You plan on depositing $10,000 at the end of each year for 30 years into a retirement account that pays 5% interest. How much could you withdraw annually in equal beginning of year amounts starting at the time you make your last deposit and continuing for a total of 20 years, assuming balances continue to earn 5% until withdrawn?
To calculate the amount you could withdraw annually, you need to find the future value of your deposits over 30 years and then calculate the annuity payment for 20 years.
Let's break it down step by step:
1. Calculating the future value of your deposits (over 30 years):
- The deposit is $10,000 per year.
- The interest rate is 5% per year.
- Since the deposits are made at the end of each year, this is an ordinary annuity.
- Using the future value of an ordinary annuity formula:
FV = P * ((1 + r)^n - 1) / r
P = $10,000 (annual deposit)
r = 5% (interest rate)
n = 30 (number of years)
FV = $10,000 * ((1 + 0.05)^30 - 1) / 0.05
= $10,000 * ((1.05)^30 - 1) / 0.05
= $10,000 * (1.05^30 - 1) / 0.05
= $10,000 * (1.05^30 - 1) / 0.05
= $10,000 * (4.3219 - 1) / 0.05
= $432,191.75
Therefore, the future value of your deposits after 30 years is $432,191.75.
2. Calculating the annuity payment for 20 years:
- You want to withdraw an equal amount annually for 20 years.
- The interest rate is 5% per year (since balances continue to earn interest until withdrawn).
- Using the future value of an annuity formula:
PMT = FV * r / ((1 + r)^n - 1)
FV = $432,191.75 (future value from step 1)
r = 5% (interest rate)
n = 20 (number of years)
PMT = $432,191.75 * 0.05 / ((1 + 0.05)^20 - 1)
= $432,191.75 * 0.05 / (1.05^20 - 1)
= $432,191.75 * 0.05 / (2.6533 - 1)
= $432,191.75 * 0.05 / 1.6533
= $86,438.35
Therefore, you could withdraw approximately $86,438.35 annually for 20 years.