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Chapter 8: Problem 5

Suppose that you are thinking about buying a car and have narrowed down your choices to two options: The new-car option: The new car costs \(\$ 28,000\) and can be financed with a four-year loan at \(6.12 \%\). The used-car option: A three-year old model of the same car costs \(\$ 16,000\) and can be financed with a four-year loan at \(6.86 \%\). What is the difference in monthly payments between financing the new car and financing the used car?

Short Answer

Expert verified
The difference in monthly payments between financing the new car and financing the used car can be found by subtracting the used car's monthly payment from the new car's monthly payment, after calculating individual payments using their respective finance amounts and interest rates.

Step by step solution

01

Calculate the Monthly Payment for the New Car

Firstly, plug the values into the monthly payment equation for the new car. \(P = \$28,000 \cdot \frac{(0.0612/12)(1 + 0.0612/12)^{4*12}}{(1 + 0.0612/12)^{4*12} - 1}\)
02

Calculate the Monthly Payment for the Used Car

Next, plug the values into the monthly payment equation for the used car. \(P = \$16,000 \cdot \frac{(0.0686/12)(1 + 0.0686/12)^{4*12}}{(1 + 0.0686/12)^{4*12} - 1}\)
03

Determine the Difference Between the Monthly Payments

Finally, subtract the used car's monthly payment from the new car's monthly payment to find out the difference between the two options.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Calculating Monthly Payments
Understanding how to calculate monthly car loan payments is a fundamental skill in financial literacy. It's essential to get a clear picture of what to expect before committing to a loan agreement. Calculating monthly payments involves a few mathematical variables such as the total loan amount, the interest rate, and the term of the loan. To simplify, monthly payment \( P \) can be found using the formula:
\[ P = P_r \cdot \frac{r(1 + r)^n}{(1 + r)^n - 1} \]
where \( P_r \) is the principal amount, \( r \) is the monthly interest rate, and \( n \) is the number of payments (loan term in months). By replacing the variables with the given data, you can calculate exactly how much you will need to pay each month.
In our exercise, you plug in the cost of the car as the principal amount, convert the annual interest rate into a monthly rate by dividing by 12, and multiply the number of years by 12 to find the total number of monthly payments.
Interest Rates
Interest rates are the cost of borrowing money expressed as a percentage. In our car loan scenario, the rate influences the monthly payments significantly鈥攈igher rates mean higher monthly payments. Interest rates can be simple or compounded. Car loans typically involve compounded interest, meaning the interest is calculated on the initial principal and also on the accumulated interest from previous periods.

Interest rates affect the total cost of the loan over its term, and they can be fixed or variable. In our case, the loan terms provided fixed interest rates, making it easy to foresee the total interest impact by simply following the monthly payment calculation.
Financial Mathematics
Financial mathematics encompasses the mathematical concepts used in the financial world to assess and manage monetary decisions. The field combines the principles of time value of money (TVM), interest rates, loan amortization, and cash flow analysis to provide a quantitative basis for economic decisions.

Calculating the monthly payment for a car loan illustrates the practical application of financial mathematics. By representing the loan's details as mathematical variables and using the formula, one can quickly determine future payments and responsibly plan personal finances.
Loan Amortization
Loan amortization is the process of paying off a debt over time through regular payments. An amortization schedule is a table that details each periodic payment on an amortizing loan as generated by an amortization calculator. It typically includes:
  • Payment dates
  • Principal amounts paid
  • Interest amounts paid
  • The remaining balance after each payment

Amortization schedules show the interest costs decline and the principal amounts increase over the term of the loan. The schedule reflects the steady progress one makes in paying down the loan balance with each monthly payment. In the car loan scenario, the amortization schedules for both loan options would visually demonstrate how the loan principal decreases over the four years.

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Most popular questions from this chapter

Suppose that you are buying a car for \(\$ 56,000\), including taxes and license fees. You saved \(\$ 8000\) for a down payment. The dealer is offering you two incentives: Incentive \(\mathrm{A}\) is \(\$ 10,000\) off the price of the car, followed by a four-year loan at \(12.5 \%\). Incentive \(\mathrm{B}\) does not have a cash rebate, but provides free financing (no interest) over four years. What is the difference in monthly payments between the two offers? Which incentive is the better deal?

If an investor sees that the return from dividends for a stock is lower than the return for a no-risk bank account, should the stock be sold and the money placed in the bank account? Explain your answer.

Describe two aspects of responsible credit card use.

In Exercises 11-18, a. Determine the periodic deposit. Round up to the nearest dollar. b. How much of the financial goal comes from deposits and how much comes from interest? $$ \begin{array}{|l|l|l|l|} \hline \$ \text { ? at the end of each month } & 7.25 \% \text { compounded monthly } & 40 \text { years } & \$ 1,000,000 \\ \hline \end{array} $$

Cellphone Plans If credit cards can cause financial woes, cellphone plans are not far behind. Group members should present a report on cellphone plans, addressing each of the following questions: What are the monthly fees for these plans and what features are included? What happens if you use the phone more than the plan allows? Are there higher rates for texting and Internet access? What additional charges are imposed by the carrier on top of the monthly fee? What are the termination fees if you default on the plan? What can happen to your credit report and your credit score in the event of early termination? Does the carrier use free T-shirts, phones, and other items to entice new subscribers into binding contracts? What suggestions can the group offer to avoid financial difficulties with these plans?
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