So a couple purchases a home with a $180,000 mortgage at a 4% rate for 30 years with monthly payments. Uh…what will the remaining balance in our mortgage be after 5 years? Now to answer this, we’re gonna have to do it in two parts. First thing we’re gonna have to do is figure out what are their monthly payments? Uh…so for that, we know that our interest rate is 4%. We know we’re doing monthly payments, uh…based on a 30-year repayment schedule, uh…with an initial amount of $180,000, and we’re looking for d–our payment amount. So we pull out our loan formula, and we set it up. So 1 plus .04, divided by 12 to the negative 12, times 30… all over .04 over 12. Uh…and since we’ve done a lot of these, I’m gonna go ahead and skip, uh…a bunch of steps here.
So I pull out my calculator, and I calculate all of that. It comes out to be about 209, uh…562, uh…and then divide by that gives me a payment of about $858.93. So the monthly payments on this loan will be $858.93. So now that takes us to the second part of the question. Now we know the monthly payments, we can determine the remaining balance. So we’re trying to find the balance after 5 years. So after 5 years, there are 25 years left to pay on the loan. So what we’re really asking is how much loan balance can be paid off in 25 years with monthly payments of this size? In other words, keeping the interest rate the same, if we have a loan for 25 years with payments of $858.93, what is the loan balance at the beginning of those 25 years? That is the current balance of the loan after 5 years. So our P0 here,
where 0 now…
Times 0’s now 5 years in, will be… uh…let’s see, $858.93 is my d, times 1, minus 1, plus .04 over 12, to the negative 12, times 25– because remember we’re going for 25 years now– over .04 over 12, and again, skipping several steps here, uh…we come up with 155,793, uh…91. So in other words, after 5 years, uh…the remaining balance on the loan will be 155,793. Now it’s worth thinking about this number a little bit. So think about how much money have we paid, uh…so far? So we’ve paid, uh…$858.93 for 5 years, 12 times a year, gives us a total of, uh… let’s see…51,535.80 that we’ve paid, uh…to the loan company, right? That we’ve made in payments.
Uh…how much of the loan have we actually paid off? Well, we started with a $183,000 loan. After 5 years, our loan balance is now 155,793, which means we’ve paid off, uh…$24,206.09, uh…paid off of the loan. Well, if we’ve paid $51,535.80 and only $24,206.09 of it has gone to the loan, then the other 27,329.71 must have gone to interest so far. And of course, the next 25 years, we’ll be adding more interest, but over those first 5 years, uh…we’ve paid $27,000 in interest..