Question

2) A 30 year $100, 000 loan with i = 9% is to be paid off with yearly payments beginning one year after the loan is made. The first 20 payments are K and the last 10 payments are 4K. Find K. Also for the first four full years make an amortization table and explain why for each year principle repaid is negative.

Answer 1

Loan amount = 100,000

Loan duration = 30 years

Interest rate = 9%

Present value of all payment made must be 100,000

Present value of payment made in 1st year [K / (1 + 0.09)^¹]

Present value of payment made in 2nd year [K / (1 + 0.09)^²]

.....

Present value of payment made in 21st year [2K / (1 + 0.09)^²¹]

Present value of payment made in 22nd year [2K / (1 + 0.09)^²²]

......

Present value of payment made in 30th year [2K / (1 + 0.09)^³⁰]

Sum of present value of payments: [K / (1 + 0.09)^¹] + [K / (1 + 0.09)^²] + ............ + [2K / (1 + 0.09)^³⁰]

I will divide the series in two parts:

First part:

[K / (1 + 0.09)^¹] + [K / (1 + 0.09)^²] + .................... + [K / (1 + 0.09)^¹⁹] + [K / (1 + 0.09)^²⁰] whose sum can be calculated using G.P. formuls which is [a * (1 - r^ⁿ) / (1 - r)]

where a = [K / (1 + 0.09)^¹]

r (ratio of two consecutive terms) = 0.9174

n = 20

Sum of first part: [K / (1 + 0.09)^¹] * (1 - 0.9174^²⁰) / (1 - 0.9174) = 9.13K

Second part:

[2K / (1 + 0.09)^²¹] + [2K / (1 + 0.09)^²²] + .................... + [2K / (1 + 0.09)^²⁹] + [2K / (1 + 0.09)^³⁰] whose sum can be calculated using G.P. formuls which is [a * (1 - r^ⁿ) / (1 - r)]

where a = [2K / (1 + 0.09)^²¹]

r (ratio of two consecutive terms) = 0.9174

n = 10

Sum of first part: [2K / (1 + 0.09)^²¹] * (1 - 0.9174^¹⁰) / (1 - 0.9174) = 2.29K

Sum of payment made = 9.13K + 2.29K = 11.42K which must be equal to 100,000

Thus, K = 8,757.51

For the first 4 years, interest accrued is more than amount paid which result in negative principal.

Year	Pending amount	Interest till end of the year	Amount paid	Interest paid	Principal paid	Pending balance
1	100,000.00	9,000.00	8,757.51	8,038.04	719.47	100,242.49
2	100,242.49	9,021.82	8,757.51	7,973.29	784.22	100,506.80
3	100,506.80	9,045.61	8,757.51	7,902.71	854.80	100,794.91
4	100,794.91	9,071.54	8,757.51	7,825.78	931.73	101,108.94