# Step-by-Step Solution: Yield to Maturity (YTM) Let's break down the problem: - **Current Price (\(P\))**: $1,047.40 (since 104.74% of $1,000) - **Coupon Rate**: 6.2% (annual payment) - **Coupon Payment (\(C\))**: $62.00 per year - **Par Value (\(F\))**: $1,000 - **Years to Maturity (\(N\))**: 22 The yield to maturity (YTM) is the interest rate (\(r\)) that equates the present value of the bond's future cash flows to its price. ## 1. Write the Bond Price Formula \[ P = \sum_{t=1}^{N} \frac{C}{(1+r)^t} + \frac{F}{(1+r)^N} \] Substitute the values: \[ 1047.40 = \sum_{t=1}^{22} \frac{62}{(1+r)^t} + \frac{100}{(1+r)^{22}} \] ## 2. Set Up for YTM Calculation This equation cannot be solved algebraically for \(r\). We'll use either **trial and error**, **financial calculator**, or **Excel/iterative method**. ### Shortcut: Use Approximate YTM Formula \[ \text{YTM} \approx \frac{C + \frac{F - P}{N}}{\frac{F + P}{2}} \] Plug in the values: - \(C = 62\) - \(F = 100\) - \(P = 1047.40\) - \(N = 22\) \[ \text{YTM} \approx \frac{62 + \frac{100 - 1047.40}{22}}{\frac{100 + 1047.40}{2}} \] \[ = \frac{62 + \frac{-47.40}{22}}{1047.40} \] \[ = \frac{62 - 2.1545}{1047.40} \] \[ = \frac{59.8455}{1047.40} \] \[ \approx .0572 \] This is **5.72%** (approximate). Let's check using the exact formula with trial and error. ## 3. Trial and Error Calculation The bond is selling at a **premium** (\(P > F\)), so YTM < coupon rate. ### Try \( r = .0572 \): Calculate present value: \[ PV = 62 \times \left[\frac{1 - (1+r)^{-22}}{r}\right] + \frac{100}{(1+r)^{22}} \] Plug in \( r = .0572 \): - \( (1 + .0572)^{22} \approx 3.459 \) - \( (1 + .0572)^{-22} \approx 1/3.459 \approx .2889 \) \[ PV = 62 \times \left[\frac{1 - .2889}{.0572}\right] + \frac{100}{3.459} \] \[ = 62 \times \left[\frac{.7111}{.0572}\right] + 288.96 \] \[ = 62 \times 12.4371 + 288.96 \] \[ = 771.10 + 288.96 = 1,060.06 \] This is **higher** than $1,047.40. Let's try a higher rate. ### Try \( r = .06 \): - \( (1.06)^{22} \approx 3.5206 \) - \( (1.06)^{-22} \approx .2841 \) \[ PV = 62 \times \left[\frac{1 - .2841}{.06}\right] + \frac{100}{3.5206} \] \[ = 62 \times \left[\frac{.7159}{.06}\right] + 284.12 \] \[ = 62 \times 11.9317 + 284.12 \] \[ = 739.77 + 284.12 = 1,023.89 \] This is **lower** than $1,047.40. The actual YTM is between 5.72% and 6.00%. ### Interpolate \[ r_1 = .0572, \quad PV_1 = 1,060.06 \] \[ r_2 = .06, \quad PV_2 = 1,023.89 \] \[ P = 1,047.40 \] Linear interpolation: \[ r = r_1 + \frac{(P - PV_1)}{(PV_2 - PV_1)} \times (r_2 - r_1) \] \[ = .0572 + \frac{1047.40 - 106.06}{1023.89 - 106.06} \times (.06 - .0572) \] \[ = .0572 + \frac{-12.66}{-36.17} \times .0028 \] \[ = .0572 + .3502 \times .0028 \] \[ = .0572 + .00098 \] \[ \approx .0582 \] ## **Final Answer** ### Yield to Maturity \[ \boxed{.0582} \] **(rounded to 4 decimal places)** --- ## Summary Table | Parameter | Value | |-------------------|-------------| | Par Value (\(F\)) | $1,000 | | Coupon (\(C\)) | $62 | | Price (\(P\)) | $1,047.40 | | Years (\(N\)) | 22 | | YTM (decimal) | **.0582** | --- ## Note - The exact YTM is best found with a financial calculator or spreadsheet, but interpolation as shown above is typically accepted for exam/assignment purposes. - Enter your answer as **.0582**.