Nominal and effective rates: why 1% a month is not 12% a year

"Interest of 1% a month" and "interest of 12% a year" look like the same thing. They are not. The difference is in how interest accumulates, and understanding it keeps you from comparing offers by the wrong rate.

Compound interest: interest on interest

In a loan, each month's interest is charged on the outstanding balance, which already includes earlier months' interest that has not been paid. That is why rates accumulate by compounding rather than by adding up: effective annual rate = (1 + monthly rate)^12 − 1.

With 1% a month, the result is (1.01)^12 − 1 = 12.68% a year, not 12%. The extra 0.68 percentage point is interest on interest.

The effect shows up strongly when nothing is paid. A debt of R$ 1,000.00 at 2% a month reaches R$ 1,268.24 after 12 months and R$ 1,608.44 after 24 months: the second year adds more interest than the first, because the base it is charged on has already grown.

Effective rate and nominal rate

• The effective rate is the one that really accumulates over the stated period. "1% a month effective" is the rate that actually grows the debt in one month, and it equals 12.68% a year effective.
• The nominal rate is a rate declared by convention and split proportionally. "12% a year nominal" is divided into 12 months of 1%, and only then does the interest accumulate month by month.

In practice, "12% a year nominal" and "1% a month effective" are the same rate, and both equal 12.68% a year effective. "12% a year effective", on the other hand, is a lower rate: it corresponds to 0.9489% a month.

How to convert

• Monthly to annual: (1 + monthly rate)^12 − 1.
• Annual to monthly: (1 + annual rate)^(1/12) − 1. For 12% a year effective, it gives 0.9489% a month.
• Nominal annual to monthly: divide the rate by 12. For 12% a year nominal, it gives 1% a month.

The most common mistake

It is multiplying the monthly rate by 12 to get the annual one. With 1% a month the error is small, but it grows fast: 2% a month is not 24% a year but 26.82%, and 5% a month is not 60% a year but 79.59%. The other frequent mistake is comparing one offer's monthly rate with another's annual rate without converting one to the other's basis.

How to read an offer's rate

Before comparing two offers, confirm four things in each:

• The period: per day, per month or per year. A daily rate converts to monthly with (1 + daily rate)^30 − 1, counting 30 calendar days.
• The kind: nominal or effective. When it is not written, ask.
• The basis of comparison: convert all offers to the same period, preferably the effective year, and only then compare.
• The total cost: the interest rate does not include fees, insurance and taxes. For that, look at the CET.

The effect on the installment

The difference between the bases changes what you pay. For a loan of R$ 10,000.00 in 12 installments under the PRICE table:

The same "12%" changes the installment by R$ 2.87 a month and the interest by R$ 34.42 over a year. With larger amounts and longer terms, the difference grows. To learn about the PRICE table, see the guide PRICE or SAC.

How Finasst handles the rate

In the simulator, the "Rate basis" field has three options: Nominal per year, which divides the rate by 12 to get the monthly one, Effective per year and Effective per month. Pick the one the contract uses.

If the contract only says "interest of 12% a year", ask whether the rate is nominal or effective. Many contracts state both, the monthly and the annual, and the annual is usually the result of (1 + monthly)^12 − 1. When the contract gives both, using the effective monthly one removes doubt.