We can break this problem into two parts:

Part A: Finding the Pretax Cost of Debt

We are given:

• Debt-Equity Ratio $\text{D/E} = 0.65$
• Weighted Average Cost of Capital (WACC) $WACC = 10.4\%$
• Tax Rate $T = 23\%$
• Cost of Equity $r_E = 14\%$

The WACC formula is:

$WACC = \left(\frac{E}{V}\right) r_E + \left(\frac{D}{V}\right) r_D (1 - T)$

Where:

• $E$ is the market value of equity,
• $D$ is the market value of debt,
• $V = E + D$ is the total value of the firm (equity + debt),
• $r_E$ is the cost of equity,
• $r_D$ is the pretax cost of debt,
• $T$ is the tax rate.

To express the debt-equity ratio in terms of $D$ and $E$:

$\frac{D}{E} = 0.65 \implies \frac{D}{V} = \frac{0.65}{1 + 0.65} = \frac{0.65}{1.65} = 0.3939$ $\frac{E}{V} = \frac{1}{1.65} = 0.6061$

Now, substitute into the WACC formula:

$10.4\% = (0.6061)(14\%) + (0.3939)(r_D)(1 - 0.23)$

$10.4\% = 8.49\% + (0.3939)(r_D)(0.77)$

$1.91\% = (0.3939)(r_D)(0.77)$

$r_D = \frac{1.91\%}{0.3939 \times 0.77} = \frac{1.91\%}{0.3033} \approx 6.30\%$

So, the pretax cost of debt is approximately 6.30%.

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Part B: Finding the Cost of Equity

Here, we are given:

• After-tax cost of debt $r_D(1 - T) = 5.8\%$
• Tax Rate $T = 23\%$

To find the pretax cost of debt $r_D$, we use:

$r_D(1 - T) = 5.8\% \implies r_D = \frac{5.8\%}{1 - 0.23} = \frac{5.8\%}{0.77} \approx 7.53\%$

Now, substitute $r_D = 7.53\%$ into the WACC formula:

$10.4\% = (0.6061)(r_E) + (0.3939)(7.53\%)(0.77)$

$10.4\% = (0.6061)(r_E) + 2.28\%$

$8.12\% = (0.6061)(r_E)$

$r_E = \frac{8.12\%}{0.6061} \approx 13.40\%$

So, the cost of equity is approximately 13.40%.

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Do you want further details or have any questions?

Here are 5 related questions:

1. How does the WACC change with an increase in the debt-equity ratio?
2. Why does the after-tax cost of debt matter in the WACC calculation?
3. How would the company’s WACC be affected if the tax rate changes?
4. What assumptions are made in using the WACC formula?
5. How does the cost of equity typically compare to the cost of debt?

Tip: Remember, the cost of debt is typically lower than the cost of equity because debt holders face less risk than equity holders.