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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)?

Question

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)?

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Solution

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=(Number of shares)×(Price per share)×(Margin requirement) 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: Proceeds=200×35=7000 \text{Proceeds} = 200 \times 35 = 7000 \text{元}
    • Initial equity required: Equity=7000×0.70=4900 \text{Equity} = 7000 \times 0.70 = 4900 \text{元}
  2. Calculate Total Value of Short Position When the Stock Price Increases:

    • Let P P be the new share price when a margin call occurs.
    • Total value of the short position (liability at price P P ): Liability=200×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: Equity=ProceedsLiability=7000200P \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: Equity0.50×Liability \text{Equity} \leq 0.50 \times \text{Liability}
    • Plugging in the values: 7000200P0.50×(200P) 7000 - 200P \leq 0.50 \times (200P)
    • Simplifying further: 7000200P100P 7000 - 200P \leq 100P 7000300P 7000 \leq 300P P700030023.33 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.

This problem has been solved

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