## 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. P is the rate per compounding period where P = R/m.

Effective interest rate per period,

Effective interest rate for t periods,

substituting the very first equation into i in the 2nd equation

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, inject 7% and 12 and 1. If you are getting interest compounded quarterly on your investment, come in 7% and Four 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 comeback will be in Five 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 Five.

Very 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 Five years: it = (1 + i) t – 1

it = (1 + 0.0722901) Five – 1 = 0.417625 = 41.76%

And we would also get it = ( 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 enlargened 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 [Two.7182818284. ] and r is the interest rate in decimal form equal to R/100. So,