To solve this problem, we need to approach it in two steps:

1. Determine the amount needed at retirement to support annual withdrawals of $30,000 for 30 years.
2. Find the amount that needs to be deposited each year for 20 years to achieve that goal, considering the account earns 7% interest.

Step 1: Calculate the required retirement amount

The withdrawals form an annuity, and we need to find the present value of this annuity at the time of retirement. The formula for the present value of an annuity is:

$PV = P \times \frac{1 - (1 + r)^{-n}}{r}$

Where:

• $P = 30,000$ (annual withdrawal)
• $r = 0.07$ (interest rate per year)
• $n = 30$ (number of years for withdrawals)

Substituting the values:

$PV = 30,000 \times \frac{1 - (1 + 0.07)^{-30}}{0.07}$

Step 2: Calculate the amount needed to be saved each year

Now, we need to determine how much needs to be deposited annually for 20 years to reach the amount calculated in Step 1. This involves a future value of an annuity calculation. The formula is:

$FV = D \times \frac{(1 + r)^n - 1}{r}$

Where:

• $FV$ is the future value (the amount we need at retirement)
• $D$ is the annual deposit
• $r = 0.07$ (interest rate per year)
• $n = 20$ (number of years until retirement)

Let me calculate both steps for you.To achieve your retirement goal, you will need to deposit $9,080.78 each year for 20 years.

Would you like more details or have any questions?

Here are 5 related questions:

1. How would the deposit amount change if the interest rate were 6% instead of 7%?
2. What happens to the required deposit if the number of withdrawal years increases to 35 years?
3. How much would you need to deposit annually if you wanted to withdraw $40,000 each year instead of $30,000?
4. What effect would a delay in starting deposits by 5 years have on the required annual deposit?
5. How would you adjust your savings if you expected to retire in 25 years instead of 20?

Tip: The earlier you start saving for retirement, the lower your required annual deposits, thanks to the power of compound interest!