Nominal, periodic, and effective annual rates
The effective annual rate includes within-year compounding. The nominal rate is an annualized quotation divided into the selected number of periods. The periodic rate is what applies during one of those periods. Continuous compounding has an annual continuous quote and no discrete periodic rate. The table holds annual growth fixed while changing the quotation convention; it does not hold the nominal rate fixed. Changing Rate entered carries a valid equivalent quote when available.
To turn the converted rate into a balance projection with principal and contributions, use the Compound Interest Calculator and match its rate and compounding convention.
The formula
Rates in the equations are decimals; displayed results are percentages. m is the number of compounding periods per year. Daily mode uses 365 equal periods. The conversion uses logarithms and exponential differences to retain precision for very small rates. Custom frequency accepts 1–3,660 whole periods per year. A periodic quote compounds its growth factor m times. Continuous compounding has no per-period input or output; choose a nominal or effective annual quote for that method.
Monthly compounding of a nominal 12%
A nominal annual rate of 12% compounded monthly means 1% per month. Compounding that monthly factor twelve times gives an effective annual rate of approximately 12.682503%. One hundred grows to approximately 112.68 before fees and taxes.
How to use this calculator
- Select the kind of rate entered and type the percentage.
- Choose equal periods or continuous compounding. For equal periods, select a frequency or enter a custom whole number of periods per year.
- Read the converted annual rate and, where applicable, the discrete periodic rate.
- Compare the table’s equivalent quotations, which all produce the same annual growth.
Compare rates on the same basis
Two quoted percentages cannot always be compared directly when their compounding conventions differ. An effective annual conversion places them on a one-year mathematical basis, provided principal stays invested and interest is reinvested throughout the year.
This conversion is not a regulatory APR calculation. Loan APR and deposit APY disclosures can incorporate specific cash-flow, fee, calendar, or legal conventions. A lender’s all-in borrowing cost cannot be reconstructed from a nominal rate and frequency alone.
Continuous quotes and discrete periods
A continuous annual quote of 12% produces effective annual growth of approximately 12.749685%. Its equivalent effective quote reverses with the natural logarithm. There is no single discrete compounding period in this model, so the per-period result is hidden. Choosing a per-period quote with continuous compounding produces an actionable input error.
Assumptions & limitations
What this calculation assumes
- Interest compounds at equal intervals for a complete year.
- The periodic rate stays fixed.
- No cash is added or removed between compounding events.
What to keep in mind
- Fees, taxes, changing balances and actual-day accrual rules are excluded.
- These are mathematical conversions; jurisdiction-specific APR and APY disclosure conventions are not reconstructed.
Common questions
When are nominal and effective rates equal?
They are equal with annual compounding, or when the rate is zero. With a positive rate and multiple periods, the effective rate is higher.
Can I use the effective rate divided by twelve as a monthly rate?
Not if you want the original effective annual growth. Take the twelfth root of its growth factor instead, or use the reverse conversion with monthly frequency.
Sources & further reading
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