What is the effective annual rate of interest corresponding to a nominal rate of 6% per annum when the interest is payable half-yearly (that is, compounded twice a year)?

Introduction:
This question asks you to convert a nominal annual interest rate with semiannual compounding into an effective annual rate. The nominal rate is 6% per annum, but because interest is compounded twice a year, the actual annual growth in value is slightly more than 6%.

Given Data / Assumptions:

• Nominal rate (R) = 6% per annum.
• Compounding frequency = half yearly, that is two times per year.
• We want the effective annual rate r_eff that describes the actual percentage increase over one full year.

Concept / Approach:
For a nominal annual rate R compounded m times per year, the effective annual rate is found by:
r_eff = (1 + R/(100 * m))^m - 1Here R is in percent, so we divide by 100 * m to get the periodic rate. In this problem, m = 2 and R = 6.

Step-by-Step Solution:
Step 1: Identify the periodic rate per half year: R/(100 * 2) = 6 / 200 = 0.03, which is 3% per half year.Step 2: Write the effective rate formula: r_eff = (1 + 0.03)^2 - 1.Step 3: Calculate (1 + 0.03) = 1.03.Step 4: Square this factor for two periods: (1.03)^2 = 1.03 * 1.03 = 1.0609.Step 5: Subtract 1 to find the effective rate in decimal form: r_eff = 1.0609 - 1 = 0.0609.Step 6: Convert to percentage: r_eff = 0.0609 * 100 = 6.09%.Therefore the effective annual rate of interest is 6.09%.

Verification / Alternative check:
We can imagine investing Rs. 100 at this rate. After six months, the amount becomes 100 * 1.03 = 103. After another six months, it becomes 103 * 1.03 = 106.09. The effective gain in one year is 6.09%, confirming the calculated effective rate.

Why Other Options Are Wrong:
5% is too low and ignores part of the compounding effect. 6% corresponds only to the nominal rate and does not allow for interest on interest within the year. 7% and 6.25% are overestimates and do not match the exact compound factor (1.03)^2. Only 6.09% corresponds to the correct computation.

Common Pitfalls:
Students often confuse nominal and effective rates and may mistakenly assume the effective rate is equal to the nominal rate. Another error is to simply multiply 6% by the number of periods, which is incorrect for compound interest. Always apply r_eff = (1 + R/(100 * m))^m - 1 when converting nominal rates with regular compounding to effective annual rates.

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