# Calculation of Yield to Maturity (YTM) forco Company's Bonds ## Data: - **Bond maturity:**5 years - **Issue date 1 year ago (remaining maturity = years) - **Coupon rate:** 5.4% - **Par value:** $1,000 - **Semiannual payments:** Yes - **Bond price:** $1,200 --- ## Step 1: Determine the semiannual coupon payment The annual coupon payment is calculated as: \[ \text{Coupon payment} = \text{Par value} \times \text{Coupon rate} \] \[ = \$1,000 \times .054 = \$54 \] Since payments are semiannual: \[ \text{Semiannual coupon} = \frac{\$54}{2} = \$27 \] --- ## Step 2: Determine the number of periods remaining Remaining maturity is 4 years, with 2 payments per year: \[ \text{Total periods} = 4 \text{ years} \times 2 = 8 \text{ periods} \] --- ## Step 3: Set up the bond pricing formula The bond's current price is the present value of future cash flows: \[ P = \sum_{t=1}^{n} \frac{C}{(1 + r/2)^t} + \frac{F}{(1 + r/2)^n} \] Where: - \( P = 120 \) - \( C = 27 \) (semiannual coupon) - \( F = 100 \) (par value) - \( n = 8 \) (periods remaining) - \( r \) = annual YTM (unknown) --- ## Step 4: Solve for YTM The equation becomes: \[ 120 = \sum_{t=1}^{8} \frac{27}{(1 + r/2)^t} + \frac{100}{(1 + r/2)^8} \] This equation is complex to solve algebraically, so typically, **financial calculator or iterative methods** are used. --- ## Step 5: Approximate YTM using trial and error or financial calculator ### Using a financial calculator or Excel's `RATE` function: - **Inputs:** - \( N = 8 \) - \( PV = -120 \) (price paid, negative because it's cash outflow) - \( PMT = 27 \) - \( FV = 100 \) - **Excel formula:** ```excel =RATE(8, 27, -120, 100) * 2 ``` *(Multiply by 2 to annualize the semiannual rate)* ### Approximate calculation: - Trial with \( r/2 = 2.\% \): \[ PV \approx 27 \times \left(\frac{1 - (1 + .02)^{-8}}{.02}\right) + \frac{100}{(1+.02)^8} \approx 1154.75 \] which is less than 120, so YTM is lower. - Trial with \( r/2 = 1.5\% \): \[ PV \approx 27 \times 7.33 + 100 / 1.126 = 27 \times 7.33 + 887.4 \approx 197.91 + 887.4 = 1085.31 \] which is less than 120, so YTM is higher. - Trial with \( r/2 \approx 1.\% \): \[ PV \approx 27 \times 6.87 + 100 / 1.083 = 185.5 + 923.4 = 1108.9 \] Still less than 120; YTM is higher. - Trial with \( r/2 \approx .5\% \): \[ PV \approx 27 \times 6.69 + 100 / 1.041 = 180.8 + 960.4 = 1141.2 \] Getting closer. - Trial with \( r/2 \approx .25\% \): \[ PV \approx 27 \times 6.60 + 100 / 1.021 = 178.2 + 980.4 = 1158.6 \] Close but still below 120. - Trial with \( r/2 \approx .1\% \): \[ PV \approx 27 \times 6.58 + 100 / 1.008 = 177.7 + 992.1 = 1169.8 \] Closer but still below 120. --- ## **Final approximation:** The YTM (annualized) is roughly **around .3% semiannual**, which gives: \[ \text{YTM} \approx .6\% \] **Thus, the approximate annual YTM:** \[ \boxed{ \textbf{YTM} \approx .6\% } \] --- ## **Summary:** - The bond's YTM is approximately **.6% annually**. - This low yield reflects the premium price ($1,200) over the par value ($1,000), indicating the bond's fixed coupon payments are attractive compared to the current market rate. --- **Note:** For precise calculation, use financial calculator or software like Excel for iterative solving.