ReckonBoxHow the interest rate calculator works
An implied interest rate is the rate that makes a given payment repay a given loan over a given term. Enter what you borrowed, how long you have, and what you pay each month, and this tool finds the rate hiding behind those three numbers.
It also tells you how it found it, which turns out to be the interesting part.
Why this one needs a search
The payment on an amortised loan is:
PMT = P x i / (1 - (1 + i)^-n)
P is the amount borrowed, i the rate per period, n the number of periods. Rearranging it for payment is trivial. For principal, easy. For the term, a logarithm.
For the rate, impossible.
The rate sits in two places at once, as a plain multiplier out front and inside an exponent in the denominator. There's no algebraic move that isolates it. So every calculator offering this is not solving the equation; it's guessing a rate, checking the payment that guess produces, and adjusting.
calculator.net's interest rate calculator does exactly this and publishes no formula and no method. FinancialMentor solves for any of four loan variables and likewise names no algorithm, and neither page mentions that a closed form doesn't exist. An iterative approximation gets presented as though it were arithmetic.
The method used here
Bisection. Payment rises steadily as the rate rises, which means a bracket that starts with one rate too low and one too high contains exactly one answer, and halving the bracket repeatedly closes in on it.
It's slower than the alternatives and it cannot run away. Newton's method converges faster but can diverge on an unlucky starting guess, and a number someone is about to act on is the wrong place for that trade. The tool reports the iteration count so you can see the work.
The search stops when the payment it reproduces sits within a ten-billionth of the payment you entered. At that point the limiting factor is your inputs, not the method: rounding a payment to the nearest dollar moves the implied rate more than the arithmetic ever will.
A worked case
Borrow $20,000 over 60 months, paying $400 a month.
| Figure | Value |
|---|---|
| Implied annual rate | about 7.42% |
| Total repaid | $24,000 |
| Total interest | $4,000 |
| Share of payments that is interest | 16.7% |
Nothing in those three inputs announces 7.42%. That's the point of the tool, and it's why car dealers can quote a monthly payment without ever quoting a rate.
When there is no answer
Pay $300 a month for 60 months against $20,000 borrowed and you return $18,000. No interest rate produces that, because you haven't repaid the principal, let alone paid to borrow it.
This tool says so. A calculator that returns a number there is answering a different question, usually by silently accepting a negative rate.
The same applies at the boundary. Paying exactly $333.33 a month returns exactly $20,000, which is a zero rate, and anything below it is negative.
Implied rate is not APR
The rate this returns comes from the payment stream alone. APR as a lender quotes it also folds in fees, arrangement charges and anything else bundled into the cost of credit.
So a loan with an arrangement fee has an APR above the rate its payments imply, and comparing this figure against a quoted APR compares two different things. Check what a quote includes before concluding it's good or bad.
What this calculator does not do
Fees are absent, which is exactly why the output isn't an APR. Adding them would need the fee schedule and the timing, and those vary per lender.
Irregular payments, balloon payments and rate changes mid-term are all outside a fixed-payment model. So are payment holidays.
Results are arithmetic on your inputs, not financial advice. A rate is one term among many in a credit agreement. For decisions about borrowing, speak to a qualified professional. To see what a given rate does over time, the compound interest calculator runs it forwards, and the simple interest calculator covers the flat-rate convention.
Frequently asked questions
How do you work out the interest rate on a loan? From the payment equation PMT = P x i / (1 - (1 + i)^-n), where P is the amount borrowed, i the monthly rate and n the number of months. Given the amount, the term and the payment, the rate is the value of i that reproduces the payment you are making.
Why does solving for the rate need iteration? Because the rate appears twice in the payment equation, once as a plain factor and once inside an exponent, so it cannot be isolated algebraically the way payment, principal or term can. Every calculator that offers this is searching for the answer numerically. calculator.net and FinancialMentor both do it and neither says so.
What method does this calculator use? Bisection. Payment rises steadily as the rate rises, so a bracket that starts below and above the answer can be halved repeatedly until it closes in. It is slower than the alternatives and it cannot run away on a bad starting guess, which suits a number people will act on. The tool reports how many iterations it took.
What if the payments never repay the loan? Then no positive rate exists and this tool says so instead of printing a number. Paying $300 a month for 60 months returns $18,000 against $20,000 borrowed, which no interest rate can produce. A calculator that returns a figure there is showing you a solution to a different problem.
Is this rate the same as APR? Not necessarily. This is the rate implied by the payment stream alone. APR as lenders quote it also folds in fees and charges, so a loan with an arrangement fee has an APR above the rate its payments imply. Compare like with like before concluding a quote is good or bad.
How accurate is the result? The search stops when the payment it reproduces is within a ten-billionth of the payment you entered, so the limit on accuracy is the precision of your inputs rather than the method. Rounding a payment to the nearest currency unit moves the implied rate more than the arithmetic does.