Hannah sold short 100 shares of ABC stocks at its current price of $45. Assuming the initial margin is 60% and the maintenance margin is 30%.

a. A month later, ABC stock dropped to $40. What is her return on equity for this investment?

b. Assuming that ABC instead rose to $50. Calculate her new margin. What is the maximum price of ABC before she will receive a margin call?

Let's tackle the problem step by step.

### Part (a) - ABC stock dropped to $40. What is Hannah's return on equity?

#### **Initial Setup**:
- Number of shares sold short: 100
- Initial stock price (when sold): $45
- Stock price after one month: $40
- Initial margin requirement: 60%
- Maintenance margin: 30%

When Hannah sold the stock short at $45, she received $ 100 \times 45 = 4500 $ dollars. However, because of the initial margin requirement of 60%, she must deposit 60% of the value of the short sale as margin. 

The margin required:
\[
\text{Initial Margin} = 60\% \times 4500 = 2700
\]

Thus, the total value in her margin account at the start is $2700.

#### **When the price drops to $40**:
Now, the stock price is $40, so her short position has changed in value. The new value of the short position:
\[
\text{New Value of Short Position} = 100 \times 40 = 4000
\]

Since Hannah sold at $45 and the price dropped to $40, she made a profit of:
\[
\text{Profit} = 4500 - 4000 = 500
\]

Her equity (the amount she now has in her margin account) is:
\[
\text{Equity} = \text{Initial Margin} + \text{Profit} = 2700 + 500 = 3200
\]

Now, the **return on equity** (ROE) is the profit relative to her initial margin:
\[
\text{Return on Equity} = \frac{\text{Profit}}{\text{Initial Margin}} = \frac{500}{2700} = 0.1852 \text{ or } 18.52\%
\]

---

### Part (b) - ABC stock rose to $50. What is her new margin, and what is the maximum price before a margin call?

#### **When the price rises to $50**:
Now, the new value of the short position is:
\[
\text{New Value of Short Position} = 100 \times 50 = 5000
\]

Hannah is losing money now because the stock price has increased. Her loss is:
\[
\text{Loss} = 5000 - 4500 = 500
\]

Her new equity is:
\[
\text{Equity} = \text{Initial Margin} - \text{Loss} = 2700 - 500 = 2200
\]

To calculate her **new margin** (which is equity as a percentage of the short position's value):
\[
\text{New Margin} = \frac{\text{Equity}}{\text{Value of Short Position}} = \frac{2200}{5000} = 0.44 \text{ or } 44\%
\]

So, her new margin is **44%**.

#### **Maximum price before a margin call**:
A margin call occurs when her equity falls below the maintenance margin of 30%.

Let $ P $ be the price at which Hannah will receive a margin call. At that point, her equity must be 30% of the value of the short position. The equity is also $ 2700 - (P \times 100 - 4500) $, which represents her initial margin minus her losses.

At the margin call point:
\[
\frac{2700 - (P \times 100 - 4500)}{P \times 100} = 30\%
\]

Solving for $ P $:

\[
2700 - (P \times 100 - 4500) = 0.3 \times P \times 100
\]
\[
2700 - 100P + 4500 = 30P
\]
\[
7200 = 130P
\]
\[
P = \frac{7200}{130} = 55.38
\]

So, the maximum price of ABC stock before Hannah receives a margin call is **$55.38**.

---

Do you want more details or have any questions? 

Here are 5 follow-up questions related to this problem:
1. What would be the return on equity if the stock price dropped to $35 instead?
2. How much additional capital would Hannah need to deposit if the stock price reached $56?
3. How would the initial margin requirement change if it were set at 50% instead of 60%?
4. How does the maintenance margin affect the risk of margin calls for short sellers?
5. What strategies could Hannah use to mitigate the risk of a margin call if the stock price rises?

**Tip**: When short selling, always monitor stock price movements carefully, as a rising price can quickly lead to losses and margin calls.

Hannah sold short 100 shares of ABC stocks at its current price of $45. Assuming the initial margin is 60% and the maintenance margin is 30%. a. A month later, ABC stock dropped to $40. What is her return on equity for this investment? b. Assuming that ABC instead rose to $50. Calculate her new margin. What is the maximum price of ABC before she will receive a margin call?

Understand how to calculate return on equity and determine the margin call threshold in a short selling scenario. Explore key concepts like initial margin, maintenance margin, and profit or loss calculation. Learn step-by-step how to solve this finance problem, with detailed examples. Suitable for high school students in Grades 10-12.

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