How to work out mortgage interest

A standard mortgage is a loan we take out when we want to buy property – typically a house or a flat. Let's see how the interest you will have to pay on a mortgage is worked out.

Annual interest vs. monthly compounding

Every mortgage has an annual interest rate, which is given as a percentage. So you might see an advert for a mortgage with a rate of 3% p.a. The “p.a.” is an abbreviation of the Latin “per annum”, which means “per year” or “a year”. Put simply, and not quite accurately, if you borrow £200,000, you pay 3% interest over a year, which is £6,000.

In reality, though, you won't pay exactly six thousand pounds – you will pay less. You normally repay a mortgage monthly, and the bank adds interest just as often. Instead of working out one annual interest charge, the bank works out twelve monthly ones. And because you gradually repay the mortgage, you owe the bank less and less money, so the amount of interest is lower and lower too. Let's show this with an example. You borrow £200,000 at 3% annual interest and repay £2,000 a month. What will the first interest charge be?

The first repayment

First we work out the monthly interest rate. We do this by dividing the annual rate by twelve:

$$\frac{3\mathrm{\%}}{12}=0.25\mathrm{\%}$$

So the monthly interest rate is 0.25%. For the first month we pay the bank 0.25% of what we still owe the bank, that is, of £200,000. We work out 0.25% as

$$200000\cdot 0.25\mathrm{\%}=200000\cdot 0.0025=500$$

For the first month we pay the bank £500 in interest. Since our monthly repayment is £2,000, after the first month we owe the bank

$$200000+500-2000=198500$$

that is, £198,500. So our first repayment is made up of £500 that went on interest and £1,500 that paid off part of our debt to the bank. We call this £1,500 the capital repayment.

The second repayment

What would the second month look like? The monthly interest rate doesn't change; it is still 0.25%. What changes is the amount we owe the bank on the mortgage. We call this amount the principal, and after the first month it is £198,500. We work out the new monthly interest in exactly the same way as for the first month, only instead of £200,000 we use £198,500, that is, we work out 0.25% of £198,500:

$$198500\cdot 0.25\mathrm{\%}=198500\cdot 0.0025=496.25$$

So in the second month the interest is only £496.25 instead of £500 – because we have already paid off part of the debt and have a smaller principal. The capital repayment will be

$$2000-496.25=1503.75$$

So in the second month we paid off more of the mortgage than in the first, and in total we now owe only

$$198500-1503.75=196996.25$$

Further repayments…

The further repayments are worked out in the same way until we reach zero. In the third month we would work out the interest as

$$196996.25\cdot 0.25\mathrm{\%}=196996.25\cdot 0.0025=492.49$$

We can see that the interest of £492.49 is again a little smaller than the previous month, and the capital repayment of £1,507.51 is again a little larger. Each following month the interest will be smaller and smaller, because we owe the bank less and less money. And the capital repayment will be larger and larger – over time we repay the mortgage faster and faster, because we pay less money in interest and more money goes towards paying off the principal.

Notice that although the annual interest rate was 3%, we won't pay £6,000 in interest. To pay £6,000 in interest, we would have to pay £500 in interest every month. But every month we pay less and less, so in the end we pay less than £6,000 in interest.

That is really everything you need to know about how mortgage interest is worked out. But let's look at a few more interesting points.

The exact calculation of monthly interest

At the beginning we said that we work out the monthly interest as

$$\frac{3\mathrm{\%}}{12}=0.25\mathrm{\%}$$

This is the calculation that banks use, but it isn't mathematically correct. Banks use it more for historical reasons, because dividing by twelve is simply easier than the mathematically correct method. What are we actually trying to achieve when we work out the monthly interest? An annual interest rate of 3% on £200,000 tells us that over a year the client would pay £6,000 in interest. Imagine that the client didn't repay the mortgage for a whole year and just let the interest be added. What monthly interest rate would we need so that after twelve months we collect exactly £6,000 in interest? If we use our “divide by twelve” formula, we actually get more than £6,000. Let's work it out to see:

First we show how to add interest to the principal in a simple way. If we worked out the monthly interest rate as 0.25%, then we get the resulting amount after adding the interest by multiplying the principal by the number 1.0025:

$$200000\cdot 1.0025=200500$$

After two months without repayments, the principal will therefore be

$$200000\cdot 1.0025\cdot 1.0025=201001.25$$

So we multiply the £200,000 by the number 1.0025 as many times as we want interest to be added. To make this simpler, we can use powers, because we know that

$$1.0025\cdot 1.0025={1.0025}^2$$

so we can write

$$200000\cdot {1.0025}^2=201001.25$$

We want to know how much the amount would be after 12 months:

$$200000\cdot {1.0025}^{12}=206083$$

We can see that after 12 months the principal would be about £206,083, which is about £83 more than it should be. Well… it isn't a huge difference, let's admit, but in short it isn't the correct result. How do we fix it?

Let's go back to the annual interest rate of our mortgage. If we want to find out how large the principal would be if we applied the annual three per cent interest to it, we can write it like this:

$$200000\cdot 1.03=206000$$

Now we want to split the number 1.03 into a product of twelve equal numbers, which will give us one plus the monthly interest rate. So the product of which twelve equal numbers gives the result 1.03? It is the twelfth root of 1.03:

$$1.03=\sqrt[12]{1.03}\cdot \sqrt[12]{1.03}\cdot \dots \cdot \sqrt[12]{1.03}$$

We can work out that the twelfth root of 1.03 is 1.0024662 and a bit, which is truly not a big difference from the number 1.0025 that we got using our divide-by-twelve method. But it is the correct result: if our monthly interest rate were $\sqrt[12]{1.03}-1$, the principal would grow by exactly three per cent after a year without repayments.

Daily vs. monthly interest

Interestingly, some banks work out the monthly interest on mortgages a little differently. Working out the monthly interest by dividing the annual interest by twelve has one more side effect: we get the same interest rate for every month, even though months are different lengths. February and July get the same rate. How do we get around that? Some banks solve it by dividing the annual interest by 360, which gives them a daily rate, and then multiplying this daily rate by the number of days in the month.

Why do they divide by 360 and not 365 or 366? Because it is, once again, easier to calculate with.