Periodic Interest Rate Calculator: Nominal to Effective

How the periodic rate works

Three numbers describe one rate. The nominal rate is the headline: 12% a year. The periodic rate is the nominal rate divided by the number of compounding periods: 12% divided by 12 is 1.00% a month. The effective annual rate is what the year actually delivers once each period's interest starts earning interest of its own: (1 + 1.00%) raised to the 12th power, minus 1, is 12.6825%. The value after one year is the amount times (1 + EAR): $10,000 times 1.126825 is $11,268.25. Simple interest at the nominal rate would give $10,000 times 1.12, which is $11,200.00 — so compounding adds $68.25 over the year. The periodic rate is the one that matters in practice: a monthly loan payment divides the nominal rate by 12, and a savings account that compounds monthly credits 1.00% a month, not 12% a year.

The worked example, line by line

Setup: nominal rate 12%, compounding monthly (m = 12), starting amount $10,000.00.

Line 1 — periodic rate. 12% ÷ 12 = 1.0000% a month.

Line 2 — EAR. (1 + 1.00%)^12 − 1 = 1.126825 − 1 = 12.6825%.

Line 3 — value after 1 year. $10,000.00 × 1.126825 = $11,268.25.

Line 4 — simple interest. $10,000.00 × 1.12 = $11,200.00.

Line 5 — the compounding bonus. $11,268.25 − $11,200.00 = $68.25.

Branch — quarterly at 12%: periodic 12% ÷ 4 = 3.00%; EAR (1.03)^4 − 1 = 12.5509%; $10,000.00 becomes $11,255.09.

Branch — daily at 12%: periodic 12% ÷ 365 = 0.0329%; EAR (1 + 0.12/365)^365 − 1 = 12.7475%; $10,000.00 becomes $11,274.75.

Branch — semi-annual at 6%: periodic 6% ÷ 2 = 3.00%; EAR (1.03)^2 − 1 = 6.09%; $10,000.00 becomes $10,609.00.

Branch — weekly at 12%: periodic 12% ÷ 52 = 0.2308%; EAR (1 + 0.12/52)^52 − 1 = 12.7341%; $10,000.00 becomes $11,273.41.

Branch — reverse, EAR 12.6825% monthly: nominal = 12 × ((1 + 12.6825%)^(1/12) − 1) = 12 × 0.01 = 12.00%.

More compounding is worth less than it looks

The branches tell a story about diminishing returns. Quarterly at 12% gives an EAR of 12.5509%; monthly gives 12.6825%; weekly gives 12.7341%; daily gives 12.7475%. The jump from quarterly to monthly is worth 0.1316 percentage points, but the jump from monthly to daily is worth only 0.0650 — and going from daily to continuous compounding would add only about 0.0059 more, to 12.7497%. On $10,000 the difference between monthly and daily compounding is $6.50 a year. Frequency matters, but the nominal rate matters far more: a 13% nominal rate compounded quarterly beats a 12% nominal rate compounded daily by a wide margin. Compare nominal rates first, and treat the frequency as a tiebreaker.

Reverse mode: from the effective rate back to the nominal

Sometimes the number you have is the effective rate — a savings account advertises an APY of 12.6825%, or a loan quotes an effective rate — and you need the nominal rate behind it. The reverse formula is nominal = m × ((1 + EAR)^(1/m) − 1). At the defaults, an EAR of 12.6825% compounded monthly gives 12 × ((1.126825)^(1/12) − 1) = 12 × 0.01 = 12.00% nominal. The reverse is exact, not approximate: run the forward calculation on the answer and you get the EAR back. Use it to check whether a quoted APY matches the nominal rate on the paperwork — if it does not, something in the quote is not what it seems.

Why loan payments use the periodic rate

A mortgage quoted at 6% nominal with monthly payments does not charge 6% a month — it charges 0.50% a month, the nominal rate divided by 12, applied to the balance each month. That monthly rate is the periodic rate, and it is the number inside the amortisation formula: the monthly payment is built from 0.50% a month over 360 months, not from 6% a year. The same logic runs for savings: a 12% nominal rate compounded monthly credits 1.00% a month. The nominal rate is the label on the box; the periodic rate is the rate that actually moves the money, period by period. Nominal equals effective only when m = 1 — when the rate compounds once a year, there is nothing to divide and nothing to compound within the year.

Rounding convention, stated plainly: rates are shown to four decimals — the periodic rate and the EAR — because the differences between frequencies live in the third and fourth decimals: monthly 12.6825% against daily 12.7475% is a gap of 0.0650 points, and two decimals would hide it. The value after one year is computed from the unrounded EAR and then rounded to cents, so $10,000 times (1.12682503) is $11,268.25. The reverse mode rounds the EAR to four decimals before computing the nominal rate, so an EAR entered as 12.6825% returns exactly 12.00% nominal at monthly compounding.

Output · formula · meaning

Who uses this

• The saver comparing accounts: sees 12% nominal compounded monthly and learns the EAR is 12.6825% — and that $10,000 becomes $11,268.25, not $11,200.00, because each month's interest earns interest after it.
• The borrower checking a loan: takes a 12% nominal mortgage and learns the monthly payment uses a 1.00% periodic rate — the nominal rate divided by 12 — and that the EAR of 12.6825% is the true yearly cost.
• The APY checker: sees an advertised APY of 12.6825% and runs reverse mode to learn the nominal rate behind it is 12.00% compounded monthly — and that a quoted APY that does not match the nominal rate deserves a second look.
• The frequency shopper: compares quarterly, monthly, weekly, and daily compounding at 12% and learns the EARs are 12.5509%, 12.6825%, 12.7341%, and 12.7475% — and that the nominal rate matters far more than the frequency.

Accuracy & limitations

The math is exact for the numbers you enter: periodic rate is nominal divided by m, EAR is (1 + periodic)^m minus 1, and the value after one year is the amount times (1 + EAR). But the model assumes the nominal rate stays fixed for the whole year and the interest compounds on schedule — real accounts change rates, charge fees, and compound on their own calendars, so the EAR on a statement can differ by a few cents from the figure here. It assumes no deposits or withdrawals during the year: add money mid-year and the value after one year is higher, take money out and it is lower. It also assumes the rate compounds at the stated frequency for the full year — a loan that quotes a nominal rate but compounds differently in the fine print is a different calculation. The reverse mode is exact for the EAR you enter, but an EAR rounded to two decimals on an advertisement will return a nominal rate that is approximate to the same precision. This is an educational tool, not financial advice.