Question

Automobile lease vs. purchase \(A\) 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 \(\$ 25,000\) or leased for a start-up fee of \(\$ 1200\) plus monthly payments of \(\$ 350\). 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 \(\$ 10,000,\) which is the amount that you could pay to purchase the car. Assuming no other costs, should you lease or buy?

Short answer

Answer: After 48 months, buying the car outright is a better option than leasing it.

Step-by-step solution

Find the Linear Function for Lease Option

The total amount paid on the lease option after m months can be determined by adding the start-up fee and multiplying the monthly payments by the number of months. The linear function can be written as: $$y=f(m) = 1200 + 350m$$

Evaluate the Total Amount Paid on Lease Option after 48 Months

Plug m=48 into the linear function to find the total amount paid after 48 months: $$y=f(48) = 1200 + 350(48)$$ You can simplify it further and calculate the amount: $$y = 1200 + 16800 = 18000$$ After 48 months, the student would have paid \(18,000\) on the lease option.

Evaluate the Total Cost of Lease Option with Residual Value

After 48 months, the residual value of the car is \(10,000\). To decide whether to lease or buy, we need to add this amount to the total cost of the lease option: $$\text{Total cost of lease option} = 18000 + 10000 = 28000$$

Compare the Total Cost of Lease Option with Purchase Option

We now have the total cost to buy (\(25,000\)) and the total cost to lease (\(28,000\)). To determine the better option, we need to compare the two costs. The purchase option is cheaper as: $$25000 < 28000$$ Solution: a. The linear function that gives the total amount paid on the lease option after m months is: $$y=f(m) = 1200 + 350m$$ b. After comparing the total cost of the lease option (\(28,000\)) to the purchase option (\(25,000\)), it is clear that purchasing the car outright is a better option than leasing it.

Key concepts

Lease vs. Buy Analysis

When deciding whether to lease or buy a car, it's important to conduct a thorough analysis to see which option is more financially advantageous. Leases are often preferred by those who enjoy having a new vehicle every few years and want lower monthly payments. Buying a car, on the other hand, is generally better for those who plan to keep the car for a long time and wish to avoid mileage limitations found in lease agreements.
To conduct a lease vs. buy analysis, consider the following:

• Compare initial costs: Buying usually requires a larger upfront payment, while leasing may offer lower starting fees.
• Calculate monthly costs: Lease payments are typically lower, but they don’t contribute to ownership.
• Evaluate long-term investment: Buying a car means you can eventually own it outright, potentially saving money over an extended period.

These factors can help you decide whether leasing or buying fits your personal and financial circumstances better.

Cost Comparison

Cost comparison is a key part of deciding to lease or buy a car. By examining all the expenses involved in both options, you can make a more informed decision. Start with the total cost analysis of leasing versus buying.
In our example, the purchase price of the car is $25,000. On the other hand, the total cost of leasing comes to $18,000 for the monthly payments over four years, plus a residual value cost of $10,000 if you decide to purchase the vehicle at the end of the lease. This totals to $28,000 when combined.

• Total cost to lease: $18,000 (monthly payments) + $10,000 (residual value) = $28,000
• Total cost to buy: $25,000 outright

Clearly, buying is cheaper in this scenario by $3,000 ($28,000 - $25,000), making it the financially better option over leasing.

Residual Value

Residual value is a crucial concept in leasing agreements. This term refers to the estimated value of the car at the end of the lease period. It impacts your financial decision-making as it is the amount you may need to pay if you choose to keep the car after the lease ends.
The agreement in this case states a residual value of $10,000. This means, after making regular payments for 48 months, you would need to pay this amount in order to own the car.

• Why Residual Value Matters: It affects the total cost of leasing if ownership at the end is desired.
• Determination of Residual Value: Typically set by the leasing company, based on market trends and expected depreciation.

Understanding residual value is key to evaluating the full cost of leasing and helps you decide if purchasing the car at the end of the lease term is worth it.