15.7: Car Loans

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Learning Objectives

Evaluate the basics of car purchasing.
Calculate purchase payments and identify related fees and interest.
Determine installment payments for car loans.

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

Obtaining a car can be daunting. The variety of models, features, additional costs, and securing financing are all steps that must be considered. This section will address some of the key issues associated with car ownership.

The Basics of Car Purchasing

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

Does it have to be new? Does it need to be a make and model you are familiar with? Does it need assisted driving features? What other details are important to you? For a new vehicle, every feature beyond the standard options adds to the cost. This leads to the question that influences all your car-buying decisions: How much can you afford to spend?

What you can afford must include not only the purchase price but also insurance (discussed later in this section), maintenance, and upkeep. Once you have this in mind, you can begin searching for a car that closely matches both your wants and your budget. Most, if not all, dealerships have websites that allow you to browse available vehicles. If new cars are out of reach, used cars tend to be more affordable but may come with wear and tear from prior use.

The sticker price—called the Manufacturer’s Suggested Retail Price (MSRP)—or the price you negotiate isn’t the final cost of buying a car. Many additional fees may be involved, along with possible sales tax. These can include, but are not necessarily limited to:

Title and registration fee: Covers registering your car with the state, obtaining license plates, and assigning the title—usually to your lender. This is mandatory.
Destination fee: Covers the cost of transporting the vehicle to the dealer.
Documentation fee (sometimes called a processing or handling fee): Covers the paperwork the dealer completes to finalize your purchase.
Dealer preparation fee: Charged for washing and preparing the car for sale. You should try to negotiate this fee out if the dealer includes it.
Extended warranties and maintenance plans: Optional add-ons that can help cover the cost of future maintenance and repairs.
Sales tax: Varies by location and can significantly affect the total cost.

While you could pay these fees upfront, they are often rolled into the financing, meaning they become part of the principal of your loan.

Example $\PageIndex{1}$: Total Cost to Purchase a Car

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

Solution

We add the car’s sale price, sales tax, and all other fees to arrive at the total cost.
The sales tax is 6.75% of the negotiated price, so: $21,800 × 0.0675 = $1,471.50.

Adding everything together: $21,800 + $1,471.50 + $31.00 + $1,000 + $175 = $24,477.50.

Try It $\PageIndex{1}$

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

Answer: $33,838.48

One way to reduce car payments is by making a down payment or offering a trade-in. This amount is applied to the purchase price before financing begins. Be aware: sales tax applies to the full purchase price, not the reduced amount! However, reducing the amount financed will lower your monthly payments. This often becomes part of the negotiation process.

Example $\PageIndex{2}$: Total Cost to Purchase a Car with Down Payment

Sophia negotiates a price of $19,800 for her new car. The sales tax in her area is 9.5%, and the dealership charges her $300 in documentation fees. Her title, plates, and registration cost $321.50. The dealership also adds a destination fee of $1,100. If she makes a down payment of $5,000, what is the total amount she will finance for the car?

Solution

The sales tax of 9.5% is based on the car's price of $19,800.
The sales tax comes to: $19,800 × 0.095 = $1,881.

Adding all the fees to the price and sales tax gives the total cost of the car: $19,800 + $1,881 + $300 + $321.50 + $1,100 = $23,402.50.

Her down payment of $5,000 is subtracted from $23,402.50.
The amount to be financed is $18,402.50

Try It $\PageIndex{2}$

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 also pay for the title, registration, and license plates, which cost $21.50. If he makes a $7,500 down payment, how much will he need to finance?

Answer: $32,777.75

When purchasing a car, the total cost of obtaining the car is not the only factor in determining your monthly payment. You will also pay interest on the loan you take out. The interest rate you receive depends on your credit score (see The Basics of Loans). However, you can choose from different lenders. The dealership will likely offer to finance your car loan. Often, dealerships provide special financing with very low rates to help move inventory and encourage sales, which could make negotiations easier. Even if the dealership offers financing, it’s a good idea to check with your bank or credit union to compare interest rates. To reduce your payments, aim to choose the lowest rate you can find.

Monthly Payments and Interest

Whether you buy a new car or a used car, if you finance the purchase, you are taking out a loan. The interest rates for used cars are often higher than those for new cars. The loan payments work the same way as other loans in terms of how payments are made. The payment function is based on compound interest formulas. The key difference between financing a new car and a used car is that financing a new car typically comes with a lower interest rate and a longer term than financing a used car.

FORMULA: PMT

The payment per period (installment payment, $PMT$) is

\[ PMT=\frac{P\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{(-nt)}\right]} \label{1} \]

where $P$ is the loan principal, $r$ is the annual interest rate (APR) in decimal form, $t$ is the term in years, and $n$ is the number of payments per year (typically, loans are paid monthly, making $n=12$).

Note, payments to lenders are always rounded up to the next penny.

Example $\PageIndex{3}$: New Car Payments

Calculate the monthly payments for the following car loans.

(a) $31,885 is being financed at an annual interest rate of 2.9% for five years.

(b) $22,778 is being financed at an annual interest rate of 4.5% for six years.

Solution

(a) $P=\$31,885$, $r=0.029$, and $t=5$ years. These are monthly payments, so $n=12$.

Substituting and calculating, we find the monthly payment to be $571.52.

\[PMT = \frac{31,885\left(\frac{0.029}{12}\right)}{\left[1-\left(1+\frac{0.029}{12}\right)^{(-12\times 5)}\right]}\approx 571.52 \nonumber\]

(b) $P=\$22,778, ~r=0.045$, and $t=6$ years. These are monthly payments, so $n=12$.

Substituting and calculating, we find the monthly payment to be $361.58.

\[PMT = \frac{22,778\left(\frac{0.045}{12}\right)}{\left[1-\left(1+\frac{0.045}{12}\right)^{(-12\times 6)}\right]}\approx 361.58 \nonumber\]

Try It $\PageIndex{3}$

Calculate the monthly payments for the following car loans. Round your answers up to the nearest cent.

(a) $18,325 is to be financed at an annual interest rate of 6.75% for four years.

(b) $41,633 is to be financed at an annual interest rate of 3.9% for six years.

Answer: (a) $436.70 (b) $649.47

Example $\PageIndex{4}$

Calculate the monthly payment for the used car if the total amount to be financed is $16,990, the interest rate is 7.5%, and the loan term is three years.

Solution

$P =\$16,990, ~r=0.075$, and $t=6$ years. These are monthly payments, so $n=12$.

Substituting and calculating, we find the monthly payment to be $528.50.

\[PMT = \frac{16,990\left(\frac{0.075}{12}\right)}{\left[1-\left(1+\frac{0.075}{12}\right)^{(-36)}\right]}\approx 528.50 \nonumber\]

Try It $\PageIndex{4}$

Calculate the monthly payment for a car loan of $21,845, financed at an interest rate of 6.3% for four years. Note, payments to lenders are always rounded up to the next penny.

Answer: $516.05

Calculating Payoff Amount

Suppose you have a five-year car loan but plan to purchase a new car after owning your current car for four years. Since there is still one year remaining on the loan, you must pay off the remaining balance before purchasing another vehicle. Using the payoff formula provided in the Installment Loans section, you can calculate the remaining principal (the payoff amount).

FORMULA: PAYOFF

The payoff amount for an installment loan, $P$, is given by

\[P=\frac{PMT\left[1-\left(1+\frac{r}{n}\right)^{(-U)}\right]}{\left(\frac{r}{n}\right)} \label{6.4.2} \]

where

$PMT$ is the installment payment,
$r$ is the annual interest rate (APR),
$n$ is the number of payments per year,
$t$ is the loan term in years, and
$U$ is the number of remaining (future) payments.

Note: Payments to lenders are always rounded up to the next penny. Do not round numbers during the computation, as rounding introduces errors. You may round only once at the end to report the final answer.

Example $\PageIndex{5}$: Payoff

Mr. Patel wants to pay off his car loan. His monthly payment is $365, and he has 16 payments remaining. If the loan was financed at 6.5%, how much does he still owe?

Solution

\[P=\frac{365\left[1-\left(1+\frac{0.065}{12}\right)^{(-16)}\right]}{\left(\frac{0.065}{12}\right)} \approx 5,579.64 \nonumber \]

He owes the lender $5,579.64.

Try It $\PageIndex{5}$

Fourteen months after Dan bought his new car, he lost his job. The car was repossessed by the lender after he had made only 14 monthly payments of $376 each. If the loan was financed over four years at an interest rate of 6.3%, how much did the car cost the lender? In other words, how much did Dan still owe on the car? Round your answer to the nearest cent.

Answer: $1,1680.01

Example $\PageIndex{6}$

You determine that you can afford to pay $400 per month for a car. What is the maximum price you can afford if the interest rate is 11% and you want to repay the loan over four years?

Solution

To compute the present value of the car ($P$), we can use the payoff formula with $PMT=\$400, ~r=0.11, ~t= 4$ and $~n=12$ ;

\[P=\frac{400\left[1-\left(1+\frac{0.11}{12}\right)^{(-48)}\right]}{\left(\frac{0.11}{12}\right)} \approx 15,476.57 \nonumber \]

Therefore, the price of the car should not exceed $15,476.57.

Monthly Cost of Owning a Car

Car insurance is designed to cover costs associated with accidents involving vehicles. Most states require some form of insurance. Without it, you may not be able to register your car or obtain a license plate. While your state’s insurance requirements can be complex, insurance companies and brokers typically ensure that your policy meets the legal minimums. They’ll also alert you if you’re not in compliance. However, they may recommend more coverage than necessary, so it’s your responsibility to decide how much coverage you want, as long as the legal minimum is met.

The cost of insurance should be factored in when assessing the affordability of buying or leasing a car. Whether your car is leased or owned, insurance is required - and it contributes significantly to the overall cost of owning a vehicle.

Cars are not “buy it and forget it” items. They require ongoing maintenance, which adds to the total cost of ownership. Tires, brakes, wipers, oil changes, and inspections are just a few of the regular expenses - aside from fuel. When creating a budget, you should account for these expected costs. Additionally, it’s wise to set aside extra money each month to cover unexpected - and potentially expensive - repairs.

Example $\PageIndex{7}$: Monthly Cost of Owning a Car

If your car payment is $287.50 per month and your car insurance costs $930 every six months, what is the monthly cost of the car when insurance is included?

Solution

The total cost of the car, including insurance, is the monthly payment of $287.50 plus the monthly insurance cost.

The insurance cost per month is $930 ÷ 6 = $155, since the insurance is for six months.

Adding these together, the total cost with insurance is $442.50.

Try It $\PageIndex{7}$

If your car payment is $410.86 per month and your car insurance costs $2,190 per year, what is the total cost of the car per month when accounting for the insurance?

Answer: $593.36

Check Your Understanding

What is a destination fee?
What is a title and registration fee?
What is a documentation fee?
What is a dealer preparation fee?
What is a down payment?
Calculate the monthly payment if the total to be financed is $34,570, the annual interest rate is 3.5%, and the loan term is five years.
What might be different between a used car loan and a new car loan?

Display Answers

1. Cost of delivering the vehicle to the dealer

2. Registers your car with the state, gets the license plate, and assigns the title of the car to the lender

6. $628.89

7. Higher interest rate and shorter loan term

Exercises

1. 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?

2. 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?

3. Matthew negotiates a purchase price of $35,100 for his 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?

4. 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 (5-8), calculate the monthly car payment based on the total financed and the interest rate.

5. The total to be financed is $36,775, the interest rate is 2.75%, for six years.

6. The total to be financed is $29,350, the interest rate is 3.9%, for five years.

7. The total to be financed is $27,180, the interest rate is 1.99%, for seven years.

8. The total to be financed is $15,489, the interest rate is 6.75%, for four years.

9. Sonya bought a car for $15,000. Find the monthly payment if the loan is to be amortized over five years at an interest rate of 10.1%.

10. You determine that you can afford $250 per month for a car. What is the maximum amount you can afford to pay for a car if the interest rate is 9% and you want to repay the loan in five years?

11. Friendly Auto offers Jennifer a car with a $2,000 down payment and $300 per month for five years. Jason wants to buy the same car but wants to pay cash. How much must Jason pay if the interest rate is 9.4%?

12. Jackie wants to buy a $19,000 car, but she can afford to pay only $300 per month for five years. If the interest rate is 6%, how much does she need to put down?

13. You review your budget and determine that you can afford $250 per month for a car. What is the maximum amount you can afford to pay for the car if the interest rate is 8.6% and you want to finance the loan over five years?

14. Lisa buys a car for $16,500 and receives $2,400 for her old car as a trade-in. Find the monthly payment for the balance if the loan is amortized over five years at 8.5%.

15. A car is sold for $3,000 cash down and $400 per month for the next four years. Find the cash value of the car today if money is worth 8.3% compounded monthly.

16. When Jose bought his car, he amortized his loan over six years at a rate of 9.2%, and his monthly payment was $350. He has been making these payments for the past 40 months and now wants to pay off the remaining balance. How much does he owe?

17. Name three common fees when buying a car.

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Example \(\PageIndex{1}\): Total Cost to Purchase a Car

Try It \(\PageIndex{1}\)

Example \(\PageIndex{2}\): Total Cost to Purchase a Car with Down Payment

Try It \(\PageIndex{2}\)

FORMULA: PMT

Example \(\PageIndex{3}\): New Car Payments

Try It \(\PageIndex{3}\)

Example \(\PageIndex{4}\)

Solution

Try It \(\PageIndex{4}\)

FORMULA: PAYOFF

Example \(\PageIndex{5}\): Payoff

Solution

Try It \(\PageIndex{5}\)

Example \(\PageIndex{6}\)

Solution

Example \(\PageIndex{7}\): Monthly Cost of Owning a Car

Try It \(\PageIndex{7}\)