# Financial mathematics question about mortgage loans

The following question is what I was working on.

A bank gives a mortgage of $\$450,000$dollars for a$30$year loan with$6$% annual interest which requires the person to pay monthly. They require the person to pay$\$2700$ dollars each month. However, the borrower decides to pay $\$3500$per month. How long will it take for this person to pay off the mortgage? I was thinking that this is a problem with annuity and stuff, but I was not able to really understand the problem. So, it would be helpful to have the following. 1. Using the present value formula for annuity, I got a much smaller number than$\$2700$ so that the money builds up to $\$450,000$. So, I am not sure where the$\$2700$ came from.

2. Using the final value formula for annuity, paying $\$2700$dollars each month for$30$years does not match anything close to$\$450,000$. I am not quite understanding how banks are making money off this loan!

3. I did try the following.

The cost that the bank charges equals

$$(1.06)\big(450K+(450K-42K)+(450K-2*42K)+\cdots+(450K-(n-1)42K)\big)$$

which simplifies to

$$(1.06)\big(450K(n)-42K(n)(n-1)/2\big)$$

which has to equal to

$$3500n$$

Where $n$ is the number of years that it takes to pay off the cost.

I got about $22$ years and numerically it sounds plausible, but I am not certain. Can someone clarify this for me?

• Hint: $450000 \dfrac{0.005\cdot 1.005^{360}}{1.005^{360}-1} = 2697.98... \approx 2700$ Commented Oct 24, 2013 at 4:39
• This is an annuity problem, but it looks like you are using simple interest. There are very few problems where we use annuities and simple interest so I urge you to rethink the problem in terms of compound interest. If you are still confused about annuity concepts I'd encourage you to ask another question about that specifically. Commented Oct 26, 2013 at 1:14
• @Macavity: Thank you so much. I was actually able to solve it. >>Tyler: It turns out that I was not thinking about paying the borrowed money. I was only paying for the charged money. The number being plausible seems like it was a coincidence. Commented Oct 26, 2013 at 2:27

$450000 =3500a_{\overline{n}|i=.005}$
Solve for $n$.