Question

Suppose payments were made at the end of each month into an ordinary annuity earning interest at the rate of 2.5%/year compounded monthly. If the future value of the annuity after 10 years is $65,000, what was the size of each payment? (Round your answer to the nearest cent.)

Answer 1

Solution:

The formula for calculating the Future value of annuity at the end of n years with monthly compounding is :

FV = P * [ [ ( 1 + r ) ⁿ- 1 ] / r ]

Where FV = Future value of annuity   ; P = Periodic Deposit i.e., Fixed amount of Monthly deposit

r = monthly rate of interest   ; n = no. of months

A per the information given in the question we have

FV = $ 65,000   ; Annual rate of Interest = r = 2.5 % = 0.025   ;  

Thus monthly interest rate = 0.025 / 12 = 0.002083

n = 10 * 12 = 120 months ; ( Since number of years = 10 years )

P = To find    ;

Applying the above values in the formula we have:

65,000 = P * [ [ ( 1 + 0.002083 ) ¹²⁰   - 1 ] / 0.002083 ]         

65,000 = P * [ [ ( 1.002083 ) ¹²⁰   - 1 ] / 0.002083 ]

65,000 = P * [ [ 1.283692 – 1 ] / 0.002083 ]

65,000 = P * [ 0.283692 / 0.002083 ]

65,000 = P * 136.171940

P * 136.171940 = 65,000

P = 65,000 / 136.171940

P = 477.337694

P = $ 477.34 ( when rounded off to the nearest cent )

Thus the size of each payment = $ 477.34

Note: The value of ( 1.002083 ) ¹²⁰   is calculated using the Excel formula =POWER(Number,Power)

=POWER(1.002083,120)= 1.283692