Anna has been saving $450 in her retirement account each month for the last 20 years and plans to continue contributing $450 each month for the next 20 years. Her account has been earning a 9 percent annual interest rate and she expects to earn the same rate for the next 20 years. Her twin brother, Anthony, has not saved anything for the last 20 years. Due to sibling rivalry, he wants to have as much as Ana is expected to have at the end of 20 years. If Anthony expects to earn the same annual interest rate as Anna, how much must Anthony save each month to achieve his goal?
To find out how much Anthony must save each month to achieve his goal, we can use the future value formula for an ordinary annuity.
The formula for the future value of an ordinary annuity is:
FV = P * ((1 + r) ^ n - 1) / r
Where:
FV = Future Value
P = Monthly payment
r = Interest rate per period
n = Number of periods
Given:
Anna's monthly payment (P) = $450
Anna's interest rate (r) = 9% = 0.09
Anna's number of periods (n) = 20
First, let's calculate the future value (FV) that Anna would have at the end of 20 years:
FV = $450 * ((1 + 0.09) ^ 20 - 1) / 0.09
FV = $450 * (1.09 ^ 20 - 1) / 0.09
FV ≈ $288,068.74
Now, let's find out how much Anthony must save each month to achieve the same future value.
Given:
Anthony's interest rate (r) = 9% = 0.09
Anthony's number of periods (n) = 20
We'll rearrange the future value formula to solve for Anthony's monthly payment (P):
FV = P * ((1 + r) ^ n - 1) / r
Rearranging the formula, we get:
P = FV * r / ((1 + r) ^ n - 1)
Plug in the values:
P = $288,068.74 * 0.09 / ((1 + 0.09) ^ 20 - 1)
P ≈ $288,068.74 * 0.09 / (1.09 ^ 20 - 1)
P ≈ $288,068.74 * 0.09 / 1.90232
P ≈ $15,778.95
Therefore, Anthony must save approximately $15,778.95 each month to achieve the same future value as Anna after 20 years.