​(Present **value of complex cash**​ flows)  How much do you have to deposit today so that beginning 11 years from now you can withdraw 

​$12 comma 00012,000 

a year for the next 

88 

years​ (periods 11 through 

1818​) 

plus an *additional* amount of 

​$24 comma 00024,000 

in the last year​ (period 

1818​)? 

 Assume an interest rate of 

99 

percent.

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Part 1

The amount of money you have to deposit today is 

​$enter your response here. 

 ​(Round to the nearest​ cent.) 

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.

(Present **value of complex cash**​ flows)  How much do you have to deposit today so that beginning 11 years from now you can withdraw $12,000 a year for the next 8 years​ (periods 11 through 18​) plus an additional amount of ​$24,000 in the last year​ (period 18​)? Assume an interest rate of 9 percent.

Learn how to calculate the present value of complex cash flows, including an annuity of $12,000 for 8 years starting 11 years from now, plus a $24,000 lump sum in year 18, all discounted at 9% interest. This detailed step-by-step solution explains the key formulas and concepts in time value of money, annuities, and discounting.

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