Turn an APR into the APY it actually pays at any compounding frequency, and turn an advertised APY back into the APR behind it.
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APY rises with more compounding, but only to a ceiling: past daily, the gain is a rounding error.
APR to APY: divide the APR by the number of compounding periods a year, add one, raise that to the power of the periods, subtract one, and turn it back into a percent. Continuous compounding replaces the power with the exponential function, so the APY is e to the APR minus one. The reverse direction takes the APY, adds one, raises it to one divided by the number of periods, subtracts one, and multiplies by the periods. For continuous the reverse is the natural log of one plus the APY.
Year-one interest is the APY applied to $10,000. The effective monthly rate is the twelfth root of one plus the APY, minus one — the rate that, paid once a month, produces the same result.
A 5% APR becomes a 5.1162% APY compounded monthly, 5.1246% weekly, 5.1267% daily and 5.1271% continuously. On $10,000, daily compounding therefore pays $512.67 in year one against $511.62 monthly — $1.06 apart. Going the other way, an advertised 5% APY compounded monthly is a 4.8889% APR, and the same 5% APY continuously is 4.8790% APR.
Savings accounts, CDs and money-market funds advertise APY, because compounding is included and the number looks bigger. Most credit-card and many loan quotes are APR, the periodic rate times the periods in a year, with compounding left out. Comparing a 5% APY savings account with a 5% APR loan, or feeding one figure into the other product's quote, compares two different things.
Nothing about fees, minimum balances, rate changes, taxes or withdrawals — all of which move the real answer more than the compounding frequency does. A card APR compounds on the average daily balance and a bonus APY often expires. These are conversions, not forecasts.
Informational arithmetic. Confirm a quoted rate, its compounding basis and its term in the disclosures before acting on it.