Equivalent Interest Rate Calculator: Convert Frequencies

How the conversion works

The conversion runs through the effective annual rate — the one number that does not depend on frequency. Step one: take the nominal rate and divide it by the number of compounding periods in a year to get the periodic rate. At the defaults, 12% divided by 12 months is 1.00% a month. Step two: compound the periodic rate for a full year to get the effective annual rate: (1 + 0.12/12)^12 − 1 = 1.01^12 − 1 = 0.126825, or 12.6825%. Step three: unwind the effective rate at the target frequency. The equivalent nominal rate at frequency n is n × ((1 + EAR)^(1/n) − 1). For quarterly, that is 4 × ((1.126825)^(1/4) − 1) = 4 × 0.030301 = 0.121204, or 12.12%. The quarterly loan has to quote a higher nominal rate than the monthly one — 12.12% against 12.00% — because it compounds less often, so each compounding has to do more work.

The full conversion table — 12% nominal monthly

Every row is the same rate, written at a different frequency. The pattern to notice is that the nominal rate falls as the frequency rises: the more often interest compounds, the lower the nominal rate that delivers the same effective return — and the annual nominal rate is the effective rate itself, because compounding once a year adds nothing on top.

The worked example, line by line

Setup: nominal rate 12%, compounding monthly (12×), converting to quarterly (4×).

Line 1 — periodic rate. 12% / 12 = 1.00% a month.

Line 2 — effective annual rate. (1 + 0.12/12)^12 − 1 = 1.01^12 − 1 = 12.6825%.

Line 3 — equivalent quarterly nominal. 4 × ((1.126825)^(1/4) − 1) = 4 × 0.030301 = 12.12%.

Line 4 — equivalent semi-annual nominal. 2 × ((1.126825)^(1/2) − 1) = 2 × 0.061520 = 12.30%.

Line 5 — equivalent annual nominal. 1 × ((1.126825)^1 − 1) = 12.68% — the effective rate itself.

Line 6 — equivalent daily nominal. 365 × ((1.126825)^(1/365) − 1) = 11.94%.

Line 7 — equivalent weekly nominal. 52 × ((1.126825)^(1/52) − 1) = 11.95%.

Every branch, checked

12% quarterly: the periodic rate is 12% divided by 4, which is 3.00% a quarter, and (1.03)^4 − 1 = 12.5509% effective. Quarterly compounding is worth less than monthly compounding at the same nominal rate — 12.5509% against 12.6825% — which is exactly why the quarterly equivalent has to be higher.

12% daily: the periodic rate is 12% divided by 365, which is about 0.0329% a day, and (1 + 0.12/365)^365 − 1 = 12.7475% effective. Daily compounding is worth more than monthly — 12.7475% against 12.6825% — so the daily equivalent nominal is lower, 11.94%.

8% monthly: the periodic rate is 8% divided by 12, which is about 0.6667% a month, and (1 + 0.08/12)^12 − 1 = 8.3000% effective. The compounding premium is smaller at lower rates: 0.30 points on top of 8%, against 0.68 points on top of 12%.

6% semi-annual: the periodic rate is 6% divided by 2, which is 3.00% half a year, and (1.03)^2 − 1 = 6.0900% effective. Two compoundings a year add only 0.09 points — the less often interest compounds, the closer the effective rate sits to the nominal one.

Reverse — EAR 12.6825%: work the effective rate back into a monthly nominal: 12 × ((1.126825)^(1/12) − 1) = 12 × 0.010000 = 12.00% nominal monthly. The round trip lands exactly where it started, which is how you know the conversion is consistent.

Nominal vs effective: the one mistake to avoid

The one mistake people make with interest rates is comparing two nominal rates with different frequencies as if they were the same thing. 12% monthly and 12% quarterly are not the same deal: the monthly one is worth 12.6825% over a year and the quarterly one is worth 12.5509% — a gap of 0.13 points, which on a $100,000 loan is $130 a year. The number that makes rates comparable is the effective annual rate, and that is what this calculator runs through: convert both rates to their effective rates, and the cheaper loan is the one with the lower effective rate, whatever the nominal quotes say. And the annual nominal rate is always the effective rate itself — compounding once a year adds nothing on top — which is why an annual quote is the only nominal rate you can read at face value.

Rounding convention, stated plainly: effective rates are shown to four decimals and nominal equivalents to two, and the conversion runs on the unrounded effective rate — so 12% nominal monthly is 12.6825% effective (the exact figure is 12.682503%), and the quarterly equivalent is 12.12% (the exact figure is 12.1204%). Daily compounding uses 365 periods a year and weekly uses 52.

Output · formula · meaning

Who uses this

• The loan shopper: is offered 12% monthly from one lender and 12% quarterly from another, and learns they are not the same deal — the monthly loan is worth 12.6825% a year and the quarterly one 12.5509% — and that the quarterly lender has to quote 12.12% to match.
• The saver comparing accounts: sees a savings account quoting 12% compounded daily and learns the daily equivalent of their monthly 12% is 11.94% — so the daily account at 12% is the better deal, worth 12.7475% effective.
• The student checking homework: has 12% nominal monthly and learns the periodic rate is 1.00%, the effective rate is 12.6825%, and the quarterly equivalent is 12.12% — with every step of the working shown.
• The effective-rate converter: is given an effective rate of 12.6825% and learns it is 12.00% nominal monthly in reverse mode — the round trip lands exactly where it started.

Accuracy & limitations

The conversion is exact for the numbers you enter: the effective annual rate is (1 + r/m)^m − 1 by definition, and the equivalent nominal rate at frequency n is n × ((1 + EAR)^(1/n) − 1) — the two formulas are inverses, so the round trip lands exactly where it started, as the reverse branch shows. The only approximations are the display — effective rates to four decimals, nominal equivalents to two — and the calendar: daily compounding uses 365 periods a year and weekly uses 52, which is the standard convention, not the exact count in a leap year or a 53-week year. The calculator assumes the rate is fixed for the whole year and that interest compounds at perfectly regular intervals — a real loan may compound on business days, charge fees, or change its rate, and none of that is in the math. This page converts rates, not loan costs: it does not count fees, minimum balances, or payment timing, and a lower effective rate does not always mean a cheaper loan once fees are added. Compare effective rates first, then read the fees.