Question

21. Dr. Rock takes out a loan for 100.000. The annual effective interest rate on the loan is 10%. Dr. Rock repays the loan by making payments at the end of each year. With the exception of the final payment, the payment amount is equal to three times the interest due. Dr. Rock's final payment is the balance due at that time, and the loan is arranged so that final payment does not exceed 1000. (a) How many payments does Dr. Rock make? (b) What is Dr. Rock's final payment?

Answer 1

(a): Dr. Rock makes a total of 23 payments over a period of 23 years.

(b): The final payment amount = $811.66. Of this $73.79 is the interest amount and $737.87 is the principal amount. After this payment the loan outstanding becomes nil.

Calculations and explanations: For this purpose I have made a loan amortization table. In the 1^st year balance at the start = $100,000 and interest amount = 10% of 100,000 = $10,000. Thus payment amount = 10000*3 = $30,000. Of this principal repayment = 30,000 – interest = 30,000 – 10,000 = $20,000. Thus balance due at the end of year 1 = 100,000 – 20,000 = 80,000. We keep on doing this for subsequent years until we arrive at the 23^rd year. Here balance due at start = 737.87 and so interest = 10% of 737.87 = 73.79. So total payment = 737.87 + 73.79 = 811.66. This is less than 1,000.

The table is shown below:

Year	Balance due at the start of the year	Interest amount @ 10%	Total payment	Principal payment	Balance due at the end of the year
1	100,000.00	10,000.00	30,000.00	20,000.00	80,000.00
2	80,000.00	8,000.00	24,000.00	16,000.00	64,000.00
3	64,000.00	6,400.00	19,200.00	12,800.00	51,200.00
4	51,200.00	5,120.00	15,360.00	10,240.00	40,960.00
5	40,960.00	4,096.00	12,288.00	8,192.00	32,768.00
6	32,768.00	3,276.80	9,830.40	6,553.60	26,214.40
7	26,214.40	2,621.44	7,864.32	5,242.88	20,971.52
8	20,971.52	2,097.15	6,291.46	4,194.30	16,777.22
9	16,777.22	1,677.72	5,033.16	3,355.44	13,421.77
10	13,421.77	1,342.18	4,026.53	2,684.35	10,737.42
11	10,737.42	1,073.74	3,221.23	2,147.48	8,589.93
12	8,589.93	858.99	2,576.98	1,717.99	6,871.95
13	6,871.95	687.19	2,061.58	1,374.39	5,497.56
14	5,497.56	549.76	1,649.27	1,099.51	4,398.05
15	4,398.05	439.80	1,319.41	879.61	3,518.44
16	3,518.44	351.84	1,055.53	703.69	2,814.75
17	2,814.75	281.47	844.42	562.95	2,251.80
18	2,251.80	225.18	675.54	450.36	1,801.44
19	1,801.44	180.14	540.43	360.29	1,441.15
20	1,441.15	144.12	432.35	288.23	1,152.92
21	1,152.92	115.29	345.88	230.58	922.34
22	922.34	92.23	276.70	184.47	737.87
23	737.87	73.79	811.66	737.87	-