# Effective annual interest rate

Effective annual interest rate | |
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**The effective annual interest rate** is the annual rate of interest equivalent to a given interest rate. It depends on the nominal interest rate and periods in which interest is capitalized, i.e. on the frequency of capitalization.

The effective annual interest rate is the percentage by which the value of capital is *increased* in *one year*.

## Formula for effective annual interest rate

We calculate the effective annual interest rate from the formula

\(r_{ef}=(1+\frac{r}{m})^{m}-1\)

Where:

- \(r_ef\) - effective annual interest rate
- \(r\) - nominal annual interest rate
- \(m\) - number of capitalization per year (e.g. for semi-annual capitalization m = 2, quarterly m = 4, monthly m = 12)

As it results from the formula, the **effective** annual interest rate **depends** primarily on:

- nominal annual interest rate \(r\)
- the number of capitalization in a year \(m\) - present both in the exponent of power and in the denominator of a fraction

## Conclusions resulting from pattern analysis

At the fixed nominal rate, taking into account the dependence of the effective rate on the annual interest rate, we can formulate the following conclusions:

- the effective rate is equal to the nominal rate only with annual capitalization
- the effective rate is higher than the nominal rate if the capitalization period is shorter than one year
- the effective rate is the higher the more interest is capitalized
- the effective rate is the highest with continuous capitalization.

## References

- Clark, M. W. (1994).
*A note on the calculation of the effective annual rate for cash discounts*. Journal of Accounting Education, 12(1), 77-79. - Thomas, A. L. (1968).
*Estimating the Effective Interest Rate*. The Accounting Review, 43(3), 589-591. - Pournara, C. (2015).
*Effective interest rates: making sense or cents?*. Learning and Teaching Mathematics, 2015(18), 46-50.