EAR Calculator — Effective Annual Rate from a Nominal Rate
Who this is for: For students untangling APR vs APY on an exam, and for savers and borrowers comparing products quoted with different compounding frequencies on the same nominal rate.
Not the right tool for: Fee-inclusive borrowing costs — APR with origination fees is not the nominal rate this converts · Variable rates — the conversion assumes the nominal rate holds all year
Nominal rate plus compounding frequency in — effective annual rate (APY) out, with the full frequency table from annual to continuous.
Quick answer: EAR = (1 + nominal/m)^m − 1 — what the nominal rate becomes after a year of compounding m times. 12% nominal is 12.5509% quarterly, 12.6825% monthly, 12.7475% daily, and 12.7497% continuous. Compare products on EAR, never on nominal rate.
| Compounding | Effective annual rate | vs chosen |
|---|---|---|
| Annual (1) | 12.0000% | -0.6825% |
| Semiannual (2) | 12.3600% | -0.3225% |
| Quarterly (4) | 12.5509% | -0.1316% |
| Monthly (12) ← selected | 12.6825% | — |
| Weekly (52) | 12.7341% | 0.0516% |
| Daily (365) | 12.7475% | 0.0650% |
| Continuous | 12.7497% | 0.0672% |
| Formula | EAR = (1 + nominal/m)^m − 1; continuous: e^r − 1 |
|---|---|
| 12% nominal | Quarterly 12.5509% · Monthly 12.6825% · Daily 12.7475% · Continuous 12.7497% |
| Reverse | EAR → nominal supported (12.6825% EAR = 12% nominal monthly) |
| Compiled | October 2026 |
Why the nominal rate lies a little
A nominal rate is a headline; the effective annual rate (EAR, banks say APY) is what you actually earn or pay once compounding frequency does its work: EAR = (1 + nominal/m)^m − 1. Take 12% nominal: compounded quarterly it's 12.5509%, monthly 12.6825%, daily 12.7475%, and continuously e^0.12 − 1 = 12.7497%. The gap between nominal and effective grows with both the rate and the frequency — at 2% nominal, monthly compounding adds under 2 basis points; at 12% it adds over two-thirds of a point. That's why comparing a monthly-compounding loan against a yearly-compounding bond on nominal numbers is rigged: convert everything to EAR first. The reverse direction works too — this page's engine can state what nominal rate with monthly compounding matches a given EAR (12.6825% effective is exactly 12% nominal monthly).
Common uses
- Comparing savings accounts, CDs, or loans quoted at different frequencies
- Exam problems: convert APR to EAR (or back) without a slip
- Pricing homework where the discount rate must be an effective annual rate
- Seeing the ceiling: how close daily compounding gets to continuous
Where these numbers come from
All results are computed in your browser from the standard closed-form formulas (and a numerical root-finder where no closed form exists — rates, IRR, YTM). Formulas follow the ordinary-annuity (END) convention used by the BA II Plus and HP 12C. Educational reference only — not investment, tax, or accounting advice.