This post breaks down that formula, shows the JavaScript behind it, and runs a real example so you can see where every dollar goes.

The Formula: Amortization in One Line

A car loan is an amortizing loan: each monthly payment covers that month's interest first, and the rest reduces the principal. The monthly payment M is:

M = P × r(1 + r)^n / ((1 + r)^n − 1)

• P — the principal (amount borrowed)
• r — the monthly interest rate (annual APR ÷ 12)
• n — the number of monthly payments

That is the entire engine. Every car loan calculator on the internet — including the free Car Loan Calculator on Sigma Calculator — runs this formula.

The JavaScript (10 Lines)

function monthlyPayment(principal, annualRate, months) {
  const r = annualRate / 12;
  if (r === 0) return principal / months;
  const factor = Math.pow(1 + r, months);
  return (principal * r * factor) / (factor - 1);
}

// $25,000 at 6.35% APR, 60 months
monthlyPayment(25000, 0.0635, 60); // 487.40

One edge case matters: when the rate is zero, the formula divides by zero, so you return a straight split of the principal. Real calculators handle this; naive ones crash on 0% APR deals.

A Real Example, Worked Out

A $28,000 car with $3,000 down means you finance $25,000. At 6.35% APR:

60 months: $487.40/month — total paid $29,244, interest $4,244 72 months: $418.47/month — total paid $30,129, interest $5,129

The 72-month loan saves about $69 a month but costs $885 more overall. The formula does not care about your budget; it only tells the truth. That is why you run the numbers before you negotiate, not after.

And the rate itself is worth real money: the same $25,000 loan at 5% instead of 6.35% is $471.78 a month over 60 months — roughly $937 less across the loan. A single point of APR is not a rounding error.

Three Things the Formula Teaches You

1. Down payment is leverage. Every extra $1,000 down cuts roughly $19 a month off a 60-month loan at 6.35%. It reduces P, and everything downstream shrinks with it.

2. Term is a trade-off, not a discount. A longer term lowers M but raises n, and total interest grows with n. The calculator shows both numbers for a reason.

3. Check affordability backwards. Start from the monthly payment you can afford and solve for the car price that fits it. The Car Affordability Calculator does exactly that — same formula, run in reverse.

Try It Yourself

The formula above is everything you need to build your own version. Or skip the coding and use the free one on Sigma Calculator — no signup, instant results, and 540+ free calculators for math, finance, health, and everyday life.

The person who knows the numbers controls the negotiation. Now you know the formula behind them.