To solve this, we'll break the problem down into two parts: first, how much you need to have in your account at year 28 to make withdrawals of $80,000 per year forever, and second, how much you need to deposit each year for the next 15 years to accumulate that amount.

Step 1: How much should be in the account at year 28?

You want to make withdrawals of $80,000 per year forever, starting from year 28. This is a perpetuity, which can be calculated using the formula for the present value of a perpetuity:

$PV = \frac{PMT}{r}$

Where:

• $PV$ is the present value (the amount you need at year 28).
• $PMT$ is the annual payment ($80,000).
• $r$ is the interest rate (9% or 0.09).

Substituting the values:

$PV = \frac{80,000}{0.09} = 888,888.89$

So, you need $888,888.89 in your account at year 28 to be able to withdraw $80,000 per year forever.

Step 2: How much to deposit each year for the next 15 years?

Now, we need to figure out how much to deposit each year for the next 15 years to accumulate $888,888.89 by year 28. The future value of an annuity formula will be used for this:

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

Where:

• $FV$ is the future value ($888,888.89).
• $P$ is the annual deposit (this is what we're solving for).
• $r$ is the interest rate (0.09).
• $n$ is the number of years of deposits (15 years).

Rearranging the formula to solve for $P$:

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

Substituting the known values:

$P = \frac{888,888.89 \times 0.09}{(1 + 0.09)^{15} - 1}$

Now, let's compute this step by step:

$P = \frac{88,888.889}{(1.09)^{15} - 1}$ $P = \frac{88,888.889}{3.6424 - 1}$ $P = \frac{88,888.889}{2.6424}$ $P \approx 33,694.760$

So, you must deposit approximately $33,694.760 per year for 15 years.

Final Answer:

The amount to be deposited each year is $33,694.760.

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Would you like further details on the calculations, or have any specific questions on how to approach these types of problems?

Here are some related questions to expand your understanding:

1. How does the length of time affect the amount you need to deposit each year?
2. What would happen if the interest rate were higher or lower than 9%?
3. How would the solution change if the annual withdrawal amount were $100,000 instead of $80,000?
4. What are the tax implications of withdrawals in a retirement account like this?
5. How does compounding frequency (e.g., quarterly vs. annually) affect this calculation?

Tip: Understanding the concept of perpetuities and annuities is key in retirement planning problems. The present value of a perpetuity allows you to determine how much you need at the beginning, and then the future value of an annuity helps you figure out how much to save annually.