Let's break down each part of this problem step by step.

a. What is the total market value of the firm without leverage?

To find the total market value of the firm without leverage, we use the information that investors are willing to invest $3.8 million in exchange for 50% of the equity in the firm.

If 50% of the firm is worth $3.8 million, then the total value of the firm (100%) is:

$\text{Total market value} = \frac{3.8 \, \text{million}}{0.5} = 7.6 \, \text{million}.$

So, the total market value of the firm without leverage is $7.6 million.

b. Suppose you borrow $0.4 million. According to MM, what fraction of the firm’s equity will you need to sell to raise the additional $3.4 million?

Now, you need a total of $3.4 million in equity financing after borrowing $0.4 million. Since the total market value of the firm remains $7.6 million, we calculate the fraction of the firm’s equity you need to sell.

The fraction of the firm's equity to be sold is:

$\frac{3.4 \, \text{million}}{7.6 \, \text{million}} = 0.4474 \approx 44.74\%.$

Rounding to the nearest whole percentage, the fraction of equity you need to sell is 45%.

c. What is the value of your share of the firm's equity in cases (a) and (b)?

Case (a):

In this case, you are not using any leverage and the total value of the firm is $7.6 million. Since you sold 50% of the firm’s equity to raise the initial capital, your share is the remaining 50%. Therefore, your share of the equity is:

$\text{Your share in case (a)} = 0.5 \times 7.6 \, \text{million} = 3.8 \, \text{million}.$

So, the value of your share of the firm’s equity in case (a) is $3.8 million.

Case (b):

In this case, you have sold 45% of the firm’s equity to raise additional financing, so you will own the remaining 55% of the firm. The total value of the firm is still $7.6 million, so your share is:

$\text{Your share in case (b)} = 0.55 \times 7.6 \, \text{million} = 4.18 \, \text{million}.$

So, the value of your share of the firm’s equity in case (b) is $4.18 million.

Final Summary:

• (a) The total market value of the firm without leverage is $7.6 million.
• (b) You will need to sell 45% of the firm’s equity to raise the additional $3.4 million.
• (c)
  • Your share of the firm’s equity in case (a) is $3.8 million.
  • Your share of the firm’s equity in case (b) is $4.18 million.

Would you like more details or clarification?

Here are 5 related questions:

1. How does leverage impact the overall value of a firm according to MM theory?
2. What risks are associated with borrowing versus selling equity?
3. How would the required equity sale change if the firm's market value increased?
4. If the cost of debt increased, how would this affect the equity sale requirement?
5. What happens to your ownership share if the firm’s valuation changes after raising funds?

Tip: Leverage can increase the value of your equity when the firm’s return on assets exceeds the interest rate on debt. However, excessive debt can lead to financial distress.