APY

APY answers one question. If a balance is left alone for a year and nothing about the account changes, what fraction more will be there at the end? Compounding is already inside the number, so two accounts can be compared on their APY without knowing how often either of them pays interest. In the United States, deposit accounts disclose returns as an APY under the Truth in Savings Act, which is why a savings page almost always shows a yield rather than a rate.

The arithmetic is $APY = (1 + r/n)^{n} - 1$, where $r$ is the nominal rate written as a decimal and $n$ is how many times a year interest is added. Paying more often raises the result, but only up to a ceiling. At a 5 percent nominal rate, monthly compounding produces 5.116 percent and daily produces 5.127 percent, and no frequency gets past the continuous limit just above that. The size of the rate matters far more than the frequency does.

The mistake is comparing an APY against an APR. In the United States an APR is quoted as a nominal rate, so the compounding is left out of it, while in the United Kingdom and the European Union the same three letters mean a rate that already compounds. An account advertising 5.05 percent APY is the worse of the two against one advertising 5 percent APR compounded monthly, because that second account is worth 5.116 percent once a year of compounding is applied. Convert everything to a yield before comparing anything. The APR against APY calculator converts in both directions, and the effective annual rate is the same quantity under its general name.

One thing an APY does not promise is that the rate lasts. A variable-rate account quoting a yield is annualising the rate it pays today, not committing to it for the next twelve months. Left in place over many years, the same mechanism is what the compound interest calculator runs forward.