How extra mortgage payments work

A scheduled amortising payment is sized to clear the loan on a fixed date. Part of it is that month's interest, charged on the principal still owed, and the rest reduces the balance. An extra amount paid against principal does not change the scheduled figure. It just makes that month's principal reduction larger, so next month's interest is calculated on a smaller number.

On $300,000 at 6.5 percent over 30 years the scheduled payment is $1,896.20. The first month's interest is $300000 \times 0.065 / 12 = 1625$, so $1,625.00, and the rest of that scheduled payment is principal. Add $200 extra and that month's principal reduction is larger by exactly that extra. The extra does not sit in a side account. It comes off the balance before the next interest charge is computed.

That is the same loop how amortisation works already runs, with a higher payment. The reason this page exists is the comparison: the original schedule against the faster one, in months and in interest, so the extra has a price in time rather than only a price in cash. The calculator above opens on those figures because they are the first worked example.

Written as a formula, the number of months $n$ until a balance $B$ clears at a constant total payment $p$, with monthly rate $i$, is

$$n = \frac{\ln\left(1 - \frac{iB}{p}\right)}{-\ln(1+i)}$$

The logarithm is only defined when $p$ is larger than that month's interest $iB$. If the combined payment does not cover the interest, the balance grows, and there is no payoff date to compute. The calculator returns that case as a statement rather than as an infinite number of months, because infinity months is not an answer a reader can use.

The closed-form month count is a logarithm. Each extra dollar raises $p$ and shortens $n$, but the shortening is not linear. On the default loan, $200 extra cuts 83 months off 360. Another $200, taking the extra to $400, cuts 132 months off the original term, which is 49 months more, not another 83. The second $200 still helps. It helps less than the first $200, because the easy months to remove are the last ones, where the scheduled payment is almost all principal already.

That flattening is the picture on the extra payment explorer. Drag the extra along the curve and months remaining fall fast at first and then flatten. Doubling the extra is not twice the decision. The first increment takes the longest, highest-interest tail off the loan. The next increment is working on a shorter remaining tail.

Interest saved is the other reading of the same fact. The original schedule on this loan pays $382,633.47 of interest. With $200 extra that falls to $279,184.67, a cut of $103,448.79. With $400 extra it falls to $222,801.45, a cut of $159,832.02. Again the second increment is real and smaller.

The zero-extra row is a restatement of the original schedule, and it is the check that the identities collapse when nothing extra is paid. The $250,000 loan in the third worked example is the same check on a different principal: extra of zero leaves 360 months and $318,861.22 of interest untouched.

The logarithm is only defined when $p$ is larger than that month's interest. If the combined payment does not cover $iB$, the balance grows, and there is no finite $n$. On a 6.5 percent loan the first month's interest on $300,000 is $1,625.00. A payment that does not cover that interest, extra or not, never catches up.

This is the same trap the credit card payoff calculator names for revolving balances, and it is rarer on a mortgage only because the scheduled payment is already set above that floor. It becomes relevant if someone cuts the payment below the schedule, or if a teaser rate ends and the new interest exceeds what they had been paying.

Negative amortisation is that case given a name: the shortfall is added to what you owe, so the debt gets bigger while you are paying it. The extra-payment identities assume the total payment covers the interest every month. They have nothing to say about a payment that does not.

Paying extra on a 30-year loan and signing a 15-year loan are the same payment formula with a different $n$, as long as the rate is the same. The 15-year against 30-year comparison is that split: the shorter contract raises the required payment and cuts full-term interest, and in the United States the 15-year quote is usually a fraction of a point cheaper as well.

What extra principal on a 30-year loan cannot copy is a lower 15-year rate. What it can copy, at the same rate, is the calendar. Pay the 15-year amount each month and a 30-year loan clears in 180 months, because both loans are the same formula. The contracted payment stays the lower 30-year figure, so stopping the extra is not a default. A 15-year payment is the minimum. Miss it and it is a missed payment.

In the United States most residential mortgages can be prepaid without a penalty. That is a market fact, not a property of 30-year loans everywhere. Some consumer loans use precomputed interest, where the total interest is fixed at the outset and paying early does not remove it the same way. Read the agreement before making a habit of an extra.

Recasting is the other option some servicers offer after a lump sum: keep the original term and lower the payment, rather than keep the payment and shorten the term. Same balance, opposite benefit, and far less interest removed. This page models a constant extra that shortens the term. It does not recast.

The identities on this page assume the extra is constant, every month, until the loan clears. A one-off lump at the start is a smaller principal: put it in as a lower amount borrowed. A lump later is a shorter remaining term at a new starting balance, which is a new run of the same formula rather than an adjustment to this one.

A refinance replaces the rate and usually resets the term. Extra principal keeps the rate and cuts the term. They are sequential, not additive. The refinance break-even calculator is the tool for that second decision. Do not add the two break-evens. The first extra-payment comparison is truncated by the refinance.

A rising extra, or an extra that arrives some months and not others, is a smaller total extra and a longer remaining term than the figure on the screen. There is no penalty in the formula for an interrupted extra. There is only less of it.

How mortgages work is the surrounding machinery: the lien, loan-to-value, escrow, points. This page is only what a constant extra does to one amortising schedule.

Prepayment penalties, the lost return on the extra dollars if they would otherwise have been invested, and the value of a cash buffer all sit outside this page. The arithmetic answers one question: if this extra is paid against this loan every month until it clears, how many months and how much interest come off. Whether those dollars should be the extra is a different question, and paying down debt versus investing is the place that comparison starts.

The interest removed is a pre-tax figure. Where mortgage interest is deductible, as it can be in the United States, the interest saved is worth less than face value after tax. The deduction is a tax-year fact, not a formula one, and it does not belong inside the month count.

The extra is also assumed applied to principal when it arrives. Some lenders hold extra money against the next scheduled payment unless they are told otherwise, so check the next statement rather than assuming. Extra money is only principal if it is applied to principal.

This is educational material, not financial advice. The figures throughout are a teaching loan: $300,000 at 6.5 percent over 30 years, with extras of $200 and $400, and a $250,000 loan with extra of zero as the identity check. They are there so every published number can be re-derived. They are not a recommendation about how much extra any household should pay.