Let’s break down each part of this problem and solve them step by step.

Part a: Rate of return if the stock price increases by 10%

You are investing with leverage:

• Your own investment: $5,000
• Amount borrowed: $5,000 at 8% interest rate per year
• Total investment in the stock: $10,000
• Initial stock price: $50 per share
• Number of shares purchased: $\frac{10,000}{50} = 200$ shares.

Now, if the stock price goes up by 10% in a year, the new price of the stock is: $\text{New price} = 50 \times (1 + 0.10) = 50 \times 1.10 = 55$

The total value of your stock after the price increase: $\text{New value of stock} = 200 \times 55 = 11,000$

Next, calculate the cost of borrowing. The interest on the $5,000 loan at 8% per year: $\text{Interest} = 5,000 \times 0.08 = 400$

Now, calculate your profit: $\text{Profit} = \text{New value of stock} - \text{Loan repayment} - \text{Original investment} = 11,000 - 5,400 - 5,000 = 600$

Finally, calculate your rate of return: $\text{Rate of return} = \frac{\text{Profit}}{\text{Initial investment}} = \frac{600}{5,000} = 0.12 = 12\%$

Part b: Price at which margin call occurs

A margin call happens when the value of your equity falls below the maintenance margin, which is 30%. Your equity is the difference between the value of the stock and the amount borrowed.

Let $P$ be the stock price at which the margin call occurs. The value of the stock at this price is: $\text{Value of stock} = 200 \times P$

The equity is: $\text{Equity} = 200 \times P - 5,000$

The maintenance margin requirement is 30%, so we require that: $\frac{\text{Equity}}{\text{Value of stock}} \geq 0.30$

Substituting the expressions for equity and stock value: $\frac{200P - 5,000}{200P} \geq 0.30$

Simplifying: $1 - \frac{5,000}{200P} \geq 0.30$

$\frac{5,000}{200P} \leq 0.70$

$P \geq \frac{5,000}{200 \times 0.70} = \frac{5,000}{140} = 35.71$

So, the stock price must not fall below $35.71 for you to avoid a margin call.

Summary:

• a) Your rate of return if the stock price increases by 10% is 12%.
• b) The stock price must not fall below $35.71 to avoid a margin call.

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Would you like further details on any part of the solution?
Here are some related questions to consider:

1. How would dividends affect your rate of return in part a?
2. What if you borrowed at a different interest rate—how would that impact your return?
3. How does increasing or decreasing the maintenance margin affect the price at which a margin call occurs?
4. How would your rate of return change if the stock price decreases by 10% instead of increasing?
5. What are the risks involved in using leverage when investing?

Tip: Always be cautious when using leverage for investments—while it can magnify your gains, it can also amplify your losses.