Question

Consider a 30-year mortgage for $345,552 at an annual interest rate of 5.6%. After 13 years, the mortgage is refinanced to an annual interest rate of 3.5%. How much interest is paid on this mortgage?

Answer 1

Time(N) = 30

Principal = $ 345,552

Interest (I) = 5.6%

Annual Payment = (Principal * I * (1 + I)^N) / ((1 + I)^N - 1)

= (345552 * 5.6% * (1 + 5.6%)^30) / ((1+5.6%)^30 -1)

= 24039.04

Year	Opening loan	Interest @5.6%	Annual payment	Principal component	Ending loan
1	3,45,552.00	19,350.91	24,039.04	4,688.13	3,40,863.87
2	3,40,863.87	19,088.38	24,039.04	4,950.66	3,35,913.21
3	3,35,913.21	18,811.14	24,039.04	5,227.90	3,30,685.31
4	3,30,685.31	18,518.38	24,039.04	5,520.66	3,25,164.65
5	3,25,164.65	18,209.22	24,039.04	5,829.82	3,19,334.83
6	3,19,334.83	17,882.75	24,039.04	6,156.29	3,13,178.54
7	3,13,178.54	17,538.00	24,039.04	6,501.04	3,06,677.49
8	3,06,677.49	17,173.94	24,039.04	6,865.10	2,99,812.39
9	2,99,812.39	16,789.49	24,039.04	7,249.55	2,92,562.85
10	2,92,562.85	16,383.52	24,039.04	7,655.52	2,84,907.33
11	2,84,907.33	15,954.81	24,039.04	8,084.23	2,76,823.10
12	2,76,823.10	15,502.09	24,039.04	8,536.95	2,68,286.15
13	2,68,286.15	15,024.02	24,039.04	9,015.02	2,59,271.14
		226226.66	312507.52

The loan amount remaining at the end of 13 years is 259271.14 as per the repayment schedule calculated above

Now we need to calculate the annual payment with N =17 and I = 3.5%

Annual Payment = (Principal * I * (1 + I)^N) / ((1 + I)^N - 1)

= (259271.14 *3.5% * (1 + 3.5%)^17) / ((1+3.5%)^17 -1)

= 20493.60

Total annual payment over 17 years = 20493.60 * 17 = 348391.25

Interest paid = Total payment - Loan amount = 348391.25 - 259271.14 = 89120.11

Now we also need to sum the interest paid for the first 13 years of loan which can be seen in the amortization schedule = 226226.66

Total interest paid = 89120.11 + 226226.66 = 315346.77

Hence the correct answer is $ 315346.77