How Are Mortgage Payments Calculated?

This is one of the most frequently asked questions about mortgages, and unfortunately the answer is not simple or obvious. This article lays out a step by step explanation for the most common mortgage type in the United States – the 30 year fixed rate mortgage.

The example I will use is a $200,000 mortgage with an “interest rate” of 6.5% and a term of 30 years. The first question you might be asking is: what exactly does an interest rate of 6.5% mean? In the US, all mortgage interest rates are expressed as a monthly rate multiplied by 12; the monthly rate is used to calculate the amount of interest owed based on the outstanding loan balance at the beginning of each month.

In our example:

The quoted in interest rate is 6.5%, so the monthly rate is 6.5% / 12 = 0.5416666%
In the first month the outstanding loan balance is $200,0000
To get the amount of interest owed in the first month, multiple $200,000 x 0.005416666 = $1,083.33 (the number 0.005416666 is 0.5416666% expressed as a decimal)

So far so good. However, the mortgage payment is not $1,083.33; it turns out that it is $1,264.14. The reason is that almost all residential mortgages in the US are “fully amortizing”; this means that at the end of the term of the mortgage (30 years in our example) the loan balance is paid down to exactly zero. The difference between $1,264.14 and $1,083.33 is $180.81 and is used to pay down the loan balance.

But how is the $1,264.14 mortgage payment calculated? It is the unique dollar amount that accomplishes these things:

It is big enough to cover the amount of interest owed each month
If it is paid consistently every month for the term of the mortgage (30 years or 360 months in put example) the “left over” amount used to pay down the loan balance – the “principal portion of the payment” – is exactly enough to get the loan balance to zero at the end of the term.

There are a few key points to note:

Since a portion of each payment is used to pay down the loan balance each month, the loan balance decreases each month (it eventually gets to zero at the end of the term).
The amount of interest owed each month is based on the loan balance at the beginning of that month; since the loan balance goes down every month (and on a fixed rate loan the interest rate stays the same) the amount of interest owed also goes down each month.
Since the monthly payment is fixed and the amount of interest owed does down every month, the principal portion of the payment goes up every month.

This relationship between the fixed monthly payment, the interest portion of payment, and the principal portion of payment can be seen in the chart below.

Note that interest makes up the lion’s share of the payment in the early years of the mortgage; the higher the interest rate the slower the decline in the interest portion of the payment in the early years.

When hearing about the complexity of the fully amortizing payment, many people’s intuition tells them that the only way to calculate this “magic number” must be by trial and error. There is, in fact, a simple mathematical formula for the fully amortizing payment (those who remember their high school algebra can view it here). In addition, most financial calculators and the online calculators on my site will do the calculation for you.

There a few technical details that you should be aware of if you are trying to calculate the precise amount of your mortgage payment:

Rounding. The formula used to calculate the fully amortizing payment results in an exact payment that goes out to many decimal places. Since real-world payments are made in dollars and cents (and it is essential that the loan is completely paid off at the end of the term) lenders round the payment up to the nearest penny. The result is that the final payment (in our example in month 360) will almost always be lower than the first 359 payments
Per Diem Interest: In the discussion above I state that the interest paid in the first month is calculated by multiplying the loan amount by the monthly rate; this is true for the first scheduled payment. However, most loans close and the loan amount is disbursed at some point during the month; in these cases the lender is also owed interest for the number of days between closing and the due date of the first scheduled payment. This “daily” interest – known as per diem interest – is paid at closing and will show up as a line item on your closing documents. The calculation of per diem interest assumes a 30 day month; compute the amount of monthly income as above and divide by 30 to get the amount of daily interest. Then multiply this daily interest amount by the number of days between closing and the 1st day of the following month.