Question

You buy a car, which cost $100.000. The purchase can be financed with a payment of 20% and the remaining 80% is covered by an 8-year annuity loan. The loan bears interest rate of 3% p.a., and it has monthly terms in the following you therefore also apply a discount rate of 3% p.a.

a) Determine the size of the payment, U, and the monthly payment, Y, belonging to the loan.

You consider if it is realistic to sell the car after 3 years.

b) Calculate the present value of the paying monthly benefit, Y, for 4 years from now. What is the value of this including the payout U?

You could even rent the car for 4 years which requires a one-time payment of $8000 today and after a monthly payment of $1000

c) What is the present value of the renting deal?

If you buy the car today, you expect to sell it for $70.000 after 3 years to pay back the annuity loan

d) How much do you have left after 3 years, when you payed back the loan? What is the present value of this? What will the present value today of buying the car under these conditions?

It turns out the leasing agreement includes a service agreement costs of $100 each month

e) What will the present value of today buying the car now that you include the cost of the service agreement?

Of course, it is highly uncertain what price you can sell the car for in three years

f) Set up an equation that shows what the car with a documented service agreement must be able to sell to after three years, so that the present value of resp. purchase and lease are the same. Determine (numerically) this selling price.

- Thank you so much in advance!

Answer 1

a) Determine the size of the payment, U, and the monthly payment, Y, belonging to the loan.

Calculation of bank loan and initial investment

Purchase price of car = 100000

Bank Loan = 100000 * 80%

                    = 80,000

Initial Investments = 100000 * 20%

                                    = 20000

Calculation of EMI

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

= 80000 * 0.0025* ( 1+0.0025)^8*12/( 1+0.0025)^8*12 – 1

= 80000 * 0.0025* ( 1+0.0025)⁹⁶/( 1+0.0025)⁹⁶ – 1

= 80000 * 0.0025* 1.2709/1.2709 – 1

= 80000 * 0.0025* 1.2709/0.2709

= 254.1737/0.2709

= 938.28

Size of payment U = 80000

Monthly payment Y = 938.28

b) Calculate the present value of the paying monthly benefit, Y, for 4 years from now. What is the value of this including the payout U?

Calculation of outstanding balance at the end of 4year is

Months	Principal Amount	Monthly ROI	Interest	EMI	Principal Paid	Unpaid Principal
1	80000.00	0.0025	200.000	938.37	738.37	79261.63
2	79261.63	0.0025	198.154	938.37	740.21	78323.27
3	78323.27	0.0025	195.808	938.37	742.56	77384.90
4	77384.90	0.0025	193.462	938.37	744.90	76446.54
5	76446.54	0.0025	191.116	938.37	747.25	75508.17
6	75508.17	0.0025	188.770	938.37	749.60	74569.81
7	74569.81	0.0025	186.425	938.37	751.94	73631.44
8	73631.44	0.0025	184.079	938.37	754.29	72693.07
9	72693.07	0.0025	181.733	938.37	756.63	71754.71
10	71754.71	0.0025	179.387	938.37	758.98	70816.34
11	70816.34	0.0025	177.041	938.37	761.32	69877.98
12	69877.98	0.0025	174.695	938.37	763.67	68939.61
13	68939.61	0.0025	172.349	938.37	766.02	68001.25
14	68001.25	0.0025	170.003	938.37	768.36	67062.88
15	67062.88	0.0025	167.657	938.37	770.71	66124.51
16	66124.51	0.0025	165.311	938.37	773.05	65186.15
17	65186.15	0.0025	162.965	938.37	775.40	64247.78
18	64247.78	0.0025	160.619	938.37	777.75	63309.42
19	63309.42	0.0025	158.274	938.37	780.09	62371.05
20	62371.05	0.0025	155.928	938.37	782.44	61432.68
21	61432.68	0.0025	153.582	938.37	784.78	60494.32
22	60494.32	0.0025	151.236	938.37	787.13	59555.95
23	59555.95	0.0025	148.890	938.37	789.48	58617.59
24	58617.59	0.0025	146.544	938.37	791.82	57679.22
25	57679.22	0.0025	144.198	938.37	794.17	56740.86
26	56740.86	0.0025	141.852	938.37	796.51	55802.49
27	55802.49	0.0025	139.506	938.37	798.86	54864.12
28	54864.12	0.0025	137.160	938.37	801.21	53925.76
29	53925.76	0.0025	134.814	938.37	803.55	52987.39
30	52987.39	0.0025	132.468	938.37	805.90	52049.03
31	52049.03	0.0025	130.123	938.37	808.24	51110.66
32	51110.66	0.0025	127.777	938.37	810.59	50172.30
33	50172.30	0.0025	125.431	938.37	812.94	49233.93
34	49233.93	0.0025	123.085	938.37	815.28	48295.56
35	48295.56	0.0025	120.739	938.37	817.63	47357.20
36	47357.20	0.0025	118.393	938.37	819.97	46418.83
37	46418.83	0.0025	116.047	938.37	822.32	45480.47
38	45480.47	0.0025	113.701	938.37	824.66	44542.10
39	44542.10	0.0025	111.355	938.37	827.01	43603.74
40	43603.74	0.0025	109.009	938.37	829.36	42665.37
41	42665.37	0.0025	106.663	938.37	831.70	41727.00
42	41727.00	0.0025	104.318	938.37	834.05	40788.64
43	40788.64	0.0025	101.972	938.37	836.39	39850.27
44	39850.27	0.0025	99.626	938.37	838.74	38911.91
45	38911.91	0.0025	97.280	938.37	841.09	37973.54
46	37973.54	0.0025	94.934	938.37	843.43	37035.18
47	37035.18	0.0025	92.588	938.37	845.78	36096.81
48	36096.81	0.0025	90.242	938.37	848.12	35158.44

The present value of the paying monthly benefit, Y, for 4 years from now is 42390.27

The present value of the paying monthly benefit, Y, for 4 years from now is

including the payout U is 73577.68

c) What is the present value of the renting deal?

In absence of any information, we have assumed that onetime payment of 8000 will be refunded back at the end of lease period.

PV of lease rental 38082.27

d) How much do you have left after 3 years, when you payed back the loan? What is the present value of this? What will the present value today of buying the car under these conditions?

Calculation of terminal cashflow

= Sale proceeds from car - oustanding bank loan at the end of 3 year

= 70000 - 46418.83 ( as calculated earlier)

= - 23581.17

PV of buying car is 30710.15

e) What will the present value of today buying the car now that you include the cost of the service agreement?

Entire Cash outflow will increased by 100

Entire Cash outflow will increased by 100.

The present value of today buying the car now that you include the cost of the service agreement is 34057.39

f) Set up an equation that shows what the car with a documented ser