# Effective Interest Rate Calculator

## Calculator Use

Calculate the effective interest rate per period given the nominal interest rate per period and the number of compounding intervals per period.

Commonly the
**effective interest rate** is in terms of yearly periods and stated such as the
**effective annual rate,**
**effective annual interest rate, annual equivalent rate (AER),** or
**annual percentage yield (APY)**, however, the formula is in terms of periods which can be any time unit you want.

## Effective Interest Rate Formula

Where r is the interest rate per period in decimal form so R = r * 100 and, i is the effective interest rate in decimal form so I = i * 100. m is the compounding times per period. P is the percent rate per compounding period where P = R/m.

Effective interest rate per period,

Effective interest rate for t periods,

substituting the first equation into i in the second equation

- Period
- commonly a period will be a year but it can be any time interval you want as long as all inputs are consistent.
- Nominal Interest Rate (R)
- is the nominal interest rate or "stated rate" in percent. r = R/100
- Compounding Periods (m)
- is the number of times compounding will occur during a period.
- Continuous Compounding
- is when the frequency of compounding (m) is increased up to infinity. Enter c, C or Continuous for m.
- Effective Interest Rate (i)
- is the effective interest rate, or "effective rate".
- Number of Periods (t)
- enter more than 1 if you want to calculate an effective compounded rate for multiple periods
- Compounded Interest Rate (I)
- when number of periods is greater than 1 this will be the total interest rate for all periods.
- Periodic Interest Rate (P)
- This is the rate per compounding period, such as per month when your period is year and compounding is 12 times per period.

If you have an investment earning a nominal interest rate of 7% per year and you will be getting interest compounded monthly and you want to know effective rate for one year, enter 7% and 12 and 1. If you are getting interest compounded quarterly on your investment, enter 7% and 4 and 1.

## Example Effective Annual Interest Rate Calculation:

Suppose you have an investment account with a "Stated Rate" of 7% compounded monthly then the
**Effective Annual Interest Rate** will be about 7.23%. Further, you want to know what your return will be in 5 years. Using the calculator, your periods are years, nominal rate is 7%, compounding is monthly, 12 times per yearly period, and your number of periods is 5.

First calculating the periodic (yearly) effective rate: i = ( 1 + ( r / m ) )^{m} - 1

i = ( 1 + ( 0.07 / 12 ) )^{12} - 1 = 0.0722901 = 7.22901%

Next calculating the
compounded interest rate of i over 5 years: i_{t} = (1 + i)^{t} - 1

i_{t} = (1 + 0.0722901)^{5} - 1 = 0.417625 = 41.76%

And we would also get i_{t} = ( 1 + ( r / m ) )^{mt} - 1 = 41.76%

### Excel function EFFECT()

This calculation for effective rate is similar to Excel function EFFECT(nominal_rate,npery) where nominal_rate = r and npery = m.

### Continuous Compounding

When the frequency of compounding is increased up to infinity we get "continuous compounding". By definition, as n approaches infinity in the term [ ( 1 + ( r / m ) )^{m} ] the value of this term approaches a limit equal to [ e^{r}].[1] Where
*e* is the constant [2.7182818284....] and r is the interest rate in decimal form equal to R/100. So,

i = e^{r} - 1

## References

[1] Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae, 31st Edition New York, NY: CRC Press, 2003.