APR vs APY Calculator

Understanding the APR vs APY Calculator

An APR versus APY calculator shows what a stated interest rate actually becomes once compounding is applied. The two numbers describe the same underlying rate but answer different questions, and the gap between them is precisely why savings accounts advertise APY while loans advertise APR. Nobody chooses the flattering number by accident.

How it actually works

Enter the stated annual rate and how many times per year it compounds. The calculator raises one plus the periodic rate to the power of the number of periods, subtracts one, and reports the resulting APY alongside the original APR and the difference between them. A 5% rate compounding monthly produces an APY of 5.1162%, a difference of 0.1162 percentage points. Compounding daily at the same stated rate pushes the APY to about 5.1267%, and the more frequent the compounding, the wider the gap.

The deeper context most people miss

Notice how quickly the gains from more frequent compounding flatten out. Going from annual to monthly compounding adds about 0.12 percentage points, but going from monthly all the way to daily adds only another 0.01. There's a mathematical ceiling here: as compounding frequency approaches infinity, the APY converges on a limit given by e raised to the rate, which for 5% is about 5.1271%. So a bank advertising continuous compounding over daily compounding is offering you roughly four ten-thousandths of a percentage point, which is marketing rather than a meaningful benefit.

Why lenders quote APR and savings accounts quote APY

The asymmetry is entirely deliberate and, in most jurisdictions, regulated rather than deceptive. APR, the annual percentage rate, is a nominal figure that describes the periodic rate multiplied out over a year without accounting for compounding within that year. APY, the annual percentage yield, incorporates the effect of interest earning interest across those periods, so for any rate compounding more than once a year, APY is the larger number. A savings institution wants the largest defensible number on the marketing material, so it quotes APY. A lender wants the smallest, so it quotes APR. Truth-in-lending rules in many countries require lenders to disclose APR specifically because it's designed to include certain fees and charges alongside the interest rate, making loan offers more comparable across lenders, and savings disclosure rules often require APY for the parallel reason of comparability. The complication for consumers is that this makes cross-product comparison genuinely tricky: a loan quoted at 6% APR and a savings account quoted at 6% APY are not offering the same underlying rate, and the loan's true annualised cost, if you compounded it the way the savings account does, would be higher than 6%. Whenever comparing two financial products, the first question worth asking is whether both numbers are the same type, because comparing an APR against an APY tilts the comparison by a margin that grows with the rate.

A worked example: where the gap actually matters

At 5%, the difference between APR and APY is about 0.12 percentage points, which on a $10,000 balance is roughly $12 a year. That's real but rarely decisive. The gap widens substantially at higher rates, because compounding is multiplicative. At 12% compounding monthly, the APY is 12.6825%, a gap of 0.68 percentage points. At 20%, typical of credit card territory, monthly compounding produces an APY of 21.9391%, nearly two full percentage points above the stated APR. On a $6,000 revolving credit card balance carried for a year, that difference between the quoted 20% and the effective 21.94% is about $116 of additional cost that the headline number doesn't show. This is why the APR-APY distinction matters far more on expensive debt than on savings: at savings-account rates the compounding effect is a rounding error, while at credit card rates it's a meaningful additional cost that compounds further if the balance persists. It's also why paying down high-rate revolving debt produces such a reliably strong return, because you're avoiding an effective rate noticeably higher than the one printed on the statement.

Comparing two savings accounts or two loan offers correctly

The practical rule is to convert everything to the same basis before comparing, and the direction of conversion depends on what you're comparing. For savings products, APY is already the comparable figure, so an account advertising 4.5% APY genuinely beats one advertising 4.4% APY regardless of their compounding frequencies, since APY already accounts for that. The trap is an account advertising a 4.5% rate without specifying whether that's APR or APY, in which case you need to ask. For loans, APR is the more useful comparison because regulations typically require certain fees to be folded into it, so a 6.2% APR loan with no fees and a 5.9% APR loan with heavy origination costs may already be correctly ranked by their APRs even though the raw interest rates differ. The situation to watch for is comparing across product types, or comparing a quoted rate from a regulated disclosure against a marketing figure from somewhere else, since those are frequently not the same basis. When in doubt, computing the APY for both using this calculator puts them on identical footing, which removes the ambiguity entirely.

The limit of compounding, and why continuous compounding exists

As compounding frequency increases, the APY does not increase without bound. It converges toward a specific limit: for a nominal rate r, the maximum achievable APY as compounding becomes infinitely frequent is e raised to the power of r, minus one, where e is the mathematical constant approximately equal to 2.71828. For a 5% nominal rate, that ceiling is about 5.1271%, and daily compounding at 5.1267% already captures essentially all of it. This is why claims about hourly or continuous compounding are effectively marketing: the incremental benefit past daily compounding is measured in ten-thousandths of a percentage point and is dwarfed by any difference in the underlying rate. The concept still matters in finance, though, because continuous compounding is mathematically convenient and is used extensively in derivatives pricing, bond mathematics, and academic models, where the clean exponential form simplifies the algebra considerably compared with discrete periods. For a consumer choosing between savings accounts, the practical implication is simple and worth internalising: an account offering 4.8% compounded annually beats one offering 4.6% compounded daily by a wide margin, because the underlying rate dominates the compounding frequency at any realistic difference. Chasing compounding frequency while ignoring the base rate optimises the smaller variable.

Variations: APR with fees, effective annual rate, and negative compounding

Several related measures appear in different contexts. In lending, the regulated APR often includes not just interest but certain mandatory fees such as origination charges, which is what makes it useful for comparing loan offers, but it means a loan's APR can exceed its stated interest rate for reasons unrelated to compounding. Effective annual rate, or EAR, is essentially the same calculation as APY applied in a corporate finance context, and the two terms are largely interchangeable in meaning even though convention assigns them to different settings. On the borrowing side, the same compounding math works against you rather than for you, and credit cards that compound daily on an average daily balance produce an effective cost meaningfully above the quoted APR. There's also the case of fees that behave like interest but aren't quoted as such, including payday lending structures where a flat fee over a short term translates into an extraordinary annualised rate once expressed as an APR, which is exactly why disclosure rules require that translation.

Comparing rates without being misled

Always establish whether a quoted number is an APR or an APY before comparing it to anything, since the two describe the same underlying rate differently and the gap widens as rates rise. For savings, compare APY to APY, which already accounts for compounding frequency. For loans, compare APR to APR, since regulated APR typically folds in certain fees and makes offers genuinely comparable. Recognise that the underlying rate matters far more than the compounding frequency, because the benefit of more frequent compounding flattens out sharply and is capped by a mathematical limit that daily compounding already nearly reaches. And pay closest attention to this distinction on expensive debt, where at credit card rates the effective annual cost can run nearly two percentage points above the quoted APR.

What people get wrong

• Comparing a loan's APR against a savings account's APY as though they're the same basis, which tilts the comparison by a margin that grows with the rate.
• Chasing daily compounding over monthly, when the difference at typical savings rates is about a hundredth of a percentage point and the underlying rate dominates.
• Assuming a credit card's quoted APR is the true annual cost, when daily compounding on a carried balance produces an effective rate meaningfully higher.
• Believing continuous or hourly compounding offers a real advantage, when APY converges on a hard mathematical ceiling that daily compounding already nearly reaches.

Where the math comes from

APY = (1 + APR/n)^n - 1, where APR is the stated annual rate as a decimal and n is the number of compounding periods per year. The difference reported is simply APY - APR. As n increases, APY converges toward its theoretical maximum of e^APR - 1, which is why gains from compounding more frequently than daily are negligible.

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