Which of these gives the lowest effective rate of interest

12.35% compounded annually

11.9% compounded semi-annually

11.6% compounded monthly

12.2% compounded quarterly

To find the lowest effective rate of interest, we need to compute the **Effective Annual Rate (EAR)** for each option using the formula:

\[
\text{EAR} = \left(1 + \frac{r}{n}\right)^n - 1
\]

Where:
- \( r \) is the nominal interest rate (as a decimal),
- \( n \) is the number of compounding periods per year.

Let's calculate the EAR for each option.

### 1. 12.35% compounded annually:
\[
\text{EAR} = \left(1 + \frac{0.1235}{1}\right)^1 - 1 = 0.1235 = 12.35\%
\]

### 2. 11.9% compounded semi-annually:
\[
\text{EAR} = \left(1 + \frac{0.119}{2}\right)^2 - 1 = \left(1 + 0.0595\right)^2 - 1 = 0.12230225 = 12.23\%
\]

### 3. 11.6% compounded monthly:
\[
\text{EAR} = \left(1 + \frac{0.116}{12}\right)^{12} - 1 = \left(1 + 0.0096667\right)^{12} - 1 = 0.12261067 = 12.26\%
\]

### 4. 12.2% compounded quarterly:
\[
\text{EAR} = \left(1 + \frac{0.122}{4}\right)^4 - 1 = \left(1 + 0.0305\right)^4 - 1 = 0.12550801 = 12.55\%
\]

### Comparison:
- 12.35% compounded annually: 12.35%
- 11.9% compounded semi-annually: 12.23%
- 11.6% compounded monthly: 12.26%
- 12.2% compounded quarterly: 12.55%

The **lowest effective rate** is **11.9% compounded semi-annually** with an EAR of **12.23%**.

Would you like more details on any part of the calculations or have any questions?

Here are 5 related questions for practice:
1. How does increasing the frequency of compounding affect the EAR?
2. What is the EAR for a nominal interest rate of 10% compounded daily?
3. How would the EAR change if the nominal rate was 11.9% compounded monthly?
4. What is the nominal rate if the EAR is 10% compounded quarterly?
5. Can the nominal interest rate ever be higher than the EAR?

**Tip:** The more frequent the compounding, the higher the effective annual rate, even if the nominal interest rate remains the same.

Which of these gives the lowest effective rate of interest: 12.35% compounded annually, 11.9% compounded semi-annually, 11.6% compounded monthly, or 12.2% compounded quarterly?

Learn how to calculate the lowest effective interest rate (EAR) between 12.35% compounded annually, 11.9% compounded semi-annually, 11.6% compounded monthly, and 12.2% compounded quarterly. Detailed step-by-step solution with examples of EAR calculation using the formula (1 + r/n)^n - 1. Suitable for Grades 10-12 students.

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