...
, ,
4.1.
4.2.
M 4.3.
M 4.4.
M 4.5.
M 4.6.
4.7.
M 4.8.
4.9,
M 4.10.
4.11.
, . , , , . (time value of money, TVM) , (discounted cash flow, DCF), .
(TVM) : (, ) , , . , , , . , , , . - . .
, : . .
4.1.
. , , , (present value) , (PV) (future value) (FV). , . , , 1000 . (PV) 10% . , , , 1000 . 10% .
:
PV , . 1000 .
i , . 10% ( 0,10 ).
, .
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FV .
. -, ? 1000 ., , 100 . (10% 1000 . 0,11000 .). , , 1100 .:
FV = 1000 . 1,10 = 1100 .
1100 . , ? 10% 1100 . 0,10 1100 ., 110 . , 1210 .
, (1210 .) . 1000 . , , 100 . 100 . . , , (simple interest) (200 . ). , 10 ., , 100 ., . , , (compound interest). (210 .) (200 .) (10 .).
, 210 . , . , , , . 1,1 ( 1,10 , ) 1,1:
FV =10070 . 1,11,1 =1000 . 1,1 =1210 .
FV = 1000 . 1,1 1,11,1=1000 . 1,13 =1331 .
, ' ;
:
1000 . 1,1 1,11,1 1,1 1,1 =1000 . 1,15= 1610,51 .
, . 1000 . 10% 1610,51 . 610,51 ., 500 . 110,51 . - .
4.1
5-, ? ?
, . 4.1, . , , , , 10%. , , . 4.1, , , . (100 .), - , 10% .
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4.1.
(.)
(.)
(.)
1000,00
100,00
1100,00
1100,00
110,00
1210,00
1210,00
121,00
1331,00
1331,00
133,10
1464,10
1464,10
146,41
1610,51
610,51
. . 4.1 . 4.1. 1000 . 10% , , 100 ., . .
, i , 1000 . :
FV=1000(1+i)n (4.1)
(4.1), PV (1000 .), 1 . (future value factor). 1610,51. :
= (1 + i)n
!0% , (1610,51). , 500 ., 10% , : 500 . 1610,51 = S04.254 . , , . . 4.2 . 4.2 .
. 4.1.
4.1.1.
, 1000 . 10% .
1. 1000 1,1 :
1000 . 1,11,11,11,11,1 = 1610,51 .
, . () , . , :
1000 .1,15 =1610,51 .
4.2. 1 . .
, n
2%
4%
6%
8%
10%
12%
1,0200
1,0400
1,0600
1,0800
1,1000
1,1200
1,0404
1,0816
1,1236
1,1664
1,2100
1,2544
1,0612
1,1249
1,1910
1,2597
1,3310
1,4049
1,0824
1,1699
1,2625
1,3605
1,4641
1,5735
1,1041
1,3167
1,3382
1,4693
1,6105
1,7623
1,2190
1,4802
1,7908
2,1589
2,5937
3,1058
1,3459
1,009
2,3366
3,1722
4.1772
4,4736
1,4859
2,1911
3,2071
4,6610
6,7275
9,6463
. . 4.2 , 4,3 1 . . , . 1 .:
FV= (1 + i)n
i , , a n .
1 3 5 7 9 11 13 15 17 19
. 4.2. I
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, , . . 4.3 . (n), (i) (PV), (FV). , , . , Lotus Excel, .
. 4.3.
2. , . 4.2. , 5 i 10%. , 1,6105. 1000 . .
3. , , , . 72 (Rule of 72). , , , (" "), 72, , :
e =
, 10% 7,2 , . 1000 ., 7,2 2000 ., 14,4 4000 ., 8000 . 21,6 ..
4.1.2.
20 , 100 . 45 8% . 65 ? , ? , 9%, 65 ?
:
FV = 100 .1,0845 =3192 .
100 ., 3092 . 45 0,08 100 ., 360 ., 2732 .
, 9%, :
FV = 100 . 1,0945 = 4833 .
, 1% , 1641 . (.4833 . ~ 3192 .) 65 . 50%- (1641 ./ 3192 . = 0,514). , .
, 72 . 8% 100 . 9 . , 45 5 , 3200 .:
100 . 2 2 2 2 2 = 100 .32 = 3200 .,
3192 .
9% 8 . 45 5,5 (45/8 = 5,625). , 50% , 8% : 1,5 3200 .= -4800. 4833 .
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4.1.3.
. 10000 . . . 7% , 6% . ?
, , . (reinvestment rate), .. , , (.. ), . , , 8% .
, , . , . ;
FV =10000 . 1,07 -11449 .
:
FV =10000 . 1,061,08 =11448 .
, .
4.1.4.
, , 100 ., . , 6% . , . , . ?
, :
FV - 100 .1,0650 = 1842 .
4.2
1626 (Peter Minuit) , 24 . 6% ; , , 1996 , .. 370 ?
4.2.
, (annual percentage rate, APR), ( 6% ) (, ). , . ( ) (effective annual rate, EFF), .
, (APR) 6% , . , ; APR. , 0,5% ( 0,005 ).
EFF , . :
FV =-(1,005)n =1,0616778
.
EFF=1,0616778-1=0,0616778 6,16778%
:
EFF= (1 +
APR
)m - 1
(4.2)
m
APR , . . 4.3 , 6% .
, . , , , . , (1 + APR / )" "", 2,71828 ( ). e= 1,0618364. , , EFF = 0,0618365, 6,18365% .
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4.3. APR 6%
6,00000%
6,09000%
6,13614%
6,16778%
6,17998%
6,18313%
6,18365%
4.3
, 12% . ?
4.3.
M 4.3-4.6
:
" 10 , 8% ?" (: FV = 2159 . !)
, , , . , 15000 . , , ? , .
. , , , . , .
1000 . 10% . , , 1000 . 10%, , 1,1 . , :
1,1 = 1000 .
, :
= 1000 . /1,1= 909,09 .
, 10% , 909,09 . , 1000 . .
, 1000 . . , , 10%, , 909,09 ., 10% . , :
1000 . = PVx1,12 = PVx1,21
:
PV = 1000 . /1,12 = 826,45 .
, 826,45 ., 10% , 1000 . .
, , , , . , , . . . , , , (discounted cashflow (DCF) analysis).
1 . , i , :
PV=
(4.3)
(1+i)n
() 1 . i .
1 ., 10% , :
PV = 1/1.15=0,62092 I, I
1000 . 10%, 1000 . 620,92 .
, , . 4.2, , . , . , 1000 ., 10% , . 4.2 , 1,6105, 1000 . :
1000 ./1,6105= 620,92 .
, . 4.4, , , . 4.2. . 4-4 10% , 0,6209.
1 . :
PV=
(1+i)n
i , , .
4.4. 1 .
, i
, n
2%
4%
6%
8%
10%
0,9804
0,9615
0,9431
0,9259
0,9091
0,9612
0,9246
0,8830
0,8573
0,8264
0,9423
0,8890
0,8396
0,793
0,7513
0,9238
0,8548
0,7921
0,7350
0,6830
0,9057
0,8219
0,7473
0,6806
0,6209
, , , , 1 . . , , 10% 1 . 0,9091 ., , 20 , 0,1486 .
4.3.1. 100 100
100 ., . . "", , 100 . , 8% , ? ?
100 ., 8% . , . :
V=100 ./1,085
, 100 1,08 68. ( , , 4.3), n, i FV, , PV. 1 , . 4.4. , 8% 5 , 0,6806. 100 . , 68 .
. , . , 68 ., , , 100 . 68 . , 8% .
4.4
100 ., 6% ?
4.4.