# Mortgage question

Assume i take the mortgage out over 25 years: using the mortgage payment formula below:

$Monthly\ payment\ = L\ [{\dfrac{x(1+x)^n}{(1+x)^n -1} }]$

where L is the loan value, x is the interest % in month and n is the number of months

In my scenario if my interest payment is 3% a year, on a £330k loan over 25 years this would equal:

$£330,000[{\dfrac{0.0025(1.0025)^{(25*12)}}{(1.0025)^{(25*12)} -1} }] = £1564$ a month

Therefore in 5 years i would have paid off £93,893 of the loan including interest over those 5 years.

My question is if i sell the house at the end of the 5 year period - what amount of the mortgage would i have to pay there and then, as the interest is calculated year on year i.e. paying it off in advance would save you the interest payments over the remainder of those years.

Using typical mortgage calculators online if i figure out the balance of the mortgage left over it will give me that balance over the remaining 20 years. How do i find out the real time value of that debt if i wanted to pay it off there and then?

Thanks in advance

• You can find the balance after five years using the Mortgage tab in the PayoffDebt.xlsx spreadsheet at commonsensemathematics.net/spreadsheets - check that it's the same formula you are using. Oct 30 '17 at 23:18

## 1 Answer

you have:

$Monthly\ payment\ = L\ [{\dfrac{x(1+x)^n}{(1+x)^n -1} }]$

therefore $L\ = {\dfrac{(1+x)^n -1}{x(1+x)^n} }(Monthly\ payment)$

use the Monthly payment you have already have (and rate) and use $240$ to represent the number of payments left after $5$ years

or $L = \dfrac{(1.0025)^{300} -(1.0025)^{60}}{(1.0025)^{300} -1}330,000$

• but by doing that you find out what is left for payment inclusive of the interest rates for the next 20 years. My question is if you pay for the lump sum on year 5, you wont be borrowing the money over the next 20 years so wouldnt need to pay the YoY interest Oct 31 '17 at 17:55
• When you make your mortgage payments, you pay all of the interest due and some amount of the principal. Early in the life of the mortgage, the payments are nearly all interest and very little principle. As the time passes and the final payments come closer, a greater percentage of the payment is principle. The mortgage formula are indifferent on whether you keep making monthly payments or pay off the remainder. Oct 31 '17 at 19:22