Question

You take out a 30-year mortgage for $70,000 at 9.25%, to be paid off monthly. If you sell your house after 15 years, how much will you still owe on the mortgage? (Round your answer to the nearest cent.)

Answer 1

Assuming `to be paid off monthly` also means compounded monthly.

We will first work out the monthly payment, M. Then we'll pay A monthly for 15 years and see what is left to be paid.

However, to understand how the formulas are derived, we will go from basic principles.

We will make two amounts: the future value of the loan, and that of the payments.
A1=FV of the loan
where A1=P(1+i/12)^(12n), n=number of years, P=amount borrowed, i=annual interest
and
A2=FV of the payments.
A2=M((1+i/12)^(12n)-1)/(i/12)
note:
\(M+MR+MR^2+...+MR^{n-1}=M(R^n-1)/(R-1)\)

The difference in the two amounts, A1-A2 is what's left to be paid.

Part A:
Find monthly payment, M
Amount 1: accumulated of 70,000 over 30 years at 9.25% compounded monthly.
A1=70000(1+0.0925/12)^(30*12)=1110820.91
A2=M((1+0.0925/12)^(30*12)-1)/(0.0925/12)=1928.9345M
Since we expect the mortgage to be paid at the end of 30 years, A1=A2, or
M=1110829.91/1928.9345=575.8728
But the actual monthly payment will be to the nearest cent, or 575.87.

Part B:
We proceed the same way, but calculate for 15 years only, and substituting M=575.87, the actual monthly payment.

A1=70000(1+0.0925/12)^(15*12)=278850.25
A2=575.87((1+0.0925/12)^(15*12)-1)/(0.0925/12)=222895.31
We still owe the bank A1-A2=55954.94

You will notice that after 15 years, the capital is not even reduced by 1/4. This is perfectly normal, because the major part of the early payments are used to pay off interest on the initial loan.

Answer 2

wow!
thank you so much!!

Answer 3

welcome ^_^