The formula
APY = (1 + APR ÷ m)^m − 1 · period rate = (1 + APY)^(1/n) − 1
Where it comes from
A nominal rate is divided across the periods: 60 % nominal a year compounded monthly is 5 % each month. An effective rate already carries the compounding inside it, so to bring it down a period you take the root rather than dividing: 60 % effective a year is 3.99 % monthly.
How to work it out by hand
- Check whether your rate is nominal or effective: the contract says which
- If it is nominal, divide it by the periods to get the period rate
- Compound that rate as many times as there are periods in the year for the effective annual rate
- To go down a period from an effective rate, take the root rather than dividing
What is worth knowing
The only figure comparable between products is the effective annual rate, which is why the law requires it almost everywhere: APR and APY in the United States, TAE in Spain, TEA in Argentina and Peru. Two loans quoting the same nominal rate can cost different amounts if one compounds monthly and the other quarterly. And there is a detail that surprises people: the jump from compounding once a year to monthly is large, but from monthly to daily is barely noticeable, because the series converges towards continuous compounding. At 24 % nominal the effective annual rates are 24 %, 26.82 % and 27.11 %.