Morgan Jennings, a geography professor, invests \( \$ 99,000 \) in a parcel of land that is expected to increase in value by 15 percent per year for the next ten years. He will take the proceeds and provide himself with a 18 -year annuity. Assuming a 15 percent interest rate, how much will this annuity be? (Do not round intermediate

To determine the annual annuity Morgan Jennings will receive, we'll follow these steps: 1. **Calculate the Future Value (FV) of the Land Investment:** Morgan invests \$99,000 in land expected to appreciate at 15% annually for 10 years. The future value \( FV \) after 10 years is calculated using the formula: \[ FV = P \times (1 + r)^n \] Where: - \( P = \$99,000 \) - \( r = 0.15 \) (15%) - \( n = 10 \) years \[ FV = 99,000 \times (1.15)^{10} \approx 99,000 \times 4.04556 \approx \$400,510.215 \] 2. **Determine the Annual Annuity \( R \):** After 10 years, Morgan will convert the \$400,510.215 into an 18-year annuity with an annual interest rate of 15%. The present value \( P \) of an annuity is given by: \[ P = R \times \frac{1 - (1 + i)^{-n}}{i} \] Solving for \( R \): \[ R = \frac{P \times i}{1 - (1 + i)^{-n}} \] Where: - \( P = \$400,510.215 \) - \( i = 0.15 \) (15%) - \( n = 18 \) years First, calculate \( (1 + i)^{-n} \): \[ (1.15)^{-18} \approx \frac{1}{12.37545} \approx 0.0808 \] Then, compute the denominator: \[ 1 - 0.0808 = 0.9192 \] Now, calculate \( R \): \[ R = \frac{400,510.215 \times 0.15}{0.9192} \approx \frac{60,076.53225}{0.9192} \approx \$65,360 \] **Final Answer:** The annuity will be approximately **$65,360** per year.