Given:
- Periodic payment amount, P = \$50,000
- Periodic interest rate, r = 3.75\% = 0.0375
- Total number of payments, n = 20 years
The formula for the present value of an ordinary annuity is:
PV = P \times \left(\frac{1 - (1 + r)^{-n}}{r}\right)
Substitute the given values into the formula:
PV = \$50,000 \times \left(\frac{1 - (1 + 0.0375)^{-20}}{0.0375}\right)
Now, let's calculate step by step:
**Step 4: Calculate the discount factor**
(1 + 0.0375)^{-20} \approx 0.4789
**Step 5: Perform the subtraction**
1 - 0.4789 \approx 0.5211
**Step 6: Divide by the interest rate**
\frac{0.5211}{0.0375} \approx 13.8962
**Step 7: Multiply by the payment amount**
\$50,000 \times 13.8962 \approx \$694,810.21
**Step 8: Round to the nearest cent**
The present value of the lottery prize is approximately \$694,810.21.
\therefore The present value of the prize is approximately \$694,810.21.