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.
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.
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!
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?
In present value terms, you're trying to find where
$450000 =3500a_{\overline{n}|i=.005}$
Solve for $n$.