We will solve this step by step using Modigliani-Miller Proposition 2 (MM Proposition II) and the formula for the Weighted Average Cost of Capital (WACC).

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Step 1: Define the given values

• Cost of equity before refinancing ($r_E$) = 14% or 0.14
• Proportion of debt ($D/(D+E)$) = 45% or 0.45
• Proportion of equity ($E/(D+E)$) = 55% or 0.55
• Cost of debt ($r_D$) = 9.5% or 0.095
• Corporate tax rate ($T_c$) = 40% or 0.40

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Step 2: Use MM Proposition II to calculate the new cost of equity

The formula for the cost of equity with leverage is:

$r_E = r_U + \frac{D}{E} \cdot (r_U - r_D) \cdot (1 - T_c)$

Where:

• $r_U$ is the unlevered cost of capital (equal to the original cost of equity before leveraging, $r_E = 14\%$).
• $\frac{D}{E}$ is the debt-to-equity ratio.
• $r_D$ is the cost of debt.
• $T_c$ is the corporate tax rate.

Step 2.1: Calculate $\frac{D}{E}$:

$\frac{D}{E} = \frac{0.45}{0.55} = 0.8182$

Step 2.2: Plug values into the MM Proposition II formula:

$r_E = 0.14 + (0.8182 \cdot (0.14 - 0.095) \cdot (1 - 0.40))$

Step 2.3: Simplify step-by-step:

• $r_U - r_D = 0.14 - 0.095 = 0.045$
• $(1 - T_c) = 1 - 0.40 = 0.60$
• $0.8182 \cdot 0.045 \cdot 0.60 = 0.022045$

So: $r_E = 0.14 + 0.022045 = 0.162045$

Final $r_E$ result:

$r_E = 16.20\%$

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Step 3: Calculate the after-tax Weighted Average Cost of Capital (WACC)

The formula for WACC is: $WACC = \left( \frac{D}{D+E} \cdot r_D \cdot (1 - T_c) \right) + \left( \frac{E}{D+E} \cdot r_E \right)$

Step 3.1: Plug in the values:

$WACC = (0.45 \cdot 0.095 \cdot (1 - 0.40)) + (0.55 \cdot 0.162045)$

Step 3.2: Simplify step-by-step:

• $0.45 \cdot 0.095 = 0.04275$
• $0.04275 \cdot 0.60 = 0.02565$ (after-tax cost of debt)
• $0.55 \cdot 0.162045 = 0.089125$ (weighted cost of equity)

$WACC = 0.02565 + 0.089125 = 0.114775$

Final WACC result:

$WACC = 11.48\%$

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Final Answers:

1. Return on equity ($r_E$) = 16.20%
2. After-tax WACC = 11.48%

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Let me know if you'd like further explanation or help!
Here are related questions for further practice:

1. How does the corporate tax rate affect the WACC?
2. Why does the cost of equity increase with leverage?
3. Can WACC be higher than the unlevered cost of equity?
4. What is the significance of MM Proposition II in finance?
5. How does the proportion of debt and equity impact the WACC?

Tip: Always recalculate intermediate steps to ensure accuracy when dealing with percentages and leverage ratios!