Mortgage Calculator

How a mortgage payment is calculated

A fixed-rate mortgage is a level-payment amortizing loan: you pay the same amount every month, each payment covers the interest that has accrued since the last one, and whatever is left over reduces the balance. The payment that makes the balance land on exactly zero at the end of the term is the one the formula below returns.

The rate the formula wants is the nominal annual interest rate, the one a lender quotes on the note. It is not the APR, which is a broader cost measure that folds certain fees into a single annual figure, and it is not the effective annual rate either. Dividing the quoted rate by twelve is a convention for how monthly mortgage interest is computed in the United States, not a mathematical derivation: 6% ÷ 12 = 0.5% a month compounds to 6.168% over a year, and the true monthly equivalent of a 6% effective annual rate would be 0.4868%, not 0.5%. Mortgages use the convention. This calculator follows it, and says so.

What your numbers mean

Real-world costs: taxes, insurance, and dues

None of what follows enters the mortgage formula. Property tax, homeowners insurance, mortgage insurance, and association dues are all costs of owning the home; none of them accrues interest on the loan and none of them reduces the balance. Your lender will usually collect the first two into an escrow account and pay them on your behalf, which is why the amount debited from your account each month is larger than the payment the formula produced.

Enter the figures you have actually been quoted. This calculator will not estimate any of them, because none of them can be estimated honestly from what it knows about you.

Anything you enter above appears as a separate estimated monthly housing payment beside the results, never merged into the principal-and-interest figure. The two are different quantities and the page will not blur them.

How the balance is paid down

The payment never changes, but what it does changes every month. Early on, most of it is interest on a large balance. Late on, the balance is small, the interest it accrues is small, and almost the whole payment goes to principal. The table below is the year-by-year summary of that shift; the full month-by-month schedule is behind it.

Understanding your mortgage payment

The periodic rate is a convention, not a derivation

A 6% mortgage does not charge 6% on the balance once a year. It charges 6% ÷ 12 = 0.5% on the outstanding balance every month. Those are not the same thing: 0.5% compounded twelve times is 6.168%, so the effective annual rate on a 6% mortgage is slightly above 6%. Going the other way, the monthly rate that genuinely compounds to 6% a year is 0.4868%, not 0.5%.

United States fixed-rate mortgages use the simple division. It is a convention the industry settled on, and it is the one this calculator follows. But it is worth knowing that it is a convention, because it is the reason a mortgage’s effective rate is always a little higher than its quoted rate.

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 above, the first payment is $1,600.00 of interest and $318.56 of principal. Because that $318.56 makes the balance a little smaller, the next month’s interest is a little smaller, which leaves a little more for principal. The effect compounds in your favour, slowly at first and then very quickly.

This 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.

What each input actually does

• Principal. The payment is directly proportional to it. Double the loan and you double the payment, exactly.
• Rate. Non-linear, and steeper than most people expect. The rate affects both the numerator and the denominator of the formula, which is why a single percentage point is worth so much over thirty years.
• Term. Lowers the payment and raises the total. Extending a loan spreads the same principal over more payments, but it also leaves a larger balance outstanding for longer, so more interest accrues.

What happens when you change one input

The section above says which direction each input moves the payment. These two tables say by how much, for your loan rather than for a typical one, because a percentage point is worth a different amount of money at every principal and every term. Change anything in the calculator and both tables follow.

If the rate were different

The steps are equal but the effects are not: half a point up costs slightly more than half a point down saves. That asymmetry is the curvature of the formula, and it is the reason a rate is worth shopping for rather than estimating. A rate below 0% or above 25% is left out of the table entirely, because the calculator would refuse to compute it.

If the term were different

Two columns of the same table move in opposite directions, and that is the whole trade. Doubling the term from 15 years to 30 does not double the cost of the loan and does not halve the payment; the payment falls by roughly 29% and the interest more than doubles. Which figure matters more is a question about a budget, and the arithmetic has no opinion on it.

The principal is missing from these tables on purpose. It is the one input whose effect needs no table: the payment is directly proportional to it, so a loan 10% larger has a payment 10% larger and total interest 10% larger, at any rate and any term.

Paying extra is not modelled here, in these tables or anywhere else on the page. Prepayment changes the schedule rather than scaling it, and the honest treatment of it is a schedule of its own rather than a fifth row bolted onto this one. The question about biweekly payments below explains what it actually does.

What moves the number, and what this model can see

A payment is the output of a model, and a model is a boundary drawn around a real system. Some of what determines what a house costs you is inside the boundary, some of it arrives as a figure you type in, and some of it is outside altogether. This table says which is which, so that nothing on the page has to be taken on trust.

The four tags are the whole point of the table. In the formula means the figure is one of the four symbols above. You supply it means the page will carry it, separately labelled, if you enter it, and will not guess it if you do not. Outside the model means it is real and this arithmetic does not describe it. Not estimable means no calculator could produce it honestly from what this one is given, and a figure offered anyway would be invented rather than computed.

Common misconceptions

“The APR and the interest rate are the same thing”

They are different measures and they answer different questions. The interest rate is what the balance accrues; the APR folds certain fees and prepaid charges into a single annual figure under a regulator-defined method, so that two offers with different fee structures can be compared. Where fees exist the APR is generally the higher of the two. Enter the note rate here; entering an APR would overstate your payment.

“A 6% mortgage means I pay 6% of the loan every year”

Interest is charged monthly on a balance that falls every month, so the amount you actually pay in interest falls every year. In the default scenario the first year’s interest is about $19,093 on a $320,000 loan, and the last year’s is about $731.

“A longer mortgage is cheaper, because the payment is lower”

The monthly payment is lower and the total cost is higher. Change the term above from 30 years to 15 and watch both numbers move at once: the payment goes up by roughly 40%, and the total interest falls by more than half. Which of those matters more is a question about your budget, not about the arithmetic. But they are not the same question, and one of them is often not asked.

“PMI is just extra interest”

It is an insurance premium, and the policy it buys protects the lender against your default, not you. It does not accrue on the balance, it does not appear anywhere in the amortization formula, and it does not reduce what you owe by a cent. For many conventional loans it can be cancelled once enough equity has built up; the statutory automatic termination point is a loan-to-value of 78%, subject to conditions, and it does not apply to FHA mortgage insurance premiums.

“This is what I will actually pay my lender”

The headline figure on this page is principal and interest, which is what the formula produces and nothing more. Most lenders also collect property tax and homeowners insurance into escrow, so the amount that leaves your account is larger. That is what the additional-costs section above is for, and it is why its total is labelled “estimated monthly housing payment” rather than “monthly payment”.

“The balance drops by the same amount every month”

It drops by more every month. The schedule and the first chart above show it directly: the principal portion of the payment grows month over month for the entire life of the loan, because the interest portion it is competing with keeps shrinking.

What this calculator does not include

Being explicit about this is more useful than quietly modelling a guess. None of the following is accounted for:

• Closing costs and cash to close. Origination fees, title, appraisal, recording, and prepaid items are all paid at closing and none of them changes the payment.
• Discount points. Points buy a lower rate, so they do change the payment, but only indirectly, by changing the rate you would enter here, and at a cost paid up front.
• Prepaid per-diem interest. A real closing produces a stub period of interest between the funding date and the start of the first full period. This calculator assumes the first payment falls exactly one month after origination.
• Day-count conventions. 30/360 and actual/365 give slightly different interest figures. Which one applies is a matter of your loan documents.
• Adjustable rates. An ARM’s payment is recalculated whenever its rate resets. The formula here describes a single fixed rate for the whole term.
• Extra payments and recasts. Prepaying principal changes the schedule rather than adding to it, which is exactly why it deserves proper treatment rather than a bolted-on field.
• Escrow analysis. Tax and insurance change over time and your escrow payment is periodically re-evaluated. The figures here are held constant.
• Balloon structures, interest-only periods, and negative amortization. None of these is a level-payment fully amortizing loan.
• Eligibility of any kind. This page computes arithmetic. It has no view on what you can borrow.

Edge cases

A few inputs make the standard calculation behave differently, and one of them makes the formula undefined outright. Each is handled deliberately rather than incidentally, and each is worth knowing about.

A rate of zero

At i = 0 the formula is 0 ÷ 0: the numerator P i vanishes and so does the denominator 1 − (1+i)⁻ⁿ. There is no value to substitute in, so the calculator uses the limit the formula tends to, which is the one anybody would have guessed: the loan divided by its number of payments. Every payment is then pure principal, the balance falls by the same amount every month, and the first chart is a straight line rather than a curve. The threshold is not exactly zero but 10⁻⁹ a month — about 0.0000012% a year — below which the denominator has no significant digits left to divide by anyway.

A schedule that ends early, or a payment short

The payment is rounded to the cent once, and the schedule then uses that rounded figure for every month. Rounded up, it sends a fraction more to principal each month than the formula assumed, and the loan can retire on payment 359 of 360. Rounded down, a few cents can still be standing when the last payment arrives. The Payments figure beside the result is the number of rows the schedule actually has rather than the number that was asked for, and the final payment is shown at its true amount. Both are consequences of money being counted in cents rather than in real numbers, and neither is an error.

A payment too small to amortize

If a payment does not cover the interest that accrues in a month, the balance grows instead of shrinking and the loan never retires. The schedule builder is bounded so that this terminates rather than looping, with the whole remaining balance falling due on the last row — but the case is unreachable from this page, and provably so: the level payment is P i divided by a denominator that is always less than one, so it is always larger than P i, the first month’s interest. A level-payment loan computed this way cannot fail to amortize. The guard exists because a schedule builder that can loop forever is a schedule builder that eventually will.

Inputs at the edges of what is accepted

Rates above 15% and terms longer than 40 years are computed and flagged, because they are unusual rather than wrong. Rates above 25%, terms longer than 50 years, and amounts above $100 million are refused outright: past those, an amortization schedule is theatre rather than information. A negative rate is refused too. The arithmetic tolerates one perfectly well, and United States fixed-rate mortgages do not, so modelling one would be inventing an instrument rather than describing one.

An input that is not a number yet

While a field is empty, half-typed, or out of range, the figures on screen are the last complete set, dimmed, under a line saying so. They are never blanked and never replaced with “$NaN”. A stale number that still looks current is the one failure mode worse than no number at all, which is why the staleness is stated rather than styled.

Nothing down, and the refinance case

A down payment of zero is accepted and gives a loan-to-value of 100%. The arithmetic is unremarkable; what it implies about mortgage insurance and about whether such a loan exists at all is between you and a lender. Typing into the loan amount instead reverses the direction of the calculation: the loan becomes the input and the home price the consequence, which is what a refinance actually looks like. With the down payment expressed as a percentage that inversion is a division rather than a subtraction, and at 100% it has no solution — there is no price at which the whole of it is a down payment and a loan remains.

Frequently asked questions

How is a monthly mortgage payment calculated?

The annual interest rate is divided by twelve to get a monthly periodic rate, and the term in years is multiplied by twelve to get the number of payments. Those two numbers and the loan amount go into the level-payment formula at the top of this page, which returns the one figure that stays the same every month for a fixed-rate loan.

Is the interest rate the same thing as the APR?

No. The interest rate, sometimes called the note rate, is what the balance actually accrues, and it is what this calculator asks for. The APR is a broader cost measure that folds certain fees and prepaid charges into a single annual figure under a regulator-defined method. Where fees exist, the APR is generally higher than the note rate.

Why is my first mortgage payment almost all interest?

Interest is charged on the balance outstanding, and the balance is at its largest on the first day. The payment is a fixed amount, so whatever the interest does not consume goes to principal. As the balance falls, the interest portion falls with it and the principal portion grows to fill the gap.

Does a longer mortgage term cost more?

At the same interest rate, yes. A longer term lowers the monthly payment because the principal is spread over more payments, but it leaves a larger balance outstanding for longer, so more total interest accrues. In practice longer-term loans are also often priced at higher rates, which widens the gap further.

Are property taxes and insurance included in the monthly payment?

Not in the figure the mortgage formula produces. That figure is principal and interest only. Most lenders collect property tax and homeowners insurance alongside it into an escrow account, so the amount actually debited each month is larger. This calculator keeps the two figures visibly separate and never merges them into a single number.

How much is PMI, and why does this calculator not work it out?

There is no universal formula for private mortgage insurance. The premium depends on the loan-to-value band, the credit score, the coverage percentage, the loan purpose, the occupancy, the debt-to-income ratio, and the individual insurer’s rate card. A calculator that produced a confident PMI figure from the loan-to-value alone would be manufacturing precision it does not have, so this one asks for the figure your lender quoted instead.

Do biweekly payments really save money?

They save money, but not for the reason usually given. Twenty-six half-payments a year is thirteen monthly payments rather than twelve, so a biweekly schedule quietly makes one extra payment a year. The saving comes from prepaying principal, not from any change in how interest is calculated. Sending one-twelfth extra each month achieves almost exactly the same thing.

Why is the last payment on the schedule a different amount?

Each month’s interest is rounded to the cent before it is subtracted from the payment, so the principal reduction is a fraction of a cent away from what the unrounded formula assumed. Over hundreds of payments that difference accumulates, and it has to land somewhere. It lands on the final payment, which is why this schedule shows that payment at its true amount rather than repeating the level figure and claiming the balance reached zero.

Is anything I type here sent anywhere?

No. The calculation runs entirely in your browser, there is no server request behind it, and nothing is saved between visits. The page itself uses Google Analytics, which collects information about the visit in the way described in the privacy policy, but no figure you enter into the calculator is ever included in that.

Assumptions

• A fixed rate for the whole term, compounded monthly.
• The periodic rate is the annual nominal rate divided by twelve, which is the United States convention for fixed-rate mortgages.
• The number of payments is the term in years multiplied by twelve.
• The first payment falls exactly one period after origination, with no stub period of prepaid interest.
• Every payment is made in full and on time; nothing is prepaid.
• The payment is rounded once, to the cent, and that rounded figure is what the schedule uses. Each month’s interest is rounded to the cent before it is subtracted, so every row adds up exactly and the final payment absorbs the accumulated difference.
• Money is carried internally as whole cents, so the schedule sums to the loan amount exactly rather than approximately.
• All figures are in United States dollars.

This page is an explanation of arithmetic, not financial advice, a rate quote, or an offer of credit. Curiosity Mapped does not track, publish, or offer mortgage rates. Use the figures your lender gives you.

Methodology

The section above lists what the model assumes about mortgages. This one lists what the implementation does about arithmetic, because several of those decisions are visible in the figures and would otherwise look like mistakes.

• Money is carried as integer cents between the moment an input is read and the moment a figure is printed. Three hundred and sixty accumulations of binary fractions is precisely the situation where 0.1 + 0.2 stops being a curiosity and starts being a column that does not add up.
• The payment is rounded once, to the cent, and the schedule then uses that rounded figure — because that is what you would actually pay. Each month’s interest is rounded to the cent before it is subtracted, which is what a servicer does, and which is what makes interest + principal = payment hold exactly in every row rather than nearly.
• The schedule is built by recurrence, not from a closed form. Balance, interest, principal, and the new balance, one month at a time. A closed form would be faster and would disagree with its own totals by a few cents; the table on this page adds up because it is the thing that was computed rather than a rendering of something else.
• The closed form appears once, in the explanation of why the balance curve bends, and it is implemented alongside the recurrence so that the claim can be checked rather than asserted.
• The formula is implemented as P i ÷ (1 − (1+i)⁻ⁿ) rather than as its algebraic twin with (1+i)ⁿ in both halves. The two are equal on paper. In floating point the second puts a very large number in the numerator and the denominator at once, which is where cancellation error comes from.
• The charts are drawn from the schedule, one point per year, by the same functions that generated the paths sitting in this page’s markup. The chart you see before any JavaScript runs is not a drawing of the real chart; it is the real chart, rendered ahead of time.
• Everything runs in the browser. No network request, no storage, no dependencies, no build step. The file the tests exercise is the file the browser loads, not a copy of it.
• The arithmetic has unit tests that assert, across a matrix of rates, terms, and principals, that the schedule’s principal column sums to the loan amount exactly in integer cents and that every row balances. The invariant is the point: a payment that is a cent out is a bug that no amount of eyeballing a table will find.

Related calculators and concepts

• Calculator Loan Calculator The same amortization engine without the housing-specific parts, for car loans, personal loans, and anything else with a level payment. Coming soon
• Calculator Amortization Schedule The schedule on its own, for a loan already in progress, starting from a payoff balance and a payment rather than from a purchase. Coming soon
• Calculator Compound Interest The other direction of the same arithmetic: what a balance grows to rather than what it takes to retire one. 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

These calculators are not yet published and are shown as plain text rather than as links, because there is nowhere for a link to go.

Every tool on this site is listed at Tools.

Concepts this page relies on

Six ideas do most of the work above. None of them has a page of its own yet; until one does, each link goes to the place on this page where the idea is explained rather than to a definition that does not exist.

• Amortization. Retiring a debt through a schedule of equal payments, each one part interest and part principal, with the split moving from the first toward the second.
• Nominal, periodic, and effective rates. Three different numbers that all describe the same loan. Dividing the first by twelve to get the second is a convention, and the third is what the convention actually costs.
• The time value of money. Interest accrues on what is outstanding, so when a dollar is repaid matters as much as how many of them there are. It is why the payment composition shifts and why the balance curve bends.
• Loan-to-value. The loan as a fraction of the property’s value, reported beside the down payment in the calculator. Not the same as the down-payment percentage once a property has been owned for a while, and it is the figure lenders price against and the one that governs mortgage insurance.
• Escrow. The account a lender uses to collect property tax and insurance alongside the payment and pay them on your behalf. It is the reason the amount debited each month is larger than the amount this formula produces.
• APR. A regulator-defined annual figure that folds certain fees into the rate so that two offers can be compared. A comparison tool, not the rate your balance accrues.