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Chapter 8: Problem 32

The unpaid balance of an installment loan is equal to the present value of the remaining payments. The unpaid balance, \(P\), is given by $$ P=P M T \frac{\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right]}{\left(\frac{r}{n}\right)} $$ where \(P M T\) is the regular payment amount, \(r\) is the annual interest rate, \(n\) is the number of payments per year, and \(t\) is the number of years remaining in the loan. a. Use the loan payment formula to derive the unpaid balance formula. b. The price of a car is \(\$ 24,000\). You have saved \(20 \%\) of the price as a down payment. After the down payment, the balance is financed with a 5 -year loan at \(9 \%\). Determine the unpaid balance after three years. Round all calculations to the nearest dollar.

Short Answer

Expert verified
The unpaid balance formula is derived from the regular payment formula of an installment loan by substituting PMT with its value in terms of P, r, n, and t. The unpaid balance of the car loan after 3 years needs manual calculations according to the given values and derived formula.

Step by step solution

01

Derive the unpaid balance formula

The regular payment PMT of an installment loan can be expressed as: \( PMT = P \frac{(r/n)}{1-(1+r/n)^{-nt}} \) where P is the loan amount, r is the annual interest rate, n is the number of payments per year, t is the duration in years. By isolating P from the formula and substituting PMT into the unpaid balance formula we get the same formula: \( P=PMT \frac{\left[1-\left(1+\frac{r}{n}\right)^{-nt}\right]}{\left(\frac{r}{n}\right)} \)
02

Calculate the initial loan amount

The initial loan amount is the price of the car subtracted by the down payment. For a \(\$ 24,000\) car and with a \(20\%\) down payment, the initial loan amount is \(\$ 24,000 * (1-0.2) = \$ 19,200\).
03

Calculate the regular payment amount

In order to calculate the unpaid balance after 3 years, we first need to know the regular payment amount for this loan. To do that we apply the PMT calculation formula where \(r=9/100, n=12, t=5\), and P=\$19,200. Then we get \( PMT = \$19,200 \frac{(9/100/12)}{1-(1+9/100/12)^{12*-5}} \). Calculate the right side to get the PMT value.
04

Calculate the unpaid balance after three years

After having the regular payment amount PMT, we can now calculate the unpaid balance \(P\) after 3 years using the unpaid balance formula where \(r=9/100, n=12, t=2\) (since the remaining term is 2 years after 3 years have passed) and \(PMT\) equals to the value got from the last step. The unpaid balance after 3 years is \(P=PMT * \frac{\left[1-\left(1+\frac{r}{n}\right)^{-n*t}\right]}{\left(\frac{r}{n}\right)} \). Calculate the right side to get the P value and round it to the nearest dollar.

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Key Concepts

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

Installment Loan
An installment loan is a type of loan that is repaid over time with a set number of scheduled payments. They are commonly used for vehicle financing, personal loans, and mortgages. The loan amount, or principal, is disbursed to the borrower in a lump sum. Then, the borrower pays back the loan through regular monthly payments, also known as installments. The payments include both the principal and interest on the loan. With each installment, a part of the payment is applied to the interest, and the remaining portion reduces the principal balance.
  1. Type of Loan: Standard borrowing for personal use.
  2. Repayment Schedule: Fixed monthly payments spread over a predetermined period.
  3. Interest and Principal: Both are included in monthly payments, reducing unpaid balance progressively.
In this style of loan, it's easy to track what you've paid and what is still owed, making it a manageable way to borrow.
Present Value
The present value (PV) is a financial concept that calculates the current worth of a sum of money to be received in the future. It's important because it helps determine how much you need today to achieve a future financial goal when factoring in interest rates over time. In installment loans, PV allows borrowers and lenders to understand the value of future loan payments in today's terms. For example, if you have payments due over several years, the present value gives insight into what those future amounts are worth now. In practical terms, it's the loan's initial principal amount when you begin to repay.
  • Importance: Helps assess loan repayment value.
  • Calculation: Considers interest rate and time period.
The present value acts like a snapshot of your loan's initial indebtedness and serves as a foundation for understanding subsequent payments.
Interest Rate
The interest rate is the cost of borrowing money, typically expressed as an annual percentage of the loan amount. For installment loans, the interest rate can greatly impact the total amount paid over the lifetime of the loan. It affects both your monthly payment and the overall cost of the loan. Most installment loans have a fixed interest rate, meaning that the rate remains constant over the loan term. This stability allows borrowers to plan their finances more accurately because the monthly payment remains the same.
  • Impact: A higher interest rate increases overall loan cost.
  • Fixed Rate Loans: Offer predictability with stable payments.
Choosing loans with competitive interest rates is crucial to minimize borrowing costs and ensure affordable monthly payments.
Payment Formula
The payment formula for an installment loan is a key tool that determines how much each installment will be. This formula takes into account the loan amount (principal), the annual interest rate, the number of payments per year, and the total number of payments. It ensures your payments are evenly spread across the loan term.The formula is written as:\[ PMT = P \frac{(r/n)}{1-(1+r/n)^{-nt}} \]where:
  • PMT: Regular payment amount.
  • P: Principal loan amount.
  • r: Annual interest rate.
  • n: Number of payments per year.
  • t: Total number of years.
The payment formula helps you calculate precise monthly payments, ensuring that the loan's balance reaches zero by the end of the term. Understanding this formula is essential for both planning and managing an installment loan effectively.

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Most popular questions from this chapter

What are credit scores?

In Exercises 1-10, use $$ P M T=\frac{P\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right]} . $$ Round answers to the nearest dollar. 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 A is $10,000 off the price of the car, followed by a four-year loan at 12.5%. Incentive 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?

Exercises 3-4 involve credit cards that calculate interest using the average daily balance method. The monthly interest rate is \(1.2 \%\) of the average daily balance. Each exercise shows transactions that occurred during the June 1 -June 30 billing period. In each exercise, a. Find the average daily balance for the billing period. Round to the nearest cent. b. Find the interest to be paid on July 1, the next billing date. Round to the nearest cent. c. Find the balance due on July 1 . d. This credit card requires a \(\$ 30\) minimum monthly payment if the balance due at the end of the billing period is less than \(\$ 400\). Otherwise, the minimum monthly payment is \(\frac{1}{25}\) of the balance due at the end of the billing period, rounded up to the nearest whole dollar. What is the minimum monthly payment due by July 9? $$ \begin{array}{|l|c|} \hline \text { Transaction Description } & \text { Transaction Amount } \\ \hline \text { Previous balance, } \$ 4037.93 & \\ \hline \text { June } 1 \quad \text { Billing date } & \\ \hline \text { June } 5 \quad \text { Payment } & \$ 350.00 \text { credit } \\\ \hline \text { June } 10 \text { Charge: Gas } & \$ 31.17 \\ \hline \text { June } 15 \text { Charge: Prescriptions } & \$ 42.50 \\ \hline \text { June } 22 \text { Charge: Gas } & \$ 43.86 \\ \hline \text { Charge: Groceries } & \$ 112.91 \\ \hline \text { June } 29 \text { Charge: Clothing } & \$ 96.73 \\ \hline \text { June } 30 \text { End of billing period } & \\ \hline \text { Payment Due Date: July } 9 & \\ \hline \end{array} $$

In Exercises 1-10, \((n)\) a. Find the value of each annuity. Round to the nearest dollar. b. Find the interest. $$ \begin{array}{|l|l|l|} \hline \begin{array}{l} \$ 50 \text { at the end of } \\ \text { each month } \end{array} & \begin{array}{l} 6 \% \text { compounded } \\ \text { monthly } \end{array} & 30 \text { years } \\ \hline \end{array} $$

S枚lve for \(P\) : $$ A=\frac{P\left[\left(1+\frac{r}{n}\right)^{n t}-1\right]}{\left(\frac{r}{n}\right)} . $$ What does the resulting formula describe?
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