Financial Leverage and Capital Structure Problems

Theoretical Foundation: This problem demonstrates the impact of financial leverage on earnings per share (EPS). In a world with no taxes, M&M Proposition I suggests that the overall value of the firm's assets remains unchanged. However, financial leverage magnifies the returns to shareholders, increasing EPS in good times and decreasing it in bad times.

a. All-Equity Plan (No Debt):

• Shares outstanding = 7,400
• Recession EBIT = $18,000 \times (1 - 0.30) = $12,600
• Normal EBIT = $18,000
• Expansion EBIT = $18,000 \times (1 + 0.25) = $22,500

Since there are no taxes or interest, Net Income = EBIT.

EPS = Net Income / Shares Outstanding

• Recession EPS = \( \frac{\$12,600}{7,400} = \$1.70 \)
• Normal EPS = \( \frac{\$18,000}{7,400} = \$2.43 \)
• Expansion EPS = \( \frac{\$22,500}{7,400} = \$3.04 \)

Percentage change in EPS:

• Recession: \( \frac{\$1.70 - \$2.43}{\$2.43} \approx -30.0\% \)
• Expansion: \( \frac{\$3.04 - \$2.43}{\$2.43} \approx +25.0\% \)

b. Levered Plan (with Debt):

First, calculate the new number of shares and interest expense.

• Shares repurchased = \( \frac{\$60,000}{\$222,000 / 7,400} = \frac{\$60,000}{\$30} = 2,000 \text{ shares} \)
• New shares outstanding = \( 7,400 - 2,000 = 5,400 \text{ shares} \)
• Interest expense = \( \$60,000 \times 0.07 = \$4,200 \)

EPS = (EBIT - Interest) / Shares Outstanding

• Recession EPS = \( \frac{\$12,600 - \$4,200}{5,400} = \$1.56 \)
• Normal EPS = \( \frac{\$18,000 - \$4,200}{5,400} = \$2.56 \)
• Expansion EPS = \( \frac{\$22,500 - \$4,200}{5,400} = \$3.39 \)

Percentage change in EPS:

• Recession: \( \frac{\$1.56 - \$2.56}{\$2.56} \approx -39.1\% \)
• Expansion: \( \frac{\$3.39 - \$2.56}{\$2.56} \approx +32.4\% \)

Observation: We observe that the percentage change in EPS is more dramatic with financial leverage. The 30% drop in EBIT led to a 39.1% drop in EPS, while the 25% increase in EBIT led to a 32.4% increase in EPS. Financial leverage magnifies both gains and losses for shareholders.

Theoretical Foundation: This problem introduces the **interest tax shield**. Interest payments are tax-deductible, which lowers the firm's tax expense and increases the net income available to investors. This is a key component of M&M Proposition I with corporate taxes.

a. All-Equity Plan (No Debt):

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

EPS = Net Income / Shares Outstanding

• Recession Net Income = \( \$12,600 \times (1 - 0.21) = \$9,954 \); EPS = \( \frac{\$9,954}{7,400} = \$1.35 \)
• Normal Net Income = \( \$18,000 \times (1 - 0.21) = \$14,220 \); EPS = \( \frac{\$14,220}{7,400} = \$1.92 \)
• Expansion Net Income = \( \$22,500 \times (1 - 0.21) = \$17,775 \); EPS = \( \frac{\$17,775}{7,400} = \$2.40 \)

Percentage change in EPS:

• Recession: \( \frac{\$1.35 - \$1.92}{\$1.92} \approx -29.7\% \)
• Expansion: \( \frac{\$2.40 - \$1.92}{\$1.92} \approx +25.0\% \)

b. Levered Plan (with Debt):

Shares outstanding = 5,400; Interest expense = $4,200 (from Problem 1)

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

EPS = Net Income / Shares Outstanding

• Recession NI = \( (\$12,600 - \$4,200) \times (1 - 0.21) = \$6,636 \); EPS = \( \frac{\$6,636}{5,400} = \$1.23 \)
• Normal NI = \( (\$18,000 - \$4,200) \times (1 - 0.21) = \$10,822 \); EPS = \( \frac{\$10,822}{5,400} = \$2.00 \)
• Expansion NI = \( (\$22,500 - \$4,200) \times (1 - 0.21) = \$14,467 \); EPS = \( \frac{\$14,467}{5,400} = \$2.68 \)

Percentage change in EPS:

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

Observation: The introduction of taxes does not change the magnifying effect of leverage. The percentage change in EPS is still more pronounced for the levered firm, although the absolute values are lower due to tax payments.

Theoretical Foundation: ROE is a measure of a firm's profitability relative to its equity. This problem shows how financial leverage, with and without taxes, magnifies the return to equity holders. The value of equity changes with leverage, which is a key part of the calculation.

a. No Debt, No Taxes:

Total Equity = $222,000. ROE = Net Income / Total Equity

• Recession ROE = \( \frac{\$12,600}{\$222,000} = 5.68\% \)
• Normal ROE = \( \frac{\$18,000}{\$222,000} = 8.11\% \)
• Expansion ROE = \( \frac{\$22,500}{\$222,000} = 10.14\% \)

Percentage change in ROE:

• Recession: \( \frac{5.68\% - 8.11\%}{8.11\%} \approx -29.96\% \)
• Expansion: \( \frac{10.14\% - 8.11\%}{8.11\%} \approx +25.03\% \)

b. With Debt, No Taxes:

Total Equity = $222,000 - $60,000 = $162,000. ROE = Net Income / Total Equity

• Recession ROE = \( \frac{\$8,400}{\$162,000} = 5.19\% \)
• Normal ROE = \( \frac{\$13,800}{\$162,000} = 8.52\% \)
• Expansion ROE = \( \frac{\$18,300}{\$162,000} = 11.30\% \)

Percentage change in ROE:

• Recession: \( \frac{5.19\% - 8.52\%}{8.52\%} \approx -39.11\% \)
• Expansion: \( \frac{11.30\% - 8.52\%}{8.52\%} \approx +32.63\% \)

c. With and Without Debt, With Taxes:

Using Net Income values from Problem 2:

No Debt: Total Equity = $222,000

• Recession ROE = \( \frac{\$9,954}{\$222,000} = 4.48\% \)
• Normal ROE = \( \frac{\$14,220}{\$222,000} = 6.41\% \)
• Expansion ROE = \( \frac{\$17,775}{\$222,000} = 8.01\% \)

Percentage change in ROE:

• Recession: \( \frac{4.48\% - 6.41\%}{6.41\%} \approx -29.95\% \)
• Expansion: \( \frac{8.01\% - 6.41\%}{6.41\%} \approx +25.00\% \)

With Debt: Total Equity = $162,000

• Recession ROE = \( \frac{\$6,636}{\$162,000} = 4.10\% \)
• Normal ROE = \( \frac{\$10,822}{162,000} = 6.68\% \)
• Expansion ROE = \( \frac{\$14,467}{\$162,000} = 8.93\% \)

Percentage change in ROE:

• Recession: \( \frac{4.10\% - 6.68\%}{6.68\%} \approx -38.62\% \)
• Expansion: \( \frac{8.93\% - 6.68\%}{6.68\%} \approx +33.68\% \)

Observation: The results confirm that leverage magnifies both positive and negative percentage changes in ROE. The presence of taxes does not eliminate this effect, although it does reduce the absolute values of ROE.

Theoretical Foundation: This problem uses break-even analysis to find the level of EBIT at which a firm is indifferent between two capital structures. Above this point, leverage is beneficial; below it, it is not. This is a practical application of the concepts introduced in Section 16.2.

Interest Expense (Plan II): \( \$716,000 \times 0.08 = \$57,280 \)

EPS Formula:

• Plan I (All-equity): \( \text{EPS}_I = \frac{\text{EBIT}}{145,000} \)
• Plan II (Levered): \( \text{EPS}_{II} = \frac{\text{EBIT} - \$57,280}{125,000} \)

a. EBIT = $300,000:

• \( \text{EPS}_I = \frac{\$300,000}{145,000} = \$2.07 \)
• \( \text{EPS}_{II} = \frac{\$300,000 - \$57,280}{125,000} = \$1.94 \)

Plan I results in a higher EPS.

b. EBIT = $600,000:

• \( \text{EPS}_I = \frac{\$600,000}{145,000} = \$4.14 \)
• \( \text{EPS}_{II} = \frac{\$600,000 - \$57,280}{125,000} = \$4.34 \)

Plan II results in a higher EPS.

c. Break-even EBIT:

Set the two EPS formulas equal and solve for EBIT:

\[ \frac{\text{EBIT}}{145,000} = \frac{\text{EBIT} - \$57,280}{125,000} \] \[ 125,000 \times \text{EBIT} = 145,000 \times (\text{EBIT} - \$57,280) \] \[ 125,000 \times \text{EBIT} = 145,000 \times \text{EBIT} - \$8,305,600,000 \] \[ 20,000 \times \text{EBIT} = \$8,305,600,000 \] \[ \text{EBIT} = \frac{\$8,305,600,000}{20,000} = \$415,280 \]

The break-even EBIT is $415,280.

Theoretical Foundation: This problem applies **M&M Proposition I (no taxes)**. The key takeaway is that in the absence of taxes and other market imperfections, the total value of the firm is independent of its capital structure. The value of a firm is the sum of its equity and debt (\(V = E + D\)).

First, we need to find the value of the unlevered firm (Plan I) by using the break-even EBIT from Problem 4 ($415,280) and the corresponding EPS.

• Break-even EPS = \( \frac{\$415,280}{145,000} = \$2.864 \)
• Total Earnings = \( \text{EBIT} = \$415,280 \)
• Value of Equity (Plan I) = \( \$2.864 \times 145,000 = \$415,280 \)

Since Plan I is all-equity, the value of the firm is equal to the value of its equity:

\[ \text{Firm Value} = V_I = E_I = \$415,280 \]

Now, let's find the share price and firm value for the levered plan (Plan II).

According to M&M Proposition I (no taxes), the value of the levered firm (\(V_{II}\)) should be the same as the unlevered firm (\(V_I\)).

\[ V_{II} = V_I = \$415,280 \]

The value of equity under Plan II is the total firm value minus the value of debt:

\[ E_{II} = V_{II} - D_{II} = \$415,280 - \$716,000 \]

However, this results in a negative equity value, which is not possible. This indicates that the break-even EBIT of $415,280 is an arbitrary point of indifference and not necessarily the total value of the firm. Let's re-evaluate using the given information more directly.

Since we cannot determine the firm's true value from the break-even point alone, we need to use a single EBIT scenario. Let's use the EBIT of $300,000 from Problem 4. This gives us a total value for the unlevered firm of $300,000. The price per share is then:

• Price per share (Plan I) = \( \frac{\text{Value of Equity}}{\text{Shares}} = \frac{\$300,000}{145,000} = \$2.07 \)

Now for Plan II, we know the total value of the firm must be the same, so \( V_{II} = \$300,000 \). The value of the equity is then:

• Value of Equity (Plan II) = \( V_{II} - \text{Debt} = \$300,000 - \$716,000 = -\$416,000 \)

This is still impossible. The problem's premise seems flawed as the debt value ($716,000) is too high relative to the firm's value implied by the EBIT figures. A firm's debt cannot exceed its total value. Assuming the debt value is a typo and should be less than the firm's value, the principle remains that the firm's total value remains constant, and the price per share is adjusted to reflect the leverage. Let's re-read the original problem to ensure no misinterpretations.

Theoretical Foundation: This problem reinforces the concept of EPS indifference points and the magnification effect of leverage, first without taxes (M&M no-tax case) and then with taxes (M&M with taxes), where the tax shield on interest is a factor.

a. Compare EPS with EBIT = $70,000 (No Taxes):

• All-Equity Plan: \( \text{EPS} = \frac{\$70,000}{15,000} = \$4.67 \)
• Plan I (Debt = $100,050): Interest = \( \$100,050 \times 0.10 = \$10,005 \). EPS = \( \frac{\$70,000 - \$10,005}{12,700} = \$4.72 \)
• Plan II (Debt = $226,200): Interest = \( \$226,200 \times 0.10 = \$22,620 \). EPS = \( \frac{\$70,000 - \$22,620}{9,800} = \$4.83 \)

Highest EPS is Plan II ($4.83). Lowest EPS is the All-Equity Plan ($4.67).

b. Break-even EBIT (No Taxes):

• All-Equity vs. Plan I: \( \frac{\text{EBIT}}{15,000} = \frac{\text{EBIT} - \$10,005}{12,700} \implies \text{EBIT} \approx \$65,247 \)
• All-Equity vs. Plan II: \( \frac{\text{EBIT}}{15,000} = \frac{\text{EBIT} - \$22,620}{9,800} \implies \text{EBIT} \approx \$65,247 \)

The break-even EBIT is the same for both plans. This is because the interest rate (10%) and the cost of unlevered equity (EBIT/Value of Equity = $70,000/$70,000 = 10%) are the same. When the return on the firm's assets equals the cost of its debt, the firm is indifferent to leverage.

c. When will EPS be identical for Plans I and II (No Taxes)?

\[ \frac{\text{EBIT} - \$10,005}{12,700} = \frac{\text{EBIT} - \$22,620}{9,800} \] \[ 9,800(\text{EBIT} - \$10,005) = 12,700(\text{EBIT} - \$22,620) \] \[ 9,800\text{EBIT} - \$98,049,000 = 12,700\text{EBIT} - \$287,274,000 \] \[ 2,900\text{EBIT} = \$189,225,000 \implies \text{EBIT} = \$65,250 \]

The break-even EBIT is $65,250.

d. Repeat with a 21% tax rate:

• All-Equity Plan: \( \text{EPS} = \frac{\$70,000(1 - 0.21)}{15,000} = \$3.68 \)
• Plan I: \( \text{EPS} = \frac{(\$70,000 - \$10,005)(1 - 0.21)}{12,700} = \$3.73 \)
• Plan II: \( \text{EPS} = \frac{(\$70,000 - \$22,620)(1 - 0.21)}{9,800} = \$3.82 \)

The break-even EBIT is the same as without taxes because the \( (1 - T_C) \) term cancels out from both sides of the equation. The tax shield makes leverage more beneficial overall, but does not change the point of indifference.

Theoretical Foundation: This problem applies **M&M Proposition I (no taxes)**. The key is that the total value of the firm's assets remains constant, regardless of the capital structure. The value of equity is the firm's total value less its debt.

First, find the value of the unlevered firm (which is the total value of all three plans since M&M holds). We can use the information from the All-Equity Plan in Problem 6, which has 15,000 shares outstanding. The value of the firm is its market value. For a consistent valuation, we can use the break-even EBIT. From Problem 6, we found the break-even EBIT for all plans is $65,247. Let's use this to find the firm value. For the all-equity firm, \(V_U = E_U\). We can't determine the firm's value from the problem alone, but we can assume the price per share is consistent across all plans. A simpler way to approach this is to assume the total value of the firm is based on the initial all-equity plan, so \(V = E\). The problem doesn't give a share price for the all-equity plan, so we will use the implied price from the EBIT and the break-even analysis from the previous problem. From Problem 6, we know that at EBIT = $70,000, EPS for Plan I is $4.72 and for Plan II is $4.83. Since the total value of the firm must be the same under all plans, we can calculate the price per share based on a single assumption. Let's assume the all-equity firm's total value is $700,000 (since EBIT is $70,000 and \(R_A\) is likely 10%). This implies a share price of \( \$700,000 / 15,000 = \$46.67 \).

• Firm Value (V) = $700,000 (assumed)
• Price per share (All-Equity) = \( \frac{\$700,000}{15,000} = \$46.67 \)

Price per Share under Plan I:

• Value of Equity ($E_I$) = Firm Value - Debt = \( \$700,000 - \$100,050 = \$599,950 \)
• Price per share = \( \frac{\$599,950}{12,700} = \$47.24 \)

Price per Share under Plan II:

• Value of Equity ($E_{II}$) = Firm Value - Debt = \( \$700,000 - \$226,200 = \$473,800 \)
• Price per share = \( \frac{\$473,800}{9,800} = \$48.35 \)

The principle illustrated is that as a firm increases its use of debt, the price per share of equity increases. This is because the overall value of the firm is constant (in the absence of taxes), but the value of the equity is reduced by the value of the debt, which is used to repurchase shares. Fewer shares outstanding means a higher price per share, even though the total value of the equity slice of the pie is smaller.

Theoretical Foundation: This problem is a direct illustration of **homemade leverage**. It shows that in a world without taxes, an individual investor can use personal borrowing or lending to replicate any corporate capital structure decision. This is the foundation of M&M's irrelevance proposition.

a. Allison's cash flow (All-Equity):

• Total Earnings = EBIT = $41,000
• EPS = \( \frac{\$41,000}{6,500} = \$6.31 \)
• Allison's cash flow = \( 100 \times \$6.31 = \$631 \)

b. Allison's cash flow (Levered):

First, find the total value of the firm (V) and the value of debt (D) and equity (E) in the new structure.

• Firm Value (V) = \( 6,500 \times \$51 = \$331,500 \)
• New Debt (D) = \( 0.30 \times \$331,500 = \$99,450 \)
• New Equity (E) = \( \$331,500 - \$99,450 = \$232,050 \)
• Shares repurchased = \( \frac{\$99,450}{\$51} = 1,950 \text{ shares} \)
• New shares outstanding = \( 6,500 - 1,950 = 4,550 \text{ shares} \)
• Interest expense = \( \$99,450 \times 0.08 = \$7,956 \)
• New EPS = \( \frac{\$41,000 - \$7,956}{4,550} = \$7.26 \)
• Allison's cash flow = \( 100 \times \$7.26 = \$726 \)

c. Homemade Unleveraging:

Allison needs to undo the firm's leverage by selling some of her stock and lending the money. The firm's D/V ratio is 30%. To unlever, she needs to hold equity and debt in the same proportions as the all-equity firm, so she needs a 0% debt position. She can sell her shares and lend the money. Let's assume she holds the same amount of the firm's assets. The firm has a 30% debt-to-value ratio. Allison has a 100% equity position. To get to an unlevered position, she needs to sell some of her stock and buy some of the firm's debt. The firm's new D/E ratio is \( \frac{0.30}{0.70} = 0.4286 \). Allison needs to sell shares and lend money to have a personal D/E ratio of 0. A simpler way is to sell enough shares to offset the corporate debt. A firm with a D/E of 0.4286 means that for every $1 in equity, there is $0.4286 in debt. The firm's assets are $331,500. Allison owns 100 shares worth $5,100. The firm is 30% debt, so she can sell 30% of her equity, or 30 shares, for $1,530. She then lends this $1,530. Her cash flow will be her dividends on her remaining 70 shares plus the interest on her loan. A more direct way is to sell shares so that her personal D/E ratio is 0. This is the correct way to "unlever." She would sell 30 shares for $1,530 and lend this at 8%. Her dividends from 70 shares would be \( 70 \times \$7.26 = \$508.20 \). Interest from lending would be \( \$1,530 \times 0.08 = \$122.40 \). Her total cash flow would be \( \$508.20 + \$122.40 = \$630.60 \), which is approximately the same as her original cash flow.

d. Why capital structure is irrelevant:

The company's choice of capital structure is irrelevant because Allison can use homemade leverage to re-create her original cash flow. The total value of the firm is unaffected by the capital restructuring, as shown by the fact that Allison's total payoff is essentially unchanged regardless of the firm's decision. Because any investor can create their own desired level of leverage, the corporate capital structure decision does not create or destroy value for shareholders in this tax-free world.

Theoretical Foundation: This problem ties together **homemade leverage**, the **cost of equity**, and the **WACC** under M&M Proposition I and II (no taxes). The goal is to demonstrate that identical firms, even with different capital structures, have the same total value and WACC, and that investors can replicate payoffs through personal actions.

a. Rico's expected return on XYZ stock:

• XYZ Total Value = \( E + D \). To find D, we need the total value of the firm, which must be the same as ABC. So \( V = \$680,000 \).
• XYZ Debt = \( V - E = \$680,000 - \$340,000 = \$340,000 \).
• Interest on XYZ debt = \( \$340,000 \times 0.07 = \$23,800 \)
• XYZ Net Income = \( \text{EBIT} - \text{Interest} = \$67,000 - \$23,800 = \$43,200 \)
• Return on Equity ($R_{E}$) for XYZ = \( \frac{\text{Net Income}}{\text{Equity Value}} = \frac{\$43,200}{\$340,000} = 12.71\% \)
• Rico's cash flow = \( \$41,500 \times 12.71\% = \$5,270.65 \)
• Rico's expected rate of return = 12.71%

b. Homemade leverage in ABC:

Rico wants to replicate the leveraged position of XYZ by investing in the unlevered firm ABC. XYZ has D/E = 1.0 ($340k debt and $340k equity). Rico can do this by borrowing an amount equal to his equity investment in ABC.

• Rico borrows $41,500 at 7%.
• He invests his own $41,500 plus the borrowed $41,500, for a total of $83,000, in ABC stock.
• ABC's cost of equity = \( \frac{\text{EBIT}}{\text{Equity Value}} = \frac{\$67,000}{\$680,000} = 9.85\% \)
• Return on his ABC investment = \( \$83,000 \times 9.85\% = \$8,175.5 \)
• Less interest paid = \( \$41,500 \times 0.07 = \$2,905 \)
• Rico's net cash flow = \( \$8,175.5 - \$2,905 = \$5,270.5 \)

The cash flows and return are identical, demonstrating homemade leverage.

c. Cost of Equity:

• ABC: \( R_E = \frac{\$67,000}{\$680,000} = 9.85\% \)
• XYZ: \( R_E = \frac{\$43,200}{\$340,000} = 12.71\% \)

d. WACC and the principle:

• ABC WACC = \( 9.85\% \) (since it is all equity)
• XYZ WACC = \( (\frac{\$340k}{\$680k}) \times 12.71\% + (\frac{\$340k}{\$680k}) \times 7\% = 6.36\% + 3.5\% = 9.86\% \)

The WACC for both firms is essentially the same (the minor difference is due to rounding). This illustrates **M&M Proposition I (no taxes)**, which states that the value of the firm and its WACC are independent of its capital structure.

Theoretical Foundation: This problem applies **M&M Proposition I and II (no taxes)**. For an all-equity firm with no taxes, the WACC is equal to the cost of equity, which is the required return on the firm's assets. The value of the firm is the value of its equity, which can be found by discounting its perpetual cash flows (EBIT) at the WACC.

Given: All-equity firm, WACC = 7.9%, \(E = \$15.6M\), no taxes.
Since there is no debt and no taxes, \( WACC = R_E = R_A = 7.9\% \).

The value of the firm is the market value of its equity. We can find the perpetual EBIT as follows:

\[ V = \frac{\text{EBIT}}{\text{WACC}} \] \[ \$15,600,000 = \frac{\text{EBIT}}{0.079} \] \[ \text{EBIT} = \$15,600,000 \times 0.079 = \$1,232,400 \]

The EBIT is $1,232,400.

Theoretical Foundation: This problem applies **M&M Proposition I and II with taxes**. For an all-equity firm, the WACC is equal to the unlevered cost of capital (\(R_U\)). The value of the firm is found by discounting the perpetual after-tax cash flow (\(\text{EBIT} \times (1-T_C)\)) at the unlevered cost of capital.

Given: All-equity firm, \(V = E = \$15.6M\), \(T_C = 22\%\), \(WACC = 7.9\%\).

Since the firm is all-equity, \(WACC = R_U = R_E = 7.9\%\). We can solve for EBIT using the perpetuity formula with the tax shield:

\[ V = \frac{\text{EBIT} \times (1 - T_C)}{R_U} \] \[ \$15,600,000 = \frac{\text{EBIT} \times (1 - 0.22)}{0.079} \] \[ \$15,600,000 = \frac{\text{EBIT} \times 0.78}{0.079} \] \[ \text{EBIT} = \frac{\$15,600,000 \times 0.079}{0.78} = \$1,580,000 \]

The EBIT is $1,580,000. The WACC is 7.9%, because for an all-equity firm, WACC is simply the unlevered cost of capital. This is not affected by taxes as there is no interest tax shield to consider in this case.

Theoretical Foundation: This problem applies the **WACC formula with taxes** and **M&M Proposition II with taxes** in reverse. It demonstrates how to find the individual components of the WACC when the overall rate is known, and how the cost of equity changes with the level of leverage.

Given: \(D/E = 1.25\), \(WACC = 7.8\%\), \(R_D = 4.7\%\), \(T_C = 21\%\).
We first find the capital structure weights: \( E/V = \frac{1}{1+1.25} = 0.4444 \) and \( D/V = \frac{1.25}{1+1.25} = 0.5556 \).

a. Cost of equity (\(R_E\)):

Use the WACC formula and solve for \(R_E\):

\[ WACC = (\frac{E}{V})R_E + (\frac{D}{V})R_D(1-T_C) \] \[ 0.078 = (0.4444)R_E + (0.5556)(0.047)(1-0.21) \] \[ 0.078 = 0.4444R_E + 0.02058 \] \[ R_E = \frac{0.078 - 0.02058}{0.4444} = 0.1292, \text{ or } 12.92\% \]

b. Unlevered cost of equity (\(R_U\)):

Use M&M Proposition II with taxes and solve for \(R_U\):

\[ R_E = R_U + (R_U - R_D)(D/E)(1-T_C) \] \[ 0.1292 = R_U + (R_U - 0.047)(1.25)(1-0.21) \] \[ 0.1292 = R_U + 0.9875(R_U - 0.047) \] \[ 0.1292 = R_U + 0.9875R_U - 0.0464 \] \[ 0.1756 = 1.9875R_U \] \[ R_U = \frac{0.1756}{1.9875} = 0.0883, \text{ or } 8.83\% \]

c. Cost of equity at different D/E ratios:

Use M&M Proposition II with taxes and the unlevered cost of capital we just found (\(R_U = 8.83\%\)).

• \(D/E = 2\): \( R_E = 0.0883 + (0.0883 - 0.047)(2)(1-0.21) = 0.1528, \text{ or } 15.28\% \)
• \(D/E = 1\): \( R_E = 0.0883 + (0.0883 - 0.047)(1)(1-0.21) = 0.1206, \text{ or } 12.06\% \)
• \(D/E = 0\): \( R_E = 0.0883 + (0.0883 - 0.047)(0)(1-0.21) = 0.0883, \text{ or } 8.83\% \)

Theoretical Foundation: This problem applies the concepts of **unlevered cost of capital** and **M&M Proposition II with taxes**. For an all-equity firm, the WACC is equal to the cost of equity, which is also the unlevered cost of capital. We then use this to calculate the cost of equity at different leverage levels.

Given: All-equity firm, \(R_D = 5.9\%\), \(WACC = 9.2\%\), \(T_C = 21\%\).
Since the firm is all-equity, \(R_E = WACC = R_U = 9.2\%\).

a. Cost of equity:

The cost of equity is 9.2%.

b. Cost of equity at 25% debt:

This means \(D/V = 0.25\). So, \(E/V = 0.75\), and \(D/E = 0.25/0.75 = 0.3333\).

Use M&M Proposition II with taxes:

\[ R_E = R_U + (R_U - R_D)(D/E)(1-T_C) \] \[ R_E = 0.092 + (0.092 - 0.059)(0.3333)(1-0.21) = 0.092 + 0.00867 = 0.1007, \text{ or } 10.07\% \]

c. Cost of equity at 50% debt:

This means \(D/V = 0.5\). So, \(E/V = 0.5\), and \(D/E = 1.0\).

\[ R_E = 0.092 + (0.092 - 0.059)(1.0)(1-0.21) = 0.092 + 0.02617 = 0.1182, \text{ or } 11.82\% \]

d. WACC in parts (b) and (c):

For part (b), with \(D/E = 0.3333\):

\[ WACC = (\frac{E}{V})R_E + (\frac{D}{V})R_D(1-T_C) \] \[ WACC = (0.75)(0.1007) + (0.25)(0.059)(1-0.21) = 0.0755 + 0.01165 = 0.0872, \text{ or } 8.72\% \]

For part (c), with \(D/E = 1.0\):

\[ WACC = (0.5)(0.1182) + (0.5)(0.059)(1-0.21) = 0.0591 + 0.0233 = 0.0824, \text{ or } 8.24\% \]

Theoretical Foundation: This problem is a direct application of **M&M Proposition I with corporate taxes**. It shows 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.

Given: \(EBIT = \$125,000\), \(R_U = 12\%\), \(T_C = 24\%\), \(R_D = 7\%\).

Value of the unlevered firm (\(V_U\)):

Since the cash flow is a perpetuity, the value is the after-tax EBIT divided by the unlevered cost of capital.

\[ V_U = \frac{\text{EBIT} \times (1 - T_C)}{R_U} = \frac{\$125,000 \times (1 - 0.24)}{0.12} = \frac{\$95,000}{0.12} = \$791,666.67 \]

Value of the levered firm (\(V_L\)):

The value of the levered firm is the value of the unlevered firm plus the present value of the tax shield. The present value of the tax shield is \( T_C \times D \).

\[ V_L = V_U + T_C \times D = \$791,666.67 + 0.24 \times \$205,000 = \$791,666.67 + \$49,200 = \$840,866.67 \]

Theoretical Foundation: This problem uses the values from Problem 14 and applies **M&M Proposition II with taxes** and the **WACC formula with taxes**. It demonstrates how the cost of equity and WACC change as a result of leverage.

From Problem 14: \(R_U = 12\%\), \(R_D = 7\%\), \(T_C = 24\%\), \(D = \$205,000\), \(V_L = \$840,866.67\).
First, find the value of equity after the recapitalization: \( E = V_L - D = \$840,866.67 - \$205,000 = \$635,866.67 \).

Cost of Equity (\(R_E\)):

Use M&M Proposition II with taxes:

\[ R_E = R_U + (R_U - R_D)(D/E)(1-T_C) \] \[ R_E = 0.12 + (0.12 - 0.07)(\frac{\$205,000}{\$635,866.67})(1 - 0.24) \] \[ R_E = 0.12 + (0.05)(0.3224)(0.76) = 0.12 + 0.01225 = 0.13225, \text{ or } 13.23\% \]

WACC:

Use the WACC formula with taxes:

\[ WACC = (\frac{E}{V})R_E + (\frac{D}{V})R_D(1-T_C) \] \[ WACC = (\frac{\$635,866.67}{\$840,866.67})(0.13225) + (\frac{\$205,000}{\$840,866.67})(0.07)(1-0.24) \] \[ WACC = (0.7562)(0.13225) + (0.2438)(0.0532) = 0.1000 + 0.0130 = 0.1130, \text{ or } 11.30\% \]

Implications:

The cost of equity has increased from 12% to 13.23% as a result of the increased financial risk. However, the WACC has decreased from 12% to 11.30% because of the tax shield benefit of debt. The implication is that the firm should use debt financing to lower its WACC and increase its total value, up to the point where the benefits of the tax shield are offset by the costs of financial distress (the static theory).