Question

After deciding to buy a new car, you can either lease the car or purchase it on a four-year loan. The car you wish to buy costs $40,000. The dealer has a special leasing arrangement where you pay $109 today and $509 per month for the next four years. If you purchase the car, you will pay it off in monthly payments over the next four years at an APR of 7 percent. You believe you will be able to sell the car for $28,000 in four years. a. What is the present value of leasing the car? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.) b. What is the present value of purchasing the car? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.) c. What break-even resale price in four years would make you indifferent between buying and leasing? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)

Answer 1

Answer:

a. In order to determine the present value of lease we can use the same APR as the car loan (7%). We can use the present value of an annuity formula:

PV = monthly payment x annuity factor

• monthly payment = $509
• PV annuity factor, 0.58333%, 48 periods = 41.76344

PV of the annuity = $509 x 41.76344 = $21,257.59

total present value of lease contract = $21,257.59 + $109 = $21,366.59

b. the present value of purchasing the car is $40,000 - $28,000/1.07⁴ = $40,000 - $21,361.07 = $18,638.93

c. the break even resale price = (sales price - PV of lease) x (1 + 7%/12)⁴⁸ = ($40,000 - $21,366.59) x (1 + 0.07/12)⁴⁸ = $18,633.41 x 1.32205 = $24,634.37