(, , , ). .
- , m (m > 1). j. , j / m.
. j = 20 % 4 , ()
20 %: 4 = 5%.
(8) :
S = P ( 1 + j / m) N, (10)
N - , N= m×t, t - . m , , .
m , .
i - , , , . . i - , , m - i = j/ m.
:
1 + i = ( 1 + j / m) m.
,
i = ( 1 + j / m) m - 1(11)
. , , j =18 %, (m =4).
. (11) :
i = (1 + 0,18 / 4)4 - 1 = 0, 1925 ( 19, 25 %).
. , 25 % .
. i = (1 + 0,25 / 12)12 - 1 = 0,2807 28,07 %.
, 25 % ( ) 28,07 %.
. , , 24% .
. j 2- , , j 12 - .
:
(1 + j 2 / 2)2 = (1 + j 12 / 12)12,
1 + j 2 / 2 = (1 + j 12 / 12)6 Þ j 2 = 2[(1 + j 2 / 12)6 - 1] =
= 2 [(1 + 0,24/12)6 - 1 ] = 0,25 j 2= 25 %.
, t (10) jm m , m S = Sm .
d. t
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S = Pe d t . (12)
d .
. d=8 % 20 . . 5 . .
. (12) ,
S = 20 000 e0,08×5 = 20 000 × e0,4 = 20 000 × 1,49182 = 29 836,49 .
3.1. 400 . . 2 30 % . .
3.2. 500 . . 1 10 % . .
3.3. , 70 . . 5 % .
3.4. $200 6 % . 2, 3, 4 5 : ) ; ) ; ) .
3.5. , 36 %, . : 42,6 %.
3.6. 12 % , .
, . . , . .
, () . t S(t).
JP t 1 t 2
JP = S(t 1 ) / S(t 2 ),
:
h = = JP - 1.
JP = 1 + h.
n , h,
JP = ( 1 + h)n.
, i - hi, n
JP = ( 1 + h 1 ) ( 1 + h 2 )( 1 + hn).
JP , h - .
JD :
JD = 1 / J P = 1 / ( 1 + h).
. 140 . . , , .. JP = 2. JD = 1/2. , 140 . . 140 × 1/2 = 70 . . .
h - , 1 + h,
S = P ( 1 + i)n = P (13)
, h i, S = P, .. : . h > i, . h < i .
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. 10% JP = 1,112 = 3,14. , h = JP - 1 = 2,14 214%.
. .
-. , r.
- , r :
1 + n× r = ( 1 + n × i) JP = ( 1 + n × i)( 1 + h) n ,
(14)
- (n = 1):
1 + r = ( 1 + i)( 1 + h),
r = i + h + h×i (15)
(14), (15) : i %, h r %.
. 6 - 5 . 2 %, 10 % . , .
. JP = (1 + 0,02)6 = 1,1262. (14) -:
r = =0,365 ( 36,5 %).
S= P( 1 + n r) = 5 (1 + 0,5×0,365) = 5,9126 . .
( )
I = 5,9126 - 5,0 = 0,9126 . .
. 1 . . . 11% ( ). 16% . , .
. (15) :
r = 0,11+0,16+ 0,11× 0,16 = 0,2876;
S = 1,0 (1 + 0,2876)2 = 1,658 . .
4.1. 500 . . 20.06.98. 15.09.98. , 1,3. - .
: R = 134%; S R = 658 194 .
4.2. 5 . 3 . 3 % . 10% . - . : R = 13,3 %; SR = 7 272 098 .
4.3. 100 . . 100 % 5 . h = =50 % . , : ) ; ) .
4.4. , 11% 6 %.
, . , .. .
t 0, tn. Rk (k = 1,2,.., n) tk. t 0 = 0 (. 1).
Rk, R k > 0, .
Rk . , .. Rk = R, .
d - , n - , nd . (. 1), , , (. 2).
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, (). S , n , .. . . n R . i% (. 3).
S n .
S = R + R( 1 + i) + R( 1 + i) 2 +...+ R( 1 + i)n- 1
n R 1 + i.
(16)
s(n;i) . (16)
S = R s(n; i)
A , . :
S = A( 1 + i)n A = S( 1 + i)-n .
,
. (17)
a(n;i) . ,
A = R × a(n; i).
. 320 . . . 24 % .
. 24 %: 12 = 2 % (17):
A = 320 = 6052, 4619 . .
(14):
S = = 9734,9952 . .
. . 5 100 . . 20 % , .. . .
. R = 100 . . n = 5 . i = 20%. (16) :
S = 100 = 744,160 . .
, R ( . 4 4).
. 4
, , .
SP (1 + i) :
SP = S (1 + i) sP (n; i) = s (n; i) (1 + i).
P:
P = (1 + i), P (n; i) = a(n; i) (1 + i). (18)
. 5 . 12 . i =3 % . R :
) ( ),
) ( ).
. ) R × a(12;0,03) = 5 .
a(12; 0,03) = = 9,95400.
R = 5 ./ 9,95400 = 502311 .
) : R × a(12;0,03) = 5 . (18):
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P (12;0,03) = a(12;0,03) × (1+ i) = 9,954 × 1,03 = 10,25262;
R = 5 ./10,25262 = 487680 .
, . . . 1,5 (. 5).
. 5 3 (1,5 ) . . , . - .
k ( ) . .5 0, - . k k ,
k = ( 1 + i)-k= R (n;i) ( 1 + i)-k. (19)
. 100 . . , , - . 20 % .
. . , - . 7 . (18) k = 3; n = 7; i = 0,2, :
3 = 100 = 208599 .
. 5 , 430 . . 21 % .
. (19) :
R = k (1+ i) k /(n; i).
k = 2; n = 5; i = 0,21, :
R = 430 1,212 = 215163 .
, . ( ) , .
5.4. m
1 . j / m, j - () . (16),
i = (1+ j / m) m - 1 (. (11)).
:
(20)
. , 3 , 500 . . 15% . , .
. (20) m = 2; n = 3; R = 500; j = 0,15, :
S = 500 = 1 746 500 .
5.5. P -
P , (m = 1). R / P, (16), iP ( P n):
(1 + i) = (1 + iP) P, i P = (1+ i)1/ P 1.
iP (16), :
(21)
. 500 . . 3 . , , 15 % . , .
. R = 500; n = 3; P = 2; m = 1. (21) :
S = = 1779 . .
. , , A = R / i. (17) :
A = R [1-(1+ i)- n ]/ i.
n ¥, , A = R / i.
: $5 000 . 10 %?
. A = R / i = 50 000 .
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: () , - .
. : 5 1000, - 8 800.
8 %.
. :
A 1 = R 1 × a (5;0,08)= 1000 × 3,993 = 3993; A 2 = R × a (8;0,08) = =800×5,747=4598.
= 1 + 2 = 3993 + 4598 = 8591.
, = 8591. , , .
5.1. 80 % 5 500 . . . .
5.2. 12,5 . ., 10 % . 20 . : 3 104 783 .
5.3. , , 10 . . 5 % .
(: ).
5.4. $10 000 . 5 %. : $200 000.
5.5. , - 100 . . (3 % ). - .
5.6. $1000 . 5 % . 5 .
5.7. $1000 $600. 5 %.
5.8. $700 . 8 %.
5.9. , , 4 $420. 8 % .
()
().
. . , , . . . . , .. .