Effective Annual Rate vs Nominal Interest Rate
Two banks quote 12% per annum. One compounds monthly, the other compounds annually. Are they offering the same deal? They are not — and the gap between the number printed in the advertisement and the number that actually applies to your money is exactly what the effective annual rate (EAR) measures. This is the formula, why compounding frequency changes the true cost or return, and four worked comparisons.
The formula
Here i is the nominal (quoted) annual rate, written as a decimal, and m is the number of times it compounds in a year. The nominal rate is the headline figure a bank advertises; the effective rate is what you actually earn or pay over one full year once compounding within the year is accounted for. EAR is always greater than or equal to the nominal rate, and the two are equal only when m = 1 (compounding once a year).
Worked example 1 — 12% compounded monthly
Question: A savings account advertises 12% per annum, compounded monthly. Find the effective annual rate.
i = 0.12, m = 12, so the monthly factor is (1 + 0.12 ÷ 12) = 1.01
(1.01)¹² = 1.126825
EAR = 1.126825 − 1 = 0.126825 = 12.6825%
The nominal rate says 12%; the true annual return is closer to 12.68% because each month's interest starts earning interest of its own for the remaining months.
Worked example 2 — 8% compounded quarterly
i = 0.08, m = 4, quarterly factor = 1.02
(1.02)⁴ = 1.02 × 1.02 × 1.02 × 1.02 = 1.08243216
EAR = 1.08243216 − 1 = 8.2432%
Worked example 3 — 6% compounded half-yearly
i = 0.06, m = 2, half-yearly factor = 1.03
(1.03)² = 1.0609
EAR = 1.0609 − 1 = 6.09%
Notice how the gap between the nominal rate and the EAR grows with m: 6% monthly-scale (m = 2) adds only 0.09 percentage points, while 12% (m = 12) adds nearly 0.68 — more frequent compounding widens the gap faster at a higher rate, because interest is earning interest on interest more often.
Worked example 4 — comparing two real offers
Question: Account P offers 12% per annum compounded monthly. Account Q offers 12.75% per annum compounded annually. Which actually pays more in a year?
Account P's EAR was already found above: 12.6825%.
Account Q compounds once a year (m = 1), so its EAR equals its nominal rate exactly: 12.75%.
Even though Account P compounds far more often, Account Q's higher nominal rate wins: 12.75% > 12.6825%. Compounding frequency alone never decides the winner — only the effective annual rate, computed for each offer, can.
Common mistakes
- Comparing two nominal rates directly when their compounding frequencies differ. A lower nominal rate compounded more often can beat a higher nominal rate compounded less often — always convert both to EAR first, as in example 4.
- Dividing i by m for the base of the power but forgetting to raise it to the power m. Both steps are needed; skipping the exponent just returns the periodic rate, not the annual one.
- Assuming "compounded monthly" means multiply the nominal rate by 12. The nominal rate already is the annual figure; m only decides how it is split into periods, not how large it becomes.
- Forgetting EAR = nominal rate when m = 1. A rate "compounded annually" needs no conversion at all — that is the one case where the headline figure is already the true figure.
- Treating EAR and total interest paid as the same question. EAR is a rate, valid for exactly one year; comparing loans over several years also needs the tenure, not EAR alone.
Where this appears in exams
| Context | Typical use |
|---|---|
| Class 11–12 Commerce/Applied Mathematics | Compound interest with different compounding periods |
| CUET Commerce/General Test | Quantitative-aptitude interest-rate questions |
| Banking and SSC competitive exams | Comparing quoted rates, compound-interest numericals |
| Everyday financial literacy | Reading a loan or deposit offer correctly before comparing it to another |
Try your own rates and compounding frequencies. The Finance converter handles the compounding maths for you, so a comparison like example 4 takes seconds instead of two separate hand calculations.
Open the Finance Converter →This calculator suite is built mainly for Class 11–12 science students, and the same team teaches them: ABC Chemistry runs Class 11–12 chemistry coaching at its Gurugram centre plus online classes across India — abcchemistry.in.