# Calculating the Implied Interest Rate (Yield to Maturity) for a Bond ## Given Data - **Face Value (\(FV\))**: \$1,000 - **Coupon Rate**: 12.8% (annual payments) - **Coupon Payment (\(C\))**: \(.128 \times \$1,000 = \$128\) - **Years to Maturity (\(n\))**: 10 - **Purchase Price (\(P\))**: \$1,195 ## Implied Interest Rate: The Yield to Maturity (YTM) The implied interest rate, or **YTM**, is the rate \(r\) that solves: \[ P = \sum_{t=1}^{n} \frac{C}{(1 + r)^t} + \frac{FV}{(1 + r)^n} \] Plug in the values: \[ 1195 = \sum_{t=1}^{10} \frac{128}{(1 + r)^t} + \frac{100}{(1 + r)^{10}} \] This equation cannot be solved algebraically; an iterative or numerical approach (trial and error, financial calculator, or spreadsheet) is needed. ## Approximate YTM Formula For a quick estimate: \[ YTM \approx \frac{C + \frac{FV - P}{n}}{\frac{FV + P}{2}} \] Plug in the numbers: \[ YTM \approx \frac{128 + \frac{100 - 1195}{10}}{\frac{100 + 1195}{2}} \] \[ YTM \approx \frac{128 - 19.5}{1097.5} \] \[ YTM \approx \frac{108.5}{1097.5} \approx .0989 \text{ or } 9.89\% \] ## More Precise Calculation (Interpolation) Let’s check with two trial rates: - At **8%**: \[ PV = 128 \times \frac{1 - (1 + .08)^{-10}}{.08} + \frac{100}{(1 + .08)^{10}} \] \[ PV \approx 128 \times 6.7101 + 463.19 \approx 859.69 + 463.19 = 1322.88 \] (Above the price) - At **10%**: \[ PV = 128 \times \frac{1 - (1 + .10)^{-10}}{.10} + \frac{100}{(1 + .10)^{10}} \] \[ PV \approx 128 \times 6.1446 + 385.54 \approx 786.51 + 385.54 = 1172.05 \] (Below the price) The actual YTM is between 9% and 10%. Let’s interpolate: \[ YTM \approx 9\% + \frac{1322.88 - 1195}{1322.88 - 1172.05} \times 1\% \] \[ = 9\% + \frac{127.88}{150.83} \times 1\% \approx 9\% + .85\% = 9.85\% \] ## **Final Answer** **The implied interest rate (YTM) of this investment is approximately:** \[ \boxed{9.9\%} \]