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Financial Math: Calculating Fixed-Rate Mortgage Payments

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Your monthly home mortgage payments depend on many factors. The price of the home minus the down payment equals the principal. And the principal, annual interest rate, and loan period all determine how much you pay each month. Most people use online mortgage calculators to compute their monthly home loan payments, but anyone with a scientific calculator can apply the formula that banks and lenders use to figure monthly payments.

(1) The first step is to determine the amount you have financed, and call this number "P." Usually, this is the selling price of the home minus your down payment. But depending on the cost of legal fees, you may have opted to finance some of those closing fees as well. Check your mortgage statement to double check.

(2) The next step is to determine your monthly interest rate. Since most mortgages are paid in 12 installments per year, the monthly interest rate is simple the annual fixed rate divided by 12. Call the monthly interest rate "R." Remember to express R as a decimal, not a percent.

(3) Now the next piece of info to gather is the length of the loan period. For mortgages, this is measured in months, so simply multiply the number of years by 12. Call the number of months "M."

(4) Lastly, we need to compute a number called "W" which is defined by this equation W = (1+R)^M , where M is the exponent (or power).

And now we are ready to combine these separate pieces of information to calculate your monthly house payments "H" with this formula:

H = (P x R x W)/(W - 1)

Let's use the home loan formula to compare two different home loan offers. Suppose Lender A offers you 8.25% for 30 years, and suppose Lender B offers you 6% for 15 years. Assume the amount of the home loan is $120,000 in both cases.

Lender A
In this home mortgage example, we have
P = 120000
R = 0.0825/12 = 0.006875
M = 30x12 = 360
W = (1.006875)³⁶⁰ = 11.7815
and so using the monthly payment formula, we have
H = (120000 x 0.006875 x 11.7815)/(10.7815)
= $901.52

Lender B
In this home mortgage example, we have
P = 120000
R = 0.06/12 = 0.005
M = 15x12 = 180
W = (1.005⁾¹⁸⁰ = 2.4541
and so using the monthly payment formula, we have
H = (120000 x 0.005 x 2.4541)/(1.4541)
= $1012.63

On a month-to-month basis, the home loan offer by Lender A yields slightly lower monthly house payments. So if you only consider your monthly budget, that is the better offer. However, over the course of 30 years you end up paying 360 x $901.52 = $324,547. But with lender B, you end up paying only 180 x $1012.63 = $182,273 over the course of 15 years.

With home loan offers, you must also consider how much you pay over the lifetime of the loan. The longer it takes to pay down the loan, the more you pay in interest, and the longer it takes to build equity. Even though shorter loan periods mean higher monthly payments, you can save in the long run. Always compare offers from multiple lenders and banks. Also consider your long term financial goals and current budget restrictions.