Step 1: The effective annual rate \( r_{{eff}} \) for semi-annual compounding is determined by the formula: \[ r_{{eff}} = \left(1 + \frac{r}{n}\right)^n - 1, \] where \( r \) is the nominal annual interest rate and \( n \) represents the number of compounding periods annually.
Step 2: Input the values \( r = 0.10 \) (10%) and \( n = 2 \) into the formula: \[ r_{{eff}} = \left(1 + \frac{0.10}{2}\right)^2 - 1 = (1.05)^2 - 1. \]
Step 3: Calculate the result: \[ r_{{eff}} = 1.1025 - 1 = 0.1025 \quad {or} \quad 10.25\%. \]