1. **Problem Statement:** Eastern Shallow, Ltd. operates a gold mine producing 50,000 ounces per year. Gold price is 300 per ounce, production cost is 250 per ounce, and the mine will last 7 years at this rate. The required rate of return is 10%. We need to find: a. The value of the mine under current conditions. b. The value of the mine if production costs increase by 10% annually due to inflation, price remains constant, and required return rises to 21%. c. Whether NPV can be used if the company can shut, reopen, or abandon the mine based on gold price fluctuations. --- 2. **Formulas and Concepts:** - Net cash flow per year = (Price per ounce - Cost per ounce) × Production quantity - Value of the mine = Present Value (PV) of net cash flows over 7 years - PV of annuity formula: $$PV = C \times \frac{1 - (1 + r)^{-n}}{r}$$ where $C$ is annual net cash flow, $r$ is discount rate, $n$ is number of years - For inflation in costs, costs grow by 10% annually, so cost in year $t$ is $250 \times 1.1^{t-1}$ - NPV method assumes cash flows are known or can be estimated; option to shut or abandon introduces flexibility (real options). --- 3. **Part a: Value of the mine under current conditions** - Annual net cash flow: $C = (300 - 250) \times 50,000 = 50 \times 50,000 = 2,500,000$ - Discount rate: $r = 0.10$ - Number of years: $n = 7$ - Calculate PV: $$PV = 2,500,000 \times \frac{1 - (1 + 0.10)^{-7}}{0.10}$$ Calculate $(1 + 0.10)^{-7} = 1.10^{-7} \approx 0.51316$ $$PV = 2,500,000 \times \frac{1 - 0.51316}{0.10} = 2,500,000 \times \frac{0.48684}{0.10} = 2,500,000 \times 4.8684 = 12,171,000$$ --- 4. **Part b: Value of the mine with inflation in costs and higher required return** - Price per ounce remains 300 - Cost per ounce in year $t$: $250 \times 1.1^{t-1}$ - Net cash flow in year $t$: $50,000 \times (300 - 250 \times 1.1^{t-1})$ - Required rate of return: $r = 0.21$ - Calculate PV of each year's net cash flow and sum: Calculate each year's net cash flow: Year 1: $50,000 \times (300 - 250 \times 1.1^{0}) = 50,000 \times (300 - 250) = 2,500,000$ Year 2: $50,000 \times (300 - 250 \times 1.1^{1}) = 50,000 \times (300 - 275) = 50,000 \times 25 = 1,250,000$ Year 3: $50,000 \times (300 - 250 \times 1.1^{2}) = 50,000 \times (300 - 302.5) = 50,000 \times (-2.5) = -125,000$ From year 3 onwards, net cash flow is negative, so the company would likely stop production. We consider only years 1 and 2. Calculate PV: $$PV = \frac{2,500,000}{(1 + 0.21)^1} + \frac{1,250,000}{(1 + 0.21)^2} = \frac{2,500,000}{1.21} + \frac{1,250,000}{1.4641} \approx 2,066,116 + 853,242 = 2,919,358$$ --- 5. **Part c: Can NPV method be used if the company can shut, reopen, or abandon the mine?** - The NPV method assumes fixed cash flows and does not account for managerial flexibility. - When the company can respond to price fluctuations by shutting or abandoning, the project has real options. - Real options valuation methods are more appropriate in this case. - Therefore, NPV alone is not sufficient to value the mine under these conditions. --- **Final answers:** a. Value of the mine = $12,171,000$ b. Value of the mine with inflation and higher return = $2,919,358$ c. NPV method alone is not appropriate when the company can shut, reopen, or abandon the mine.