What rate are you
actually paying?

Field guide

How to calculate the interest rate of a loan.

For an amortising loan, the relationship between principal, payment, term, and interest rate is captured by the standard amortisation formula:

where P is the principal, r is the monthly rate (decimal), and n is the number of months. Three of the four variables can be solved algebraically for the fourth — but not r. The exponent (1 + r)−n stops you from isolating it. So when a lender hands you P, PMT, and n and refuses to tell you the rate, you have to solve numerically.

Why Newton-Raphson?

We rearrange the equation so one side is zero:

and find the r that makes f(r) = 0. Newton-Raphson works on any smooth function: from a starting guess r₀ you draw the tangent line at (r₀, f(r₀)) and use the intercept as the next guess r₁. The update rule is r₁ = r₀ − f(r₀) / f′(r₀). Convergence is quadratic — every iteration roughly doubles the number of correct digits. Most loan inputs solve to 12 decimals in under 10 iterations.

The starting guess matters

Newton can race off in the wrong direction from a bad seed , particularly toward the r ≤ 0 region where the equation behaves badly. This calculator seeds with a coarse estimate based on the over-payment fraction:

and adds a back-off step that halves the Newton step whenever the next iterate would land on or below zero. The net effect is a solver that is both fast for sane inputs and stable for absurd ones.

Worked example: $20,000 over 5 years at $400/month

Total payments are $400 × 60 = $24,000, so the implicit interest is $4,000. Plug P = 20,000, n = 60, and PMT = 400 into the residual and iterate: r ≈ 0.006183 per month, or APR ≈ 7.420%. Effective annual rate (compounding monthly) is (1.006183)12 − 1 ≈ 7.678%. Newton-Raphson gets there in four iterations.

The three failure modes

• Payment too low: if PMT < P / n, the payment doesn't cover the principal in n months even at zero interest. The implied rate is negative; the calculator rejects this with an explicit message rather than hand back nonsense.
• Exact zero-interest: if PMT = P / n exactly, the rate is zero. The formula has a removable singularity there, so we return 0 directly instead of letting the solver divide by zero.
• Non-convergence: for absurd inputs (rate > 100% per month), the cap kicks in. In practice this only fires for typos.

Why a 1-percentage-point swing matters

Rate is the most leveraged input on a long loan. A 30-year mortgage at 6% versus 7% on $300,000 is the difference between $1,799/month and $1,996/month — and $71,069 in extra interest over the life of the loan. The comparison card above shows the same one-point symmetry for whatever rate you just solved for: same principal, same term, payment moves with the rate.

Disclaimer

This calculator solves the textbook amortisation equation. Real-world APR may differ from the nominal rate this tool recovers if the lender bundles origination fees, points, or mandatory insurance into the financing. Those are added into the legally-disclosed APR but are not part of the simple PMT-versus-principal arithmetic. For a regulated APR figure, refer to the Truth-in-Lending disclosure on the loan documents.