Let's break down the problem and solve it step by step. #### **Given Data** - Construction starts in **1 year** if rent levels make it feasible. - **Construction cost (paid at t = 1):** \$1,300,000 - **NOI at t = 2** (first year after construction): - 60% chance: **\$172,500** - 40% chance: **\$87,000** - **NOI growth rate after t = 2:** **2% per year** - **Required rate of return:** **12%** We are to find: 1. **Land value at t = 1** in each NOI scenario. 2. **Maximum price to pay for land today** (t = 0). --- ### **Step 1: Calculate Value of Building at t = 1 in Each Scenario** For both scenarios, the building will begin to generate NOI at t = 2 (one year after construction). The value of the building at t = 1 (immediately after construction is completed and before NOI starts) is the **present value at t = 1** of all future NOIs (t = 2 onward). The NOI grows at 2% per year, so this is a growing perpetuity starting at t = 2. #### **Formula for Present Value of a Growing Perpetuity (starting at t = 2):** \[ PV_{t=1} = \frac{NOI_{t=2}}{r - g} \] Where: - \( NOI_{t=2} \) = first year's NOI after construction - \( r = 12\% \) (required rate of return) - \( g = 2\% \) (NOI growth rate) --- #### **Scenario 1: NOI = \$172,500 (60% probability)** \[ PV_{t=1} = \frac{172,500}{0.12 - 0.02} = \frac{172,500}{0.10} = \$1,725,000 \] #### **Scenario 2: NOI = \$87,000 (40% probability)** \[ PV_{t=1} = \frac{87,000}{0.12 - 0.02} = \frac{87,000}{0.10} = \$870,000 \] --- ### **Step 2: Value of Land at t = 1 in Each Scenario** **Land value at t = 1** is the value of the completed property minus construction cost: \[ \text{Land Value}_{t=1} = \text{PV of building at } t=1 - \text{Construction Cost} \] #### **Scenario 1:** \[ \text{Land Value}_{t=1} = \$1,725,000 - \$1,300,000 = \boxed{\$425,000} \] #### **Scenario 2:** \[ \text{Land Value}_{t=1} = \$870,000 - \$1,300,000 = \boxed{ -\$430,000 } \] --- ### **Step 3: Maximum Price to Pay for Land Today (t = 0)** We need to find the **expected value** of the land at t = 1, then discount it back one year at 12% to get the present value at t = 0. #### **Expected Value at t = 1:** \[ E[\text{Land Value}_{t=1}] = 0.6 \times \$425,000 + 0.4 \times (-\$430,000) \] \[ = \$255,000 - \$172,000 = \$83,000 \] #### **Discounting to t = 0:** \[ \text{Present Value at t = 0} = \frac{E[\text{Land Value}_{t=1}]}{1.12} \] \[ = \frac{\$83,000}{1.12} = \$74,107.14 \] --- ## **Final Answers** ### **1. Land value at the end of the year (t = 1) in both scenarios:** - **If NOI = \$172,500:**    **\$425,000** - **If NOI = \$87,000:**     **-\$430,000** ### **2. Maximum price the developer should be willing to pay for the land (today):** - **\$74,107.14** --- **Summary Table:** | Scenario | Probability | Land Value at t=1 | |----------|-------------|------------------| | NOI = \$172,500 | 60% | \$425,000 | | NOI = \$87,000 | 40% | -\$430,000 | **Maximum price to pay today (t = 0):** **\$74,107.14** --- **If you have follow-up questions or need sensitivity analysis (e.g., different discount rates), let me know!**