4. The following are two popular approaches used by automobile dealers: (a) Cash

Question

4. The following are two popular approaches used by automobile dealers:

(a) Cash Rebate Versus Low Rate Dealer Financing You are given two mutually exclusive options from the dealer on a $20,000 car:

(i) $1,500 cash rebate or

(ii) 36-month low rate loan at 3% APR. The prevailing APR on 36-month auto loan from a typical bank is 8%.

Which option is a better deal?

(b) Buying Versus Leasing You are interested in a $25,000 car. A simplified leasing contract includes the following:

(i) up-front cost of $3,000, (ii) $400 monthly lease payment over a 36-month period, and

(iii) purchase cost of $12,000 at the end of the lease. What are the “implied” APR and EAR of the lease?

Should you lease the car or buy and finance the car with a loan from the bank in (a)? <Suggested answer 8.45%; 8.79%>

Explanation / Answer

a)Monthly Payments: Formula - Pmt = Lr / (1-(1+r)-t)
The amount you borrow is L, the interest rate per period is r, the number of periods is t, and P is the payment per period.

3% APR –
L = $20,000
r = 3%/12 = 0.25%
t = 36
= [($20,000 x 0.25%) / (1-(1+0.25%)-36)] = $581.62

8% APR –
L = $20,000
r = 8%/12 = 0.667%
t = 36
= [($20,000 x 0.667%) / (1-(1+0.667%)-36)] = $626.76

Savings per month = $626.76 - $581.62 = $45.12

Present Value of all savings: Formula - PV = Pmt x ((1-((1+r)-n )) / r)
Payment per period (PMT) = $45.12
Discount Rate per period= 8% (Normal Bank Rate)
Number of periods (n) = 36
PV = $45.12 x ((1-((1+0.08)-36)) / 0.08) = $528.68
As the PV of savings is lesser than one time cash rebate, cash rebate of $1,500 is a better deal.

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