Problem 41 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The monthly payment that amortizes a loan of \(A\) dollars in \(t\) yr when the interest rate is \(r\) per year, compounded monthly, is given by $$P=f(A, r, t)=\frac{A r}{12\left[1-\left(1+\frac{r}{12}\right)^{-12 t}\right]}$$ a. What is the monthly payment for a home mortgage of $$\$ 300,000$$ that will be amortized over \(30 \mathrm{yr}\) with an interest rate of \(6% /\) year? An interest rate of \(8 \% /\) year? b. Find the monthly payment for a home mortgage of $$\$ 300,000$$ that will be amortized over 20 yr with an interest rate of \(8 \% /\) year.

Step-by-Step Solution

Answer

a. The monthly payment for a home mortgage of \$300,000 with a 6% interest rate is approximately \$1,798.65, and with an 8% interest rate, it is approximately \$2,201.29. b. The monthly payment for a home mortgage of \$300,000 with an 8% interest rate amortized over 20 years is approximately \$2,507.34.

Step 1: a. Monthly Payment with 6% and 8% Interest Rates

We are given a home mortgage of \$300,000 that will be amortized over 30 years and asked to find the monthly payment with interest rates of 6% and 8% per year, respectively. First, we'll need to convert the interest rates into decimals. Then, we can plug the values into the formula and calculate the monthly payment for each interest rate. 1. For the 6% interest rate: A = 300,000, r = 0.06, t = 30 2. For the 8% interest rate: A = 300,000, r = 0.08, t = 30 3. Plug the values into the formula and solve for \(P\): For the 6% interest rate: \(P = \frac{300,000 \cdot 0.06}{12\left[1-\left(1+\frac{0.06}{12}\right)^{-12 \cdot 30}\right]}\) For the 8% interest rate: \(P = \frac{300,000 \cdot 0.08}{12\left[1-\left(1+\frac{0.08}{12}\right)^{-12 \cdot 30}\right]}\) After the calculations, we get the monthly payments for the 6% and 8% interest rates, respectively.

Step 2: b. Monthly Payment with 8% Interest Rate Over 20 Years

We are given a home mortgage of $300,000 that will be amortized over 20 years with an interest rate of 8% per year. First, we'll convert the interest rate into a decimal and plug the values into the formula. Then, we can calculate the monthly payment. 1. A = 300,000, r = 0.08, t = 20 2. Plug the values into the formula and solve for \(P\): \(P = \frac{300,000 \cdot 0.08}{12\left[1-\left(1+\frac{0.08}{12}\right)^{-12 \cdot 20}\right]}\) After the calculation, we get the monthly payment for the loan with an 8% interest rate amortized over 20 years.

Key Concepts

Loan Interest Calculation

Calculating loan interest is essential for understanding how much you'll pay over time on a loan. Interest is the money paid regularly at a specific rate for borrowing money. Here's a simple breakdown:

• Annual interest rate is divided by 12 to find the monthly interest rate.
• The formula incorporates this monthly rate to compute the monthly payments.
• Higher interest rates mean you'll pay more over the life of the loan.

For example, if you have an annual interest rate of 6%, as in the exercise above, convert it to 0.06 and divide by 12 to get the monthly rate of 0.005.
This step is crucial to determine the precise financial obligations attached to your loan.

Monthly Payment Calculation

Once the interest rate is calculated, the next step is to find out how much you'll pay each month. Monthly payments consist of both principal and interest.
The formula given in the problem helps to determine these payments accurately:

• Enter the loan amount, interest rate, and term into the formula.
• Simplify to solve for the monthly payment \(P\).

The key component is the formula: \[P = \frac{A r}{12\left[1-\left(1+\frac{r}{12}\right)^{-12 t}\right]}\]This formula considers how interest accrues over time. Using it, you can determine monthly incomings and outgoings like in the provided solution. For example, when calculating the monthly payment for a 6% rate over 30 years, this formula aligns how banks structure their offerings with your repayment capabilities.

Financial Mathematics

Financial mathematics focuses on using mathematical techniques to solve real-world financial problems. It combines principles from various domains to aid in decision-making.

• It enables you to evaluate different loan options and understand the implications of interest rates and payment terms.
• It helps structure loans in ways that are logical and manageable over time.

Using financial math in the given problem, you can predict not only your monthly payments but also how much interest you'll ultimately pay. Applying the formula helps reveal the total cost of borrowing, empowering more informed financial decisions.

Home Mortgage Computation

A home mortgage involves long-term financial commitments, making accurate computations crucial. Mortgages bring together a series of factors like loan amount, interest rate, and term length.

• These variables are plugged into formulas to calculate monthly payments.
• Amortization formula provides the structure for paying down the loan over time.

In the provided example, we see how the mortgage’s term (20 or 30 years) impacts the calculations.
This process exemplifies why home-buyers must think carefully about loan term duration. Calculations not only help estimate payments but also allow borrowers to see the big picture for budgeting current and future expenses. With such insights, home mortgage computation becomes a foundational tool for savvy financial planning.