buhvalov a v finansovye vychisleniya dlya professionalov nas
DESCRIPTION
Определение уровня корпоративного управления, сложившегося в среднемсреди российских компаний, включенных в исследование, и проведениесравнительного анализа динамики развития основных компонентов практикикорпоративного управления в данных компаниях.2. Выявление основных аспектов практики корпоративного управления, вкоторых наиболее заметно проявилось ее улучшение.3. Выявление наиболее проблемных аспектов практики корпоративногоуправления, требующих усиления внимания к ним со стороны акционеров,менеджмента, потенциальных инвесторов и других заинтересованныхсторон.4. Сопоставление уровня развития корпоративного управления по выборке вцелом и в компаниях, относящихся к различным группам в рамках выборки(в компаниях, имеющих листинг, и в компаниях IPO).TRANSCRIPT
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-, 2001
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14
1. 15
1.1. . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2. . . . . . . . . . . . . . . 17
1.3. . . . . . . . . . . . . . . . . . . . . . 22
1.4. . . . . . . . . . . . . . . . . . . . . . . . . 24
1.5. Excel . . . . . . . . . . . . . . . . . . . . . . . . . . 26
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2. 33
2.1. . . . . . . . . . . . . . . . . 33
2.2. . . . . . . . . . . 35
2.3. , . . . . . . . . . 36
2.4. . . . . . . . . . . . . . . . . . . . . . 37
2.5. . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.7. . . . 39
2.8. . . . . . . . . . . . . . 42
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3. 47
3.1. . . . . . . . . . . . . . . . . 47
3.2. . . . . . . . . . . . . 50
3.3. . . . . . . . . . 51
3.4. . . . . . . . . . . . 52
3.5. . . . . . . . . . . . . . . 53
3.6. . . . . . . . . . . . . . . . . 59
-
4
3.7. . . . . . . . . . . . . . . . . . . . . . . 60
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4. 68
4.1. . . . . . . . . . . . . . . . . . . 68
4.2. . . . . . . . . . . . . . . . . . . . . 69
4.3. . . . . . . . . . . . . . . . . . . . . . . . . 70
4.4. . . . . . . . . . . . . . . . . 72
4.5. . . . . . . . . . . . . . . . . . . . . . 74
4.6. . . . . . . . . . . 76
4.7. Excel . . . . . . . . . . . . . . . . . . . . . . . . . . 80
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5. 91
5.1. . . . . . . . . . . . 91
5.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.3. . . . . . . . . . . . . . . . . . . 99
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6. 108
6.1. . . . . . . . . . . . . . . . . . . . . 108
6.2. . sn; i . . . . . . . . . . . . . . . . 109
6.3. . . . . . . . . . . . 111
6.4. . . . . . . . . . . . . . . . . . . . . . . 111
6.4.1. . . . . . 111
6.4.2. m . . . . . 115
6.4.3. . . . . 120
6.5. . . . . . . . . . . . . . . . . . . . . 125
6.6. , . . . . . . . . . . . . . . . . . . . . . 126
6.7. Excel . . . . . . . . . . . . . . . . . . . . . . . . . . 127
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7. 134
7.1. . an;i . . . . . . . . . . . . . . . . 134
7.2. . . . . . . . . . 136
-
5
7.3. . . . . . . . . . . . . 137
7.3.1. . . . . . 137
7.3.2. m . . . . . 138
7.3.3. . . . . 140
7.3.4. . . . . . . . . . . . . . . . . . . . . . . . . 145
7.4. . . . . . . . . . . . . . . . . . . . . . 152
7.5. . . . . . . . . . . . . . . . . . . 153
7.6. . . . . . . . . . . . . . 155
7.7. Excel . . . . . . . . . . . . . . . . . . . . . . . . . . 158
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
8. 165
8.1. . . . . . . . . . . . . . . . . . . . . . . . 165
8.2. , . . . 167
8.3. . . . . . . . . . . . . . 171
8.4. . . . . . 179
8.5. . . . . . . . . . . . . . . . . . . . . 181
8.6. . . . . . . . . . . . . . . . . 183
8.7. . . . . . . . . . . . . . . . . 183
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
9. 192
9.1. . . . . . . . . . . . . . . . . . . . . . . 192
9.2. . . . . . . . . . . . . . . . . . . . . . . 195
9.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
9.4. . . . . . . . . . . . . 205
9.5. . . . . . . . . . . . . . . . . . . . . . 208
9.6. Excel . . . . . . . . . . . . . . . . . . . . . . . . . . 219
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
10. 230
10.1. . . . . . . . . . . . . . . . . . . . . 230
10.2. . . . . . . . . . . . . 233
10.3. . . . . . . . . . . . . . . . . . . . . . . . . 235
10.4. . . . . . . . . . . . . . . 236
10.5. . . . . . . . . . . . . . . 245
10.6. NPV IRR . . . . . . . . . . . . . . . . 247
-
6
10.7. . . . . . . . . 24910.8. Excel . . . . . . . . . . . . . . . . . . . . . . . . . 252
10.8.1. . . . . . . . . . 25310.8.2. . . . . . . . . . . . . . . . . . . . . 25710.8.3. . . . . . . . . . 259
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
267
A. Excel 296
A.1. Microsoft Excel . . . . . . . . . . . . . . . 296A.2. . . . . . . . . . . . . . . . . 297A.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299A.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302A.5. . . . . . . . . . . . . . . . . . . . . . 304
. 307
1. . . . . 307 2. . . . . . . . 309 3. . . . . . . . 311 4. Kp; i . . . . . . . . . . . . . . . 313
. - 314
-
? - , - -, . , 1917 . , , -, , - .
? , -, , , , ... - . , - ? - , ? , - . , , - MS Excel, Microsoft Oce, -
. ,
, - . ; - , , , .
? - . . 130
-
8
250 . CD ROM . CD ROM .
? - , - . -, , .- , -, - -.
.. .. [1], - . . . - . , -, . - MS Excel. , . ., , , . . , .
? - , , . . , - , - . , , -. , -
-
9
. , , - .
, , , - -. . - -.
, .., (. [79]). , . , , - , -, , . [2, 5, 6] Excel -. , . - Excel [4].
? - , , - - . , - , - . , , . , .
, -, . - : (PV Present Value), - (NPV Net Present Value), (IRR
-
10
Internal Rate of Return). , , - (., , [10]). , , , . - -.
. XXI- - (, , - -) . . . , - .
- MS Excel. . , Excel , -. , , ( ), ( ), 2.32. , ( ) .
- ? - MS Word Adobe Page Maker . MiKTEX, - , TEX , (- ), . LATEX 2 -
-
11
(. [3]). PS PDF , . MS Windows -. -.
-? ? - - ( ) , -, - . , - .
-, , - , . , ( ) - LATEX. , , : - , , , ..
, , - . . -. , . , , ( , - !) ( ) . , . - , ( !). 1990- , - , , .
-
12
- , - - , -. - . -. - Excel. - . Excel - .
.., - -, - , - . .
? - . . , . -.
, , . - . -,
,
, , , , .
: , - : , . 3.2.
. . . , - , - .
-
13
1. .., .. -. .: , -, 1997. 176 .
2. . MS Excel VBA . .: -, 1999. 336 .
3. ., ., . LATEX LATEX2. .: , 1999. 606 .
4. .., .. Microsoft Excel 2000. .:-, 1999. 1088 .
5. .. . , , - . .: , , 1998. 400 .
6. .., .., .. -- Excel. 3- . .: , 1999. 324 .
7. .. . 2- ., . . .: , 1995. 320 .
8. .. . .: , 1998. 255 .
9. .. . .: , 2000. 400 .
10. ., ., . . . . .:-, 1997. XII+1024 .
-
- (1985, ); - - .- - . -, , -.
. ,
Positivity (Kluwer Academic Publishers). -
, : , , - , , - - . (, ).
- (1995, ), - - .-- . - , . , , . CD ROM, , - , . , - CD ROM.
- (1965, ), - .- - . , , . - . - . - .
-
1.
1.1.
.
: %.
-
pro centum , . , ,
. , -
.
. -, , - . , - - - . , 78% 1998 . ( ). - , 20% 35%. -, , - ( ) .
, .
, 100%. . , 302.4 . . 78% ,
302.478/100 = 236 . .
-
16 1.
, - , . , - A B, A 20% , B 15% , -, 35% , , A - 5% , B. .
, .
1. , n% A.
. x - :
A 100%,
x n%.
, :
A
x=
100
n,
x:
x =An100
. (1.1)
2. , n% A.
. x - :
x 100%,
A n%.
:
x
A=
100
n,
-
1.2. 17
x:
x =A100
n. (1.2)
3. , B A.
. x :
B 100%,
A x%.
:
B
A=
100
x,
x:
x =A
B100% . (1.3)
- .
1.2.
1.1. - 1 200 . 20%. ?
. x :
1 200 . 100%,
x . 20%.
1. (1.1), :
x =120020
100= 240 .
-
18 1.
1.2. 180 , 60% . ?
. x - :
x 100%,
180 60%.
2. x (1.2):
x =180100
60= 300 .
1.3. 210 70 . ?
. x :
210 100%,
70 x%.
3. x (1.3):
x =70
210100% = 331
3%.
.
1.4. 800 . 600 . , 20% , 15% . - ?
.
800 + 600 = 1400 .
, . - (1.1),
8000.20 = 160 .,
-
1.2. 19
6000.15 = 90 . .
160 + 90 = 250 .
, 250 - 1 400. x - (1.3):
x =250
1 400100% = 176
7%.
1.5. , , 1.4 , , , .
. , - 160 ., 90 . 160 90. (100%) - 90. , 160 90. x. (1.3):
x =160
90100% = 1777
9%.
, , , 7779% , , .
1.6. , , 1.4 , , , - .
. , 90 160. (100%) 160. -, 90 160. x (1.3):
x =90
160100% = 56.25%.
, , , 43.75%, , .
-
20 1.
1.7. 1.4 - 800 . . , , , , - .
. 1.4 , , - , 160 .
8000.15 = 120 . 160 120. x (1.3):
x =160
120100% = 1331
3%.
, , , 3313%, .
, . - . , .
1.8. 10% - 10%. , .
. 100%.
100% + 100%0.1 = 110%. 10%, 110%0.1 = 11% . ,
110% + 11% = 121%
. , - 21%.
. :
10% + 10% = 20%.
, .
-
1.2. 21
1.9. 800 . 512 . , .
. x. -
800 x100
.,
800 800 x100
= 800
(1 x
100
).
800
(1 x
100
)x
100.
:
800
(1 x
100
) 800
(1 x
100
)x
100= 800
(1 x
100
)2.
512, - x :
800
(1 x
100
)2= 512,
(1 x
100
)2= 0.64.
, 100%,
x 100, 1 x100
0,
:
1 x100
= 0.8, x = 20%.
, , - 3.
. - . :
800 512 = 288 .
-
22 1.
, 288 - 800 . (1.3):
288
800100% = 36%.
2 18%. , , .
1.3.
( ) () . , , , () . - , . ( ) . - .
1.10. 50 , 100 ., 30 , 50 . 10% - , 15%. ?
.
50100 + 3050 = 6500 . . 50 -
501000.1 = 500 ., 30 30500.15 = 225 . -. 500+225 = 725 . . , 725 6 500 (1.3):
725
6 500100% = 11 2
13%.
-
1.3. 23
1.11. - 1.10 , ?
. , - 500 . , 225 . (1.3), 500 225:
500
225100% = 2222
9%.
225 . 100%, 12229% , .
1.12. 1.10 12% ., .
. x , . 30 + x.
50100 + (30 + x)50 .
(30 + x)500.15 = 7.5(30 + x) . 501000.1 = 500 .
(1.3) :
7.5(30 + x) + 500
5000 + 50(30 + x)100 = 12,
, , x = 3623 . , 37 .
1.13. -. , , , . () ?
-
24 1.
. , - , (- ) , ( ). , . - , . , .
, , .
1.4.
, : - , , , , - , . . - . . -.
- . , - . - , - .
- :
(.) (%)
50 000 15
50 000 75 000 25
75 000 34
: - ,
-
1.4. 25
. - 4. .
1.14. $ 140 000, $ 55 000. $ 50 000, 8% . $ 12 000. , .
. , ,
$ 50 0000.08 = $4 000.
$ 140 000 $ 55 000 $ 4 000 $ 12 000 = $69 000. 25%.
$ 69 0000.25 = $17 250.
, , ( , ). , .
( ) :
(.) (%)
30 000 15
30 000 28
- . . . .
-
26 1.
: $ 30 000 15% . $ 30 000, - $ 72 000, $ 30 000 15% , 28%. $ 72 000, 28%.
1.15. - $ 28 000, - $ 68 000, - $ 105 000. , .
. - 15% ,
$ 28 0000.15 = $4 200.- $ 30 000 15%,
$ 30 0000.15 = $4 500. ($ 68 000 $ 30 000 = $38 000) - 28%,
$ 38 0000.28 = $10 640. -
$ 4 500 + $10 640 = $15 140.
- $ 72 000, - 28%.
$ 105 0000.28 = $29 400.
1.5. Excel
, - Excel. -, .
, Excel, . .
-
1.5. Excel 27
. 1.
-
28 1.
. Excel . - , , , .
, ( ), . ( Enter), , , .
() - : -; ; - . - .
(), - . - .
, - . - , . .
. , . , Ctrl+Backspace.
, . : - , - . , . , .
, , -: , , . , , . , , - . , , - .
, , - (. 1).
1.14 1.15 (. 2)
-
1.5. Excel 29
. 2.
-
30 1.
() ( B17, B27). - , , , , - , :
=(; _; _)
- , () . -.
1. 180
2 . 80 . 17% . () ?
2. 1 140 , - 70% . ?
3. 1 215 170 . , - .
4. 80
45 . 55 72 . . , 22% , 15% . ?
5. , , -
4 , , .
6. 4 , , - , , , - .
7. , 4, -
40% 80%. ?
-
31
8. 4 -
50% 100%. , .
9. 30 . . 36.3 . . , .
10. 9 ?
11. 9 ?
12. 20 , - 50 . 70 , 100 . 12% , 9% . , .
13. , 12, 80 - .
14. - 12 , - ?
15. , 12 , , ?
16. , - 12, 10%.
17. , ( ) 12, .
18. , ( ) 12, .
19. $ 800 000, $ 180 000. - $ 300 000 5% .
-
32 1.
$ 22 000. , -.
20. () $ 120 000, 10% . 22% . $ 190 000. , .
21. - : $ 38 000 - ; , - $ 2 100 $ 1 200 . , - $ 12 500. $ 5 500, - $ 6 500. - ?
22. - $ 62 000, $ 12 000, , , $ 8 700. - , , , ?
-
2.
2.1.
, - , , , . - , .
: - , . -. : . - , . r%, r r% . , 9% P P0.09.
P r%, - S 1:
S = P + Pr = P (1 + r) .
P , Pr - . - .
2.1. 17% - (). , - 1 200 .?
1 , , , r = 10%, (1 + r) (1 + 10), (1 + 0.1).
-
34 2.
. :
S = P (1 + r) = 1 200(1 + 0.17) = 1 404 .
, - P r%, , P - . S, n r% P , :
S = P + Prn S = P (1 + rn) .
, , , n - . - ( ) , t:
S = P (1 + rt) . (2.1)
- . , - r% (), , , 3 - 3 , 3.25 . ? - P , - , , Prt, , t . P + Prt = P (1 + rt).
(2.1), , .
, - : , (1/360 , ) (1/52 , ). t (2.1)
-
2.2. 35
k/360 k/52 (k ). , 2 16 , :
t = 2 +16
360=
736
360,
:
t = 2 +2
52=
106
52.
-. . , rm% , r = 12rm%. , rm = 6%, r = 126 = 72%.
2.2.
, .
2.2. 2.1, , , , () ?
. (2.1) :
S1 = 1200(1 + 0.500.17) = 1 302 .,S2 = 1200(1 + 3.000.17) = 1 812 .,S3 = 1200(1 + 5.250.17) = 2 271 .
2.3. 6% . 2 6 10 000 . 16 -. ?
. S = 10000 ., t = 2.5 . - r = 6% = 0.06. (2.1) :
P =S
1 + rt. (2.2)
-
36 2.
, :
P = 10000/(1 + 0.062.5) = 10 000/1.15 = 8695.65 .
2.4. 1 500 . 1 3 1 631.25 . ?
. P = 1500 . t = 1.25 S = 1631.25 . r. (2.1) r:
r =1
t
(S
P 1
). (2.3)
:
r =1
1.25(1 631.25
1 500 1
)=
0.0875
1.25= 0.07 = 7% .
2.3. ,
- . .
2.5. - - , 9 800 . , 10 000 . . r ?
. (2.3) P = 9800, S = 10000, t = 0.25:
r =1
0.25(10 000
9 800 1
)= 0.0816 = 8.16% .
-
2.4. 37
2.4.
. -, , P , ( , ). t r. , (2.1), S = P (1 + rt). , , q, m . :
q =S
tm.
m = 12, .
2.6. , 2 000., , 500 . 6 , . , 6% ?
. 1 500 . , . (2.1) S, r = 6% t = 0.5 :
S = P (1 + rt) = 1 500(1 + 0.060.5) = 1 545 .,
1 545
6= 257.5 .
2.5.
, . r%, S ( ), P - , , t , , Srt P = S Srt. -, :
P = S(1 rt). (2.4)
-
38 2.
2.7. 5 000 . 3 , 5% . ?
. P (2.4) S = 5000, r = 5%, t = 3:
P = 5000(1 0.053) = 5 0000.85 = 4250 .
2.8. 10 000 . . , - 8% ?
. (2.4) P = 10000, r = 0.08, t = 0.25:
10 000 = S(1 0.080.25). S:
S = 10204.08 .
2.9. 2.8, 8% ?
. (2.1) P = 10000, r = 0.08 t = 0.25 - :
S = 10000(1 + 0.080.25) = 10 200 . 2.8, -
, , .
2.6.
, - : S - P , , S. . P :
P = S(1 td) , (2.5)
-
2.7. 39
t , -, d% , .
2.10. ( ) 10 000 . 15 . 15 - 10%. ? , 15 ?
. 15 15 60. -, 360 ( ), t = 60/360 = 1/6. (2.5) S = 10000, d = 0.1, t = 1/6 :
P = 10000(1 160.1) = 10 000 59
60= 9833.33 .
15 15 360 + 60 = 420 , . .,
t = 420/360 = 7/6.
(2.5) S = 10000, d = 0.1, t = 7/6 :
P = 10000(1 760.1) = 10 000 53
60= 8833.33 .
2.7.
, , - . . - : S P -, , S - . , -,
-
40 2.
. , . , , , .
- ( , . .) r%, S t , (2.1), P (1+ rt). () P S, t , () (2.2):
P =S
1 + rt.
- .
-
. ( -). , .
, - . - . - .
2.11. 10 000 . 5 5 000 . - 10 . : 6 000 . 3 , 7 ( ). 7, 8% ?
. ( . .) , . - .
-
2.7. 41
-
60 1 2 3
S3 = 64 5
S1 = 10
6 7S4 = x
8 9 10
S2 = 5
11 t
0, - 0 :
P1 =S1
1 + 5r=
10
1 + 50.08 =10
1.4= 7.143 ,
P2 =S2
1 + 10r=
5
1 + 100.08 =5
1.8= 2.778 ,
P3 =S3
1 + 3r=
6
1 + 30.08 =6
1.24= 4.839 ,
P4 =S4
1 + 7r=
x
1 + 70.08 =x
1.56.
, :
P1 + P2 = P3 + P4.
:
7.143 + 2.778 = 4.839 +x
1.56,
, x:
x = (7.143 + 2.778 4.839)1.56 = 7.928 .
, , 15 000 .
6 000 + 7928 = 13 928 .
-
42 2.
2.12. - , 55 000 . -: 5 11 000 . , 15 000 ., - 10 000 . , , 8% .
. :
-
-
11 000 11 000 11 000 11 000 11 000
10 000 10 000 10 000 10 00015 000
54
4
3
3
2
2
1
1
0
0
t
t
1- :
2- :
P1 P2 0:
P1 =11000
1 + 10.08 +11 000
1 + 20.08 +11 000
1 + 30.08 +
+11 000
1 + 40.08 +11 000
1 + 50.08 = 44 729.39 .,
P2 = 15000 +10 000
1 + 10.08 +10 000
1 + 20.08 +
+10 000
1 + 30.08 +10 000
1 + 40.08 = 48 520.22 .
0 , , , .
2.8.
- r i . ,
-
2.8. 43
(CPI Consumer Price Index), 1+i . P S, :
S = P (1 + i) .
- , - ( ) ., - . - ( ); , - .
r . - :
P = S1 + r
1 + i= S
(1 +
r i1 + i
).
, 1+ i 1, :
P = S(1 + r i).
, , - i , .
r1 , , r1 - . -, r r1 - :
1 + r = (1 + r1)(1 + i) . (2.6)
, - . - , , - . , ,
-
44 2.
. - , . , , , - .
- , , , 1993. (. 1994 .). .
2.13. 1996 . - i = 50%. 1 . . r = 90% ( , ). .
. (2.6):
r1 =r i1 + i
=0.90 0.501 + 0.50
= 0.267 = 26.7%.
, - ( 10% ), .
2.14. - 30%. - , r1 5%.
. (2.6):
r = (1 + r1)(1 + i) 1 = (1 + 0.05)(1 + 0.30) 1 = 0.365 = 36.5%.
1. , 3 000 , , 7% . : ) 3 ,) 1 , ) 3 5 ?
-
45
2. , 4% , 5 000 .: ) 4 , ) 1 , ) 2 9 ?
3. 100 000 . 2 6 120 000 . () ?
4. , 6% , 6 000 . 6 540 .?
5. , 50 000 . 20 000 ., 1 6 4% (), . ?
6. - , 450 . , . , 6% ()?
7. , 15 000 ., 6 000. 2 6 , . , 8% ()?
8. 90 000 . , 12% . ?
9. 8 90 000 . ?
10. 9, 12% ()? : ?
11. - 15 000 ., 1 . 1 , - 7%. - ? - , 1 ?
12. 20 20 000 . , 10 , 1 10%?
13. 1 1992 40 000 ., - 1 , 38 790 . ?
-
46 2.
14. - - 20 000 . -: 5 000 . 2 , 5 000 . 3 10 000 . 5 , . - , 10 000 . 3 10 000 . 4 . , 5% (). , - ?
15. - 14 - 4 , ?
16. - , 200 . . , , - . - , 8% ()?
17. - 16 - . , ?
18. , 120 000 . . - 3 , - 40 000 . - 30 000 ., 90 000 . , . ? 8% ().
19. i = 20%. , r 3%?
20. i = 20%. , 25%.
-
3.
3.1.
2 , , - , . -, . , 2 , i%, i i% .
, P i n , , - , i% .
P Pi. S1 :
S1 = P + Pi = P (1 + i).
P (1 + i) P (1 + i)i . S2 :
S2 = P (1 + i) + P (1 + i)i = P (1 + i)2.
,
S3 = P (1 + i)3,
n- Sn :
Sn = P (1 + i)n .
(1 + i)n . - n . ,
-
48 3.
. , , , - , . ,, (n = 0.5) 3 2 (n = 19/6) . . ( , ) t :
St = P (1 + i)t . (3.1)
i t (. , 1). - .
, , . -. , rm% , P S = P (1 + rm)
12. - i :
P (1 + rm)12 = P (1 + i) ,
i:
i = (1 + rm)12 1.
, rm = 6%, , :
i = (1 + 0.06)12 1 = 101.2%
, . 2.1 , , , .
3.1. 8% . 20 000 . :) 5 , ) 6 ?
. ) (3.1), S P = 20000, i = 0.08, t = 5:
S = 20000(1 + 0.08)5 = 200001.469328 = 29 386.56 .
-
3.1. 49
, 8% , 5 :
S = 20000(1 + 0.085) = 20 0001.4 = 28 000 .) (3.1) S P = 20000,
i = 0.08, t = 6.25:
S = 20000(1 + 0.08)6.25 = 200001.617702 = 32 354.04 .
, - , , , , - , -, . , 1/m , i/m . (3.1) :
S = P
(1 +
i
m
)tm,
t , - (t ). , t = 0.25.
, i - m i/m jm. :
S = P
(1 +
jmm
)tm. (3.2)
3.2. 3.1(), j4 = 8% j12 = 8%.
. (3.2) j4 = 8%:
S = 20000
(1 +
0.08
4
)54= 200001.4859474 = 29 718.95 .
(3.2) j12 = 8%:
S = 20000
(1 +
0.08
12
)512= 200001.4898457 = 29 796.91 .
, , , .
-
50 3.
3.2.
-, . 3.1 3.2. .
3.3. , - j12 = 7%. , 3 000 . 4 6 ?
. (3.2) j12 = 0.07, m = 12, t = 4.5:
3 000 = P
(1 +
0.07
12
)124.5,
3 000 = P (1 + 0.0058)54 .
P :
P = 3000(1 + 0.0058)54 = 30000.7317655 = 2195.30 .
, - , S t . S. :
P = S(1 + i)t, (3.3)
i% t , :
P = S
(1 +
jmm
)tm
, (3.4)
jm t . (1 + i)t .
3.4. 5 000 ., - 2 7 000 . j1 ?
-
3.3. 51
. j1 1 . (3.1) S = 7000, P = 5000, t = 2 :
7 000 = 5000(1 + i)2. i:
(1 + i)2 = 1.4, 1 + i = 1.183; i = 0.183 = 18.3% .
3.5. -, 8 .
. (3.1) S = 2P , t = 8:
2P = P (1 + i)8.
i :
1 + i =82 = 1.09051, i = 0.09051 = 9.051% .
3.3.
( 3.2), , t - jm = j, m -. , m S = Sm :
limm
Sm = limm
P
(1 +
j
m
)mt.
j/m = h. m, h 0, :
limm
Sm = limh0
P (1 + h)jht = P ( lim
h0(1 + h)
1h )jt.
, limh0(1 + h)1/h = e, e = 2.718281828 . . .,
, :
limm
Sm = Pejt.
-
52 3.
- . t :
S = Pet. (3.5)
. j. e
x x ex :
ex = 1 +x
1!+x2
2!+x3
3!+ . . . .
3.6. 3.1, j = 8%.
. (3.5) P = 20000, = j = 0.08,t = 5, :
S = 20000 e0.085 = 20000 e0.4 = 200001.49182 = 29 836.49 .
3.2, , , , , j12.
(3.5) :
P = Set. (3.6)
3.4.
, . 2.6, - dc, , fm, m fm/m%. P , S, :
P = S(1 dc)t, (3.7)
P = S
(1 fm
m
)tm, (3.8)
t ( ).
-
3.5. 53
3.7. 10 000 . 15 . - 15 10%. ? -, 15 ?
. 15 15 60. - (3.7) S = 10000, dc = 0.1, t = 60/360 = 1/6 = 0.1(6):
P = 10000(1 0.1)0.1(6) = 0.98259310 000 = 9825.93 .
15 15 360 + 60 = 420 , t = 420/360 = 7/6 = 1.1(6):
P = 10000(1 0.1)1.1(6) = 0.884333810 000 = 8843.34 .
2.10, , , , - , , .
3.5.
. , . - , . : .
, : , ( is ic); jm, m jm/m ; - ( ); ds dc fm, m . - S :
(1) S = P (1 + tis),
-
54 3.
(2) S = P (1 + ic)t,
(3) S = P
(1 +
jmm
)tm,
(4) S = Pet,
(5) S =P
1 tds ,
(6) S =P
(1 dc)t ,
(7) S =P(
1 fmm)tm .
t ( ).
, .
- - - , - t . 21 , -, 42 - . .
(1) (2), :
P (1 + tis) = P (1 + ic)t,
is ic, :
is =(1 + ic)
t 1t
, (3.9)
ic =t1 + tis 1. (3.10)
, t , .
-
3.5. 55
(1) (3) is jm:
is =
(1 + jmm
)tm 1t
, (3.11)
jm = m(tm1 + tis 1). (3.12)
(1), (4) is :
is =et 1
t, (3.13)
=ln(1 + tis)
t. (3.14)
(1), (5) is ds:
is =ds
1 tds , (3.15)
ds =is
1 + tis. (3.16)
(1), (6) is dc:
is =(1 dc)t 1
t, (3.17)
dc = 1 1t1 + tis. (3.18)
(1), (7) is fm:
is =
(1 fmm
)tm 1t
(3.19)
fm = m
(1 1
tm1 + tis
). (3.20)
(2), (3) ic jm:
ic =
(1 +
jmm
)m 1, (3.21)
jm = m(m1 + ic 1). (3.22)
-
56 3.
, ic jm t , -, (3.21) (3.22) ic jm, t. - ic ( (3.23) (3.24)), ic dc ( (3.27) (3.28)), ic fm ( (3.29) (3.30)), jm ((3.31) (3.32)), jm dc ( (3.35) (3.36)), jm fm ( (3.37) (3.38)), dc ( (3.41) (3.42)), fm ( (3.43) (3.44)), dc fm( (3.49) (3.50)). . (2), (4) ic :
ic = e 1, (3.23)
= ln (1 + ic). (3.24)
(2), (5) ic ds:
ic =1
t1 tds
1, (3.25)
ds =1 (1 + ic)t
t. (3.26)
(2), (6) ic dc:
ic =dc
1 dc , (3.27)
dc =ic
1 + ic. (3.28)
(2), (7) ic fm:
ic =1(
1 fmm)m 1, (3.29)
fm = m
(1 1
m1 + ic
). (3.30)
(3), (4) jm :
jm = m(me 1), (3.31)
= m ln
(1 +
jmm
). (3.32)
-
3.5. 57
(3), (5) jm ds:
jm = m
(1
tm1 tds
1), (3.33)
ds =1
(1 + jmm
)tm
t. (3.34)
(3), (6) jm dc:
jm = m
(1
m1 dc
1), (3.35)
dc = 1 1(1 + jmm
)m . (3.36) (3), (7) jm fm:
jm =fm
1 fmm, (3.37)
fm =jm
1 + jmm. (3.38)
(4), (5) ds:
= ln(1 tds)t
, (3.39)
ds =1 et
t. (3.40)
(4), (6) dc:
= ln (1 dc), (3.41)dc = 1 e. (3.42)
(4), (7) fm:
= m ln(1 fm
m
), (3.43)
fm = m(1 em ). (3.44)
-
58 3.
(5), (6) dc ds:
ds =1 (1 dc)t
t, (3.45)
dc = 1 t1 tds. (3.46)
(5), (7) ds fm:
ds =1
(1 fmm
)tmt
, (3.47)
fm = m(1 tm1 tds ). (3.48)
(6), (7) dc fm:
dc = 1(1 fm
m
)m, (3.49)
fm = m(1 m
1 dc
). (3.50)
- . .
3.8. 5% - 8 . , , - ( ). - ?
. is, ic - . (3.9) :
is =(1 + 0.05)8 1
8= 0.05969 = 5.97% ,
5.97% .
3.9. , 7% -, . - ( ) , ?
-
3.6. 59
. , ic . (3.24) :
= ln(1 + ic) = ln(1 + 0.07) = 0.0677 = 6.77%
3.6.
, - , ic, - .
. 3.5, jm, , dc, fm. (3.21),(3.23), (3.27) (3.29). - . . .
3.10. 10% (). : ) , ) , ) -, ) ?
. , ic : ) j12 =0.1/12 , ) j4 = 0.1/4 , ) j2 = 0.1/2 , ) = 0.1. - :) (3.21) :
ic = (1 + 0.1/12)12 1 = 0.1047 = 10.47% ;
) (3.21) :
ic = (1 + 0.1/4)4 1 = 0.1038 = 10.38% ;
) (3.21) :
ic = (1 + 0.1/2)2 1 = 0.1025 = 10.25% ;
) (3.23) :
ic = e0.1 1 = 0.1052 = 10.52% .
-
60 3.
- 7.6.
3.7.
, - - , - . , ,
. . . .. .
() , , - . , ( !) - , . , - , , . , , . , , .
, - . , , , , , , -.
, - . : . , , . , 4 , .
-
3.7. 61
. , - , (- ) 25% . , - , 4 ( ). 25% , . , - m. , : ?
Mt , t. 6 :
M1 = m,
M2 = m+34m =
74m,
M3 = m+34m+
12m =
94m,
M4 = m+34m+
12m+
14m =
104 m,
M5 = m+34m+
12m+
14m+ 0 =
104 m,
M6 = m+34m+
12m+
14m+ 0 14m = 94m.
, 4 , 5- -, .
. st , t. - (t 1) :
st = m 14m(t 1). (3.51)
(3.51) , st - , m, , 14m.
, - t- , st > 0. -, - . st > 0 t, : t < 5. : 4 .
, .
-
62 3.
st > 0 :
m m(t 1) > 0. t, :
t < 1 +1
, (3.52)
: , .
, 1 , t = 1+1 st = 0.
t = 1 - , m, , m:
Mt =s1 + st
2t =
m
2(1 + 1
).
= 0.05. 80% . , - , 5% . (3.52) , , - , 20(!) -. . :
M20 =m
221 = 10.5m.
, , , , 20 . , - , , 20 .
. -, : - .
, , , > 1. , t t1m .
, t, - :
-
3.7. 63
t1m mt2i=0
i > 0.
:
t2i=0
i =t1 1 1 ,
:
t1m mt1 1 1 > 0.
> 1 t > 1, t1 > 1 t. , t1 1:
1
t1 1 >
1 1. (3.53) (3.53) . . .
< 1 ( ), 1 1 < 0, (3.53) t. , , .
> 1 ( ), 1 1 > 0, t (3.53) . (3.53):
t1 1.
t = st1 st (t 2). , t 2 = m,
= 1 + .
, :
st1 st = t2m mt3
i=0 i t1m+ mt2i=0 i =
= t2m t1m+ mt2 = (1 + )t2m = t2m.
-
65
(3.55) -, :
s1 = m; M1 = s1;
2 = m; s2 = s1 2; M2 = M1 + s2;3 = 2; s3 = s2 3; M3 = M2 + s3;...
......
t = t1; st = st1 t;Mt = Mt1 + st.
(3.56)
(3.56), , = 1.20 = 0.25 9 . -, , m = 1. , , Mt. , 9 6.1m :
t t st Mt
1 1.00 1.00
2 0.05 0.95 1.95
3 0.06 0.89 2.84
4 0.07 0.82 3.66
5 0.09 0.73 4.39
6 0.10 0.63 5.02
7 0.12 0.51 5.53
8 0.14 0.37 5.90
9 0.17 0.20 6.10
10 0.21 -0.01 6.09
- , , .
-
66 3.
1. 8 000 . , 6% (). : ) 1 ,) 8 , ) 4 , ) 6 6 ?
2. 1, j3 = 6%.
3. , - j6 = 10%. , - 20 000 . 3 3 ?
4. 30 000 ., 5 40 000 . j12 ?
5. 1 ., , j1 = 10% 1 000 000 .?
6. 1 000 . 1 , : ) j1 = 5%, ) j6 =5%, ) j12 = 5%, ) j360 = 5%, ) j = 5%?
7. ) 8 ?
8. , - j = 7%, 10 5 000 .?
9. -: 1990 . 12%, 1991 . 18%, 1992 1993 . 24%. 31 1993 , 1 1990 3 000 .?
10. - 15 000 ., 1 - , 7%. , : ) 1 , ) 1 ? 11 2.
11. 10 7% (), . , ?
12. - , 7% ?
13. ic , : ) j2 =10%, ) j6 = 10%, ) j12 = 10%, ) j = 10%.
14. 8% (). jm , -, : ) m = 2, ) m = 6, ) m = 12, ) m =?
-
67
15. j4 = 6% . , ?
16. 60 ds = 6%. , ?
17. f3 = 8% dc. dc, ?
18. dc = 6%. fm , - , : ) m = 2, ) m = 4, ) m = 12?
19. 6% (). , : ) , ) , ) , ) ?
20. dc = 8%. - ?
21. f4 = 8%. - ?
-
4.
4.1.
- : (1) - , , ; (2) , - ; (3) ; (4) .
- ; , - . - , (1) . - -. : , -. ( - ), . , - . , - , .
, . , ,
-
4.2. 69
( ) . - , , - - . , , - , , , . , - , . -.
4.2.
, , , , .
n - S, (100/n)%, , S/n. , S/n. k - Sk :
Sk = S k Sn, (4.1)
k = 1, . . . , n . :
S1 = S Sn, S2 = S 2 S
n, . . . , Sn = S n S
n= 0 .
, - (S/n). .
4.1. 58 000 . 8 . -, 8 4 000 .
. (4.1) , . , :
58 000 4 000 = 54 000 .
-
70 4.
(100/8)%, 12.5%. -,
54 0000.125 = 6750 . :
(.) (.)
0 0 58 000
1 6 750 51 250
2 6 750 44 500
3 6 750 37 750
4 6 750 31 000
5 6 750 24 250
6 6 750 17 500
7 6 750 10 750
8 6 750 4 000
- 2. . - .
4.3.
, - , , , . , - , . , , .
2 - : ,. .: , 1999, 96.
-
4.3. 71
. .
- . n , Kn - :
Kn = 1 + 2 + 3 + + n = n(n+ 1)2
. (4.2)
i- -:
Ai = Sn (i 1)
Kn, (4.3)
S .
, - . . - :
d = Ai+1 Ai = S n (i+ 1 1)Kn
S n (i 1)Kn
=
=S
Kn(1) = 2S
n(n+ 1).
.
4.2. 4.1 . - , .
. 4.1, 4 000 . 58 000 . K8:
K8 = 1 + 2 + 3 + + 8 = 1 + 82
8 = 36.
n (i 1) i = 1, . . . , 8:
8, 7, 6, 5, 4, 3, 2, 1.
-
72 4.
8/36 - , 7/36 - . 1/36 - . :
(.) (.)
0 0 0 58 000
1 8/36 12 000 46 000
2 7/36 10 500 35 500
3 6/36 9 000 26 500
4 5/36 7 500 19 000
5 4/36 6 000 13 000
6 3/36 4 500 8 500
7 2/36 3 000 5 500
8 1/36 1 500 4 000
, - 4.1, , 4.2 - . , 4.1 - 31 000 ., 4.2 19 000 .
- . .
4.4.
. , , .
k- -. r (
-
4.4. 73
), . S. - Sr
S Sr = S(1 r). S(1 r)r
S(1 r) S(1 r) r = S(1 r)2. , , - k- Sk :
Sk = S(1 r)k. (4.4) S,
n , S0, - r, , :
S(1 r)n = S0. r, :
r = 1 nS0S. (4.5)
, (4.4) - , , .
- , - , S, , (1 r).
, . -, r . -, , Sk(1 r)k 0 < r < 1 . , .
- .
-
74 4.
4.3. - 4.1 .
. (4.5) -:
r = 1 8
4 000
58 000= 1 0.71586 = 0.28414 = 28.414%
- :
(.) (.)
0 0 58 000.00
1 58 000.000.2841 = 16 477.80 41 522.202 41 522.200.2841 = 11 796.46 29 725.743 29 725.740.2841 = 8 445.08 21 280.664 21 280.660.2841 = 6 045.84 15 234.825 15 234.820.2841 = 4 328.21 10 906.616 10 906.610.2841 = 3 098.57 7 808.047 7 808.040.2841 = 2 218.26 5 589.788 5 589.780.2841 = 1 588.06 4 001.72
, , , : 15 235 ., 19 000 .
4.5.
, - r . - , .
-
4.5. 75
, . : - . .
4.4. - 4.1 .
. - 12.5%. , , 25%. :
(.) (.)
0 0 58 000.00
1 58 000.000.25 = 14 500.00 43 500.002 43 500.000.25 = 10 875.00 32 625.003 32 625.000.25 = 8 156.25 24 468.754 24 468.750.25 = 6 117.19 18 351.565 18 351.560.25 = 4 587.89 13 763.676 13 763.670.25 = 3 440.92 10 322.757 10 322.750.25 = 2 580.69 7 742.068 7 742.060.25 = 1 935.52 5 806.54
, , 5 806.54 ., , . - 4 000 ., - 1 935.52 . :
1 935.52 + (5 806.54 4 000) = 3 742.06 .
( 4.2 4.3), , - , , , - . .
-
76 4.
4.6.
- , , , . , - -. .
4.5. , 4.1, ( ), 60 000 . 25%. , , - .
. :
- -
(.) (.) (.) (%) (.)
1 60 000 6 750 53 250 25 13 312.5
2 60 000 6 750 53 250 25 13 312.5
3 60 000 6 750 53 250 25 13 312.5
4 60 000 6 750 53 250 25 13 312.5
5 60 000 6 750 53 250 25 13 312.5
6 60 000 6 750 53 250 25 13 312.5
7 60 000 6 750 53 250 25 13 312.5
8 60 000 6 750 53 250 25 13 312.5
: 106 500 .
4.6. -, 4.5, ( 4.2).
. :
-
4.6. 77
- -
(.) (.) (.) (%) (.)
1 60 000 12 000 48 000 15 7 200
2 60 000 10 500 49 500 15 7 425
3 60 000 9 000 51 000 25 12 750
4 60 000 7 500 52 500 25 13 125
5 60 000 6 000 54 000 25 13 500
6 60 000 4 500 55 500 25 13 875
7 60 000 3 000 57 000 25 14 250
8 60 000 1 500 58 500 25 14 625
: 96 750 .
, , 8 , - , - (. 4.5). :
106 500 96 750 = 9750 .
, :
9 750
106 500100% = 9.15%.
, , 0. - , 6%, - . 0 :
13 312.5(
1
1 + 0.06+
1
(1 + 0.06)2+
1
(1 + 0.06)3+
+1
(1 + 0.06)4+
1
(1 + 0.06)5+
1
(1 + 0.06)6+
+1
(1 + 0.06)7+
1
(1 + 0.06)8
)= 82666.19 .
-
78 4.
- 0 :
7 200 11 + 0.06
+ 7425 1(1 + 0.06)2
+ 12750 1(1 + 0.06)3
+
+13125 1(1 + 0.06)4
+ 13500 1(1 + 0.06)5
+ 13875 1(1 + 0.06)6
+
+14250 1(1 + 0.06)7
+ 14625 1(1 + 0.06)8
= 73025.78 .
, :
82 666.19 73 025.78 = 9640.41 .
9 640.41
82 666.19100% = 11.66%.
, , , , (9.15%).
4.7. , , 4.5, - .
. , , - , , , .
- 0 . 6% ( 4.6), - : 73 298.97 . , , , .
-
4.6. 79
- -
(.) (.) (.) (%) (.)
1 60 000 16 477.80 43 522.20 15 6 528.33
2 60 000 11 796.46 48 203.54 15 7 230.53
3 60 000 8 445.08 51 554.92 25 12 888.73
4 60 000 6 045.84 53 954.16 25 13 488.54
5 60 000 4 328.21 55 671.79 25 13 917.95
6 60 000 3 098.57 56 901.43 25 14 225.36
7 60 000 2 218.26 57 781.74 25 14 445.44
8 60 000 1 588.06 58 411.94 25 14 602.99
: 97 327.88 .
4.8. , , 4.5, - .
. :
- -
(.) (.) (.) (%) (.)
1 60 000 14 500.00 45 000.00 15 6 825.00
2 60 000 10 875.00 49 125.00 15 7 386.75
3 60 000 8 156.25 51 843.75 25 12 960.94
4 60 000 6 117.19 53 882.81 25 13 470.70
5 60 000 4 587.89 55 412.11 25 13 853.03
6 60 000 3 440.92 56 559.08 25 14 139.77
7 60 000 2 580.69 57 429.31 25 14 354.83
8 60 000 1 935.52 58 064.48 25 14 516.12
: 97 489.14 .
- 0
-
80 4.
. - 6% ( 4.6), : 73 524.77 .
:
8 (.) 0 -
(.)
106 500.00 82 666.19
96 750.00 73 025.78
97 327.88 73 298.97
97 489.14 73 542.77
, - 0. , , . - 0 - . , . - , .
4.7. Excel
Excel -. : . - ( ) -. , Excel .
-
4.7. Excel 81
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4.2 , -. , - 4.2, . / . - , .
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-
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( A3, A9) ( A11) - ( B15:B22).
. 5.
1. 600 000 . 5 . , .
-
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-
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2. 1 - , - 50 000 .
3. 1 - , .
4. 1 - , 50 000 .
5. 1 - , 50 000 .
6. 1 - , - 50 000 .
. 7 19 :
(.) (%)
200 000 15
200 000 30
7. 1 ( -) 350 000 . , , , - .
8. , - 1 7, , .
9. , , 5 1 7, , .
10. 0 , - 1 7 , - . 10%.
-
90 4.
11. 0 , - 1 7 , . - 10%.
12. 0 , 1 7 , , , ? - 10%.
13. 1 7 , - .
14. , 1 7, - , , ?
15. 0 , - 1 7 , - . - 10%.
16. 0 , 1 7, , , ? 10%.
17. 1 7 , - .
18. 0 , - 1 7 , - . - 10%.
19. 1 7.
-
5.
5.1.
, , -, , . - , ? ,
?
,
, , . , , - . , , - . , -
. . 2.6
. , .
S, t , P , t S, i. P = PV S (3.3):
PV = S(1 + i)t. (5.1)
PV Present Value, - .
jm, (3.4):
PV = S
(1 +
jmm
)tm
.
-
92 5.
, , - . i ( jm) .
5.1. , , - 5 000 . , - j4 = 6%?
. (3.4) 5 000 . S = 5000, m = 4, jm = 0.06, t = 2, :
P = 5000(1 +
0.06
4
)8
= 50000.8877 = 4438.55 .
, 5 000 . 4438.55 . .
t P i%, S = FV (3.1):
FV = P (1 + i)t.
FV Future Value, - .
jm, - (3.2):
FV = P
(1 +
jmm
)tm.
, , , S, t , (3.6):
PV = Set,
P , t , (3.5):
FV = Pet.
-
5.1. 93
5.2. , j2 = 8%, 3 10 000 . ?
. ( ) - (3.2) P = 10000, j2 = 0.08, m = 2, t = 3:
FV = 10000
(1 +
0.08
2
)23= 100001.26532 = 12 653.2 .
10 000 . 12 653.2 .
- , . , t . , - : .
, -, , , - , , . - , - .
- . , . . , - . 5.1:
-0
P
1 2 3 4 5 6 7 8
5 000
tm
5.2:
-
94 5.
--8 -7 -6
10 000
-5 -4 -3 -2 -1 0
S
tm
, - -. , 5.1 5 000 . t = 4 :
PV = 5000(1 +
0.06
4
)4
= 4710.92 .,
5.2 10 000 . t = 3
PV = 10000(1 +
0.08
2
)3= 11248.64 .
T 6= 0, S - PVT (S).
5.2.
, . S1, . . . , Sn R1, . . . , Rk ( ). W . , -:
PVT (S1) + . . .+ PVT (Sn) = PVT (R1) + . . .+ PVT (Rk) + PVT (W ).
- , n k. -, . .
-
5.2. 95
V1 V2 (, ) t, T :
PVT (V1 + V2) = PVT (V1) + PVT (V2).
, T V , - t, :
PVT (V ) = V (1 + i)Tt.
V = V1 + V2, :
PVT (V1 + V2) = (V1 + V2)(1 + i)Tt =
= V1(1 + i)Tt + V2(1 + i)
Tt =
= PVT (V1) + PVT (V2).
5.3. , i% - , : 2 S; 5 R, W . , :
PV0(S) = PV0(R) + PV0(W ).
. , , , , :
-0 1 2
S
3 4 5R + W
6 7 t
R+W S. (3.1), :
R+W = S(1 + i)3.
(3.3), :
PV0(R +W ) = (R+W )(1 + i)5 = S(1 + i)3(1 + i)5 =
= S(1 + i)2 = PV0(S).
-
96 5.
,
PV0(R+W ) = PV0(R) + PV0(W ),
, :
PV0(S) = PV0(R) + PV0(W ).
5.4. , i% - , : 3 S1; 7 S2; 5 R1; 11 R2. W . , :
PV0(S1) + PV0(S2) = PV0(R1) + PV0(R2) + PV0(W ).
. , , , , :
-0 1 2 3
S1
4 5R1
6 7
S2
8 9 10 11R2 + W
t
R2 +W S1, S2 R1. t = 5 ,
S1(1 + i)2 R1.
, t = 7:
(S1(1 + i)2 R1)(1 + i)2 + S2 = S1(1 + i)4 R1(1 + i)2 + S2.
t = 11 :
(S1(1 + i)4 R1(1 + i)2 + S2)(1 + i)4 = S1(1 + i)8 R1(1 + i)6 + S2(1 + i)4.
t = 11 R2 +W , :
R2 +W = S1(1 + i)8 R1(1 + i)6 + S2(1 + i)4.
-
5.2. 97
0:
PV0(R2 +W ) = PV0(S1(1 + i)8 R1(1 + i)6 + S2(1 + i)4) =
= (S1(1 + i)8 R1(1 + i)6 + S2(1 + i)4)(1 + i)11 =
= S1(1 + i)3 R1(1 + i)5 + S2(1 + i)7 =
= PV0(S1) PV0(R1) + PV0(S2).
PV0(R2 +W ) = PV0(R2) + PV0(W ),
:
PV0(R2) + PV0(W ) = PV0(S1) PV0(R1) + PV0(S2),
, :
PV0(S1) + PV0(S2) = PV0(R1) + PV0(R2) + PV0(W ).
, . .
5.5. 700 . - j4 = 6%. 6 300 ., 2 . ?
. , , ., , , , , . ( ).
-0
700
1 2300
3 4 5 6 7 8 9 10x
11 t
-
98 5.
, x. -, , -:
700 = 300(1 + i)2 + x(1 + i)10.
i = 0.06/4 = 0.015, x:
x = 700(1 + i)10 300(1 + i)8 == 700(1.015)10 300(1.015)8 == 474.43 .
5.6. - 2 600 . , - j12 = 5%. 400 ., 100 . 3 , - , 500 . ?
. , , :
24
600
23....9 8
400
7....2 1 0
100
1 2 3....
x
11 12
-
500
t
, , - , , . ( ) . -, , x. , , , , :
600(1 + i)27 + x = 400(1 + i)11 + 100(1 + i)3 + 500(1 + i)9.
-
5.3. 99
i = 0.05/12 = 0.00417 x:
x = 4001.0041711 + 1001.004173 + 5001.004179 6001.0041727 == 4001.04684 + 1001.01256 + 5000.96324 6001.11891 == 330.265 .
, 3 330.27 .
5.3.
- . , - . , . , - ( i ), , , - . .
t , S1 S0 :
S1 = S0(1 + i)t, (5.2)
t , S1 S0 :
S1 = S0(1 + i)t, (5.3)
5.7. 12 000 . 1 1998 -. : ) 1 1997 ; ) 1 - 1999 ? 8% ().
. ) 1.5 , :
S = 12000(1 + 0.08)1.5 = 120000.8910 = 10 692.00 .
-
100 5.
) 0.5 , , , 12 000 .:
S = 12000(1 + 0.08)0.5 = 120001.0392 = 12 470.40 .
() : S1, . . . , Sk t1, . . . , tk, , S0. - : S0, t0; t0 S0. :
0 t1
S1
t2..........
S2
t0..........
S0
tk
-Sk
t
, 0 S0 ( ) - S1, . . . Sk, :
S0(1 + i)t0 =
kl=1
Sl(1 + i)tl . (5.4)
S0, - S0:
S0 = (1 + i)t0
kl=1
Sl(1 + i)tl . (5.5)
t0 S0, (5.4) t0. :
lnS0(1 + i)t0 = ln
kl=1
Sl(1 + i)tl ;
lnS0 t0 ln(1 + i) = lnk
l=1
Sl(1 + i)tl ;
-
5.3. 101
t0 =lnS0 ln
kl=1 Sl(1 + i)
tl
ln(1 + i). (5.6)
5.8. 800 000 ., 1 500 000 . 8 1 300 000 . . - , 6% (-)?
. , - (5.5):
S0 = (1 + 0.06)1[ 0.8(1 + 0.06)0.5 + 1.5(1 + 0.06)1 + 1.3(1 + 0.06)1.(6)] =
= 3574 118.97 .
5.9. , 3 600 000 . ?
. (5.6) t0:
t0 =ln 3.6 ln(0.81.060.5 + 1.51.061 + 1.31.061.(6))
ln 1.06=
= 1.1238 = 1 45 .
, 3 600 000 . 1 45 .
, - . - , .
5.10. - - 1 000 . 1 306 . 3 . - , - : 4 , . , j2 = 6%?
-
102 5.
. , , . x :
-0
1 000
1 2x
3 4 5 6
1 306
7 8x
9 t
, 0 ( ) - , , , , . , :
1 000 + 1306(1 + i)6 = x(1 + i)2 + x(1 + i)8.
i = 0.06/2 = 0.03 x:
x(1.032 + 1.038) = 1 000 + 13061.036;x = 1208.87 .
, - 1 208.87 .
, , - . , 4 - , :
1 000(1 + i)4 + 1306(1 + i)2 = x(1 + i)2 + x(1 + i)4.
(1+i)4, , .
, - . . - ( ) , - . ( ), . ? , , . .
-
103
5.11. , -, 1 000 . 27 3 000 . 5 . , , , j4 = 8%. ?
. , (3 ). 27 - 9 , 5 20 . 0 x:
-0x
9
1 000
20
3 000
t
0,, :
x =1000
(1 + i)9+
3000
(1 + i)20.
i = 0.08/4 = 0.02 x:
x =1000
(1 + 0.02)9+
3000
(1 + 0.02)20=
= 10000.8368 + 30000.6730 == 2855.8 .
- :
mq=1
SqVpq =
lk=1
RkVtk . (5.7)
, :
-
104 5.
R1, . . . , Rl ;t1, . . . , tl , -
;S1, . . . , Sm ;p1, . . . , pm , -
;V = 1 + i, -
, - ;
V = (1 + i)1, - , - ;
i , -, .
1. 10 000 ., :) 3 6 ,) 2 9 ,) ?
8% ( , , - 8% ()).
2. j4 = 6%. 25 000 ., :) 5 4 ,) 1 8 ?
3. 5.3, . 5.2, , :
) PV2(S) = PV2(R) + PV2(W ),
) PV3(S) = PV3(R) + PV3(W ),
) PV5(S) = PV5(R) + PV5(W ),
) PV6(S) = PV6(R) + PV6(W ).
4. 2 S1, 4 S2, 7 S3, 14 S4. 6 R1, 10 R2 17 R3. W . ,
PV0(S1) + PV0(S2) + PV0(S3) + PV0(S4) =
= PV0(R1) + PV0(R2) + PV0(R3) + PV0(W ).
-
105
5. , i = 5% (), 12 000 . 1 6 4 500 ., 2 2 000 . 3 6 . ?
6. , - j6 = 5%.
7. 3 5 000 . , - j2 = 8%. 2 000 ., 3 6 3 500 . 6 , 10 000 . ?
8. . 2 , 1 6 . 2 6 25 000 . , i = 5%()?
9. , j12 = 5%.
10. 8, = 5%.
11. 5 . . 8% (-). 3 . ., 2 . . 2 . ?
12. , j4 = 10%, 2 000 . 1 6 3 000 . 1 6 ?
13. , - .
14. , 25 000 ., 12% (). 1 6 15 000 ., 6 . -?
15. 12 . . 15% (). 6 4 500 ., 6 - 2 500 . 6 3 .., 2 . ?
-
106 5.
16. 80 . ., - 12% (). - - . 20 . . 3 ,30 . . 6 , 10 . . 1.5 . ?
17. - . 30 . ., 20 . ., 20 . . 2.5 . ?
18. - - 16, - ( ). ?
19. 16 , - - 12% (). - 2 , , 25 . . 30 . . 2 , -, , . - ?
20. - , , 30 .. 30 . . . , 2 , 10 . . 2 ?
21. 2 000 . 1 1994 . 15% (). , : ) 1 1993 , ) 1 1995 , ) 1 1996 ?
22. 3 500 . 2 , 3 000 2 5 200 3 . 5 . , 8% ?
23. , 12 000 . , -, ?
-
107
24. 22 3 6 . ?
25. - - 25 000 . - . - 30 000 . 2 , 2 . , 9% ()?
26. - - , , 15 000 . 8 20 000 . 18 -. - , 6% () . , ?
27. , , ?
-
6.
6.1.
. , -, , - , - , . . : , , - . - . .
, , n - . ,
b1, b2, . . . , bn,
, , , - q, -. Sn n , q 6= 1, :
Sn =b1(q
n 1)q 1 . (6.1)
-
6.2. . sn; i 109
6.2. . sn; i
, - .
: n , , , R; , - i. :
0 1
R
2
R
3..........
R
n1R
n-
R
n , S.
, n, -, R. , n , n1, R(1+i). , n , n2, R(1+ i)2 . . , - n , 2, R(1+ i)n2., n , 1, - R(1+ i)n1. , n :
S = R+R(1 + i) +R(1 + i)2 + . . .+R(1 + i)n2 +R(1 + i)n1.
, b1 = R, q = 1+ i, n. (6.1) n :
S =b1(q
n 1)q 1 =
R[(1 + i)n 1]1 + i 1 = R
(1 + i)n 1i
.
, R -, :
sn; i =(1 + i)n 1
i. (6.2)
-
110 6.
:
S = Rsn; i. (6.3)
(6.3) - sn; i. , - n i ( 2). , , , Excel.
6.1. , - 2 500 . , 5% (). 8 ?
. ( ), R = 2500 ., n = 8, i = 5%. (6.3):
S = 2500s8; 5%.
2 , s8; 5% = 9.54911. :
S = 25009.54911 = 23 872.77 .
6.2. 100 . , j12 = 9%. 2 ?
. , , , R = 100, n = 24 (2 12 ),i = 0.09/12 = 0.0075. (6.3):
S = 100 (1 + 0.0075)24 1
0.0075= 100s24; 0.75% .
2 , s24; 0.75% = 26.18847. :
S = 10026.18847 = 2618.85 .
-
6.3. 111
6.3.
R S. R S (6.3):
R =S
sn; i. (6.4)
.
6.3. - 8 5 000 ., , - i = 5%(). ?
. : S = 5000, n = 8, i = 0.05. 2, s8; 5%=9.54911. (6.4) :
R =5000
9.54911= 523.61 .
, - 523.61 .
6.4.
, .
6.4.1.
, , R, , i. n - S R (6.3) (6.4), .
p -
, p - , R/p. i. :
-
112 6.
0....
Rp
Rp
1
Rp
....
Rp
Rp
2
Rp
....
Rp
Rp
3....
Rp
n1
Rp
....
Rp
Rp
n
-
Rp
n np . , n .
, R/p. i , 1/p , , n
R
p(1 + i)
1p .
, n ,
R
p(1 + i)
2p .
, n ,
R
p(1 + i)
np1p ,
np 1 i , 1/p . ( - :
np 1p
= n 1p,
1 n i , n 1/p .)
n :
S =R
p+R
p(1 + i)
1p +
R
p(1 + i)
2p + . . .+
R
p(1 + i)
np1p .
b1 = R/p
q = (1 + i)1p .
-
6.4. 113
k = np. (6.1), k , S:
S =b1(q
k 1)q 1 =
R
p((1 + i)
1p
)np 1
(1 + i)1p 1
= R(1 + i)n 1p[(1 + i)
1p 1]
.
:
s(p)n; i =
(1 + i)n 1p[(1 + i)
1p 1]
. (6.5)
p - :
S = Rs(p)n; i . (6.6)
s(p)n; i :
s(p)n; i = sn; iKp; i , (6.7)
Kp; i =i
p[(1 + i)1p 1]
.
p - (6.6), 2 4 sn; i Kp; i .
, , Rr, r (r > 1). i - . :
0 1....r1 r
Rr
r+1....
2r
Rr
....3r....
Rr
n2r
Rr
....nr
Rr
....n
-
Rr
n n/r 4. n S .
4 , , n/r .
-
114 6.
, Rr. - r n, , n Rr(1 + i)
r. 2r n, , n Rr(1 + i)
2r. n r n,, n Rr(1+ i)
nr. n :
S = Rr +Rr(1 + i)r +Rr(1 + i)
2r + . . .+Rr(1 + i)nr.
b1, Rr, q, (1+i)
r k = n/r. k :
S =b1(q
n 1)q 1 =
Rr[((1 + i)r)
nr 1]
(1 + i)r 1 = Rr(1 + i)n 1(1 + i)r 1 .
i - (6.2), :
S = Rr
(1 + i)n 1i
(1 + i)r 1i
,
S = Rrsn; isr; i
. (6.8)
.
6.4. - 4 000 ., . 6 , 5% ()?
. - p -- , R = 4000, p = 4, i = 0.05, n = 6. (6.6):
S = Rs(p)n; i = 4000s
(4)6; 5%.
-
6.4. 115
s(4)6; 5% (6.7) ( 2
4):
s(4)6; 5% = s6; 5%K4; 5% = 6.801911.01856 = 6.92815.
, :
S = 40006.92815 = 27 712.61 .
6.5. , - 3 4 . ., , 6% (). 12 ?
. . Rr = 4 . ., r = 3, i = 0.06, n = 12. (6.8):
S = 4000 000s12; 6%s3; 6%
.
sn; i, - (6.2):
s12; 6% =(1 + 0.06)12 1
0.06= 16.8699,
s3; 6% =(1 + 0.06)3 1
0.06= 3.1836.
, :
S = 4000 000 16.86993.1836
= 21 196 004.52 .
6.4.2. m
R , m jm, jm/m%. :
-
116 6.
0 1
R
1+ 1m
.....
2 1m
2.....
R
n1R
n1+ 1m
.....
n 1m
n-
R
n .
. - 1 n m jm, - n
R
(1 +
jmm
)m.
2 n, n
R
(1 +
jmm
)2m.
n 1 n, , n
R
(1 +
jmm
)(n1)m.
(3.2). :
S = R+R
(1 +
jmm
)m+R
(1 +
jmm
)2m+ . . .+R
(1 +
jmm
)(n1)m.
b1 = R, q = (1 + jm/m)
m k = n. :
S =b1(q
n 1)q 1 =
R[((
1 + jmm
)m)n 1](1 + jmm
)m 1 = R(1 + jmm
)mn 1(1 + jmm
)m 1 .
-
6.4. 117
jm/m (6.2), :
S = R
(1 + jmm
)mn 1jmm(
1 + jmm
)m 1jmm
= Rsmn; jm
m
sm; jmm
. (6.9)
(6.9) - 2.
p -
p . R/p. m jm, jm/m%. , . 6.4.1. n .
, - , R/p. jm , 1/p , n (3.2)
R
p
(1 +
jmm
)mp
.
jm , 2/p , n (3.2)
R
p
(1 +
jmm
) 2mp
.
n 1p n, n , (3.2), :
R
p
(1 +
jmm
)m(n 1p)
.
:
S =R
p+R
p
(1 +
jmm
)mp
+R
p
(1 +
jmm
) 2mp
+ . . . +R
p
(1 +
jmm
)m(n 1p
).
-
118 6.
b1 = R/p ,
q = (1 +jmm
)mp
k = np. :
S =b1(q
k 1)q 1 =
R
p
((1 + jmm
)mp
)np 1(
1 + jmm
)mp 1
=
=R
p(1 + jmm
)mn 1(1 + jmm
)mp 1
.
jm/m, :
S =R
p
(1 + jmm
)mn 1jmm(
1 + jmm
)mp 1
jmm
,
S =R
psmn; jm
m
smp
; jmm
. (6.10)
, sn; i (6.2) n (n ). (6.10) m/p , sn; i - (6.2) n. (6.10) . (6.10) 2.
p - p = m
(6.10) :
S =R
msmn; jm
m
s1; jmm
.
-
6.4. 119
:
s1; jmm
=(1 + jmm )
1 1jmm
= 1.
, :
S =R
msmn; jm
m
. (6.11)
, Rr, r (r > 1). jm, m jm/m%. - , . 6.4.1. n .
, Rr. r n m jm/m%, n , (3.2),
Rr(1 +jmm
)mr.
2r n, n
Rr(1 +jmm
)2mr.
n r n, n
Rr(1 +jmm
)m(nr).
:
S = Rr +Rr
(1 +
jmm
)mr+Rr
(1 +
jmm
)2mr+
+ . . .+Rr
(1 +
jmm
)m(nr).
-
120 6.
b1 = Rr,
q = (1 +jmm
)mr
k = n/r. :
S =b1(q
k 1)q 1 =
Rr
[((1 + jmm
)mr)nr 1
](1 + jmm
)mr 1 =
= Rr
(1 + jmm
)mn 1(1 + jmm
)mr 1 . jm/m (6.2), :
S = Rr
(1 + jmm
)mn 1jmm(
1 + jmm
)mr 1jmm
,
S = Rrsmn; jm
m
smr; jmm
. (6.12)
(6.12) - 2.
6.4.3.
R - ( ) . n . , . 6.2.
n -. , n ,
-
6.4. 121
(3.5), Re. , , (3.5) Re2 . , n , (3.5) Re(n1) . :
S = R+Re +Re2 + . . .+Re(n2) +Re(n1) .
b1 = R, q = e
k = n. k :
S =b1(q
k 1)q 1 =
R((e)n 1)e 1 = R
en 1e 1 ,
S = Ren 1e 1 . (6.13)
p -
p R/p .
, .6.4.1. - , (1+i)1/p e/p, :
S =R
p+R
pep +
R
pe2
p + + R
pe
(np1)p .
b1 = R/p, q = e
/p k = np. k :
S =b1(q
k 1)q 1 =
R
p
(ep
)np 1
ep 1
=R
p e
n 1ep 1
,
S =R
p e
n 1ep 1
. (6.14)
, Rr, r (r > 1). . - , . 6.4.1.
-
122 6.
, (1+ i)r er n:
S = Rr +Rrer +Rre
2r + . . . +Rre(nr).
b1 = Rr, q = e
r k = n/r. :
S =b1(q
k 1)q 1 =
Rr((er)
nr 1
)er 1 = Rr
en 1er 1 ,
S = Rren 1er 1 . (6.15)
.
6.6. - 10 . ., , 12%. , - 6 , : ) , ; ) , ; ) - , ; ) , .
. ) j4 = 12%. (6.9) m = 4, n = 6, jm/m = 12%/4 = 3% :
S = Rsmn; jm
m
sm; jmm
= 10000s24; 3%s4; 3%
.
2 sn; i:
s24; 3% = 34.4265; s4; 3% = 4.18363.
, :
S = 100 000 34.42654.18363
= 82 288.574 .
-
6.4. 123
) p - j2 = 12%. (6.10) m = 2, n = 6, p = 4:
S =R
psmn; jm
m
smp
; jmm
=10000
4 s12;6%s0.5;6%
.
2 sn;i, - (6.2):
s12; 6% =(1 + 0.06)12 1
0.06= 16.8699,
s0.5; 6% =(1 + 0.06)0.5 1
0.06= 0.49271.
:
S =10000
4 16.86990.49271
= 85 597.51 .
) - = 0.12. - (6.13) = 0.12, n = 6:
S = Ren 1(e 1) = 10 000
e0.126 1(e0.12 1) = 82 702.69 .
) p - = 0.12. - (6.14) = 0.12, n = 6, p = 12:
S = Ren 1
p
(ep 1
) = 10000 e0.126 112(e
0.1212 1
) = 87430.82 .
6.7. - , 6 100 . . , .
-
124 6.
. ) (6.9) R S = 100000:
R = 100 000s4; 3%s24; 3%
= 100 000 4.1836134.4265
= 12 152.29 .
) (6.10) R S = 1000 000 p = 4:
R = 100 0004 s0.5; 6%s12; 6%
= 400 000 0.4927216.8699
= 11 682.58 .
) (6.13) R S = 100 000:
R = 100 000e0.12 1e0.72 1 = 100 000
0.12750
1.05443= 12 091.84 .
) (6.14) R S = 100 000 p = 12:
R = 100 00012 e0.01 1e0.72 1 = 100 00012
0.01005
1.05443= 11 437.50 .
6.8. 10 . . 10 , : ) j4 = 12%, ) = 12%?
. ) - (6.12) r = 2, m = 4, n = 10, jm/m = 12%/4 = 3%:
S = Rrsmn; jm
m
smr; jmm
= 10000s40; 3%s8; 3%
.
2 :
s40; 3% = 75.40126, s8; 3% = 8.89234.
:
S = 10000 75.401268.89234
= 84 793.54 .
) (6.15) = 12% = 0.12, n = 10, r = 2:
S = Rren 1er 1 = 10 000
e1.2 1e0.24 1 = 85 534.53 .
-
6.6. 125
6.5.
- .
, S ., n . q. ( -) , - . , , i% . - , .
, - Sq, R. R , n , - n S. (6.4) R = S/sn;i. :
= Sq +S
sn; i. (6.16)
6.9. 1 . . 8% 4 . , . , , 5% . - .
. 2 ,
s4; 5% = 4.310125.
(6.16) S = 1000 000, q = 0.08, n = 4, i = 0.05 :
= 1000 0000.08 + 1000 0004.310125
= 312 011.83 .
- , (6.2) (6.15).
- . 7.4.
-
126 6.
6.6. ,
, - , , . , - - , ( , ), , . .
6.10. - , 10 200 000 . , . - 18% . , . , 10% ?
. S. - R , S. (6.4) :
R =S
sn; i=
S
s10; 10%.
, 200 000 ., - , 18% , 0.18S . , S -:
0.18S +S
s10; 10%= 200 000.
2 s10; 10% = 15.9374. :
0.18S +S
15.9374= 200 000.
, :
S = 823 908.50 .
-
6.7. Excel 127
6.7. Excel
, , - :
, , , ( ), .
Excel -. , Excel, , , 5.
-, . , -, .
_
( 0 _)
_
_
(1 , 0 )
, ( - ) , :
5 Excel , .
-
128 6.
(;_; FV [;_][;])
(;;_ NPER [;_][;])
(_;;_ RATE [;_][;][;])
(_;;_ PMT [;_][;])
(;_;; PV [;_][;])
(;;_; IPMT _;_[;])
(;;_; PPMT _;_[;])
(;_;_; CUMIPT _;_;_;)
(;_;_; CUMPRINC _;_;_;)
, -, . - - , . .
6.11. - 100 000 . 8% 2 000 . . ?
. , - . :
=(8%/12;-2000;100000)
, 61 . - (-
-
129
, , ), , - .
. 7 6.1, 6.3 6.11, , .
, . , - . : # !. , , . - 666 .
, - , . - .
(), . , . , 6.1 (. 7) - B7 , 6.11 B31 .
( 6.3 . 7), . , .
1. 25 000 . , - 5% (). : ) 3 , ) 10 ?
2. 1 , , j4 = 5%.
3. 6 40 000 . , , - i = 10% (). ?
-
130 6.
. 7.
-
131
4. 3 , , j12 = 5%.
5. , 300 000 . , 5% (). 4 ?
6. -, 4 2 . .?
7. , 4 15 . . , 5% (). 16 ?
8. - , 20 120 . ., - ?
9. 120 000 . , j6 = 8%. 10 ?
10. - , 10 2 . .?
11. 9 30 000 . 10 ?
12. 9, 15 3 . .?
13. j4 =3%. 800 ., - . 5 ?
14. -, 6 6 000 .?
15. ( ) = 8%. - 500 . 7 6 ?
16. - . 7 6 ?
17. 2 1.5 . . 15, . 6 ?
-
132 6.
18. 17, 6 7 . .?
19. 5 . . 15% 5 . , 10% . .
20. , , - , 8% .
21. 19 , - j4 = 10%.
22. 19 , - 10% .
23. 19 , - , j6 = 10% -.
24. 2 . . 3 . 12% . - , j4 = 8%. ?
25. 24, , , = 8%?
26. , - 20 5 . . , . 12% - . , , . , , 8% . -?
27. 26 50 . . ? - ?
28. S(p;m) p - ( p = 1 ),
-
133
jm ( m = 1 i). , i = j = , R, n :
S(1; 1) < S(1;m) < S(1;) < S(p; 1)|p>1 m>1 < S(p;m)|p=m>1 p>1 < S(p;).
- R, n, i, jm, p m.
-
7.
7.1.
. an;i
2.6 5.1 . - . , , , - . , .
, n , - R . i, - , (6.2), S = Rsn; i. , , , - (3.3), :
PV = S(1 + i)n = Rsn; i(1 + i)n = R
(1 + i)n 1i
(1 + i)n =
= R1 (1 + i)n
i.
PV 0(. (5.1)), , - n , R, PV (R,n), R n , PV .
, R
-
7.1. 135
, :
an; i =1 (1 + i)n
i. (7.1)
, n -, R , i , :
PV (R,n) = PV = Ran; i (7.2)
, an; i, - n i. , 3 - . , , , -. , (7.2) Excel.
, (7.2) , - (6.3). , n , :
0 1
R
2
R
3..........
R
n1R
n-
R
(3.3) : 0 R(1 + i)1; 0 R(1 + i)2 ..; n-, , 0 R(1+ i)n. - 0 ( ):
PV (R,n) = R(1 + i)1 +R(1 + i)2 + . . . +R(1 + i)n.
n - b1 = R(1 + i)
1 q = (1 + i)1. , (7.2):
-
136 7.
PV (R,n) =b1(q
n 1)q 1 =
R(1 + i)1[(1 + i)n 1](1 + i)1 1 =
=R(1 + i)1[(1 + i)n 1]
1 (1 + i)1 + i
=
= R1 (1 + i)n
i= Ran; i.
7.2.
(7.2) : i%, - R n ? , .
1 R, 0 R(1 + i)1. 2 R, - 0 R(1 + i)2 . . n R, 0 R(1 + i)n. - . 7.1. , , , (7.2). , 0, n -, R .
7.1. , 10% (), , - 10 000 . . - ( ). ?
. 0 , - 10 000 . , i = 10% = 0.1. (7.2) :
PV = Ran; i .
3 : a5; 10% = 3.79079.
-
7.3. 137
, :
PV = 100003.79079 = 37 907.90 .
7.3.
-: , m , , .
7.3.1.
(7.2), n , i .
p -
(3.3) (6.6), PV :
PV = S(1 + i)n = Rs(p)n; i(1 + i)
n =
= R(1 + i)n 1p[(1 + i)
1p 1]
(1 + i)n = R1 (1 + i)np[(1 + i)
1p 1]
.
:
a(p)n; i =
1 (1 + i)np[(1 + i)
1p 1]
.
PV :
PV = Ra(p)n; i . (7.3)
a(p)n; i :
a(p)n; i =
1 (1 + i)ni
ip[(1 + i)
1p 1]
,
-
138 7.
:
a(p)n; i = an; iKp; i, (7.4)
Kp; i , (6.7). ,
a(p)n; i ,
an; i ( 3) Kp; i ( 4).
(3.3) S, - (6.8), - :
PV = S(1 + i)n = Rr(1 + i)n 1(1 + i)r 1 (1 + i)
n =
= Rr1 (1 + i)n(1 + i)r 1 .
i, :
PV = Rr
1 (1 + i)ni
(1 + i)r 1i
= Rran; isr; i
.
, :
PV = Rran; isr; i
. (7.5)
(7.5) 2 3.
7.3.2. m
(3.4) S, - (6.9), - :
PV = S
(1 +
jmm
)nm
= R
(1 + jmm
)nm 1(1 + jmm
)m 1(1 +
jmm
)nm
=
-
7.3. 139
= R1
(1 + jmm
)nm
(1 + jmm
)m 1 . jm/m, :
PV = R
1(1 + jmm
)nm
jmm(
1 + jmm
)m 1jmm
= Ranm; jm
m
sm; jmm
.
, - :
PV = Ranm; jm
m
sm; jmm
. (7.6)
p -
(3.4) S, - (6.10):
PV = S
(1 +
jmm
)nm
=R
p(1 + jmm
)nm 1(1 + jmm
)mp 1
(1 +
jmm
)nm
=
=R
p1
(1 + jmm
)nm
(1 + jmm
)mp 1
.
jm/m, :
PV =R
p
1(1 + jmm
)nm
jmm(
1 + jmm
)mp 1
jmm
=R
panm; jm
m
smp
; jmm
.
, :
PV =R
panm; jm
m
smp
; jmm
. (7.7)
-
140 7.
p - p = m
. 6.4 , s1; jm/m = 1, (7.7) - :
PV = Ranm; jm
m
m. (7.8)
(3.4) S, - (6.12):
PV = S
(1 +
jmm
)nm
= Rr
(1 + jmm
)nm 1(1 + jmm
)mr 1(1 +
jmm
)nm
=
= Rr1
(1 + jmm
)nm
(1 + jmm
)mr 1 . jm/m, :
PV = Rr
1(1 + jmm
)nm
jmm(
1 + jmm
)mr 1jmm
= Rranm; jm
m
smr; jmm
.
, :
PV = Rranm; jm
m
smr; jmm
. (7.9)
7.3.3.
(3.6) , (6.13), :
PV = Sen = Ren 1e 1 e
n = R1 ene 1 .
, :
PV = R1 ene 1 . (7.10)
-
7.3. 141
p -
(3.6) , - (6.14), :
PV = Sen =R
p e
n 1ep 1
en = Rp 1 e
n
ep 1
.
, :
PV =R
p 1 e
n
ep 1
. (7.11)
(3.6) - (6.15):
PV = Sen = Rren 1er 1 e
n = Rr1 ener 1 .
:
PV = Rr1 ener 1 . (7.12)
, - .
7.2. - , 8 - 2 500 ., . : ) - i = 5%, ) j4 = 5%, ) = 5%?
. - . ) . (7.2):
PV = Ran; i = 2500 a8; 5%.
3 a8; 5% = 6.46321 :
PV = 25006.46321 = 16 158.78 .
-
142 7.
) 4 . - (7.6) n = 8, m = 4, jm/m = 5%/4 = 1.25%:
PV = Ranm; jm
m
sm; jmm
= 2500a32; 1.25%s4; 1.25%
(7.1) :
a32; 1.25% =1 (1 + 0.0125)32
0.0125= 26.2413.
(6.2) :
s4; 1.25% =(1 + 0.0125)4 1
0.0125= 4.07563.
:
PV = 2500 26.24134.07563
= 16 096.47 .
) (7.10) n = 8, = 0.05:
PV = R1 ene 1 = 2500
1 e0.058e0.05 1 = 16 075.33 .
7.3. , - 10 3 000 ., - . 2 . - : ) i = 5%, ) j4 = 5%, ) = 5%.
. p - p = 6.
) . - (7.3) n = 10, i = 5%, p = 6:
PV = Ra(p)n; i = 3000a
(6)10; 5%.
a(6)10; 5% (7.4) 3 4:
a(6)10; 5% = a10; 5%K6; 5% = 7.721731.02064 = 7.88111.
-
7.3. 143
:
PV = 30007.88111 = 23 643.30 .
) 4 . - (7.7) n = 10, m = 4, jm = 5%, p = 6:
PV =R
pamn; jm
m
smp
; jmm
=30000
6 a40; 1.25%s 2
3; 1.25%
.
(6.2) :
s 23; 1.25% =
(1 + 0.0125)23 1
0.0125= 0.66529.
(7.1) :
a40; 1.25% =1 (1 + 0.0125)40
0.0125= 31.3269.
:
PV = 30000 31.326960.66529 = 23 543.79 .
) . - (7.11) n = 10, p = 6, = 0.05:
PV =R
p 1 e
n
ep 1
=3000
6 1 e
0.0510
e0.056 1
=
= 50047.0199 = 23 509.95 .
7.4. , 12 3 2 500 ., . - : ) i = 5%, ) j4 = 5%, ) = 5%?
-
144 7.
. . ) - . (7.5) r = 3, n = 12, i = 5%:
PV = Rran; isr; i
= 2500a12; 5%s3; 5%
.
a12; 5% s3; 5% 3 2, -:
a12; 5% = 8.86325, s3; 5% = 3.1525.
:
PV = 2500 8.863253.1525
= 7028.75 .
) 4 . (7.9) n = 12, m = 4, r = 3,jm/m = 5%/4 = 1.25%:
PV = Rranm; jm
m
smr; jmm
= 2500a48; 1.25%s12; 1.25%
.
2 3 s12; 1.25% a48; 1.25%, (6.2) (7.1):
s12; 1.25% =(1 + 0.0125)12 1
0.0125= 12.8604,
a48; 1.25% =1 (1 + 0.0125)48
0.0125= 35.9315.
:
PV = 2500 35.931512.8604
= 6984.91 .
) =5% = 0.05. (7.12) n = 12, r = 3, = 0.05:
PV = Rr1 ener 1 = 2500
1 e0.6e0.15 1 = 6969.92 .
-
7.3. 145
7.3.4.
. , . , , , ,
. ,
, R, , . . PV - , , ( ) R. . , , limn a
n = 0 a > 1. .
, i
- (7.2). - n:
PV = limn
PV = limn
Ran; i = R limn
1 (1 + i)ni
=R
i.
, :
PV =R
i. (7.13)
p -
, i
- (7.3):
PV = limn
PV = limn
Ra(p)n; i = R limn
1 (1 + i)np[(1 + i)
1p 1]
=
-
146 7.
=R
p[(1 + i)1p 1]
.
(6.6) :
p((1 + i)1/p 1) = iKp; i
,
, PV :
PV = RKp; ii
. (7.14)
, i
- (7.5):
PV = limn
PV = limn
Rran; isr; i
=
= Rr limn
1 (1 + i)n(1 + i)r 1 =
Rr(1 + i)r 1 .
(6.2) ,
(1 + i)r 1 = isr; i , :
PV =Rr
isr; i (7.15)
m jm
- (7.6):
PV = limn
PV = limn
Ranm; jm
m
sm; jmm
=
= R limn
1(1 + jmm
)nm
(1 + jmm
)m 1 =R(
1 + jmm
)m 1 .
-
7.3. 147
(6.2) :(1 +
jmm
)m 1 = jm
msm; jm
m
,
:
PV =R
jmm sm; jm
m
. (7.16)
p - m
jm
- (7.7):
PV = limn
PV = limn
R
panm; jm
m
smp
; jmm
=R
p lim
n
1(1 + jmm
)nm
(1 + jmm
)mp 1
=
=R
p 1(
1 + jmm
)mp 1
.
(6.2) :
(1 +
jmm
)mp 1 = jm
msm
p; jmm
,
:
PV =R
p jmm smp
; jmm
. (7.17)
, m = p, :
s1; jm/m = 1.
(7.17) :
PV =R
m jmm=
R
jm. (7.18)
-
148 7.
m jm
- (7.9):
PV = limn
PV = limn
Rranm; jm
m
smr; jmm
=
= Rr limn
1(1 + jmm
)nm
(1 + jmm
)mr 1 == Rr
1
(1 + jmm )mr 1 .
(6.2) :(1 +
jmm
)mr 1 = jm
msmr; jm
m
.
PV :
PV =Rr
jmm smr; jm
m
. (7.19)
- (7.10):
PV = limn
PV = limn
R1 ene 1 = R limn
1 ene 1 =
R
e 1 .
:
PV =R
e 1 . (7.20)p -
- (7.11):
PV = limn
PV = limn
R1 en
p
(ep 1
) = Rp
(ep 1
) .
-
7.3. 149
, :
PV =R
p
(ep 1
) . (7.21)
- (7.12):
PV = limn
PV = limn
Rr1 enp(er 1) =
Rrer 1 .
, :
PV =Rr
er 1 . (7.22)
7.5. 1 200 000 . -. , - , : ) 12% (), ) j4 = 12%, ) = 12%?
. ) . (7.13) R = 1200 000, i = 0.12:
PV =R
i=
1200 000
0.12= 10 000 000 .
) m jm. (7.16) m = 4, jm = 12%, jm/m = 12%/4 = 3%:
PV =R
jmm sm; jm
m
=1200 000
0.03s4; 3%.
2 s4; 3% = 4.18363. - :
PV =1200 000
0.034.18363 = 9561 074.95.
-
150 7.
) - . - (7.20) R = 1200 000, = 12% = 0.12:
PV =R
e 1 =1200 000
e0.12 1 = 9411 997.13 .
7.6. , ( ).
. ) p - . - (7.14) p = 12, i = 12%:
PV = RKp; ii
= 1200 000K12; 12%0.12
.
4 : K12; 12% = 1.053874826. , :
PV = 1200 000 1.053870.12
= 10 538 700 .
) p - 4 . (7.17) m = 4, p = 12, jm = 12% = 0.12, jm/m = 0.12/4 = 0.03 = 3%:
PV =R
p jmm smp
; jmm
=1200 000
120.03s 412
; 3%
.
s1/3; 3%, - (6.2):
s 13; 3% =
(1 + 0.03)13 1
0.03= 0.33005.
PV:
PV =1200 000
120.030.33005 = 10 099 479.88 .) p -
. (7.21) p = 12, = 12% = 0.12:
PV =R
p
(ep 1
) = 1200 00012(e
0.1212 1)
= 9 950 083.42 .
-
7.4. 151
7.7. 7.5 , - 2 2 400 000 .
. .
) . - (7.15) r = 2, Rr = 2400 000, i = 12% = 0.12:
PV =R
isr; i =2400 000
0.12s2; 12%.
s2; 12% , - (6.3):
s2; 12% =(1 + 0.12)2 1
0.12= 2.12.
, :
PV =2400 000
0.122.12 = 9433 962.26 .
) - . (7.19) r = 2, Rr =2400 000, m = 4, jm/m = 12%/4 = 3% = 0.03:
PV =Rr
jmm smr; jm
m
=2400 000
0.124 s42; 12%
4
=
=2400 000
0.03s8; 3%.
2 : s8; 3% = 8.89234 PV:
PV =2400 000
0.038.89234 = 8996 507.11 .
) - . (7.22) r = 2, Rr = 2400 000, = 12% = 0.12:
PV =Rr
er 1 =2400 000
e0.122 1 = 8847 953.99 .
-
152 7.
7.4.
, . 6.5. , , -, . - . .
, S ., , n . , - q .
, n -, S. , - (7.2) S = an; q, :
=S
an; q. (7.23)
- , , - . . t- Stq, St t- S1 = S. t-
t = Stq. t- :
St = St1 t1 , t = 2, . . . , n. .
7.8. 300 . . 5 , . - q = 5%. .
. (7.23). n = 5, S =S1 = 300 000, q = 0.05. 3 :
a5; 5% = 4.32948.
-
7.5. 153
:
=S
a5; 5%=
300 000
4.32948= 69 292.39 .
:
- (t) ()
t- t- t- (.) dt = St = (.) St0.05
= St1 dt1 St0.051 300 000 69 292 15 000 54 292
2 245 708 69 292 12 285 57 007
3 188 701 69 292 9 435 59 857
4 128 844 69 292 6 442 62 850
5 65 994 69 292 3 300 65 994
: 300 000 .
7.5.
S - . n (7.23):
S =(1 (1 + q)n)
q.
:
(1 + q)n = 1 Sq,
n:
n ln(1 + q) = ln(1 Sq
),
n = ln(1 Sq
)ln(1 + q)
. (7.24)
-
154 7.
n -. - , , - . .
7.9. , - , , , , 70 000 . .
. (7.24) :
n = ln
(1 300 000 0.05
70 000
)ln(1 + 0.05)
= 4.94284.
, 4.94 -. , 4 - 70 000 ., , . - :
= 62943 + 3147 = 66 090 .
:
- (t) ()
t- t- t- (.) dt = St = (.) St0.05
= St1 dt1 St0.051 300 000 70 000 15 000 55 000
2 245 000 70 000 12 250 57 750
3 187 250 70 000 9 362 60 638
4 126 612 70 000 6 331 63 669
5 62 943 66 090 3 147 62 943
: 300 000 .
-
7.6. 155
7.6.
- sn; i an; i i n. , - S , n R . . : ? , S, R, n, sn; i, - (6.3), i.
. S , n , R . - , . S, R n an; i (7.2), i.
i q, n, sn; i an; i:
sn; i = q an; i = q. (7.25)
. - 7.6 9.5.
(7.25) , ( n > 2), . , - . , n = 5 - sn; i = q :
(1 + i)5 1i
= q, (1 + i)5 qi 1 = 0,
i5 + 5i4 + 10i3 + 10i2 + (5 q)i 1 = 0. -
. : .
f(x) [x1;x2] - ( )
-
156 7.
, - f(x) , ( ) - . :
x = x1 f(x1)(x2 x1)f(x2) f(x1) . (7.26)
. : Lotus 1-2-3, Quattro Pro, Excel. .7.6 , Excel.
7.10. 10 2 . . 150 . . - , ?
. 10 - S = 2 . . (6.3), - :
s10; i =S
R=
2000 000
150 000= 13.(3).
, i (7.25). :
(1 + i)10 1i
= 13.(3).
.
f(i) = 0,
f(i) =(1 + i)10 1
i 13.(3).
2 i, f(i) : i = 5% i = 10%. , i :
s10; 5% = 12.5779, f(0.05) = 12.5779 13.(3) = 0.7554,
-
7.7. Excel 157
s10; 10% = 15.9374, f(0.10) = 15.9374 13.(3) =