投資人今天融券放空200股每股價值35元的FB股票，經紀商的要求保證金比例(margin requirement)為70%，假設融券收入全數存入經紀商，不可用來做其他投資。維持保證金比例(maintenance margin)為50%，FB股價漲到每股多少元時投資人會收到保證金催繳(margin call)?

1. Break Down the Problem

• Initial short sale details:
  • Number of shares sold short: 200
  • Price per share: 35元
• Margin requirement: 70%
• Maintenance margin: 50%
• Identify the stock price at which a margin call occurs.

2. Relevant Concepts

• Initial Equity (E) from margin requirement can be calculated with: $E = \text{(Number of shares)} \times \text{(Price per share)} \times \text{(Margin requirement)}$
• After selling the shares short, the investor has the short sale proceeds and must maintain a certain equity to avoid a margin call based on the maintenance margin.

3. Analysis and Detail

1. Calculate Initial Proceeds and Equity Required:

   • Initial proceeds from short sale: $\text{Proceeds} = 200 \times 35 = 7000 \text{元}$
   • Initial equity required: $\text{Equity} = 7000 \times 0.70 = 4900 \text{元}$

2. Calculate Total Value of Short Position When the Stock Price Increases:

   • Let $P$ be the new share price when a margin call occurs.
   • Total value of the short position (liability at price $P$): $\text{Liability} = 200 \times P$

3. Equity After Price Increase:

   • Equity after price increase when a margin call is triggered can be expressed as: $\text{Equity} = \text{Proceeds} - \text{Liability} = 7000 - 200P$

4. Condition for Margin Call:

   • The margin call occurs when the equity falls below the maintenance margin: $\text{Equity} \leq 0.50 \times \text{Liability}$
   • Plugging in the values: $7000 - 200P \leq 0.50 \times (200P)$
   • Simplifying further: $7000 - 200P \leq 100P$ $7000 \leq 300P$ $P \geq \frac{7000}{300} \approx 23.33$

4. Verify and Summarize

After verifying calculations, the critical price per share at which a margin call is triggered has been accurately estimated.

Final Answer

The investor will receive a margin call when the FB stock price increases to at least 23.33元.