Annual Percentage Rate (APR) = Periodic rate X number of periods in a year
Typically the periods would be daily, weekly, 15 days, monthly, quarterly, 6 months, or annually depending upon the agreement you signed.
Annual Percentage Yield (APY) = (100%+Periodic rate)^number of periods in a year - 100%
You can do a mathematical proof if you want, but APY will always > APR due to inter-period compounding (exponent of # of periods).
For most loans, usually the APR will get converted to a daily periodic rate and that daily rate is then applied to the loan balance at the end of the day. As mentioned above, the APY depends on your balance, and may or may not be higher than the APR depending upon mid-year transfers.
When shopping for loans, banks will often quote you APR because this number is lower. When shopping for investments, they will quote you APY because this is higher.
I personally like to convert the APR to a daily periodic rate regardless of the compounding period because this gives you a more accurate estimate of the true cost of capital on your loans and the true rate of return on your investments.
I have almost never seen accounts that are compounded more frequently than daily.
Of course you can compare interest rates for 2 accounts with different compounding periods, but you have to normalize the interest rate for both to a common compounding period, and take into account how the bank calculates the average periodic balance.
It can get complicated quickly
