Find the effective rate which is equivalent to a normal rate of 10% p.a. compounded:
Given:
\[ (1.05)^2 = 1.1025, \quad (1.025)^4 = 1.1038. \]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: \[ r_{{eff}} = 1.1025 - 1 = 0.1025 \quad {or} \quad 10.25\%. \]
Step 1: The formula for the effective annual rate \( r_{{eff}} \) from a nominal rate \( r \) compounded \( n \) times per year is: \[ r_{{eff}} = \left(1 + \frac{r}{n}\right)^n - 1. \] Here, \( r = 0.10 \) (10% nominal rate), and \( n \) represents the number of compounding periods annually.
Step 2: Calculate for semi-annual compounding (\( n = 2 \)): \[ r_{{eff}} = \left(1 + \frac{0.10}{2}\right)^2 - 1 = (1.05)^2 - 1. \]
Step 3: Determine the value: \[ r_{{eff}} = 1.1025 - 1 = 0.1025 \quad {or} \quad 10.25\%. \]
Step 4: Now, calculate for quarterly compounding (\( n = 4 \)): \[ r_{{eff}} = \left(1 + \frac{0.10}{4}\right)^4 - 1 = (1.025)^4 - 1. \]
Step 5: Determine the value: \[ r_{{eff}} = 1.1038 - 1 = 0.1038 \quad {or} \quad 10.38\%. \]