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.
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 x y 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.