-

-

.., .., .., .., ..

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 .