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.
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,
\( i = (1 + \dfracEffective interest rate for t periods,
\( i_t = (1 + i)^substituting the first equation into i in the second equation
\( i_t = \left(1 + \dfracPeriod 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.
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: it = (1 + i) t - 1
it = (1 + 0.0722901) 5 - 1 = 0.417625 = 41.76%
And we would also get it = ( 1 + ( r / m ) ) mt - 1 = 41.76%
This calculation for effective rate is similar to Excel function EFFECT(nominal_rate,npery) where nominal_rate = r and npery = m.
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,
[1] Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae, 31st Edition New York, NY: CRC Press, 2003.