1. **State the problem:** We need to find the total value, WACC, and cost of equity for Aradel Inc. after it borrows an additional 20 million to repurchase equity. 2. **Given data:** - Current firm value $V_0 = 120$ million - Debenture (debt) $D_0 = 40$ million at 10% interest - Risk-free rate $r_f = 4\%$ - Covariance of equity return with market $\text{Cov}(R_e, R_m) = 1.86\% = 0.0186$ - Variance of market return $\text{Var}(R_m) = 1.25\% = 0.0125$ - Tax rate $T_c = 30\% = 0.3$ - Current equity return average $R_e^{avg} = 7.2\%$ - Additional debt to borrow $\Delta D = 20$ million 3. **Calculate beta of equity $\beta_e$ using CAPM:** $$\beta_e = \frac{\text{Cov}(R_e, R_m)}{\text{Var}(R_m)} = \frac{0.0186}{0.0125} = 1.488$$ 4. **Calculate cost of equity $r_e$ using CAPM:** $$r_e = r_f + \beta_e (r_m - r_f)$$ We need market risk premium $r_m - r_f$. Since $R_e^{avg} = 7.2\%$ is given and management feels shares are underpriced, we use CAPM formula to find implied market risk premium. Rearranging CAPM for market risk premium: $$7.2\% = 4\% + 1.488 (r_m - 4\%) \Rightarrow r_m - 4\% = \frac{7.2\% - 4\%}{1.488} = \frac{3.2\%}{1.488} = 2.15\%$$ So, $$r_e = 4\% + 1.488 \times 2.15\% = 7.2\%$$ (consistent) 5. **Calculate current cost of debt after tax:** $$r_d = 10\%$$ $$r_d (1 - T_c) = 10\% \times (1 - 0.3) = 7\%$$ 6. **Calculate current equity value $E_0$:** $$E_0 = V_0 - D_0 = 120 - 40 = 80 \text{ million}$$ 7. **After borrowing additional 20 million, new debt:** $$D_1 = 40 + 20 = 60 \text{ million}$$ 8. **New equity value after repurchase:** $$E_1 = V_1 - D_1$$ We need to find $V_1$ first. 9. **Calculate unlevered cost of equity $r_u$ (cost of capital if no debt):** Using Modigliani-Miller with taxes: $$r_u = \frac{E_0}{V_0} r_e + \frac{D_0}{V_0} r_d (1 - T_c) = \frac{80}{120} \times 7.2\% + \frac{40}{120} \times 7\% = 4.8\% + 2.33\% = 7.13\%$$ 10. **Calculate new beta of equity $\beta_{e1}$ after leverage change:** Using formula: $$\beta_{e1} = \beta_u + (\beta_u - \beta_d) \frac{D_1}{E_1} (1 - T_c)$$ Assuming debt beta $\beta_d = 0$ (risk-free debt), and unlevered beta $\beta_u$ is: $$\beta_u = \frac{E_0}{V_0} \beta_e + \frac{D_0}{V_0} \beta_d = \frac{80}{120} \times 1.488 + 0 = 0.992$$ We don't know $E_1$ or $V_1$ yet, so use value formula: 11. **Calculate new firm value $V_1$ using tax shield:** $$V_1 = V_0 + T_c \times \Delta D = 120 + 0.3 \times 20 = 120 + 6 = 126 \text{ million}$$ 12. **Calculate new equity value:** $$E_1 = V_1 - D_1 = 126 - 60 = 66 \text{ million}$$ 13. **Calculate new equity beta:** $$\beta_{e1} = 0.992 + (0.992 - 0) \times \frac{60}{66} \times (1 - 0.3) = 0.992 + 0.992 \times 0.909 \times 0.7 = 0.992 + 0.631 = 1.623$$ 14. **Calculate new cost of equity $r_{e1}$:** $$r_{e1} = r_f + \beta_{e1} (r_m - r_f) = 4\% + 1.623 \times 2.15\% = 4\% + 3.49\% = 7.49\%$$ 15. **Calculate new WACC:** $$WACC = \frac{E_1}{V_1} r_{e1} + \frac{D_1}{V_1} r_d (1 - T_c) = \frac{66}{126} \times 7.49\% + \frac{60}{126} \times 7\% = 3.92\% + 3.33\% = 7.25\%$$ **Final answers:** - Total value after leverage $V_1 = 126$ million - WACC $= 7.25\%$ - Cost of equity $r_{e1} = 7.49\%$