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Effective Annual Interest Rate
About this Calculator
Tells you effective annual interest rate, or APY (annual percentage yield), given the nominal annual interest rate and the number of compounding periods. If you have an investment earning a nominal interest rate of 7% per year and you will be getting interest compounded monthly, enter 7% and 12. If you are getting interest compounded quarterly on your investment, enter 7% and 4.
Effective Annual Interest Rate Formulas:
If the Nominal Interest Rate (also known as the "Stated Rate") is stated as 7% compounded monthly then the Effective Annual Interest Rate will be about 7.22%. A nominal interest rate of 7% will become 7%/12 months = 0.583% per month (0.07/12 months = 0.00583 per month). Compounding monthly (1 + 0.00583)^12 = 1.0722 which becomes 1.0722 - 1 = 0.0722 = 7.22%.
The formula can be written as:
r = [ (1+i/n)^n ] - 1,
where r is the effective rate, i is the stated rate and n is the number of compounding periods.
Excel function EFFECT()
This calculation is equivilant to the Excel function EFFECT(nominal_rate,npery) where nominal_rate = i and npery = n.
Continuous Compounding
When the frequency of compounding is increased up to infinity we get "continuous compounding". By definition, as n approaches infinity in the term [ (1+i/n)^n ] the value of this term approaches a limit equal to [e^i ].[2] Where e is the constant [2.7182818284....] and i is the interest rate in decimal form. So,
r = e^i - 1.[1]
References
[1] http://www.answers.com/topic/effective-interest-rate-1
[2] Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae, 31st Edition
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