Alpha Inc. which is a publicly traded company uses a mix of debt and equity to finance its operations, with an equity portion of 70%, the rest is debt. Shares of the common stock are currently selling at $120. The bonds of the firm are currently trading at $1200, have an annual coupon rate of 6%, 20 years to maturity and a face value of $1000. The company has a beta of 1.60 and a 35% marginal tax rate. The market risk premium is 5.5% and the risk-free rate is 5%. The cost of equity is 13.8 %.

What is the after-tax cost of debt?

Question 13 options:

2.903%

3.247%

4.467%

4.280%

To calculate the after-tax cost of debt, we first need to compute the pre-tax cost of debt and then adjust it for taxes.

### Step 1: Compute the Pre-tax Cost of Debt
The pre-tax cost of debt is the yield to maturity (YTM) on the firm's bonds. This can be found using the bond's current price, face value, coupon rate, and years to maturity.

Given:
- Annual coupon payment = $ 6\% \times 1000 = 60 $
- Face value = $1,000
- Current price = $1,200
- Years to maturity = 20

We use the following formula for YTM approximation:

\[
\text{YTM} \approx \frac{C + \frac{F - P}{n}}{\frac{F + P}{2}}
\]

Where:
- $ C $ = Annual coupon payment = 60
- $ F $ = Face value of the bond = 1,000
- $ P $ = Current price of the bond = 1,200
- $ n $ = Years to maturity = 20

Substituting the values:

\[
\text{YTM} \approx \frac{60 + \frac{1000 - 1200}{20}}{\frac{1000 + 1200}{2}}
= \frac{60 + \frac{-200}{20}}{\frac{2200}{2}}
= \frac{60 - 10}{1100}
= \frac{50}{1100} \approx 0.04545 \text{ or } 4.545\%
\]

So, the pre-tax cost of debt (YTM) is approximately 4.545%.

### Step 2: Compute the After-tax Cost of Debt
The after-tax cost of debt is adjusted by the tax rate. The formula is:

\[
\text{After-tax cost of debt} = \text{YTM} \times (1 - \text{Tax rate})
\]

Given that the tax rate is 35%, we substitute the values:

\[
\text{After-tax cost of debt} = 4.545\% \times (1 - 0.35) = 4.545\% \times 0.65 \approx 2.954\%
\]

Therefore, the after-tax cost of debt is approximately **2.954%**.

Since this is closest to the provided option **2.903%**, that would be the correct answer.

### Answer: **2.903%**

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Do you want further details or have any questions? Here are 5 related questions:

1. How is the yield to maturity (YTM) on a bond calculated?
2. What is the impact of the tax rate on the cost of debt?
3. Why does a higher bond price lead to a lower yield to maturity?
4. How does a company’s beta affect its cost of equity?
5. Why is the market risk premium important in calculating the cost of equity?

**Tip:** The cost of debt is generally lower than the cost of equity because interest payments on debt are tax-deductible, making debt financing more attractive.

Alpha Inc. which is a publicly traded company uses a mix of debt and equity to finance its operations, with an equity portion of 70%, the rest is debt. Shares of the common stock are currently selling at $120. The bonds of the firm are currently trading at $1200, have an annual coupon rate of 6%, 20 years to maturity and a face value of $1000. The company has a beta of 1.60 and a 35% marginal tax rate. The market risk premium is 5.5% and the risk-free rate is 5%. The cost of equity is 13.8%. What is the after-tax cost of debt?

Learn how to calculate the after-tax cost of debt for Alpha Inc., a company using debt and equity financing. Explore bond pricing, yield to maturity, and tax-adjusted cost of debt using detailed formulas and a step-by-step solution. This is suitable for university-level finance students.

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