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%?

To find the present value of Jack and Diane's lottery winnings, we need to calculate the present value of an annuity, considering the exception in the eleventh payment.

### Step 1: Break the problem into two parts
- 19 payments of $50,000 (except the 11th year).
- 1 payment of $20,000 at the end of year 10 (the 11th payment).

We will use the present value formula for an annuity to calculate the 19 payments of $50,000 and handle the adjustment for the 11th payment separately.

### Step 2: Present value of 19 payments of $50,000 (except year 11)
The present value of an annuity can be calculated using the formula:

\[
PV = P \times \left(1 - \left(1 + r\right)^{-n}\right) \div r
\]

Where:
- $P = 50,000$ (annual payment),
- $r = 0.06$ (interest rate),
- $n = 20$ (total number of payments).

However, to account for the exception in year 11, we split this into two parts:
- Payments 1 through 10: $n = 10$,
- Payments 12 through 20: $n = 9$.

### Step 3: Present value of the first 10 payments
For the first 10 payments, starting immediately:

\[
PV_{1-10} = 50,000 \times \left(1 - \left(1 + 0.06\right)^{-10}\right) \div 0.06
\]

### Step 4: Present value of payments 12 to 20
These payments are also $50,000 but start from year 12 (which is 11 years into the future). The present value for payments 12 through 20 can be calculated similarly, but we discount them back to the present:

\[
PV_{12-20} = 50,000 \times \left(1 - \left(1 + 0.06\right)^{-9}\right) \div 0.06 \times \left(1 + 0.06\right)^{-11}
\]

### Step 5: Present value of the 11th payment
For the 11th payment, the amount is reduced to $20,000, and we discount it back to the present value:

\[
PV_{11} = 20,000 \times \left(1 + 0.06\right)^{-10}
\]

### Step 6: Sum up the present values
The total present value is the sum of the present values of all components:

\[
PV_{\text{total}} = PV_{1-10} + PV_{12-20} + PV_{11}
\]

Let's calculate this.The present value of Jack and Diane's lottery prize, given a 6% interest rate and the structure of the payments, is approximately **$558,324.58**.

Would you like more details or have any questions?

Here are five related questions to expand your understanding:
1. How would the present value change if the interest rate were higher or lower?
2. What is the significance of discounting future payments when calculating present value?
3. How does the 11th payment affect the overall present value compared to a uniform annuity?
4. What are other methods for calculating the present value of irregular payments?
5. Can the same approach be applied to monthly payments instead of annual ones?

**Tip:** When calculating present values, always account for the timing of the first payment (immediate vs. at the end of the period).

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%?

Calculate the present value of Jack and Diane's lottery prize with 20 annual payments of $50,000, considering a lower 11th payment of $20,000. Learn how to use the present value formula for annuities and discount future payments at a 6% interest rate. Suitable for advanced high school and college-level finance students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation