Question

Shaylea, age​ 22, just started working​ full-time and plans to deposit ​$5,500 annually into an IRA earning 9 percent interest compounded annually. Deposits will be made at the end of each year. How much would she have in 20 ​years, 30 ​years, and 40 ​years? If she changed her investment period and instead invested ​$458.33 monthly and the investment also changed to monthly​ compounding, how much would she have after the same three time​ periods? Comment on the differences over time.

Answer 1

1)

Future value after 20 years:

Future value = Present value ( 1 + r)ⁿ

Future value = 5,500 ( 1 + 0.09)²⁰

Future value = 5,500 * 5.604411

Future value = $30,824.59

Future value after 30 years:

Future value = Present value ( 1 + r)ⁿ

Future value = 5,500 ( 1 + 0.09)³⁰

Future value = 5,500 * 13.267678

Future value = $72,972.32

Future value after 40 years:

Future value = Present value ( 1 + r)ⁿ

Future value = 5,500 ( 1 + 0.09)⁴⁰

Future value = 5,500 * 31.40942

Future value = $172,751.81

Future value after 20 years if compounded monthly:

Rate = 9% /12 = 0.75%

Number of periods = 20 * 12 = 240

Future value = Monthly payment * [ ( 1 + r)ⁿ - 1] / r

Future value = 458.33 * [ ( 1 + 0.0075)²⁴⁰ - 1] / 0.0075

Future value = 458.33 * 667.88687

Future value = $306,112.59

Future value after 30 years if compounded monthly:

Rate = 9% /12 = 0.75%

Number of periods = 30 * 12 = 360

Future value = Monthly payment * [ ( 1 + r)ⁿ - 1] / r

Future value = 458.33 * [ ( 1 + 0.0075)³⁶⁰ - 1] / 0.0075

Future value = 458.33 * 1,830.743483

Future value = $839,084.66

Future value after 40 years if compounded monthly:

Rate = 9% /12 = 0.75%

Number of periods = 40 * 12 = 480

Future value = Monthly payment * [ ( 1 + r)ⁿ - 1] / r

Future value = 458.33 * [ ( 1 + 0.0075)⁴⁸⁰ - 1] / 0.0075

Future value = 458.33 * 4,681.320273

Future value = $2,145,589.52

Comment:

There is a direct relationship between future value and time. When time increases, future value also increases considerably because of compounding effect.

When the compounding period increases as in this case where the compounding period was increased to monthly, the future value also increases. This happens because of compounding effect where interest gets added to interest.