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?

Question

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?

Answer 1

To find out how much Anthony must save each month to achieve his goal, we first need to calculate the future value of Anna's retirement account after 20 years.

Anna has been saving $450 per month for the last 20 years. We can use the formula for the future value of an ordinary annuity to calculate the future value of her contributions:

FV = P * [(1 + r)^n - 1] / r
where:
FV = future value
P = monthly payment (in this case, $450)
r = monthly interest rate (9% divided by 12, or 0.09/12)
n = number of months (20 * 12, or 240)

Using this information, we can calculate the future value of Anna's retirement account:

FV = 450 * [(1 + (0.09/12))^240 - 1] / (0.09/12)
FV ≈ $ 307,243.62

Therefore, at the end of 20 years, Anna is expected to have approximately $307,243.62 in her retirement account.

Now, let's proceed to calculate how much Anthony must save each month to achieve the same amount.

Anthony has not saved anything for the last 20 years, so he has $0 in his retirement account currently. He wants to have the same amount as Anna, which is $307,243.62.

We can use the same formula as before to solve for the monthly payment that Anthony needs to make:

307,243.62 = X * [(1 + (0.09/12))^240 - 1] / (0.09/12)

Simplifying the equation, we can isolate X (the monthly payment Anthony needs to make):

X = 307,243.62 * (0.09/12) / [(1 + (0.09/12))^240 - 1]
X ≈ $208.59

Therefore, Anthony must save approximately $208.59 each month for the next 20 years to achieve the same amount as Anna at the end.