Question

Rachel purchased a car for $21,500 three years ago using a 4-year loan with an interest rate of 9.0 percent. She has decided that she would sell the car now, if she could get a price that would pay off the balance of her loan.

What is the minimum price Rachel would need to receive for her car? Calculate her monthly payments, then use those payments and the remaining time left to compute the present value (called balance) of the remaining loan.

Answer 1

Loan amount = $21,500. r = 9% and n = 4 years or 4*12 = 48 months

Monthly payment =  [P x R x (1+R)^N]/[(1+R)^N-1]

= [21500*(0.09/12)*(1+9%/12)^48]/(1+9/12)^48 -1)

= $535.03

Tha monthly payment can also be computed using the PMT function in excel and the same result will be obtained. The loan table is shown below:

Month	Principal at the start of the period	Monthly payment	Interest @ 9%/12	Principal at the end of the month
1	21,500.00	535.03	161.25	21,126.22
2	21,126.22	535.03	158.45	20,749.64
3	20,749.64	535.03	155.62	20,370.23
4	20,370.23	535.03	152.78	19,987.98
5	19,987.98	535.03	149.91	19,602.86
6	19,602.86	535.03	147.02	19,214.86
7	19,214.86	535.03	144.11	18,823.94
8	18,823.94	535.03	141.18	18,430.09
9	18,430.09	535.03	138.23	18,033.29
10	18,033.29	535.03	135.25	17,633.51
11	17,633.51	535.03	132.25	17,230.73
12	17,230.73	535.03	129.23	16,824.93
13	16,824.93	535.03	126.19	16,416.09
14	16,416.09	535.03	123.12	16,004.19
15	16,004.19	535.03	120.03	15,589.19
16	15,589.19	535.03	116.92	15,171.08
17	15,171.08	535.03	113.78	14,749.83
18	14,749.83	535.03	110.62	14,325.43
19	14,325.43	535.03	107.44	13,897.84
20	13,897.84	535.03	104.23	13,467.05
21	13,467.05	535.03	101.00	13,033.02
22	13,033.02	535.03	97.75	12,595.74
23	12,595.74	535.03	94.47	12,155.18
24	12,155.18	535.03	91.16	11,711.32
25	11,711.32	535.03	87.83	11,264.12
26	11,264.12	535.03	84.48	10,813.57
27	10,813.57	535.03	81.10	10,359.65
28	10,359.65	535.03	77.70	9,902.32
29	9,902.32	535.03	74.27	9,441.56
30	9,441.56	535.03	70.81	8,977.34
31	8,977.34	535.03	67.33	8,509.64
32	8,509.64	535.03	63.82	8,038.43
33	8,038.43	535.03	60.29	7,563.69
34	7,563.69	535.03	56.73	7,085.39
35	7,085.39	535.03	53.14	6,603.51
36	6,603.51	535.03	49.53	6,118.00
37	6,118.00	535.03	45.89	5,628.86
38	5,628.86	535.03	42.22	5,136.05
39	5,136.05	535.03	38.52	4,639.54
40	4,639.54	535.03	34.80	4,139.31
41	4,139.31	535.03	31.04	3,635.32
42	3,635.32	535.03	27.26	3,127.56
43	3,127.56	535.03	23.46	2,615.99
44	2,615.99	535.03	19.62	2,100.58
45	2,100.58	535.03	15.75	1,581.31
46	1,581.31	535.03	11.86	1,058.14
47	1,058.14	535.03	7.94	531.05
48	531.05	535.03	3.98	0.00

Thus present value of monthly payments of $535.03 for the last 12 months can be computed using the present value of annuity formula.

Present value of annuity = P*[1-(1+r)^(-n)]/r

= 535.03*(1-(1+0.0075)^(-12)/0.0075

= 535.03*(1-0.9142)/0.0075

= $6,118

Thus minimum price Rachel would need to receive for her car = $6,118