To find the market value of equity after the repurchase, we can follow these steps:

1. Initial Market Value of Equity: The firm is an all-equity firm, meaning its value before the repurchase is based purely on the stock. Since the company plans to repurchase 40,000 shares by borrowing $2.2 million, the price per share can be calculated.

   $\text{Price per share} = \frac{\text{Amount borrowed}}{\text{Number of shares repurchased}} = \frac{2.2 \text{ million}}{40,000} = 55 \text{ dollars per share}.$

2. Total Market Value of the Firm before Repurchase: The total market value of the firm's equity can be calculated by multiplying the price per share by the total number of shares outstanding before the repurchase:

   $\text{Market Value of Equity before Repurchase} = 55 \times 500,000 = 27.5 \text{ million dollars}.$

3. Market Value of Equity after the Repurchase: After the repurchase, the company will owe $2.2 million in debt. Therefore, the market value of equity will be the total value of the firm minus the new debt:

   $\text{Market Value of Equity after Repurchase} = 27.5 \text{ million} - 2.2 \text{ million} = 25.3 \text{ million}.$

Thus, the market value of equity after the repurchase is $25.3 million.

Correct Answer:

$25.3 million

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Do you want further clarification or details on any step? Here are five related questions to extend the understanding:

1. How does the stock repurchase affect the firm's debt-to-equity ratio?
2. What would be the impact of taxes on the market value of equity?
3. How would a change in interest rates affect the market value after the repurchase?
4. What happens to the firm's share price after the repurchase?
5. How is shareholder wealth affected by a share repurchase?

Tip: Share repurchases can increase earnings per share (EPS) by reducing the number of shares outstanding, but they also increase the firm's financial leverage.