APR and APY describe the same annual rate from two sides: APR is the nominal rate before compounding, APY is what the balance actually earns once compounding is folded in. Enter either number with its compounding frequency, and this calculator converts it to the other and shows what a balance really does over time, including how much the compounding adds over simple interest.
APR, the annual percentage rate, is a nominal figure: it is the stated yearly rate with no adjustment for how often interest is added to the balance. APY, the annual percentage yield, is the effective figure: it answers the question of what one dollar grows to after a full year of compounding at whatever frequency the account uses. A 5.00 percent APR compounded daily and a 5.13 percent APY are the same account described two ways.
The distinction is enforced by regulation on both sides of the balance sheet. Banks must disclose the APY on deposit accounts under Regulation DD, the rule that implements the Truth in Savings Act, and lenders must disclose the APR on loans under the Truth in Lending Act. Each number is the honest one for its audience, which is why you will almost never see them swapped: savers see the yield, borrowers see the nominal rate.
Variables in plain terms. n is the number of compounding periods per year: 1 for annual, 2 for semiannual, 4 for quarterly, 12 for monthly, 365 for daily. APR is the nominal annual rate and APY is the effective annual rate, and the first two formulas convert in either direction. The balance formulas show what a starting balance does over a number of years, with the simple-interest line deliberately ignoring intra-year compounding so the last row isolates what compounding is worth.
Take $10,000 at a 5.00% APR compounded daily for 3 years. The equivalent APY is 5.1267%, so the first year turns the deposit into $10,512.67 even though the stated rate is only 5.00%. After 3 years the balance reaches $11,618.22, of which $1,618.22 is interest. Had the same 5.00% been applied as simple interest with no compounding, the 3-year interest would be $1,500.00 exactly. Compounding itself contributed $118.22, and over 10 years the same setup pays $1,486.65 more than the simple-interest version.
Frequency matters less than people expect at low rates and more at high ones. The table below holds the rate fixed at 5.00% and varies only how often interest is added.
| Compounding | Equivalent APY | 1-year interest |
|---|---|---|
| Annually | 5.0000% | $500.00 |
| Semiannually | 5.0625% | $506.25 |
| Quarterly | 5.0945% | $509.45 |
| Monthly | 5.1162% | $511.62 |
| Daily | 5.1267% | $512.67 |
From annual to daily the APY climbs only about 0.13 points, because the rate is small. The climb scales with the rate: at 25 percent, daily versus annual compounding is worth about 1.3 points of extra yield a year, which is why the compounding question is far from academic on credit cards and other expensive debt.
On deposits, compounding is a quiet bonus. On revolving debt it is a quiet tax. Credit cards quote an APR and then compound interest daily on any carried balance. A $5,000 balance left untouched for a year at a 24.99% APR accumulates $1,418.94 in interest, an effective annual cost of 28.38%. Simple interest on the same figure would have been $1,249.50, so daily compounding added $169.44 purely through frequency. A loan's APR also folds in most lender fees, which pushes it above the note rate, yet it still leaves out the compounding effect, so the true cost of carrying a balance sits above the advertised number.
That asymmetry explains the advertising pattern. The bank quotes the yield where it wants the number big and the rate where it wants the number small, and both quotes are perfectly legal because each is the disclosure the regulation requires. The defense is to convert everything to one measure before comparing, which is what the calculator above does in both directions.
Rate literacy pays off most around the products that quote rates aggressively. The CD interest calculator already works directly from the APY, because that is the number banks advertise on certificates of deposit, and it projects maturity values month by month. The compound interest calculator runs the same growth math over decades, where the compounding boost stops being pocket change and becomes the majority of the return. And on the debt side, the credit card minimum payment calculator shows how long a balance survives at a card's daily-compounded effective rate, which is the 28.38% story above told month by month. On the loan side, the loan payment calculator shows how the note rate and the APR diverge once fees enter, and how much that gap costs if you do not hold the loan to maturity.
For the same underlying rate, APY is always at least as large as APR, and the two are equal only when compounding happens once a year. A 5.00 percent APR compounded daily works out to a 5.13 percent APY. The gap grows with both the compounding frequency and the rate itself: at 5 percent the daily gap is about 0.13 points, while at 25 percent it is about 1.3 points.
Not quite. Under the Truth in Lending Act, a loan's APR must include most lender fees spread over the loan, so on a mortgage the APR runs slightly above the note rate. What APR does not include is the effect of intra-year compounding on a balance you carry. Credit cards compound interest daily, so a card advertising a 24.99 percent APR effectively charges about 28.38 percent a year on money that stays borrowed.
Regulation DD, which implements the Truth in Savings Act, requires deposit accounts to disclose APY because it is the honest number for what a saver earns. Lenders disclose APR under the Truth in Lending Act for the same reason on the borrowing side. Marketing follows the rule that sounds better in each position: a saver is more impressed by the larger effective yield, while a borrower is more impressed by the smaller nominal rate.
Most savings accounts, money market accounts, and credit cards compound daily, using either 365 or 360 as the day count. Certificates of deposit often compound monthly or quarterly, and some bonds pay semiannually. The frequency is always disclosed in the account agreement, and it is the only piece of information needed to convert between APR and APY.
Convert both to the same measure first. Enter the APR, its compounding frequency, and the calculator returns the equivalent APY, which you can then compare directly against the other account's advertised APY. Comparing an APR against an APY without converting systematically favors whichever number is larger, which is exactly the trap when a lender quotes 5.00 percent APR against a savings account advertising 5.13 percent APY.
The gap says nothing about quality. It is determined entirely by the compounding frequency, so a daily-compounding account at 4.90 percent APR (5.04 percent APY) beats a monthly-compounding account at 5.00 percent APR (5.12 percent APY) even though its gap is smaller. Always compare the effective annual numbers against each other and ignore the gap itself.
All calculation logic on this page is implemented from the primary documents below. Each one is listed with the figure it supplies, so any rate or threshold in the result can be traced back to its published source.
Same compounding math, different starting question.