Let's solve each part **step by step**. We are asked to calculate the firm's value under different growth scenarios. This is a classic **perpetuity with growth** (Gordon Growth Model): \[ \text{Firm Value} = \frac{P_0}{r - g} \] Where: - \(P_0\) = current annual profits = **$4.1 \text{ billion}** - \(r\) = required rate of return (interest rate) = **7% = 0.07** - \(g\) = growth rate of profits (varies by scenario) --- ### **a. Profits grow at an annual rate of 8%** - \(g = 8\% = 0.08\) Plug into formula: \[ \text{Firm Value} = \frac{4.1}{0.07 - 0.08} \] \[ \text{Firm Value} = \frac{4.1}{-0.01} \] \[ \text{Firm Value} = -410 \] **Interpretation:** When the growth rate \(g\) exceeds the required rate of return \(r\), the denominator becomes negative or zero, and the theoretical firm value becomes negative or infinite. **The formula breaks down in this case and the value is technically infinite** (not meaningful in reality). **Final answer (a):** > The firm's value is infinite (the formula does not work for \(g \geq r\)). --- ### **b. Profits grow at an annual rate of 3%** - \(g = 3\% = 0.03\) \[ \text{Firm Value} = \frac{4.1}{0.07 - 0.03} \] \[ \text{Firm Value} = \frac{4.1}{0.04} \] \[ \text{Firm Value} = 102.50 \] **Final answer (b):** > The firm's value is **$102.50 billion**. --- ### **c. Profits grow at an annual rate of 0%** - \(g = 0\) \[ \text{Firm Value} = \frac{4.1}{0.07 - 0} \] \[ \text{Firm Value} = \frac{4.1}{0.07} \] \[ \text{Firm Value} \approx 58.57 \] **Final answer (c):** > The firm's value is **$58.57 billion**. --- ### **d. Profits decline at an annual rate of 2%** - \(g = -2\% = -0.02\) \[ \text{Firm Value} = \frac{4.1}{0.07 - (-0.02)} \] \[ \text{Firm Value} = \frac{4.1}{0.09} \] \[ \text{Firm Value} \approx 45.56 \] **Final answer (d):** > The firm's value is **$45.56 billion**. --- ## **Summary Table** | Scenario | Growth Rate (\(g\)) | Firm Value (billion $) | |----------|---------------------|-------------------------------| | (a) | 8% | Infinite | | (b) | 3% | $102.50 | | (c) | 0% | $58.57 | | (d) | -2% | $45.56 | **All values rounded to two decimal places.**