Let's break down the problem step by step: ### **Given:** - Original bond maturity: 16 years - Coupon rate: 6% (semiannual) - Issued: 2 years ago (so 14 years left to maturity) - Current price: 91% of face value - Tax rate: 21% --- ## **Step 1: Find the Pretax Cost of Debt (Yield to Maturity)** ### **Bond Details** - Face value (FV): \$1,000 (assumed for calculation) - Coupon payment (C): \( 0.06 \times 1,000 = \$60 \) per year, so $30 every 6 months - Years to maturity: 16 - 2 = 14 years - Periods (n): 14 years × 2 = **28 semiannual periods** - Current price (P): 91% of \$1,000 = \$910 #### **Yield to Maturity (YTM) Equation (Semiannual):** \[ 910 = \sum_{t=1}^{28} \frac{30}{(1 + r)^t} + \frac{1000}{(1 + r)^{28}} \] where \( r \) = semiannual yield #### Let's solve for \( r \) (using a financial calculator or Excel): **In Excel:** \[ \text{=RATE}(28, -30, 910, 1000) \] Let's calculate manually for reasonable accuracy: #### **Trial and Error/Approximation:** Alternatively, use the approximate YTM formula for bonds: \[ YTM \approx \frac{C + \frac{F - P}{n}}{\frac{F + P}{2}} \] Where: - C = annual coupon = \$60 - F = face value = \$1,000 - P = price = \$910 - n = years to maturity = 14 \[ YTM \approx \frac{60 + \frac{1,000 - 910}{14}}{\frac{1,000 + 910}{2}} \] \[ YTM \approx \frac{60 + 6.43}{955} \] \[ YTM \approx \frac{66.43}{955} = 0.0696 = 6.96\% \] But this is **annual**, and the bond pays semiannually, so this is the annualized YTM. But the bond actually pays 28 periods of $30 each, so let's use the **RATE** function for accuracy. #### **Using RATE formula:** - n = 28 - PMT = -30 - PV = 910 - FV = 1000 \[ \text{RATE}(28, -30, 910, 1000) \approx 0.0351 \] So semiannual yield \( r \) = **0.0351** or **3.51%** Annual pretax cost of debt (multiply by 2): \[ 3.51\% \times 2 = 7.02\% \] --- ## **Step 2: Aftertax Cost of Debt** \[ \text{Aftertax Cost} = \text{Pretax} \times (1 - T) \] \[ = 7.02\% \times (1 - 0.21) \] \[ = 7.02\% \times 0.79 = 5.55\% \] --- ## **Step 3: Which is More Relevant?** **Answer:** The **aftertax cost of debt** is more relevant because interest is tax-deductible. --- ## **Final Answers:** ### 1. Pretax cost of debt: \[ \boxed{7.02\%} \] ### 2. Aftertax cost of debt: \[ \boxed{5.55\%} \] ### 3. Which is more relevant? \[ \boxed{\text{Aftertax cost of debt}} \] --- **Summary Table:** | Calculation | Value | |--------------------------|---------| | Pretax cost of debt | 7.02% | | Aftertax cost of debt | 5.55% | | More relevant | Aftertax cost of debt |

### 1. Introduction Understanding the concepts of bond valuation, yield to maturity (YTM), and the tax implications of debt is essential for accurately assessing a company's cost of debt. Bonds are fixed-income securities that obligate the issuer to make periodic interest payments (coupons) and repay the face value at maturity. The yield to maturity represents the internal rate of return (IRR) earned by an investor if the bond is held until maturity, considering current market prices, coupon payments, and face value. The pretax cost of debt reflects the yield before accounting for taxes, which influence the net cost of debt due to the tax deductibility of interest payments. The aftertax cost of debt adjusts the pretax cost to reflect the actual expense after considering the company's tax savings. This distinction is critical because, in financial decision-making, the aftertax cost provides a more accurate measure of the true cost of borrowing. The concepts of bond pricing, YTM calculation, and tax effects are interconnected; understanding how to derive the yield from bond information and adjust it for taxes enables firms and investors to make sound financial decisions. The calculation involves solving for the bond’s yield based on its current price, coupon payments, and maturity, then adjusting that yield for the company’s tax rate to determine the aftertax cost of debt. #### Explanation: This introduction establishes foundational knowledge about bonds, yields, and tax effects, which are crucial to interpreting the problem. It clarifies that the key tasks involve deriving the bond’s yield to maturity from its current price and then adjusting this yield for tax implications. Recognizing these concepts facilitates understanding the subsequent steps and ensures accurate application of formulas in bond valuation and tax adjustments. --- ### 2. Presentation of Relevant Formulas Required To Solve The Question **a) Bond Price Formula (Present Value of Cash Flows):** \[ P = \sum_{t=1}^{n} \frac{C}{(1 + r)^t} + \frac{F}{(1 + r)^n} \] - *Where:* - \( P \) = current bond price - \( C \) = coupon payment per period - \( F \) = face value of the bond - \( r \) = yield per period (semiannual in this case) - \( n \) = total number of periods **Relevance & Rationale:** This formula calculates the present value of all future cash flows (coupon payments and face value repayment). Solving for \( r \) given \( P \), \( C \), \( F \), and \( n \) yields the bond’s yield to maturity (YTM). It is fundamental in bond valuation and necessary to determine the pretax cost of debt. --- **b) Approximate Yield to Maturity (YTM) Formula:** \[ YTM \approx \frac{C + \frac{F - P}{n}}{\frac{F + P}{2}} \] - *Where:* - \( C \) = annual coupon payment - \( F \) = face value - \( P \) = current price - \( n \) = number of periods **Relevance & Rationale:** This approximation provides a quick estimate of YTM when solving the exact formula is complex. It is particularly useful for initial calculations or verification purposes, especially when solving iteratively or using financial calculators. --- **c) Aftertax Cost of Debt Formula:** \[ \text{Aftertax Cost of Debt} = \text{Pretax Cost of Debt} \times (1 - T) \] - *Where:* - \( T \) = corporate tax rate **Relevance & Rationale:** Interest payments are tax-deductible, reducing the effective cost of debt. This formula adjusts the pretax yield to reflect the actual expense to the firm after considering tax savings. It is vital for evaluating the true cost of debt financing in financial analysis. --- ### 3. A Detailed Step-by-Step Solution **Step 1: Determine the bond's remaining maturity and relevant cash flows** - The bond has an original maturity of 16 years and was issued 2 years ago, so remaining maturity is: \[ 16 - 2 = 14 \text{ years} \] - The bond pays semiannual coupons, so total periods: \[ 14 \times 2 = 28 \text{ periods} \] - Coupon payment per period: \[ C = 6\% \times \$1,000 = \$60 \text{ annually} \] \[ \text{Semiannual coupon} = \frac{\$60}{2} = \$30 \] - Current price of the bond: \[ P = 91\% \times \$1,000 = \$910 \] **Explanation:** The remaining maturity and coupon payments define the cash flow schedule necessary for calculating YTM. The semiannual coupons are crucial because the bond's payments occur twice a year, affecting the calculation of periodic yield. --- **Step 2: Calculate the semiannual yield to maturity (r)** - Using the bond price formula: \[ \$910 = \sum_{t=1}^{28} \frac{\$30}{(1 + r)^t} + \frac{\$1,000}{(1 + r)^{28}} \] - To solve for \( r \), utilize a financial calculator or Excel’s RATE function: \[ r = \text{RATE}(n=28, \text{PMT}=-30, \text{PV}=-910, \text{FV}=1000) \] - Computing this yields: \[ r \approx 0.0351 \text{ or } 3.51\% \] **Explanation:** Using the present value formula and a financial calculator allows for precise determination of the semiannual yield, which directly corresponds to the bond’s yield to maturity on a semiannual basis. --- **Step 3: Convert semiannual yield to annual pretax cost of debt** - Since the yield \( r \) is semiannual: \[ \text{Annual pretax YTM} = r \times 2 = 3.51\% \times 2 = 7.02\% \] **Explanation:** Doubling the semiannual yield provides the annualized pretax cost of debt, aligning with standard financial reporting and comparison practices. --- **Step 4: Calculate the aftertax cost of debt** - Given the tax rate: \[ T = 21\% \] - The aftertax cost: \[ \text{Aftertax cost} = 7.02\% \times (1 - 0.21) = 7.02\% \times 0.79 = 5.55\% \] **Explanation:** Adjusting the pretax yield for the tax rate accounts for the tax shield benefits of interest payments, reflecting the actual cost to the company. --- ### **Conclusion** The precise calculation of the bond’s yield to maturity yields a **pretax cost of debt of approximately 7.02%**. Adjusted for taxes, the **aftertax cost of debt is approximately 5.55%**. Since interest expense is tax-deductible, the aftertax cost provides a more relevant measure for financial decision-making, making it the primary figure in evaluating the company's debt financing costs. --- **Final summary:** | Measure | Value | |--------------------------------|---------| | Pretax cost of debt | 7.02% | | Aftertax cost of debt | 5.55% | | More relevant measure | Aftertax cost of debt |