# Nominal Interest Rate Calculator

## Calculator Use

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, i_{t} = ( 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.

- 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" in percent. i = I/100
- Number of Periods (t)
- enter more than 1 if you want to calculate an effective compounded rate for multiple periods
- Compounded Interest Rate (i
_{t}) - 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.

## Nominal Annual Interest Rate Formulas:

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.

## Continuous Compounding

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 [1], we can solve for r to get the formula:

r = ln(i + 1).

## References

[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.