Download - Текст лекций
-
-
-
.., .., .., .., ..
5
2002
-
2
1. , ...................................................................................................... 4
1.1. .......................................................................................... 4 1.2. ............................. 5 1.3. .......................................................................... 7 1.4. ............................................................................... 12
2. . ............ 15
2.1. ........................................ 15 2.2. ............... 16 ........................................................................................ 16
2.3. ......... 18 ........................................................................................ 18
2.4. ...... 20 ................................................ 20
3. ............................................................................................................................. 22
3.1. ... 22 3.2. .............. 24 ...................................................................................................... 24 3.3. ............................................................................................................... 29
3.4. ............. 38
4. . ............................................................................................................. 40
4.1. . .................... 40 4.2. . ............................. 45 4.3. . .................................................................... 50
4.4 . . ......................................................................................... 67
4.5 1 EXCEL . ......................................................................... 69
. ................................................................................................................ 81
-
3
............................................................................. 83
-
4
, , , . . .
.
1. ,
. . .
1.1.
. , , .. , .
. , , , , .
, , , .
"". . .
. , (). , .
, () :
1) ; 2) ; 3)
. :
- ;
-
5
- ; - . , , ,
, , , .. . , , , , .
. , , , . . , .
, . "" , , . .
1.2.
, . . , , , , ..
, , , . , , , .
, , .
, , , ,
-
6
. , , , .. , - , , , , , .
, , , :
1. , .
2. . - , , .. . .
3. , . : .
, , ..
, , .. . .
4. . , .. , .
5. , , , , , , .
, ; , ; , .. , . ,
-
7
, .
6. : ) , , ,
. ; )
; )
; , ..; )
, , - , - .
1.3.
, , , , , , . , 1.
(. . 1.4) . , , , (. . 1.1) . - .
, 1.1.
1 ,
: - ; - ; - ; - ; - . , ,
. .
-
8
, ( ) : .
. -, , , (, , ).
, . , .
( ), : -, , , .
- , - : , , , , . .
, .
. - , 2.
, : , , , , . .
- . .
, , , .
, / ,
2 : )
, , , , , , - . .; ) () (, . .); ) ; ) . , .
-
9
, , . .
, , , (, , . .), , .
, , , . .
. 1.1.
.
:
-
10
1) , ; 2) , (
). , ,
: 3 , , .
, , . .
, , .
-, , .
, - .
:
1) ; 2) ; 3) .
() ( ) - (, , , . .).
, .
- . porto foglio , .
-: .
3 , , . ,
, , , .
-
11
, , , , , , . , , , .
, . , , , . , . , , , .
, . , , .
.
, . , , , .
.
: , , , .
- . : -
, . .
-
12
( . selectio , ) , .
,
. .
. , . , , .
1.4.
, , .
, , , , . , , , , .
- , , , , .
: () ; ; ; ; , , .
:
, ;
-
13
;
. , : -
; - , .
. , , ( ) (.. ).
() .
, , , .
, . , .
. , , .
. .
:
- , ;
- , .. , ;
- , .. , .
, .
(. . 1.2) :
1. . 2.
, , ; ..
-
14
1 , 2 , 3 ; 4 , 5,11 , 6, 12 , 8 , 7 , 9, 13 , 10 , 14 , 15 ( ), 16 (, )
. 1.2. - 3.
, .
.
, , , . .
-
15
.
4. :
- ; -
; - () .
2. .
. . . . . . . . .
2.1.
; , ().
, . i = 1,2,...,, j = 1,2,..., n. , i-e , j- , qij. Q = (qij) ( ) 4.
.
i- . , , j-, ,
qj = ijiqmax . , i-
. , qj , qij. , , rij,
4 , , ; ( ), . .
-
16
rij = qj - qij. (2.1)
R = (rij) 5.
2.1. (2.1),
R = (rij)
8241
10358
12432
4825
Q .
. , q1 = 1max iiq = 8; q2 = 5, q3 = 8, q4 =
12 . ,
4617
2500
0426
8033
R .
2.2.
(), , ; . (). .
() . , - , . () ,
, ,
j
iji
qmaxmax .
i- , ,
ijj
i qa max ,
ai.
5 , .
-
17
2.2. 2.1 .
. ijj
i qa max : a1=8,
a2=12, a3=10, a4=8. : a2=12. , (i=2).
( , ). i-e , , , ..
: bi = min qij. i0 0ib . ,
i0 , 0ib = i
ibmax =
j
iji
qminmax .
2.3. 2.1 .
. 2.1 b1 = 2, b2 = 2, b3 = 3, b4 = 1. b3 = 3. , 3- (i=3).
( ). , , Q, R = (rij). ,
, ..
j
iji
rmaxmin .
i-e , ri = ijjrmax
i0 0ir = iibmin =
j
iji
rmaxmin .
2.4. 2.1 .
. R,
ri = ijjrmax : r1 = 8, r2 = 6, r3 = 5, r4 = 7.
: r3 = 5. , - 3- (i=3). , .
( ).
-
18
, ci= {minqij + (1 )maxqij}, 0 1. , . =0 , =1 . () -.
2.5. 2.1 =1/2.
. Q , i ci= 1/2minqij + 1/2maxqij. ,
1=1/22+1/28=5; 2=7; 3=6,5; 4= 4,5. 2=7. , =1/2 (i=2).
2.3.
pj , j, , . ().
() . . pj , , i- , Qi
qi1 qi2 qin
p1 p2 pn
M[Qi ] Qi , iQ :
iQ = M[Qi ] =
n
j
ijjqp1
.
i- iQ ,
,
n
j
ijji
ii
qpQ1
maxmax
2.6. 2.1
-
19
, : p1 =1/2, p2=1/6, p3=1/6, p4=1/6. ,
.
. i- : 1Q =1/2*5+1/6*2+1/6*8+1/6*4= 29/6, 2Q = 25/6, 3Q = 7,
4Q = 17/6. 7
. (
). , ,
i- Ri
ri1 ri2 rin
p1 p2 pn
M[Ri] ,
iR : iR = M[Ri] =
n
j
ijjrp1
..
, :
n
j
ijji
ii
rpR1
minmin .
2.7. , 2.6. , , ().
. i- . R :
1R = 1/2*3+1/6*3+1/6*0+1/6*8=20/6, 2R = 4, 3R = 7/6, 4R = 32/6.
, 7/6
: iii
RR min = 7/6.
. () (), . , .
() ().
-
20
-, . , ( ) . , pj 1/n. , ,
n
j
iji
ii
qn
Q1
1maxmax .
,
n
j
iji
ii
rn
R1
1minmin .
2.8. 2.1, : ) ; ) . . ) 1Q = (5+2+8+4)/4=19/4, 2Q = 21/4, 3Q = 26/4, 4Q = 15/4.
, , 26/4.
)
: 1R = (3+3+0+8)/4 = 14/4, 2R = 3,
3R = 7/4, 4R = 18/4. , ,
7/4.
2.4.
, ( ) , : . , . 6.
6 . , , .
-
21
.
(), r() (, ); , r .
. . , . - , , r .
, b, > b, () (b) r(a) r(b) . , b . , . , . () 7. : , r , .. .
. 2.3, R Q . ,
R , Q ,
(R ,Q )
. , ; , . , ( ) . , 2.6 2.7 .
, R Q
, , . , , i ( iR , iQ ) f(i) = 3 iQ - 2 iR ,
7 , ,
, .. , . , .
-
22
f(i). , , , . , .
2.9. , 2.6 2.7, .. 2.1 : p1 =1/2, p2=1/6, p3=1/6, p4=1/6. , , . .
. f(i) = 2 iQ - iR .
2.6 2.7, : f(1) = 2*29/6 20/6 = 6,33; f(2) = 2*25/6 4 = 4,33;
f(3) = 2*7 7/6 = 12,83; f(4) = 2*17/6 32/6 = 0,33 , ,
.
3.
. . .
, , , , .
3.1.
, . , :
1) , , , 8 k: = {k};
2) 2k
k (,
8 k ( ), ..
, .
-
23
22k )( kE );
3) k ;
4)
k ,
. . n !
nkkk ,..., 21 , ,
. , () ,
,
n
i
ii pkk1
,
i ik k.
ik ,
n
i
ikn
k1
1.
, ( ) :
2)()( kkpkVar ii 2
)(1
)( kkn
kVar i .
.
2
ks ,
22 )(1
1kk
ns i .
, ( )
2)( kkps iik
2)(1
1kk
ns ik .
,
k
sk .
.
1. k (, ).
2. 2s (
) ks .
-
24
3. k 2s
ks .
. k , : mp , IRR (Internal Rate of Return) () 9 ..
.
3.2.
, , . . , , . 1. , , 100000 . . . 3.1. -, - 8%, , . . , .
3.1
i
-,%
, %
1
2
0.05 8.0 12.0 -3.0 -2.0
0.20 8.0 10.0 6.0 9.0
0.50 8.0 9.0 11.0 12.0
0.20 8.0 8.5 14.0 15.0 9 (IRR)
, .
-
25
0.05 8.0 8.0 19.0 26.0
8.0 9.2 10.3 12.0
. , , , . , 10%- , .
-, ( ), 9% (. . 100000 . 9000 . ) 10 . . , . : , , , ; . -, 1, 100000 . , . . , , 2,
1 .
, ( ). . 3.1 , . - . 8% .
-
26
1. - ?
: ) ; ) , , ; ) .
). . 1.
1. - , . , .. . , , - : -, , , . , , , , -, . , , , , , .
, , . , .
. ; , . 3.1 . - , . . , , - 8%, 1.00, , 9%, 0.50.
, , . ,
-
27
. (Internal Rate of Return, IRR), (Expected Rate of Return, ERR), :
ERR =
n
i
ip1
IRRi , (3.1)
IRRi, i- ; pi i- ; .
(3.1), , , , 2 12.0%:
ERR = -2.0% 0.05 + 9.0% 0.20 + 12.0% 0.50 + 15% 0.20 + 6.0% 0.05 = = 12.0%.
1. (1) . 3.1.
, , . . 3.1 ( ) 1 2.
. 3.1. .
1; 2.
1 -3.0 +19.0%, 2 -2.0 +26.0%. , ,
-
28
1.00. , 2 , 1 .
2. - . .
: , , .
, . , . , 2000 . , 10 000 .
(10 0000,5 + 00,5) 2000 = 3000 (.) , , -
. . , , , 2000 . . , , , . , . , , .
, , . , , .
-
29
3.3.
, , . , , .
3.3.1. : ,
. , , , ? , .
, 5 (. . 3.1). . () , (), () , ( ). , , , , . . 3.2.
, , , . , .
. , , , .
-
30
2. . , ?
: ) ; ) ; ) .
). . 2.
2. , , 89% .
ERR
. 3.2.
3.1 1 2. 2.
,
1. , (IRR ) . 3.2
3.2.
-
31
, IRR
, IRR
P1 = 0,25
2 = 0,5
3 = 0,25
90%
20%
- 50%
25%
20%
15% ERR. .
: (3.1) :
ERR = 0,25 90% + 0,5 20% + 0,25 (-50%) = 20%.
:
ERR = 0,25 25% + 0,5 20% + 0,25 15% = 20%
, , , IRR : -50% 90%, 15% 25%. 3.3 , ( ).
. 3.3. ,
50%, 90%. 15% 25% . , ERR (20%) , . , .
-
32
. 3.4.
, , , - (ERR), , ERR. , . .
, -
- -
n
i
ii pERRIRR1
2
(3.2)
, , .
ERR 68,26%.
.
:
5,4925,0)2050(5,0)2020(25,0)2090( 222 A %
:
5,325,0)2015(5,0)2020(25,0)2025( 222 B %
, 68,26% IRR = 20% 3,5%, .. 16,5% 23,5%. . . 68,26% -29,5% 69,5%.
, 30%.
20 90 ERR
-
33
- . , .
1.
, , 2 . 3.1. , , 12.0%. ,
n
i
ii pERRIRR1
22 = (-2,0 12,0)20,05 + (9,0 12,0)20,20 + (12,0-12,0)20,50
+
+(15,0-12,0)20,20 + (26,0-12,0)
20,05 = 23,20,
2 =4,82% ,
. , , , 68.3% , 95.4% (99.7%) .
. 3.3 , 1, , . , - , 2 . . 3.3 , , , - , 2 . ; , , , , , .
3.3
-
34
-
1
2
1. , % 2. 3. , % 4.
8.00
0.00
0.00
0.00
9.20
0.71
0.84
0.09
10.30
19.31
4.39
0.43
12.00
23.20
4.82
0.40
3. , ?
, , 3.
3. , , , , . , , . , . , , , . , , (semivariance, SV),
m
i
ii pERRIRRSV1
2 , (3.3)
, . , , (. 3.1). , 9.2%, (3.3)
SV = (8,0 9,2)0,52 + (8,5 9,2)20,20 + (9,0 9,2)20,50 = 0,19. ,
. 2.1, : 0,00; 0,19; 12,54
-
35
11,60. , . 2. 1 - 1 , . , . , , , , , , , .
3.3.2. , ,
CV. :
CV = /ERR (3.4) . ,
CV, .
2. 1,
= 49,5% = 3,5%. .
: CVA = 49,5/20 = 2,475; CVB = 3,5/20 = 0,175.
1, 2, , : 2,475/ 0,175 = 14. 14 .
, ERR.
2. D 3.4:
3.4.
D
-
36
, IRR
D, IRR
1 = 0,2
2 = 0,6
3 = 0,2
30%
20%
10%
115%
80%
45%
3. ERR, CV. .
(3.1) : ERRC = 300,2 + 200,6 + 100,2 = 20%;
ERRD= 1150,2 + 800,6 + 450,2 = 80%.
(3,2):
%3,62,0)2010(6,0)2020(2,0)2030( 222 C
%14,222,0)8045(6,0)8080(2,0)80115( 222 D
, D , ERR. ,
CV, ERR (. . 3.5).
3.5. C D
(3.4) : CV = 6,3/20 = 0,315; CVD = 22,14/80 = 0,276.
, , CV D , .. , ERRD.
-
37
CV .
4. 1. 1 2 ? . .
: 4- . 3.2 . , , : 2 , 1, , .
, ,
: CV. , . .
3. , L (. . 3.5).
, . 3.5 L
L
1995
1996
1997
1998
20%
15%
18%
23%
40%
24%
30%
50%
. 1-2 1 . , .
-
38
3.
n
i
i ARRIRRn 1
21 (3.5)
n , , a ARR (ARR Average Rate of Return, ) IRR n :
n
i
iIRRn
ARR1
1 (3.6)
, (3.6) : ARRK = (20 + 15 + 18 + 3)/4 = 19%; ARRL = (40 + 24 + 30 + 50)/4 =
36%.
(3.5)
%9,24:})1923()1918()1915()1920{( 2222 K
%9,94:})3650()3630()3624()3640{( 2222 L
, L ,
. CV. (3.4) : CVK = 2,9/19= 0,15; CVL = 9,9 / 36 = 0,275.
L 2 , , .
3.5 , L . , L .
CV , L.
3.4.
, (), Y . Y , . 1 = Y/,
1. , 11 , . Y 2 = Y/,
2 (, 12). /Y /(Y),
-
39
. /Y
/(Y) 1/1 1/2. ,
:
,
- . .
-
40
4. .
4.1. .
, . , , , , , . .
, . . , , . (fair price) . , , , . .
(1952). . . , .
(1964) (1965), () .
( ) . , , , , . . , .
, . . .
:
-
41
1. mp- .
mp = xi mi , (4. 1.)
xi - , ( ) XT=(1, 2, n);
mi - .
2. - p - . i ( Vp), :
2p =V p = X
T*COV*X
1
1 11
22 2N
i
N
ij
jiijji
N
i
ii rxxx ,
(4.2.) COV- ii n.
- , , , . , . x y :
n
iii yyxx
nyxCov
1
))((1
),(
(4.3)
, , iii. . i j :
ri,j = COVi,j /i j, (4.4)
, , . ri,j , . ,
-
42
, 1.0 +1.0. , (4.4) : ri,j :
COVi,j = ri,j i j, i - i , rij i- j- .
(. 4.1. ) , , , . , , . , .
(. 4.1. ), , , .. .
, : 12% , 5.1%. 21.2%, 8.3%.
, , . 4.1.
. 4.1
-
43
4.1.
:
r=-1.00 r=-0.7
r=0 r=0.18 =1,00
0 0.00 1.00 5.10 8.3 8.3 8.3 8.3 8.3
1 0.05 0.95 5.45 6.825 7.183 7.956 8.143 8.945
2 0.10 0.90 5.79 5.35 6.174 7.765 8.124 9.59
3 0.15 0.85 6.14 3.875 5.336 7.739 8.244 10.235
4 0.20 0.80 6.48 2.4 4.759 7.878 8.497 10.88
5 0.25 0.75 6.83 0.925 4.544 8.176 8.872 11.525
6 0.28 0.72 7.03 0.04 9.15 11.91
7 0.30 0.70 7.17 0.55 4.741 8.614 9.355 12.17
8 0.35 0.65 7.52 2.025 5.303 9.174 9.928 12.815
9 0.40 0.60 7.86 3.5 6.131 9.834 10.579 13.46
10 0.45 0.55 8.21 4.975 7.133 10.576
11.293 14.105
11 0.50 0.50 8.55 6.45 8.246 11.383
12.059 14.75
12 0.55 0.45 8.90 7.925 9.431 12.244
12.868 15.395
13 0.60 0.40 9.24. 9.4 10.663 13.146
13.712 16.04
14 0.65 0.35 9.59 10.875 11.928 14.083
14.586 16.685
15 0.70 0.30 9.93 12.35 13.217 15.047
15.483 17.33
16 0.75 0.25 10.28 13.825 14.523 16.035
16.401 17.975
17 0.80 0.20 10.62 15.3 15.842 17.041
17.336 18.62
18 0.85 0.15 10.97 16.775 17.172 18.063
18.285 19.265
19 0.90 0.10 11.31 18.25 18.508 19.098
19.247 19.91
-
44
20 0.95 0.05 11.66 19.725 19.852 20.144
20.219 20.555
21 1.00 0.00 12.00 21.2 21.2 21.2 21.2 21.2
0 , 21 - . , , , 5,1%, 8,3%. , , 12%, 21,2%. , 60% 40% , 9,24%, 13,71%, ( = 0,18). ( = 1,00), , 60% . , , . , , ( = -1,00), 9,4%. , 28% 72% - ( 6), . , , , , 7,03%.
(. 4.1) .
-
45
. 4.2. .
4.2. .
[1, 7, 8, 9] , , . , , , . , , . , , , , , . . , .
5.1 5.79 6.48 7.17 7.86 8.55 9.24 9.93 10.62 11.31 120.55
4.68
8.81
12.94
17.07
21.2
0.55
sko1i 1
sko1i 2
sko1i 3
sko1i 4
sko1i 5
125.1 mpi
-
46
1) . , :
1n
i
ix
( ) , , . 2) ( ). , :
1n
i
ix ;
xi 0 i.
, (long) . ( ). . , , . , (short sale). (), , . ( ). , .
, , .
3) --. , .
4..1.
- 12% 21.1 5.1% 8.3 ,
-
47
(mp = xi mi) 8.9%. 0.18.
.
. = (X1, X2),
p. X1 - ; X2 - ,
p= XCOVXT = 22
222,12121
21
21 2 xrxxx =
= 222
21
2
1
2 3.818.03.82.2122.21 xxxx min
: X1 + X2 =1
12 X1 + 5.1 X2 8.9 X1,X2 0
.
. 4.3. 12.88% (1=0.55 2=0.45),
X1 + X2 =1 12 X1 + 5.1 X2 8.9 . EXCEL
. (.4.4. . 4.2.).
-
48
.4.4. 5
.
. 4.2. . Microsoft Excel 8.0
$E$5 0 12.8799
$A$3 0 0.550
$B$3 0 0.449
: 12.88% , 0.55, 0.45.
4.2.
REXX, SNS LIKX :
REXX SNS LIKX
mi (%) 12 10.5 11
i 25 10 20
-
49
REXX SNS LIKX
REXX 1 0.52 0.27 SNS 0.52 1 0.75
LIKX 0.27 0.75 1 16. . . = (X1, X2 , X3),
mp. X1 REXX, X2 SNS, X3 LIKX.
mp=12X1+7X2+11X3max X1 + X2 + X3 =1
p= XCOVXT
16
X1,X2, X3 0
iv
COVi,j= ri,j i j,.
COV=
400150135
150100130
135130625
EXCEL 10. 11.29 , $G1:$I1 (. 4.5.)
10
.
-
50
. 4.5. L (X).
: 11.324% , REXX, SNS LIKX 0.47, 0.29 0.25
4.3. .
, - , .
11 ( -, Standard & Poors).
11
rm .
Standard & Poors: S&P-100 (500, 1000 .) -.
(), 1995 . - , . ,
( 20 1999 . 46 , 27 ). , ,
, .
(1 1995 .).
-
51
. ,
(, ) , .. , , , . , , , , , . (market model)[9]:
mi=ai+imr+i (4.5.) mi - i ( );
mr - (, );
ai - , i , ;
i - , , i , ;
i - . (4.5)
v. -
, . - (beta) :
i =
21
1
xxN
yyxxN
2
1
x
yyxxN
2xx
yyxx
)(
),(
xVar
yxCov= 2
mr
ir
=
2
1
1
mr
k
irrii mmmm
k
(4.6)
. , .
, &, (), Moscow Times, ROS, ( , ). , ( .
-
52
i = rii mm
ir- i- , mr
2-
. -
. , , , . - . , . - .
(4.5), i,
i2= Var(mi),
12: (1)
( ) (market risk); (2) ( ) (unique risk)vi.
Var(mi)= Var(ai+imr+)= =Var(ai)+Var(imr)+Var(i)=
= 2i Var(mr)+ Var(i).
, Var(mi)= i2 :
i2 = i
2mr2+
2,
N
rm
rm
N
imr
1
2
2 ,
2 = Nt /)(
2 ,
12
=
2 yyi =
, +
2
yyi
+
2
ii yy
=
i2 =
+
i2mr
2 +
2
-
53
imr i(
), i, i (4.5.).
, : , , ,
. i2mr
2/i
2 Ri2
R-squared ( Ri2
)vii. , . , R-squared , - , .
, , . , - , 1, . 1, , . . , - 1 ( 0).
. , ( ) . , i- xi, :
mp = xi(i + i mr + ). (4.7) , :
. , , , . - , . (, ).
-
54
(4.7) , : .
p2 = p
2mr2+p
2 ,
(4.8.)
2
1
2 )(
n
iiip x ,
N
i
iip x1
222
( ) . , .
[1, .172]:
= (X1, X2, Xn),
p .
p =
iiimr
iii xx
222
2
pi
riii mmax ) (
1n
i
ix
-
55
, :
1) n , , N , .
2) (, &) rm .
3) ( ).
4) mr2,
2mr
iri
5) 2i
6)
,
i . - mp, , .
- (Capital Assets Pricing Model, CAPM). , , .
(P). (P)
. (P) [16]. C , -
im~ ,
. ( , , , .), , , , .
-
56
:
ifrfi mmmm )(~
, (4.9)
im~ -
; mf - , . - , , .
i - i (i) .
. - , ,
i- . - , , i- . ( fr mm ) - .
- (Security Market Line SML). SML :
ifrfi mmmm )(~
SML , . SML . , SML, (), , / , /. 4.3. 4.7.
(SML) . ,
- :
. .
, mf.
-
57
, , , . , , .
mf.
mi = ai + i mr = mf + i(mr mf)+ i, i,= ai + (i-1) mf.
mf . , , ,
. , , , =0.
. , > 0,
, a < 0, . . ,
( fr mm ) 8% (
50 ). , , , ( ) 5%
, - 0,65, , , :
ifrfi mmmm )(~ = 5% + 0,65 x 8% = 10,2% , .
, , .
: ? ,
, m0 , (treasure bills), .
. 1998 .: ( ) 200% , ( ) 15%. , , , .
: , .. ( fr mm ) ?
-
58
. -, - , . , , , . , , , , .
-, , .
, . 50 . , , . . : ; , , . , .
, 1995-1997 . , 80% ( ). , , , .
(CML) , . . , .
, , , , . ,
, p. ,
(CML),
, . . m p (mp rm , p mr )
-
59
mp= mf +
mr
fr mm
p , (4.10)
mp - () ; mf - ;
mr - ; p - .
:
.
.
. , , , , 100, . ? , . 1, 1. 20%, 20%.
, 1.5. . , 30%, 1.5 , . 1.5 50% 150% .
, 0.5 , , . : , , . , .
. . , , , .
, . : 1)
, 2) ; 3) copa .
-
60
, , . .
, , , . . , Merril Lynch, - , . , . {PRIVATE "TYPE=PICT;ALT="}
CAPM .
, - 2.5, . 6.25%, , S&P 500, 14%. , , 25.625% (6.25 + 2,5 * (14 6.25)). 19.375%. , , , : 25.625%. NPV , . - 1.5, 11.625% (1,5 * (14 6.25)), 17.875%.
-
61
. -
; - () .
4.3. GLSYTR (mi)
(mr) :
1 2 3 4 5 6 7 8 9 10
mi 23 21 20 22 23 24 25 27 25 20
mr 10 9 9 10 10 11 11 12 10 8
, 4%. ( ,
(), (SML) ). :
1) , mi , mr - ;
2) : ( ) , ( ) , R2
, . 3) ;
2.5
mf = 6.25%
-
62
4) (SML).
1)
13 EXCEL. 1. (. 4.4. 4.5.).
. 4.4. - .
13
.. - . EXCEL.
-
63
. 4.5. .
2. (. 4.3 4.5). 4.3.
Y-
4.667
mf 1.833
4.4.
df SS MS
1 40.3333 40.333 8 7.667 0.9533 9 48
4.5.
mi
1 23.000 0.000
2 21.167 -0.167
3 21.167 -1.167
4 23.000 -1.000
5 23.000 0.000
-
64
6 24.833 -0.833
7 24.833 0.167
8 26.667 0.333
9 23.000 2.000
10 19.333 0.667
4.3,
mi = 4.667 + 1.833 mr. , - GLSYTR 1.833.
.
i
=
2
1
1
mr
k
irrii mmmm
k
=2.2/1.2=1.833,
N
m
m
N
ii
i1 230/10=23,
N
m
m
N
ir
r
1 =100/10=10,
N
mmN
i
rr
mr
1
2
2 = 1.2,
N
irrii mmmm
N 1
1=2.2
2
2 = Nt /)(
2 .
2 = 7.667/10 = 0.77 (7.667 . 4.)
4.
df SS MS
1 40.3333 40.333 8 7.667 0.9533 9 48
4.
Df
SS
MS
k =1 2
yyt
2
yyt
/k
n-k-1 = 8 2)(te 2)(te /(n-k-1)
-
65
n-1 = 9 2
yyt
( )
i2 = 1.833*1.833=3.36,
: i2mr
2 = 3.36*1.2= 4.03.
i2 = i
2mr2+
2=4.03+0.77=4.8
R-squared 0.840 ( . 5) .
Ri2 =i
2mr2/
2i = 4.03/4.8=0.84
, . GLSYTR 84% .
5.
R
0.917
R- 0.840 R-
0.820
0.979
10.000
i,= ai + (i - 1)mf= 4.667 +(1.833 1) 4=8 GLSYTR
, . . 1.833.
GLSYTR . 8.
3) GLSYTR 4.6.
-
66
mi = 4.67 + 1.833mf
18
19
20
21
22
23
24
25
26
27
28
7 8 9 10 11 12 13
mf -
mi
-
GL
SY
TR
. 4.6.
4) . 4.7. (SML).
-
67
4.4 . .
( ) (factor models) , ( ).
. ,
- , . , . , . , , . , (Arbitrage Pricing Theory, ). , , .
, , . , , , .
, :
1. . 2. . 3. . 4. .
, . . , , .
, , . , , .
-
68
BARRA, 1970- . . BARRA ( , ) . BARRA, 2, 68 . BARRA , , .
(1976). . , , , . SML
:
ijjifi mmmm )(...)( 0101 ,
j -
j )( 1.00j
)( 0j .
: , . : , , , ..
, , . , (, , ). . , .
-
69
4.5 1 EXCEL
.
. 1. : a0, ,
( ) , R2; 2.
GLSYTR TRUW , (mp) () .
.
GLSYTR
TRUW
(mf)
m1 m2
1 10 3 23 14 2 9 6 21 12 3 9 6 20 11 4 10 5.5 22 15 5 10 8 23 14 6 11 9 24 16 7 11 6 25 16 8 12 5.5 27 17 9 10 4.5 25 15
10 8 6.5 20 12
.
-
70
. 1. .
. TRUW . EXCEL.
:
1) . (. 2)
2) (. 3),
3) Y , . , (. 4).
4) , .
5) .
6) .
7) .
-
71
.2.
.3.
-
72
.4. . .
. 1-4 .
. 3
a0, a1. , - t-, .
t-
Y-
-1.633 2.412 -0.677
(mf) 1.583 0.240 6.605
-
73
TRUW (m2 ) mr
m2 = -1.63 + 1.58mr
R 0.919 R- 0.845 R-
0.826
0.830 10
df SS MS F F
1 30.083 30.083 43.625
0.000
8 5.517 0.690 9 35.6
( ) TRUW
22 = 2/N = 5.517/10 = 0.5517
GLSYTR .
m1 = 4.667 + 1.833 mr 12 = 2/N = 7.667/10 = 0.767
. =
(X1, X2), p. EXCEL .
:
= (X1, X2, Xn),
p.
p =
i
iimr
i
ii xx222
2
p
i
riii mmax ) (
-
74
1n
i
ix
- . X1 - GLSYTR; X2 - TRUW. ,
, .. 6% (60/10=6%).
p=
i
iimr
i
ii xx222
2
=
552.0767.02.1)58.158.183.1283.1( 2221
22221
221 xxxxxx mi
n
x1 + x2 = 1
) (i
riiip maxm
101.581.631083.167.4 21 xx 6 x1 , x2 0
.5.
-
75
.6. . D25 E25 1 2 ( ).
:
p=
i
iimr
i
ii xx222
2
=
552.0767.02.1)58.158.183.1283.1( 2221
22221
221 xxxxxx
.7.
2
mr .
19.
-
76
.
.8. (1).
.9. :
-
77
(D25*D25*B24*B24+2*B24*B25*E25*D25*+E25*E25*B25*B25)*A19+D25
*D25*B27+E25*E25*B28) ( 2).
.10.
-
78
.11. (G27), (D25:E25), (.12)
.12.
-
79
.13. .
.14. .
1.83 X1 X2
1.58 0.056 0.944
-
80
. 1
0.767 1.88
. 2
0.552
a01 4.67 1 1 1.000 1 a02 -1.63 23 14.2 14.692 6
m( ) 6
: 1.88 % , GLSYTR 5.6%, TRUW 94.4%.
-
81
.
.
1.
(.*) .
. 3. : a0, ,
( ) , (
) , R2 , .; 4.
(.**) , (mp) () .
5. (ML); 6. (SML).
*
1 2 3 4 5
1 5 3.1 10 8.1 16 6 5
2 0 1.8 -1 3 6 4 4
3 12 1 8 5.3 1 7 7
4 5 3 7 1 -3 6 12
5 -4.6 3 -5 -3.1 -5 -9 -2
6 -8.9 2.1 -10 -12 -17 -12 -5
7 12 3.8 14 5 15 16 8
8 5 4.1 3 3.2 6 3 7
9 6 3.2 1 1.2 -5 9 9
10 4 3 5 1.3 -4 2 8
11 -3 1.9 -7 5 5 -7 5
12 -7 3.2 -8 3 8 8 -8
13 4 1.6 5 -6 9 9 6
14 6.5 3 9 5 -6 7 -5
-
82
15 9 2.9 8.7 3 15 14 4
:
**
1 2 3 4 5 6 7 8 9 10
1;
2
1;
3
1;
4
1;
5
2;
3
2;
4
2;
5
3;
4
3;
5
4;
5
2.
( %) yt - xt 10 . 75%
25% . 25%
75% .
: 1.
11 12 , .
2. 11 12 :
) ; ) , .
3. , .
(t = 1, 2, , 10)
1 2 3 4 5 6 7 8 9 10
I yt 2,97 3,06 2,85 1,88 1,90 2,00 2,22 2,11 2,16 2,34
xt -8,77 -6,03 14,14 24,96 3,71 10,65 -0,22 0,27 -3,08 -6,72
II yt 3,06 2,85 1,88 1,90 2,00 2,22 2,11 2,16 2,34 2,44
xt -6,03 14,14 24,96 3,71 10,65 -0,22 0,27 -3,08 -6,72 8,58
III yt 2,85 1,88 1,90 2,00 2,22 2,11 2,16 2,34 2,44 2,40
xt 14,14 24,96 3,71 10,65 -0,22 0,27 -3,08 -6,72 8,58 1,15
IV yt 1,88 1,90 2,00 2,22 2,11 2,16 2,34 2,44 2,40 1,89
xt 24,96 3,71 10,65 -0,22 0,27 -3,08 -6,72 8,58 1,15 7,87
-
83
V yt 1,90 2,00 2,22 2,11 2,16 2,34 2,44 2,40 1,89 1,94
xt 3,71 10,65 -0,22 0,27 -3,08 -6,72 8,58 1,15 7,87 5,92
VI yt 2,00 2,22 2,11 2,16 2,34 2,44 2,40 1,89 1,94 1,72
xt 10,65 -0,22 0,27 -3,08 -6,72 8,58 1,15 7,87 5,92 -3,10
VII yt 2,22 2,11 2,16 2,34 2,44 2,40 1,89 1,94 1,72 1,75
xt -0,22 0,27 -3,08 -6,72 8,58 1,15 7,87 5,92 -3,10 13,61
VIII yt 2,11 2,16 2,34 2,44 2,40 1,89 1,94 1,72 1,75 2,01
xt 0,27 -3,08 -6,72 8,58 1,15 7,87 5,92 -3,10 13,61 -5,86
IX yt 2,16 2,34 2,44 2,40 1,89 1,94 1,72 1,75 2,01 2,11
xt -3,08 -6,72 8,58 1,15 7,87 5,92 -3,10 13,61 -5,86 -2,94
X yt 2,34 2,44 2,40 1,89 1,94 1,72 1,75 2,01 2,11 1,91
xt -6,72 8,58 1,15 7,87 5,92 -3,10 13,61 -5,86 -2,94 13,77
1. .. : . . .: -, 1999. - 247 .
2. .. : , , : . .: , 1999. 112 .
3. .. - . L / : . - .: , 2000.
1. Mathcad: . .:
" ", 1999. 656. 2. ., . , , -
, 1996 . 3. .. -
.: "", 1998 4. .., .., ..
, , . - ., . "". 1998.
5. .. . . - ., . " ". 1996.
6. ., ., . .: -, 1997
-
84
7. ., . : .: , 1997
8. / . . . .: -, 1996
9. . , , Catallaxy, , 1994 .
10. .. . . - ., . "", 1997..
11. . . - ., . " ". 1994. 12. .. . -.:
" ", 1998 13. ., . : :
2- / . . . .. .: , 1997.
14. . : .:,1998.576. .
15. .. - -.: , 2001.120 .
-
85
i
.
2x .
:
22x )( xE
2x x - -
.
n nxxx ,..., 21 :
2)(
1)( xxn
xVar i .
.
2s :
22 )(1
1xx
ns i .
,
n
iix
nx
1
1
, 2
22 )(
1
1xx
ns i
ii
. x y
n
iii yyxx
nyxCov
1
))((1
),(
iii
.
.
x y :
22
,
yx
yx
x,y
.
:
22,
xxyy
xxyyr xy
yx
yxCov
),(.
-
86
iv 2p =V
p = X
T*COV*X
1
1 11
22 2N
i
N
ij
jiijji
N
i
ii rxxx
323,232
313,131212,121
2
3
2
3
2
2
2
2
2
1
2
1
2
22
rxx
rxxrxxxxx
v .
(), a,
; y , . . :
n
iii yy
1
2 min.
:
a = (X
X )-1
X
Y Y .
xaaY 10
.
1 :
1=
21
1
xxN
yyxxN
2
1
x
yyxxN
)(
),(
xVar
yxCov=
x
y
yxr
,
0= xay 1 mi = ai + i mf :
i = 2mf
im
=
2
1
1
mf
k
iffii mmmm
k
fiii mma
vi .
,
- iy
ie ;
iii eyy
(*)
iy
, i ,
, , . ,
, , . ie , y. , .
(*) :
)()()( eVaryVaryVar (**)
, Var () : )(yVar
,
, Var(e) .
-
87
n (**), :
n
i
n
iiii
n
ii yyyyyy
1 1
22
1
2 ,)()( (***)
ty
- y, ;
Se
2 =
n
iii yy
1
2=
n
iie
1
2 - ;
Sy2 =
n
ii yy
1
2)( -
,
2
1
)( yyi
n
i
- , .
vii
- R2.
2
2
2
2
2)(
1yy
yy
yy
teR
t
t
tt
, , . . , Y .
. , -
. R2, 2R , :
1
111 22
kn
nRR ,
n ; k .