## Calculate the Effective Interest Rate

The effective interest rate is calculated using the following formula:

\(\text{Effective Interest Rate} = \left(1 + \frac{i}{n}\right)^{nt} – 1 \)

Where:

- i = the nominal interest rate (the stated interest rate)
- n = the number of compounding periods per year (for example, 12 for monthly compounding, 4 for quarterly compounding)
- t = the time the money is invested or borrowed for, in years

This formula accounts for the impact of compounding, which is the process by which interest is added to the principal amount and then itself earns interest.

Remember, this formula is used when interest is compounded. If interest is not compounded, the effective interest rate will be the same as the nominal interest rate.

## Example to Calculate the Effective Interest Rate

Let’s take a specific example to understand how to calculate the effective interest rate.

Let’s say you’re considering a loan that has a nominal interest rate of 5% compounded semi-annually. You want to know the effective annual rate (EAR) or the annual percentage yield (APY).

Firstly, we need to understand the terms in the formula:

- i = nominal interest rate = 5% = 0.05 (when expressed as a decimal)
- n = number of compounding periods per year. Since the interest is compounded semi-annually, n = 2.
- t = time in years. In this case, since we are calculating the effective annual rate, t = 1 year.

Substituting these values into the formula:

\(\text{Effective Interest Rate} = \left(1 + \frac{i}{n}\right)^{nt} – 1 \)

\(= \left(1 + \frac{0.05}{2}\right)^{2*1} – 1 \)

\(= \left(1.025\right)^{2} – 1 \)

\(= 1.050625- 1 \)

\(= \text{0.050625 or 5.0625%} \)

So, for this loan, the effective annual rate is approximately 5.0625%, which is slightly higher than the nominal rate of 5% due to the semi-annual compounding.