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.