Galvatron Metals has a bond outstanding with a coupon rate of 6.1 percent and semiannual payments. The bond currently sells for $1,933 and matures in 19 years. The par value is $2,000 and the company's tax rate is 22 percent. What is the company's aftertax cost of debt?

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BlackTom AI · Accepted Answer

We start by clearly restating the problem and the answer choices to anchor our analysis. The question asks for the company's aftertax cost of debt for a bond with a semiannual coupon, given price, maturity, par value, coupon rate, and tax rate. The answer options are: 3.20%, 5.29%, 4.61%, 2.99%, 5.00%.

First, assess the structure of the bond and the pre-tax cost. The bond has a coupon rate of 6.1% on a $2,000 par, so annual coupon is 0.061 × 2000 = $122, or $61 every six months. The bond sells for $1,933 and will mature in 19 years (38 semiannual periods). The tax rate is 22%. The aftertax cost of debt is typically computed as the yield to maturity (YTM) on the pre-tax debt, times (1 − tax rate).

Now, evaluate each option, step by step:

Option 1: 3.20%.
- If 3.20% per half-year were the yield, the annual nominal YTM would be about 6.40% (2 × 3.20%), and the effective annual YTM would be (1.0320)^2 − 1 ≈ 6.54%. This is lower than the coupon rate of 6.10%? Actually it's higher than 6.10%; however, we must check consistency with the given price. A higher yield would imply a price below par; here the price is 1933, which is below 2000, so a yield above 6.10% is plausible. But we need precise alignment with the price. When solving the exact YTM for price 1933 with 38 periods and $61 semiannual coupons, the per-half-year yield comes out around 3.21%, giving an annual nominal of about 6.42%, not exactly 6.40%, and the aftertax cost would be around 6.42% × (1 − 0.22) ≈ 5.01%. Therefore, 3.20% per half-year is slightly low in the precise calculation, leading to an aftertax cost a bit below 5.0%. Conceptually plausible, but the exact calculation nudges us away from this precise figure.

Option 2: 5.29%.
- This would imply a pre-tax YTM somewhat around 6.58% (depending on whether we use nominal or effective). If the pre-tax YTM is this high, the corresponding price would be lower than 1933, not matching the given price. The bond’s price being 1933 suggests a pre-tax YTM in the mid 6% range, not as high as 6.58% nominal, so 5.29% after tax would correspond to too high a pre-tax yield to reconcile with the bond price. Thus this option is unlikely.

Option 3: 4.61%.
- An aftertax cost of 4.61% would correspond to a pre-tax YTM of about 4.61% / (1 − 0.22) ≈ 5.93% pre-tax nominal annually (or roughly 5.93% as a semiannual yield doubled would be ~2.97% per half-year). However, the price calculation with a semiannual yield near 2.97% per half-year would produce a price closer to par than 1933, since the coupon rate is 6.1% and the yield is near the coupon. The mismatch in price indicates this option is inconsistent with the given data.

Option 4: 2.99%.
- A 2.99% per half-year yield would imply a pre-tax annual yield of about 5.98% (nominal) or roughly (1.0299)^2 − 1 ≈ 6.00% effective annual. Using such a yield would produce a price closer to or above par, not as low as 1933, given a 6.1% coupon. Since the actual price is 1933, this option underestimates the required yield, making it inconsistent with the provided price.

Option 5: 5.00%.
- If we interpret this as the aftertax cost of debt, the corresponding pre-tax yield would be about 5.00% / (1 − 0.22) ≈ 6.410%, which is a plausible pre-tax yield given the bond trades at 1933. A per-half-year yield around 3.205% leads to a nominal annual YTM around 6.41% and an effective annual YTM around (1.03205)^2 − 1 ≈ 6.62%. This aligns well with price 1933 for a coupon of 6.1% on 2000 par with 38 periods left, making 5.00% a consistent aftertax cost.

In summary, the key factors are: the bond’s price below par implies a pre-tax yield above the 6.1% coupon, and applying the tax shield (1 − tax rate) of 0.78 to that pre-tax yield yields the aftertax cost. Among the options, 5.00% matches the aftertax cost that corresponds to the calculated pre-tax YTM necessary to justify the price 1933, when considering semiannual coupon payments and the 22% tax rate.

Galvatron Metals has a bond outstanding with a coupon rate of 6.1 percent and semiannual payments. The bond currently sells for $1,933 and matures in 19 years. The par value is $2,000 and the company's tax rate is 22 percent. What is the company's aftertax cost of debt?

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