How do I work out the interest rate from a monthly payment?

The rate is the one at which the monthly payment, repeated for the whole loan term and discounted back to today, is worth exactly the amount borrowed. There is no way to rearrange the equation with ordinary algebra — the unknown sits inside a power — so it has to be solved by trial and error, which is what this page does in a few dozen iterations. It is the same calculation a lender performs to produce the annual percentage rate, run from the payment side instead of the rate side.

Why is there no field for the interest rate?

Because the point of the page is to derive the rate from what you will actually pay. If the rate were an input, the answer would be an echo of whatever was typed there. Leaving it out means the figure on the panel depends only on the three things you can verify — what was borrowed, what leaves your account each month, and how many times — and those are also the three things a lender cannot easily dispute.

Is the answer the same as the annual percentage rate?

It is the same equation, and whether the two numbers agree depends on one input. The annual percentage rate relates the value actually received to the payments made, so it accounts for fees; this page relates whatever you type as the amount borrowed to the payments. Type the contract amount and you get the note rate, because the fees are not in the arithmetic. Type the amount that arrived in your account after the fees came out and you get the annual percentage rate, because the fee is now in the arithmetic. Same calculation, different principal.

Why is the rate lower than the rate I would get by compounding?

Because the panel multiplies the monthly rate by twelve rather than compounding it. A loan solved at 6.771 percent a year is 0.564242 percent a month, and twelve of those multiplied together come to 6.985 percent — the difference is the interest on the interest, which does not arise on an amortising loan because the balance is falling rather than accumulating. Multiplying the monthly rate by the number of periods is the convention credit disclosures use to annualise a periodic rate, so the figure here matches what a disclosure would print.

What if the loan has a balloon payment or interest-only months?

Then a single monthly payment does not describe it, and the answer is an average rather than a rate. The solver assumes the same payment every month for the whole term; a schedule with a large final payment or a run of interest-only months has a different shape, and forcing it into one monthly figure produces a rate that is too high or too low depending on which payments you averaged. On those loans, ask for the annual percentage rate, which is computed from the actual payment schedule rather than from an average of it.

Can the page always find a rate?

No, and it says so rather than guessing. If the payments are too small to repay the principal over the term at any conceivable rate — 200,000 repaid at 555 a month over thirty years never retires the debt, whatever the rate — the equation has no solution and the page reports an error. The same happens in the other direction, when a single payment exceeds the whole principal and the term is one month, and when the number of payments is not a positive whole number of months. In those cases the inputs describe something that is not a loan, and no rate would make them one.

Interest Rate Calculator

Result

6.771%

Nominal annual rate (%)

Total interest: 268,000.00
Total paid: 468,000.00

An interest rate calculator works the equation backwards. Nearly every page in this group starts with a rate and computes a payment; this one starts with the payment and computes the rate. Enter the amount borrowed, what you pay each month and how many months you pay it, and the page solves for the annual rate those three numbers imply, then reports the total interest on top of it. It is the rate at which the payments, discounted back to today, are worth exactly the amount advanced — nothing more and nothing less. On 200,000 repaid at 1,300 a month for thirty years, that rate is 6.771 percent a year. Pay 1,199.10 a month instead and it is exactly 6 percent, because 1,199.10 is the payment a 6 percent loan of that size and term requires. Two things make the figure worth understanding rather than only reading. The first is what it measures. The federal definition of the annual percentage rate calls it a measure of the cost of credit, expressed as a yearly rate, that relates the amount and timing of value received by the consumer to the amount and timing of payments made — which is exactly the relation this page solves, approached from the payment side. The second is that there is no rate field on the panel. There is nowhere to type what someone told you the rate was, so the answer cannot be contaminated by it: everything on the panel comes from the cash flow you entered. That is what makes this the honest way to check a loan. If a lender quotes 6 percent and the payment on 200,000 over thirty years implies 7.4, then 7.4 is what the payments actually cost, and the gap is either a fee that was rolled into the balance or an error in the quote. Both are worth knowing before signing. The one convention to keep straight is how the answer is annualised. The panel reports a nominal annual rate: the monthly rate the solver found, multiplied by twelve. That is the convention credit disclosures use, and it is why a 6 percent loan prints 6.000 rather than the 6.168 that compounding would produce if the interest were left to accrue. A second convention is the schedule. The solver assumes the same payment every month for the whole term, which is the ordinary amortising loan. If the schedule is anything else — a balloon at the end, a fixed period followed by a larger one, interest-only payments that switch to principal — then the monthly figure is an average of a schedule the page is not modelling, and the rate it reports describes that average rather than the loan. If you want the rate on a loan that charges fees, the annual percentage rate page takes the fee as an input and solves the same equation with it; if you already know the rate and want the payment, the loan calculator is the same relationship run forwards.

200,000 over 360 months, at five different monthly payments

Monthly payment	Annual rate	Total interest	Total paid
1000	4.387	160000	360000
1199.1	6	231676	431676
1300	6.771	268000	468000
1500	8.232	340000	540000
2000	11.627	520000	720000

The term is fixed at thirty years in every row, so the table is a single question asked five ways: what does a given monthly payment imply? Read the first two columns together and the relationship looks violent — an extra 300 a month, from 1,000 to 1,300, moves the rate by more than two percentage points, 4.387 to 6.771. That is not a quirk of the table but of the equation: at long terms the payment is dominated by interest, so a small change in what you pay maps to a large change in the rate that justifies it. The last two columns show why the rate is worth solving for at all. Every row repays the same 200,000, and the difference between the top row and the bottom one is 360,000 of interest — more than the loan itself. If you are choosing between offers, the payment is what you will feel each month and the rate is what the difference is actually worth; this table is the conversion between the two, at one term and one principal.

Formula

Amount advanced = monthly payment × [1 − (1 + monthly rate)^(−number of payments)] ÷ monthly rate Annual rate = monthly rate × 12

Amount borrowed: The sum advanced at the start, which the payments have to repay exactly over the term — with a level payment schedule this equation has one and only one rate that balances it
Monthly payment: The amount paid every month for the whole term, principal and interest together. It is the one figure most borrowers know for certain, which is why solving from it is more reliable than solving from a quoted rate
Number of months: How many payments are made — the loan term expressed in months, so thirty years is 360. It is also, multiplied by the payment, the whole of what leaves your account
Monthly rate: The rate per month that makes the discounted payments equal the amount borrowed: the quantity the page solves for, and the only unknown in the equation above
Annual rate: The monthly rate multiplied by twelve, which is the nominal annual rate — the same annualisation a credit disclosure performs, and not the effective rate you would get by compounding twelve times

Use it when you know the payment rather than the rate, which is the situation a borrower is actually in. Advertisements quote rates, but the number that arrives on a statement every month is the payment, and the payment is what has to fit a budget. Two of the most common uses are checking a quote and comparing two of them. Checking: a lender says 6 percent on 200,000 over thirty years, which should be 1,199.10 a month; if the paperwork says 1,250, enter 1,250 and the page returns 6.53 percent, and the difference is now a number you can ask about rather than a suspicion. Comparing: two offers at the same quoted rate but different payments are not the same offer, and one at a higher quoted rate with a lower payment is not automatically worse — enter each payment and compare the solved rates, which is the only figure that accounts for both the rate and whatever else the payment is carrying. A third use is the one that surprises people: entering the amount actually handed over rather than the amount on the contract. If 200,000 is borrowed but 197,000 arrives after an origination fee, solving on 197,000 with the same payments returns a rate that includes the fee — which is what the annual percentage rate page computes from the other direction, and the two should agree to the last decimal.

Worked examples

200,000 repaid at 1,300 a month for thirty years
1. Payments leave the account: 1,300 × 360 = 468,000 repaid in total
2. Interest is the difference: 468,000 − 200,000 = 268,000
3. Monthly rate: the rate at which 1,300 × [1 − (1 + i)^−360] ÷ i = 200,000, which is i = 0.564242% a month
4. Annual rate: 0.564242% × 12 = 6.771%
The default case. Two figures are worth reading together: the 6.771 percent the payments imply, and the 468,000 that actually leaves the account — more than twice the amount borrowed. The interest is not a small add-on to a large loan; on a thirty-year term at this rate it is larger than the principal. Note also what the page does not show: if this loan compounded the interest rather than amortising it, the annualised cost would be 6.985 percent rather than 6.771, because the panel annualises the monthly rate linearly. For a loan whose interest is paid rather than accumulated, the linear figure is the right one.
The same loan at exactly 6 percent
1. The payment a 6% loan requires: 200,000 × 0.005 × 1.005^360 ÷ (1.005^360 − 1) = 1,199.10 a month
2. Solving that payment back: the monthly rate is 0.5%
3. Annual rate: 0.5% × 12 = 6%, which is the rate we started from
4. Total repaid: 1,199.10 × 360 = 431,676, of which 231,676 is interest
The round trip, and the useful check: the payment a 6 percent loan produces is the payment that solves back to 6 percent. The 36.30 a month that separates this case from the one above is 6.771 percent versus 6 percent — a difference of 0.771 of a percentage point that costs 36,324 over the term. This is also the case to use when a quote gives a rate and you want to know what the payment should be before the paperwork arrives.
100,000 over two years at 5,000 a month
1. Payments leave the account: 5,000 × 24 = 120,000
2. Interest is the difference: 120,000 − 100,000 = 20,000, or a fifth of the principal
3. Monthly rate: the rate at which 5,000 × [1 − (1 + i)^−24] ÷ i = 100,000, which is i = 1.51312% a month
4. Annual rate: 1.51312% × 12 = 18.157%
A short term changes the arithmetic in a way the total can hide: 20,000 of interest on 100,000 looks like 20 percent, but it is paid over two years on a balance that is falling, so the rate is 18.157 percent a year rather than 10. The shorter the term, the less of the principal is outstanding on average, which is why doubling the rate and halving the term do not cancel out. The monthly rate here is 1.513 percent, and a card charging 1.5 percent a month is charging almost exactly this.

Limitations

It assumes one rate for the whole term. A variable-rate loan, a fixed period that resets to a different rate, or a teaser rate followed by the real one all break the assumption, and the single rate this page reports is an average of a schedule it cannot see — useful for comparison and wrong as a description of any particular month. It assumes the payment is the same every month. A balloon payment, an interest-only period, or any step-up schedule makes the monthly figure an average over a shape the page does not model, and the solved rate will be pulled toward whatever the early payments were. It models no fees. Everything the payments carry is folded into the solved rate, so if the amount advanced on the contract is not the amount that arrived in your account, the answer is neither the note rate nor the annual percentage rate until you enter the right principal — and if you enter what arrived, the answer is a rate that includes the fees, which is the annual percentage rate by another route. It annualises linearly, multiplying the monthly rate by twelve, so the figure it prints is not the effective annual rate and will be lower than a compounded figure at the same monthly rate; that is the disclosure convention and not an error, but the two numbers are not interchangeable. It assumes payments are made exactly on time and that none are missed or prepaid, both of which change what the money actually cost. It cannot solve every set of inputs: if the payments are too small to ever repay the principal at any rate, or if a single payment exceeds the whole principal, there is no rate that balances the equation and the page reports an error rather than inventing one. Finally, it reports a rate and not a verdict: whether the rate is legal, fair, or better than the alternative is a question about a market, not about arithmetic.

Frequently asked questions

How do I work out the interest rate from a monthly payment?: The rate is the one at which the monthly payment, repeated for the whole loan term and discounted back to today, is worth exactly the amount borrowed. There is no way to rearrange the equation with ordinary algebra — the unknown sits inside a power — so it has to be solved by trial and error, which is what this page does in a few dozen iterations. It is the same calculation a lender performs to produce the annual percentage rate, run from the payment side instead of the rate side.
Why is there no field for the interest rate?: Because the point of the page is to derive the rate from what you will actually pay. If the rate were an input, the answer would be an echo of whatever was typed there. Leaving it out means the figure on the panel depends only on the three things you can verify — what was borrowed, what leaves your account each month, and how many times — and those are also the three things a lender cannot easily dispute.
Is the answer the same as the annual percentage rate?: It is the same equation, and whether the two numbers agree depends on one input. The annual percentage rate relates the value actually received to the payments made, so it accounts for fees; this page relates whatever you type as the amount borrowed to the payments. Type the contract amount and you get the note rate, because the fees are not in the arithmetic. Type the amount that arrived in your account after the fees came out and you get the annual percentage rate, because the fee is now in the arithmetic. Same calculation, different principal.
Why is the rate lower than the rate I would get by compounding?: Because the panel multiplies the monthly rate by twelve rather than compounding it. A loan solved at 6.771 percent a year is 0.564242 percent a month, and twelve of those multiplied together come to 6.985 percent — the difference is the interest on the interest, which does not arise on an amortising loan because the balance is falling rather than accumulating. Multiplying the monthly rate by the number of periods is the convention credit disclosures use to annualise a periodic rate, so the figure here matches what a disclosure would print.
What if the loan has a balloon payment or interest-only months?: Then a single monthly payment does not describe it, and the answer is an average rather than a rate. The solver assumes the same payment every month for the whole term; a schedule with a large final payment or a run of interest-only months has a different shape, and forcing it into one monthly figure produces a rate that is too high or too low depending on which payments you averaged. On those loans, ask for the annual percentage rate, which is computed from the actual payment schedule rather than from an average of it.
Can the page always find a rate?: No, and it says so rather than guessing. If the payments are too small to repay the principal over the term at any conceivable rate — 200,000 repaid at 555 a month over thirty years never retires the debt, whatever the rate — the equation has no solution and the page reports an error. The same happens in the other direction, when a single payment exceeds the whole principal and the term is one month, and when the number of payments is not a positive whole number of months. In those cases the inputs describe something that is not a loan, and no rate would make them one.

References

12 CFR Part 1026, Appendix J (Annual Percentage Rate Computations for Closed-End Credit Transactions), paragraph (b)(1) General Rule — the annual percentage rate shall be the nominal annual percentage rate determined by multiplying the unit-period rate by the number of unit-periods in a year, which is the annualisation this page's output performs; paragraph (a)(2) describes the actuarial method this page inverts — Consumer Financial Protection Bureau regulation, via the Electronic Code of Federal Regulations (United States)
12 CFR 1026.22(a)(1)–(3) — the annual percentage rate is a measure of the cost of credit, expressed as a yearly rate, that relates the amount and timing of value received by the consumer to the amount and timing of payments made; the rate is accurate if within 1/8 of 1 percentage point for a regular transaction and 1/4 of 1 percentage point for an irregular one — Consumer Financial Protection Bureau regulation, via the Electronic Code of Federal Regulations (United States)
12 CFR 1026.14(c)(1) and its accompanying commentary — the annual percentage rate may be computed by multiplying each periodic rate by the number of periods in the year, the linear annualisation that converts this page's solved monthly rate into the annual figure it prints — Consumer Financial Protection Bureau regulation, via the Electronic Code of Federal Regulations (United States)

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