Saldo Devedor no Mês M (Price)

Outstanding balance at month m

How much you still owe after the m-th payment comes down to which amortization system you're on. Under Price, the balance is SD_m = PV·(1+i)^m − PMT·((1+i)^m − 1)/i, where PV is the financed amount, i the monthly rate and PMT the fixed installment. Under SAC it's SD_m = PV·(1 − m/n), which drops in a straight line with the month index. Price doesn't.

Take R$ 100,000 financed at 1% per month over 100 months. After 50 payments, Price still has roughly R$ 73,000 on the books, since those early installments are mostly interest. SAC, which pays down R$ 1,000 of principal every month, sits at exactly R$ 50,000 at that same moment. That gap is why the two systems hand you such different numbers when you ask to settle early.

Real-world applications

People reach for this when reconciling an annual bank statement, weighing an early settlement (quitação antecipada), sizing up refinancing (portabilidade de crédito), booking the liability in accounting, or checking insurance coverage tied to the outstanding balance (DFI/MIP in housing loans). It's also where any "shorten term vs. lower installment" simulation begins once you've made an extra payment.

FAQ

Why does Price leave so much balance after half the term? The fixed installment loads the interest up front. Real principal amortization only picks up speed toward the end of the schedule.

Is the SAC formula always linear? Yes. In pure SAC the principal portion is PV/n month after month, so the balance steps down by the same amount no matter what the rate is.

Does the bank quote match this calculation? It should come out close. The small differences usually trace back to pro-rata interest between the last payment date and the settlement date, along with a few residual fees.