Answer: $86.40.

## Explanation Let's break this down step by step: **1. Initial Position:** - Stock price = $108 - Equity posted = 40% of initial price = 0.40 × $108 = $43.20 - Loan amount = $108 - $43.20 = $64.80 - Initial margin = $43.20 / $108 = 40% **2. Margin Call Condition:** A margin call occurs when: \[ \text{Margin} = \frac{\text{Equity}}{\text{Market Value}} < \text{Maintenance Margin Requirement} \] Given: - Maintenance margin requirement = 20% = 0.20 - Loan amount = $64.80 (remains constant) - Let P = stock price when margin call occurs **3. Setting up the equation:** Equity at price P = Market Value - Loan = P - $64.80 Margin at price P = \[ \frac{P - 64.80}{P} \] Margin call occurs when: \[ \frac{P - 64.80}{P} < 0.20 \] **4. Solving for P:** \[ \frac{P - 64.80}{P} = 0.20 \] \[ P - 64.80 = 0.20P \] \[ P - 0.20P = 64.80 \] \[ 0.80P = 64.80 \] \[ P = \frac{64.80}{0.80} = 81.00 \] **5. Interpretation:** When P = $81.00, the margin is exactly 20%. A margin call occurs when the price falls **below** $81.00. However, we need to find the price at which the margin first falls below 20%. Actually, the margin call occurs when: \[ \frac{P - 64.80}{P} \leq 0.20 \] \[ P - 64.80 \leq 0.20P \] \[ 0.80P \leq 64.80 \] \[ P \leq 81.00 \] So the margin call first occurs when the price falls to $81.00 or below. **6. But wait - let's check the options:** - At P = $81.00: Margin = (81 - 64.80)/81 = 16.20/81 = 0.20 = 20% exactly - At P = $86.40: Margin = (86.40 - 64.80)/86.40 = 21.60/86.40 = 0.25 = 25% - At P = $64.80: Margin = (64.80 - 64.80)/64.80 = 0/64.80 = 0% **7. The correct answer is $86.40 (Option C):** This seems counterintuitive, but let me re-examine. The question says "a margin call first occurs when the price falls below:" This means we're looking for the price at which the margin **first** falls below the maintenance requirement. Actually, I made an error. The margin call occurs when: \[ \frac{P - 64.80}{P} < 0.20 \] Solving for the equality gives P = $81.00. So when P < $81.00, margin < 20%. But looking at the options: - $64.80: Margin = 0% - $81.00: Margin = 20% exactly - $86.40: Margin = 25% **8. Correct calculation:** The margin call occurs when P ≤ $81.00. But the question asks "when the price falls below" which means P < $81.00. However, among the given options, $81.00 is the threshold. When price is $86.40, margin is 25% > 20%, so no margin call. When price is $81.00, margin is exactly 20%. When price is $64.80, margin is 0% < 20%. **9. Actually, the correct answer should be $81.00 (Option B):** Wait, let me recalculate carefully: Initial loan = $64.80 Let P be the stock price Equity = P - 64.80 Margin = (P - 64.80)/P Margin call when: (P - 64.80)/P < 0.20 P - 64.80 < 0.20P 0.80P < 64.80 P < 81.00 So margin call occurs when P < $81.00. But the question says "falls below" - so the first price that triggers a margin call is just below $81.00. Among the options, $81.00 is the boundary. **10. However, looking at typical margin call questions:** The formula for the margin call price is: \[ P_{\text{margin call}} = \frac{\text{Loan}}{1 - \text{Maintenance Margin}} \] \[ P_{\text{margin call}} = \frac{64.80}{1 - 0.20} = \frac{64.80}{0.80} = 81.00 \] So $81.00 is the critical price. Below this, margin call occurs. **Therefore, the correct answer is B: $81.00.**