Question

A car dealer offers a purchase option and a lease option on all new cars. Suppose you are interested in a car that can be bought outright for 25,000 dollar or leased for a start-up fee of 1200 dollar plus monthly payments of 350 dollar. a. Find the linear function \(y=f(m)\) that gives the total amount you have paid on the lease option after \(m\) months. b. With the lease option, after a 48-month (4-year) term, the car has a residual value of 10,000 dollar, which is the amount that you could pay to purchase the car. Assuming no other costs, should you lease or buy?

Short answer

Answer: Purchasing the car outright is more cost-effective, as it will save 3,000 dollars over leasing and then purchasing.

Step-by-step solution

Create the linear function for total amount under the lease option

To find the total amount paid after \(m\) months, we'll take the start-up fee and add it to the product of the monthly payment and the number of months. The linear function will have the form \(y=f(m) = a + b * m\). The start-up fee is 1200 dollar, and the monthly payment is 350 dollar. So the function will be \(y = 1200 + 350m\).

Calculate the total amount for lease option after 48 months

To find the total amount paid after 48 months, replace \(m\) in the function with 48: \(y = 1200 + 350(48)\). Calculate the value of this expression: \(y = 1200 + 16800 = 18000\).

Include the residual value of the car in the lease option

After 48 months, the residual value of the car is 10,000 dollar. To determine the total amount needed to buy the car outright at the end of the lease, add this amount to the total paid during the lease term: \(18000 + 10000 = 28000\).

Compare purchasing and leasing options

Now comparing both options: - Buying outright: 25,000 dollar - Lease and purchase: 28,000 dollar since the total cost for the lease and subsequent purchase is 28000 Based on these amounts, it is more cost-effective to buy the car outright for 25,000 dollar, as it will save you 3,000 dollar over leasing and then purchasing.

Key concepts

Lease Payments

Lease payments involve regular installments made to a leasing company for the use of a vehicle over a specified period. In our example, the car dealer offers a lease with a start-up fee and subsequent monthly payments. To calculate your total lease cost, you need to determine the total of all these payments over the leasing period.

Here's how this works:

• The initial start-up fee is \(1200.
• The monthly payment is \)350.

If the lease is structured for a 48-month term, the total payment over this period can be calculated using a simple linear function. This is usually expressed as:\[y = a + b \cdot m\]where \(a = 1200\) is the start-up fee, \(b = 350\) is the monthly payment, and \(m\) represents the number of months. For this scenario, you calculate the total cost for 48 months:\[y = 1200 + 350 \cdot 48 = 18000\] This calculation tells you how much you have spent on lease payments by the end of the lease term.

Cost Comparison

When considering a lease or purchase, comparing total costs is essential. You need to account for all expenses associated with each option.

In our scenario, the comparison is between buying a car outright for $25,000 or choosing the lease option, then purchasing the car at its residual value.

For the lease, you first need to sum up all your lease payments. After calculating that you spend $18,000 during the lease, you must consider the car's residual value if you opt to purchase it once the lease is over. The residual value in this case is $10,000. Here’s how to tally up the costs:

• Total lease payments for 48 months: $18,000
• Residual value (to buy the car): $10,000

This results in a total cost of $28,000 for the lease plus purchase scenario. Comparatively, buying the car directly costs $25,000. Thus, purchasing the car outright saves $3,000.

Residual Value

Residual value is the expected value of a car at the end of a lease term, which you can typically choose to pay if you decide to keep the car. This value plays a crucial role when deciding whether to lease or buy a car, as it affects the overall cost of acquiring the vehicle through a lease.

In our example, after the 48-month lease, the car's residual value is $10,000, meaning you can purchase the car for that amount. This becomes an additional cost if you decide to own the car after leasing it.

Proper assessment of the residual value is vital, especially if it’s less than the car’s market value at lease-end; it might be a good deal to buy it. However, in the example, even with the residual value of $10,000 added to the leasing cost of $18,000, the total $28,000 surpasses the new purchase price of $25,000. Thus, understanding residual value helps in making a smart financial decision whether to lease and then buy or purchase outright.