APR vs APY: nominal rate against yield

Under US rules an APR is a nominal rate. Take the rate charged in one period and multiply by the number of periods in a year: 2 percent a month becomes a 24 percent APR. That multiplication assumes interest never earns interest, which stops being true the moment anything is added to the balance more than once a year.

An APY is the effective rate. It answers what a balance grows by over a full year with the compounding counted:

$$APY = \left(1 + \frac{APR}{n}\right)^{n} - 1$$

Here $n$ is how many times a year interest is added. At a 5 percent nominal rate, monthly compounding produces 5.116 percent and daily produces 5.127 percent. Push $n$ higher still and the result converges on a continuous limit rather than running away, so at ordinary deposit rates the level of the rate moves the answer far more than the frequency does. At card rates the frequency is worth whole percentage points instead, which the next section shows.

That leaves the two numbers with different completeness. An APY stands alone, because the compounding is already inside it. An APR does not: with no frequency attached it cannot be turned into what a year actually costs or pays, and two accounts quoting the same APR at different frequencies are not the same account. The APR against APY calculator converts in both directions, and the general name for the APY side is the effective annual rate.

One qualification, and it bites the moment a comparison crosses a border. Reading APR as a nominal rate is a US convention rather than a fact of arithmetic. Consumer credit rules in the United Kingdom and the European Union define their APR as an effective annual rate, with the compounding already inside it, so a loan rate labelled APR outside the US may already be closer to what a US page would call an APY. Find out which convention a quote was written under before setting it against another.

On a deposit the two terms differ by compounding alone. On US borrowing there is a second difference, and it pushes the other way.

In the United States, the APR a lender discloses on a mortgage, a car loan or a personal loan is required by the Truth in Lending Act to fold in certain charges beyond the interest on the note. A low note rate carrying heavy setup fees therefore shows up in the APR even when the advertised rate does not give it away, which is why a loan advertisement carries two figures: the note rate, and an APR at or above it. Deposit accounts run under a different rule, the Truth in Savings Act, and that is why a US savings page quotes a yield rather than a rate.

The fee rule is narrower than it sounds. It covers closed-end credit, meaning a fixed amount on a fixed schedule. A credit card is open-end credit, and its disclosed APR is the periodic rate multiplied up, with the annual fee disclosed beside it rather than folded into it. So on a card only the compounding difference is in play. Take a card quoting a 24.99 percent APR and compounding monthly: interest alone comes to about 28.1 percent of a balance carried for the whole year, roughly three points above the advertised number, and any annual fee sits on top of that. Many issuers apply a daily periodic rate instead, which widens the gap a little further.

So the two adjustments run in opposite directions wherever the fee rule bites. An instalment loan APR is pushed up by fees and left flat by compounding, while an APY is pushed up by compounding and touched by no fees at all. Neither number on its own is the whole cost of borrowing.

For deposits the rule is short: put every quote into APY terms, then compare. The trap is a headline that looks bigger and is not. A 5.05 percent APY is worth less over a year than a nominal 5 percent compounded monthly, because the second works out to 5.116 percent. Those levels are illustrative, and the reversal does not depend on them: it comes from the compounding, so it is available wherever rates happen to sit.

Inside the United States that particular pairing is rare in the open, because Truth in Savings makes a deposit account quote its yield. It turns up constantly elsewhere, and on anything quoted as a nominal rate with a compounding frequency printed beside it, where the conversion is yours to do.

For loans the answer depends, and what it depends on is how long you keep the loan. Between two loans of the same size and the same term, disclosed under the same rules and both held to the end, the lower APR is the cheaper loan, because that is the comparison the disclosure was built for. Repay early and it bends: an APR spreads upfront charges across the whole term, so a loan with large fees looks better under an APR than it proves to be when it is cleared in a fraction of that term. Loans of different lengths are not directly comparable on APR for the same reason.

What never works is setting a loan APR against a savings APY. One carries fees and skips compounding, the other does the reverse, so part of the distance between them is contents rather than rate. Put both in the same terms first. The loan payment calculator works in note-rate terms, which is what drives the payment schedule.

Both figures answer precise questions, and both carry assumptions that are easy to miss.

An APY annualises the rate an account pays today. On a variable-rate account that is no commitment for the next twelve months, and it also assumes the balance is left alone, since money taken out partway through stops compounding and an early withdrawal from a term deposit can carry a penalty the yield never shows. It counts interest only, so a monthly maintenance charge sits outside the yield while still reducing what you keep.

An APR assumes the loan runs its full term. It levels the setup costs over every year of the schedule, so a mortgage disclosed over thirty years spreads its fees across thirty years of arithmetic while a borrower who refinances or sells in five pays them out of five. The shorter the holding period, the further the disclosed APR sits below the cost actually borne. It also assumes competing offers package their charges the same way, and the rules decide which charges count, so two lenders can quote one APR between them and still not cost the same.

• Same APR, different compounding frequency: not the same cost.
• Same APY, different account charges: not the same return.
• Same APR, different loan terms or different holding periods: not a like-for-like comparison.
• Same three letters, different country: check whether that APR is nominal or effective first.

Read that way, the pair stops being two names for one idea. The compound interest calculator runs the same mechanism forward over many years, where a fraction of a percentage point turns into a gap worth reading.