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Chapter 5: Problem 41

Use the amortization formulas given in this section to find (a) the monthly payment on a loan with the given conditions and (b) the total interest that will be paid during the term of the loan. \(\$ 125,000\) is amortized over 30 years with an interest rate of \(7.25 \%\)

Short Answer

Expert verified
Monthly payment: $852.90; Total interest paid: $182,044.

Step by step solution

01

Understand the loan conditions

The exercise provides a loan of \( \$125,000 \) with a term of 30 years and an annual interest rate of \( 7.25\% \). We aim to find the monthly payment and the total interest paid.
02

Convert annual interest rate to monthly

Since the loan payments are monthly, we first convert the annual interest rate to a monthly rate by dividing by 12. Thus, the monthly interest rate is \( 0.0725 / 12 \approx 0.0060417 \).
03

Calculate the number of payments

There are \(30\) years to pay off the loan, with monthly payments, so the total number of payments \( n = 30 \times 12 = 360 \).
04

Apply the monthly payment formula

Use the formula for the monthly payment on an amortized loan: \[ M = P \times \frac{r(1+r)^n}{(1+r)^n - 1} \]where \( M \) is the monthly payment, \( P = 125,000 \) is the principal, \( r = 0.0060417 \) is the monthly interest rate, and \( n = 360 \) is the number of payments.Substitute the values: \[ M = 125,000 \times \frac{0.0060417(1+0.0060417)^{360}}{(1+0.0060417)^{360} - 1} \]
05

Calculate the monthly payment

Calculate the expression from Step 4 to determine the monthly payment \( M \). You should find that: \[ M \approx 852.90 \]
06

Calculate the total of all payments

Multiply the monthly payment by the number of payments to find the total cost of the loan: \[ 852.90 \times 360 \approx 307,044 \]
07

Calculate the total interest paid

To find the total interest paid over the term of the loan, subtract the principal from the total cost of the loan: \[ 307,044 - 125,000 = 182,044 \]

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Key Concepts

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

monthly payment calculation
To understand the monthly payment calculation for an amortized loan, first recognize the loan amount, the interest rate, and the loan term. With this loan structure, borrowers pay down the loan over time in equal installments. Let鈥檚 break this down.
The monthly payment formula for an amortized loan is:
  • \( M = P \times \frac{r(1+r)^n}{(1+r)^n - 1} \)
Where:
  • \( M \) is the monthly payment.
  • \( P \) is the loan principal, which is the original amount borrowed \(\(125,000\,\\)\) in this exercise.
  • \( r \) is the monthly interest rate, calculated as the annual rate divided by 12. In our example, \(0.0725 / 12 \approx 0.0060417\).
  • \( n \) is the total number of payments, calculated as the loan term in years multiplied by 12. For a 30-year loan, \( n = 30 \times 12 = 360 \) payments.
Calculating the monthly payment ensures you know how much you need to pay the lender each month, until the loan's term completes.
loan interest calculation
Understanding how loan interest works is crucial in managing your finances. Loan interest is a fee charged for borrowing money, and it can add significantly to the total cost of your loan.
To calculate loan interest paid over the life of a loan, follow these simple steps:
  • First, compute the total cost of the loan by multiplying the monthly payment \( M \) by the number of payments \( n \).
  • In our original exercise, the total amount paid over 30 years is \( 852.90 \times 360 \approx 307,044 \).
  • Then, determine the interest paid by subtracting the principal \( P \) from the total cost of the loan, giving us \( 307,044 - 125,000 = 182,044 \) in interest over the term.
By understanding these calculations, you can clearly see the impact of interest rates over long-term loans and make informed choices about loan terms and rates.
amortization formula
The amortization formula is a fundamental tool for managing loans. It structures loans in a way that allows equal payments over time, gradually reducing the principal balance.
This formula involves understanding the relationship between principal, interest rate, and number of payments:
  • The primary goal is to pay off both the principal and the interest over time with equal monthly installments.
  • The formula for the monthly payment has been set to accommodate these conditions, ensuring all obligations are met by the end of the loan term.
  • It utilizes the compound interest principles within its calculation, allowing each payment to cover both the interest accrued during the period and a portion of the principal.
Using the amortization formula helps borrowers keep track of what part of their payment contributes to interest and what part reduces the principal, aiding in better financial planning and management.

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