Introduction / Context:
For a fixed nominal annual rate, increasing compounding frequency increases the effective annual yield. Comparing half-yearly (2 times) to quarterly (4 times) over one year highlights the frequency effect while keeping the nominal rate unchanged.
Given Data / Assumptions:
- P = ₹ 800
- Nominal rate = 20% p.a.
- Case H: half-yearly compounding (2 periods at 10% each)
- Case Q: quarterly compounding (4 periods at 5% each)
- Time = 1 year in both cases
Concept / Approach:
Compute amounts separately, then subtract to get the difference in interest. With n compounding periods: A = P (1 + r/n)^(n t). For 1 year, t = 1.
Step-by-Step Solution:
Half-yearly: A_H = 800 * (1 + 0.20/2)^2 = 800 * (1.10)^2 = 800 * 1.21 = 968; CI_H = 168.Quarterly: A_Q = 800 * (1 + 0.20/4)^4 = 800 * (1.05)^4 ≈ 800 * 1.21550625 ≈ 972.405; CI_Q ≈ 172.405.Difference = CI_Q − CI_H ≈ 172.405 − 168 = ₹ 4.405 ≈ ₹ 4.40.
Verification / Alternative check:
Recompute with more decimals to confirm ≈ ₹ 4.40. The sign is positive since quarterly compounds more frequently.
Why Other Options Are Wrong:
- Nil, ₹ 2.50, ₹ 6.60, ₹ 3.20 do not match the computed frequency effect at 20% nominal on ₹ 800 over one year.
Common Pitfalls:
- Using 20% for each subperiod instead of dividing by the number of periods.
- Comparing amounts instead of interest and then forgetting to subtract principal.
Final Answer:
Rs. 4.40.
Discussion & Comments