Calculate the nominal interest rate per period given the effective interest rate per period and the number of compounding intervals per period. Also calculates the interest rate per compounding interval.
Where i = I/100 and r = R/100; nominal interest rate per period, r = m × [ ( 1 + i)1/m - 1 ]. Effective interest rate for t periods, it = ( 1 + i )t - 1. The rate per compounding period P = R / m, in percent. Periods which can be any time unit you want such as years.
Suppose If the Effective Interest Rate or APY is 8.25% compounded monthly then the Nominal Annual Interest Rate or "Stated Rate" will be about 7.95%. An effective interest rate of 8.25% is the result of monthly compounded rate x such that i = x * 12.
The formula can be written as:
r = m × [ ( 1 + i)1/m - 1 ],
where i is the effective rate, r is the stated rate and m is the number of compounding periods.
When the frequency of compounding is increased up to infinity we get "continuous compounding". Using our formula from our Effective Annual Interest Rate Calculator, where i = e^r - 1 becomes e^r = i + 1. And, by definition ln(e^r) = r , we can solve for r to get the formula:
r = ln(i + 1).
 Algebra and Trigonometry: A Functions Approach; M. L. Keedy and Marvin L. Bittinger; Addison Wesley Publishing Company; 1982.
Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae, 31st Edition New York, NY: CRC Press, 2003.