Let's solve each part step by step. --- ### **Given Data** - **Original Maturity:** 16 years - **Time passed:** 2 years - **Time remaining:** 14 years (16 - 2) - **Coupon rate:** 6% (semiannual) - **Face value (par):** Assume $1,000 - **Current price:** 91% of face value = $910 - **Tax rate:** 21% --- ### **1. Pretax Cost of Debt (Yield to Maturity, YTM)** #### **Step 1: Cash Flows** - **Coupon Payment:** 6% annual on $1,000 = $60 per year Since semiannual: $60 / 2 = $30 every 6 months - **Periods:** 14 years × 2 = 28 periods - **Face value paid at end:** $1,000 #### **Step 2: YTM Calculation** The bond price formula: \[ P = \sum_{t=1}^{N} \frac{C}{(1 + r)^t} + \frac{F}{(1 + r)^N} \] Where: - \( P = $910 \) - \( C = $30 \) - \( F = $1,000 \) - \( N = 28 \) - \( r = \) periodic YTM (semiannual) \[ 910 = \sum_{t=1}^{28} \frac{30}{(1 + r)^t} + \frac{1000}{(1 + r)^{28}} \] This requires either a financial calculator or Excel. Let's estimate: #### **Use Excel/Yield Formula** You can use the **RATE** function in Excel: \[ \text{RATE}(nper, pmt, -pv, fv) \] - nper = 28 - pmt = 30 - pv = -910 - fv = 1000 Let's estimate manually: - Try \( r = 3.5\% \) per period (7% annual). - Try \( r = 3.75\% \) per period (7.5% annual). But let's interpolate: #### **Quick Approximate Calculation:** The current yield is: \[ \text{Current Yield} = \frac{60}{910} \approx 6.59\% \] But YTM will be higher because the bond is sold at a discount. Let's use a YTM calculator (or formula): \[ YTM \approx \frac{C + \frac{F - P}{N}}{\frac{F + P}{2}} \] \[ YTM \approx \frac{60 + \frac{1,000 - 910}{14}}{\frac{1,000 + 910}{2}} \] \[ YTM \approx \frac{60 + \frac{90}{14}}{955} \] \[ \frac{90}{14} \approx 6.43 \] \[ YTM \approx \frac{60 + 6.43}{955} \approx \frac{66.43}{955} \approx 0.0696 \text{ or } 6.96\% \] This is a **simple yield**, but since interest is paid semiannually, let's adjust it: Let's estimate the semiannual yield (\( r \)), and double it for the annual yield. #### **Try r = 3.5% semiannual (7% annual):** \[ PV = 30 \times \left(1 - (1 + 0.035)^{-28}\right)/0.035 + 1000 \times (1 + 0.035)^{-28} \] \[ = 30 \times (1 - 0.375)/0.035 + 1000 \times 0.375 \] \[ = 30 \times 0.625/0.035 + 375 \] \[ = 18.75 / 0.035 + 375 = 535.71 + 375 = 910.71 \] This is extremely close to the price of $910. So the semiannual rate is **3.5%**, which means the annual YTM is: \[ YTM = 3.5\% \times 2 = 7.00\% \] #### **Rounded to 2 decimal places:** **Pretax cost of debt = 7.00%** --- ### **2. Aftertax Cost of Debt** \[ \text{Aftertax cost of debt} = YTM \times (1 - \text{tax rate}) \] \[ = 7.00\% \times (1 - 0.21) = 7.00\% \times 0.79 = 5.53\% \] --- ### **3. Which is more relevant?** The **aftertax cost of debt** is more relevant, because interest is tax-deductible and the true cost to the company is after taxes. --- ## **Summary Table** | Calculation | Value | |--------------------------|------------| | Pretax cost of debt | 7.00% | | Aftertax cost of debt | 5.53% | | More relevant | Aftertax cost of debt | --- ### **Multiple Choice Answer** **Aftertax cost of debt**