Pin It

Financial Math: Calculating the Mortgage Balance after X Number of Years

E. Klein | Sep 19, 2009 |

0 Comments | Rating: 0

When you have a long-term, fixed-rate mortgage, it is useful to know how much of the principal balance will be due at the end of 5, 10, 15 years, etc. (The principal is the amount borrowed.) For example, if you want to refinance your home at a later date, you need to know the remaining balance on your mortgage.

The balance due is related to the number of years left on the mortgage, but it is related by a complex equation that accounts for amortization. Thus, the formula used to compute the mortgage balance due isn't intuitive.

Amortization is the way in which interest is factored into your monthly payments. Even though you pay the same dollar amount each month, you do not pay the same amount of interest every month. During the first years of your mortgage, as much as 90%-95% of your house payment goes toward interest, and only 5%-10% goes to the principal. But during the final years of the lending period, only 5%-10% of your bill goes toward paying interest, and the rest goes toward paying off the principal balance.

For instance, suppose you have a 30 year mortgage. After 10 years, you might logically assume that you have paid off 1/3 of the principal.But in reality, you may only have paid off 15%-20% of the original loan amount. That's amortization at work! So, if you want to know how the balance paid and the balance due are calculated, grab a calculator, paper, pencil, and keep reading.

Variables

To compute the mortgage balance after a certain number of years, you need to gather three pieces of info about your loan: the principal (P), the dollar amount of your monthly loan payments (L), and the monthly interest rate (R). The monthly rate is the annual rate divided by 12. For example, if you have an annual rate of 7.5%, your monthly rate is 0.075/12 = .00625

Computing the Mortgage Balance Paid

The amount you have paid after X months is given by the equation

Amt Paid = (L/R - P)[(1+R)^X - 1]

Remember to keep X in terms of months, not years for this formula.

Computing the Mortgage Balance Due

To find the amount of principal still due on your home loan, simply subtract the balance paid from the principal.

Amt Due = P - (L/R - P)[(1+R)^X - 1]
= L/R - (L/R - P)(1+R)^X

First Example

Suppose you have 30 year mortgage on $125,000, and the annual fixed rate is 8.25%. According to these terms, your monthly house payments are $939. Thus, P = 125000, L = 940, and R = 0.006875. How much will you still owe on the mortgage after 7 years = 84 months? Well, using 84 as the value for X and the formula above, we get

Amt. Due = 939/0.006875 - (939/0.006875 - 125000)(1.006875)⁸⁴
= 136581 - (11727)(1.77806)
= 115730.

So, after 7 years you still owe $115,730 on the $125,000 loan. Even though you have already made 84 x $939 = $78,876 worth of monthly payments so far, most of that has been applied to the interest.

Alternative Formula for Computing Mortgage Balance Due

Here is an alternative formula that gives you the same answer. You need to fill in values for the principal (P), the monthly rate (R), and the total loan term (N) expressed in months. Then

Amt. Due = P[(1+R)^N – (1+R)^X]/[(1+R)^N – 1]

Second Example

Suppose you have a 15 year mortgage on $80,000 and a fixed annual rate of 6%. According to these terms, we have P = 80000, N = 180, and R = 0.005. If you want to find how much you will owe after 10 years, use X = 120. Then,

Amt Due = 80000[(1.005)¹⁸⁰ – (1.005)¹²⁰]/[(1.005)¹⁸⁰ – 1]
= 80000[0.63469]/[1.45409]
= 34918

So after 10 years you owe $34,918.