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.
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 ]. Where e is the constant [Two.7182818284. ] and r is the interest rate in decimal form equal to R/100. So,