# Effective interest rate

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The effective interest rate (EIR), effective annual interest rate, annual equivalent rate (AER) or simply effective rate is the percentage of interest on a loan or financial product if compound interest accumulates in periods different than a year. It is the compound interest payable annually in arrears, based on the nominal interest rate. It is used to compare the interest rates between loans with different compounding periods. In a situation where a 10% interest rate is compounded annually, its effective interest rate would also be 10%.

## Calculation

The effective interest rate is calculated as if compounded annually. The effective rate is calculated in the following way, where r is the effective annual rate, i the nominal rate, and n the number of compounding periods per year (for example, 12 for monthly compounding):
    $r\ =\ \left(1+{\frac {i}{n}}\right)^{n}-1$
For example, a nominal interest rate of 6% compounded monthly is equivalent to an effective interest rate of 6.17%. 6% compounded monthly is credited as 6%/12 = 0.005 every month. After one year, the initial capital is increased by the factor (1 + 0.005)$^{12}$ ≈ 1.0617. Note that the yield increases with the frequency of compounding.
When the frequency of compounding is increased up to infinity (as for many processes in nature) the calculation simplifies to:
    $r\ =\ e^{i}-1$
where $e\approx 2.72$ is Euler's mathematical constant.
Effective annual rates at different frequencies of compounding :
| Nominal annual rate | Frequency of compounding | Frequency of compounding | Frequency of compounding | Frequency of compounding | Frequency of compounding |
| Nominal annual rate | Semi-annual | Quarterly | Monthly | Daily | Continuous |
| 1% | 1.003% | 1.004% | 1.005% | 1.005% | 1.005% |
| 5% | 5.063% | 5.095% | 5.116% | 5.127% | 5.127% |
| 10% | 10.250% | 10.381% | 10.471% | 10.516% | 10.517% |
| 15% | 15.563% | 15.865% | 16.075% | 16.180% | 16.183% |
| 20% | 21.000% | 21.551% | 21.939% | 22.134% | 22.140% |
| 30% | 32.250% | 33.547% | 34.489% | 34.969% | 34.986% |
| 40% | 44.000% | 46.410% | 48.213% | 49.150% | 49.182% |
| 50% | 56.250% | 60.181% | 63.209% | 64.816% | 64.872% |

## Comparison with APR

The primary difference between annual percentage rate (APR) and effective interest rate, is that the effective interest rate includes the compounding effect, while APR assumes the payee has paid off all interest on a loan each month. Additionally, the APR method, depending on legal jurisdiction, reflects other factors that may effect the cost of a loan such as including fees that may be charged as a part of a loan. Effective interest is the standard in the European Union and many other countries, while APR is often used in the United States.

## Comparison with APY

Annual percentage yield or effective annual yield is the analogous concept for savings or investments, such as a certificate of deposit. Since a loan by a borrower is an investment for the lender, both terms can apply to the same transaction, depending on the point of view. For a zero-coupon bond such as a US treasury bill, an annual effective discount rate may be specified instead of an effective interest rate, because zero coupon bonds trade at a discount from their face values.

## Effective interest rate (accountancy)

In accountancy, the term effective interest rate is used to describe the rate used to calculate interest expense or income under the effective interest method. This is not the same as the effective annual rate, and is usually stated as an APR rate.