How it works
This tool answers a reverse question. A lender or a car dealer hands you a quote: borrow a certain sum, pay a fixed amount each month for a set number of months, and the balance hits zero. What they sometimes leave out — or bury in fine print — is the actual annual percentage rate being charged. That rate is the single most important figure for comparing offers, and this calculator extracts it from the other three knowns.
You supply the amount financed, the monthly installment, and the repayment term in months. The engine then iterates to find the periodic rate that, when amortized across the full term, reproduces the exact payment you entered. The output is an implied annual rate.
One caveat: the result is a mathematical estimate, not professional financial advice. It does not account for origination fees, prepayment penalties, or escrow items that may be bundled into a real-world offer.
The formula
0 = P × r × (1 + r)^n / ((1 + r)^n - 1) − M
Here, P is the principal, M is the monthly installment, n is the term in months, and r is the monthly rate. The equation cannot be rearranged algebraically for r, so the calculator uses numerical root-finding (Newton-Raphson) to converge on the value that drives the left side to zero. The annual rate is then r × 12.
Worked example
A furniture store offers a $15,000 financing plan. The payment is $300 every month for 5 years (60 months). The stated "promotional rate" is vague, so you want to verify the true cost of borrowing.
Term in months: 5 × 12 = 60
The solver tests candidate monthly rates until the amortization formula returns a payment of exactly $300 on a $15,000 balance over 60 months.
Monthly rate found: r ≈ 0.0049196
Annual implied rate: 0.0049196 × 12 ≈ 0.0590 (5.90%)
So the financing plan carries an implied annual rate of roughly 5.90%. That is the number to compare against competing offers or a bank personal loan.
| Input | Value |
|---|---|
| Amount financed | $15,000 |
| Monthly installment | $300 |
| Term | 60 months |
| Solved monthly rate | 0.49196% |
| Implied annual rate | ≈ 5.90% |
Things to watch
Escrow contamination. If the $300 includes property taxes, gap insurance, or a warranty, the solver treats all of it as interest-bearing debt. Strip out non-interest portions before entering the payment, or the backed-out rate will look far higher than reality.
Origination and documentation fees. A lender may charge $500 in fees on a $15,000 advance. If those fees are financed, the true principal is $15,500 and the rate derived from a $15,000 entry will be understated. If the fees are paid out of pocket, the effective borrowing cost is still higher than the nominal rate shown.
Balloon payments. This tool assumes the balance fully amortizes to zero over the stated term. If the contract has a final lump-sum payment, the standard equation no longer applies and the solver will return a rate that does not reflect the deal.
Rate vs. APR. The figure produced here is a nominal annual rate. A formal APR calculation rolls in certain mandatory fees, so it can exceed the rate you derive from principal, payment, and term alone. Use the result as a close approximation for comparison, not a regulatory disclosure.