Question

You have taken out a standard $250,000, 30-year mortgage, at a nominal annual rate of 4.9%, with monthly compounding. Determine how much you will still owe on your mortgage, as a percent of the original $250,000, after 15 years (180 payments) if you only make the required monthly payments.

A) 65.56% B) 66.23% C) 66.90% D) 67.56% E) 68.21%

Pleasse do not use excel to explain this to me thank you.

Answer 1

Answer is 67.56%

Calculation of monthly payments:

Amount borrowed = $250,000
Annual interest rate = 4.90%
Monthly interest rate = 0.4083%
Period = 30 years or 360 months

Let monthly payment be $x

$250,000 = $x/1.004083 + $x/1.004083^2 + ... + $x/1.004083^359 + $x/1.004083^360
$250,000 = $x * (1 - (1/1.004083)^360) / 0.004083
$250,000 = $x * 188.42952
$x = $1,326.76

Calculation of loan outstanding:

Period outstanding = 15 years or 180 months

Loan outstanding = $1,326.76/1.004083 + $1,326.76/1.004083^2 + ... + $1,326.76/1.004083^179 + $1,326.76/1.004083^180
Loan outstanding = $1,326.76 * (1 - (1/1.004083)^180) / 0.004083
Loan outstanding = $1,326.76 * 127.29554
Loan outstanding = $168,890.63

Percentage of original = Loan outstanding / Amount borrowed
Percentage of original = $168,890.63 / $250,000
Percentage of original = 0.6756 or 67.56%