APR vs APY: the rate before and after compounding

APR (annual percentage rate) is a yearly rate that does not include the effect of compounding within the year. APY (annual percentage yield) does: it is what you actually earn or pay over a year once interest is added to the balance and starts earning interest itself. Whenever interest compounds more than once a year, the APY is higher than the APR. The compound interest calculator lets you change the compounding frequency and watch the difference appear.

The short answer

In the US, lenders must disclose an APR under the Truth in Lending Act, and banks must quote an APY on savings accounts under the Truth in Savings Act. That split is convenient for whoever is quoting the number: the APR makes borrowing look slightly cheaper, and the APY makes saving look slightly more generous, because the APY is always the larger figure for the same underlying rate. Neither is misleading once you know which is which, but comparing an APR with an APY is comparing different things.

The conversion formula

To turn an APR into an APY, divide the APR by the number of compounding periods a year (n), add 1, raise it to the power n and subtract 1:

• APY = (1 + APR ÷ n) to the power n, minus 1

At 5% compounded monthly, that is (1 + 0.05 ÷ 12) to the power 12, minus 1, or about 5.12%. Compounded quarterly it is about 5.09%, and daily about 5.13%. The gap grows with the rate: a credit card at 24.9% APR compounded monthly costs about 27.95% a year in effective terms. That is the number to keep in mind when a card balance sits for a year; How credit card interest works, and how to pay less of it covers the daily mechanics.

How much compounding frequency matters

Less than people think, at ordinary savings rates. In the compound interest calculator, 10,000 left for 10 years at 5% grows to about 16,289 with yearly compounding, 16,470 with monthly compounding and 16,487 with daily compounding. Moving from yearly to monthly is worth about 181 over a decade; monthly to daily adds only about 17 more. The rate and the length of time matter far more than the compounding schedule, which is the main lesson of Compound interest, explained with real numbers. The calculator steps the balance month by month and applies the compounding factor at each period, and like any calculator it assumes the rate stays fixed, which variable savings rates rarely do.

Why a loan's APR includes fees

For borrowing, "APR" often means more than the nominal rate. Disclosure rules in many countries require the APR to include certain upfront fees, so it reflects the real cost of the credit. Take a 20,000 loan at 8.5% over 60 months: the payment is about 410 a month. If a 600 arrangement fee is deducted upfront, you receive only 19,400 but still repay 410 a month, and the APR calculator finds the true APR is about 9.80%. Compounded monthly, that is an effective annual rate of about 10.26%. This is why two loans with the same headline rate can have different APRs, and why the fee-inclusive APR is the better starting point for comparison.

Names used in different countries

• United States: APR for credit (a nominal monthly rate × 12, including certain fees) and APY for deposits.
• United Kingdom: AER (annual equivalent rate) for savings, which works like APY; APR for credit, which is already an effective annual figure including compulsory fees. A "representative APR" is the rate at least 51% of successful applicants receive, so yours may be higher. Mortgages quote an APRC.
• European Union: the APRC (annual percentage rate of charge) for consumer credit, also an effective annual rate including fees.

So a UK or EU APR behaves more like a US APY than a US APR. When comparing offers from different countries, check which convention is being used.

Which number to compare

• Savings: compare APY (or AER) with APY, and check whether the rate is fixed, variable or introductory.
• Loans: compare fee-inclusive APRs over the same term, then the total amount repayable, which the loan comparison calculator shows side by side.
• Cards: the purchase APR matters only if you carry a balance; if you pay in full each month, fees and rewards matter more.

To see how a target balance builds from regular deposits at a given rate, the savings goal calculator works backwards from the amount you want.