Rates / effective annual yield

APY and effective annual rate

A nominal rate hides how often interest compounds. This engine converts any quoted rate into the effective annual yield you actually earn, and shows how little extra the final few compounding periods add.

Compounding12× / yrAPY − nominal0.168 ptsExtra vs simple interest$488.50

Effective annual rate

6.1678%

APY on the selected frequency

Nominal rate

6.000%

The quoted or stated rate

Periodic rate

0.50000%

Applied 12× per year

Balance after horizon

$13,489

5.0 years at the effective rate

Compounding frequency comparison

The same nominal rate expressed as an effective annual yield at every common frequency.

6.00% nominal

Compounding frequency comparison
Frequency	Periods / yr	Periodic rate	APY	Balance after horizon
Annually (1)	1	6.0000%	6.0000%	$13,382
Semi-annually (2)	2	3.0000%	6.0900%	$13,439
Quarterly (4)	4	1.5000%	6.1364%	$13,469
Monthly (12)	12	0.5000%	6.1678%	$13,489
Weekly (52)	52	0.1154%	6.1800%	$13,496
Daily (365)	365	0.0164%	6.1831%	$13,498
Continuous (∞)	∞	6.0000%	6.1837%	$13,499

A nominal rate is only comparable once it is converted to an effective annual yield. Daily and continuous compounding differ by a few basis points at typical rates.

APY by compounding frequency

How the effective yield converges as compounding accelerates.

Balance growth at the effective rate

Monthly path of the balance when the APY is applied continuously in time.

Canonical equations

The exact relationships implemented by this engine.

APY = (1 + r/n)ⁿ − 1 · n compounding periods per year

APY = eʳ − 1 · the n → ∞ limit of the periodic form

i = r / n · the rate actually applied each period

FV = P(1 + APY)^t · equivalent to P(1 + r/n)^(nt)

Show model notation and computationOpen the symbolic form, the variable definitions, and the step-by-step derivation when you need the detail.

A clear path from inputs to results.

A nominal rate tells you the annual rate before compounding; APY tells you what you actually earn over a year. The difference is small at low rates and short horizons, but it is the only number that makes two savings products genuinely comparable.

Step-by-step methodWorked exampleAssumptions explained

How to use

Enter the nominal annual rate exactly as quoted.

Choose the compounding frequency. Monthly is standard for savings; daily is common for money market accounts.

Set a principal and horizon to see the balance the effective rate produces.

Compare the APY across frequencies — the gains shrink quickly after daily compounding.

Use the continuous case as the theoretical ceiling for this nominal rate.

Check the periodic rate: that is the number actually applied to the balance each period.

The formula

\mathrm{APY} = \left(1 + \frac{r}{n}\right)^{n} - 1, \qquad \mathrm{APY}_{\infty} = e^{r} - 1, \qquad i = \frac{r}{n}

Reference: Truth in Savings Act (Regulation DD) definition of annual percentage yield; standard effective-rate algebra.

Worked example

6% nominal rate compounded monthly (default inputs)

Inputs

Nominal annual rate 6%.
Compounded monthly (n = 12).
Principal $10,000 over 5 years.

Calculation path

1
Periodic rate: 6% / 12 = 0.5% per month.
2
Compound twelve periods: 1.005^12 − 1 = 6.1678% APY.
3
Daily compounding gives 6.1831%; continuous gives 6.1837%.
4
Apply to the principal: 10,000 × 1.061678^5 = $13,488.50.

Result

APY 6.1678% · continuous ceiling 6.1837% · balance after five years $13,488.50.

Interpreting the result

Monthly compounding adds 16.8 basis points to a 6% nominal rate. Moving from monthly to daily adds only 1.5 more, and daily to continuous adds 0.06 — the sequence converges fast. If a product quotes a nominal rate, convert it before comparing; if it quotes an APY, that is already the effective number.

Assumptions & limitations

The rate is fixed for the whole horizon. Variable-rate products can only quote an APY for the current rate.

No deposits or withdrawals occur during the year; those change the effective yield.

Fees are excluded. A fee reduces the realised APY and is not part of the quoted rate.

Interest is reinvested at the same rate for every period.

Compounding periods are assumed equal in length; some accounts use actual-day conventions that shift the result slightly.

Frequently asked questions

Is APY the same as APR?

No, and the difference matters. APY capitalises interest, so it is the rate you earn on savings. APR is the nominal rate on borrowing and does not include compounding. A 6% APR loan costs more than 6% a year once interest compounds.

How do I compare a 5.9% APY with a 6.0% nominal monthly rate?

Convert both to APY. The 6.0% nominal monthly rate is 6.168% APY, so it beats the 5.9% APY. Never compare a nominal rate against an effective one.

What is the periodic rate used for?

It is the rate applied to the balance each compounding period, and it is what you use to compute a single period’s interest. At 6% monthly it is 0.5% — not 6% divided by twelve in any approximate sense, but exactly that.

Why does continuous compounding not add much?

Because the sequence (1 + r/n)^n converges to e^r quickly. By daily compounding you are already within a basis point of the limit. Continuous compounding is mainly a modelling convenience for derivatives pricing.

Does APY include fees?

The regulatory definition requires the APY to reflect the effect of compounding but not all fees. Always check the account terms for maintenance or withdrawal charges that reduce the realised yield.

How does this apply to loans?

The same algebra runs in reverse: the effective cost of a loan quoted at a nominal rate with monthly compounding is higher than the quoted rate. For loans, the relevant regulatory number is usually APR, which also folds in fees.