Contemporary Mathematics

# 6.11Buying or Leasing a Car

Contemporary Mathematics6.11 Buying or Leasing a Car

Figure 6.25 The choice to lease or buy is based on cost and other concerns. (credit: “Car Insurance” by Pictures of Money/Flickr, CC BY 2.0)

## Learning Objectives

After completing this section, you should be able to:

1. Evaluate basics of car purchasing.
2. Compute purchase payments and identify the related interest cost.
3. Evaluate the basics of leasing a car.
4. Identify and contrast the pros and cons of purchasing versus leasing a car.
5. Investigate the types of car insurance.
6. Solve application problems involving owning and maintaining a car.

There are people who don’t need a car and won’t purchase one. But for many people, whether or not to have a car is not a question. Having a car is a basic necessity for these people.

Obtaining a car can be daunting. The models, the features, the additional costs, and finding funding are all steps that need to be taken. One of the big decisions is whether to buy the car or to lease the car. This section will address some of the issues associated with each option.

## The Basics of Car Purchasing

The biggest questions you will answer before purchasing a car are, what do you want and what do you need?

Does it have to be new? Does it have to be a make and model you are familiar with? Does it have to have assisted driving? What other details are important to you? For a new vehicle, every feature beyond standard features comes with additional cost, which leads to the question that constrains all of your decisions about a car. How much can you afford to spend on a car?

What you can afford must include insurance costs (discussed later in this section) and maintenance and upkeep. Once you have this in mind, you can search for a car that matches, as closely as possible, what you want and can afford. Most, if not all, dealers have websites that you can search through to identify the car you want. If new cars are not affordable, used cars cost less but come with the wear and tear of use.

The sticker price of the car, called the manufacturer’s suggested retail price (MSRP), or the negotiated price you arrive at, isn’t the end of the cost to buying a car. There are many fees that accompany the purchase of the car, and perhaps even sales tax. These include but aren’t necessarily limited to the following:

• the title and registration fee, which includes registering your car with the state, getting the license plate, and assigning the title of the car to the lender. This cannot be avoided.
• a destination fee, which covers the cost of delivering the vehicle to the dealer
• a documentation fee, sometimes referred to a processing fee of handling fee, is the cost of all the paperwork the dealer did to get you the car
• a dealer preparation fee, which is for washing the car and other preparation of that sort. You should try to negotiate that out of the cost of the dealer tries to charge for that
• extended warranties and maintenance plans, which help cover some of the costs of caring for the car.
• Sales tax.

You could pay for these immediately, but they are often added to the financing of the car, meaning they become part of the principal of your loan.

## Example 6.98

### Total Cost to Purchase a Car

Nichole negotiates with her car dealership so that the price is $21,800. She needs to pay the 6.75% sales tax on the car. Other fees are$31.00 for title and registration, $1,000 in destination fees, and a documentation fee of$175. What is the total cost of Nichole’s car?

1.
Luther negotiates for the price of his car, reaching agreement at $28,975. He needs to pay 8% in sales tax, 2.1% in ownership tax, a$950 destination fee, processing fees totaling $370, and registration fee of$617. What is the total cost for Luther’s purchase?

One way to bring down payments on a car is to provide a down payment or a trade in. This is money applied to the purchase price before financing happens. Be warned, the sales tax applies to the full purchase price! If you reduce the amount financed, the payments respond by going down. This often becomes part of the negotiating process.

Sophia negotiates a $19,800 price for her new car. The sales tax is 9.5% in her area, and the dealership charges her$300 in documentation fees. Her title, plates, and registration come to $321.50. The dealership adds to this a destination fee of$1,100. If she places a down payment of $5,000 on the car, what is the total she will finance for the car? ## Your Turn 6.99 1. Carlos buys a car with a negotiated price of$36,250. The sales tax in his region is 6.5%. The dealer charges a $1,200 destination fee and a$450 documentation fee. He must pay for title, registration, and license plates, which come to $21.50. If he has a$7,500 down payment, how much will he have to finance?

When purchasing a car, the total cost to obtain the car is not the only factor in your monthly price. You will also pay an interest rate for the loan you obtain. The interest rate you will get is dependent on your credit score (see The Basics of Loans). But you can choose from different lenders. The dealership will likely offer to finance your car loan. Frequently, dealerships offer special financing with very low rates. This is to help move inventory, and may indicate their desire to make sales. This might make negotiating easier. Even if the dealership offers financing, check with your bank or credit union to determine the interest rates they are offering. To reduce your payments, choose the lowest rate you can find.

## Purchase Payments and Interest

Whether or not you buy a new car or a used car, if you finance the purchase, you are taking out a loan. The interest rates available for used cards are frequently higher than those for new cars. These loan payments work exactly the same way as other loans do as far as payments are concerned. The payment function comes from The Basics of Loans. The difference between financing a new car or a used car is that financing a new car typically comes with a lower interest rate and a longer term that financing a used car.

## FORMULA

The payment, $pmtpmt$, per period to pay off a loan with beginning principal $PP$ is $pmt=P×(r/n)×(1+r/n)n×t(1+r/n)n×t−1pmt=P×(r/n)×(1+r/n)n×t(1+r/n)n×t−1$, where $rr$ is the annual interest rate in decimal form, $tt$ is the term in years, and $nn$ is the number of payments per year (typically, loans are paid monthly making $nn$ = 12).

Note, payment to lenders is always rounded up to the next penny.

## Checkpoint

Often, the formula takes the form $pmt=P×(r)×(1+r)n(1+r)n−1pmt=P×(r)×(1+r)n(1+r)n−1$, where $rr$ is the interest rate per period (annual rate divided by the number of periods per year), and $nn$ is the total number of payments to be made.

## Example 6.100

### New Car Payments

In the following, calculate the monthly payment using the given total to be financed, the interest rate, and the term of the car loan.

1. Total to be financed is $31,885, interest rate is 2.9%, for 5 years. 2. Total to be financed is$22,778, interest rate is 4.5%, for 6 years.

In the following, calculate the monthly payment using the given total to be financed, the interest rate, and the term of the car loan.
1.
Total to be financed is $18,325, interest rate is 6.75%, for 4 years. 2. Total to be financed is$41,633, interest rate is 3.9%, for 6 years.

## The Basics of Leasing a Car

Leasing a car is an alternative to purchasing a car. It is still a loan, and acts like one in many respects. They typically last either 24 months or 36 months, though other terms are available. Leases also come with mileage limits, frequently 10,000, 12,000, or 15,000 miles per year. When the lease is over, the car is returned to the dealer. At that time there may be fees that have to be paid, such as for damage to the car or for extra miles driven over the limit.

There are two components to lease costs. One is the monthly payment for the lease. The other is the fees for leasing, These often are paid before the lease is complete. These include:

• a down payment, which is your initial payment that is applied to the price of the car. It reduces the amount you finance, much the same as when you purchase a car. It is recommended that this be negotiated away.
• the acquisition fee, sometimes called the bank fee. This is the money charge for the company to set up the lease. It is essentially a paperwork fee. It is not likely that this can be negotiated.
• a security deposit, which might be required. It is about the same as 1 month’s payment for the lease. The deposit is returned to you if the car is in good shape at the end. This can be negotiated away.
• disposition fees, which cover the cost the company will incur when they take your car back and are typically between $200 and$450.
• the title, registration, and license fees, just as with the purchase of a car.
• sales tax, which will likely be applied. The sales tax only covers the depreciated portion of the car (more on depreciation later) in many states. Since this depends on the state in which the car is leased, you should determine the sales tax rules for where you lease the car.

As you can imagine, this can come to a fairly high dollar amount.

## People in Mathematics

### Zollie Frank

Zollie Frank and Armund Shoen founded one of the original leasing companies, Four Wheels, in 1939. Their company leased automobiles to corporations. They began by leasing 5 cars to the Petrolager pharmaceutical company in year one. This saved Petrolager money and provided a steady cash flow to the Four Wheels business. In year two, the number of cars leased to Petrolager was 75. Their new idea was to lease cars directly to companies for one year. Previously, such companies might pay for mileage, gas, and a partial down payment. Sadly, the salesmen who were being so helped often left the company before the car was paid for, and so the company lost the down payment money.

1.
Natalie needs to budget for her car maintenance. She expects to buy new tires each 5 years, which will cost her $390 to replace them all. Oil changes near her cost$59.99, and she believes she will get one every 3 months. Her inspection costs $25 per year. Wipers for her car cost$115 for all three and she anticipates changing them every year. She drives less than 30,000 miles per year, so she plans to replace the air filter once per year. The air filter for her car costs $46.25. How much should she budget per month to cover these costs? ## Check Your Understanding 69. What is destination fee? 70. What is a title and registration fee? 71. Calculate the monthly payment if the total to be financed is$34,570, the interest rate is 3.5%, for 5 years.
72.
What might be different for a used car loan and a new car loan?
73.
What fees are similar between leasing a car and buying a car?
74.
If the cost of a car is $30,000, and the residual value of the car after 3 years is$18,000, what is the monthly depreciation for the car?
75.
If the money factor for a lease is 0.000015625, what is the annual interest rate?
76.
For people who do not mind driving an older car or maintaining a car, which is preferable, a lease or purchasing?
77.
What does collision insurance cover?

## Section 6.11 Exercises

1 .
What are the two main questions to answer before buying a car?
2 .
What is a documentation fee?
3 .
What is a dealer preparation fee?
4 .
What is a down payment?
5 .
How long do leases typically last?
6 .
Can you drive a leased car any mileage?
7 .
What are three fees associated with leasing a car?
8 .
What is the monthly depreciation for a leased car?
9 .
How are the money factor and the annual percentage rate related?
10 .
11 .
12 .
What does liability insurance cover?
13 .
What does collision insurance cover?
14 .
Many expenses associated with a car can be anticipated. Name three maintenance expenses that can be anticipated.
In the following exercises, find the total cost to purchase the car.
15 .
Alexia negotiates a purchase price of $17,850 for her new car. The sales tax in her area is 6.5%. Her license, plates, and registration come to$285.00. The dealership charges her a $600 destination fee and a$150 processing fee. How much will she finance in total for the car?
16 .
Stephanie negotiates a purchase price of $25,670 for her new car. The sales tax in her area is 8.0%. Her license, plates, and registration come to$389.00. The dealership charges her a $700 destination fee and a$345 processing fee. How much will she finance in total for the car?
17 .
Matthew negotiates a purchase price of $35,100 for her new car. The sales tax in his area is 7.25%. His license, plates, and registration come to$325.00. The dealership charges him a $900 destination fee and a$125 processing fee. How much will he finance in total for the car?
18 .
Madisyn negotiates a purchase price of $45,800 for her new car. The sales tax in her area is 7.25%. Her license, plates, and registration come to$199.00. The dealership charges her a $1,000 destination fee and a$275 processing fee. How much will she finance in total for the car?
In the following exercises, calculate the car payment based on the total financed and the interest rate.
19 .
Total to be financed is $36,775, interest rate is 2.75%, for 6 years. 20 . Total to be financed is$29,350, interest rate is 3.9%, for 5 years.
21 .
Total to be financed is $27,180, interest rate is 1.99%, for 7 years. 22 . Total to be financed is$15,489, interest rate is 6.75%, for 4 years.
In the following exercises, find the total cost to obtain the lease.
23 .
A $2,000.00 down payment,$120 in acquisition fees, a security deposit of $350.00,$200.00 in disposition fees, and sales tax of $2,860. 24 . No down payment,$260 in acquisition fees, a security deposit of $450.00,$400.00 in disposition fees, and sales tax of $3,155.00. 25 . A$4,000 down payment, $360 in acquisition fees, a security deposit of$900.00, $1,000.00 in disposition fees, and sales tax of$4,275.
26 .
A $7,000.00 down payment,$225 in acquisition fees, a security deposit of $800.00,$675.00 in disposition fees, and sales tax of $3,673. In the following exercises, find the monthly depreciation of the car. 27 . The purchase price of a car is$34,000. Its residual price is $23,500. What is its monthly depreciation for a 24-month lease? 28 . The purchase price of a car is$23,500. Its residual price is $11,750. What is its monthly depreciation for a 36-month lease? In the following exercises find the APR based on the MF. 29 . MF = 0.00004125 30 . MF = $0.000046458\overline 3$ In the following exercises find the MF based on the APR. 31 . 8.75% 32 . 6.25% In the following exercises, find the lease payment based on the given information. 33 . Price is$41,700, residual price is $27,105, 24-month lease, money factor is 0.000025. 34 . Price is$22,165, residual price is $12,855.70, 24-month lease, money factor is 0.0000275. 35 . Price is$30,650, residual price is $16,857.50, 36-month lease, APR is 8.25%. 36 . Price is$24,800, residual price is $14,384, 36-month lease, APR is 5.85%. 37 . Sara needs to budget for her car maintenance. She expects to buy new tires each 3 years, which will cost her$540 to replace them all. Oil changes near her cost $39.99, and she believes she will get one every 4 months. Her inspection costs$20 per year. Wipers for her car cost $115 for all three and she anticipates changing them every year. She drives less than 30,000 miles per year, so she plans to replace the air filter once per year. The air filter for her car costs$36.25. She plans to get new brake pads every year, which cost her $267. How much should she budget per month to cover these costs? 38 . Mwibeleca needs to budget for his car maintenance. He expects to buy new tires each 3 years, which will cost her$560 to replace them all. Oil changes near his cost $69.99, and he believes he will get one every 4 months. His inspection costs$25 per year. Brake pads will cost $215 each year. Wipers for his car cost$143.50 for all three and he anticipates changing them every year. He drives less than 30,000 miles per year, so he plans to replace the air filter once per year. The air filter for his car costs $62.88. How much should Mwibeleca budget per month to cover these costs? 39 . John will either lease or buy a car. The total cost to purchase the car is$35,830, and he would finance the car for 5 years at 2.99%. If he leases, he would pay $3,287.50 for the lease, and then his payments would be based on a price of$32,750, a residual price of $20,305, 36 months, with a money factor of 0.00001725. Compare the payments to purchase the car to the payments of the lease plus the lease cost divided by 36. 40 . Zachary will either lease or buy a car. The total cost to purchase the car is$22,945, and he would finance the car for 6 years at 1.99%. If he leases, he would pay $2,387.75 for the lease, and then his payments would be based on a price of$21,350, a residual price of \$12,390.30, 36 months, with a money factor of 0.000018375. Compare the payments to purchase the car to the payments of the lease plus the lease cost divided by 36.
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