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: $\text{Return on Equity} = \frac{\text{Profit}}{\text{Initial Margin}} = \frac{500}{2700} = 0.1852 \text{ or } 18.52\%$

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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.

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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.