# Effective interest rate

(Redirected from Effective annual rate)

The effective interest rate (EIR), effective annual interest rate, annual equivalent rate (AER) or simply effective rate is the interest rate on a loan or financial product restated from the nominal interest rate and expressed as the equivalent interest rate if compound interest was payable annually in arrears.

It is used to compare the interest rates between loans with different compounding periods, such as weekly, monthly, half-yearly or yearly. The effective interest rate sometimes differs in one important respect from the annual percentage rate (APR): the APR method converts this weekly or monthly interest rate into what would be called an annual rate that (in some parts of the world) doesn't take into account the effect of compounding.[1]

By contrast, in the EIR, the periodic rate is annualized using compounding. It is the standard in the European Union and many other countries around the world.

The EIR is precise in financial terms, because it allows for the effects of compounding, i.e. the fact that for each period, interest is not calculated on the principal, but on the amount accumulated at the end of the previous period, including capital and interest. This reasoning is easily understandable when looking at savings: if interest is capitalized every month, then in every month the saver earns interest on the entire sum, including interest from the previous period. Thus if one starts with $1000 and earns interest at 2% every month, the accumulated sum at the end of the year is$1268.24, giving an effective interest rate of about 26.8%, not 24%.

The term nominal EIR or nominal APR can (subject to legislation) be used to refer to an annualized rate that does not take into account front-fees and other costs can be included.

Annual percentage yield or effective annual yield is the analogous concept used for savings or investment products, such as a certificate of deposit. Since any loan is an investment product for the lender, the terms may be used to apply to the same transaction, depending on the point of view.

Effective annual interest or yield may be calculated or applied differently depending on the circumstances, and the definition should be studied carefully. For example, a bank may refer to the yield on a loan portfolio after expected losses as its effective yield and include income from other fees, meaning that the interest paid by each borrower may differ substantially from the bank's effective yield.

## 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):

${\displaystyle r\ =\ \left(1+{\frac {i}{n}}\right)^{n}-1}$

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.

When the frequency of compounding is increased up to infinity the calculation will be:

${\displaystyle r\ =\ e^{i}-1}$

The yield depends on the frequency of compounding:

Effective annual rates at different frequencies of compounding
Nominal
annual rate
Frequency of compounding
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%

The effective interest rate is a special case of the internal rate of return.

If the monthly interest rate j is known and remains constant throughout the year, the effective annual rate can be calculated as follows:

${\displaystyle r\ =\ (1+j)^{12}-1}$

The annual percentage rate (APR) is calculated in the following way, where i is the interest rate for the period and n is the number of periods.

APR = i × n

## 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.