- 155 -

COMPOUND INTEREST

Compund Interest

. 100 3 12%

=

= = 12

[ = 100 1 = 12 ]

36

[ 3 12 = 36 ]

3

[ = + = 100 + 36 = 136 ]

(12 ) (100 + 12 = 112) 112 :

= 112 1 = 13.44

112 + 13.44 = 125.44 :-

= 125.44 1 = 15.05

= 125.44 + 15.05 = 140.49

100 3 .

:

:

=

(1)

:: .

: .

: .

: .

(1 + ) : () ().

:

Compounded annually Compounded semiannually Compounded quarterly Compounded monthly Compounded daily Compounded continuously . .

.

( )

( )

12% 112% 1 = 12%

12% 212% 2 = 6%

12% 412% 4 = 3%

12% 1212% 12 = 1%

12% 36012% 360 = 0.000333333

12% 360 24 = 864012% 8640 = 0.000013889

(1) :

36% . () ().

: () = 12

() = = 3%

(2) :

2000 10 16% :

- .

- .

- .

- .

- .

- .

: = 2000

- :

() = 10 1 = 10

() = 16%

=

= 2000 (1 + 0.16)10

= 8822.87

= -

= 8822.87 - 2000 = 6822.87

- :

() = 10 2 = 20

() = 0.16 2 = 0.08

=

= 2000 (1 + 0.08)20 = 2000 (1.08)20

= 9321.91

= -

= 9321.91 - 2000 = 7321.91

- :

() = 10 4 = 40

() = 0.16 4 = 0.04

=

= 2000 (1 + 0.04)40 = 2000 (1.04)40

= 9602.04

= -

= 9602.04 - 2000 = 7602.04

- :

() = 10 12 = 120

() = 0.16 12 = 0.01333333

=

= 2000 (1 + 0.01333333)120

= 2000 (1.0133333)120

= 9801.88

= -

= 9801.88 - 2000 = 7801.88

- :

() = 10 360 = 3600

() = 0.16 360 = 0.000444444

=

= 2000 (1 + 0.000444444)3600

= 2000 (1.000444444)3600

= 9902.54

= -

= 9902.54 - 2000 = 7902.54

- :

() = 10 8640 = 86400

() = 0.16 8640 = 0.0000185185

=

= 2000 (1 + 0.0000185185)86400

= 2000 (1.0000185185)86400

= 9905.90

= -

= 9905.90 - 2000 = 7905.90

:

8822.87

9321.91499.04

9602.04280.13

9801.88199.84

9902.54100.66

9905.903.36

.

: Continuous Compounding

.

=

(2)

:: .

: .

: .

-: .

: 2.718282

(3) :

3000 18 8% :

- .

- .

- .

- .

: = 3000 .

- :

() = 2

() = 2 = 3

() = 0.08 2 = 0.04

=

= 3000 ( 1 + 0.04)3

= 3000 ( 1.04 )3

= 3000 1.0124864 = 3374.59

= -

=3374.59 - 3000 = 374.59

- :

() = 4

() = 4 = 6

() = 0.08 4 = 0.02

=

= 3000 ( 1 + 0.02)6

= 3000 ( 1.02 )6

= 3000 1.1261624 = 3378.49

= -

=3378.49 - 3000 = 378.49

- :

() = 12

() = 12 = 18

() = 0.08 12 = 0.0066667

=

= 3000 ( 1 + 0.0066667)18

= 3000 ( 1.0066667 )18

= 3000 1.1270479 = 3381.14

= -

=3381.14 - 3000 = 381.14

- :

- = 0.08

() =

=

= 3000

=3000 0.12

=3000 1.127497 = 3382.49

= -

=3382.49 - 3000 = 382.49

: Effective Interest Rate

. .

:

=

(3)

:

: .

: .

: .

(4) :

12% ()

: () = 4

() = 0.12 4 = 0.03

=( 1 + ) - 1

=(1 + 0.03)4 -1

=(1.03)4 - 1

= 1.125509 - 1

= 0.12551

= 12550%

(5) :

14% :

: .

: .

: .

: .

: : :

=2

=0.14 2 = 0.07

=(1 + ) - 1

=(1 + 0.07)2 - 1

=(1.07)2 - 1

=1.1449 - 1

=0.1449

=14.49 %

: :

=4

=0.14 4 = 0.035

=(1 + ) - 1

=(1 + 0.035)4 - 1

=(1.035)4 - 1=1.147523 - 1

=0.147523

=14.752 %

: :

=12

= 0.14 12 = 0.0116666

=(1 + ) - 1

=(1 + 0.116666)12 - 1

=(1.0116666)12 - 1

=1.14934 - 1

=0.14934

=14.934%

: :

=360

=0.14 360 = 0.000388889

=(1 + ) - 1

=(1 + 0.000388889)360 - 1

=(1.000388889)360 - 1

=1.150243 - 1

=0.150243

=15.024%

(6)

: ( ) 8% () 8.5%

:

.

: ( ) :

= 12

= 0.08 12 = 0.0066667

=(1 + ) - 1

=(1 + 0.0066667)12 - 1

=(1.0066667)12 - 1

=1.0829995 - 1

= 0.0829995 = 8.299995%

: () : =4

=0.085 4 = 0.02125

=(1 + ) - 1

=(1 + 0.02125)4 - 1

=(1.02125)4 - 1

=1.087748 - 1

=0.087748 =8.7748%

SYMBOL 92 \f "Symbol" () ( )

:

() ( )

:

( ) :-

=

- 1

(4)

:

=

-=

=

=2.718282

(7)

18% ( ) .

:

-=18%

=1

=

- 1

=

- 1

=

- 1

=1.197217 - 1

=0.197217 = 19.722%

(8) :

8000 15% :

: .

: .

: .

:

: =8000

-=15%

=3

=

= 8000

=8000

= 8000 1.5683122

=12546.497

:

=8000

-= 15%

=5

=

=8000

=8000

= 8000 2.117

=16936

:

=8000

-= 15%

=7

=

=8000

=8000

= 8000 2.857651

=22861.209

(9) :

20000 . :

( )10% .

()14% .

.

: :

: :

=20000

=4

5 =5 4 = 20

=0.10 4 = 0.25

= (1 + )

=20000 (1 + 0.025)20

=20000 (1.025)20

=20000 1.638616= 32772.329

: .

=32772.329

=12

5 = 5 12 = 60

=0.14 12 = 0.0116666

= (1 + )

=32772.329 ( 1 + 0.0116666)60

=32772.329 (1.0116666)60

=32772.329 2.0056098

=65728.504

SYMBOL 92 \f "Symbol" 65728.504 10 .

(10) :

8500 12 . :-

( ) 12% .

() 15% .

() 18% .

.

: : :

=8500

=2

5 =5 2 = 10

=0.12 2 = 0.06

= (1 + )10

=8500 (1 + 0.06)10

=8500 (1.06)10

=8500 1.7908477

=15222.205

: :

=15222.205

=4

5 = 5 4 = 20

=0.15 4 = 0.0375

= ( 1 + )

=15222.205 (1 + 0.0375)20

=15222.205 (1.0375)20

=15222.205 2.088152

=31786.278

: () :

=31786.278

=12

=2 12 = 24

=0.18 12 = 0.015

= (1 + )

=31786.278 (1 + 0.015)24

=31786.278 (1.015)24

=31786.278 1.4295028

=45438.574

SYMBOL 92 \f "Symbol" 12 45438.574

(11) :

3500 6 3 5 12% .

: . 3 3 :-

1- 3500 3 12% .

2- (1) 12% 5 .

: 3500 :

=3500

=2

3 =3 2 = 6

= 0.12 2 = 0.06

= (1 + )

=3500 (1 + 0.06)6

=3500 (1.06)6

=3500 1.4185191

=4964.817

: 4964.817 12% :

= 4964.817

=12%

=5

= ( 1 + )

=4964.817 (1 + )

=4964.817 ( 1 + 0.05 )

=4964.817 ( 1.05 )

=5213.058

: 3500 4964.817 3 12% 4964.817 5 ( 6 ) 12% .

Present Value at Compound Interest

.

(1) ( ) :

=

=

(5)

:

: .

: .

: .

: .

: .

:

=

(12) :

7200 3 8% .

:=7200

=2

3 =3 2 = 6

=0.08 2 = 0.04

=

= 7200 (1 + 0.04)-6

=7200 (1.04)-6

=7200 0.07903145

=5690.265

.

(13) :

12800 5 15% .

:=12800

=12

5 =5 12 = 60

=0.15 12 = 0.0125

=

=12800 (1 + 0.0125)-60

=12800 (1.0125)-60

=12800 0.4745676

=6074.465

(14) :

11.4% 3 22000 4

:=22000

=12 3 = 4

4 =4 4 = 16

=0.114 4 = 0.0285

=

=22000 (1 + 0.285)-16

=22000 (1.0285)-16

=22000 0.6378686

=14033.11

(15) :

9000 3 8 . 9%

:

3 6 (3 2 = 6) 8 7 :

1- .

2- (1) :

: 9000 :

=9000

=9%

=

=

=

=

=

= 8866.995

: 8866.995 :

=8866.995

=7

=0.09 2 = 0.045

=

= 8866.995 (1 + 0.045)-7

= 8866.995 (1.045)-7

=6515.720

(16) :

:

: 4500 .

: 2000 3000 3 .

6% .

: .

: = 4500

: 3000

=2

=3 2= 6

=0.06 2 = 0.03

=

=3000 (1 + 0.03)-6

=3000 (1.03)-6

=3000 0.8374842

=2512.453

= 2512.453 + 2000 = 4512.453

(4500) (4512.453) .

= 4512.453 - 4500 = 12.453

:

(17) :

3200 3 6400 . :

: .

: .

: .

: : :

=3200

=6400

=3 1 = 3

= (1 + )

6400=3200 (1 + )3

(1 + )3=

(1 + )3= 2

(1 + )3= 2

3 ( + )= 2

(1 + )

=

(1 + )

=0.100343

1 +

= 0.100343

1 +

=1.259921

=1.259921 - 1

=0.259921

=25.99%

: :

=3200

=6400

=3 2 = 6

= (1 + )

6400=3200 (1 + )6

(1 + )6=

(1 + )6=2

:

(1 + )6 = 2

6 (1 + )= 2

(1 + )

=

(1 + )

=0.0501716

1 + = 0.0501716

1 + =1.122462

=1.122462 - 1

=0.122462= 12.25%

SYMBOL 92 \f "Symbol" =12.25 2 = 24.5%

: :

=3200

=6400

=3 4 = 12

= (1 + )

6400=3200 (1 + )12

(1 + )12=

(1 + )12=2

:

(1 + )12= 2

12 (1 + )= 2

(1 + )

=

(1 + )

=0.0250858

1 + = 0.0250858

1 + =1.0594631

=1.0594631 - 1

=0.059463 = 5.946%

SYMBOL 92 \f "Symbol" = 5.496 4 = 21.984%

(18) :

3000 5 19000 . :

: .

: .

: .

: : :

=3000

=19000

=5 12 = 60

= ( + )

19000=3000 (1 + )60

(1 + )60=

(1 + )60=6.333333

:

(1 + )60= 6.333333

60 (1 + )= 6.333333

(1 + )

=

(1 + )

=0.007952

1 + = 0.007952

1 + =1.0184788

=1.0184788 - 1

=0.0184788

=1.84788%

SYMBOL 92 \f "Symbol" = 1.84788 12 = 22.175%

: :

=3000

=19000

=5 360 = 1800

= (1 + )

19000=3000 (1 + )1800

(1 + )1800=

(1 + )1800=6.333333

:

(1 + )1800

= 6.333333

1800 (1 + )= 6.333333

(1 + )

=

(1 + )

=

(1 + )

=0.0004453

1 + = 0.0004453

1 + =1.001026

=1.001026 - 1 = 0.001026

SYMBOL 92 \f "Symbol" = 0.001026 360

= 0.36935 = 36.94%

: :

=3000

=19000

=5

=2.7182818

=

19000=3000

=

=6.33333

(Ln)

= 6.33333

5 - = 6.33333

5 - 1= 6.33333

-

=

=

= 0.369165

=36.92%

(19) :

2500 5600 16% . :

: .

: .

: . : .

_1156586378.unknown

_1156586704.unknown

_1156586899.unknown

_1156586981.unknown

_1156587046.unknown

_1156587075.unknown

_1156587131.unknown

_1156587149.unknown

_1156587111.unknown

_1156587057.unknown

_1156587020.unknown

_1156587035.unknown

_1156587001.unknown

_1156586947.unknown

_1156586967.unknown

_1156586933.unknown

_1156586823.unknown

_1156586862.unknown

_1156586881.unknown

_1156586843.unknown

_1156586782.unknown

_1156586803.unknown

_1156586757.unknown

_1156586552.unknown

_1156586630.doc

9000

=

=

7

_1156586662.unknown

_1156586691.unknown

_1156586651.unknown

_1156586595.unknown

_1156586614.unknown

_1156586575.unknown

_1156586471.unknown

_1156586505.unknown

_1156586532.unknown

_1156586490.unknown

_1156586417.unknown

_1156586438.unknown

_1156586397.unknown

_1156585520.unknown

_1156586100.unknown

_1156586299.unknown

_1156586337.unknown

_1156586359.unknown

_1156586318.unknown

_1156586131.unknown

_1156586276.unknown

_1156586116.unknown

_1156585738.unknown

_1156585889.unknown

_1156586076.unknown

_1156585769.unknown

_1156585567.unknown

_1156585714.unknown

_1156585535.unknown

_1156585223.unknown

_1156585332.unknown

_1156585486.unknown

_1156585505.unknown

_1156585466.unknown

_1156585269.unknown

_1156585290.unknown

_1156585253.unknown

_1156585135.unknown

_1156585186.unknown

_1156585206.unknown

_1156585166.unknown

_1156585082.unknown

_1156585099.unknown

_1156585045.unknown