Question

Janet buys a $20,000 car. Prevailing market rates are nominal 8% annual, convertible monthly. The dealership offers her the choice of a rebate upon purchase of the car for cash, or alternatively, Janet can make no down payment and 60 monthly payments based on a nominal 2.5% annual interest, convertible monthly. The first payment would be due one month after the purchase of the car. The amount of the rebate is set so that the dealership is indifferent as to whether Janet takes the rebate or finances the car at the offered below market interest rate. Determine the amount of the rebate.

Please write the full process, thank you so much!

Answer 1

Monthly payments on $20,000 loan at 2.5% is:

Monthly payment	=	[P * R * (1+R)^N ] / [(1+R)^N -1]
Using the formula:
Loan amount	P	20,000.00
Rate of interest per period:
Annual rate of interest		2.500%
Frequency of payment	=	Once in 1 month period
Numer of payments in a year	=	12/1 =	12
Rate of interest per period	R	0.025 /12 =	0.2083%
Total number of payments:
Frequency of payment	=	Once in 1 month period
Number of years of loan repayment	=	5
Total number of payments	N	5*12 =	60
Period payment using the formula	=	[ 20000*0.00208*(1+0.00208)^60] / [(1+0.00208 ^60 -1]
Monthly payment	=	354.95

Monthly payment is $354.95

Prevailing market rate is 8% but loan rate offered is only 2.5%.

Present value of $354.95 per month paid for 60 months at 8% is:

a	Present value of annuity=	P* [ [1- (1+r)-n ]/r ]
P=	Periodic payment	354.95
r=	Rate of interest per period
	Annual interest	8.00%
	Number of interest payments per year	12
	Interest rate per period	0.08/12=
	Interest rate per period	0.667%
n=	number of periods:
	Number of years	5
	Periods per year	12
	number of payments	60
	Present value of annuity=	354.95* [ (1- (1+0.00667)^-60)/0.00667 ]
	Present value of annuity=	17,505.58
b	Face value of car	$                20,000.00
c	Less: present value	$                17,505.58
	Discount	$                  2,494.42

Discount given is $2,494.42.

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