Question

Semi-annually.

Answer 1

Step 1: For compounding semi-annually, the effective annual rate \( r_{{eff}} \) is calculated using the formula: \[ r_{{eff}} = \left(1 + \frac{r}{n}\right)^n - 1, \] where: - \( r \) is the nominal annual interest rate, - \( n \) is the number of compounding periods per year. 
Step 2: Substitute the given values \( r = 0.10 \) (10%) and \( n = 2 \): \[ r_{{eff}} = \left(1 + \frac{0.10}{2}\right)^2 - 1 = (1.05)^2 - 1. \] 
Step 3: Compute the value: \[ r_{{eff}} = 1.1025 - 1 = 0.1025 \quad {or} \quad 10.25\%. \]

Answer 2

Step 1: The effective annual rate \( r_{{eff}} \) for a given nominal rate \( r \) compounded \( n \) times per year is given by: \[ r_{{eff}} = \left(1 + \frac{r}{n}\right)^n - 1. \] where: - \( r = 0.10 \) (10% nominal rate), - \( n \) is the number of compounding periods per year. 
Step 2: First, compute for semi-annual compounding (\( n = 2 \)): \[ r_{{eff}} = \left(1 + \frac{0.10}{2}\right)^2 - 1 = (1.05)^2 - 1. \] 
Step 3: Compute the value: \[ r_{{eff}} = 1.1025 - 1 = 0.1025 \quad {or} \quad 10.25\%. \] 
Step 4: Now, compute for quarterly compounding (\( n = 4 \)): \[ r_{{eff}} = \left(1 + \frac{0.10}{4}\right)^4 - 1 = (1.025)^4 - 1. \] 
Step 5: Compute the value: \[ r_{{eff}} = 1.1038 - 1 = 0.1038 \quad {or} \quad 10.38\%. \]