Problem 16 - Precalculus: Mathematics for Calculus

Question

Mortgage \(\quad\) What is the monthly payment on a 30 -year mortgage of \(\$ 80,000\) at \(9 \%\) interest? What is the monthly payment on this same mortgage if it is to be repaid over a 15-year period?

Step-by-Step Solution

Answer

$643.28 for 30 years; $812.15 for 15 years.

Step 1: Identify the Formula

To solve this problem, we need to use the formula for the monthly payment on an amortizing loan, which is \( M = P \frac{r(1+r)^n}{(1+r)^n-1} \), where \( M \) is the monthly payment, \( P \) is the principal amount, \( r \) is the monthly interest rate, and \( n \) is the total number of payments.

Step 2: Calculate Monthly Interest Rate

Since the interest rate is annual, we first convert it to a monthly rate. The annual interest rate is \(9\%\), so the monthly interest rate \( r \) is \( \frac{9}{12}\% = 0.0075 \) (as a decimal).

Step 3: Calculate Total Payments for 30-Year Mortgage

For a 30-year mortgage, \( n \) is the number of total monthly payments, which is \( 30 \times 12 = 360 \) payments.

Step 4: Calculate Monthly Payment for 30-Year Mortgage

Plug the values into the formula: \( M = 80000 \frac{0.0075(1+0.0075)^{360}}{(1+0.0075)^{360}-1} \). After calculating, the monthly payment is approximately \( M \approx 643.28 \).

Step 5: Calculate Total Payments for 15-Year Mortgage

Now, for a 15-year mortgage, \( n \) is \( 15 \times 12 = 180 \) payments.

Step 6: Calculate Monthly Payment for 15-Year Mortgage

Plug the new \( n \) value into the formula: \( M = 80000 \frac{0.0075(1+0.0075)^{180}}{(1+0.0075)^{180}-1} \). After calculating, the monthly payment is approximately \( M \approx 812.15 \).

Key Concepts

Amortizing Loan

An amortizing loan is a type of loan that is paid off over time through regular payments. With each payment, not only is the interest covered, but also a portion of the loan's principal amount. This is different from interest-only loans, where you only pay the interest initially and the principal remains unchanged. When you're dealing with a mortgage, most often it is this type of amortizing loan format. The beauty of these loans is that by the time you finish paying all your scheduled payments, you have completely settled both the principal and the interest.

In a typical amortizing loan, like a mortgage, as the loan matures, a larger portion of each payment goes towards the principal. At first, you might feel like you're only seeing the interest, but over time, the balance begins to decrease more noticeably.

• Each payment reduces both principal and interest.
• Regular payments mean predictability in budgeting.
• Over the loan period, the principal reduces, leading to less accumulated interest over time.

This makes amortizing loans a popular choice for homeowners who want a stable, predictable payment schedule and wish to eventually own their home outright.

Monthly Interest Rate

Understanding the monthly interest rate is crucial when calculating mortgage payments. Typically, interest rates are given annually, but for the purpose of monthly mortgage calculations, you need to convert this rate to a monthly one. This involves dividing the annual interest rate by 12, as there are 12 months in a year.

Consider a situation where the annual interest rate is 9%. You would calculate the monthly interest rate as follows: divide 9 by 12, which equals 0.75%. However, to use it in the formula, you should convert it to a decimal: 0.0075.

• Annual interest rate divided by 12 gives the monthly rate in percentage.
• The rate is then converted to decimal form to apply in formulas.
• This conversion is essential for accurate payment calculation.

Being familiar with converting interest rates helps in assessing different mortgage plans and aids in better financial planning when understanding your interest costs over time.

Total Number of Payments

The total number of payments refers to how many individual payments you'll make throughout the lifespan of your loan. When looking at a mortgage, this is determined by the loan's term. For example, a 30-year mortgage results in 360 monthly payments, since you multiply 30 years by 12 months per year. Similarly, for a 15-year mortgage, it's 180 payments.

Understanding how these work:

• A longer loan term results in more payments, which might reduce individual payment amounts but could increase total interest paid.
• A shorter term has fewer payments, which generally results in higher individual payment amounts but potentially lower overall interest costs.
• Knowing the number of payments helps in planning your budget and prepares you for the financial commitment by giving clear insight into loan duration.

Therefore, when deciding on a mortgage, it is important to balance the desire for smaller payments with the goal of minimizing total interest paid over the life of the loan.