We are asked to determine the present value of complex cash flows where you withdraw a specific amount each year for a certain period and an additional lump sum in the last year. To solve this, we will break it down step by step using the present value formulas for both an annuity and a lump sum.

Step-by-Step Breakdown:

1. Annual Withdrawals: You will withdraw $13,000 per year starting 11 years from now for 55 years (periods 11 through 15). This is an annuity that starts at year 11.
2. Lump Sum: You will also withdraw an additional $26,000 in year 15.

We are given an interest rate of 5% per year.

Steps to Calculate:

Step 1: Present Value of Annuity ($13,000 per year for 5 years)

To find the present value of the annuity starting from year 11, we use the present value of an annuity formula: $PV_{\text{annuity}} = C \times \frac{1 - (1 + r)^{-n}}{r}$ Where:

• $C = 13,000$ (the annual withdrawal)
• $r = 0.05$ (the interest rate)
• $n = 5$ (the number of periods)

This will give us the present value at year 10 (since payments start at year 11), and we will discount it back to today.

Step 2: Present Value of Lump Sum ($26,000 in year 15)

The present value of the lump sum can be calculated using the formula: $PV_{\text{lump sum}} = \frac{F}{(1 + r)^n}$ Where:

• $F = 26,000$ (the lump sum to be received in year 15)
• $r = 0.05$ (the interest rate)
• $n = 15$ (the number of years)

Step 3: Combine the Results

Once we have the present value at year 10 of the annuity and the present value of the lump sum at year 15, we need to discount both amounts back to today.

I'll perform these calculations now.The amount you would need to deposit today is $47,059.44 to ensure that starting 11 years from now, you can withdraw $13,000 annually for 5 years and an additional $26,000 in the 15th year, assuming a 5% interest rate.

Would you like further clarification or more details? Here are five related questions to deepen your understanding:

1. How does changing the interest rate affect the present value?
2. What is the difference between an ordinary annuity and an annuity due?
3. How would the calculations change if the payments were made monthly instead of annually?
4. What if the lump sum amount was withdrawn earlier, say in year 13 instead of year 15?
5. How does the length of the withdrawal period affect the total deposit needed?

Tip: Always remember to use the correct present value formula based on the timing of the cash flows—whether it's an annuity or a single lump sum.