Question

A lottery winner is entitled to receive $50,000 every year for the next 10 years. If the interest rate is 3 percent per year, what is the present value of the lottery prize? Round the answer to the nearest hundredths.

Answer 1

To calculate the present value (PV) of the lottery prize, we can use the formula for the present value of an annuity, since the winner will receive a fixed amount each year for a specified number of years.

The formula for the present value of an annuity is:

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

where:
- \( P \) is the annual payment (in this case, $50,000),
- \( r \) is the annual interest rate (3% or 0.03), and
- \( n \) is the number of years (10 years).

Substituting the values into the formula:

\[
PV = 50000 \times \left(1 - (1 + 0.03)^{-10}\right) / 0.03
\]

Let's calculate it step by step.

1. Calculate \( (1 + r)^{-n} \):

\[
(1 + 0.03)^{-10} = (1.03)^{-10} \approx 0.744094
\]

2. Use this value to compute \( 1 - (1 + r)^{-n} \):

\[
1 - 0.744094 \approx 0.255906
\]

3. Now, substitute this value back into the PV formula:

\[
PV = 50000 \times \frac{0.255906}{0.03}
\]

4. Calculate \( \frac{0.255906}{0.03} \):

\[
\frac{0.255906}{0.03} \approx 8.59686667
\]

5. Finally compute the present value:

\[
PV = 50000 \times 8.59686667 \approx 429843.3333
\]

Rounding this to the nearest hundredths, the present value of the lottery prize is:

\[
\boxed{429843.33}
\]