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You want to buy a \(\$ 25,000\) car. The company is offering an interest rate of \(2 \%\) APR for 48 months (4 years). What will your monthly payments be?

Short Answer

Expert verified
The monthly payment will be approximately \$543.57.

Step by step solution

01

Calculate Monthly Interest Rate

The Annual Percentage Rate (APR) is given as 2%. To find the monthly interest rate, divide the APR by 12 (since there are 12 months in a year):\[\text{Monthly Interest Rate} = \frac{2\%}{12} = \frac{0.02}{12} = 0.0016667\]
02

Determine Total Number of Payments

Since the loan term is 48 months (4 years), the total number of monthly payments will be:\[N = 48\]
03

Apply the Monthly Payment Formula

To find the monthly payment for a fixed-rate loan, we use the formula:\[M = P \times \frac{r(1 + r)^N}{(1 + r)^N - 1}\]where:- \(M\) is the monthly payment,- \(P\) is the loan principal (\$25,000),- \(r\) is the monthly interest rate (0.0016667),- \(N\) is the total number of payments (48).Plug the values into the formula:\[M = 25000 \times \frac{0.0016667(1 + 0.0016667)^{48}}{(1 + 0.0016667)^{48} - 1}\]
04

Simplify and Compute Monthly Payment

Calculate the expression:1. First calculate \((1 + r)^{48}\): \[(1 + 0.0016667)^{48} \approx 1.083\]2. Substitute back into the formula and compute: \[M = 25000 \times \frac{0.0016667 \times 1.083}{1.083 - 1} \] \[M \approx 25000 \times \frac{0.0018028941}{0.083} \approx \frac{45.0723525}{0.083} \approx 543.57\]
05

Finalize Your Answer

After computing using the formula, we find that the monthly payment \(M\) is approximately \$543.57. This is the amount you will need to pay every month for the duration of the loan term (48 months).

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Key Concepts

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

Understanding Annual Percentage Rate (APR)
The Annual Percentage Rate, or APR, is a crucial concept to grasp when dealing with loans. APR represents the yearly cost of borrowing after all fees and interest rates are considered. It gives you a more detailed view of what a loan will truly cost each year. It's an all-inclusive rate that helps to compare different loan offers.
When borrowing, expect to hear about the APR, as it's often advertised to attract borrowers. For the car loan example, a 2% APR signifies that borrowing the money over a full year costs 2% of the principal. This means if you're borrowing $25,000, the yearly cost in interest would be $500.
Understanding APR is key to making informed financial decisions. A lower APR means less money paid over the lifetime of the loan, making it a way to save money.
Fixed-Rate Loan Simplified
A fixed-rate loan is one where the interest rate remains constant throughout the life of the loan. This means that your monthly payments will not change. The predictability of fixed-rate loans appeals to many borrowers who prefer not to deal with fluctuating payments. In the car loan example, the 2% APR represents the interest for the entire loan term of 48 months, and because it is fixed, you will know exactly how much you owe each month.
  • The major advantage: it protects you from rising interest rates.
  • It's easier to budget: same monthly payment enables consistent financial planning.
When making a financial plan, a fixed-rate loan offers stability, helping you manage your other expenses with confidence.
Understanding the Interest Rate Formula
The interest rate formula is crucial for calculating the monthly payment on a fixed-rate loan like a car loan. It simplifies the complex interaction between the borrowed principal, interest rate, and loan term into one formula:\[M = P \times \frac{r(1 + r)^N}{(1 + r)^N - 1}\]Where:
  • \(M\) is the monthly payment.
  • \(P\) is the principal amount (i.e., the loan amount).
  • \(r\) is the monthly interest rate, computed by dividing the APR by 12.
  • \(N\) is the total number of payments.
This formula considers the time-value of money, ensuring that the payments cover both the interest and a portion of the principal every month. In our example of a \(25,000 car loan, each monthly payment of \)543.57 covers part of both the interest and the principal.Mastering this formula helps you anticipate how much you'll pay over time, enabling better financial choices.
Grasping the Loan Term
The loan term is the length of time over which you agree to repay your loan. It is divided into months for the sake of monthly installments. For instance, the car loan example uses a loan term of 48 months, or 4 years, meaning you are obligated to pay every month for those four years. It's important because:
  • A longer loan term can mean smaller monthly payments, but higher total interest costs.
  • A shorter term means larger monthly payments, but you'd pay less in interest overall.
When considering a loan's term, think about your monthly budget and long-term financial goals. While a longer term can seem appealing due to lower payments, it might not be the most cost-effective. Assess your financial situation to choose a loan term that aligns with your financial objectives.

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Most popular questions from this chapter

Determine which formula from sections \(2.2-2.4\) to use and solve the problem. You loan your sister \(\$ 500\) for two years and she agrees to pay you back with \(3 \%\) simple interest per year. How much will she pay you back

Assume you take out a 30 -year mortgage loan for \(\$ 250,000\) at a fixed \(4.5 \%\) APR. a. What will be the amount of your monthly payment? b. After the first ten years of payments, how much will remain on your loan balance? c. After the first twenty years of payments, how much will remain on your loan balance? d. Notice that the amount of the loan balance reduction during the second ten years, was very considerably bigger than the amount of the loan balance reduction during the first ten years. Why does the loan balance decrease at a faster and faster pace, the longer that the loan has been in repayment?

You deposit \(\$ 300\) in an account earning \(5 \%\) APR compounded annually. How much will you have in the account in 10 years? a. How much will you have in the account in 10 years? b. How much interest will you earn? c. What percent of the balance is interest?

You deposit \(\$ 1,000\) into an account earning \(5.75 \%\) APR compounded continuously. a. How much will you have in the account in 15 years? b. How much total interest will you earn? c. What percent of the balance is interest?

Suppose another mortgage lender offers you a fixed-rate 15 -year mortgage at \(2.95 \%\) APR. You have \(\$ 20,000\) saved as a down payment, and you can afford a maximum mortgage payment of \(\$ 950\) per month. You are interested in a certain house for sale, with firm selling price of \(\$ 200,000\). a. Find the monthly payment for this house. Can you afford it, under the terms of this lender? b. (Challenge): Suppose this same lender offers to increase the APR by only \(0.05 \%,\) for each additional year added to the loan period beyond 15 years (so that a 16-year loan would have \(3.00 \%\) APR, and a 17-year loan would have \(3.05 \%\) APR, and so on ), up to a maximum loan period of 25 years. Given these terms, does any combination of APR and loan period exist that would let you afford the house? If so, state the minimum number of additional years needed, the total resulting loan period, the resulting APR, and the resulting monthly payment.

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