Yearly ↔ Monthly Rate Equivalence

How to convert between annual and monthly rates

To convert equivalent compound rates, use i_year = (1 + i_month)^12 − 1, or run it backwards with i_month = (1 + i_year)^(1/12) − 1. Watch out for the nominal rate (annual divided by 12). That one is simple interest, and it understates the real monthly effective rate whenever the contract actually compounds.

Take a concrete case. Selic at 15% per year comes out to (1.15)^(1/12) − 1 ≈ 1.17% per month. It is not 1.25%, which is what you would get from 15/12. And a CDB advertised at "1% per month" is really paying (1.01)^12 − 1 = 12.68% per year.

Practical applications

Say you want to compare a monthly CDB against an annual LCI, work out the Total Effective Cost (CET) of a loan, or figure out if it pays to move money between products. None of those comparisons mean anything until you have pulled every rate onto the same period.

FAQ

Can I divide an annual rate by 12 to find the monthly? Only when the contract spells out "nominal rate". If you need compound equivalence, reach for the 1/12 exponent instead.

Why is the effective annual rate higher than 12× the monthly? Each month's interest goes on to earn its own interest in the months that follow. That compounding slowly drags the annual figure above the simple multiple.

Does this work for daily and quarterly rates too? It does. The general formula is i_b = (1 + i_a)^(a/b) − 1, where a and b count how many periods per year each rate has.