What is the compound interest on ₹10,000 for 1 year at 10% per annum, compounded half-yearly?
₹1,025. Half-yearly compounding means 5% per half-year for 2 periods: 10,000 x 1.05 x 1.05 = ₹11,025, so CI = ₹1,025, which is ₹25 more than yearly compounding.
Explain it simply
When interest is added more often, the interest from the first half-year starts earning its own interest in the second half. The trick is to convert both the rate and the time into the new period. Half-yearly: halve the rate, double the number of periods. Quarterly: divide the rate by 4 and multiply the periods by 4.
Example
Question: P = ₹10,000, R = 10% p.a., T = 1 year, compounded half-yearly.
- Rate per half-year = 10 / 2 = 5%.
- Number of half-years = 1 x 2 = 2.
- Amount = 10,000 x (1.05)^2 = 10,000 x 1.1025 = ₹11,025.
- CI = ₹1,025.
Compare: yearly compounding gives ₹1,000. Quarterly gives 10,000 x (1.025)^4 - 10,000 = ₹1,038.13 (rounded).
const amt = (n) => 10000 * Math.pow(1 + 0.1 / n, n);
console.log(Math.round(amt(2) - 10000)); // Output: 1025
console.log((amt(4) - 10000).toFixed(2)); // Output: 1038.13This is why bank deposits quote an "effective annual yield" slightly above the nominal rate.
What to say in the interview
"For half-yearly compounding I halve the rate and double the periods, so 5% for 2 periods. That gives 11,025, so the interest is 1,025, slightly more than the 1,000 from yearly compounding."
Common mistakes
- Using 10% for 2 periods, which treats it like 2 years.
- Halving the rate but forgetting to double the number of periods.