Which of the following schemes of computing interest yields the maximum interest for a year?
Interest compounded monthly at 2% per month.
Comparing Interest Schemes for Maximum Yield
To determine which scheme yields the maximum interest over a year, we calculate the Effective Annual Rate (EAR) for each option. The EAR represents the actual annual rate of return taking compounding into account.
Calculating Effective Annual Rate (EAR)
The formula for EAR is:
$ \text{EAR} = \left(1 + \frac{r}{n}\right)^n - 1 $
Where:
- r is the nominal annual interest rate.
- n is the number of times interest is compounded per year.
Alternatively, if the rate per compounding period (i) is given, and the number of periods per year is n:
$ \text{EAR} = (1 + i)^n - 1 $
Option 1: Annual Compounding
Rate = 24% per year (compounded annually).
$ \text{EAR} = (1 + 0.24)^1 - 1 = 0.24 $
EAR = 24.00%
Option 2: Monthly Compounding
Rate = 2% per month.
Number of compounding periods per year (n) = 12.
Rate per period (i) = 0.02.
$ \text{EAR} = (1 + 0.02)^{12} - 1 \approx 1.26825 - 1 $
EAR ≈ 26.83%
Option 3: Quarterly Compounding
Rate = 6% per quarter.
Number of compounding periods per year (n) = 4.
Rate per period (i) = 0.06.
$ \text{EAR} = (1 + 0.06)^4 - 1 \approx 1.26248 - 1 $
EAR ≈ 26.25%
Option 4: Semi-annual Compounding
Rate = 12% per 6 months.
Number of compounding periods per year (n) = 2.
Rate per period (i) = 0.12.
$ \text{EAR} = (1 + 0.12)^2 - 1 = 1.2544 - 1 $
EAR = 25.44%
Comparison of EARs
Comparing the calculated EARs:
| Scheme | Compounding Frequency | Rate per Period | Effective Annual Rate (EAR) |
| 1 | Annually | 24% | 24.00% |
| 2 | Monthly | 2% | ≈ 26.83% |
| 3 | Quarterly | 6% | ≈ 26.25% |
| 4 | Semi-annually | 12% | 25.44% |
Conclusion
The scheme with the highest Effective Annual Rate (EAR) yields the maximum interest. Based on the calculations, Option 2 (Interest compounded monthly at 2% per month) provides the highest EAR of approximately 26.83%.
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