Question

Marvin would like to have $50,000 set aside 15 years from now. Assuming a 5% return, how much does Marvin need to put away on a monthly basis to ensure a future goal of $50,000? Assume those monthly payments are made at the beginning of the month. You must show your work!

Answer 1

Solution:

The formula for calculating the Future value of savings at the end of “ n” months with monthly compounding , where the first payment is made at the beginning of the first period

FV = P * [ ( ( 1 + r ) ⁿ   – 1 ) / r ] * ( 1 + r )

Where FV = Future value of savings   ; 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 = $ 50,000 ;

Annual Interest rate = 5 % = 0.05

Thus Monthly interest rate = 0.05 / 12 =   0.0041667                ( Since the compounding is monthly )

Thus r = 0.0041667

The monthly deposits are to be made for a period of 15 years

Converting the same into months we have

= 15 * 12 months = 180 months

Thus n = 180

P = To find    ;

Applying the available information in the formula we have

$ 50,000 = P * [ ( ( 1 + 0.0041667 ) ^{( 180 )} – 1 ) / 0.0041667 ] * ( 1 + 0.0041667 )

$ 50,000 = P * [ ( ( 1.0041667 ) ^{( 180 )} – 1 ) / 0.0041667 ] * ( 1.0041667 )

$ 50,000 = P * [ ( 2.1137039 – 1 ) / 0.0041667 ] * ( 1.0041667 )

$ 50,000 = P * [ ( 1.1137039 / 0.0041667 ] * ( 1.0041667 )

$ 50,000 = P * [ 267.2889438 ] * ( 1.0041667 )

$ 50,000 = P * 268.4026477

P * 268.4026477 = $ 50,000

P = $ 50,000 / 268.4026477

P = $ 186.2872830

Thus the monthly payments to be made by Marvin = $ 186.2873 ( when rounded off to four decimal places )

= $ 186.29 ( when rounded off to two decimal places )

Note: The value of ( 1.0041667 ) ¹⁸⁰   is calculated using the Excel formula =POWER(Number,Power)

=POWER(1.0041667,180) = 2.1137039