determine how much Larry should deposit each month into his account so that both men will have the same amount of money in their accounts at age 65

Question

Since he was 23 years old, Ben has been depositing $200 at the end of each month into a tax-free retirement account earning interest at the rate of 3.5%/year compounded monthly. Larry, who is the same age as Ben, decided to open a tax-free retirement account 5 years after Ben opened his. If Larry's account earns interest at the same rate as Ben's, determine how much Larry should deposit each month into his account so that both men will have the same amount of money in their accounts at age 65. (Round your answer to the nearest cent.)

Answer 1

We need to start by determining the Future Value of Ben's investment, which is the amount of money he will have at age of 65 in the tax-free retirement account.

Since Ben in paying $200 at the end of every month that means this is an ordinary annuity ( ordinary annuity is when the same amount od money is being paid at the end of every period, Annuity due is used when the same amount is being paid at the beginning of the period), we can either use the Future Value annuity tables for easy solution or apply the formula. Since the tables are not provided here we will use the Future Value formula.

The formula to determin Future Value is

P = PMT [((1 + r)ⁿ - 1) / r]

where:

P is the Future Value

PMT is the payment made every period

r is the interest rate

n is number of period or payments

For Ben

PMT = $200

r=0.00291667 (we get this number by dividing the interest rate by 12 to get the monthly rate 0.035/12, 3.5% or 0.035 is the annual rate we need to get the monthly rate since the periods are on monthly basis)

n =504 ( we get this number by identifying how many payments Ben will make he started at age of 23 and will stop at age of 65 meaning he will pay for 42 years and since he is paying monthly we multiply the 42 by 12 month 42*12).

Now we can get the Future Value by using the formula P=200*(((1+0.00291667)⁵⁰⁴)-1)/0.00291667

Future Value=$229,024.42

The problem is asking us the determine the monthly payments for Larry. If Larry and Ben will have the same amount of money at the age of 65. This means that Larry's investment Future Value is equal to Ben's investment Future Value which is equal to $229,024.42

We have the interest rate which is the same and we can get the number of periods by either subtracting 60 period from total Ben's periods which is (5years * 12 month per year) or by calculating that Larry will start paying at age of 28 (5 years after Ben) and stop at age of 65 (65-28)*12=444)

Now we can use the same Future Value formula to solve for the unknown part which is the monthly payment and our formula will look like this

229,024.42=PMT*(((1+0.00291667⁾⁴⁴⁴⁾-1)/0.00291667)

229024.42=PMT*906.555694

PMT=229024.42/906.555694

PMT=252.63

Larry should pay every month $252.63 to get the same amount of money as Ben at age of 65.