To calculate the effective interest rate, remember to account for the compounding frequency. The formula \( \text{Effective Rate} = \left( 1 + \frac{r}{n} \right)^n - 1 \) takes into account how often interest is compounded within a year. The more frequently the interest is compounded (i.e., larger values of \( n \)), the higher the effective rate will be, even if the nominal rate remains constant. Always ensure you substitute the values for \( r \) and \( n \) correctly and simplify step by step.
To calculate the effective interest rate for a nominal rate of \(10\%\) compounded half-yearly, use the effective rate formula:
\(r_{eff} = \left(1 + \frac{r_{nom}}{n}\right)^n - 1\)
Where:
Substitute the values:
\(r_{eff} = \left(1 + \frac{0.10}{2}\right)^2 - 1\)
\(r_{eff} = \left(1 + 0.05\right)^2 - 1\)
\(r_{eff} = (1.05)^2 - 1\)
\(r_{eff} = 1.1025 - 1\)
\(r_{eff} = 0.1025\)
Converting to a percentage:
\(r_{eff} = 10.25\%\)
The effective interest rate is \(10.25\%\).