Mortgage Amortization Formula

P is the principal: the amount actually financed at closing. Not the purchase price, not the appraised value. Price minus down payment, plus anything rolled in — financed closing costs, an upfront funding fee on a VA loan, an FHA upfront mortgage insurance premium. If a fee is financed, it is principal, and it accrues interest for the life of the loan like every other dollar.

r is the periodic interest rate — the rate for one payment period, not for one year. For a standard monthly mortgage, r = annual rate ÷ 12. A 6.5% loan has r = 0.065 ÷ 12 = 0.00541667. Note that this is the note rate, not the APR. APR blends in fees to make offers comparable and is a disclosure figure; it is never what the payment is computed from.

n is the total number of payment periods, again not years. A 30-year monthly mortgage has n = 360. A 15-year has n = 180.

The single most common error in applying the mortgage amortization formula by hand is a units mismatch, and it is worth seeing how badly it fails so you recognize the symptom. On a $400,000 loan at 6.5% for 30 years, the correct payment is $2,528.27. Substitute the annual rate 0.065 for r while leaving n at 360 and the equation returns $26,000.00 a month — a suspiciously round number, because at that absurd rate the payment collapses to almost pure interest. Keep the monthly rate correctly but set n to 30 instead of 360 and you get $14,481.99, the payment for a thirty-month loan.

Both mistakes produce a number so far off that the error announces itself. The rule of thumb: r and n must be measured in the same time unit, and that unit must be the payment period. Convert both, or convert neither.

The level payment that retires principal P at periodic rate r over exactly n periods is:

M = P × [ r(1+r)ⁿ ] / [ (1+r)ⁿ − 1 ]

Equivalently, and often easier to punch into a calculator:

M = P × r / [ 1 − (1+r)⁻ⁿ ]

The two forms are algebraically identical — divide the numerator and denominator of the first by (1+r)ⁿ and you get the second — and either will do. The quantity (1+r)ⁿ is the growth factor: what one dollar becomes after n periods of compounding at rate r. It appears twice because the equation is balancing two compounding processes against each other, which is the subject of the next section.

Two edge cases are worth noting. If r = 0, the formula is undefined — the denominator goes to zero — and the correct payment is simply P ÷ n, a zero-interest loan repaid in equal slices. And M scales linearly with P: double the loan amount at the same rate and term and the payment doubles exactly. It does not scale linearly with r or with n, which is why a one-point rate move and a five-year term change have such different effects on affordability.

M as computed here is principal and interest only. Property taxes, homeowners insurance, mortgage insurance, and HOA dues are collected alongside it in escrow and are not part of the amortization arithmetic at all — they change what leaves your bank account without changing a single row of the schedule. Our principal and interest calculator isolates the P&I component if you need to separate it from an all-in quote.

The formula is not arbitrary, and you can reconstruct it from one idea without any calculus.

Start with the lender's position. They hand over P today. In exchange they receive n equal payments of M, spread out over the future. For the deal to be fair at rate r, the value of that stream of payments, measured in today's dollars, must equal exactly P.

What is a future dollar worth today? If money compounds at r per period, a dollar received one period from now is worth 1/(1+r) today, because that is the amount which, invested at r, becomes one dollar. A dollar two periods out is worth 1/(1+r)², and so on. So the present value of the whole stream is:

P = M/(1+r) + M/(1+r)² + M/(1+r)³ + … + M/(1+r)ⁿ

That is a finite geometric series — every term is the previous one multiplied by the constant factor 1/(1+r). Geometric series have a closed-form sum, which is the piece of algebra doing all the work here. Applying it collapses the whole right-hand side to:

P = M × [ 1 − (1+r)⁻ⁿ ] / r

Solve that for M and you have the second form of the payment equation from the previous section, and therefore the first as well. Nothing else is needed. A mortgage is a present-value-of-an-annuity problem with a bank on one side of it, and the mortgage amortization formula is what falls out when you insist that the loan and the payments be worth the same amount on day one.

This framing also explains something that otherwise looks like a coincidence: the bracketed term [1 − (1+r)⁻ⁿ]/r is called the annuity factor, and it is the same quantity used to price bonds, value pension obligations, and evaluate lease-versus-buy decisions. A mortgage is simply the most familiar annuity most people will ever encounter.

The payment equation tells you the size of M. It says nothing about where each payment goes. That requires a second, much simpler rule, applied one period at a time.

Let Bₜ₋₁ be the balance owed at the start of period t. Then:

interestₜ = Bₜ₋₁ × r

principalₜ = M − interestₜ

Bₜ = Bₜ₋₁ − principalₜ

That is the whole per-period engine, and it is worth noticing what it does not contain. There is no schedule of principal amounts fixed in advance, no percentage table, no lender discretion. Interest is charged on what you currently owe. Principal is the leftover. The balance falls by that leftover. Repeat n times and it reaches zero — which is exactly the property M was solved for.

Because M is constant while Bₜ₋₁ shrinks every period, the interest term shrinks too, and the principal term grows by precisely the amount the interest term fell. The split rotates continuously, and it accelerates: each dollar of principal retired permanently removes r dollars per period of future interest, which frees up more principal next period, which removes more interest. That compounding-in-reverse is the mechanism behind every strategy for paying a mortgage off early.

It also means the interest column is not something the lender front-loads by choice. It is the arithmetic consequence of charging interest on an outstanding balance that starts at its maximum. There is no alternative accounting under which the early payments look different.

Iterating the three lines above will give you the balance at any month, but you do not have to. There is a closed form that reaches month k directly:

Bₖ = P(1+r)ᵏ − M × [ ((1+r)ᵏ − 1) / r ]

Read it as a difference of two competing quantities. The first term, P(1+r)ᵏ, is what the debt would have grown to after k periods if you had paid nothing at all and interest had simply compounded. The second term is the future value of the payments you actually made, each one compounded forward from the date you made it to period k. What remains is the gap between the debt that grew and the payment stream that chased it. At k = n the two terms are equal by construction and the balance is zero.

An equivalent form some references prefer expresses the balance as the present value of the payments still outstanding:

Bₖ = M × [ 1 − (1+r)^(k−n) ] / r

Both return the same number. The first is more intuitive if you think of a loan as a debt being chipped away; the second is more intuitive if you think of a loan as a stream of remaining obligations. Use whichever matches how you reason about it.

This formula is the workhorse behind several everyday questions. Payoff quote at any date: evaluate Bₖ. Equity at sale: home value minus Bₖ. Balloon payoff: Bₖ at the maturity month. Break-even on a refinance: compare the balance and remaining payment stream on the old loan against a new one starting at Bₖ. Cumulative principal paid through month k is just P − Bₖ, and cumulative interest through month k is M×k − (P − Bₖ).

Take P = $400,000, an annual rate of 6.5%, and a 30-year term. Convert first: r = 0.065 ÷ 12 = 0.00541667, and n = 30 × 12 = 360.

The growth factor (1 + 0.00541667)³⁶⁰ works out to 6.991798. Substituting into the payment equation:

M = 400,000 × [ 0.00541667 × 6.991798 ] / [ 6.991798 − 1 ] = $2,528.27

Now apply the per-period split. Month one: interest = $400,000 × 0.00541667 = $2,166.67. Principal = $2,528.27 − $2,166.67 = $361.61. New balance = $399,638.39. Month two charges interest on the new, slightly smaller balance: $399,638.39 × 0.00541667 = $2,164.71, so principal rises to $363.56 and the balance falls to $399,274.83. Month three: $2,162.74 interest, $365.53 principal, balance $398,909.30.

Notice the pattern already forming across three rows: interest falls by about $1.96 a month and principal rises by almost exactly the same amount. The payment never moves. Over 357 more rows that drift compounds into a complete reversal.

Now use the balance formula to skip ahead rather than iterating. At k = 12:

B₁₂ = 400,000(1.00541667)¹² − 2,528.27 × [ ((1.00541667)¹² − 1) / 0.00541667 ] = $395,529.10

Twelve payments totaled $30,339.27. Only $4,470.90 of that reduced the debt; $25,868.36 was interest. After a full year of paying on time, the borrower has retired 1.1% of the principal. Iterating twelve rows by hand returns the same $395,529.10, which is the check worth running the first time you use the closed form.

The long view: 360 payments total $910,177.95, of which $510,177.95 is interest — more than the house cost to finance. Milestones along the way: $339,104.51 owed at year 10, $290,236.56 at year 15, $222,661.13 at year 20, and $129,216.65 at year 25. Every one of those figures came from the same two equations. Run your own inputs through the calculator above and the 30-year mortgage amortization schedule page to see the full row-by-row version.

People often read the early rows as evidence of something unfair. They are not. They are a direct consequence of comparing two numbers.

In month one the lender is owed interest on the entire principal: P × r. On the example above that is $2,166.67. The payment is $2,528.27. The difference — $361.61 — is all that is left for principal. The ratio between P × r and M is what determines the shape of the whole schedule, and on a long-term loan at a meaningful rate, P × r is very nearly as large as M.

This gives a useful diagnostic you can run on any quote in your head. Multiply the loan amount by the monthly rate. If that number is close to the quoted payment, the loan will amortize slowly at first. If it is a small fraction of the payment, principal reduction starts fast. It also explains why the crossover — the month when principal first exceeds interest — sits so late on a 30-year note. On our $400,000 loan at 6.5%, that crossover does not arrive until month 233, in year 20, when the payment splits $1,262.00 interest to $1,266.27 principal. For nearly two-thirds of the term, the majority of every payment is interest.

Shorten the term and everything shifts. A larger M against the same P × r means more principal from the very first row, an earlier crossover, and dramatically less total interest — which is the entire argument for a 15-year note, at the cost of a much higher required payment. Raise the rate and it moves the other way: P × r grows, M grows less than proportionally, and the schedule flattens.

Every equation on this page assumes three things: a fixed periodic rate, a constant payment M, and payments made exactly on schedule. Break any of those and the closed form no longer applies. You have to iterate instead — which is precisely what a calculator or a spreadsheet does for you.

The most common break is an extra principal payment. Add X per month and the per-period recursion still works fine — interest is still Bₜ₋₁ × r, principal is now M + X − interest — but there is no clean expression for the new payoff month, because the term itself has become the unknown. You solve it by stepping forward until the balance crosses zero. On the $400,000 example, adding $200 a month retires the loan in 293 payments instead of 360 — 24 years and 5 months, 67 months early — and cuts total interest from $510,177.95 to roughly $398,286, a saving of about $111,892. That result cannot be read off a formula; it comes from the iteration, which our mortgage calculator with extra payments runs for you.

Three other structures break it in different ways. An adjustable-rate mortgage holds the closed form only within each fixed-rate period; at every reset the lender re-solves the payment equation using the new rate, the current balance as P, and the remaining months as n. A recast does the same thing voluntarily after a large principal payment — same rate, same maturity date, smaller balance, so a new and smaller M. A balloon keeps the formulas intact but truncates the schedule early, so the answer you want is Bₖ at maturity rather than a zero balance.

What survives all of these is the per-period rule. Interest on the current balance, principal as the remainder, balance reduced accordingly. That recursion is always true. The closed forms are shortcuts that exist only when the inputs happen to stay constant.

Nothing on this page is proprietary. The payment equation is the standard present-value-of-an-ordinary-annuity result from financial mathematics, the same relation implemented by the PMT function in every spreadsheet program and by the amortization routines in loan servicing systems. The remaining-balance expression is its direct companion, obtained by applying the same present-value logic to the payments not yet made. Any two people working the algebra correctly will land on identical figures.

The figures in the worked example were computed at full floating-point precision and rounded only for display. That is why a hand recomputation can differ by a cent: twelve payments of the displayed $2,528.27 total $30,339.27, and subtracting the $4,470.90 of principal leaves $25,868.37, while carrying the unrounded payment of $2,528.2721 through gives $25,868.36. Real servicers resolve this by rounding the payment to the cent and adjusting the final payment up or down to clear the balance exactly, which is why your last payment will differ slightly from all the others.

Other real-world details the formulas deliberately exclude: escrow for taxes and insurance, mortgage insurance premiums, day-count conventions on loans that accrue daily rather than monthly, and the treatment of a payment received mid-cycle. None of these change the amortization arithmetic; they change what you write on the check. For the rate context that makes an example realistic, Freddie Mac publishes weekly national averages in its Primary Mortgage Market Survey, and the Consumer Financial Protection Bureau's loan options guide explains how term and rate structure interact with the variables above.

The most useful thing you can do with the mortgage amortization formula is stop taking payment quotes on faith. Compute M yourself from P, r, and n; if a lender's number differs, something has been added — financed fees, mortgage insurance, escrow — and it is fair to ask what. The calculator at the top of this page and every other tool on the mortgage amortization calculator homepage runs on exactly the equations printed above, so you can verify any figure either produces with a pocket calculator and five minutes.