Interest Compounding - Responses and Explanations
In a comparison of compound interest on a principal of Rs. 10,000 at an annual interest rate of 10%, over a period of 3.5 years, the results show a significant difference when interest is compounded yearly versus half-yearly.
For yearly compounding (n=1), the final amount is calculated as follows:
\[ A = 10,000 \times (1.10)^{3.5} \]
Calculating \((1.10)^{3.5}\) gives us approximately 1.396, resulting in a final amount of Rs. 13,960. The compound interest for this case is Rs. 3,960 (Final Amount - Principal).
For half-yearly compounding (n=2), the interest rate per period is reduced to 5% (10%/2), and the total number of compounding periods is \(2 \times 3.5 = 7\). The final amount and compound interest are calculated as follows:
\[ A = 10,000 \times (1.05)^{7} \]
Calculating \((1.05)^7\) gives us approximately 1.4071, resulting in a final amount of Rs. 14,071. The compound interest for this case is Rs. 4,071.
The difference in compound interest between the two cases is Rs. 111 (Rs. 4,071 - Rs. 3,960). This difference arises due to the more frequent compounding of interest in the half-yearly compounding scenario, earning "interest on interest" more often.
Here's a summary of the results:
| Compounding Frequency | Final Amount (Rs.) | Compound Interest (Rs.) | |----------------------|--------------------|------------------------| | Yearly (n=1) | 13,960 | 3,960 | | Half-Yearly (n=2) | 14,071 | 4,071 |
In conclusion, the compound interest is Rs. 111 higher when interest is compounded half-yearly compared to yearly for the same principal, rate, and time period. This difference emphasises the importance of considering the compounding frequency when calculating compound interest.
Engaging in personal-finance planning, one might consider the impact of compounding frequency on investing in a scenario involving math. For instance, if an individual invests Rs. 10,000 at a 10% annual interest rate, they could see a difference in compound interest when comparing yearly and half-yearly compounding. Specifically, half-yearly compounding yields Rs. 111 more in compound interest over the same period compared to yearly compounding, demonstrating the importance of understanding the compounding frequency in finance.
