1. The Mathematics of Amortization: The PMT Equation
Amortization is the process of spreading out a loan into a series of equal periodic payments. Each payment is divided into paying off the interest accrued for that period and reducing the principal balance.
To solve the monthly payment (PMT) of a fixed-rate loan, financial institutions use the standard annuity equation. Using P (Principal loan amount), r (Monthly interest rate, which is the annual rate divided by 12), and n (Total number of monthly payments), the monthly payment is solved using standard algebra.
2. The Principal vs. Interest Shifting Curve
At the beginning of a loan term, the principal balance is at its highest point, meaning the calculated interest share of your monthly payment is also at its peak. Only a small fraction of the monthly payment goes toward reducing the principal.
As the principal balance is gradually paid down, the interest charge for each subsequent month decreases. Consequently, a larger percentage of your fixed monthly payment is directed toward principal reduction. This creates an exponential curve where equity accumulation starts slowly and accelerates rapidly toward the end of the loan term.
// Standard monthly amortization step resolver
function resolveAmortizationStep(balance: number, monthlyRate: number, pmt: number) {
const interestPayment = balance * monthlyRate;
const principalPayment = pmt - interestPayment;
const newBalance = balance - principalPayment;
return { interestPayment, principalPayment, newBalance };
}3. The Compounding Power of Prepayments
Prepayments are additional payments made directly to the loan principal, outside of the required monthly payment schedule. Because prepayments reduce the outstanding principal balance immediately, they alter the compounding curve.
Subsequent monthly interest charges are recalculated based on the lower balance, meaning more of your regular payment goes toward principal. Over time, making consistent prepayments significantly reduces the total interest paid and shortens the loan term by years.