TB MC Qu. 60 USA Manufacturing issued...

USA Manufacturing issued 30-year, 7.2 percent semiannual bonds 6 years ago. The bonds currently sell at 101 percent of face value. What is the firm's aftertax cost of debt if the tax rate is 21 percent?

Question

TB MC Qu. 60 USA Manufacturing issued...

USA Manufacturing issued 30-year, 7.2 percent semiannual bonds 6 years ago. The bonds currently sell at 101 percent of face value. What is the firm's aftertax cost of debt if the tax rate is 21 percent?

BlackTom AI · Accepted Answer

We start by restating the problem in our own words and then examine each answer choice in turn, using both the coupon information and the price to gauge the yield.

- The bond is a 30-year issue with a 7.2% annual coupon, semiannual payments. This means a semiannual coupon of 7.2%/2 = 3.6 dollars per 100 face value every six months. The bond was issued 6 years ago, so there are 30 - 6 = 24 years remaining, i.e., 24 × 2 = 48 semiannual periods left until maturity. The current price is 101% of face value, or 101 per 100 face.
- The yield to maturity (YTM) for a standard fixed-rate bond with price near par and with a coupon above/below price is approximated by solving for the rate y in the present-value equation: 101 = 3.6 × [1 − (1 + y)^(-48)] / y + 100 × (1 + y)^(-48). Here y is the semiannual yield. The annualized YTM would be 2y.
- Because the price is above par (101 > 100), the YTM will be slightly less than the annual coupon rate. The annual coupon rate is 7.2%, so we expect the pre-tax YTM to be a bit less than 7.2%, and the after-tax cost of debt will be (1 − tax rate) × pre-tax YTM.

Now evaluate each option for reasonableness:

Option A: 3.76%
- This is far below the coupon rate and far below the price-adjusted YTM we expect for a bond priced at 101 with a 7.2% coupon. A semiannual yield corresponding to an annualized rate around 3.76% would imply a much higher price (above 101) or a much lower coupon than 7.2%, which contradicts the given numbers. This is not plausible as the overall after-tax cost given the coupon and price.

Option B: 4.82%
- An after-tax estimate of 4.82% suggests a pre-tax yield around 4.82% / (1 − 0.21) ≈ 6.10% (if we were to reverse the tax adjustment). While this is closer to the expected range than 3.x% options, it still sits noticeably below the coupon rate adjusted for price. Since the price is only mildly above par (101), the YTM should be fairly close to the coupon rate but a bit lower; 6.1% pre-tax is a bit low for this situation. This is unlikely to be the correct pre-tax yield when the price is 101 and the coupon is 7.2%.

Option C: 5.62%
- A pre-tax yield around 5.62% implies a pre-tax YTM close to 5.62% (annualized) and after tax would be 5.62% × (1 − 0.21) ≈ 4.43%, which does not align with the standard convention of quoting after-tax cost as the after-tax rate on debt. However, if the option list is interpreted as giving the after-tax rate directly, 5.62% would correspond to a pre-tax yield of about 7.11% (since 7.11% × (1 − 0.21) ≈ 5.62%). This would be somewhat higher than expected given a price of 101 and a coupon of 7.2%, making it a plausible but not exact match. Overall, this option is within the realm of possibility when considering the rounding and the relationship between price, coupon, and YTM, and it is the closest to the expected after-tax cost for a bond priced at 101 with a 7.2% coupon.

Option D: 4.40%
- This would correspond to a pre-tax yield around 4.40% / (1 − 0.21) ≈ 5.56%, which is notably lower than the coupon rate and would imply a price well above par if the YTM were that low. The given price of 101 does not support such a low yield on a high coupon bond; therefore this is unlikely.

Option E: 3.59%
- This is very close to Option A and far below the coupon and reasonable YTM range for a bond trading at 101 with a 7.2% coupon. It would imply a pre-tax yield around 4.5% or less, which is inconsistent with a price of 101 and a 7.2% coupon. This is not consistent with the market price provided.

Overall, the most consistent choice with a 7.2% coupon, a price of 101, and a 21% tax rate is the option that yields a pre-tax rate just below 7.2% and an after-tax cost around the mid-5%s range. Among the options, 5.62% stands out as the closest after-tax value that corresponds to the expected pre-tax YTM given the bond’s characteristics and price. The remaining options either imply an implausibly low YTM given the coupon and price, or require inconsistent interpretations of after-tax versus pre-tax values.

TB MC Qu. 60 USA Manufacturing issued... USA Manufacturing issued 30-year, 7.2 percent semiannual bonds 6 years ago. The bonds currently sell at 101 percent of face value. What is the firm's aftertax cost of debt if the tax rate is 21 percent?

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