8%
8% · 5000 = 0, 08 · 5000 = 400
25% 250
x - 100%
250 - 125%
250
x = · 100% = 200
125%
25%
x - 100%
250 - 75%
250
x = · 100% = 333, 33
75%
 K0
O = r · K0
10% 20%
50%
10
r
rr
r - ri
r = rr + ri + rr · ri rr =
1 + ri
ri
5% 3%
0,05-0,03 0,02
rr = = = 0, 0194
1+0,03 1,03
rr = 1, 94%
rf
rf = r · (1 - T ) T
T = 19%
r = 4%
rf = 4% · (1 - 0, 19) = 3, 24%
p
Kn On n
p p
Kn = K0 · (1 + n · r), On = K0 · n · r, Kn = K0 + On.
p
Kn +n2+...+nm = K0 · (1 + n1 · r1 + n2 · r2 + . . . nm · rm)
1
r = 4%
5000
rf = 4% · (1 - 0, 19) = 3, 24%
p
K1 = 5000(1 + 1 · 0, 0324) = 5162
p
K2 = 5000(1 + 2 · 0, 0324) = 5324
p
1
K1/2 = 5000(1 + · 0, 0324) = 5081
2
p
9
K9/12 = 5000(1 + · 0, 0324) = 5121, 5
12
1 9
· 0, 0324 · 0, 0324
2 12
 ik = a·r r
a
108
Kp = 5000(1 + · 0, 0324) = 5048, 6
360
12% 10% 9%
r1 = 12% · 0, 81 = 9, 72% r2 = 10% · 0, 81 = 8, 1% r3 = 9% · 0, 81 = 7, 29%
4 2 6
n1 = , n2 = , n3 = .
12 12 12
1 1 6
Kp = 1000(1 + · 0, 0972 + · 0, 081 + · 0, 0729) = 1082, 35
3 6 12
r n = n1 + n2 + . . . + nm
r1, r2, . . . , rm
K0(1 + rn) = K0(1 + n1r1 + n2r2 + . . . + nmrm)
m
"
1 n1r1 + n2r2 + . . . + nmrm
r = njrj =
n n1 + n2 + . . . + nm
j=1
r
1 1 1
· 0.12 + · 0, 10 + · 0, 09
3 6 2
r = = 0, 1017
1 1 1
+ +
3 6 2
r = 10, 17%
 K0 Kn
p
Kn
K0 =
1 + nr
D = Kn - K0
r = 32%
2000
K0 = = 1948, 05
30
1+ ·0,32
360
d
DH = Kn · d · n
K0 = Kn - DH = Kn(1 - dn)
Kn K0
d
DH = Kn - K0 = 2000 - 1948, 05 = 51, 95.
DH 51, 95
d = = = 0, 3117
30
Kn · n 2000 ·
360
31, 17%
 D = DH
1 1
n = - .
d r
n = 1
r d
d = r = .
1 + r 1 - d
d < r r
d
0 0 K0
1 K0r K1 = K0 + K0r = K0(1 + r)
2 K1r = K0(1 + r)r K2 = K1 + K1r = K1(1 + r) =
= K0(1 + r)(1 + r) = K0(1 + r)2
3 K2r = K0(1 + r)2r K3 = K0(1 + r)3
s
n Kn-1r = K0(1 + r)n-1r Kn = K0(1 + r)n
"
s
Kn
n
r = - 1
K0
Kn
log
lnKn - lnK0
K0
n = =
log(1 + r) ln(1 + r)
{Kn}
 K0 = 2000 r = 8%
rf = 8% · 0, 81 = 6, 48%
s
K1 = 2000 · (1 + 0, 0648) = 2129, 6
s
K3 = 2000 · (1 + 0, 0648)3 = 2414, 53
( )nk
r
s
Knk = K0 1 +
k
1000
6%
rf = 6% · 0.81 = 4, 86%
( )2·2
s 0.0486
K2·2 = 1000 1 + = 1100, 8
2
( )2·4
s 0.0486
K2·4 = 1000 1 + = 1101, 43
4
( )2·12
s 0.0486
K2·12 = 1000 1 + = 1101, 86
12
( )2·360
s 0.0486
K2·360 = 1000 1 + = 1102, 07
360
K0
r = 7, 1%
2K0
ln
ln 2
K0
n = = H" 10
ln 1, 071 ln 1, 071
 n
r %
70
n H"
r
n
70
n = H" 10
7, 1
[( ) ]nr
k
( )nk
r r
r
s
Kk" = lim K0 1 + = K0 lim 1 + = K0enr
k" k"
k k
s s
Knk = Knk
1 2
( )nk ( )nk
1 2
r1 r2
K0 1 + = K0 1 +
k1 k2
( )k ( )k
1 2
r1 r2
1 + = 1 +
k1 k2
8%
9%
r1f = 8% ·) 81 = 6, 48% r2f = 9% 0, 81 = 7, 29%
0,
( ( )·
4 1
0,0648 0,0729
1 + = 1, 0664 1 + = 1, 0729
4 1
 11% 12%
11, 5% 12, 5%
( )4 ( )1
0,11 0,115
1 + = 1, 1146 1 + = 1, 115
4 1
( )4 ( )1
0,12 0,125
1 + = 1, 1255 1 + = 1, 12
4 1
s
1 2 m
Kn +n2+...+nm = K0(1 + r1)n (1 + r2)n . . . (1 + rm)n
1
r
1 2 m
K0(1 + r)n = K0(1 + r1)n (1 + r2)n . . . (1 + rm)n ,
"
n
1 2 m
r = (1 + r1)n (1 + r2)n . . . (1 + rm)n - 1
7%
6%
K0 = 1000 r.
r1f = 7% · 0, 81 = 5, 67% r2f = 6% · 0, 81 = 4, 86%
s
K1+2 = 1000(1 + 0, 0567)(1 + 0, 0486)2 = 1161, 90
"
3
r = 1, 07 · 1, 062 - 1 = 0, 0633 6, 33%
 re
( )k
r
re = 1 + - 1
k
k
r
re
24%
25%
( )12
0,24
re = 1 + - 1 = 0, 268
12
( )4
0,25
re = 1 + - 1 = 0, 274
4
rd
%
"
( )
k
rd = k 1 + r - 1
8%
"
" rd = 2( 1 + 0, 08 - 1) = 0, 0785
4
rd = 4( 1 = 0, 08 - 1) = 0, 0777
7, 85% 7, 77%
s
K0 Kn
s
Kn
K0 =
(1 + r)n
Kn - K0 = D
r = 5%
5000
K0 = = 4319, 19
(1+0,05)3
S
T = ,
N
S N
Sn = S - nT
( )
n-1
On = Sn-1 · r = S 1 - r
N
r
An = T + On = T [1 + (N - n + 1)r]
n {1, 2, . . . , N}
N + 1
O = O1 + O2 + . . . + ON = r · S ·
2
(
N + 1)
A = A1 + A2 + . . . + AN = S 1 + r
2
24000
25%
 An
1 24000 6000 6000 12000 18000
2 18000 6000 4500 10500 12000
3 12000 6000 3000 9000 6000
4 6000 6000 1500 7500 0
"
24000 15000 39000
N + 1 5
O = r · S · = 0, 25 · 24000 · = 15000
2 2
( (
N + 1) 5)
A = S 1 + r = 24000 1 + 0, 25 · = 39000
2 2
m mN
r
rm =
m
N mN r
rm
S
T =
mN
Sn = S - nT n = 1, 2, . . . , mN.
(
n - 1) r
On = Sn-1 · rm = S 1 - · n = 1, 2, . . . , mN.
mN m
r mN + 1
O = O1 + O2 + . . . + OmN = · S ·
m 2
An = T + On n = 1, 2, . . . , mN.
(
mN + 1)
A = A1 + A2 + . . . + AmN = S 1 + rm
2
2 · 4 = 8
24000
T = = 3000
8
0,25
rm = = 0125
2
An
1 24000 3000 3000 6000 21000
2 21000 3000 2625 5625 18000
3 18000 3000 2250 5250 15000
4 15000 3000 1875 4875 12000
5 12000 3000 1500 4500 9000
6 9000 3000 1125 4125 6000
7 6000 3000 750 3750 3000
8 3000 3000 375 3375 0
"
24000 13500 37500
8 + 1
O = 0, 125 · 24000 · = 13500
2
(
8 + 1)
A = 24000 1 + 0, 125 · = 37500
2
(
r n - 1)
Ok,n = S 1 - k = 1, 2, . . . , m, n = 1, 2, . . . , N.
m N
N + 1
O = r · S ·
2
(
N + 1)
A = S 1 + r
2
An
1 24000 3000 3000 6000 21000
2 21000 3000 3000 6000 18000
3 18000 3000 2250 5250 15000
4 15000 3000 2250 5250 12000
5 12000 3000 1500 4500 9000
6 9000 3000 1500 4500 6000
7 6000 3000 750 3750 3000
8 3000 3000 750 3750 0
"
24000 15000 39000
4 + 1
O = 0, 25 · 24000 · = 15000
2
(
4 + 1)
A = 24000 1 + 0, 25 · = 39000
2
N + 1 mN + 1 m - 1 1
"O = rS - rmS = rmS = 0, 125 · 24000 · = 1500
2 2 2 2
(
S m + 1)
On = · r · N - n + n = 1, 2, . . . , N.
N 2m
r mN + 1
O = O1 + O2 + . . . + ON = · S ·
m 2
n
S
Sn = S - n n = 1, 2, . . . , N.
mN
S
An = + On n = 1, 2, . . . , N.
mN
S
A =
mN
24000
25% 26%
1%
 24000
x
x - 1% · x = 24000
x · 99% = 24000
24000
x = = 24242
0, 99
24242
T = = 3030.25
8
A
1 24242 3030 25 3030 25 21211 75
2 21211 75 3030 25 3030, 25 + 2651 47 8711 97 18181 5
3 18000 3030 25 3030 25 15151 25
4 15000 3030 25 2272, 69 + 1893 91 7196 85 12121 25
5 12000 3030 25 3030 25 9090 75
6 9000 3030 25 1515, 13 + 1136 35 5681 73 6060 5
7 6000 3030 25 3030 25 3030 25
8 3000 3030 25 787, 87 + 393 94 4211 06 0
"
242420 13681 61 37923 61
m
r
( )m m re =
r
1 + - 1
m
S
T =
N
S
Sn = (N - n)
N
S
On = (N - n + 1)re
N
N + 1
O = S re
2
S
An = [1 + (N - n + 1)re]
N
N
A1 = A2 = . . . = AN = A
N S
S(1 + r)N = SqN, q = 1 + r (1)
qN - 1
A + A(1 + r) + . . . + A(1 + r)N-1 = A (2)
q - 1
q - 1
A = SqN q = 1 + r.
qN - 1
qn - 1
Sn = Sn-1 · q - A = Sqn - A n = 1, 2, . . . , N
q - 1
On = Sn-1 · r n = 1, 2, . . . , N
Tn = A - On = Sn-1 - Sn n = 1, 2, . . . , N
( )
q - 1
O = NA - S = S NqN - 1
qN - 1
1, 25 - 1
A = 24000 · 1, 254 · = 10162, 60163 H" 10162, 602
1, 254 - 1
O1 = S · r = 24000 · 0, 25 = 6000
T1 = A - O1 = 4162, 602
S1 = S - T1 = 24000 - 4162, 602 = 19837, 398
O2 = S1 · r = 4959, 35
T2 = A - O2 = 10162, 602 - 4959, 35 = 5203, 26
S2 = S1 - T2 = 19837, 398 - 5203, 26 = 14634, 146
O3 = S2 · r = 14634, 13 · 0, 25 = 3658, 537
T3 = A - O3 = 10162, 602 - 3658, 537 = 6504, 065
S3 = S2 - T3 = 14634, 146 - 6504, 065 = 8130, 081
O4 = S3 · r = 8130, 081 · 0, 25 = 2032, 52
T4 = A - O4 = 10162, 602 - 2032, 52 = 8130, 082
S4 = S3 - T4 = 0
Tn A
1 24000 4162 602 6000 10162 602 19837 398
2 19837 39 5203 252 4959 35 10162 602 14634 146
3 14634 13 6504 065 3658 537 10162 602 8130 081
4 8130 081 8130 082 2032 52 10162 602 0
"
24000 001 16650 407 40650 408
 m
2 q - 1
A = SqN ·
2m + (m - 1)r qN - 1
m
( )mN (1 + r
)m - 1
q - 1 r
m
A = SqN = S · 1 + ·
r
qN - 1 m (1 + )mN - 1
m
( )m
r
q = 1 + re, re = 1 + - 1
m
p
S(1 + r)p
Op = Sr p
S
S = 24000(1 + 0, 25)3 = 46875
 A
1 24000 - - - 30000
2 30000 - - - 37500
3 37500 - - - 46875
4 46875 11718 75 11718 75 23437 5 35156 25
5 35156 25 11718 75 8789 06 20507 81 23437 5
6 23437 5 11718 75 5859 38 17578 13 11718 75
7 11718 75 11718 75 2929 69 14648 44 0
"
46875 29296 88 76171 88
N + 1 4 + 1
O = Sr = 46875 · 0, 25 · = 29296, 88
2 2
S · r = 24000 · 0, 25 = 6000
3 · 6000 + 15000 = 33000
Ke = [A1(1 + r)N-1 + A2(1 + r)N-2 + . . . + AN-1(1 + r) + AN] - S
( )m
r
re = 1 + - 1,
m
m r
 K r
n
( )
n + 1
+
Pn = K · n · 1 + r
2
n
( )
n - 1
-
Pn = K · n · 1 + r
2
%
12%
3 + 1
+
P3 = K · 3 · (1 + r)
2
K
2000
K = = 537, 64
3 · (1 + 2 · 0, 12)
 K r
n
(1 + r)n - 1
Q+ = K(1 + r)
n
r
n
(1 + r)n - 1
Q- = K
n
r
12000
6%
(1 + 0, 06)5 - 1
Q+ = 12000(1 + 0, 06) = 71703, 82
5
0, 06
(1 + 0, 06)5 - 1
Q- = 12000 = 67645, 16
5
0, 06
K0 n
1 - (1 + r)-n (1 + r)n - 1
K0 = K(1 + r) = K
r r(1 + r)n-1
1 - (1 + r)-n (1 + r)n - 1
K0 = K = K
r r(1 + r)n
16%
16%
rm = = 4%
4
S
1622
1, 04 - 1
1622 = S · 1, 048 ·
1, 048 - 1
1, 048 - 1
S = 1622 = 10920, 51
0, 04 · 1, 048
S K0
K0 > 10000 10000
5%
(1 + r)5 - 1
Q5 = K0(1 + r)5 + K
r
(1 + 0, 05)5 - 1
20000 = 10000(1 + 0, 05)5 + K
0, 05
20000 - 10000(1, 05)5
K = · 0, 05 = 1309, 75
(1, 05)5 - 1
 m · n n
( )(1
m + 1 + r)n - 1
Q+ = K m + r
nm
2 r
( )(1
m - 1 + r)n - 1
Q- = K m + r
nm
2 r
4, 8%
4,8%
r = = 1, 2% m = 3 n = 4 · 3 = 12
4
( )(1 + 0, 012)12 - 1
3 + 1
Q+ = 200 3 + 0, 012 = 7756, 29
3·12
2 0, 012
( )(1 + 0, 012)12 - 1
3 - 1
Q- = 200 3 + 0, 012 = 7725, 51
3·12
2 0, 012
m
( )m (1 + r
)mn - 1
(1 + re)n - 1 r
m
Q+ = K(1 + re) = K 1 +
n
r
re m (1 + )m - 1
m
r
(1 + )mn - 1
(1 + re)n - 1
m
Q- = K = K
n
r
re (1 + )m - 1
m
( )m
r
re = 1 + - 1
m
 4%
re = (1 + 0, 04)4 - 1 = 0, 1699
(1 + 0, 1699)4 - 1
Q+ = 1000(1 + 0, 1699) = 6013, 01
4
0, 1699
(1 + 0, 1699)4 - 1
Q- = 1000 = 5139, 76
4
0, 1699
4%
4%
r = = 1%
4
re = (1 + 0, 01)2 - 1 = 0, 0201
(1 + 0, 0201)16 - 1
20000 = K ·
0, 0201
20000 · 0, 0201
K = = 1072, 17
(1, 0201)16 - 1