The effective annual rate formula converts a nominal annual rate with intra-year compounding into the actual percentage change over one year. Divide the nominal rate by the number of compounding periods, compound that periodic rate for one year, and subtract one.
EAR = (1 + r_nom / m)^m - 1
— effective annual rate, expressed as a decimal before converting to percent
— nominal or stated annual interest rate, expressed as a decimal
— number of compounding periods per year
EAR_continuous = e^(r_nom) - 1
— effective annual rate under continuous compounding
— mathematical constant approximately equal to 2.71828
— nominal annual rate as a decimal
The nominal rate is the quoted annual rate before intra-year compounding. A 7.35% nominal rate compounded monthly does not mean the balance grows by exactly 7.35% over the year. It means each month receives one-twelfth of 7.35%, and later months earn interest on earlier interest. Enter 7.35% as 0.0735 in the formula.
m counts compounding periods in one year. Use 1 for annual, 2 for semiannual, 4 for quarterly, 12 for monthly, 52 for weekly, and a convention stated by the problem for daily compounding. Daily conventions can use 365 or another specified day count, so do not assume when a contract gives one.
EAR is a one-year effective rate. It places offers with different compounding frequencies on the same annual basis. For a saver, the higher EAR produces more one-year growth if fees, risk, and other terms are equal. For a borrower, the lower EAR produces less interest under the same simplifying assumptions.
With annual compounding, m is 1, so the effective rate equals the nominal rate. With more frequent compounding and a positive nominal rate, EAR is higher because interest credited in one period earns interest in later periods. The nominal rate is divided across the periods; it is not applied in full every month or quarter.
EAR isolates compounding, not every cost of a financial product. Loan fees, teaser periods, changing rates, and payment timing can affect actual borrowing cost. Similarly, a deposit's taxes, withdrawal restrictions, and account fees are outside the basic formula. Use EAR to compare compounding mechanics, then examine the remaining terms.
| Step | Offer A: 7.35% monthly | Offer B: 7.25% quarterly |
|---|---|---|
| Nominal annual rate | 0.0735 | 0.0725 |
| Compounding periods | 12 | 4 |
| Periodic rate | 0.0735 / 12 = 0.006125 | 0.0725 / 4 = 0.018125 |
| One-year growth factor | (1.006125)^12 = 1.076027 | (1.018125)^4 = 1.074495 |
| Subtract opening principal | 1.076027 - 1 | 1.074495 - 1 |
| Effective annual rate | 7.6027% | 7.4495% |
| Year-end value of $12,400 | $13,342.74 | $13,323.74 |
Offer A's EAR is 7.6027%, while Offer B's EAR is 7.4495%. On a $12,400 opening balance with no deposits or withdrawals, those rates produce about $13,342.74 and $13,323.74 after one year. The example keeps more digits during the calculation and rounds displayed currency only at the end.
The result also gives a useful check: multiplying $12,400 by each one-year growth factor reproduces the year-end balance. If your calculated balance instead applies 7.35% every month, the answer will be far too large because the annual rate has been treated as a monthly rate.
In many finance exercises, APR refers to a nominal annual rate and EAR is the compounded annual equivalent. APY commonly describes an effective annual yield on a deposit, so APY and EAR may represent the same mathematical one-year rate. Real-world disclosure rules and labels vary by jurisdiction and product, however. Read how the quoted figure is defined instead of relying only on the acronym.
An EAR comparison is cleanest when the offers have the same risk, fees, cash-flow timing, and rate stability. If one loan charges an origination fee, the basic EAR formula does not absorb that fee. If one savings account changes its rate after three months, a one-rate EAR projection does not describe the entire year.
Sometimes a question gives an effective annual rate and asks for the equivalent nominal rate at a stated compounding frequency. Reverse the compounding process: take the m-th root of one plus EAR, subtract one to get the periodic rate, and multiply by m. Keep the same definition of m throughout. This is an algebraic rearrangement of the standard formula, not a second economic assumption.
Entering 7.35 instead of 0.0735. Percentages must be converted to decimals before exponentiation. Convert the final decimal back to a percentage for reporting.
Dividing EAR by m. The divided input is the nominal annual rate. EAR is the result after one year of compounding.
Using the number of months in the whole investment. In the EAR formula, m is periods per year. A three-year monthly investment still uses m equal to 12 when calculating its effective annual rate.
Comparing nominal rates directly. Two quoted rates with different compounding frequencies are not fully comparable until both are converted to effective rates.
Calling EAR a real rate. “Effective” means adjusted for compounding here. It does not mean adjusted for inflation. A real interest rate answers a different question.
Ignoring continuous compounding wording. When the problem explicitly says continuously compounded, use the exponential version rather than inserting an extremely large arbitrary m.
Excel and Google Sheets provide EFFECT(nominal_rate, npery). Enter the nominal annual rate as a decimal and the number of compounding periods per year. For Offer A, EFFECT(7.35%, 12) returns approximately 7.6027%. The function expects an integer number of periods and does not represent continuous compounding; use the exponential formula for that case.
A quote such as “7.35% compounded monthly” usually means 7.35% is a nominal annual rate allocated across 12 monthly periods. It does not mean 7.35% is earned every month. By contrast, “7.35% effective annually” already states the one-year compounded result, so running it through the nominal-rate formula would compound it twice. When wording is ambiguous, look for labels such as nominal, stated, APR, effective, APY, and the stated compounding frequency.
The number of compounding periods is not always the number of payments. A monthly loan can compound daily while collecting monthly payments. EAR uses the interest compounding frequency because its job is to measure one-year balance growth before separate payment cash flows. Payment timing belongs in a full cash-flow analysis, not in m.
EAR is the actual one-year compounded rate implied by a nominal annual quote. Divide the nominal rate by periods per year, compound for exactly that many periods, and compare offers only after putting them on the same effective basis.
Divide the nominal annual rate by the number of compounding periods per year, add one, raise the result to that number of periods, and subtract one. Convert the decimal result to a percentage.
A nominal rate is an annual quote that does not itself show intra-year compounding. EAR is the actual one-year percentage change after that compounding is included.
They often describe the same effective one-year compounded rate, especially for deposit products. Check the product's disclosure because terminology and included costs can vary.
For a positive nominal rate compounded more than once per year, yes. With annual compounding they are equal. Other rate signs or product features require separate analysis.
Raise e to the nominal annual rate written as a decimal, then subtract one. This is the limit of increasingly frequent compounding.
Not in the basic compounding formula. It converts the stated rate for compounding frequency; fees and other cash flows must be analyzed separately.
By the FinanceBrain Team · Last verified July 12, 2026 · How we produce and verify articles