Question

Please show work!

You have decided to acquire a new car that costs $30,000. You are considering whether to lease it for three years or to purchase it and financing the purchase with a three-year installment loan. The lease requires no down payment and lasts for three years. Lease payments are $400 monthly starting immediately, whereas the installment loan will require monthly payments starting a month from now at an annual percentage rate (APR) of 8%. The discount rate (APR) is also 8%.

If you expect the resale value of the car to be $20,000 three years from now, should you buy or lease it?

What is the break-even resale price of the care three years from now, such that you would be indifferent between buying and leasing it?

Answer 1

First, we need to find out the monthly instalment to be paid, if the loan is taken.

GIven are the following details:

P or principal amount = $30,000

R or monthly interest rate is 0.08/12 or 0.00667

N or number of instalments = 3 * 12 = 36

The formula for EMI is (P * R) * (1+R)^N)/(((1+R)^N)-1)

= (30,000*0.00667) *(1+0.00667)^36 / ((1+0.00667)^36)-1

= 254.047 / 0.270237 = $940.09

If the car is bought, the monthly payment towards the service of the loan will be $940.09. However, $20,000 can be salvaged at the end of three years. If it is leased, however, the monthly payment is only $400, however, no money will be received at the end of 3 years.

The next step is to bring back these future cash flows in present terms, using 8% discount rate to make the call.

If you buy the car on loan the present value will of the payments will be:

-940.09/(1 + 0.08/12)^1 - 940.09/(1 + 0.08/12)^2 - 940.09/(1 + 0.08/12)^3 and so on until - 940.09/(1 + 0.08/12)^36 + 20,000/(1 + 0.08/12)^36

The present value of the cash flow if you buy the car is -$14,254.907

Similarly, you can calculate the present value of the lease payments for 36 months

-400/(1 + 0.08/12)^1 - 400(1 + 0.08/12)^2 - 400/(1 + 0.08/12)^3 and so on until - 400/(1 + 0.08/12)^36

The present value of the cash flow if you lease the car is -$12764.72

Since the present value of outflow for the lease payment is lower than buying the car on loan, leasing the car is the better option.

Now, we need to find out the resale value of the car at the end of three years for us to be indifferent about the two options.

The present value of the cash flow of the monthly instalments of the loan is

-940.09/(1 + 0.08/12)^1 - 940.09/(1 + 0.08/12)^2 - 940.09/(1 + 0.08/12)^3 and so on until - 940.09/(1 + 0.08/12)^36 = -$30,000

The present value of the salvage value of the car should be the different between the present value of the monthly instalments and the present value of lease payments = 30,000 - 12764.72 = $17,235.278

The salvage value of the car after 3 years or 36 months should be $17,235.278 * (1 + (8/100)/12)^36 = $21,892.89