Let's break each part down step by step. --- ### 1. **Value of Debt-Generated Tax Shield** - **Permanent Debt, Corporate Tax Rate**: \( D = \$45 \text{ million} \) \( T_C = 21\% = 0.21 \) - **Value of Tax Shield** (perpetuity): \( \text{Tax Shield} = T_C \times D = 0.21 \times \$45 = \$9.45 \) million **Answer:** **Debt-generated tax shield = \$9.45 million** --- ### 2. **After-tax WACC Calculation** #### **Market Value Weights:** - Total firm value (\( V \)): \( V = D + E = \$45 + \$200 = \$245 \) million - Debt Weight (\( w_D \)): \( w_D = D / V = 45/245 \approx 0.1837 \) - Equity Weight (\( w_E \)): \( w_E = E / V = 200/245 \approx 0.8163 \) #### **WACC Formula with Taxes:** \[ \text{WACC} = w_E \cdot r_E + w_D \cdot r_D \cdot (1 - T_C) \] Given: - \( r_D = 7.1\% \) - \( r_E = 15.9\% \) - \( T_C = 21\% \) #### **Plug in the numbers:** \[ \begin{align*} \text{WACC} &= 0.8163 \times 0.159 + 0.1837 \times 0.071 \times (1 - 0.21) \\ &= 0.8163 \times 0.159 + 0.1837 \times 0.071 \times 0.79 \\ &= 0.129790 + 0.1837 \times 0.05609 \\ &= 0.129790 + 0.010302 \\ &= 0.140092 \\ &= 14.01\% \end{align*} \] **Answer:** **After-tax WACC = 14.01%** --- ### 3. **Value of the Firm if Interest Deductibility is Repealed After 5 Years** #### **Step 1: Value of Tax Shield for 5 Years** - Annual tax shield: \( 0.21 \times \$45 = \$9.45 \) million per year - Discount rate for tax shield = borrowing rate = 7.1% - Present value of **5-year annuity**: \[ \text{PV} = \text{Tax Shield} \times \left[ \frac{1 - (1 + r)^{-n}}{r} \right] \] \[ \text{PV} = \$9.45 \times \left[ \frac{1 - (1 + 0.071)^{-5}}{0.071} \right] \] First, calculate \( (1 + 0.071)^{-5} \): \[ (1.071)^{-5} \approx 0.7084 \] \[ 1 - 0.7084 = 0.2916 \] \[ \frac{0.2916}{0.071} \approx 4.107 \] \[ \text{PV} = \$9.45 \times 4.107 \approx \$38.82 \text{ million} \] #### **Step 2: Value of the Firm Without Tax Shield (All-Equity Value)** - Market value of assets = \$245 million - Current value includes the permanent tax shield (\$9.45 million). - New value = All-equity value + value of tax shields for only 5 years All-equity value = Market value of assets **minus** value of permanent tax shield: \[ \text{All-equity value} = \$245 - \$9.45 = \$235.55 \text{ million} \] **New total value = All-equity value + 5-year tax shield value** \[ \text{New value} = \$235.55 + \$38.82 = \$274.37 \text{ million} \] **But this can't be correct, because the 5-year tax shield value should replace the perpetual tax shield, not be added on top of an all-equity value. The market value of the firm (\$245 million) already includes the perpetual tax shield (\$9.45 million). So:** - Remove the perpetual tax shield (\$9.45 million) - Add the present value of the 5-year tax shield (\$38.82 million) \[ \text{New value} = (\$245 - \$9.45) + \$38.82 = \$235.55 + \$38.82 = \$274.37 \text{ million} \] But this seems to **increase** the value, which cannot be correct. Let's reconsider: - The market value (\$245 million) **already includes** the value of the tax shield (\$9.45 million). If the law changes, the value of the tax shield **drops to the present value of the 5-year annuity (\$38.82 million)**. So, the change in firm value is: \[ \text{New value} = (\text{All-equity value}) + (\text{PV 5-year tax shield}) \] \[ \text{All-equity value} = \$245 - \$9.45 = \$235.55 \text{ million} \] \[ \text{New value} = \$235.55 + \$38.82 = \$274.37 \text{ million} \] **However**, this suggests the value goes up, which can't be. Let's check MM logic: - The **current value** = Value if all-equity-financed + Value of tax shield - Value if all-equity-financed = \$245 - \$9.45 = \$235.55 million - Current value = \$235.55 + \$9.45 = \$245 million (matches balance sheet) - If the tax shield only lasts for 5 years, the value = Value if all-equity-financed + PV of 5-year tax shield - Value = \$235.55 + \$38.82 = \$274.37 million But the **PV of a 5-year annuity** (\$38.82 million) is higher than the value of a perpetual tax shield (\$9.45 million)! That can't be—the perpetual tax shield must be worth more. Let's check the math: - **Perpetual tax shield:** \[ \text{PV} = \frac{T_C \cdot r_D \cdot D}{r_D} = T_C \cdot D = 0.21 \times \$45 = \$9.45 \text{ million} \] - **5-year annuity:** \[ \text{PV} = \$9.45 \times \left[ \frac{1 - (1.071)^{-5}}{0.071} \right] \] \[ (1.071)^{-5} \approx 0.7084 \] \[ 1 - 0.7084 = 0.2916 \] \[ 0.2916 / 0.071 = 4.107 \] \[ \$9.45 \times 4.107 = \$38.82 \text{ million} \] **This means the value of 5-year shield (\$38.82 million) is greater than the perpetual shield (\$9.45 million), which cannot be.** **What's wrong?** - The annual tax shield is the tax savings from interest **each year**. - For a perpetuity: Value = \( T_C \times D \) - For an n-year annuity: Value = \( T_C \times D \times \left[ \frac{1 - (1 + r_D)^{-n}}{r_D} \right] \) But this formula applies **if the debt is replaced each year for n years**—i.e., the debt is rolled over and stays at \$45 million for 5 years, then goes to zero. **But in practice, after 5 years, the tax shield is zero, not the debt. So the value of the tax shield is the PV of a 5-year annuity of \$9.45 million, discounted at 7.1%.** So, **the new value of the firm is:** \[ \text{All-equity value} + \text{PV of 5-year tax shield} \] \[ = \$245 - \$9.45 + \$38.82 = \$235.55 + \$38.82 = \$274.37 \text{ million} \] But this is illogical. The perpetual tax shield should be worth more than a 5-year annuity. **Let's double-check:** - Perpetual tax shield: \$9.45 million is the annual benefit, discounted as a perpetuity at the debt rate: \[ \text{PV} = \frac{\$9.45}{0.071} = \$133.10 \text{ million} \] But the standard MM approach is just \( T_C \times D \) if debt is permanent and riskless. But if the debt is riskless and permanent, the value of the tax shield is **not** the present value of the annual tax shield discounted at the cost of debt; it's just \( T_C \times D \). But if the tax shield disappears after 5 years, the present value is: \[ \text{PV} = \sum_{t=1}^{5} \frac{\$9.45}{(1.071)^t} \] Let’s calculate it explicitly: - Year 1: \( \$9.45 / 1.071 = \$8.83 \) - Year 2: \( \$9.45 / (1.071)^2 = \$8.24 \) - Year 3: \( \$9.45 / (1.071)^3 = \$7.70 \) - Year 4: \( \$9.45 / (1.071)^4 = \$7.19 \) - Year 5: \( \$9.45 / (1.071)^5 = \$6.72 \) Sum: \( 8.83 + 8.24 + 7.70 + 7.19 + 6.72 = \$38.68 \) million So, **PV of 5-year tax shield = \$38.68 million** So, **current value = all-equity value + perpetual shield = all-equity value + \$9.45 million** **If the shield lasts for only 5 years, value = all-equity value + \$38.68 million** **But this still says the 5-year shield is worth more than the perpetual, which cannot be!** **The issue is that the annual shield is not \$9.45 million if we use the perpetuity formula.** Let’s clarify: - If the shield is \$9.45 million **per year forever**, its present value at 7.1% is \( 9.45 / 0.071 = \$133.10 \) million. - The standard MM formula simplifies the value of the tax shield to \( T_C \times D \) **if the debt is risk-free and perpetual**. The **correct way** is: - **Perpetual shield (risk-free debt):** Value of tax shield = \( T_C \times D \) = \$9.45 million - **5-year shield:** PV of annual shield (\$7.1\% \) of \$45m = \$3.195 million interest per year Tax shield per year = \( 0.21 \times 3.195 = \$0.671 \) million per year PV of 5-year annuity of \$0.671 million per year at 7.1%: \[ \text{PV} = \$0.671 \times \left[ \frac{1 - (1.071)^{-5}}{0.071} \right] \approx \$0.671 \times 4.107 = \$2.76 \text{ million} \] **Wait, this is much less, which makes sense.** **So the error before was taking the entire debt as the tax shield per year, not the interest.** **CORRECT FORMULA:** - **Annual tax shield = T_C \times r_D \times D** - **Perpetual Value = T_C \times D (\( r_D \) cancels out)** - **5-year Value = T_C \times r_D \times D \times [Annuity factor]** So, - \( T_C = 0.21 \) - \( D = \$45 \) million - \( r_D = 7.1\% = 0.071 \) - Annual shield = \( 0.21 \times 0.071 \times 45 = 0.21 \times 3.195 = \$0.671 \) million - Annuity factor for 5 years at 7.1%: 4.107 (as above) - PV = \$0.671 \times 4.107 = \$2.76 \) million --- ### **Final Answers:** #### **1. Value of the firm's debt-generated tax shield:** \[ \boxed{\$9.45 \text{ million}} \] #### **2. United Frypan’s after-tax WACC:** \[ \boxed{14.01\%} \] #### **3. New value of the firm if interest deductibility is repealed after 5 years: