Financial Leverage and Capital Structure Problems

Theoretical Foundation: This problem applies **M&M Proposition I with taxes**. This proposition states that the value of a levered firm is equal to the value of an unlevered firm plus the present value of the interest tax shield. The present value of the tax shield is calculated as the tax rate multiplied by the amount of debt (\(T_C \times D\)).

First, we must find the value of the unlevered firm (\(V_U\)). The value is the perpetual after-tax EBIT discounted at the unlevered cost of capital (\(R_U\)).

\[ V_U = \frac{\text{EBIT} \times (1 - T_C)}{R_U} = \frac{\$57,000 \times (1 - 0.21)}{0.103} = \frac{\$45,030}{0.103} = \$437,184.47 \]

Now, we can find the value of the levered firm (\(V_L\)) using M&M Proposition I with taxes.

\[ V_L = V_U + T_C \times D = \$437,184.47 + 0.21 \times \$134,000 = \$437,184.47 + \$28,140 = \$465,324.47 \]

Analysis: According to M&M Proposition I with taxes, the value of the firm increases with every dollar of debt because of the tax shield. The theory, in its simplest form, suggests that the optimal capital structure is 100% debt. Therefore, to maximize the value of the firm, the company should increase its debt-equity ratio. However, this model does not account for the costs of financial distress (bankruptcy costs), which would eventually offset the benefits of the tax shield.

Theoretical Foundation: This problem applies **M&M Proposition I with taxes** to different scenarios of leverage. It highlights how the firm's value increases as it adds debt due to the tax shield, but it also demonstrates the difference between adding debt as a percentage of unlevered value versus levered value.

Given: \(EBIT = \$37,700\), \(R_U = 11\%\), \(T_C = 22\%\), \(R_D = 6\%\).

a. Current value of the company (\(V_U\)):

Since the firm is all-equity, its value is the present value of its after-tax perpetual cash flows, discounted at the unlevered cost of capital (\(R_U\)).

\[ V_U = \frac{\text{EBIT} \times (1 - T_C)}{R_U} = \frac{\$37,700 \times (1 - 0.22)}{0.11} = \frac{\$29,406}{0.11} = \$267,327.27 \]

b. Debt as a percentage of unlevered value:

Use the formula: \( V_L = V_U + T_C \times D \).

• 50% unlevered value: \( D = 0.50 \times V_U = 0.50 \times \$267,327.27 = \$133,663.64 \). \[ V_L = \$267,327.27 + 0.22 \times \$133,663.64 = \$267,327.27 + \$29,406 = \$296,733.27 \]
• 100% unlevered value: \( D = 1.0 \times V_U = \$267,327.27 \). \[ V_L = \$267,327.27 + 0.22 \times \$267,327.27 = \$267,327.27 + \$58,812 = \$326,139.27 \]

c. Debt as a percentage of levered value:

The amount of debt \(D\) is a function of the levered value \(V_L\). We set up the equation and solve for \(V_L\).

\[ V_L = V_U + T_C \times D \] \[ V_L = V_U + T_C \times (0.50 \times V_L) \] \[ V_L (1 - 0.50 \times T_C) = V_U \] \[ V_L = \frac{V_U}{1 - 0.50 \times T_C} \]

• 50% levered value: \[ V_L = \frac{\$267,327.27}{1 - 0.50 \times 0.22} = \frac{\$267,327.27}{1 - 0.11} = \frac{\$267,327.27}{0.89} = \$300,367.72 \]
• 100% levered value: This is not possible. If \(D=V_L\), then \( E=0 \) and the firm has no equity. The value of the firm would be \( V_L = V_U + T_C \times V_L \), which implies \( V_L (1-T_C) = V_U \). This would only be possible if the firm's value were finite, but it would have no equity, which violates the premise of a levered firm.

Theoretical Foundation: This problem is a direct application of the **M&M irrelevance proposition with homemade leverage**. It shows that in a world without taxes, investors can use personal borrowing to replicate the returns of a levered firm, and this arbitrage opportunity will force the firms to have the same total value.

a. Comparing returns:

Strategy 1: Invest in Knight (levered firm)

• Cost of investment: \( 0.05 \times \$1,500,000 = \$75,000 \)
• Knight's Net Income: \( \text{EBIT} - \text{Interest} = \$325,000 - \$48,000 = \$277,000 \)
• Return on investment: \( 0.05 \times \$277,000 = \$13,850 \)
• Rate of return: \( \frac{\$13,850}{\$75,000} = 18.47\% \)

Strategy 2: Invest in Day (unlevered firm) with homemade leverage

To replicate the investment in Knight, the investor must borrow a proportional amount of debt. The debt-to-equity ratio for Knight is \( \frac{\$800,000}{\$1,500,000} = 0.5333 \). The investor needs to borrow to achieve this same leverage ratio. The cost of the homemade strategy must equal the cost of the first strategy, which is $75,000. So the investor will have $75,000 in equity and some amount of debt. The debt-equity ratio of Knight is \( \frac{800,000}{1,500,000} = 0.5333 \). So for every dollar of equity, the firm has $0.5333 in debt. The investor's equity stake is $75,000. To replicate Knight, the investor needs to borrow \( \$75,000 \times 0.5333 = \$40,000 \). The total investment in Day's stock is then \( \$75,000 + \$40,000 = \$115,000 \). The cost of the homemade strategy is the same as the cost of the first strategy, $75,000, so we have a contradiction here. Let's reconsider. The investor wants to get a return on a $75,000 equity investment. So let's re-evaluate the costs. The cost of investing in Knight is the equity investment of $75,000. The cost of the homemade leverage strategy is the equity investment in Day plus the debt borrowed. The investor has to have an initial net cost that is the same. So the net cost of the investment must be $75,000. The investor purchases Day's stock and borrows. The interest rate on the borrowing is 6%. Let \(x\) be the value of Day stock purchased. The borrowed amount is \( x - \$75,000 \). The cash flows must be equal. The earnings from Knight's stock are \( \$13,850 \). The earnings from Day's stock are \( x \times \frac{\text{Day's Net Income}}{\text{Day's Equity}} - \text{Interest on loan} \). Day's Net Income = Day's EBIT = $325,000. Day's Equity = $2,100,000. So Day's return on equity = \( \frac{\$325,000}{\$2,100,000} = 15.48\% \). Let \(E_P\) be the amount of equity to be invested in Day. Let \(D_P\) be the amount of debt the investor borrows. We need \( E_P - D_P = \$75,000 \). The earnings on the homemade strategy will be \( E_P \times 0.1548 - D_P \times 0.06 \). This looks too complicated for a basic problem. Let's use the irrelevance proposition directly. The market value of Day is its equity value, which is \( \$2,100,000 \). The market value of Knight is \( \$1,500,000 + \$800,000 = \$2,300,000 \). Since the firms are identical, their total market values should be the same. But they are not. This means there is an arbitrage opportunity. The overvalued firm is Knight. The undervalued firm is Day. Investors will want to sell Knight and buy Day, using homemade leverage to replicate Knight's returns. So, an investor should sell 5% of Knight's stock, receiving \( 0.05 \times \$1,500,000 = \$75,000 \). The investor then buys a proportional amount of Day's stock and borrows to create the leverage. The D/E ratio for Knight is \( \frac{\$800,000}{\$1,500,000} = 0.5333 \). The investor wants to replicate this leverage ratio on their own. The investor should purchase Day's stock with an amount equal to their equity investment in Knight plus the amount they would need to borrow to replicate Knight's leverage. The investor's equity investment is $75,000. To replicate Knight, they should borrow \( 0.5333 \times \$75,000 = \$40,000 \). The total amount to invest in Day's stock is \( \$75,000 + \$40,000 = \$115,000 \). The initial cost of this strategy is the same: investor's own money, $75,000. The return on this homemade strategy:

• Earnings on Day's stock: \( \$115,000 \times \frac{\$325,000}{\$2,100,000} = \$17,812.50 \)
• Less interest on loan: \( \$40,000 \times 0.06 = \$2,400 \)
• Total return: \( \$17,812.50 - \$2,400 = \$15,412.50 \)

This is higher than the $13,850 return from investing in Knight directly. The investor should choose the homemade leverage strategy.

b. Investor choice and when the process ceases:

Investors will sell shares of the overvalued firm (Knight) and use homemade leverage to buy shares of the undervalued firm (Day). This will drive down the price of Knight's stock and drive up the price of Day's stock. The process will continue until the market values of the two identical firms are equal, thus eliminating the arbitrage opportunity. This demonstrates the market-based argument for M&M's irrelevance theorem.

Theoretical Foundation: This problem applies the **M&M Proposition I with taxes** to determine the new firm value and share price, and then calculates EPS and ROE under the new capital structure. The key is that the firm's value is the unlevered value plus the tax shield. The price per share is the new equity value divided by the new number of shares.

Given: \(V_U = \$222,000\), \(D = \$60,000\), \(T_C = 21\%\).
Original shares outstanding = 7,400.
Original stock price = \( \frac{\$222,000}{7,400} = \$30 \).

New firm value (\(V_L\)):

\[ V_L = V_U + T_C \times D = \$222,000 + 0.21 \times \$60,000 = \$222,000 + \$12,600 = \$234,600 \]

New equity value (E):

\[ E = V_L - D = \$234,600 - \$60,000 = \$174,600 \]

New number of shares:

The firm repurchases shares at the old price. Number of shares repurchased = \( \frac{\$60,000}{\$30} = 2,000 \). New shares outstanding = \( 7,400 - 2,000 = 5,400 \).

New stock price (\(P\)):

\[ P = \frac{\text{New Equity Value}}{\text{New Shares Outstanding}} = \frac{\$174,600}{5,400} = \$32.33 \]

a. EPS and ROE (relevered firm):

Interest expense = \( \$60,000 \times 0.07 = \$4,200 \).

Net Income = \( (\text{EBIT} - \text{Interest}) \times (1 - T_C) \).

b. Percentage change in EPS:

• Recession: \( \frac{\$1.23 - \$2.00}{\$2.00} = -38.5\% \)
• Expansion: \( \frac{\$2.68 - \$2.00}{\$2.00} = +34.0\% \)

Theoretical Foundation: This problem requires calculating ROE with M&M Propositions with taxes. We need to use the net income from Problem 19 and the new equity value, which is based on the levered firm value minus debt.

Given: Net Income from Problem 19. New equity value = \( \$174,600 \).

ROE = Net Income / New Equity Value

• Recession ROE = \( \frac{\$6,636}{\$174,600} = 3.80\% \)
• Normal ROE = \( \frac{\$10,822}{\$174,600} = 6.20\% \)
• Expansion ROE = \( \frac{\$14,467}{\$174,600} = 8.29\% \)

Percentage change in ROE:

• Recession: \( \frac{3.80\% - 6.20\%}{6.20\%} \approx -38.71\% \)
• Expansion: \( \frac{8.29\% - 6.20\%}{6.20\%} \approx +33.71\% \)

Analysis: The results are slightly different from those in Problem 3 because the total equity value is now different due to the tax shield. The core observation that leverage magnifies the percentage changes in ROE remains the same, but the absolute values are affected by the more accurate valuation of the firm's equity.

Theoretical Foundation: This is a derivation of the WACC formula with corporate taxes, based on M&M Proposition II with taxes. It shows that the WACC declines as the firm takes on more debt because of the tax shield.

We start with the standard WACC formula with taxes:

\[ WACC = (\frac{E}{V})R_E + (\frac{D}{V})R_D(1-T_C) \]

And we use M&M Proposition II with taxes, \(R_E = R_U + (R_U - R_D)(D/E)(1-T_C)\).
Substitute \(R_E\) into the WACC formula:

\[ WACC = (\frac{E}{V})[R_U + (R_U - R_D)(D/E)(1-T_C)] + (\frac{D}{V})R_D(1-T_C) \]

Now, distribute the \((E/V)\) term:

\[ WACC = (\frac{E}{V})R_U + (\frac{E}{V})(R_U - R_D)(\frac{D}{E})(1-T_C) + (\frac{D}{V})R_D(1-T_C) \]

The \(E\) in the second term cancels out:

\[ WACC = (\frac{E}{V})R_U + (\frac{D}{V})(R_U - R_D)(1-T_C) + (\frac{D}{V})R_D(1-T_C) \]

Factor out the \((D/V)(1-T_C)\) term from the second and third terms:

\[ WACC = (\frac{E}{V})R_U + (\frac{D}{V})(1-T_C)[(R_U - R_D) + R_D] \] \[ WACC = (\frac{E}{V})R_U + (\frac{D}{V})(1-T_C)[R_U] \]

Now, factor out the \(R_U\) term:

\[ WACC = R_U[(\frac{E}{V}) + (\frac{D}{V})(1-T_C)] \] \[ WACC = R_U[(\frac{E}{V}) + (\frac{D}{V}) - T_C(\frac{D}{V})] \]

Since \(E/V + D/V = 1\), the formula simplifies to:

\[ WACC = R_U[1 - T_C(\frac{D}{V})] \]

Conclusion: This derivation shows that the WACC is directly related to the unlevered cost of capital and the tax shield. The second term, \( T_C(D/V) \), represents the value of the tax shield per dollar of capital, which reduces the overall cost of capital as debt increases.

Theoretical Foundation: This is a derivation of **M&M Proposition II with corporate taxes**. It shows that the cost of equity rises as the firm takes on more debt because of the increased financial risk to shareholders, but the increase is offset by the tax shield.

We start with the WACC formula with corporate taxes and rearrange it to solve for \(R_E\).

\[ WACC = (\frac{E}{V})R_E + (\frac{D}{V})R_D(1-T_C) \]

Since \(WACC = R_U[1 - T_C(D/V)]\) (from Problem 21), we can set the two expressions for WACC equal to each other:

\[ (\frac{E}{V})R_E + (\frac{D}{V})R_D(1-T_C) = R_U[1 - T_C(\frac{D}{V})] \]

Now, isolate the term with \(R_E\):

\[ (\frac{E}{V})R_E = R_U[1 - T_C(\frac{D}{V})] - (\frac{D}{V})R_D(1-T_C) \] \[ (\frac{E}{V})R_E = R_U - R_U T_C(\frac{D}{V}) - R_D(\frac{D}{V}) + R_D T_C(\frac{D}{V}) \]

Rearrange the terms and factor out \((D/V)\) and \(T_C\):

\[ (\frac{E}{V})R_E = R_U - (\frac{D}{V})(R_U T_C + R_D - R_D T_C) \]

This looks a bit messy. Let's go back and try another approach. Let's start with the total value of the levered firm from M&M Proposition I with taxes.

\[ V_L = E + D = V_U + T_C \times D \]

The unlevered firm's value is the unlevered cash flow discounted at the unlevered cost of capital. So, \(V_U = \frac{\text{EBIT}(1-T_C)}{R_U}\). The cash flow to equity holders of the levered firm is \((\text{EBIT} - \text{Interest})(1-T_C)\). The value of equity is this cash flow discounted at the cost of equity \(R_E\).

\[ E = \frac{(\text{EBIT} - R_D \times D)(1-T_C)}{R_E} \]

From M&M Proposition I, \( E = V_U + T_C \times D - D \). \[ \frac{(\text{EBIT} - R_D \times D)(1-T_C)}{R_E} = V_U + T_C \times D - D \]

Substitute \(V_U\) back in:

\[ \frac{\text{EBIT}(1-T_C) - R_D \times D(1-T_C)}{R_E} = \frac{\text{EBIT}(1-T_C)}{R_U} + T_C \times D - D \]

This is getting complicated. Let's go back to the first method and be more careful.

\[ (\frac{E}{V})R_E = R_U[1 - T_C(\frac{D}{V})] - (\frac{D}{V})R_D(1-T_C) \]

Divide by \((E/V)\):

\[ R_E = \frac{R_U[1 - T_C(\frac{D}{V})] - (\frac{D}{V})R_D(1-T_C)}{(\frac{E}{V})} \] \[ R_E = \frac{V}{E}[R_U - R_U T_C\frac{D}{V} - R_D\frac{D}{V} + R_D T_C\frac{D}{V}] \] \[ R_E = \frac{V}{E}[R_U - \frac{D}{V}(R_U T_C + R_D - R_D T_C)] \]

Since \(V = E+D\), then \( V/E = 1 + D/E \). \[ R_E = (1+\frac{D}{E})[R_U - \frac{D}{E+D}(R_U T_C + R_D - R_D T_C)] \]

This is still not straightforward. Let's use a simpler, more direct method. We know that the value of the firm's assets is the sum of the value of debt and equity. The return on assets (\(R_A\)) is the WACC. We have two cash flows to the firm's investors: interest payments and dividends. The total value of the firm is \(V_L = V_U + T_C D\). We also know that \(E = V_L - D\). The cash flow to equity is \((\text{EBIT} - R_D D)(1 - T_C)\). The value of equity is the present value of these cash flows. \[ E = \frac{(\text{EBIT} - R_D D)(1 - T_C)}{R_E} \]

We can solve for \(R_E\):

\[ R_E = \frac{\text{EBIT}(1 - T_C) - R_D D (1-T_C)}{E} \]

From M&M Proposition I, we know \( \text{EBIT}(1-T_C) = V_U R_U \). Substitute this in:

\[ R_E = \frac{V_U R_U - R_D D(1-T_C)}{E} = \frac{V_U R_U}{E} - \frac{R_D D(1-T_C)}{E} \] \[ R_E = \frac{V_U}{E}R_U - \frac{D}{E}R_D(1-T_C) \]

From M&M Proposition I, \(V_U = V_L - T_C D = E + D - T_C D\). Substitute this in:

\[ R_E = \frac{E + D - T_C D}{E} R_U - \frac{D}{E}R_D(1-T_C) \] \[ R_E = (1 + \frac{D}{E} - T_C \frac{D}{E})R_U - \frac{D}{E}R_D(1-T_C) \] \[ R_E = R_U + \frac{D}{E}R_U - T_C \frac{D}{E}R_U - \frac{D}{E}R_D(1-T_C) \] \[ R_E = R_U + \frac{D}{E}(R_U - R_U T_C - R_D + R_D T_C) \] \[ R_E = R_U + \frac{D}{E}[R_U(1-T_C) - R_D(1-T_C)] \] \[ R_E = R_U + (R_U - R_D)(1-T_C)(\frac{D}{E}) \]

This is the correct formula. The final step is to rearrange the terms to match the form in the book, which they have done incorrectly. The correct formula is what I derived above.

Theoretical Foundation: This problem connects the **CAPM** with **M&M Proposition II (no taxes)**. It shows that the beta of a firm's equity is a function of its asset beta and its financial leverage, which quantifies the relationship between business risk and financial risk.

We begin with M&M Proposition II (no taxes):

\[ R_E = R_A + (R_A - R_D)(\frac{D}{E}) \]

Now, let's use the Capital Asset Pricing Model (CAPM). The CAPM states that the expected return on a security is equal to the risk-free rate plus a risk premium, which is the security's beta times the market risk premium. So, for a firm's equity, assets, and debt, we have:

• Equity: \( R_E = R_f + \beta_E \times (R_M - R_f) \)
• Assets: \( R_A = R_f + \beta_A \times (R_M - R_f) \)
• Debt: \( R_D = R_f + \beta_D \times (R_M - R_f) \)

The problem states that the debt is risk-free, so \(R_D = R_f\). This implies that the beta of debt (\(\beta_D\)) is 0.

Now, substitute the CAPM expressions into the M&M Proposition II equation:

\[ R_f + \beta_E \times (R_M - R_f) = [R_f + \beta_A \times (R_M - R_f)] + [R_f + \beta_A \times (R_M - R_f) - R_f](\frac{D}{E}) \] \[ R_f + \beta_E \times (R_M - R_f) = R_f + \beta_A \times (R_M - R_f) + \beta_A \times (R_M - R_f) (\frac{D}{E}) \]

Subtract \(R_f\) from both sides and factor out \((R_M - R_f)\) from the right-hand side:

\[ \beta_E \times (R_M - R_f) = (R_M - R_f)[\beta_A + \beta_A (\frac{D}{E})] \]

Divide both sides by \((R_M - R_f)\):

\[ \beta_E = \beta_A + \beta_A (\frac{D}{E}) \] \[ \beta_E = \beta_A (1 + \frac{D}{E}) \]

Conclusion: This derivation shows that the equity beta (\(\beta_E\)) is directly related to the firm's asset beta (\(\beta_A\)) and its leverage. The term \( \beta_A \) represents the business risk, and the term \( \beta_A(\frac{D}{E}) \) represents the financial risk. As financial leverage increases, the equity beta increases, which makes the equity more risky for investors.

Theoretical Foundation: This problem uses the formula derived in Problem 23 to demonstrate the **magnifying effect of financial leverage on equity risk**. It shows that even a firm with average business risk can have a very high equity beta (and thus high required return) if it is highly levered.

We are given that the firm's asset beta is \( \beta_A = 1.0 \). We use the formula from Problem 23: \( \beta_E = \beta_A (1 + D/E) \).

• \(D/E = 0\): \( \beta_E = 1.0 \times (1 + 0) = 1.0 \)
• \(D/E = 1\): \( \beta_E = 1.0 \times (1 + 1) = 2.0 \)
• \(D/E = 5\): \( \beta_E = 1.0 \times (1 + 5) = 6.0 \)
• \(D/E = 20\): \( \beta_E = 1.0 \times (1 + 20) = 21.0 \)

Conclusion: This illustrates that as a firm increases its financial leverage (its debt-to-equity ratio), the systematic risk of its equity increases dramatically. For a firm with an asset beta of 1.0, meaning its business risk is the same as the overall market, a high D/E ratio can lead to an extremely high equity beta, making the stock far more volatile than the market. According to the CAPM, a higher beta leads to a higher required return on equity. This means that as a firm adds debt, shareholders demand a higher return to compensate for the increased financial risk they are bearing. This is a direct consequence of M&M Proposition II, which states that the cost of equity rises with leverage.