Let's break down the problem step-by-step: **Given:** - Current Price (\(P\)) = $1,297.58 - Coupon Rate = 10% annual - Coupons paid **semiannually**, so each payment = \( \frac{10\% \times 1,000}{2} = \$50 \) - Par Value (\(FV\)) = $1,000 - Number of years = 10, so total periods (\(N\)) = 10 × 2 = 20 - We are to find the **yield to maturity (YTM)**, annualized. --- ### The Bond Price Formula \[ P = \sum_{t=1}^{N} \frac{C}{(1 + r)^t} + \frac{FV}{(1 + r)^N} \] Where: - \(C\) = coupon payment per period = $50 - \(N\) = number of periods = 20 - \(r\) = periodic yield (per 6 months) - \(FV\) = face value = $1,000 - \(P\) = price = $1,297.58 --- ### Plug in Values \[ 1,297.58 = \sum_{t=1}^{20} \frac{50}{(1 + r)^t} + \frac{1,000}{(1 + r)^{20}} \] --- ### Solving for \(r\) (periodic rate): This equation can't be solved algebraically, so let's use **trial and error (or a financial calculator)**: #### Let's try a few values: #### If r = 3% (per period, so annual = 6%): \[ (1 + 0.03)^{20} = (1.03)^{20} \approx 1.8061 \] \[ \text{PV of coupons} = 50 \times \frac{1 - (1.03)^{-20}}{0.03} \] \[ = 50 \times \frac{1 - 1/1.8061}{0.03} = 50 \times \frac{1 - 0.5536}{0.03} = 50 \times \frac{0.4464}{0.03} = 50 \times 14.879 = \$743.96 \] \[ \text{PV of face value} = \frac{1,000}{1.8061} = \$553.68 \] Total PV = \$743.96 + \$553.68 = **\$1,297.64** This is almost exactly the bond price, so the semiannual rate is **3%**. --- **Annual yield to maturity = 2 × 3% = 6%** --- ### **Final Answer** \[ \boxed{6\%} \] --- ### **Summary Table** | Choice | Calculation | |--------|-------------| | 8% | Too high | | 6% | **Correct** | | 4% | Too low | | 2% | Too low | | 1% | Too low |