How to calculate interest on a mortgage

A conventional mortgage is a loan we take out when we want to buy a property - typically a house or flat. Let's show you how to calculate the interest you will have to pay on your mortgage.

Annual interest vs. monthly interest

Every mortgage has a set annual interest rate, which is given as a percentage. So you may see an advert for a mortgage with a rate of 3% p.a. This "p. a." stands for the Latin "per annum" and means "per year" or "per annum". Simply and inaccurately put, if you borrow one million crowns, you will pay 3% per year in interest, which is 30 000 crowns.

But in reality you don't pay exactly 30,000, you pay less. You pay your mortgage monthly and the bank often charges you interest anyway. Instead of calculating one year's interest, the bank calculates twelve months' interest. And as you pay off the mortgage over time, you owe the bank less and less money and therefore the amount of interest is lower and lower. Let's illustrate this with an example. You borrow one million crowns at 3% annual interest and pay 10,000 crowns a month. What will be the first interest rate?

The first instalment

The first thing to do is to calculate the monthly interest rate. This is calculated by dividing the annual rate by twelve:

$$\frac{3 \%}{12} = 0{,}25 \%$$

The monthly interest rate is therefore 0.25%. For the first month, we will pay the bank 0.25% of what we still owe the bank, i.e. one million crowns. We calculate 0.25% as

$$1\ 000\ 000\cdot 0{,}25\%=1\ 000\ 000\cdot0{,}0025=2500$$

For the first month, we pay the bank CZK 2,500 in interest. Since our monthly payment is 10,000 crowns, we owe the bank after the first month

$$1\ 000\ 000+2500-10\ 000=992\ 500$$

992,500 crowns. So our first instalment consists of 2500 crowns that went to interest and 7500 crowns that we used to retire our debt to the bank. So we call this 7500 crowns a mortgage.

The second instalment

What would the second month look like? The monthly interest rate is unchanged, it is still 0.25%. But the amount we owe the bank on the mortgage changes. We call this amount the principal and after the first month it is CZK 992 500. We will calculate the new monthly interest in exactly the same way as the first month, but instead of one million we will calculate 992 500 crowns, i.e. we will calculate 0.25% of 992 500 crowns:

$$992\ 500\cdot 0{,}25\%=992\ 500\cdot0{,}0025=2481{,}25$$

So the second month interest is only 2481.25 crowns, instead of 2500 crowns - because we have already redeemed part of the debt and have a smaller principal. The interest will be equal to

$$10\ 000 - 2481{,}25 = 7518{,}75$$

Thus, we have redeemed more of the mortgage in the second month than in the first month, and in total we owe only

$$992\ 500-7518{,}75=984\ 981{,}25$$

More repayments...

The next payments are still calculated the same way until we get to zero. So the third month, we would calculate the interest as

$$984\ 981{,}25\cdot0{,}25\%=984\ 981{,}25\cdot0{,}0025=2462.45$$

We can see that the interest of 2462.45 crowns is again a little less than the previous month, and conversely the death of 7537.55 is again a little more. Each subsequent month the interest will be less and less as we owe less and less money to the bank. And the mortgage will get bigger and bigger - over time, we pay off the mortgage faster and faster because we pay less money in interest and more money goes towards paying down the principal.

We may notice that even though the annual interest was 3%, we are not paying 30,000 crowns in interest. In order to pay 30,000 in interest, we would have to pay 2,500 crowns in interest every month. However, we pay less and less each month and, as a result, we pay less than CZK 30 000 in interest.

In fact, that's all you need to know about how mortgage interest is calculated. But let's take a look at some other interesting facts.

The exact calculation of monthly interest

At the beginning, we said that we would calculate the monthly interest as

$$\frac{3 \%}{12} = 0{,}25 \%$$

This is a calculation that banks use, but it is not mathematically correct. Banks use it more for historical reasons, because dividing by twelve is simply easier than the mathematically correct method. What do we actually want to achieve when we calculate the monthly interest? An annual interest of 3% on one million tells us that in a year the client would pay 30,000 crowns in interest. Let us imagine that the client did not pay the mortgage for a whole year and just let the interest accrue. What monthly interest rate would we need to have to collect exactly CZK 30 000 in interest after twelve months? If we use our formula "divided by twelve", we get more than 30 000 crowns. Let's do the math on purpose:

First, we'll show you how to simply add the interest to the principal. If we have calculated the monthly interest rate as 0.25%, the resulting amount after adding the interest is obtained by multiplying the principal by 1.0025:

$$1\ 000\ 000\cdot1{,}0025=1\ 002\ 500$$

After two months without payments, the principal will be equal to

$$1\ 000\ 000\cdot1{,}0025\cdot1{,}0025=1\ 005\ 006{,}25$$

Thus, we multiply the million by 1,0025 as many times as we want to add interest. To simplify things, we can use powers of one, because we know that

$$1{,}0025\cdot1{,}0025=1{,}0025^2$$

so we can write

$$1\ 000\ 000\cdot1{,}0025^2=1\ 005\ 006{,}25$$

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

$$1\ 000\ 000\cdot1{,}0025^{12}=1\ 030\ 416$$

We can see that after 12 months the principal would be equal to 1,030,416 crowns, which is 416 crowns more than it should be. Like... it's not much of a difference, let's face it, but in short, it's not the right result. How to fix it?

Let's go back to our annual interest rate on our mortgage. If we want to figure out how much the principal would be if we applied an annual interest rate of three percent to it, we can write it like this:

$$1\ 000\ 000\cdot1{,}03=1\ 030\ 000$$

We now want to decompose the number 1.03 into the product of twelve equal numbers to get the monthly interest rate. So, the product of what twelve equal numbers gives us the result 1.03? It's the 12th root of 1.03:

$$1{,}03 = \sqrt[12]{1{,}03}\cdot\sqrt[12]{1{,}03}\cdot\ldots\cdot\sqrt[12]{1{,}03}$$

We can calculate that the twelfth root of 1.03 is 1.0024662 and some change, which really isn't much difference from the result of 1.0025 we got by our method of dividing by twelve. But it is the correct result: if our monthly interest rate were equal to $\sqrt[12]{1,03}$, the principal would increase by exactly three percent after a year of default.

Daily vs. monthly interest

Interestingly, some banks calculate the monthly interest on mortgages a little differently. In fact, calculating the monthly interest by dividing the annual interest by twelve has another side effect: we get the same interest rate for all months, even though the months are different in length. February and July get the same rate. How to get out of this? Some banks solve this by dividing the annual interest rate by 360 to get the daily rate and then multiplying that daily rate by the number of days in the month.

Why do they divide 360 and not 365 or 366? Because, again, it's easier to calculate.

Links

• How to pay off a mortgage