Watanabe, Incorporated, is trying to determine its cost of debt. The firm has a debt issue outstanding with 19 years to maturity that is quoted at 97 percent of face value. The issue makes semiannual payments and has an annual coupon rate of 7 percent. If the tax rate is 23 percent, what is the aftertax cost of debt?Note: Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.Blank.xlsx

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BlackTom AI · Accepted Answer

We need to determine the aftertax cost of debt for a bond with these given features.

First, restate the key data: the bond has 19 years to maturity, is priced at 97% of face value, pays semiannual coupons, and has an annual coupon rate of 7%. The corporate tax rate is 23%. The aftertax cost of debt is typically calculated as the yield to maturity (YTM) on the debt, multiplied by (1 − tax rate).

Step 1: Identify cash flows and pricing convention. Since the bond pays semiannually, the coupon per half-year is (0.07 × 100) / 2 = 3.50 dollars. There are 19 × 2 = 38 half-year periods. The bond’s price is 97% of face value, i.e., 97.

Step 2: Set up the present value equation for the yield per half-year. Let i be the yield per half-year. The price equals the present value of coupons plus the principal:
97 = 3.50 × [1 − (1 + i)^−38] / i + 100 × (1 + i)^−38.
This is the standard bond pricing formula with semiannual compounding.

Step 3: Solve for i (the semiannual yield). This requires either a financial calculator or an iterative numerical method. The result gives the yield per half-year.

Step 4: Convert to an annual yield. The nominal annual yield with semiannual compounding is YTM_annual = 2 × i.

Step 5: Apply tax adjustment. The aftertax cost of debt is YTM_aftertax = YTM_annual × (1 − tax_rate) = YTM_annual × 0.77.

Step 6: Numerical approximation. A quick approximation can be done by comparing the coupon rate to the price: the bond pays 7% annual coupon while trading at 97, implying a yield slightly above 7% due to the discount. A rough estimate for YTM_annual is around 7.2%. Applying the tax shield: 7.2% × 0.77 ≈ 5.54%. A more precise calculation (solving the PV equation for i and converting to annual) yields a value around 5.62%.

Step 7: Finalizing. The calculated aftertax cost of debt, based on the given price, coupon, maturity, and tax rate, is approximately 5.62% when rounded to two decimals, as required. Remember not to round intermediate steps during the calculation; the final value should be presented to two decimals.

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