Question

Find the future value for each of the following scenarios, where m is the periodic deposit and r is the interest rate.

		compounding	time	future	interest
m	r	frequency	in years	value	earned
$275	5.7%	annually	9	$	$
$275	4.9%	semiannually	11	$	$
$450	2.1%	quarterly	14	$	$
$375	3.7%	monthly	14	$	$
$200	7.5%	weekly	8	$	$

Answer 1

Scenario 1:

Periodic deposit = $275

Annual interest rate = 5.70%
Compounding frequency = 1

Periodic interest rate = Annual interest rate / Compounding frequency
Periodic interest rate = 5.70% / 1
Periodic interest rate = 5.70%

Time in years = 9

Number of periods = Time in years * Compounding frequency
Number of periods = 9 * 1
Number of periods = 9

Future value = $275*1.057^8 + $275*1.057^7 + …. + $275*1.057 + $275
Future value = $275 * (1.057^9 - 1) / 0.057
Future value = $275 * 11.349632
Future value = $3,121.15

Total deposit = Periodic deposit * Number of periods
Total deposit = $275 * 9
Total deposit = $2,475.00

Interest earned = Future value - Total deposit
Interest earned = $3,121.15 - $2,475.00
Interest earned = $646.15

Scenario 2:

Periodic deposit = $275

Annual interest rate = 4.90%
Compounding frequency = 2

Periodic interest rate = Annual interest rate / Compounding frequency
Periodic interest rate = 4.90% / 2
Periodic interest rate = 2.45%

Time in years = 11

Number of periods = Time in years * Compounding frequency
Number of periods = 11 * 2
Number of periods = 22

Future value = $275*1.0245^21 + $275*1.0245^20 + …. + $275*1.0245 + $275
Future value = $275 * (1.0245^22 - 1) / 0.0245
Future value = $275 * 28.701646
Future value = $7,892.95

Total deposit = Periodic deposit * Number of periods
Total deposit = $275 * 22
Total deposit = $6,050.00

Interest earned = Future value - Total deposit
Interest earned = $7,892.95 - $6,050.00
Interest earned = $1,842.95

Scenario 3:

Periodic deposit = $450

Annual interest rate = 2.10%
Compounding frequency = 4

Periodic interest rate = Annual interest rate / Compounding frequency
Periodic interest rate = 2.10% / 4
Periodic interest rate = 0.525%

Time in years = 14

Number of periods = Time in years * Compounding frequency
Number of periods = 14 * 4
Number of periods = 56

Future value = $450*1.00525^55 + $450*1.00525^54 + …. + $450*1.00525 + $450
Future value = $450 * (1.00525^56 - 1) / 0.00525
Future value = $450 * 64.905217
Future value = $29,207.35

Total deposit = Periodic deposit * Number of periods
Total deposit = $450 * 56
Total deposit = $25,200.00

Interest earned = Future value - Total deposit
Interest earned = $29,207.35 - $25,200.00
Interest earned = $4,007.35

Scenario 4:

Periodic deposit = $375

Annual interest rate = 3.70%
Compounding frequency = 12

Periodic interest rate = Annual interest rate / Compounding frequency
Periodic interest rate = 3.70% / 12
Periodic interest rate = 0.3083%

Time in years = 14

Number of periods = Time in years * Compounding frequency
Number of periods = 14 * 12
Number of periods = 168

Future value = $375*1.003083^167 + $375*1.003083^166 + …. + $375*1.003083 + $375
Future value = $375 * (1.003083^168 - 1) / 0.003083
Future value = $375 * 219.674492
Future value = $82,377.93

Total deposit = Periodic deposit * Number of periods
Total deposit = $375 * 168
Total deposit = $63,000.00

Interest earned = Future value - Total deposit
Interest earned = $82,377.93 - $63,000.00
Interest earned = $19,377.93

Scenario 5:

Periodic deposit = $200

Annual interest rate = 7.50%
Compounding frequency = 52

Periodic interest rate = Annual interest rate / Compounding frequency
Periodic interest rate = 7.50% / 52
Periodic interest rate = 0.1442%

Time in years = 8

Number of periods = Time in years * Compounding frequency
Number of periods = 8 * 52
Number of periods = 416

Future value = $200*1.001442^415 + $200*1.001442^414 + …. + $200*1.001442 + $200
Future value = $200 * (1.001442^416 - 1) / 0.001442
Future value = $200 * 569.456376
Future value = $113,891.28

Total deposit = Periodic deposit * Number of periods
Total deposit = $200 * 416
Total deposit = $83,200.00

Interest earned = Future value - Total deposit
Interest earned = $113,891.28 - $83,200.00
Interest earned = $30,691.28