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Category Finance & Calculations

Annual Percentage Rate (APR)

Enter a value between −100 and 10,000

Compounding Frequency

Reference Principal (optional)

Used to show dollar interest difference

APY —

Input Rate —

Compounding —

Rate Difference —

All Compounding Frequencies

Frequency	APY	Difference

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About

A quoted APR of 5% does not mean you earn 5% on your deposit. The actual yield depends on how often interest compounds. With daily compounding (n = 365), that same 5% APR becomes an APY of approximately 5.127%. On a $100,000 balance, that difference is $127 per year. Banks in the United States are required by the Truth in Lending Act (TILA) to disclose APR on loans and by Regulation DD to disclose APY on deposit accounts. Confusing the two leads to mispriced comparisons. This calculator converts between APR and APY for any compounding frequency, including continuous compounding.

Note: the formula assumes a fixed rate over the entire year and does not account for variable-rate adjustments, fees, or intra-period withdrawals. For loan products, the quoted APR may include origination fees per Regulation Z, making it higher than the nominal rate. This tool treats APR as the nominal annual rate without fees. Pro tip: when comparing savings accounts, always compare APY values directly. When comparing loan offers, compare APR values that include fees.

Formulas

The core conversion from APR (nominal annual rate) to APY (effective annual yield) for discrete compounding:

APY = (1 + APRn)ⁿ − 1

For continuous compounding (n → ∞):

APY = e^APR − 1

The reverse conversion from APY to APR:

APR = n × ((1 + APY)¹ⁿ − 1)

For continuous compounding reverse:

APR = ln(1 + APY)

Where APR = Annual Percentage Rate (nominal, as a decimal), APY = Annual Percentage Yield (effective, as a decimal), n = number of compounding periods per year, e ≈ 2.71828 (Euler's number), ln = natural logarithm.

Reference Data

Compounding Frequency	Periods per Year (n)	APY for 5% APR	APY for 10% APR	APY for 15% APR	APY for 24% APR
Annually	1	5.000%	10.000%	15.000%	24.000%
Semi-Annually	2	5.063%	10.250%	15.563%	25.440%
Quarterly	4	5.095%	10.381%	15.865%	26.248%
Monthly	12	5.116%	10.471%	16.076%	26.824%
Semi-Monthly	24	5.122%	10.494%	16.132%	26.973%
Bi-Weekly	26	5.124%	10.498%	16.142%	27.000%
Weekly	52	5.125%	10.506%	16.158%	27.049%
Daily	365	5.127%	10.516%	16.180%	27.116%
Continuous	∞	5.127%	10.517%	16.183%	27.125%
Federal Reserve benchmark rate range (2020-2024): 0.25% - 5.50%
Typical Savings Account (US)	365	APY usually 0.01% - 5.30%
Typical Credit Card (US)	365	APR usually 15% - 30%
Typical Mortgage (US, 30yr fixed)	12	APR usually 3% - 8%
Typical Auto Loan (US)	12	APR usually 4% - 12%

Frequently Asked Questions

Each compounding period adds earned interest to the principal before the next period begins. With n = 12 (monthly), interest earned in January accrues interest in February. With n = 1 (annually), no intra-year compounding occurs. The difference grows as APR increases. At 5% APR, the gap between annual and daily compounding is about 0.127%. At 24% APR, it exceeds 3.1%.

Negligible. For 5% APR, daily compounding yields APY of 5.12675% while continuous yields 5.12711%. The difference is 0.00036%. On $100,000 that is $0.36 per year. Continuous compounding is primarily a theoretical construct used in options pricing (Black-Scholes) and academic finance.

No. Regulation Z requires lenders to include certain fees (origination, discount points) in the disclosed APR, making it higher than the note rate. This calculator treats APR as the pure nominal annual interest rate without fee loading. To compare loan offers accurately, use the lender-disclosed APR that already includes fees, then convert to APY to see the true annual cost.

Most US credit cards compound daily (n = 365). A card advertising 24% APR actually costs 27.116% APY if balances carry month to month. On a $10,000 balance, that is $2,712 in annual interest rather than the $2,400 the APR figure suggests. A difference of $312.

Yes, only when compounding occurs once per year (n = 1). The formula reduces to APY = (1 + APR)¹ − 1 = APR. For any n > 1, APY > APR (assuming positive rates). Also, at 0% rate, APY = APR = 0 regardless of compounding.

The formula mathematically supports negative rates. A −0.5% APR with monthly compounding yields APY of approximately −0.4994%. Negative rates have been implemented by the European Central Bank and Bank of Japan. The calculator accepts negative inputs and produces correct results. Note that with negative rates, more frequent compounding results in a less negative (closer to zero) APY.