APR is the yearly cost of borrowing as US federal rules define it under Regulation Z; APY is what a deposit earns in a year, compounding included, under Regulation DD. On a credit card, the APR is simply the periodic rate times the number of periods, so it leaves compounding out. On a car loan or mortgage, the APR folds certain fees into the rate. APY always includes compounding and never includes fees. That is why a 5% savings rate shows as a 5.13% APY with daily compounding, and why a 24.99% card APR compounds to about 28.4% on a balance left unpaid all year.
Educational only, not financial advice. This page explains how US disclosure rules define and compute two numbers; it does not recommend any account or lender. Regulations cited are as published in the eCFR, current to October 27, 2026. Worked examples use labeled hypothetical inputs.
The two definitions, from the regulations themselves
Most explanations of APR and APY start from a formula. The regulations start from a purpose, and the purpose explains the formula.
Regulation Z, the Truth in Lending rule at 12 CFR part 1026, defines the APR for installment credit in §1026.22(a)(1) as "a measure of the cost of credit, expressed as a yearly rate, that relates the amount and timing of value received by the consumer to the amount and timing of payments made." The words that matter are "amount and timing." An APR answers the question: if I receive this much money on this date and pay back these amounts on these dates, what yearly interest rate makes the two sides equal?
Regulation DD, the Truth in Savings rule at 12 CFR part 1030, defines the annual percentage yield in §1030.2 as "a percentage rate reflecting the total amount of interest paid on an account, based on the interest rate and the frequency of compounding for a 365-day period." The APY answers a different question: if I leave this deposit alone for a year, how much interest will it earn as a percentage of what I put in?
| APR (Regulation Z) | APY (Regulation DD) | |
|---|---|---|
| Used for | Loans and credit cards | Savings, checking, CDs, money market deposit accounts |
| Measures | Cost of credit as a yearly rate | Interest earned over a 365-day period |
| Includes compounding? | Not on a credit card; on an installment loan it is built into the actuarial math, but the result is still quoted as a nominal rate | Yes, by definition |
| Includes fees? | Finance charges, yes (on installment loans); some fees are excluded by rule | No: bonuses and fees are outside the APY |
| Accuracy tolerance | Generally 1/8 of 1 percentage point; 1/4 point for irregular loans | 1/20 of 1 percentage point (0.05%); rounded to two decimal places |
| Where you see it | Loan disclosure, card application table, monthly statement | Account disclosure, advertising, periodic statement (as APY earned) |
The last row of that table hides the most useful practical rule. When a bank advertises a rate of return on a deposit, §1030.8(b) says it "shall state the rate as an 'annual percentage yield' using that term," and that the plain interest rate may appear only "in conjunction with, but not more conspicuously than," the APY. On the credit side, §1026.18(e) requires a closed-end loan disclosure to show the annual percentage rate "using that term," with a brief description such as "the cost of your credit as a yearly rate."
How a bank computes the APY
Appendix A to Regulation DD gives the formula. It does not start from an interest rate at all. It starts from dollars:
APY = 100 × [(1 + Interest ÷ Principal)^(365 ÷ Days in term) − 1]
"Principal" is the amount assumed to be deposited at the start, "Interest" is the total dollar interest earned on it over the term, and "Days in term" is the actual number of days. For an account with no stated maturity, such as an ordinary savings account, the appendix says to assume a 365-day term, so the exponent becomes 1 and the formula collapses to a simple ratio: APY = 100 × Interest ÷ Principal.
The appendix works its own examples. A NOW account that pays $61.68 of interest over a 365-day year on $1,000 has an APY of 6.17%, because 61.68 ÷ 1,000 is 6.168%, rounded. A six-month certificate of deposit that pays $30.37 on $1,000 over 182 days has an APY of 6.18%, because 1.03037 raised to the power 365 ÷ 182 is 1.0618. Both check exactly by script.
Three assumptions are written into that calculation, and each matters when you compare accounts:
- Everything stays put. The appendix says institutions "shall assume that all principal and interest remain on deposit for the entire term and that no other transactions (deposits or withdrawals) occur during the term." If you withdraw interest as it is paid, you earn less than the APY.
- Only interest counts. The appendix excludes the value of a bonus (and anything worth $10 or less) and leaves out amounts that depend on circumstances that may or may not occur. A cash bonus for opening an account is real money, but it is not in the APY, so compare it separately.
- The rate is assumed to hold. For a variable-rate account, the appendix bases the APY on the rate in effect when the account is opened or advertised and assumes it will not change during the year. A variable APY is a snapshot, not a promise.
Regulation DD also sets how interest is counted day to day. Section 1030.7 requires institutions to calculate interest on the full principal each day by the daily balance method or the average daily balance method, using a daily rate of at least 1/365 of the interest rate. It does not require any particular compounding frequency. Compounding is therefore a choice the bank makes, and the APY is how you see the result of that choice.
Here is what that looks like for a hypothetical 6% interest rate compounded daily on $1,000. Each day adds 6% ÷ 365 of the balance, and after 365 days the account holds $1,061.83. The interest is $61.83, and the simple formula gives an APY of 6.18%. The same 6% compounded monthly would earn $61.68 in a year (6.17%), which happens to be the appendix's NOW-account example. Two accounts with the same 6% "interest rate" can therefore disclose APYs one hundredth of a point apart, and the APY is the number that tells you which earns more.
Converting a rate to a yield, and why frequency matters less than people think
When the only thing separating the two numbers is compounding, the conversion is the familiar textbook formula, where r is the nominal annual rate and n is the number of compounding periods in a year:
Effective annual rate = (1 + r ÷ n)^n − 1
The table applies it across the range of rates people actually see on deposits.
| Interest rate (nominal) | Compounded annually | Quarterly | Monthly | Daily (365) |
|---|---|---|---|---|
| 1.00% | 1.0000% | 1.0038% | 1.0046% | 1.0050% |
| 2.00% | 2.0000% | 2.0151% | 2.0184% | 2.0201% |
| 3.00% | 3.0000% | 3.0339% | 3.0416% | 3.0453% |
| 4.00% | 4.0000% | 4.0604% | 4.0742% | 4.0808% |
| 4.50% | 4.5000% | 4.5765% | 4.5940% | 4.6025% |
| 5.00% | 5.0000% | 5.0945% | 5.1162% | 5.1267% |
| 6.00% | 6.0000% | 6.1364% | 6.1678% | 6.1831% |
| 8.00% | 8.0000% | 8.2432% | 8.3000% | 8.3278% |
| 10.00% | 10.0000% | 10.3813% | 10.4713% | 10.5156% |
Two patterns stand out. First, at savings-account rates the compounding frequency is a rounding question. At 4%, moving from monthly to daily compounding adds 0.0066 points, which is 66 cents a year on $10,000. An account paying 4.10% compounded monthly beats one paying 4.00% compounded daily by a wide margin. The nominal rate dominates, and that is the whole reason the APY exists: it lets you compare a 4.10% monthly account with a 4.00% daily account on one number (4.18% against 4.08%).
Second, the gap between the rate and the yield grows faster than the rate. Double the rate and the gap roughly quadruples, because compounding is interest on interest. At 5% the daily-compounding gap is 0.13 points; at 10% it is 0.52. That is why compounding barely matters on a savings account and matters a great deal on a credit card.
Show the effective rates
| Nominal rate | Effective, monthly | Effective, daily | Gap, daily (points) |
|---|---|---|---|
| 1% | 1.00% | 1.01% | 0.01 |
| 2% | 2.02% | 2.02% | 0.02 |
| 3% | 3.04% | 3.05% | 0.05 |
| 4% | 4.07% | 4.08% | 0.08 |
| 5% | 5.12% | 5.13% | 0.13 |
| 6% | 6.17% | 6.18% | 0.18 |
| 8% | 8.30% | 8.33% | 0.33 |
| 10% | 10.47% | 10.52% | 0.52 |
| 15% | 16.08% | 16.18% | 1.18 |
| 20% | 21.94% | 22.13% | 2.13 |
| 24.99% | 28.06% | 28.38% | 3.39 |
| 30% | 34.49% | 34.97% | 4.97 |
Credit card APR: a nominal rate by rule
Regulation Z treats open-end credit, which includes credit cards, differently from installment loans. Section 1026.14(b) says that where periodic rates are used to compute the finance charge, the APR to be disclosed "shall be computed by multiplying each periodic rate by the number of periods in a year." Nothing in that sentence compounds.
Run a hypothetical card with a 24.99% purchase APR through it. If the issuer uses a daily periodic rate, that rate is 24.99 ÷ 365 = 0.06847% a day. If it uses a monthly periodic rate, it is 24.99 ÷ 12 = 2.0825% a month. On a card statement, §1026.7(b)(4) requires each periodic rate used to compute the interest charge to be shown "expressed as an annual percentage rate," along with the balances it applies to, so the APR on your statement is this multiplied figure.
Now leave a balance on that card for a full year with no payments and no new purchases, which no real account would do but which isolates the compounding effect. Interest charged on interest pushes the effective cost above the disclosed APR:
| A $5,000 balance carried for one year at a 24.99% APR (hypothetical) | Interest | Effective annual rate |
|---|---|---|
| If interest were simple (no compounding) | $1,249.50 | 24.99% |
| Compounded monthly at 2.0825% | $1,403.03 | 28.06% |
| Compounded daily at 0.06847% | $1,418.94 | 28.38% |
The disclosed APR is 24.99% in every row. The extra $169.44 between the first and last rows is the compounding the APR leaves out by design. Real balances are paid down, so the dollar figures overstate any actual bill, but the effective rate per dollar carried is right.
Two more features of the card APR follow from the rules and catch people out:
- Some fees never reach the APR. Section 1026.4(c) lists charges that are not finance charges, including "fees charged for participation in a credit plan, whether assessed on an annual or other periodic basis" and charges for late payment or exceeding a credit limit. An annual fee is therefore outside the card's APR. A $95 annual fee on a card used for $3,000 of carried balances is a cost of about 3.2% of that balance a year that the APR does not show.
- The headline APR is set by a table. For card applications and solicitations, §1026.60 requires the disclosures in a table substantially similar to the model forms, with the APRs in bold text. That table is the right place to read the rate, not the marketing around it.
The APY calculator shows what a given rate and compounding frequency produce in a year, which is also what a carried card balance compounds to. For loans, the APR calculator solves the APR from the amount, the upfront fees, the rate and the term, which is the method in the next section.
Installment loan APR: where fees enter the rate
For a car loan, personal loan or mortgage, the APR does more work. It is computed from the finance charge, which §1026.4(a) defines as "the cost of consumer credit as a dollar amount," including "any charge payable directly or indirectly by the consumer and imposed directly or indirectly by the creditor as an incident to or a condition of the extension of credit." The examples in §1026.4(b) include interest, "points, loan fees, assumption fees, finder's fees, and similar charges," and appraisal, investigation and credit report fees, subject to the exclusions in paragraphs (c) through (e).
Fees paid up front reduce the amount financed. Section 1026.18(b) builds the amount financed from the principal loan amount, adds other amounts financed that are not finance charges, and subtracts any prepaid finance charge. The APR is then the rate that makes the payment stream worth exactly the amount financed. Appendix J to Regulation Z spells out the actuarial method: each period the unpaid balance grows by the finance charge earned and falls by the payment made, and the APR "shall be the nominal annual percentage rate determined by multiplying the unit-period rate by the number of unit-periods in a year." Even here, the final step is a multiplication, not a compounding.
A labeled hypothetical shows what a fee does. Borrow $10,000 at an 8% note rate for 36 months. The monthly payment is $313.36. Now suppose a $300 loan fee is charged as a prepaid finance charge, so the amount financed is $9,700 while the payments stay $313.36. Solve for the rate that makes 36 payments of $313.36 worth $9,700 and you get an APR of 10.08%. The total finance charge is $1,581.09: $1,281.09 of interest plus the $300 fee.
Keep the fee and the note rate and change only the term:
| Same $10,000 at 8%, same $300 prepaid fee | Monthly payment | APR |
|---|---|---|
| 12 months | $869.88 | 13.77% |
| 36 months | $313.36 | 10.08% |
| 60 months | $202.76 | 9.30% |
The fee is identical in all three rows. On the short loan it is spread over one year, so it adds almost six points to the rate; on the five-year loan it adds 1.3 points. That makes the APR a fair comparison between offers for the same term, and a misleading one across terms: in these three rows the longer loan shows the lower APR while costing more total interest.
The regulation also allows some slack. Section 1026.22(a)(2) treats an APR as accurate if it is within 1/8 of 1 percentage point of the correctly computed rate, and §1026.22(a)(3) widens that to 1/4 of a point for irregular transactions with features such as multiple advances or irregular payment amounts. Two APRs that differ by 0.1 point may not represent a real difference at all.
Mortgage APR: the full-term assumption
Mortgages add two wrinkles. First, some closing costs are carved out of the finance charge. Section 1026.4(c)(7) excludes, for transactions secured by real property or residential mortgage transactions, bona fide and reasonable fees for title examination and title insurance, property surveys, preparing loan documents, notary and credit-report fees, appraisals and pre-closing inspections, and escrow deposits that would not otherwise be finance charges. Points and loan fees stay in. Two lenders can therefore charge you the same total closing costs and disclose different APRs, depending on which of their fees fall on which side of that line.
Second, the APR assumes you keep the loan for its full term. Take a hypothetical $300,000, 30-year fixed loan at a 6.5% note rate with $4,500 of prepaid finance charges. The payment is $1,896.20 and the APR comes to 6.65%. If you sell or refinance early, the arithmetic changes. If the same loan is paid off at the end of year 7, when the balance is $271,248.73, the effective rate over the seven years you actually had it is 6.78%. The same $4,500 is spread over 84 months instead of 360, so it weighs more heavily. The shorter you expect to keep a loan, the more the up-front fees matter and the less the disclosed APR tells you.
The amortization calculator shows the balance at any month, which is the number you need to run that kind of early-exit check, and the mortgage calculator prices the payment itself.
When the two numbers meet, and when they cannot
Three situations confuse the APR and the APY in practice.
A deposit account that says "annual percentage rate." Regulation DD defines the interest rate as "the annual rate of interest paid on an account which does not reflect compounding," and adds that for account disclosures the interest rate "may, but need not, be referred to as the 'annual percentage rate'" as well. So an APR on a savings disclosure is the nominal interest rate, and the APY beside it is the number that includes compounding.
Converting a loan APR into an APY. The compounding formula only works when the APR is a pure periodic rate with no fees inside it. For a credit card you can apply it to see what an unpaid balance compounds to. For an installment loan the APR already reflects fees and the payment schedule, and turning it into an "APY" produces a number no regulation defines. If you want the effective annual cost of a loan, compute it from the monthly APR with the same formula, (1 + APR ÷ 12)^12 − 1, and call it an effective rate, not an APY: the 10.08% loan above is about 10.56% on an effective basis.
APY earned on a statement. Your periodic statement may show an "annual percentage yield earned," which Part II of the Regulation DD appendix computes from the interest actually credited and your average daily balance for the statement period. The appendix's example is a 30-day period with an average balance of $1,000 and $5.25 of interest, which annualizes to 6.58%. It moves with your balance and any rate changes, so it will not always match the advertised APY.
A quick check on any rate you are quoted
Four questions turn any quoted rate into something you can compare, and each one maps to a rule above.
First, is it a yield or a cost? Money coming to you should be quoted as an APY, and in an advertisement for a deposit account it has to be. Money going from you should be quoted as an APR. If a savings offer leads with a plain "rate," look for the APY beside it.
Second, does it compound inside the year, and how often? For a deposit the APY has already done that work. For a card, the APR has not, so divide it by 365 or 12 to see the periodic rate on your statement and run the compounding formula if you carry a balance.
Third, what fees sit inside it, and which sit outside? On an installment loan, points and loan fees raise the APR. On a card, the annual fee does not. On a mortgage, title, appraisal and document fees do not. Write the outside fees down next to the rate so they are not forgotten.
Fourth, how long will you actually hold it? A deposit APY assumes a year with nothing withdrawn. A loan APR assumes every payment is made to the end of the term. If your plan is shorter, the fee-heavy offer gets worse and the simple one gets relatively better, as the 12-month and seven-year examples showed.
What to compare, product by product
| You are comparing | Compare this | Watch for |
|---|---|---|
| Savings accounts, money market deposit accounts | APY | Variable rates (APY is a snapshot); bonuses outside the APY; fees |
| Certificates of deposit | APY for the same term | Early withdrawal penalties; whether interest must be withdrawn |
| Credit cards | Purchase APR, then the full application table | Annual fees outside the APR; penalty APRs; how long an introductory rate lasts |
| Personal and auto loans of the same term | APR | Terms that differ (fees look smaller on longer loans) |
| Mortgages | APR, plus total fees and the rate itself | Full-term assumption; excluded closing costs; how long you will keep the loan |
The rule underneath every row is the same. Use the APY when money comes to you and the APR when money goes from you, and only compare numbers built the same way. A 4.6% APY on a savings account and a 4.6% APR on a loan are not "the same rate": the first already includes a year of compounding, and the second may include fees.
Mistakes that cost money
Comparing a deposit APY with a loan APR to decide whether to pay down debt. The fair comparison is like for like. A card at 24.99% APR compounds to about 28.4% on a balance carried all year, against a savings account's APY. The gap between what the card costs and what the deposit earns is wider than the two headline numbers suggest.
Reading the APR as the total cost of a short or early-exit loan. The 12-month version of the $10,000 loan above disclosed 13.77% for exactly the same $300 fee as the 36-month loan at 10.08%. Look at the finance charge in dollars as well as the APR.
Treating a variable APY as fixed. Regulation DD tells the bank to compute it on the rate in effect when the account is opened or advertised. If rates fall, so does the yield.
Ignoring the frequency once rates are high. On savings it rarely matters. On a balance carried all year at a 24.99% APR, daily compounding adds about 3.4 points.
For the Canadian version of card interest, where lenders disclose rates under different rules, the credit card interest guide for Canada works the calculation from a statement. Compounding over longer periods, where the same arithmetic drives savings growth, is laid out in the compound interest calculator, and the debt avalanche vs snowball guide uses APRs to order debts for payoff.
Sources
All regulations read from the eCFR (Title 12, current to October 27, 2026); calculations re-run by CalculatorVillage on October 29, 2026.
- 12 CFR 1026.4, Finance charge: definition, examples in paragraph (b), exclusions in paragraph (c) including participation fees and real-estate related fees.
- 12 CFR 1026.7, Periodic statement: periodic rates and the corresponding APR on statements.
- 12 CFR 1026.14, Determination of annual percentage rate (open-end): periodic rate times periods, 1/8 point tolerance.
- 12 CFR 1026.18, Content of disclosures: amount financed and finance charge.
- 12 CFR 1026.22, Determination of annual percentage rate (closed-end): definition, actuarial or US Rule method, 1/8 and 1/4 point tolerances.
- 12 CFR 1026.60, Credit and charge card applications and solicitations: the tabular disclosure and bold APRs.
- Appendix J to Part 1026: the actuarial method.
- 12 CFR part 1030 (Regulation DD), §1030.2 definitions, §1030.3 rounding and accuracy, §1030.7 payment of interest, §1030.8 advertising and Appendix A, Annual Percentage Yield Calculation.