Question

6. Ann obtains a fully amortizing 30 year Fixed Rate Mortgage with monthly payments for $1,250,000 at 4.38%. What fraction of Ann's 40th payment goes to interest? Write your answer in percent, without the % sign.

Answer 1

Amount borrowed = $1,250,000

Annual interest rate = 4.38%
Monthly interest rate = 0.365%

Period = 30 years or 360 months

Calculation of monthly payment:

Let monthly payment be $x

$1,250,000 = $x/1.00365 + $x/1.00365^2 + …. + $x/1.00365^360
$1,250,000 = $x * (1 - (1/1.00365)^360) / 0.00365
$1,250,000 = $x * 200.16806
$x = $6,244.75

Monthly payment = $6,244.75

Calculation of loan outstanding after 39^th payment:

Loan outstanding = $6,244.75/1.00365 + $6,244.75/1.00365^2 + … + $6,244.75/1.00365^321
Loan outstanding = $6,244.75 * (1 - (1/1.00365)^321) / 0.00365
Loan outstanding = $6,244.75 * 188.89948
Loan outstanding = $1,179,630.03

Calculation of loan outstanding after 40^th payment:

Loan outstanding = $6,244.75/1.00365 + $6,244.75/1.00365^2 + … + $6,244.75/1.00365^320
Loan outstanding = $6,244.75 * (1 - (1/1.00365)^320) / 0.00365
Loan outstanding = $6,244.75 * 188.58896
Loan outstanding = $1,177,690.91

Loan repaid in 40^th payment = Loan outstanding after 39^th payment - Loan outstanding after 40^th payment
Loan repaid in 40^th payment = $1,179,630.03 - $1,177,690.91
Loan repaid in 40^th payment = $1,939.12

Interest paid = Monthly payment - Loan repaid
Interest paid = $6,244.75 - $1,939.12
Interest paid = $4,305.63

Interest paid in percent = Interest paid / Monthly payment
Interest paid in percent = $4,305.63 / $6,244.75
Interest paid in percent = 0.6895 or 68.95%