How To Calculate Mortgage Payments
Interest and Mortgage Formula Calculation
Copyright 2013 by Morris Rosenthal
All Rights Reserved
The second reason you can't use simple interest in mortgage interest rate calculations is that the interest is compounded, added to the principal, on a regular basis. That compounding period was one year in our simple example above, it's normally monthly for mortgages, but compounding periods are often daily on shorter loans, or even "continuous". Compounding interest is what makes mortgage loans seem so expensive in the long run, but it's easier to understand loans by looking first at how interest accumulates on a principal amount. Let's return to our simple example of making a deposit, or loaning money to the bank to see how compounding works to annually increase the amount of the deposit. From our first example, a $100,000 deposit for one year at 5% interest:
Interest = ($100,000) x (0.05) = $5,000
At the end of the first year, which equals the one year compounding period, the interest is added to the principal, so your new amount at the end of the year is:
New amount = Starting deposit + Interest for one year
= $100,000 + $5,000
= $105,000
During the second year of the deposit, simple interest would earn you
= ($105,000) x (0.05)
= $5,250
New amount = Starting deposit + Interest for one year
= $105,000 + $5,250
= $110,250
And if you repeat this calculation ten times for ten years, you end up with the following table which shows annual compounding at work:
| Year | Principal | Interest |
| One | 100,000 | 5,000 |
| Two | 105,000 | 5250 |
| Three | 110,250 | 5512.5 |
| Four | 115,762.5 | 5788.14 |
| Five | 121,550.64 | 6077.53 |
| Six | 127,628.17 | 6381.41 |
| Seven | 134,009.58 | 6700.48 |
| Eight | 140,710.06 | 7035.5 |
| Nine | 147745.56 | 7387.28 |
| Ten | 155132.84 | 7756.64 |
After 10 years, the principal has grown by over 50%, from $100,000 to $155,132.84. The amount of interest you are earning every year has also grown over 50%, even though the interest rate is fixed, at 5% compounded annually.
In order to illustrate the effect compound interest has on mortgage payments, let's turn the simple ten year deposit (a loan to the bank) into a mortgage (a loan to you), where you want pay off the principal so that you can own the house after 10 years. If you were only willing to pay $5,000/year at the end of each year, it would just cover the annual interest on the loan, which would cancel the annual compounding (prevent the interest from accumulating) but after ten years you would still owe the bank $100,000. But let's say you were willing to pay $6,000/year at the end of each year. Since the interest gets added back onto the principal at the end of every year, principal goes down very slowly.
Amount owed = (Principal + Interest for one year) - Annual payment
= ($100,000 + Interest for one year) - $6,000
Since we made the payment at the end of the year, the interest for that year is based on the whole amount, or
Interest for year = $100,000 x 0.05
= $5,000
Amount owed = ($100,000 + $5,000) - $6000
= $105,000 - $6,000
= $99,000
So you can already see the effect of compound interest working against your attempt to pay off the mortgage:
Principal paid = $100,000 - $99,000
= $1,000
You paid the bank $6,000, yet you only reduced the amount of money you owe on the mortgage by $1,000! Now let's compute the second year of our ten year attempt to pay off the mortgage at $6,000 a year:
Amount owed = ($99,000 + Interest for one year) - $6,000
Again, the interest for that year is based on the reduced amount, or
Interest = $99,000 x 0.05
= $4,950
Amount owed = ($99,000 + $4,950) - $6000
= $103,950 - $6,000
= $97,950
And again, compounding works against your attempt to pay off the bank:
Principal paid = $99,000 - $97,950
= $1,050
So you've now paid the bank $12,000 on your $100,000 mortgage, yet the total principal paid off after two years is:
Total principal paid = First year principal paid + Second year principal paid
= $1,000 + $1,050
= $2,050
The mortgage payments for ten years, with the interest paid and principal remaining, are shown in table on the following page:
| Year | Principal | Interest | Payment |
| One | 100,000 | 5,000 | 6,000 |
| Two | 99,000 | 4,950 | 6,000 |
| Three | 97,950 | 4,897.5 | 6,000 |
| Four | 96,847.5 | 4,842.38 | 6,000 |
| Five | 95,689.88 | 4,784.49 | 6,000 |
| Six | 94,474.37 | 4,723.72 | 6,000 |
| Seven | 93,198.09 | 4,659.9 | 6,000 |
| Eight | 91,857.99 | 4,592.90 | 6,000 |
| Nine | 90,450.89 | 4,522.54 | 6,000 |
| Ten | 88,973.43 | 4,448.67 | 6,000 |
After ten years you've paid the bank $60,000 on your $100,000 mortgage at 5%, and you still owe them $88,973.43! That's the compound interest the bank is charging fighting against your payments, and the only way to pay less interest is to pay more per year for fewer years. Let's say you were willing to pay $12,000 at the end of each year. Would that get the mortgage paid off in ten years? We'll compute the first two years again:
Amount owed = (Principal + Interest for one year) - Annual payment
= ($100,000 + Interest for one year) - $12,000
Interest = $100,000 x 0.05
= $5,000
Amount owed = ($100,000 + $5,000) - $12,000
= $105,000 - $12,000
= $93,000
Principal reduction = $100,000 - $93,000
= $7,000
Compound interest still works against your attempt to pay off the mortgage, but over half of the $12,000 payment actually goes to reducing the principal owed. In the second year of our ten year attempt to pay off the mortgage at $12,000 a year:
Amount owed = ($93,000 + Interest for one year) - $12,000
Again, the interest for that year is based on the reduced amount, or:
Interest = $93,000 x 0.05
= $4,650
Amount owed = ($93,000 + $4,650) - $12,000
= $97,650 - 12,000
= $85,650
And the amount of principal paid down the second year is:
Principal reduction = $93,000 - $85,650
= $7,350
After two years, you've paid the bank $24,000 on your $100,000 mortgage, yet the total principal paid off after two years is:
Total Principal paid = First year principal paid + Second year principal paid
= $7,000 + $7,350
= $14,350
And the remaining payments would look like this:
| Year | Principal | Interest | Payment |
| One | 100,000 | 5,000 | 12,000 |
| Two | 93,000 | 4,650 | 12,000 |
| Three | 85,650 | 4,282.5 | 12,000 |
| Four | 77,932.5 | 3,896.63 | 12,000 |
| Five | 69,829.13 | 3,491.46 | 12,000 |
| Six | 61,320.58 | 3,066.03 | 12,000 |
| Seven | 52,386.61 | 2,619.33 | 12,000 |
| Eight | 43,005.94 | 2,150.3 | 12,000 |
| Nine | 33,156.24 | 1657.81 | 12,000 |
| Ten | 22,814.05 | 1,140.7 | 12,000 |
The larger payment decreases the mortgage principal faster, so less interest is charged. Let's look at how much of the money paid in each instance went to principal reduction as a percentage of the amount paid.
% for principal reduction = [(original principal - principal paid) / original principal] x 100
So for the $6,000 per year payment
= [($60,000 - $44,422.10) / $60,000] x 100
= [$15,577.90 / $60,000] x 100
= 25.96%
While at $12,000 a year, the amount that went towards principal reduction:
= [($120,000 - $31,954.76) / $120,000] x 100
= [$88,045.24 / $120,000] x 100
= 73.37%
The scary looking formula below is for computing a monthly loan payment at a given interest rate over a number of years (measured in months) which results in the loan being completely paid off at the end of that term.
M = P [ i(1 + i)n ] / [ (1 + i)n - 1]
There are far more complex versions of the mortgage formula to deal with more exotic types of loans, but for normal fixed rate mortgages and loans, this is the one formula you need to master. It's shown in standard shorthand, where a symbol next to a bracket or parenthesis means multiplication, the "/" means division, and a superscript (the n above the parenthesis) means "raise to the power of." Don't worry if you aren't sure what all this means, we'll be showing several detailed examples.
The meaning of the letters in the equation, in order, is:
M = The monthly payment
P = The principal, or the amount of money being borrowed
i = The interest for each compounding period, or the interest per month for a standard mortgage
n = The number of compounding periods, or the number of months for a standard mortgage
So let's jump right in and do a basic mortgage calculation. For a fifteen year mortgage at 5% interest compounded monthly, we would first solve for i by dividing the annual interest percentage by the number of months in the year. Note that for mortgage purposes, all months are created equal, there is no adjustment for February being short or the difference of one day between months like November and December.
i = 5% / # months per year
= 0.05 / 12
= 0.004167
and the number of payments = n
n = (12 payments/year) x (15 years)
= 180 monthly payments
Next we would solve for (1 + i)n , the factor which is used in both the top and bottom (numerator and denominator) of our mortgage equation, which is really just a giant fraction.
(1 + i)n = (1+ 0.004167)180
= (1.004167)180
If you remember your High School math, raising a number to a power means multiplying the number by itself that many times. But multiplying 1.004167 times 1.004167 would get rather tiring by the 180th multiplication, and would likely lead to multiple mistakes. So you can use the xy key on any pocket calculator that has an xy, or, you can just use Google with the ^ character, which is above the "6" on your keyboard (Shift and 6):
which yields 2.113830. The final zero isn't significant, so the number we'll use is 2.11383.
Now we can replace the (1 + i)n in our formula with 2.11383 so it reads:
M = P [ i (2.11383)] / [ 2.11383- 1]
or doing the subtraction in the bottom of the fraction and substituting the 0.004167 we earlier computed for i
= P [0.004167 x 2.11383] / 1.11383
= P [0.0088083] / 1.11383
And since we can do division before multiplication as well as after
= P (0.0079081)
So for a fifteen year mortgage at 5% annual interest compounded monthly, the monthly payment will be equal to the principal loaned multiplied by 0.0079081. Back before computers came into common use, bankers had books filled with tables that gave, for each length of loan and a large range of possible interest rates, a factor with which to multiply the principal to get the monthly payment. Let's do a couple mortgages with the factor we just calculated:
For a $100,000 fifteen year mortgage at 5%, the payment would be:
M = $100,000 x 0.0079091 = $790.81
For a $180,000 fifteen year mortgage at 5%, the payment would be:
M = $180,000 x 0.0079091 = $1423.638
Of course, nobody charges thousandths of pennies on mortgage payments, but does that mean keeping six digits was overkill? Let's run through the same problem quickly keeping as many significant digits as Google gives us, starting way back with our interest rate calculation:
i = 5% / # months per year
= 0.05 / 12
= 0.00416666667
And the next calculation would be
(1 + i)n = (1+ 0.00416666667)180
= (1.00416666667)180
= 2.11370393 versus the 2.113830 we used earlier
Replacing the (1 + i)n in our formula with 2.11370393 we get:
M = P [ i (2.11370393)] / [2.11370393- 1] or
= P [0.00416666667 x 2.11370393] / 1.11370393
= P [0.00880709972] / 1.11370393
= P (0.00790793629)
Or on our $100,000 loan, the payment would be
= $100,000 (0.00790793629)
= $790.793629 or rounding down to pennies
= $790.79
And this is the payment you'll see if you use any of the online mortgage calculators, because they all carry more significant digits than we did in our first calculation. Of course, the total difference was two cents a month, but since it doesn't cost anything extra to calculate with lots of digits, it can't hurt.
When banks calculate mortgage payments, they may keep track of the stray hundredths of pennies. Even if they round up in order to fix the monthly payment, the final payment of the mortgage may be a little less than the rest of the payments to make up for it. At most, it's a question of a dollar or two discount in the final payment, so it's nothing to get excited about.
If you want to skip the formula and just read your monthly mortgage payment from a table, I've created fixed rate mortgage tables for 15 and 30 year mortgages, covering rates from 0.0% to 7.95%. Note, I use the same numbers from this page in my amortization formula example.