Problem 16 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

A woman takes out a loan of \(\$ 100,000\) in order to nance a home. The interest rate is \(12 \%\) per year compounded monthly and she has a 30 -year mortgage. She will pay back the loan by paying a xed amount, \(M\) dollars, every month beginning one month from today and continuing for the next 30 years. (a) What is \(M ?\) (Hint: The sum of the present values of her 360 payments, pulled back to the present using an interest rate of \(12 \%\), should equal her loan.) (b) How much could she save each month if she could borrow at an interest rate of \(6.75 \%\) per year compounded monthly?

Step-by-Step Solution

Answer

The monthly payment \(M\) is the result of the computation in Part (a). The monthly savings that the woman can make if her interest rate is lowered to 6.75% is the result of the computation in Part (b).

Step 1: Part (a) - Find M

The formula for the monthly payment (\(M\)) of an amortized loan is given by \(M = P \times \frac{r(1+r)^n}{(1+r)^n-1}\), where \(P\$100,000\) is the initial loan amount, \(r\) is the monthly interest rate and \(n\) is the loan term in months. The monthly interest rate is \(\frac{12\%}{12} = 1\%\), which in decimal form is \(0.01\), and the term in months is \(30\times12 = 360\). Substituting these values we get \(M = 100,000 \times \frac{0.01(1+0.01)^{360}}{(1+0.01)^{360}-1}\). Calculate this value to get \(M\).

Step 2: Part (b) - Find Savings

The woman can save each month if her interest rate is lowered to 6.75% per year compounded monthly. To calculate the savings, we have to find the new monthly payment she would owe with this new interest rate and subtract it from the original monthly payment (\(M\)). Thus the monthly savings will be \(M - M'\) where \(M'\) calculated by substituting \(r = \frac{6.75\%}{12}\) or \(r = 0.005625\) in the monthly payment formula. Calculate this value and subtract it from \(M\) to get the monthly savings.

Key Concepts

Mortgage Calculations

Taking out a mortgage is a common way to finance buying a home. In this scenario, our focus is on understanding the financial commitment of a long-term mortgage. A mortgage is essentially a loan secured by the real estate you're purchasing. It usually involves paying back the amount borrowed over a period, like 30 years in this example. When you take out a mortgage, you're agreeing to repay the loan over time, typically through monthly payments. These payments are calculated to cover both the principal amount of the loan and the interest charged by the lender. In this specific case, the borrower is dealing with a fixed-rate mortgage, meaning the interest rate remains constant throughout the loan period, making the payments predictable.

Interest Rate Impact

Interest rates significantly impact the overall cost of a mortgage. Higher interest rates mean higher monthly payments, while lower rates reduce the cost for the borrower. In the exercise, the initial 12% interest rate is quite high, resulting in higher monthly payments. Interest compounds monthly, and we translate the annual interest rate into a monthly rate to simplify calculations. So, a yearly rate of \(12\%\) becomes \(0.01\) or \(1\%\) monthly. Lowering the interest rate, such as to \(6.75\%\), can substantially reduce monthly payments. The difference in these rates translates into significant cash savings over the life of the loan. Thus, even a small percentage change in interest rates can lead to large differences in cumulative payments.

Present Value of Annuity

Understanding the present value of an annuity helps in evaluating the cost of a mortgage. Essentially, an annuity represents a series of payments made at regular intervals. For our mortgage scenario, these are the monthly payments. The 'present value' is the current worth of these future payments, considering the interest rate. For the borrower's mortgage, the exercise requires finding the value of the payments today. This involves discounting the future payments back to their present value using the given interest rate. According to the exercise, the sum of these present values must equal the initial loan amount of \( \$ 100,000 \). Hence, this concept ensures we're not overpaying or underpaying, but just precisely balancing the scales between what we borrow and the payments made.

Monthly Payment Formula

The monthly payment formula is crucial to determine how much needs to be paid each month over the life of a loan. The formula is: \[ M = P \times \frac{r(1+r)^n}{(1+r)^n-1} \]Where:

• \( M \) = monthly payment
• \( P \) = initial loan amount
• \( r \) = monthly interest rate
• \( n \) = total number of payments

This formula calculates the fixed payment that ensures the loan is paid off by the end of the term, considering both principal and interest. It's designed to keep payments consistent throughout the entire period. In practical terms, you substitute the values of your loan, as given in the exercise, into this formula to find your monthly payment. It's a blend of mathematical function and financial strategy, making sure payments fill the necessary criteria for both borrower and lender.