Jack and Diane are lottery winners. They hold the ticket to the Grand Prize in the “Set for Life” Prize that makes 20 consecutive annual payments of $50,000 starting immediately. There is one exception: the eleventh payment (to be received at the end of year 10) is not $50,000 but only $20,000 (that is, this one payment is $30,000 LESS than the other 19). Which of the following comes closest to the present value of the prize if interest rates are 6%?
Group of answer choices

$ 543,496

$ 970,000

$ 635,849

$ 577,906

$ 591,154

To calculate the present value (PV) of the prize, we can treat it as the sum of a series of cash flows, each discounted back to the present at the given interest rate of 6%.

### Step 1: Present value of 19 payments of $50,000
These payments are made every year for the first 9 years and from year 11 to 20. We will use the formula for the present value of an annuity to calculate this portion.

The present value of an annuity is given by:
\[
PV = P \times \left( 1 - \frac{1}{(1 + r)^n} \right) \div r
\]
Where:
- $ P $ = Payment ($50,000)
- $ r $ = Interest rate (6% or 0.06)
- $ n $ = Number of periods (19 payments)

For the first 9 payments (years 0 to 9):
\[
PV_1 = 50,000 \times \left( 1 - \frac{1}{(1 + 0.06)^{10}} \right) \div 0.06
\]

For the 10 payments after year 10 (years 11 to 20), we'll discount it back to year 10 and then discount it further back to year 0. The formula for this portion is:
\[
PV_2 = 50,000 \times \left( 1 - \frac{1}{(1 + 0.06)^{10}} \right) \div 0.06 \times \frac{1}{(1 + 0.06)^{10}}
\]

### Step 2: Present value of the $20,000 payment at year 10
This is a single payment that we need to discount back to the present. The formula for the present value of a single payment is:
\[
PV_3 = 20,000 \div (1 + 0.06)^{10}
\]

### Step 3: Sum of the present values
We will now add $ PV_1 $, $ PV_2 $, and $ PV_3 $ to get the total present value of the prize.

Let me compute these values.The present value of the prize comes to approximately **$584,664**. 

The closest answer to this value is **$577,906**.

Would you like further details on the calculation or have any questions?

Here are 5 related questions to expand your understanding:
1. How does changing the interest rate affect the present value of an annuity?
2. What is the difference between the present value of an annuity and a perpetuity?
3. How would the present value change if the $20,000 payment occurred earlier or later?
4. Can you calculate the present value of monthly payments instead of annual ones?
5. What impact does compounding frequency have on present value?

**Tip:** The further

Jack and Diane are lottery winners. They hold the ticket to the Grand Prize in the “Set for Life” Prize that makes 20 consecutive annual payments of $50,000 starting immediately. There is one exception: the eleventh payment (to be received at the end of year 10) is not $50,000 but only $20,000 (that is, this one payment is $30,000 LESS than the other 19). Which of the following comes closest to the present value of the prize if interest rates are 6%?
Group of answer choices

$ 543,496
$ 970,000
$ 635,849
$ 577,906
$ 591,154

Learn how to calculate the present value of a lottery prize with 20 annual payments of $50,000, adjusted by a reduced payment in year 10. This detailed breakdown includes annuity discounting and time value of money concepts with formulas, suitable for college-level finance students.

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