How amortisation works, payment by payment

Amortisation is the repayment of a debt in level instalments over a fixed term, where each instalment pays the interest that has built up since the last one and puts everything left over against the balance. It is spelled amortization in American English and means the same thing. Mortgages, car loans and most personal loans are built this way.

Each period, three things happen in this order, and nothing else happens at all:

• Interest is charged on the balance outstanding, at the rate for one period.
• That interest is taken out of your payment.
• Whatever remains reduces the balance.

Writing $B_k$ for the balance after payment $k$, $i$ for the period rate and $M$ for the payment:

$$\text{interest}_k = i \times B_{k-1}, \qquad \text{principal}_k = M - \text{interest}_k, \qquad B_k = B_{k-1} - \text{principal}_k$$

A 30 year mortgage is that loop run 360 times. The row-by-row table it produces is the amortisation schedule.

One convention to check before you copy any of this across a border. Everything on this page takes the period rate as the annual rate divided by the number of payments a year, which is how loans are quoted in the United States and the United Kingdom. It is not universal. Canadian mortgage rates are compounded semi-annually by statute, so a quoted 6.5 percent charges $(1.0325)^{1/6} - 1$, about 0.5345 percent, each month rather than $0.065/12$. The mechanism below is identical either way. Only the number you feed into $i$ changes.

The payment is not a figure a lender picks. It is the one amount that, repeated for the whole term, lands the balance exactly on zero at the final payment, which is what the loan payment calculator above solves for. Seen from the other end, it is the payment stream whose present value at the loan's own rate equals the amount borrowed, so it is the time value of money doing the pricing.

One warning about the word itself. In accounting, amortisation also means writing off an intangible asset across its useful life, which is depreciation for things you cannot touch. Same word, unrelated subject. Everything below is the loan sense of amortisation.

The payment is level. What the payment does is not.

Interest is charged on what you still owe. The balance falls a little each month, so the interest charge falls a little each month, and because the payment does not move, whatever the interest stops taking has nowhere to go except the balance. Interest share down, principal share up, and the two always add to the same total.

The rate at which that happens is exact, and it is the part most descriptions leave out. Set the principal part of one payment beside the principal part of the next:

$$\text{principal}_{k+1} = M - i B_k = M - i\left(B_{k-1} - \text{principal}_k\right) = \text{principal}_k(1 + i)$$

The amount coming off your balance grows by exactly one period's interest every period. Not roughly, exactly. The principal column of an amortisation schedule is a compound growth series running at the rate you are being charged, which is why the compound interest calculator describes the shape of a debt as well as the shape of a savings account.

On $250,000 at 6.5 percent over 30 years, the first payment puts $226.00 against the balance and the second puts $227.23, which is the same amount grown by one month of interest at 0.541667 percent. Repeat that 358 more times and it ends up swallowing almost the whole payment, but it starts from almost nothing:

Read the interest column downwards and you have the whole behaviour of a fixed-rate loan in six lines.

There is no trick in this and no clause in the agreement that causes it. At the start of a loan you still owe essentially all of it, so the interest charge is as big as it is ever going to be.

Month one on the loan above: interest is $250,000 times 0.541667 percent, which is $1,354.17, inside a payment of $1,580.17. That leaves $226.00, or 14.3 percent of the payment, to reduce the debt. Nobody decided that split. It is the only thing those three numbers can produce.

The result is a repayment curve that begins almost flat. The fraction of the loan actually repaid after $k$ of $n$ payments is:

$$\frac{(1+i)^k - 1}{(1+i)^n - 1}$$

Put $k = 60$ and $n = 360$ into that at 0.541667 percent a month and it returns 6.4 percent. Five years of payments on a 30 year mortgage, and 6.4 percent of the debt has gone: the balance moves from $250,000 to $234,027.44 while you have handed over the equivalent of nearly 38 percent of what you borrowed.

Two milestones on this loan are worth carrying around. The interest share does not drop below half until payment 233, which is 19 years and 5 months in. Half of the original balance is not repaid until payment 257, at 21 years and 5 months. Neither is a fixed feature of a 30 year loan, because both depend on the rate: at 3 percent the split flips at payment 84, and at 10 percent it holds out until payment 278. A higher rate does not only cost more, it postpones the point at which you start making visible progress.

This is also why moving house or refinancing a few years in can feel like starting again. A new loan does restart at the interest-heavy end of a fresh schedule. That is worth understanding rather than fearing: the interest share is a consequence of the balance and the rate, not a penalty for refinancing, so a restarted schedule is not on its own a reason against it. A new loan at a lower rate, or over a shorter term, can cost less in total than the one it replaces even though its first year looks worse. The comparison that settles it is total cost to the end including fees, not the shape of the first column.

Money paid on top of the scheduled payment, and applied to principal, is not split. All of it reduces the balance, which deletes every future interest charge that money would otherwise have carried. Whether a lender applies it that way by default is a separate question, taken up at the end of this section.

The cleanest way to see how big that is: pay next month's principal early. On the loan above, payment 2 was going to put $227.23 against the balance and leave $249,546.77 owing. Add $227.23 to payment 1 and the balance lands on $249,546.77 a month early. From there the schedule is identical to the one you were given, running one month ahead of itself, so the debt is cleared in 359 payments and the payment you never make is the 360th.

That is the whole of the claim that extra payments save interest, and it works because your extra principal compounds at the loan's own rate. $227.23 grown at 0.541667 percent a month for the remaining 359 months comes back to $1,580.17. A small amount paid 30 years early cancels a payment nearly seven times its size.

That multiple deserves a second look, because it is easy to read as something it is not. The return on the money is the loan's own rate and nothing more. The sevenfold figure is simply that rate compounded for 359 months, and the payment it removes falls due 30 years from now, in money that will buy less than $1,580.17 buys today. Seven times the cash is not seven times the purchasing power. Two further things sit between the arithmetic and the benefit: the interest removed is a pre-tax figure, and where mortgage interest is deductible, as it can be in the United States, it is worth less than face value after tax; and the same money left invested elsewhere would have compounded too, at whatever rate that investment turned out to pay.

Timing is most of the effect. The same amount paid in year 25 has only a few periods left to compound, so it removes a fraction as much. Two practical points follow from that:

• Say what the money is for. Extra money is only principal if it is applied to principal. Some lenders hold it against the next scheduled payment instead unless they are told otherwise, so check the next statement from the lender rather than assuming.
• Read the agreement before making a habit of it. Some loans charge for early repayment, and 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. What is allowed varies by country and by loan type.

There is a third option worth knowing about. Some lenders will recast a loan after a lump sum, keeping the original term and lowering the payment rather than keeping the payment and shortening the term. Same balance, opposite benefit, and far less interest removed.

Shortening the term changes neither the rate nor the loop. It changes $n$, which changes the payment, which changes every split in the schedule.

The same $250,000 at the same 6.5 percent:

Going from 30 years to 15 raises the payment by about 38 percent and cuts the interest by about 55 percent. Going from 30 to 20 raises the payment by about 18 percent and cuts the interest by about 38 percent.

At the same rate, a shorter term always costs less in total, because you owe less at every point in the schedule as well as owing it for fewer months. That part holds at every rate. What does not hold everywhere is the flattering version of it, that the percentage cut in interest always beats the percentage rise in the payment. It is true at 6.5 percent and it fails when money is cheap, because a low rate leaves little interest to save while the payment still has to rise enough to fit the term. At 3 percent, the same swap from 30 years to 15 raises the payment by about 64 percent and cuts the interest by only about 53 percent. For this loan the two lines cross at roughly 4.1 percent, and the two changes are measured against different bases in any case, so the comparison is a rough feel rather than a law.

The shape changes too, not just the total. Month one of the 15 year version charges the same $1,354.17 of interest as the 30 year version, since it is the same balance at the same rate. But that charge sits inside a payment of $2,177.77 rather than $1,580.17, so 62 percent of the payment goes to interest instead of 85.7 percent, and the split turns in favour of principal at payment 53 rather than payment 233.

The 20 year row is worth a second look, because you can reach it without signing for it. Paying $1,863.93 every month on the 30 year loan clears it in exactly 240 payments, because a level overpayment is arithmetically the same thing as the shorter term, provided overpayments are permitted and the extra is applied to principal. Choosing between them is a question about commitment rather than about arithmetic: the shorter contract is compulsory and in some markets is priced a little below the longer one, while the overpayment is optional and can stop in a bad month.

Four things catch people out.

• The payment is not the price. The payment is what leaves your account each month. The total interest is what the loan costs, and the term pulls the two apart: on the numbers above, 30 years costs $318,861.22 in interest against $141,998.31 over 15. Same debt, same rate.
• A payment that does not cover the interest makes the balance grow. If the payment is smaller than the interest charge, the shortfall is added to what you owe and the debt gets bigger while you are paying it. That is negative amortisation, and it is how a deferred or reduced-payment arrangement can leave a borrower owing more than before.
• The bill is often bigger than the payment. In the United States, a mortgage servicer usually collects property tax and insurance in the same monthly charge and holds them in escrow. Only the principal and interest part amortises, so the money leaving your account can be well above $1,580.17.
• A rate change restarts the arithmetic. On a variable or adjustable loan, the lender re-amortises: it works out the payment that clears the balance you have now over the term you have left, at the new rate. The schedule you were handed at the start is a projection, not a promise.

One more, over long terms. The payment is fixed in cash but not in purchasing power. Whenever prices rise, $1,580.17 in the final year of a 30 year loan buys less than it does in the first, and over a stretch that long the gap has historically been wide. That is why a long fixed-rate debt behaves differently from most commitments once inflation is taken into account, though the direction depends on what prices and incomes actually do, which nobody knows in advance. A schedule counts dollars, not what they buy.