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.)
To determine the amount that Larry should deposit each month into his account so that both men will have the same amount of money at age 65, we need to calculate the future value of Ben's account when he reaches age 65.
Let's break down the problem step by step:
1. Calculate the number of months that Ben deposits money into his account until he reaches age 65:
Since Ben started depositing when he was 23 years old and will deposit until he reaches 65, we have a total of (65 - 23) * 12 = 504 months.
2. Calculate the monthly interest rate:
The annual interest rate is 3.5%, compounded monthly. To convert it to a monthly interest rate, we divide it by 12: 3.5% / 12 = 0.0029.
3. Calculate the future value of Ben's account:
Using the formula for future value of monthly contributions with compound interest:
FV = P * [(1 + r)^n - 1] / r
where FV is the future value, P is the monthly deposit, r is the monthly interest rate, and n is the number of months.
Plugging in the values:
P = $200 (monthly deposit),
r = 0.0029 (monthly interest rate),
and n = 504 (number of months),
FV = 200 * [(1 + 0.0029)^504 - 1] / 0.0029.
The future value of Ben's account at age 65 is found to be approximately $235,582.28.
4. Calculate the number of months Larry will deposit money into his account:
Larry decided to open his account 5 years after Ben opened his account, so he will deposit money for a total of (65 - 28) * 12 = 444 months.
5. Determine the monthly deposit for Larry's account:
To have the same amount at age 65, Larry's future value should also be approximately $235,582.28. Using the formula mentioned above, we can substitue FV, r, and n to find P (monthly deposit):
235,582.28 = P * [(1 + 0.0029)^444 - 1] / 0.0029
Rearranging the equation to solve for P:
P = 235,582.28 * 0.0029 / [(1 + 0.0029)^444 - 1].
The value found for P is approximately $247.18.
Therefore, Larry should deposit approximately $247.18 each month into his account to have the same amount as Ben at age 65.