# Step-by-Step Solution for Bond Equivalent and Effective Annual Yield Let's solve for a 10-year bond with: - Par Value (\( F \)) = \$1,000 - Annual Coupon Rate = 8% - Coupon Payment (\( C \)) = 8% of \$1,000 = \$80 - Maturity (\( n \)) = 10 years We need to find the **Bond Equivalent Yield (BEY)** and **Effective Annual Yield (EAY)** for various bond prices. Let's walk through the solution for a generic bond price (\( P \)). You can substitute any specific price. ## 1. Finding the Yield to Maturity (YTM) The price of a bond can be written as: \[ P = \sum_{t=1}^{n} \frac{C}{(1+y)^t} + \frac{F}{(1+y)^n} \] where: - \( P \) = current price of the bond - \( C \) = annual coupon payment (\$80) - \( F \) = face value (\$1,000) - \( n \) = number of years to maturity (10) - \( y \) = yield to maturity (annual, compounded annually) To find \( y \), you typically use a financial calculator or Excel's RATE function. For illustration, let's use a trial-and-error approach (or you can use a financial calculator). ## 2. Bond Equivalent Yield (BEY) For **annual coupon bonds**, the YTM is already the BEY. BEY is mainly used for semiannual coupon bonds, where: \[ BEY = 2 \times \left( \frac{(1+\text{YTM}/2)^{2} - 1}{2} \right) \] But with **annual coupons**, BEY = YTM. ## 3. Effective Annual Yield (EAY) EAY is the true yield accounting for compounding. For annual compounding (as in this case), EAY is the same as YTM. \[ EAY = (1+\text{YTM})^1 - 1 = \text{YTM} \] For semiannual coupons, it would be: \[ EAY = (1 + \frac{\text{BEY}}{2})^2 - 1 \] ## Example Calculation Suppose the bond price \( P \) is \$950. Step 1: Set up the price formula: \[ 950 = \sum_{t=1}^{10} \frac{80}{(1+y)^t} + \frac{100}{(1+y)^{10}} \] Step 2: Estimate \( y \) (YTM) using a financial calculator or Excel: In Excel: \[ \text{=RATE}(n, -C, P, F) \] \[ \text{=RATE}(10, -80, 950, 100) \approx 8.68\% \] Step 3: Since it is an **annual coupon bond**: - BEY = YTM = 8.68% - EAY = YTM = 8.68% ## Final Answer For an **annual coupon bond**, both the **Bond Equivalent Yield** and **Effective Annual Yield** are equal to the **Yield to Maturity** calculated from the bond price. For a \$950 price, the YTM, BEY, and EAY are all approximately **8.68%**. For other prices, repeat the calculation using the same formula and method.