Difficulty: Medium
Correct Answer: 6.09%
Explanation:
Introduction / Context: This question tests understanding of the difference between nominal and effective interest rates. When a nominal annual rate is compounded more than once a year, such as half yearly, the effective annual rate becomes slightly higher than the nominal rate. We are given a nominal rate of 6% per annum with half yearly compounding and asked to find the equivalent effective annual rate.
Given Data / Assumptions:
Concept / Approach: For a nominal annual rate R compounded m times per year, the effective annual rate Reff is: Reff = (1 + (R / 100) / m)^m - 1 Here R = 6 and m = 2. So we consider half yearly rate of 3% and compound this twice to obtain the effective annual rate.
Step-by-Step Solution: Nominal annual rate R = 6%. Number of compounding periods in a year m = 2 (half yearly). Periodic rate = 6 / (2 * 100) = 3 / 100 = 0.03. Effective factor for one year: (1 + 0.03)^2. Compute (1.03)^2 = 1.0609. Effective annual rate Reff = 1.0609 - 1 = 0.0609. Convert to percent: Reff = 6.09%.
Verification / Alternative Check: You can imagine investing Rs 100 for one year at 6% nominal with half yearly compounding. After 6 months: amount = 100 * 1.03 = 103. After 12 months: amount = 103 * 1.03 = 106.09. Interest earned = 6.09, so the effective rate is 6.09%, confirming the calculation.
Why Other Options Are Wrong: 6.06%, 6.07%, and 6.08% are all slightly lower than the exact computed effective rate and arise from approximate or truncated calculations. Only 6.09% matches the precise compounding result.
Common Pitfalls: A frequent error is to assume that the effective rate equals the nominal rate and ignore compounding within the year. Some learners incorrectly add 3% twice to get 6% and do not multiply the factors, missing the extra interest on interest.
Final Answer: The effective annual rate of interest is 6.09%.
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