APR, APY and the compounding in between
A nominal annual rate — often labelled APR or simply 'interest rate' — is the rate before compounding. APY, the annual percentage yield, is what you actually earn over a year once each period's interest starts earning interest itself: APY = (1 + r ÷ m)^m − 1, where r is the nominal rate and m the number of compounding periods a year. 5% compounded monthly is 5.116% APY; compounded daily, 5.127%. US banks are required by the Truth in Savings Act to advertise deposit products by APY precisely so that different compounding schedules can be compared on one number.
The table converts common nominal rates at each frequency. The gap between nominal and effective grows with the rate: at 1% compounding adds almost nothing; at 20% daily compounding adds more than two points.
| Nominal | Annual | Quarterly | Monthly | Daily | Continuous |
|---|---|---|---|---|---|
| 1% | 1.000% | 1.004% | 1.005% | 1.005% | 1.005% |
| 3% | 3.000% | 3.034% | 3.042% | 3.045% | 3.045% |
| 5% | 5.000% | 5.095% | 5.116% | 5.127% | 5.127% |
| 7% | 7.000% | 7.186% | 7.229% | 7.250% | 7.251% |
| 10% | 10.000% | 10.381% | 10.471% | 10.516% | 10.517% |
| 15% | 15.000% | 15.865% | 16.075% | 16.180% | 16.183% |
| 20% | 20.000% | 21.551% | 21.939% | 22.134% | 22.140% |
What compounding is worth in dollars
On a $10,000 deposit at a 5% nominal rate, the first year's interest is $500 with annual compounding and $512.67 with daily. The extra $12.67 is interest on interest. Small on one year; the effect compounds over many years, and it is the whole reason APY, not the nominal rate, is the right basis for comparison.
| Compounded | Interest | Extra from compounding |
|---|---|---|
| Annually | $500.00 | $0.00 |
| Quarterly | $509.45 | $9.45 |
| Monthly | $511.62 | $11.62 |
| Daily | $512.67 | $12.67 |
| Continuously | $512.71 | $12.71 |
Comparing accounts
Banks quote in different ways. Convert everything to APY before comparing: a slightly lower rate compounded daily can beat a higher rate compounded quarterly.
| Bank | Quoted | Effective yield |
|---|---|---|
| Bank A | 4.90% rate, compounded daily | 5.022% APY |
| Bank B | 4.95% rate, compounded quarterly | 5.043% APY |
| Bank C | 5.00% APY | 5.000% APY |
APY on debt: credit cards
The same arithmetic applies to what you owe. Credit cards quote an APR and compound daily on the carried balance, so the effective rate is higher than the sticker. A 22% APR is about 24.6% effective — and on a $5,000 balance carried for a year, roughly $1,230 in interest.
| APR | Effective annual rate | Interest on $5,000 for a year |
|---|---|---|
| 18% | 19.72% | $985.82 |
| 22% | 24.60% | $1,229.97 |
| 25% | 28.39% | $1,419.58 |
| 29% | 33.63% | $1,681.37 |
APY back to a nominal rate
The reverse conversion, APR = m × [(1 + APY)^(1 ÷ m) − 1], is useful when a product advertises APY but you need the period rate — to compute a month's interest by hand, to enter a rate into a calculator that expects a nominal figure, or to match a bank's daily accrual. Switch to the APY → APR tab. For continuous compounding the formulas become APY = e^r − 1 and r = ln(1 + APY).
- APY = (1 + r ÷ m)^m − 1; APR = m × [(1 + APY)^(1/m) − 1].
- Continuous: APY = e^r − 1; r = ln(1 + APY).
- APY is the same thing as the effective annual rate (EAR) used in finance texts.
- Two accounts with the same APY pay the same over a year regardless of compounding frequency.
APR on loans is a different animal
For loans, APR by law includes certain fees — origination fees, points, some closing costs — spread over the loan's life, so a loan's APR can be higher than its interest rate for a reason unrelated to compounding. That APR is a cost-of-borrowing disclosure, not a compounding conversion; the personal loan and mortgage calculators compute it from the fees. This page is about the compounding relationship between a nominal rate and its effective yield.
Converting one nominal rate to another
Two rates compounded differently are equivalent when they give the same APY. The Rate A → rate B mode converts through the effective rate: find the APY of rate A, then the nominal rate B that compounds to the same APY, B = m × [(1 + APY)^(1 ÷ m) − 1]. Use it to compare a quarterly-compounded CD with a monthly one, or to enter a rate into a formula that expects a different period.
Worked example: 6% compounded monthly has APY (1.005)^12 − 1 = 6.168%. The quarterly rate with the same APY is 4 × [(1.06168)^(1/4) − 1] = 4 × (1.005^3 − 1) = 6.030%, and the annually compounded equivalent is 6.168%.
- Periodic rate = APR ÷ m: 5% compounded monthly accrues 5% ÷ 12 = 0.4167% a month.
- Daily rate = APR ÷ 365: 5% is 0.0137% a day, the figure a daily-accrual account applies to each day's balance.
- The compound interest calculator shows the same equivalent-rate step next to its growth projection.