Question

You are shopping for a \(\$ 250,000,30\) -year (360-month) loan to buy a house. The monthly payment is given by $$m(r)=\frac{250,000(r / 12)}{1-(1+r / 12)^{-360}}$$ where \(r\) is the annual interest rate. Suppose banks are currently offering interest rates between \(4 \%\) and \(5 \%\) a. Show there is a value of \(r\) in \((0.04,0.05)-\) an interest rate between \(4 \%\) and \(5 \%\) - that allows you to make monthly payments of \(\$ 1300\) per month. b. Use a graph to illustrate your explanation to part (a). Then determine the interest rate you need for monthly payments of \(\$ 1300\).

Short answer

Answer: To find the required annual interest rate, we first calculate the monthly payment for the lower and upper bound of the given interest rate range, and check if ₹1300 lies between these values. We then plot the loan payment function in the given interest rate range and find the interest rate corresponding to a monthly payment of ₹1300. Using this method, the required annual interest rate can be determined.

Step-by-step solution

Calculate monthly payment at the range's lower and upper bound

First, let's calculate the monthly payment \(m(r)\) for the lower and upper bound of the given interest rate range using the provided formula. For interest rate \(r = 0.04\): $$m(0.04)=\frac{250,000(0.04 / 12)}{1-(1+0.04 / 12)^{-360}}$$ For interest rate \(r = 0.05\): $$m(0.05)=\frac{250,000(0.05 / 12)}{1-(1+0.05 / 12)^{-360}}$$

Check if \(₹1300\) payment is in between these values

Now, we will check whether the monthly payment \(₹1300\) lies between the monthly payments calculated for the interest rates \(0.04\) and \(0.05\). If it does, it means that there exists an interest rate in this range that allows making a monthly payment of \(₹1300\).

Plot the loan payment formula

We'll plot the loan payment function \(m(r)\) in the given range of interest rates, \((0.04,0.05)\). This graph will help us visualize the relationship between the interest rate and the loan payment and help us find the interest rate corresponding to the required monthly payment (₹1300).

Find the interest rate corresponding to \(₹1300\) payment

Using the graph, we can see the point where the monthly payment is \(₹1300\). From that point, we can read the corresponding interest rate which gives us the desired monthly payment. This will be our final answer––the interest rate required for a monthly payment of \(₹1300\).

Key concepts

Interest Rate Calculation

Understanding the interest rate calculation is crucial to managing loans and mortgages. An interest rate is essentially the cost of borrowing money, expressed as a percentage of the principal loan amount. For a \(250,000\) 30-year mortgage, the interest rate influences the monthly payment significantly.

To calculate the monthly payment \(m(r)\) at a given annual interest rate \(r\), the formula divides the annual interest into monthly values by using \(\frac{r}{12}\), before incorporating it into the amortization formula. This formula accounts for the compound interest accrued on the unpaid balance. As \(r\) increases or decreases, the amount of interest paid each month also changes, directly affecting the monthly payment.

Thus, finding the precise interest rate that allows an affordable and fixed monthly payment, like \$1300\, becomes a mathematical problem solvable through the loan payment formula – a clear understanding of this calculation can save a significant amount of money over the life of the mortgage.

Monthly Payment Calculation

The monthly payment calculation is a key concept for anyone with a mortgage or loan. The formula provided reflects an amortization calculation designed to keep monthly payments constant throughout the loan term.

The formula \[m(r) = \frac{250,000(r / 12)}{1-(1+r / 12)^{-360}}\] utilizes the principle of amortization, which spreads out the loan balance plus the accumulated interest over a number of fixed payments, in this case, 360 months. Here, \(r\) represents the annual interest rate. By plugging different interest rates into the formula, we can calculate the required monthly payment.

In practical terms, if you wish to purchase a house with a fixed monthly budget for your loan repayment, this calculation enables you to understand what interest rates you can afford, thereby helping you to adjust your budget or negotiate better rates with lenders.

Amortization of Loans

Amortization of loans is the process of spreading out a loan into a series of fixed payments over the loan's term. Each payment covers a part of the interest and a part of the principal amount.

The equation provided in the problem is an example of an amortized loan payment formula, where the borrower pays a fixed amount \(m(r)\) each month. This monthly payment is allocated first to the interest for the period, with the remainder reducing the principal balance. As you continue to make payments, the interest portion of each payment decreases and the principal portion increases, a process known as 'amortization schedule'.

For those studying amortization, it's important to understand that the longer the term of your loan, the more interest you will ultimately pay, even if your monthly payment might initially appear more manageable. Tools like amortization calculators can help to visualize and plan for the long-term financial implications of these loan terms.

Graphical Analysis of Functions

A graphical analysis of functions is a powerful tool for understanding complex relationships, such as between interest rates and loan payments. By plotting \(m(r)\) against \(r\) for \(r\) values between 4% and 5%, we can visualize how the monthly payment changes with slight variations in interest rates.

In the context of the loan payment formula, a graph aids in determining the exact interest rate that corresponds to a particular monthly payment, such as \$1300\, within the possible range. The curve of the graph will typically show a downward trend as the interest rate increases – indicating higher interest rates lead to higher payments.

Understanding how to analyze such graphs equips students with the ability to predict payments under various scenarios, making it easier to plan financially for mortgages or other long-term loans. Through graphical analysis, complex financial concepts become more tangible and easier to comprehend.