How To Calculate Mortgage Payments

Mortgage Math Workbook

Copyright 2014 by Morris Rosenthal

All Rights Reserved

Interest and Mortgage Formula Calculation

If you loaned a bank $100,000 at a 5% interest rate, compounded annually, the bank would pay you $5,000 per year. So why can't you get a $100,000 mortgage and pay the bank $5,500 a year, let them earn a 10% profit? The reason is that traditional mortgages are designed so you end up owning the house when the mortgage is paid off. Our simple example above would apply to an "interest only" mortgage, where you are really just renting the house from the bank. After 30 years, zero equity. It's the reverse of your loaning $100,000 to the bank and earning $5,000 per year in interest. The bank doesn't get to keep your $100,000, they're just paying for the use of it. In essence, the bank is renting the principal from you, the same way you rent a house from the bank with an interest only mortgage.

The next complication in mortgage interest rate calculations is that interest is compounded. Going back to our loaning the bank money example, lets say you agreed to loan the bank $100,000 for 10 years, with the interest being compounded onto the principal annually. Using simple interest compounded annually, the situation would look like this.

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

So 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 loan into a mortgage, where you are working to pay off the principal so that you can own the house. If you were only willing to pay $5,000/year, you'd never make a dent in the principal, so it would be an interest only mortgage. But let's say you were willing to pay $6,000/year. That comes to $500 a month, but since we're keeping it simple and only compounding interest once a year, there's no reason to track the monthly payments. Since the interest gets added back onto the principal at the end of every year, principal goes down very slowly. The mortgage payments would look like this:

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

So, after ten years you've paid the bank $60,000 on your $100,000 mortgage, 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 in the long run is to pay more per year. Lets say you were willing to pay $12,000 per year, or $1,000 per month. Would that get the mortgage paid off in ten years?

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

So, after ten years you've paid the bank $120,000 on your $100,000 mortgage, and you still owe them another $22,814.05, but at least the end is in near, and in another two years the loan will be paid off.

With mortgages, we want to find the monthly payment required to totally pay down a borrowed principal over the course a number of payments.The standard mortgage formula is:

M = P [ i(1 + i)n ] / [ (1 + i)n - 1]

Where M is the monthly payment. i = r/12. The same formula can be expressed many different way, but this one avoids using negative exponentials which confuse some calculators.

For our $100,000 mortgage at 5% compounded monthly for 15 years, we would first solve for i as

i = 0.05 / 12 = 0.004167 and n as 12 x 15 = 180 monthly payments

Next we would solve for (1 + i)n = (1.004167)180 using the xy key on the calculator, which yields 2.11383

Now our formula reads M = P [ i(2.11383)] / [ 2.11383- 1] which simplifies to

M = P [.004167 x 2.11383] / 1.11383 or

M = $100,000 x 0.00790 = $790.81

All of the rounding down I did makes a 2 cent difference on the monthly payment, compared with keeping all the digits the calculator can handle. Now, one important feature of the mortgage formula is that it's the principal is multiplied last, meaning that we can develop a table of mortgage rate multipliers for any fixed time period that will yield a monthly payment simply by multiplying the principal borrowed.

If you're curious to know how much interest you'd pay the bank over the course of the mortgage,just multiply the amount of the monthly payment by the number of payments and subtract the principal:

($791.81 x 180 ) - $100,000 = $142,525.80 - $100,000 = $42,525.80

The only bright side to paying the bank all of that interest is that in most cases, it's deductible on your Federal income tax in the in the years that it's paid. The savings to you depends on what tax bracket you're in. If you're only in the 10% tax bracket to start with, you're only getting a 10% discount on your taxes for carrying a mortgage. If you're in the 25% tax bracket, you're getting a 25% discount.

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 4.0% to 5.95%. Note, I use the same numbers from this page in my amortization formula example.