Answer: 9.76%

## Explanation To solve this problem, we need to convert an effective annual rate (EAR) of 10% to a semiannual bond equivalent yield (BEY). ### Step 1: Understanding the relationships 1. **Effective Annual Rate (EAR)**: The actual annual return accounting for compounding within the year. 2. **Quarterly compounding**: The bond pays interest quarterly, so the quarterly rate compounds 4 times per year. 3. **Semiannual Bond Equivalent Yield (BEY)**: A yield quoted on a semiannual basis, which is simply double the semiannual periodic rate. ### Step 2: Convert EAR to quarterly periodic rate The relationship between EAR and quarterly rate is: \[ \text{EAR} = (1 + r_q)^4 - 1 \] Where \( r_q \) is the quarterly periodic rate. Given EAR = 10% = 0.10: \[ 1.10 = (1 + r_q)^4 \] \[ 1 + r_q = (1.10)^{1/4} \] \[ 1 + r_q = 1.10^{0.25} \] \[ 1 + r_q \approx 1.024113 \] \[ r_q \approx 0.024113 \] So the quarterly periodic rate is approximately 2.4113%. ### Step 3: Convert quarterly rate to semiannual rate A semiannual period consists of 2 quarterly periods. The relationship is: \[ 1 + r_s = (1 + r_q)^2 \] Where \( r_s \) is the semiannual periodic rate. \[ 1 + r_s = (1.024113)^2 \] \[ 1 + r_s \approx 1.048809 \] \[ r_s \approx 0.048809 \] So the semiannual periodic rate is approximately 4.8809%. ### Step 4: Calculate the semiannual bond equivalent yield The semiannual BEY is simply: \[ \text{BEY} = 2 \times r_s \] \[ \text{BEY} = 2 \times 0.048809 \] \[ \text{BEY} \approx 0.097618 \] \[ \text{BEY} \approx 9.7618\% \] ### Step 5: Compare with options - **A. 9.76%** - This matches our calculated value of 9.7618% - **B. 10.13%** - This would be incorrect - **C. 10.25%** - This would be the nominal annual rate with quarterly compounding (4 × 2.4113% = 9.645%, not 10.25%) ### Alternative calculation method: We can also solve this directly: \[ \text{BEY} = 2 \times [(1 + \text{EAR})^{1/4} - 1]^2 \] \[ \text{BEY} = 2 \times [(1.10)^{0.25} - 1]^2 \] \[ \text{BEY} = 2 \times [1.024113 - 1]^2 \] \[ \text{BEY} = 2 \times (0.024113)^2 \] \[ \text{BEY} = 2 \times 0.000581 \] \[ \text{BEY} \approx 0.097618 \] **Therefore, the correct answer is A. 9.76%.** ### Key Concept: The bond equivalent yield (BEY) is a convention used in bond markets where yields are quoted on a semiannual basis, even for bonds with different payment frequencies. It allows for easier comparison between bonds with different payment schedules.

Author: LeetQuiz .