Question

You have just purchased a home and taken out a $440,000 mortgage. The mortgage has a 30​-year term with monthly payments and an APR of 5.28%.

a. How much will you pay in​ interest, and how much will you pay in​ principal, during the first​ year?

b. How much will you pay in​ interest, and how much will you pay in​ principal, during the 20th year​ (i.e., between 19 and 20 years from​ now)?

Answer 1

Amount borrowed = $440,000

Annual interest rate = 5.28%
Monthly interest rate = 0.44%

Loan period = 30 years or 360 months

Let monthly payment be $x

$440,000 = $x/1.0044 + $x/1.0044^2 + .... + $x/1.0044^360
$440,000 = $x * (1 - (1/1.0044)^360) / 0.0044
$440,000 = $x * 180.48479
$x = $2,437.88

Monthly payment = $2,437.88

Answer a.

After first year:

Remaining loan period = 29 years or 348 months

Loan outstanding = $2,437.88/1.0044 + $2,437.88/1.0044^2 + .... + $2,437.88/1.0044^348
Loan outstanding = $2,437.88 * (1 - (1/1.0044)^348) / 0.0044
Loan outstanding = $2,437.88 * 177.95372
Loan outstanding = $433,829.81

Total Amount paid = 12 * $2,437.88
Total Amount paid = $29,254.56

Principal repaid = Amount borrowed - Loan outstanding after first year
Principal repaid = $440,000 - $433,829.81
Principal repaid = $6,170.19

Interest paid = Total Amount paid - Principal repaid
Interest paid = $29,254.56 - $6,170.19
Interest paid = $23,084.37

Answer b.

After 19 years:

Remaining loan period = 11 years or 132 months

Loan outstanding = $2,437.88/1.0044 + $2,437.88/1.0044^2 + .... + $2,437.88/1.0044^132
Loan outstanding = $2,437.88 * (1 - (1/1.0044)^132) / 0.0044
Loan outstanding = $2,437.88 * 99.96276
Loan outstanding = $243,697.21

After 20 years:

Remaining loan period = 10 years or 120 months

Loan outstanding = $2,437.88/1.0044 + $2,437.88/1.0044^2 + .... + $2,437.88/1.0044^120
Loan outstanding = $2,437.88 * (1 - (1/1.0044)^120) / 0.0044
Loan outstanding = $2,437.88 * 93.07572
Loan outstanding = $226,907.44

Total Amount paid = 12 * $2,437.88
Total Amount paid = $29,254.56

Principal repaid = Loan outstanding after 19th year - Loan outstanding after 20th year
Principal repaid = $243,697.21 - $226,907.44
Principal repaid = $16,789.77

Interest paid = Total Amount paid - Principal repaid
Interest paid = $29,254.56 - $16,789.77
Interest paid = $12,464.79