Two different rate questions, one tool
Most rate calculators only go one direction: put in a rate, get a payment. This one also runs backwards. The first tab takes a loan quote, an amount, a monthly payment and a term, and works out the annual percentage rate hiding inside it, which is the number worth comparing against any other offer. The second tab takes a plain nominal rate and works out what it actually earns or costs once compounding is applied, in either direction. They are unrelated questions that happen to share the word "rate", so each gets its own tab rather than being squeezed into one form.
Finding the real rate behind a payment
A fixed-repayment loan follows one formula. Borrow P, repay a fixed amount M every month for n months, and the monthly rate r that ties them together satisfies:
Given P, r and n, working out M is one line of arithmetic. Given P, M and n instead, working out r is not, because r appears both multiplied out front and buried inside a power on both sides of the divide. There is no algebraic rearrangement that isolates it; every textbook and finance reference that covers this formula says the same thing, solving for the rate needs a numerical, trial-and-improvement method rather than a formula. This calculator uses bisection: it tries a rate, checks whether the payment that rate produces is too high or too low against yours, halves the gap, and repeats until the two payments agree to within a few thousandths of a percent, usually inside 60 or so tries. It is slower than algebra, but it is exact, and it is the same basic approach lenders' own systems use to publish the APR on a loan.
Worked example: borrow 20,000, repay 386.66 a month for 60 months. Feed those three numbers in and the bisection settles on almost exactly 6.00% APR, which you can check by running 20,000 at 6% over 60 months through the formula the other way and landing back on 386.66. The total paid over the five years comes to 23,199.60, so 3,199.60 of it is interest on top of the 20,000 borrowed.
Why the advertised rate and the real rate can differ
A "flat rate" is not the same calculation as the APR this tool solves for, and the gap between them is not a rounding error. A flat rate charges interest on the original amount for every month of the term, even though the balance is actually falling as you repay it, so the true cost per pound still outstanding is much higher than the flat percentage suggests. A retailer or dealer advertising "3% flat" on a loan can carry a real APR closer to double that once the falling balance is accounted for properly, and a fee bolted onto the loan on top makes the gap wider still. None of that is visible from the headline percentage alone, only from the payment it actually produces.
This is exactly the situation the first tab is built for. Ignore the advertised percentage entirely, take the amount, the monthly payment and the term straight off the credit agreement or the quote, and let the bisection tell you the rate those three numbers actually imply. Do that for two competing offers and you are comparing like with like, regardless of how each lender chose to describe its own headline figure.
Reading a payment quote back into a rate before signing
Any time a lender, dealer or "buy now, pay later" scheme quotes a payment rather than a rate, the same three numbers unlock the truth: what you are borrowing, what you would pay each month, and how many months you would pay it for. Put those into the first tab and treat the resulting APR the same way you would treat a rate printed on any other offer, as the figure to compare, not the one on the advert. It is a useful check even when a rate is quoted, since it also catches arithmetic that quietly does not add up, a payment that is too small ever to clear the balance, or a fee folded into the amount without being mentioned.
Two guardrails sit behind the numbers automatically. If the payment, multiplied by the term, comes to less than the amount borrowed, no interest rate, however low, makes that loan repayable on schedule, so the calculator says so plainly rather than guessing at a rate. At the other end, if the payment barely exceeds what pure interest alone would cost each month, almost none of it is reducing the balance, and the solved rate will come back very high; the calculator flags that case too, since it usually means either a genuinely expensive product or a number typed into the wrong field.
APR and APY are not the same number
Loans are quoted in APR. Savings and investment products are quoted in APY, sometimes called the effective annual rate. Both start from a nominal annual rate, but APY folds in what happens when interest compounds more often than once a year, so it is always a little higher than the nominal rate it is built from, and the more frequently interest compounds, the bigger that gap gets. Advertising a savings account by its APY is standard practice because it is the more honest number: it is what a pound sitting in the account for a full year actually turns into, not the input the bank started from. The second tab moves in either direction: give it the nominal rate and a compounding frequency for the APY, or give it the APY and get the nominal rate a lender or provider must have started from.
Worked examples: a nominal 12% compounded monthly works out to an APY of 12.6825%, and a nominal 5% compounded daily works out to 5.1267%. Both gaps look small written down, but they compound the same way the balance itself does, so over several years the difference between quoting the nominal figure and the effective one is not a rounding footnote.
One honest note on legal APR figures
What this tool solves for is the pure interest-rate mathematics behind a payment, not any single country's legal disclosure figure. In the United States, Regulation Z defines which fees a lender must fold into the APR it discloses. The UK and the EU use a different, harmonised calculation, generally called APRC on anything secured against property, built from a formula set out in EU consumer credit law that every lender in the bloc must apply the same way. The three regimes do not always treat the same fee identically, an arrangement fee that counts in one jurisdiction's APR can sit outside another's, so a headline APR quoted in one country is not always directly comparable to one quoted in another. Put the same amount, payment and term from an actual agreement into the first tab, though, and the interest-rate maths behind it is the same everywhere; only the legal fee-folding rules differ, and this tool does not attempt to reconstruct those.
Questions people ask
Why can't the rate just be worked out with a formula, the way the payment can?
Because the rate appears twice in the payment formula, once on its own and once raised to the power of the number of payments, on both sides of the equation once you try to isolate it. There is no algebraic rearrangement that separates it out, so every calculator that solves for a rate, including this one, uses a numerical method, trying rates and narrowing in on the one that fits, rather than a single-line formula.
How accurate is the bisection method used here?
Very. It narrows the gap between a rate that is too low and one that is too high by half at every step, and this calculator keeps going until that gap is smaller than a thousandth of a percentage point, typically inside 60 or so steps. The tiny remaining uncertainty is far smaller than the rounding already present in any real payment figure, which is usually quoted to the nearest cent or penny.
What does it mean if the calculator says the payment can never repay the loan?
It means the payment, multiplied by the number of months, adds up to less than the amount borrowed. Since interest can only ever add to what is owed, not subtract from it, no positive rate makes that arithmetic work; the balance would grow every month rather than fall. Check the amount, the payment and the term are all for the same loan and the same currency before trying again.
Why does a very high solved rate get a warning?
A payment that barely exceeds the pure interest cost each month is barely touching the balance, which is the signature of either a genuinely very expensive loan or numbers typed into the wrong fields, an annual figure entered as a monthly one, for instance. The warning exists so an implausible rate gets a second look rather than being taken at face value.
Is APY always higher than the nominal rate?
Yes, whenever the rate is positive and compounding happens more than once a year. Compounding pays interest on interest already earned within the year, which can only add to the total, never take from it. The two figures are equal only in the special case of compounding exactly once a year.
Why do savings accounts quote APY but loans quote APR?
Convention rather than law in most markets, though a widely followed one. APY is the more flattering number for a saver, since it is always the larger of the two, so savings products lead with it. APR is the number regulators require lenders to disclose for credit, built to fold in fees rather than to reflect compounding frequency, which is why it is not simply the borrowing equivalent of APY.
Does compounding frequency actually matter much in practice?
Less than the headline rate itself, but not nothing. Moving the same nominal rate from monthly to daily compounding typically shifts the effective annual rate by a few hundredths of a percentage point, visible in the comparison table above. It rarely changes which of two products is the better deal on its own, but it does explain why two providers quoting what looks like the same nominal rate can publish slightly different APY figures.
These figures are estimates of the underlying interest-rate mathematics, not a substitute for the legal APR or APRC disclosed on a specific credit agreement, which may fold in fees this tool does not see. Not financial advice; check the actual agreement, or ask the lender, before relying on any figure here.