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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