lonvert an interest rate of \( 10 \% \) p.a. compounded quarterly to an annual effecti- terest rate.

To convert a nominal interest rate of **10% per annum compounded quarterly** to an **annual effective interest rate**, you can use the following formula: \[ \text{Effective Annual Rate (EAR)} = \left(1 + \frac{r}{m}\right)^m - 1 \] where: - \( r \) is the nominal annual interest rate (in decimal form), - \( m \) is the number of compounding periods per year. Given: - \( r = 10\% = 0.10 \) - \( m = 4 \) (since interest is compounded quarterly) **Step-by-Step Calculation:** 1. **Divide the annual rate by the number of compounding periods:** \[ \frac{r}{m} = \frac{0.10}{4} = 0.025 \] 2. **Add 1 to this value:** \[ 1 + 0.025 = 1.025 \] 3. **Raise this to the power of the number of compounding periods:** \[ 1.025^4 \] 4. **Calculate \( 1.025^4 \):** \[ 1.025^4 \approx 1.10381289 \] 5. **Subtract 1 from the result to find the EAR:** \[ 1.10381289 - 1 = 0.10381289 \] 6. **Convert to percentage:** \[ 0.10381289 \times 100 \approx 10.38\% \] **Conclusion:** The **annual effective interest rate** corresponding to a **10% nominal rate compounded quarterly** is approximately **10.38%**. **Final Answer:** The annual effective interest rate is approximately 10.38 %.