Let's break down the problem step by step: --- ### **1. Key Facts** - **Original Maturity:** 16 years - **Bond issued:** 2 years ago **Time Remaining (n):** 14 years (16 - 2) - **Coupon Rate:** 6% (semiannual payments) - **Current Price:** 91% of face value - **Tax Rate:** 21% - **Face Value (assume):** $1,000 (standard unless otherwise stated) --- ### **2. Find the Bond's Yield to Maturity (Pretax Cost of Debt)** #### **Cash Flows** - **Coupon Payment:** \( 6\% \times 1,000 = $60 \) per year Since it's semiannual: \( \frac{60}{2} = $30 \) every 6 months - **# of Periods:** \( 14 \) years \(\times 2 = 28 \) periods - **Current Price:** \( 0.91 \times 1,000 = $910 \) --- #### **Yield to Maturity Calculation** The price equation: \[ P = \sum_{t=1}^{28} \frac{30}{(1 + r)^t} + \frac{1000}{(1 + r)^{28}} \] Where \( r \) is the semiannual yield. We need to solve for \( r \): \[ 910 = \sum_{t=1}^{28} \frac{30}{(1 + r)^t} + \frac{1000}{(1 + r)^{28}} \] This requires trial/error or a financial calculator (or Excel's RATE function). Let's use Excel's RATE formula as a guide: \[ \text{RATE}(nper, pmt, pv, fv) \] - nper = 28 - pmt = 30 - pv = -910 - fv = 1000 \[ \text{RATE}(28, 30, -910, 1000) = ? \] Let's estimate by hand: **First, calculate approximate yield:** The bond sells at a discount, so YTM > 6%. - **Annual Coupon:** $60 - **Years to maturity:** 14 - **Price:** $910 **Approximate YTM formula:** \[ YTM \approx \frac{C + \frac{F - P}{n}}{\frac{F + P}{2}} \] Plug in the values: - C = $60 - F = $1,000 - P = $910 - n = 14 \[ YTM \approx \frac{60 + \frac{1,000 - 910}{14}}{\frac{1,000 + 910}{2}} = \frac{60 + \frac{90}{14}}{955} \] \[ \frac{60 + 6.43}{955} = \frac{66.43}{955} = 0.06954 \] So, **6.95%** annual yield (approximate). But since coupons are semiannual, and the bond's price is a bit lower, let's use the semiannual formula. --- #### **Use Financial Calculator/Excel for Precise YTM** Since you want to show all steps: 1. **Semiannual Coupon:** $30 2. **Number of Periods:** 28 3. **Present Value:** -$910 4. **Future Value:** $1,000 \[ \text{RATE}(28, 30, -910, 1000) = r \] Using a financial calculator or Excel (or by trial and error): - Try r = 3.5% per period (i.e., 7% annual): Calculate PV of coupons: \[ PV_{\text{coupons}} = 30 \times \frac{1 - (1 + 0.035)^{-28}}{0.035} = 30 \times 18.828 = 564.84 \] PV of face value: \[ PV_{\text{FV}} = 1,000 \times (1 + 0.035)^{-28} = 1,000 \times 0.3825 = 382.50 \] Total PV = 564.84 + 382.50 = 947.34 (which is higher than $910). Increase r to 4% per period (8% annual): PV coupons: \[ 30 \times \frac{1 - (1 + 0.04)^{-28}}{0.04} = 30 \times 17.292 = 518.76 \] PV FV: \( 1,000 \times (1.04)^{-28} = 1,000 \times 0.335 = 335 \) Total PV = 518.76 + 335 = 853.76 (which is lower than $910). Now interpolate between 3.5% and 4%: \[ YTM = 7\% + \left(\frac{947.34-910}{947.34-853.76}\right) \times (8\%-7\%) = 7\% + \left(\frac{37.34}{93.58}\right) \times 1\% \] \[ = 7\% + 0.399 \times 1\% = 7.399\% \] So **YTM ≈ 7.40%** (annualized, compounded semiannually). **Pretax cost of debt:** **7.40%** --- ### **3. Aftertax Cost of Debt** \[ \text{Aftertax cost of debt} = \text{Pretax cost of debt} \times (1 - \text{tax rate}) \] \[ = 7.40\% \times (1 - 0.21) = 7.40\% \times 0.79 = 5.846\% \] Rounded to two decimal places: **5.85%** --- ### **4. Which is more relevant?** **Aftertax cost of debt** is more relevant, because interest payments are tax-deductible. --- ## **Final Answers** 1. **Pretax cost of debt:** \(\boxed{7.40\%}\) 2. **Aftertax cost of debt:** \(\boxed{5.85\%}\) 3. **Which is more relevant?** \(\boxed{\text{Aftertax cost of debt}}\)