Problem 17 - Precalculus Mathematics for Calculus

Question

What is the monthly payment on a 30 -year mortgage of \(\$ 100,000\) at \(8 \%\) interest per year, compounded monthly? What is the total amount paid on this loan over the 30 -year period?

Step-by-Step Solution

Answer

Monthly payment: $733.76. Total paid: $264,153.60.

Step 1: Understand the Problem

We are asked to find the monthly payment for a 30-year mortgage with an initial loan amount of $100,000 and an annual interest rate of 8%, compounded monthly. Additionally, we have to find the total payment over the term of the loan.

Step 2: Identify the Formula

The formula for calculating the monthly payment on an amortizing loan is given by: \[ M = P \frac{r(1+r)^n}{(1+r)^n-1} \]where:- \( M \) is the monthly payment,- \( P \) is the principal amount,- \( r \) is the monthly interest rate,- \( n \) is the total number of payments.

Step 3: Calculate the Monthly Interest Rate

The annual interest rate is 8%, so the monthly interest rate \( r \) is:\[ r = \frac{0.08}{12} = 0.0066667 \]

Step 4: Calculate the Total Number of Payments

Since the mortgage term is 30 years, and payments are made monthly, the total number of payments \( n \) is:\[ n = 30 \times 12 = 360 \]

Step 5: Substitute Values into the Formula

Substitute the values for \( P \), \( r \), and \( n \) into the formula:\[ M = 100,000 \times \frac{0.0066667(1+0.0066667)^{360}}{(1+0.0066667)^{360}-1} \]

Step 6: Perform the Arithmetic

Calculate to find the monthly payment:\[ M \approx 733.76 \]

Step 7: Calculate the Total Payment Over 30 Years

Multiply the monthly payment by the total number of payments:\[ \text{Total Payment} = 733.76 \times 360 = 264,153.60 \]

Step 8: Conclusion

The monthly payment for the mortgage is approximately $733.76, and the total amount paid over the 30-year period is $264,153.60.

Key Concepts

Monthly Payment Calculation

Understanding how to calculate the monthly payment on a mortgage is crucial for managing your financial planning. A mortgage is a type of amortizing loan. Here’s how you can break down the calculation:When you take out a mortgage, the goal is to pay off both the principal and the interest over the life of the loan. With a monthly payment structure, you pay the same amount every month until the loan is paid off. This involves using a specific formula to calculate these payments.
The key formula used in a mortgage monthly payment calculation is:\[ M = P \frac{r(1+r)^n}{(1+r)^n-1} \]**Parameters in the Formula**

• **\( M \):** The monthly payment you need to make.
• **\( P \):** The initial amount of the loan, also known as the principal.
• **\( r \):** The monthly interest rate, which is the annual interest rate divided by 12.
• **\( n \):** The total number of payments you will make over the life of the loan. For a 30-year mortgage, this would be 360 payments.

By substituting the values of these parameters into the formula, you can compute the exact monthly payment required to fully amortize the loan over its term.

Amortizing Loan Formula

The concept of an amortizing loan is that it is paid off through regular, equal payments. In each payment, part of the money goes to cover the interest, while the rest decreases the balance of the loan - this is called the principal. With each subsequent payment, the amount that goes towards the interest decreases as the unpaid principal reduces. This process continues until the entire loan is paid off, which is known as full amortization.
**Amortization Schedule**

• An amortization schedule is a table that details each periodic payment on an amortizing loan. It shows how much of each payment goes towards the principal and how much goes towards the interest.
• The earlier payments cover a larger portion of interest, while later payments progressively cover more principal.

This schedule is helpful for understanding how loans are structured and can aid in the budgeting process. Ultimately, understanding amortization enables you to better grasp the long-term implications of taking out a mortgage.

Compound Interest Calculation

In mortgage calculations like the one we're discussing, compound interest is a central concept. Unlike simple interest, which is calculated only on the principal amount, compound interest considers both the principal and the accumulated interest from prior periods.
**How Compound Interest Works**

• In the context of a mortgage, interest is compounded monthly. This means each month interest is calculated on the current principal plus all accrued interest from previous months.
• This interest is then added to the principal, and the new sum becomes the basis for calculating the following month's interest.

The importance of this concept is that it ensures the total amount paid back on the loan over time is much more than just the original loan amount, particularly highlighting the significance of the interest rate on a long-term loan's overall cost. Understanding compound interest is key to comprehending how much a loan actually costs over its duration.