To solve this problem, we need to calculate the present value of a series of future cash flows. In this case, you're planning to withdraw $12,000 per year for 8 years, starting 11 years from today (years 11 through 18), and in the final year (year 18), you will also withdraw an additional $24,000.

The interest rate is 9% per year.

Step-by-Step Process

1. Break Down the Cash Flows:

• Years 11 through 17: You withdraw $12,000 each year (7 years).
• Year 18: You withdraw $12,000 plus an additional $24,000 (total $36,000 in year 18).

2. Calculate the Present Value of the Annuity:

The withdrawals from years 11 through 17 form an annuity of $12,000 per year for 7 years, starting 11 years from now. We need to calculate the present value of this annuity as of year 10 (since it starts in year 11). Then, we’ll discount that amount back to the present (year 0).

The present value of an annuity is given by the formula:

$PV_{\text{annuity}} = P \times \left[ \frac{1 - (1 + r)^{-n}}{r} \right]$

Where:

• $P$ = annual payment ($12,000)
• $r$ = interest rate (9% or 0.09)
• $n$ = number of periods (7 years)

Substituting the values:

$PV_{\text{annuity}} = 12,000 \times \left[ \frac{1 - (1 + 0.09)^{-7}}{0.09} \right]$

First, calculate the term inside the brackets.

$1 - (1 + 0.09)^{-7} = 1 - \frac{1}{(1.09)^7} = 1 - 0.54703 = 0.45297$

Now calculate the present value factor:

$PV_{\text{annuity}} = 12,000 \times \frac{0.45297}{0.09} = 12,000 \times 5.032 = 60,384$

This is the present value of the annuity as of year 10. Now, we need to discount this amount back to the present (year 0) using the discount factor $(1 + r)^{-10}$:

$PV_{\text{year 0}} = 60,384 \times \frac{1}{(1.09)^{10}} = 60,384 \times 0.42241 = 25,493.68$

3. Calculate the Present Value of the Lump Sum in Year 18:

In year 18, you will withdraw $36,000. We need to discount this back to the present (year 0):

$PV_{\text{lump sum}} = 36,000 \times \frac{1}{(1.09)^{18}} = 36,000 \times 0.23643 = 8,511.48$

4. Total Present Value:

Now, add the present value of the annuity and the present value of the lump sum to get the total present value:

$PV_{\text{total}} = 25,493.68 + 8,511.48 = 34,005.16$

So, the amount of money you have to deposit today is approximately $34,005.16.

Summary:

The amount you need to deposit today is $34,005.16.

Would you like more details or clarification on any part? Here are 5 related questions you might explore:

1. How does changing the interest rate affect the present value?
2. What if the withdrawal period started sooner (e.g., in year 5)?
3. How do different compounding periods (quarterly, monthly) affect this calculation?
4. How would the calculation change if the additional $24,000 was withdrawn earlier?
5. How is the present value affected if you receive $15,000 instead of $12,000 annually?

Tip: Always break down cash flows into annuities and lump sums when dealing with multiple types of withdrawals.