Let's break down each part of the question: **Given:** - Market value of equity (\(E\)) = \$22.6 million - Market value of debt (\(D\)) = \$11.3 million - Treasury bill rate (risk-free rate, \(R_f\)) = 4% - Expected return on market (\(R_m\)) = 10% - Beta of equity (\(\beta\)) = 1.11 - No taxes --- ### **a. What is the company's debt-equity ratio?** \[ \text{Debt-Equity Ratio} = \frac{D}{E} \] \[ \text{Debt-Equity Ratio} = \frac{11.3}{22.6} = 0.5 \] **Answer (a):** \[ \boxed{0.50} \] --- ### **b. What is the company’s weighted average cost of capital (WACC)?** First, we need the cost of equity (\(R_e\)) and cost of debt (\(R_d\)). Since the company pays no taxes, the WACC formula is: \[ \text{WACC} = \frac{E}{V} R_e + \frac{D}{V} R_d \] where \(V = E + D\). #### **Step 1: Calculate cost of equity (\(R_e\)) using CAPM** \[ R_e = R_f + \beta (R_m - R_f) \] \[ R_e = 0.04 + 1.11 \times (0.10 - 0.04) \] \[ R_e = 0.04 + 1.11 \times 0.06 \] \[ R_e = 0.04 + 0.0666 = 0.1066 \text{ or } 10.66\% \] #### **Step 2: Cost of debt (\(R_d\))** Assuming the company's debt is risk-free (since no info is given), use the T-bill rate: \[ R_d = 4.00\% \] #### **Step 3: Find weights** \[ V = E + D = 22.6 + 11.3 = 33.9 \] \[ \frac{E}{V} = \frac{22.6}{33.9} = 0.6667 \] \[ \frac{D}{V} = \frac{11.3}{33.9} = 0.3333 \] #### **Step 4: Calculate WACC** \[ \text{WACC} = 0.6667 \times 10.66\% + 0.3333 \times 4.00\% \] \[ = 0.6667 \times 0.1066 + 0.3333 \times 0.04 \] \[ = 0.07107 + 0.01333 = 0.0844 \text{ or } 8.44\% \] **Answer (b):** \[ \boxed{8.44\%} \] --- ### **c. What is the cost of capital for an otherwise identical all-equity company?** For an all-equity firm, the cost of capital is the cost of equity as if the firm were unlevered, i.e., the firm's asset beta is the unlevered beta. Since there are no taxes and the debt is risk-free, use the formula: \[ \beta_{A} = \frac{E}{V} \beta_E + \frac{D}{V} \beta_D \] But \(\beta_D = 0\) (risk-free debt): \[ \beta_{A} = \frac{E}{V} \beta_E + \frac{D}{V} \times 0 = \frac{E}{V} \beta_E \] \[ \beta_A = 0.6667 \times 1.11 = 0.740 \] Now, use CAPM to find the all-equity cost of capital: \[ R_{A} = R_f + \beta_A (R_m - R_f) \] \[ R_A = 0.04 + 0.74 \times (0.10 - 0.04) \] \[ = 0.04 + 0.74 \times 0.06 \] \[ = 0.04 + 0.0444 = 0.0844 \text{ or } 8.44\% \] **Answer (c):** \[ \boxed{8.44\%} \] --- ## **Summary Table** | Part | Answer | |------|---------------| | a | 0.50 | | b | 8.44% | | c | 8.44% |

Introduction: Understanding the valuation and capital structure of a firm involves examining how different sources of financing—equity and debt—impact the company's overall risk and return profile. The market value of equity reflects investors' valuation of the firm's future earnings and growth prospects, while the market value of debt indicates the cost and risk associated with borrowings. The debt-equity ratio measures the relative proportion of debt used to finance assets, which influences the company's leverage, risk, and cost of capital. Capital asset pricing model (CAPM) provides a framework to estimate the expected return on equity based on systematic risk, represented by beta, the risk-free rate, and the market risk premium. The weighted average cost of capital (WACC) combines the costs of equity and debt, weighted by their proportions in the firm's capital structure, to determine the overall required return for the firm’s investors. In the absence of taxes, the WACC simplifies to a weighted average of the costs of equity and debt, adjusted for their respective market values. These concepts are fundamental in assessing the company's financial structure, determining its cost of capital, and evaluating investment decisions. This foundational understanding enables accurate calculation of the debt-equity ratio, WACC, and the cost of capital for a hypothetical all-equity firm, as required in the problem. Explanation: This introduction links key financial concepts—market value, leverage, CAPM, and WACC—to the problem, providing the necessary theoretical background. Recognizing how leverage impacts risk and return, and the role of CAPM in estimating equity costs, is essential for precise calculations. The discussion emphasizes the importance of these concepts in practical valuation and capital structure analysis, setting the stage for applying specific formulas and procedures. --- ### Presentation of Relevant Formulas Required To Solve The Question 1. **Debt-Equity Ratio:** \[ \text{Debt-Equity Ratio} = \frac{D}{E} \] - *Rationale:* Measures the leverage of the firm by comparing total debt to total equity, indicating the proportion of debt financing relative to equity. 2. **Cost of Equity via CAPM:** \[ R_e = R_f + \beta (R_m - R_f) \] - *Rationale:* Estimates the return required by equity investors based on the systematic risk (beta), the risk-free rate, and the market risk premium. It reflects the compensation investors demand for bearing market risk. 3. **Weighted Average Cost of Capital (WACC):** \[ \text{WACC} = \frac{E}{V} R_e + \frac{D}{V} R_d \] - *Rationale:* Provides the average rate of return required by all providers of capital, weighted by their relative market values, assuming no taxes. It is used to evaluate investment projects and firm valuation. 4. **Unlevered Beta (for all-equity firm):** \[ \beta_A = \frac{E}{V} \beta_E + \frac{D}{V} \beta_D \] - *Rationale:* Reflects the firm's asset risk independent of its capital structure; useful for estimating the cost of capital in an unlevered scenario. --- ### A Detailed Step-by-Step Solution **Step 1: Calculate the debt-equity ratio** \[ \text{Debt-Equity Ratio} = \frac{D}{E} = \frac{11.3\, \text{million}}{22.6\, \text{million}} = 0.50 \] *Explanation:* The debt-equity ratio quantifies leverage; with debt at \$11.3 million and equity at \$22.6 million, the ratio indicates that debt constitutes half of the equity value, reflecting moderate leverage. --- **Step 2: Calculate the cost of equity (\(R_e\)) using CAPM** \[ R_e = R_f + \beta (R_m - R_f) \] \[ R_e = 0.04 + 1.11 \times (0.10 - 0.04) = 0.04 + 1.11 \times 0.06 \] \[ R_e = 0.04 + 0.0666 = 0.1066 \text{ or } 10.66\% \] *Explanation:* The CAPM formula estimates the expected return required by shareholders, considering the firm’s systematic risk (beta). The market risk premium (6%) is scaled by beta, and added to the risk-free rate, resulting in an equity cost of approximately 10.66%. --- **Step 3: Determine the weights of debt and equity in the firm's capital structure** Total value: \[ V = E + D = 22.6 + 11.3 = 33.9\, \text{million} \] Weights: \[ \frac{E}{V} = \frac{22.6}{33.9} \approx 0.6667 \] \[ \frac{D}{V} = \frac{11.3}{33.9} \approx 0.3333 \] *Explanation:* These weights reflect the proportion of equity and debt in the firm’s total capital, crucial for calculating WACC. --- **Step 4: Compute the weighted average cost of capital (WACC)** Assuming the cost of debt equals the risk-free rate (since no risk premium for debt is provided): \[ R_d = 4.00\% \] Calculate WACC: \[ \text{WACC} = (0.6667 \times 10.66\%) + (0.3333 \times 4.00\%) \] \[ = 0.6667 \times 0.1066 + 0.3333 \times 0.04 \] \[ = 0.07107 + 0.01333 = 0.0844 \text{ or } 8.44\% \] *Explanation:* The WACC combines the costs of equity and debt weighted by their market proportions, representing the overall required return on the firm’s invested capital. --- **Step 5: Determine the cost of capital for an identical all-equity company** Calculate the firm's asset beta (\(\beta_A\)): \[ \beta_A = \frac{E}{V} \beta_E + \frac{D}{V} \beta_D \] Given \(\beta_D = 0\) (risk-free debt): \[ \beta_A = 0.6667 \times 1.11 + 0.3333 \times 0 = 0.740 \] Estimate the unlevered cost of capital: \[ R_{A} = R_f + \beta_A (R_m - R_f) \] \[ = 0.04 + 0.74 \times 0.06 = 0.04 + 0.0444 = 0.0844 \text{ or } 8.44\% \] *Explanation:* This reflects the cost of capital for a hypothetical all-equity firm with the same asset risk as the leveraged firm, useful for understanding how leverage affects the cost of capital. --- ### **Conclusion** The company's debt-equity ratio is 0.50, indicating moderate leverage. The weighted average cost of capital (WACC) is approximately 8.44%, calculated based on the firm's capital structure and the costs derived via CAPM. An otherwise identical all-equity firm would have a cost of capital of approximately 8.44%, assuming no taxes and risk-free debt. These calculations provide insights into the firm's leverage and cost of capital, integral to valuation and financial decision-making.