How this instrument works
Two contracts rarely quote interest the same way. A savings account might compound monthly, a bond coupon semi-annually, a business loan quarterly — and lined up side by side, the raw percentages are not comparable, because a smaller number compounded often can beat a larger one compounded rarely. This instrument fixes that by routing any quoted percentage through its effective annual rate, the single annualized growth figure UK savings disclosures print as AER and US ones print as APY, then unwinding that figure back out to a nominal percentage at whatever second frequency you actually need — not just the annual headline, but a monthly, quarterly or daily figure to drop straight into a model or a rival quote.
The arithmetic runs in two passes. The first raises one plus the periodic slice of the known percentage to the power of its own compounding count, which folds every sub-period's growth on top of the last and lands on effective annual rate — exactly what a bank must publish once it credits interest more than once a year. The second pass reverses that: it takes the root of one plus the effective annual rate matching the target count, then multiplies back up by that count to restate growth as a nominal figure quoted the way the target convention expects. Two percentages produced this way are equivalent in the strict sense that both compound a dollar to the identical balance after a year — never mind that the printed digits differ.
What this cannot settle is which quote is the better deal once anything besides pure compounding enters the picture. Origination fees, account maintenance charges, tax treatment, credit risk, and day-count conventions that split a year into 360 or 365 days all sit outside this formula, which assumes clean, evenly spaced compounding periods and nothing else moving. It answers one narrow question precisely — what nominal figure at frequency B produces the same annual growth as this nominal figure at frequency A — and leaves every other term of the deal for the reader to weigh separately.
- Enter the percentage you already have into Known rate, % — the nominal figure exactly as printed on the quote or statement.
- Set Compounding of known rate to match how that figure actually compounds: Annually, Semi-annually, Quarterly, Monthly or Daily.
- Choose Target compounding frequency — the convention of the second quote you are comparing against, or the one your spreadsheet model expects.
- Read Effective annual rate (both are equivalent to this) — the annualized growth figure shared by both conventions.
- Read Equivalent nominal rate at the target frequency, % — the number to set beside the other quote or type into that model.
Worked example — 12% monthly measured against an annual quote
Set Known rate, % to 12 and Compounding of known rate to Monthly, so n1 = 12. The first pass computes (1 + 0.12 ⁄ 12)^12 − 1 = 0.126825030132, meaning Effective annual rate (both are equivalent to this) reads 12.6825030132% — a dollar left alone for a year under this account grows by that much, not by the flat 12% printed on the label.
Now set Target compounding frequency to Annually, so n2 = 1. Taking the first root of anything is the value itself, and multiplying by n2 = 1 changes nothing, so Equivalent nominal rate at the target frequency, % also reads 12.6825030132%. That confirms an account compounding annually would need to quote 12.6825% — not 12% — to hand a saver the exact same year-end balance as 12% compounded monthly. Two differently labeled contracts, one identical outcome.
Questions
What makes two rates "equivalent" rather than just similar?
They compound the same dollar to the identical balance after exactly one year, even though the printed percentages differ and the compounding schedules differ. This instrument checks that by routing both through effective annual rate — the shared annualized growth figure — and confirming they land on the same number to several decimal places, not merely close.
Why can't I just compare the two headline percentages directly?
Because compounding frequency changes what a headline percentage actually delivers. A quote compounded monthly earns interest on interest eleven extra times a year that an annually-compounded quote never accrues, so a 12% monthly figure and a 12.68% annual figure pay out identically despite one label reading lower. Comparing labels without converting first can make the worse-paying account look better.
How is this different from a plain APY or AER calculator?
An APY-style tool stops once it reaches the effective annual figure. This one takes that extra step: it unwinds the effective annual rate back down to a nominal percentage at whichever second frequency you name, which is what you actually need when a model, contract or rival quote is expressed monthly, quarterly or daily rather than annually.
Why does setting the same frequency on both sides return the original number?
Converting a percentage to its own frequency is an identity — nothing about the compounding schedule changed, so there is nothing for the arithmetic to adjust. Feed 12% compounded annually in with Annually chosen on both sides and Equivalent nominal rate at the target frequency, % returns 12% exactly, confirming the two-pass formula introduces no drift when n1 equals n2.
Why not just multiply a monthly figure by twelve to annualize it?
That produces a nominal, uncompounded figure — useful for a disclosure like APR, but not an equivalent rate. Multiplying 1% a month by twelve gives a flat 12% and ignores that each month's interest also earns interest going forward; this instrument's first pass, raising one plus the periodic slice to the twelfth power, captures that compounding instead, which is why 1% monthly actually equals 12.6825% compounded annually, not 12%.
Does this account for fees, tax, or day-count conventions?
No. The formula assumes clean, evenly spaced compounding periods and nothing else changing hands. Account fees, withholding tax on interest, and the 360-day versus 365-day accrual bases some bond and loan markets use can each shift a real payout a little from this clean result — treat the output as the pure compounding conversion, then weigh those other terms separately.
References
- Federal Reserve — Selected Interest Rates (H.15)
- SEC Investor.gov — Compound interest calculator
- CFPB — Bank accounts and services
Read this first: This instrument shows arithmetic, not advice. Real offers add fees, taxes and terms that vary by lender and place — verify the figures against your actual paperwork before deciding anything.