Loan Calculator

How a loan payment is calculated

One equation produces the payment. It is the only thing on this page that is not a consequence of it.

M is the payment, P is the amount borrowed, i is the periodic interest rate, and n is the total number of payments. You will also meet this written as M = P i(1+i)ⁿ / ((1+i)ⁿ − 1). The two are the same equation rearranged, and they agree to the last digit a computer can hold.

The two numbers you are never asked for

Neither i nor n is something you know about your loan. You know an annual rate, a term in years, and how often you pay. The calculator derives the other two:

i = r / f, the annual rate divided by the number of payments a year. n = years × f, the term multiplied by the same number.

Asking you for a periodic rate would be asking you to perform the conversion this page exists to show. So both derived values are displayed live, marked as derived, in the table below and again beneath the term field in the calculator. The division i = r/f is a lending convention rather than a mathematical necessity, and that distinction matters enough that the mortgage calculator gives it a section of its own: the periodic rate is a convention, not a derivation.

What your numbers mean

The payment is rounded to the cent once, and every period uses that same figure. The interest, though, is worked out on the balance that actually stands at each due date, and rounded to the cent there too. Those two roundings do not cancel, so the last payment absorbs whatever is left: $495.05 rather than $495.03. That is why the totals on this page are read off the schedule rather than multiplied out of the payment. Sixty payments of $495.03 is $29,701.80, and the schedule totals $29,701.82. Two cents is small, but a page whose headline disagrees with its own table is not one you should trust with the large numbers either.

How the balance is paid down

The payment never changes. What changes is what it is made of. Interest is charged on what you still owe, so it is largest at the start and shrinks with the balance, and whatever the interest does not consume reduces the principal. That leaves a little less to charge interest on next time. The effect compounds in your favour, slowly at first and then quickly.

Understanding what you are looking at

A payment is not a cost

The payment is the most visible number on any loan and the least informative one about what the loan costs. It is a rate of outflow, not a total. Two loans with the same payment can differ by thousands of dollars in what you hand over, because a payment can always be lowered by lengthening the term, and lengthening the term always raises the total. Stretching the default loan here from five years to seven drops the payment by $117.71 and adds $1,992.64 in interest. The payment is what you can afford. The total is what it costs. They are different questions and a single number cannot answer both.

Why the early payments are mostly interest

Interest accrues on the balance outstanding, and the balance is largest on day one. The payment is fixed, so whatever the interest does not consume is what reduces the principal. On the default scenario the first payment is $145.83 of interest and $349.20 of principal. Because that $349.20 makes the balance a little smaller, the next month’s interest is a little smaller, which leaves a little more for principal. That is the whole mechanism.

It is also why the balance curve above bends rather than falling in a straight line. The closed form for the balance after k payments is B_k = P(1+i)^k − M[((1+i)^k − 1) / i], and the exponential terms in it are exactly where the curvature comes from. On a five-year loan at 7% the bend is gentle. On a thirty-year mortgage it is dramatic, which is the single fact that surprises people most about mortgages.

What each input actually does

The principal scales everything: double it and the payment, the interest, and the total all double exactly. The rate moves the payment steeply and the total interest more steeply still, because it compounds against a balance that is itself shrinking more slowly. The term is the only input that moves the payment and the total in opposite directions, which is what makes it the input people get wrong. The frequency barely matters, and why it barely matters is more interesting than the fact that it does.

What changing the payment frequency actually does

Switch the frequency select above and four things change at once: the payment, the number of payments, the periodic rate, and the total interest. It is tempting to read the last of those as a saving. It is not one, and seeing why is worth more than the calculator itself.

Why the interest column moves at all

Under i = r/f, changing f changes two things together. A nominal rate divided by a bigger number and compounded more often is not the same annual rate: 7% divided twelve ways compounds to 7.2290% a year, and divided fifty-two ways it compounds to 7.2458%. That pushes the interest up. At the same time, paying more often at a fixed term in years retires principal sooner, which pulls the interest down. The second effect is larger, so the column falls.

Neither of those is a saving, because those four rows are four different contracts, not one loan at four cadences. The thing that makes them comparable is the fourth column. Across all four, the amount leaving your account each year differs by $11.32 on about $5,935, which is under two tenths of one percent. The total interest differs by $55.20 over five years, and it differs because the contracts differ, not because one schedule is cleverer than another.

Changing when you pay barely changes anything. You pay about the same amount each year, so you get about the same result. If a lender offers you a cadence and frames it as a saving, the fourth column is the one to ask about.

Semimonthly and biweekly are not the same thing

Semimonthly means twice a month: 24 payments a year, always on the same two dates. Biweekly means every fourteen days: 26 payments a year, drifting through the calendar. The names sound interchangeable and the schedules are not. Two extra payments a year is the entire difference, and it is the mechanism behind every claim you have read that biweekly payments retire a loan early. See what paying extra actually does, which is where that claim turns out to be true for a reason that has nothing to do with the cadence.

The related idea of how often interest is added, rather than how often you pay, belongs to the compound interest calculator, which treats it at length: compounding frequency and nominal, periodic, and effective rates. On an amortizing loan there is no separate compounding choice to make, because interest that is paid every period never compounds at all.

Interest rate, APR, and what your lender actually charges

Your paperwork will show two percentages and they will not match. The interest rate prices the money: it is the number that goes into the formula above, and the only one this calculator uses. The annual percentage rate prices the loan, folding origination fees, points, and other charges into a single figure spread across the term. A loan with no fees has one number twice. A loan with fees has an APR above its interest rate, and the gap is the honest measure of what the fees cost.

Enter the interest rate here, not the APR. Entering the APR would produce a payment slightly larger than the one you will actually be billed, because it would charge you for the fees twice: once when you paid them, and again as a higher rate on the balance. This calculator does not compute an APR, and it will not guess at one, because that would mean inventing a fee schedule it has no way to know.

One case deserves care. If a fee was financed rather than paid up front, it is part of the balance, so it belongs in the loan amount field. Then the payment this page reports is correct, and the APR is still higher than the rate, because you are paying interest on the fee.

Lenders differ in how interest accrues

The formula above assumes interest is computed on the balance standing at each due date. That is not the only convention in use, and the differences are not cosmetic.

• Monthly actuarial. Interest is computed on the balance at each due date. This is what the page models, and what standard fixed-rate installment loans and United States mortgages use.
• Daily simple interest. Interest accrues every day on the outstanding balance. Common on car loans. Paying a few days early or late changes what you owe, which a periodic model cannot see at all. Over a full term the totals land close; month to month they do not.
• Precomputed interest. The total interest is fixed when the loan is written and spread across the payments. The Consumer Financial Protection Bureau describes this as an uncommon way of calculating interest on a car loan, and notes that paying off early may earn only a partial rebate of the interest you have not yet used. This is the one case where this calculator’s early-payoff arithmetic would be actively wrong, so it is worth knowing which kind of loan you have.

Not every loan works this way

The formula above describes a loan that is fully amortizing: every payment is identical, every payment covers all the interest due plus some principal, and the last one leaves a balance of zero. Most consumer loans are like this. Several common ones are not, and for those this page will give you a confident wrong answer rather than no answer, which is why they are listed.

• Interest-only loans charge the interest and nothing else for a period, so the balance does not move. The payment is simply P × i, and the formula above does not apply until the amortizing phase begins.
• Balloon loans are amortized against a long term but come due on a short one, leaving a large final payment. You can approximate the payment here by entering the amortization term, but the payoff is not what this page will show you.
• Adjustable-rate loans have a rate that changes on a schedule. Everything here assumes a fixed rate, so the figures hold only until the first adjustment.
• Credit cards and other revolving credit have no term and no fixed payment. There is nothing for this formula to describe.
• Income-driven student loan repayment sets the payment from your income rather than from the balance, and the balance can grow. A standard ten-year repayment plan, on the other hand, is exactly the model above.

Mortgages are this loan, plus other things

A mortgage is a loan of exactly this kind: same formula, same schedule, same arithmetic. What makes it its own subject is everything attached to it, namely collateral, and costs that travel with the property rather than with the debt. Property taxes, homeowners insurance, mortgage insurance, and association dues are billed alongside the payment and are not part of it. If that is what you are working out, the mortgage calculator models those separately and keeps them visibly apart from principal and interest, which is the whole reason it is a different page.

What happens when you change one input

A percentage point is worth different money at every principal, every term, and every cadence, so a rule of thumb cannot answer this and these tables are recomputed from scratch instead. Each row is a real loan, priced by the same engine that produced the headline figure, with a real schedule behind it.

If the rate were different

If the term were different

The two columns in the term table move in opposite directions, and that is the trade people most often take only half of. A lower payment is a real benefit and a higher total is a real cost. This page will not tell you which matters more to you, because it does not know anything about you.

What paying extra actually does

Money paid above the scheduled payment goes straight to principal, which means the next period’s interest is charged on a smaller balance, which leaves more of the following payment for principal. The schedule simply ends early. Open the extra payment field in the calculator and the result panel gains a second block showing how much earlier and how much less.

There is one thing to check with your lender first: that extra money is applied to principal rather than held as a prepayment of next month’s instalment. Those two are very different, and the second saves you nothing.

The biweekly trick, and what is actually doing the work

The most repeated piece of loan advice is that paying biweekly retires a loan early. The section on payment frequency shows that switching cadence on its own does almost nothing, so something else must be going on, and it is this: the advice is not really about cadence.

Take the default loan. The monthly payment is $495.03. Pay half of that, $247.52, every two weeks, and you make 26 payments a year, which is thirteen monthly payments’ worth rather than twelve. The loan is retired in 119 payments instead of 130, which is 4 years and 7 months instead of 5 years, and the interest falls from $4,662.85 to $4,215.77.

That saving is real. What produced it is not the cadence: it is the extra $495.16 a year. Paying the same extra amount monthly would do the same work. The genuine advantage of the biweekly framing is behavioural rather than mathematical, because two of your twenty-six payments fall in months where you make three of them, and a schedule that hides the extra payment inside a rhythm is easier to keep than one that asks for it. That is a real benefit. It is just not arithmetic, and a calculator should say which is which.

What moves the number, and what this model can see

Four inputs produce the payment. A great many other things determine what your loan actually costs, and the useful distinction is not between large and small but between what is inside the formula, what sits beside it, and what this page cannot see at all.

Common misconceptions

“A lower payment means a cheaper loan”

It usually means a longer one. The payment falls because the same principal is spread over more periods, and more periods means more interest. The term table above shows both columns at once for exactly this reason.

“Total paid is the payment times the number of payments”

Close, and wrong in a way that matters more than the amount. Sixty payments of $495.03 is $29,701.80. The schedule totals $29,701.82, because the final payment absorbs the rounding. The habit of multiplying out is the same habit that leads people to trust a payment figure carried to two decimals over a schedule that actually adds up.

“Paying biweekly saves money”

Switching cadence and keeping the annual outlay the same changes very little: $11.32 a year on the default loan. Paying half the monthly payment every two weeks does save money, and the reason is that it is thirteen monthly payments a year rather than twelve. The saving comes from paying more, not from paying more often.

“The lender takes all the interest first”

Nothing is front-loaded by design. Interest is charged on the balance outstanding, the balance is largest at the beginning, and that is the entire explanation. Nobody chose to weight it that way; it falls out of charging interest on what is owed.

“The APR is the interest rate”

The APR is the interest rate plus the fees, expressed as a rate. If they are equal, the loan has no fees. Use the interest rate in this calculator, because that is the number the payment is computed from.

“The calculator includes everything I pay each month”

It includes principal and interest and nothing else. A mortgage arrives with property taxes, insurance, possibly mortgage insurance, and possibly association dues. A car loan arrives with comprehensive insurance and registration. Those are real obligations and they are not part of the loan payment, so a calculator that folded them into one figure would make the payment look larger and the loan look more expensive than it is.

“Extra payments reduce my monthly payment”

They reduce the number of payments, not their size. The payment is fixed by the contract; paying extra retires the balance sooner. Some lenders will recast a loan on request, which does lower the payment, but that is a separate thing you have to ask for and it is not what this schedule shows.

What this calculator does not include

Everything in this list is real money or a real risk. None of it is in the formula, and listing it is more useful than approximating it.

• Fees of any kind. Origination fees, application fees, points, documentation fees, and prepayment penalties are outside the model. If a fee was financed it belongs in the loan amount; if you paid it up front it does not, and the loan still cost you that money.
• The APR. Not computed, and not guessable from what you have entered.
• Costs billed alongside the loan. Property taxes, homeowners insurance, mortgage insurance, and association dues on a mortgage; insurance and registration on a car loan. The mortgage calculator models the housing set explicitly, deliberately kept separate from principal and interest.
• Accrual conventions other than periodic. Daily simple interest and precomputed interest are described above and not modelled.
• Anything that does not amortize. Interest-only periods, balloon payments, deferred interest, grace periods, and adjustable rates are different models rather than different inputs.
• Tax treatment. Mortgage interest and student loan interest can be deductible in the United States under conditions that depend entirely on your circumstances. Nothing here accounts for it.
• Whether you should borrow. This page computes a payment. It makes no judgement about affordability and is not advice.

Edge cases

The final payment is almost never equal to the others

The payment is rounded to the cent once and used unchanged; the interest is rounded to the cent at every due date. The residue lands on the last payment. Here that is $495.05 against a regular $495.03. On a thirty-year mortgage the gap can reach a few dollars. The result panel reports the final payment separately rather than letting the schedule quietly disagree with the headline.

A zero-interest loan

At 0% the formula divides by zero, so the calculator uses the limit it tends to: the principal divided by the number of payments. Enter 0 and the explanation changes to match, because a loan with no interest has no split to narrate. The threshold is a very small number rather than exact zero, since a rate of 0.0000001% is not meaningfully different from none and the closed form loses precision there.

A loan that never amortizes

Stretch a term far enough at a high enough rate and the payment converges toward the interest alone. Once rounded to the cent it can equal the first period’s interest exactly, at which point nothing comes off the balance and no amount of time retires it. $1,000 at 30% over 31 years is such a case: the payment is $25.00 and the first month’s interest is $25.00. The calculator detects this, says so, and shows the whole balance falling due on the final payment, which is what would actually happen. It is a sign the term is too long for the rate rather than a loan anyone would be offered.

A single payment

A term short enough to produce one payment is a valid loan and an unusual one. The explanation collapses accordingly rather than talking about a first payment and a later one when there is only the one.

Values the calculator refuses

Rates above 60% and terms beyond 40 years are rejected rather than computed, because past those points the schedule is theatre rather than information. Rates above 36% are computed but flagged, since 36% is where United States consumer-lending convention places the responsible-lending ceiling. Nothing is ever silently clamped: an input the calculator will not accept produces an error and the previous result stays on screen, marked as out of date, rather than being replaced by a figure for a loan you did not describe.

Frequently asked questions

Why is my last payment different from all the others?

Because the payment is rounded to the cent once and then used unchanged for every period, while the interest is worked out on the actual balance each time. Those two roundings do not cancel, and the difference has to land somewhere. It lands on the final payment. On the default scenario the regular payment is $495.03 and the sixtieth is $495.05. Over thirty years at 6.5% on $400,000 the gap reaches a few dollars. Your lender does the same thing, which is why a payoff quote is never exactly the payment you have been making.

Does paying biweekly instead of monthly save money?

Not on its own. Switch the cadence on this page and you will see the total interest move by about $55 on a five-year $25,000 loan, because a nominal rate divided by a larger number is not quite the same contract. What people usually mean by the biweekly trick is different: paying half the monthly payment every two weeks makes 26 payments a year, which is thirteen monthly payments’ worth rather than twelve. That does retire the loan sooner, and it does cut the interest, because you paid more. The cadence is not what did it.

Why is the APR higher than the interest rate I was quoted?

The interest rate prices the money. The APR prices the loan, so it folds in origination fees, points, and anything else you paid to get it, spread across the term. If the loan has no fees the two are the same. If it has fees the APR is higher, and the gap is the honest measure of what the fees cost you. This calculator uses the interest rate, because that is the number the payment is actually computed from. It does not compute an APR, and a page that guessed at one would be inventing your fee schedule.

Is this the same as what my lender will quote me?

The payment should match to the cent if your loan is an ordinary fixed-rate installment loan accruing on the monthly actuarial convention, which most are. It will not match if your lender uses daily simple interest, which is common on car loans, or precomputed interest, which is uncommon and behaves quite differently if you pay early. It also will not match if fees were financed into the balance and you entered only the amount you borrowed. What this page cannot do at all is tell you what rate you would be offered.

What is the difference between the interest rate and the periodic rate?

The interest rate is annual. The periodic rate is what actually gets applied to your balance each period, and it is the annual rate divided by the number of payments a year: 7% a year is 0.5833% each month. That division is a lending convention rather than a mathematical necessity. Compounding 0.5833% twelve times gives 7.229%, not 7%, which is why the effective annual rate on a 7% loan is slightly above 7%.

Why does the calculator not ask for my credit score?

Because a credit score is not in the formula. It is one of the things that determines what rate a lender offers you, along with income, the term, the collateral, and the lender’s own cost of funds. Once you have a rate, the score has done its work and plays no further part in the arithmetic. This page starts where the rate starts. It is a calculator, not an underwriter, and it makes no judgement about whether you should borrow.

What happens if I pay the loan off early?

On a normally amortizing loan, interest stops accruing on the part of the balance you retire, so paying early always costs less in total interest. The schedule below shows what you would owe at the end of every period. Two things can spoil it. A prepayment penalty is a fee for doing it, still legal on some loan types. Precomputed interest is worse: the total interest was fixed at origination, and paying early may earn you only a partial rebate of what you have not yet used.

Does this include insurance, taxes, or fees?

No, and that is deliberate. The payment on this page is principal and interest and nothing else. A loan often arrives attached to costs that are not part of the loan payment: property taxes, homeowners insurance, mortgage insurance, and association dues with a mortgage; comprehensive insurance and registration with a car loan. They are real money and they are not in this formula. For the housing case the mortgage calculator models them explicitly and keeps them visibly separate from principal and interest.

Why does a longer term cost more when the payment is smaller?

Because interest is charged on what you still owe, and a longer term means you still owe more for longer. Stretching the default $25,000 loan from five years to seven drops the payment by $117.71 and adds $1,992.64 in interest. Both figures are real, and neither is the whole answer: a payment you can meet is worth something, and the sensitivity table further down shows you the trade in both columns at once rather than arguing for one of them.

Assumptions

• The rate is fixed for the whole term, and every payment is made in full and on time.
• Interest accrues on the balance standing at each due date, at i = r/f. This is the monthly actuarial convention, and it is a convention rather than a derivation.
• Payments fall at the end of each period, which is the ordinary-annuity convention installment loans use. A loan paid at the start of each period would cost slightly less.
• Periods are treated as exact fractions of a year. They are not always. Twelve months and twenty-four half-months land on a calendar year, but 26 biweekly payments span 26 × 14 = 364 days and 52 weekly payments span 52 × 7 = 364, against a calendar year of 365.2425 days. So a five-year term paid biweekly or weekly is really 4.983 years of calendar time. The model treats n = t × f as exact and does not adjust for the drift.
• The payment is rounded to the cent once, and the final payment absorbs the difference.
• Extra payments, where entered, are applied to principal in the same period they are made.
• Figures are in United States dollars and formatted for a United States reader. The arithmetic is not specific to any currency.

Methodology

The payment comes from the closed form shown at the top of this page. Everything else on the page comes from a schedule built by recurrence, one period at a time, rather than from a second closed form. That is deliberate: a recurrence is what makes the rows add up to the totals printed beside them.

Money is carried as integer cents from the moment it is parsed until the moment it is formatted for display. Interest is rounded to the cent before it is subtracted from the payment, which is what a servicer does, and which is what makes interest plus principal equal the payment exactly in every row rather than nearly. The principal column sums to the amount borrowed exactly, to the cent, at every input the calculator accepts.

Total paid and total interest are read off the schedule, never computed as the payment multiplied by the number of payments. The difference is two cents on the default scenario and it is the difference between a page that agrees with itself and one that does not.

The schedule loop is bounded by a known number of payments rather than by the balance reaching zero. A loan whose payment does not cover its first period’s interest would never terminate under the latter, and such loans exist: see a loan that never amortizes.

The same functions that draw the charts generated the paths in this page’s markup, so the chart you see without JavaScript is not an imitation of the live one, it is the live one rendered ahead of time. The unit tests load the file the browser loads, rather than a copy, and every expected value in them was derived independently rather than by recording what the implementation said.

Nothing you enter leaves your browser. There is no network request in the calculator, no value written to the address bar, and nothing stored between visits. This is enforced by the test suite as well as intended.

Related calculators and concepts

• Calculator Mortgage Calculator This same loan, plus collateral and the costs that travel with a property. Taxes, insurance, mortgage insurance, and association dues, kept visibly separate from principal and interest. Open the calculator →
• Calculator Compound Interest Calculator The same arithmetic pointed the other way: what a balance grows to rather than what it takes to retire one. It also takes the nominal against effective distinction further, from the saving side. Open the calculator →
• Calculator Amortization Schedule The schedule on its own, starting from a payoff balance and a payment on a loan already in progress, with payment dates, printing, and export. Coming soon
• Calculator Refinance Break-Even How many months of a lower payment it takes to recover what the refinance cost to close. Coming soon

The tools above shown as plain text rather than as links are not yet published, because there is nowhere for a link to go.

Every tool on this site is listed at Tools.

Concepts this page relies on

• Amortization. Repaying a debt with level payments, each of which covers the interest due and reduces the principal by whatever is left.
• The periodic rate. The annual rate divided by the number of payments a year, applied to the outstanding balance. A convention, treated at length on the mortgage calculator.
• Effective annual rate. What a periodic rate actually compounds to over a year, and the reason the frequency table’s last column is not flat. The compound interest calculator derives it.
• Annual percentage rate. The interest rate with the fees folded in, which is what makes it the right number for comparing offers and the wrong number for computing a payment.
• Present value. The formula above is a present-value identity rearranged: the loan is worth what the stream of payments is worth today, and solving for the payment is what produces M.
• Ordinary annuity. A fixed payment at the end of each period, which is the structure every figure here assumes.