John has taken out a $10,000 loan, which is being repaid with 30 level annual payments....

John has taken out a $10,000 loan, which is being repaid with 30 level annual payments. The first payment will be one year from now. John pays an additional $500 during the fourth and twelfth payments. Following the two additional payments, the loan is repaid by annual payments of the original size with a larger final payment after the last full repayment.

If the effective annual interest rate is 5%, what is the amount of the larger payment?

A. 480

B. 640

C. 800

D. 970

E. 1130

Solutions

Expert Solution

Let the annual Normal payment be ''A''

Hence

Present value of the Loan =

=>A or Annual Payment = $650.514

Repayment schedule-

	A	B=A*5%	C	A+B-C
Year	Opening loan Balance	Interest@5%	Payment	Closing loan
1	10000.000	500.000	650.514	9849.486
2	9849.486	492.474	650.514	9691.446
3	9691.446	484.572	650.514	9525.505
4	9525.505	476.275	1150.514	8851.266
5	8851.266	442.563	650.514	8643.315
6	8643.315	432.166	650.514	8424.967
7	8424.967	421.248	650.514	8195.701
8	8195.701	409.785	650.514	7954.972
9	7954.972	397.749	650.514	7702.207
10	7702.207	385.110	650.514	7436.803
11	7436.803	371.840	650.514	7158.129
12	7158.129	357.906	1150.514	6365.522
13	6365.522	318.276	650.514	6033.284
14	6033.284	301.664	650.514	5684.434
15	5684.434	284.222	650.514	5318.142
16	5318.142	265.907	650.514	4933.535
17	4933.535	246.677	650.514	4529.698
18	4529.698	226.485	650.514	4105.669
19	4105.669	205.283	650.514	3660.438
20	3660.438	183.022	650.514	3192.946
21	3192.946	159.647	650.514	2702.079
22	2702.079	135.104	650.514	2186.669
23	2186.669	109.333	650.514	1645.489
24	1645.489	82.274	650.514	1077.249
25	1077.249	53.862	650.514	480.598
26	480.598	24.030	650.514	-145.887
27	-145.887	-7.294	650.514	-803.695
28	-803.695	-40.185	650.514	-1494.394
29	-1494.394	-74.720	650.514	-2219.627
30	-2219.627	-110.981	650.514	-2981.123

As we can see the loan balance is negative from year 26, due to additional payment in year 4 and 12.

hence a lumpsum or large payment to be made at year 25th End to close the loan account.

Lumpsum amount to be paid a year 25th end = 650.514+480.598 = 1130 approx

Correct Answer-E. 1130

Payment schedule if 1130 is paid at the end 25th year-

	A	B=A*5%	C	A+B-C
Year	Opening loan Balance	Interest@5%	Payment	Closing loan
1	10000.000	500.000	650.514	9849.486
2	9849.486	492.474	650.514	9691.446
3	9691.446	484.572	650.514	9525.505
4	9525.505	476.275	1150.514	8851.266
5	8851.266	442.563	650.514	8643.315
6	8643.315	432.166	650.514	8424.967
7	8424.967	421.248	650.514	8195.701
8	8195.701	409.785	650.514	7954.972
9	7954.972	397.749	650.514	7702.207
10	7702.207	385.110	650.514	7436.803
11	7436.803	371.840	650.514	7158.129
12	7158.129	357.906	1150.514	6365.522
13	6365.522	318.276	650.514	6033.284
14	6033.284	301.664	650.514	5684.434
15	5684.434	284.222	650.514	5318.142
16	5318.142	265.907	650.514	4933.535
17	4933.535	246.677	650.514	4529.698
18	4529.698	226.485	650.514	4105.669
19	4105.669	205.283	650.514	3660.438
20	3660.438	183.022	650.514	3192.946
21	3192.946	159.647	650.514	2702.079
22	2702.079	135.104	650.514	2186.669
23	2186.669	109.333	650.514	1645.489
24	1645.489	82.274	650.514	1077.249
25	1077.249	53.862	1130.000	0

jeff jeffy answered 2 years ago

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