[x];
x33x34x66x49x7fx68a 6= 0x4b,x53x68y = ax+b. x5dx25x2bx31x32x3bx19CAPM
x54x65x63x44x48x3ax25x68x19x37x34x31x32x7e
x2fx30x31x32
[1] Black, F., 1972, Capital market equilibrium with restricted borrowing,
Journal of Business, 45: 444–454.
53
[2] Cochrane, J. H., 2001, Asset Pricing, Princeton University Press, Prince-
ton, N. J..
[3] Duffie, D., 1997, Dynamic Asset Pricing Theory, Second ed., Princeton
University Press, Princeton, N.J..
[4] Dybvig, P. H. & J. E. Jr. Ingersoll, 1982, Mean-variance theory in com-
plete markets, Journal of Business, 55: 233–251.
[5] Huang, Chi-fu & R. H. Litzenberger, 1988, Foundations for Financial
Economics, Prentice–Hall, Englewood Cliffs, N. J..
[6] Ingersoll, J. E. Jr., 1987, Theory of Financial Decision Making, Rowman
& Littlefield Publishers, Maryland.
[7] Jarrow, R. A., 1988, Finance Theory, Prentice-Hall, Englewood Cliffs, N.
J..
[8] Ross, S. A., 1978, A simple approach to the valuation of risky streams,
Journal of Business, 51: 453–475.
[9] Sharpe, W., 1964, Capital asset prices: a theory of market equilibrium
under conditions of risk, Journal of Finance, 19: 425–442.
x5cx5dx2ax42x43x1dx1ex1fx201
1. x21x42x22Rx2aRx3bx27x60x27x71x68x7fx1fx78x71x68x2x7ex1fx2a [x36x7ax68)x1e
x68)x51x69x68)x71x68] x25x26x54x1fx6ax41x57x4bx1dx58x41x54x78x36x7ax68x7fx1fxdx15
x2x7dx72xdx79x52xfx67x51x54x78xfx49x69x63x44x48x10x10x26x47x75x3bxdx6ex69xfx27x62x7ax78x51
x3x54x6ex69xcx1xdPeanox6ex69xfx78x25x18x3ax3ex36x51xcx75x12x2fxdx78x24x13x54x4bx1d
x6cx79x2x1x53x74x64x4ex68x5x12x25x4ex4fx58x40x41x54x78x39x36x7ax68x19x1ex68x2x7bx1x75
x53x3fx55x5x51x23x25x4ex4fx57x39x1ex68x19x51x69x68x2x7bx1x75x53x35x55x5x51x23x25x4ex4fx78
x51x69x68x7fx1fx7fxcx1x24x3fx42x22x78 xdx74xfx67x2x5dx17x2cx25x4ex4fx0x3fx16x35x47xex5
x12x78x39x51x69x68x4bx1dx19x71x68x77x77x1x2x78x2cx4fx68x4fx2cxcx71x68x1x51x69x68x57x33
x69x68x7fx1fx48x58x33x69x68xex62x7ax1xdx33x2ax2ex1x70x16x68xfx78x17x71x33x69x68x2x7ex1f
x51x69x68x54x24x2ax27x62x7ax54x78x1x53x62x7ax24x2ax51x23x25x48x25x18x5ax47x75x70x38x6ex69x48
x10xcx1xdx7fx0x1dx6ex69xfx78x49x71x68x5ax1ax10x5ax10x26x25x4ex4fx47xex5x12x48x7bx7ex48
x7fx7fx5axcx10x1xdx71x68x74xfx78x58x71x68x74x77x51x69x68x74x21x51x3bx75x73x49x66x17x2cx25
x77x77x69x66x1ex5x3bx24x2ax25x22x78
x57x58x4fx2cx1x37x34x47x75x3bx71x68x74x27x47x29x7ex2fxdx14x27x28x44x5ax36x51x2ex4e
x2ax69x78x71x72x57x58x36x2fx2cx3bx51x69x68x19xax24x23x34x78xfx2x1x61xcxdx57x24x2ax25
54
x22x51x3x54x55x24x5fx60x48x69x63x44xcx2cx7ex3bx71x68x74x24x58x26x7ex37x36x54x1x33x4fx9
x9x78x25x70x68x7bx7cx78x10x7ex48xdx15x5ax73x25x7ex47Friedmanx20x37x74x7ax54x71x3bx78
2. x49x42x23x24x78x2ax70x2bx71x2ax6cx71x6x65x68x60x27x1f:R ! R, x10x60x27x70
x2bx36x2ax6cx1x71x68x3x7bx2ax6cx5ax2x71x68x54x65x68x48x4x10x10x59x70x2bx71x68 (x36x2a
x6c) x 2Rx5dx62x61x4dx70x2bx71x68(x7bx2ax6c) y = f(x) 2Rx57x10x7bx10x7x78x77x65
x68x21x70x71x54x25x22x2x7bx7ax78x10x23x24x10x15x79x78x2bx69x25Ax4Bx27x62x7ax78x7bx7a
g:A ! B x2x5dx10x59x70x2bx38x46x 2 Ax62x7ax61x4dx70x2bx38x46y = g(x) 2 B. x45
x31x48x25x18Ax4Bx5bx2x71x68x74x54x14x69x48x33x34gx67x2x21x70x71x54x71x2ax6cx71x6
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x17x14x69x54x74x40x78
3. x25x2dx54x27x41x26x27x2ax5dx40x41x2bx71x68x15x70x62x69x26x2bx5cx14x27x48x10x3ex54
x7fx1fx67x40x41x70x2bx24x6cx72x76x78x7ex7fx70x2bnx75x24x6cx72x76x2x5dnx2bx15x6dx28x2b
x5cx54x71x68 yi (i = 1;2;:::;n) x21x14x41x54x68x14 y = (y1;y2;:::;yn) x7fx1fx78x10
x7fx7fx44x1Rn. x58Rx2cx54x38x46xexcx1nx75x24x6cx78x49x70x2bx24x6cx72x76x2cx23x24
x4ex4fx0x3fx5x12x48xfx2ex25x4ex4fx16x35x5x12x78x7ax58x48x70x2b n x75x24x6cx57x70x2bx71
x68x2x23x24x7bx16x54x78x24x6cx72x76x54x67x4bx54x6ex69x5cx67x2x3bx25x4ex4fx25x78x64x5x12
x27x62x7ax54x78x26x70x2b 1 x4 n ? 1 x2b 0 x21x4dx41x54 n x75x24x6cxcx1Rn x54x29x3a
x25x2dx78x25x26x54x24x6cx70x63x51nx2bx78x59x2bnx75x24x6cx5bx23x60x27x1x10x3ex54x74x59
x3dx22x48x4x10x3ex16x44x71x68x13x54x4x78x31x32x51x70x14x24x6cx5ax51x25x26x54x1dx60x48x33
x34x25x70x24x6cx14x67xcx1Rn x54x37x78x70x14x24x6cx40x2cx54x24x6cx2bx68x70x62x2ex5fx66n
x2bx78x35xex70x62x51x11x2bx24x6cx2ex25x60x27x1x25x14x24x6cx54x1cx1dx14x25x48x39x58x25x14
x24x6cx2ex25x40x41x40x78x31x32x3ax3ex65x63x54x72x76x2x2cx68x14x25x54x50x27S x2bx70x71x56
x54x4ax6x21x40x41x54S x75x72x76x48x33x34S x2bx6dx6ex24x6cx67x2x21x22Arrow–Debreu
x2cx68x78x31x32x40x6fx2cx68x14x40x41x25x2bS x75x72x76x54x40x48x33x34x21x10x7x54x2cx68x42
x43x67x2x7ex7fx54x57x35xex67x2x2ex7ex7fx54x78x49x7ex7fx42x43x2cx48x59x70x50x62xbx21 (x4
Sx75x24x6c)x5bx23x3bx40x6fx2cx68x54x1cx1dx14x25x27x60x27x48x39x58x50x62xbx21x67x23x3bx40
x6fx2cx68x14x27x62x4ax78x58x49x2ex7ex7fx42x43x2cx48x67x66x49x11xbx50x62xbx21x2ex25x60x27
x1x40x6fx2cx68x54x1cx1dx14x25x48x39x58x3bx40x6fx2cx68x27x62x4ax67x51x2ax52x78xfx2x7ex1f
x7ex1ex54x33x6dx3cx61xdx48x23x3bx40x6fx2cx68x14x27x1x10x62x8x70x2bx4ax4bx48x76x78
x70x71x74x75x56x48x3ax3ex2ex70x62x5dnx2bx71x68x25x26x2cx2bx14x27x78x1x61x11x64x47
x75x48x3ax3ex5ax23x24x5dx10x2dx2bx14x27x78x49x13x70x64x74x40x56x48x3ax3ex44x = (x1;x2;:::;xn)T,
x25x18T x67x60x27x5dx25nx2bx71x68x2dx2bx14x27x48x4x21x22xdx3ax6fxfx78x3ax3excx25x64x40
x26x54x24x6cx1x2ex25x2d,x58xbx4bx40x26x54x24x6cxcx1x2fx25x2dx78 nx75x5cx24x6cx7fx1fx57
nx75x4fx24x6cx7fx1fx5bx40x41nx75x24x6cx72x76x48xfx2x51x18x51xcx75x5dx10x3ex48x33x1
x78x2bx2ex10x54x24x6cx72x76x78x10x70x2bx4fx24x6c = ( 1; 2;:::; n) x57x70x2bx5cx24x6c
x = (x1;x2;:::;xn)T x23x62x7ax10x3ex54(x2ex23x66x67x54)x16x2x48x4
￠x = 1x1 + 12x2 +￠￠￠+ nxn 2R:
x49x3ax3exbx4bx54x65x63x2cx48x4fx24x6c x54x79x7ax2x2cx68x14x25x48x5cx24x6cxx54x79x7a
55
x2x2cx68x4ax4bx48x10x3ex54x16x2x54x79x7ax67x2x2cx68x14x25x54x5x4ax6x78x25x18x48x5dx4f
x24x6cx57x5cx24x6cx48x3fx6fx27x48x49x57x58x6ex7ax44x67xfx51x28x73x78x2ex1fx49x68x4fx44x5a
x57x7fx2ex0x48x33x78x25x18x67x36x51x4fx24x6cx57x5cx24x6cx54x48x3fx78x58x7bx7x54x16x2xe
xcx1x78x2bx24x6cx54x75x53x78
x10x66x70x2b n x75x24x6cx72x76Rn x27x70x48x10x2x2x70x2bx43x7fx45x75x54x25x22x78
x10x2x7ax7bx78x2bx24x6cx75x76x54x30x3ax54x6cx3bx78x62x7ax61x10x2x54 n x75x24x6cx72x76
xcx1 n x75x31x32x33x34x54x27x6fx28xcx31x35x54x27x78x10x66Rn x2cx54x15x1fx78x2bx24x6c
x = (x1;:::;xn)x4y = (y1;y2;:::;yn),x17x10x2x7fx60x27x1
(x;y) = x1y1 +x2y2 +￠￠￠+xnyn:
kxk =
p
(x;x) = (x21 +x22 +￠￠￠+x2n)1=2
xcx1x24x6cxx54x36x37x78 kx?ykxcx1x24x6cxx57yx75x76x54x38x39x78x58x10x3ex75x76
x54x30x3a fixex26x56x26x62x7ax2a
cosfi = (x;y)kxkkyk:
x16x17x2(x;y) = 0xcx1xx57yx3ax3bx6fx55x2fx39 x49Rx2cx51x61x2ax5bx24x13x48x3ax3e
x67x23x49x17x2cx62x7ax3cx3dx25x22x78x5ax67x2x70x48Rn x2cx54x70x2bx77(x24x6c)x5cfx(k)g
x55x24x66x70x2bx35x62x77(x24x6c) a,x2x5dkx(k)?ak! 0. x26x7ex58x23x62x7ax67x69x57
x6fx69x54x25x22x78Rnx2cx54x70x2bx69x25Axcx1x3ex79x2x5dx17x2cx59x70x2bx51x24x2ax54x77
x5cx54x24x2ax5ax49x17x2cx78x3fx79xex23x62x7ax1x70x2bx67x69x54x40x69x78
4. x31x35x54x27x41Hilbertx54x27x1cx3ax3ex49xbx4bx10nx75x24x6cx72x76x19x57x62
x7ax1fx10x2x54x25x22x78x62x7ax61x10x2x54nx75x24x6cx72x76xcx1x2ex55x72x76x78x61xdx25
x2bx2ex55x72x76x1X, (x;y)x60x27x;y 2 X x75x76x54x10x2x78x33x34x2exfx52x2cx10x51
x24x56x3x38x1dx60x2ax10x66x15x1fx;y;z 2 X,
1. (x;y) = (y;x).
2. x10x66x15x1fx71x68?;? 2R, (?x+?y;z) = ?(x;z)+?(y;z).
3. (x;x) > 0,x51x6c(x;x) = 0x17x6cx4cx17x = 0.
x25x60x38x1dx60x57 X x54x75x68x36x51x3x4cx78x7bx7ex48x10x23x24x76x77x19x70x71x54x74x40x78
x2bx71x44x48xdX x2x70x71x54x24x6cx72x76x48x4x17x2cx54x38x46x23x7bx0x24x58x57x15x79x71
x68x7bx16x78x31x32x49Xx44x62x7ax70x2bx3b(x;y)x27x60x27x54x8x2ax6cx65x68x48x9xax44x29
x60x38x1dx60x48x33x34Xx67xcx1x75x53x54x27x78x57x2ex55x72x76x70x26x48x3ax3ex10x26x62x7ax
x54x3dx3bkxk = p(x;x). x2ex1fx57x7fx5dkxkxcx1xx54x41x42x78x25x26x48x31x32x51x70
x2bx69x25(x45x31x48 Mx1x70xbx69x7dx51x2ax54x28x29x2ax6cx21x4dx41x54x24x6cx72x76)x48x17
56
x44x23x62x7ax70x64x10x2(x45x31x48x10Mx2cx54x28x29x2ax6cx;yx62x7a(x;y) = E[xy])x48
x33x34x2ex55x72x76x54x2dx2ex31x63x5bx23x10x25x26x54x10x2x72x76x41x7d(x10x66Mx27x70x48
E[x2] = 0 x1fx8 x = 0 x2x79x41x42 x x2ex1x28x29x2ax6cx48x24x25x27 1 x9x66 0. x10
x18x48 Mx2cx15x1fx78x2bx28x29x2ax6cx7bx9x5ax5bx2x79x41x42x10x3ex7bx9x54x25x27x9x661,
x2ex7bx9x54x25x27x9x660)x78x2ex1fx2ex55x72x76x51x70x38xcx1x49x1ex59x54x1dx60x58x36x51
x45x6ex49x10x2a x57x58x42x7ffxng ‰ X xaxb limn;m!1kxn ? xmk = 0, x5dx5ex60
x61 ˉx 2 X, x4x5limn!1kxn ? ˉxk = 0. x26x66x3ax3ex1ax49x47x48x54x24x6cx72x76x23
x25x2x33x2ax75x54x48x10x2ex70x62x51x25x38xfx51x3bx54x1dx60x78x5dx25x38x1dx60x0x19x70x2b
x10x2x72x76x2cx12x48x53x10x21x38x5dx2ex55x72x76x48x25x67x62x7ax61x21x22Hilbertx54x27x48
x4x10x2x7ex74x54x10x2x72x76x78
x21x51x69x7dx51x2ax54x28x29x2ax6cx21x40x41x54x69x25x48x31x32x3b(x;y) = E[xy]x27
x62x7ax17x44x54x10x2x48x33x34x10x67x2x70x2bHilbertx72x76x78x7ex7fx44x2eL2(P),x17x2c
P x60x27x25x27x51x3bx78x51x2ax2bx69x7dx51x2ax54x28x29x2ax6cx0x41x54x24x6cx72x76x48x7ex1f
x10x26x54x43x55x27x62x7ax10x2x48x10x5ax2x70x2bHilbertx72x76x78xfx10x49x6fx60x44x57x2e
x55x72x76x36x51x2ex4ex78x26x78x33x2ax2ex2bx69x7dx51x2ax54x28x29x2ax6cx0x41x54x10x2x72x76
x2ex70x62x2x7ex74x54x78x31x32x10x2x7ex74x54x48x33x34x10x2Hilbertx72x76x48x51x6cx67x2
L2(P)x2cx54x70x2bx67x14x72x76x57x7ax75x48 L2(P)x54x15x1fx70x2bx67x14x72x76x5ax23x3cx2e
x40x41x69x7dx51x2ax54x28x29x2ax6cx21x0x41x54Hilbertx72x76x78x49x3ax3ex54x65x63x2cx48x70
x71x5x61x62x50x62xbx21x72x76Mx2L2(P)x54x67x14x72x76x48x4x10x5ax2Hilbertx72
x76x78
5. Hilbert x54x27x44x10x55x2fx59x1cx10x70x2b Hilbert x72x76x2cx54x38x46x27x70x48
x31x32(x;y) = 0,x33x34x57x24xbx70x26x48x3ax3excxx57yx3bx66x78x78x2bx69x25x25x7bx3b
x66xex2x5dx70x2bx69x25x2cx54x15x1fx70x2bx24x6cx57x4dx70x2bx69x25x2cx54x15x1fx70x2bx24x6c
x5bx3bx66x78x10x66x3bx66x1dx48x3ax3ex51x56x5cx3bx66x33x1ex62x69x41x7dx2a
x55x2fx27x25x36x3f x20X x1dx7axcHilbertx44x45x0 Y ‰ X x1dx15x79x7axcx45
x46x44x45x6bx1x2Y 6= X. x5dx5eX x7ax62x47x48x49x4ax1dY x14x4bx7axcx14Y x68x4cx79
x75x76x46x44x45Y ? x79x4dx71x7d
X = Y 'Y ?;
x6cx6dx6ex6fx70x 2 X,x60x61x3x7ax79yx 2 Y x71y?x 2 Y ?,x4x5x = yx +y?x .
xdx27x3dx4e x49 Y x2cx39x70x2bx57 x x54x2ax5bx15x5dx54x24x6c yx, x4 yx 2 Y
x9xakx ? yxk = miny2Y kx ? yk = d. x2bx71x44x48x3ax3ex23x24x60x19x70x2bx77x5c
fynxg ‰ Y,x53x68kx?ynxk ! d. x3ax3ex5dx8limn;m!1kynx ?ymx k = 0. x6cx79x48
x3ax3ex23x18x38x52x2cx56x5cxdx36x4fx47x17x40x55xexfx41x7dx2a
kx+yk2 +kx?yk2 = 2(kxk2 +kyk2):
x39x58x17n;m !1x18x48
kynx ?ymx k2 = k(x?ynx)?(x?ymx )k2
57
= 2(kx?ynxk2 +kx?ymx k2)?
2
x? y
n
x +y
m
x
2
?
2
! 2(d2 +d2)?4d2 = 0
x7bx7ex48 ynx x51x24x2ayx. x26x66Y x2X x54x67x14x72x76x48x46yx 2 Y:
x4dx70x69x4bx48x12 y?x = x?yx, x3ax3ex5dx8x48 (yx;y?x ) = 0. x2bx71x44x48x10x66
x15x1fy 2 Y x51kx?yxk6kx?yk,x39x58x10x66x15x1fy 2 Y x4x15x1ft > 0,x51
kx?yxk2 = (x?yx;x?yx)
6 kx?yx ?t(y ?yx)k2
= (x?yx ?t(y ?yx);x?yx ?t(y ?yx))
= (x?yx;x?yx)?2t(x?yx;y ?yx)+t2(y ?yx;y ?yx)
x7bx7ex48x10x66x15x1fy 2 Y x4t > 0x51
(x?yx;y ?yx) 6 t2ky ?yxk2:
x7bx1tx23x15x79x38x5dx7dx48 y?yx x7fx23x47x1Y x2cx54x15x1fx38x46x48x21x24x?yx xc
x62x57Y x2cx54x15x1fx38x46x3bx66x48x4y?x = x?yx 2 Y ?. 2
x25x70x3bx66x33x1ex43x7fx51x3bx78x24x13x48x3ax3ex53x3bx10x27x65x63x1cx1dx69x6ax14x78x49
x6fx51x2cx48x3ax3ex53x7fx10x27x2cx68Rieszx60x27x62x69x78
6. Hilbert x54x27x44x10 Riesz x4fx50x36x3fx1c Riesz x60x27x62x69x5dx8x48
Hilbert x72x76x2cx54x7fx0x1cx1dx65x68x70x62x23x7ex1fx11x2bx38x46x10x17x62x38x46x54x10
x2x27x60x27x78x10x10x3dx3ex57x58x4fx54x79x7ax49x66x2ax31x32xdx42x43xf(x26x69x7dx51x2ax54x11
xbx28x29x2ax6c(x2cx68x54x50x27x4ax6)x21x0x41x54Hilbertx72x76)x51x7fx0x54x1cx1dx62
x4ax65x68x48x33x34x10x70x62x23x7ex1fx11x2bxdx62x4ax2cx68xf (x4xdx28x29x13x1ax7bx14xf)x27
x60x27x78
Riesz x4fx50x36x3f x20 X x67x7axc Hilbert x44x45x0x57x58f:X ! Rx67x7a
xcx5fx60x75x76x64x65x6bx5dx5ex60x61x3x7ax79 xf 2 X, x4x5x6dx6ex6fx70 x 2 X, x73
f(x) = (xf;x).
xdx27x3dx4e xd Y = fx 2 Xjf(x) = 0g. x33x34x2exfx52x2cx48 Y x2 X x54
x67x14x72x76x78x15x47x43x7dx54z 2 Y ?,x62x7axf = zf(z)=kzk2,x2exfx52x2cx48 xf x67
x2x1dx70x54x9xax62x69x75x39x54x38x46x78 2
58
