What is the effective annual rate of interest corresponding to a nominal rate of 6% per annum when interest is payable half yearly?

Introduction / Context:
This question compares a nominal annual rate with its effective annual rate when interest is compounded more than once per year. A nominal rate of 6% per annum payable half yearly means that the interest is applied twice per year at half the nominal rate. The effective annual rate will be slightly higher than 6% because the interest from the first half year also earns interest during the second half year.

Given Data / Assumptions:

• Nominal rate r = 6% per annum.
• Compounding frequency = twice per year (half yearly).
• We want the effective annual rate R that is equivalent.
• Time horizon for the effective rate is one full year.

Concept / Approach:
When a nominal annual rate r is compounded m times per year, the periodic rate is r / m, and the effective annual rate R is given by R = (1 + r / m) ^ m - 1, expressed in decimal form. We then convert R into a percentage. In this problem, r = 0.06 in decimal and m = 2 for half yearly compounding, so the periodic rate is 0.06 / 2 = 0.03 per half year. We then compute the one year growth factor (1.03) ^ 2 and subtract 1 to find the effective annual rate.

Step-by-Step Solution:
Step 1: Convert the nominal rate to decimal: r = 6% = 0.06. Step 2: Since interest is payable half yearly, the number of compounding periods per year m = 2. Step 3: The rate per half year is r / m = 0.06 / 2 = 0.03. Step 4: The one year growth factor under half yearly compounding is (1 + 0.03) ^ 2 = 1.03 ^ 2. Step 5: Compute 1.03 ^ 2 = 1.0609. Step 6: The effective annual rate R is then R = 1.0609 - 1 = 0.0609. Step 7: Express R as a percentage: R = 6.09% approximately. Step 8: Therefore, the effective annual rate is about 6.09%.

Verification / Alternative check:
We can check reasonableness by comparing with annual compounding at 6%. Under annual compounding, the effective annual rate and nominal rate match, so it would be 6%. When compounding more frequently, such as half yearly, the effective rate should be slightly higher than 6% but not dramatically so. The value 6.09% is just a little above 6%, which fits perfectly with this intuition. Other options such as 7% or 5% would be too far from the nominal rate to be right in a half yearly compounding scenario at this moderate rate.

Why Other Options Are Wrong:
The option 5% is lower than the nominal 6% and would imply that compounding reduces the effective rate, which contradicts basic interest theory. The option 6% assumes that effective rate equals nominal rate without adjusting for compounding frequency, which is incorrect here. The option 7% is significantly higher than the correct effective rate and would require a much higher growth factor than 1.0609. The option 6.50% is still too high and does not match the exact outcome of the calculation using the formula for effective rate with m = 2.

Common Pitfalls:
Students often confuse nominal and effective rates, especially when different compounding intervals are involved. Some may mistakenly use the formula R = r * m or R = r / m directly, which are incorrect for effective rate calculations. Others forget to convert the rate from percentage to decimal before applying the formula, causing numerical errors. The key is to remember that the effective rate is based on the compound growth factor over the full year, not just a linear scaling of the nominal rate.

Final Answer:
The effective annual rate of interest corresponding to a nominal rate of 6% per annum payable half yearly is 6.09%, which is option D.