Question

The original sale price of a car is $22,194.96. The required down payment is $1,410. What is the monthly payment if the customer can apply for a 3-year auto loan with a promotional financing rate at 2.99%? Based on answer $604.36 monthly payment, if a customer decides to skip the 2.99% financing promotion loan but take the cash rebate offer, how much cash does the customer need to pay to buy the new car? We assume that regular market interest rate for auto loan is 6%. Answer is $21,275.92 but I don't know how they got it. Please show work including formula(s) used!

Answer 1

In the given case, you want to know how they got the answer of $ 21275.92, if customer take rebate offer.

The calculation is as follow:

First see the calculation to know, how the monthly payment of $ 604.36 is calculated.

The amount due after down payment = 22194.96-1410= $ 20784.96

20784.96 = Monthly Payment *PVAF(r%, n period)

PVAF(r%, n period) is an annuity factor where r% is rate of interest which is to be converted to monthly rate and n is 36 because there are monthly payments

PVAF (0.2492%, 36) = (1+r%)ⁿ - 1/ (r%* (1+r%)ⁿ)

= (1+0.002492)³⁶ - 1/ (0.002492*(1+0.002492)³⁶)

= 1.093737-1/ (0.002492*1.093737)

=0.093737/ 0.002726

= 34.39

THEREFORE:

20784.96/34.39= Monthly Payment

Monthly Payment = $ 604.39

Now, As you know, if the customer takes loan at promotional financing rate of 2.99%, he will have to pay $ 1410 as down payment and $ 604.36 monthly for 3 years.

So,we will calculate the Present value of these payment, to determine how much cash payment to be made after rebate:

Present value of Payment = PV of Down Payment + 604.36 * PVAF(r%, n period)

PVAF(r%, n period) is an annuity factor where r% is rate of interest which is to be converted to monthly rate and n is 36 because there are monthly payments

PVAF (0.5%, 36) = (1+r%)ⁿ - 1/ (r%* (1+r%)ⁿ)

= (1+0.005)³⁶ - 1/ (0.005*(1+0.005)³⁶)

= 1.196681-1/ (0.005*1.96681)

=0.196681/ 0.005983

= 32.871

As down payment is made instant, it PV will be equal to down payment.

Therefore:

PV of Cash Payment= 1410 + 604.36*32.871

= 1410+ 19865.92

= $ 21275.92

You can also refer table of it as given below:

Period	Payment (a)	PV factor @ 0.5% (b)	Present Value (a*b)
0	1410.00	1	1410.00
1	604.36	0.995	601.35
2	604.36	0.990	598.36
3	604.36	0.985	595.38
4	604.36	0.980	592.42
5	604.36	0.975	589.48
6	604.36	0.971	586.54
7	604.36	0.966	583.62
8	604.36	0.961	580.72
9	604.36	0.956	577.83
10	604.36	0.951	574.96
11	604.36	0.947	572.10
12	604.36	0.942	569.25
13	604.36	0.937	566.42
14	604.36	0.933	563.60
15	604.36	0.928	560.80
16	604.36	0.923	558.01
17	604.36	0.919	555.23
18	604.36	0.914	552.47
19	604.36	0.910	549.72
20	604.36	0.905	546.98
21	604.36	0.901	544.26
22	604.36	0.896	541.55
23	604.36	0.892	538.86
24	604.36	0.887	536.18
25	604.36	0.883	533.51
26	604.36	0.878	530.86
27	604.36	0.874	528.22
28	604.36	0.870	525.59
29	604.36	0.865	522.97
30	604.36	0.861	520.37
31	604.36	0.857	517.78
32	604.36	0.852	515.21
33	604.36	0.848	512.64
34	604.36	0.844	510.09
35	604.36	0.840	507.56
36	604.36	0.836	505.03
		Total	21275.93