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 need to calculate the future value of Anna's retirement account after 20 years.

To calculate the future value of an investment with monthly contributions and compound interest, we can use the future value of an ordinary annuity formula:

FV = P * ((1 + r)^n - 1) / r

Where:
FV = Future Value
P = Monthly contribution
r = Monthly interest rate
n = Number of months

In this case, Anna has been saving $450 each month for 20 years, which is a total of 240 months. The interest rate is 9% per year, which is 0.09/12 or 0.0075 as a monthly interest rate.

Let's calculate Anna's future value:

FV = 450 * ((1 + 0.0075)^240 - 1) / 0.0075
FV ≈ $254,252.08

So, after 20 years, Anna is expected to have approximately $254,252.08 in her retirement account.

Now that we know Anthony wants to have the same amount as Anna, we can use the same future value formula to calculate how much he needs to save each month for the next 20 years.

Let's use the same formula with the future value as $254,252.08, the monthly interest rate as 0.0075, and the number of months as 240:

254252.08 = P * ((1 + 0.0075)^240 - 1) / 0.0075

Now, we can solve this equation for P, which represents Anthony's monthly contribution:

P = 254252.08 * 0.0075 / ((1 + 0.0075)^240 - 1)

Using a calculator, we find that P is approximately $786.84.

Therefore, Anthony must save approximately $786.84 each month for the next 20 years to achieve his goal of having as much as Anna is expected to have at the end of 20 years.