A young executive deposits $400 at the end of each month for 9 years and then increases the deposits. If the account earns 6%, compounded monthly, how much (to the nearest dollar) should each new deposit be in order to have a total of $400,000 after 25 years?
Thank you for your help.
To find out how much each new deposit should be in order to have a total of $400,000 after 25 years, we need to break down the problem into several steps.
Step 1: Calculate the value of the deposits made during the first 9 years.
Since the young executive deposits $400 at the end of each month for 9 years, we need to calculate the future value of these monthly deposits.
To do this, we will use the formula for the future value of a series of deposits:
FV = P * [(1 + r)^n - 1] / r
Where:
FV = Future Value
P = Monthly deposit amount
r = Monthly interest rate
n = Number of deposits
In our case, P = $400, r = 6% / 12 = 0.005, and n = 9 * 12 = 108.
FV = 400 * [(1 + 0.005)^108 - 1] / 0.005
FV ≈ $55,367.88
So, the value of the deposits made during the first 9 years is approximately $55,367.88.
Step 2: Calculate the future value of the additional deposits made over the next 16 years.
To find out how much money needs to be deposited over the next 16 years to reach a total of $400,000, we subtract the value of the deposits made in the first 9 years from the desired total.
Total additional deposits needed = $400,000 - $55,367.88
Total additional deposits needed ≈ $344,632.12
Step 3: Calculate the future value of the additional deposits.
Using the same formula as in Step 1, we can calculate the future value of the additional deposits:
FV = P * [(1 + r)^n - 1] / r
In this case, P is the monthly deposit amount we want to find, r is still 0.005, and n is the number of deposits made over the next 16 years. Since the deposits are made annually, n is 16.
$344,632.12 = P * [(1 + 0.005)^16 - 1] / 0.005
Now, we need to solve this equation for P. We can rearrange the equation to isolate P:
P = ($344,632.12 * 0.005) / [(1 + 0.005)^16 - 1]
P ≈ $925.04
So, each new deposit should be approximately $925.04 in order to have a total of $400,000 after 25 years. Please note that this is an approximation as the final answer was rounded to the nearest dollar.