Problem 39 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The Sandersons are planning to refinance their home. The outstanding principal on their original loan is $$\$ 100,000$$ and was to be amortized in 240 equal monthly installments at an interest rate of \(10 \% /\) year compounded monthly. The new loan they expect to secure is to be amortized over the same period at an interest rate of 7.8\%/year compounded monthly. How much less can they expect to pay over the life of the loan in interest payments by refinancing the loan at this time?

Step-by-Step Solution

Answer

By refinancing their home at a 7.8% interest rate instead of the original 10% rate, the Sandersons can expect to pay approximately \(\$28,080.76\) less in interest payments over the life of the loan.

Step 1: Calculate the monthly payment for both loans

We will use the following formula to calculate the monthly payment (PMT) for each loan: \[PMT = P \times \frac{r(1+r)^n}{(1+r)^n - 1}\] Where \(P\) represents the principal, \(r\) represents the monthly interest rate (annual interest rate divided by 12), and \(n\) represents the number of monthly installments (240 in this case). For the original loan: - Principal, P = $100,000 - Annual interest rate = 10% = 0.1 - Monthly interest rate, r = 0.1/12 - Number of monthly installments, n = 240 For the new loan: - Principal, P = $100,000 - Annual interest rate = 7.8% = 0.078 - Monthly interest rate, r = 0.078/12 - Number of monthly installments, n = 240

Step 2: Calculate the total payments made for both loans

Multiply the monthly payment (from Step 1) by the number of months (240) to find the total payment for both loans. Total payment (original loan) = monthly payment (original loan) × 240 Total payment (new loan) = monthly payment (new loan) × 240

Step 3: Calculate the total interest paid for both loans

Subtract the outstanding principal ($100,000) from the total payment (from Step 2) to find the total interest paid for both loans. Total interest (original loan) = total payment (original loan) - $100,000 Total interest (new loan) = total payment (new loan) - $100,000

Step 4: Find the difference in total interest paid between the two loans

Subtract the total interest paid for the new loan (from Step 3) from the total interest paid for the original loan (from Step 3). Difference in interest payments = total interest (original loan) - total interest (new loan) Following the steps above, we can find the difference in interest payments between the original loan and the new loan after refinancing at the lower interest rate.

Key Concepts

Loan Refinancing

Refinancing a loan is a strategic financial decision that involves taking out a new loan to pay off an existing one. The objective is often to secure a lower interest rate, reduce monthly payments, or modify the loan terms. When homeowners like the Sandersons consider refinancing their mortgage, they are aiming to minimize their financial burden.

By opting for a lower interest rate, as in the Sandersons' case, the total amount of interest paid over the life of the loan significantly decreases. However, it's important to keep in mind that refinancing may also involve additional fees such as closing costs, which should be weighed against the potential savings. Analyzing the break-even point where the savings from a lower interest rate exceed the costs of refinancing is crucial for making an informed decision.

Compounded Interest Calculation

The concept of compounded interest is central to many financial transactions, including loans and investments. It refers to the process where interest is calculated on the initial principal, which also includes all accumulated interest from previous periods.

For loans, compounded interest can significantly increase the total repayment amount over time. In the provided exercise, the interest is compounded monthly, which means interest accrues on the loan balance every month and is included in the following month's interest calculation. The formula provided in the solution, \[PMT = P \times \frac{r(1+r)^n}{(1+r)^n - 1}\], is utilized to determine the monthly payment considering this compound interest. Understanding this calculation is crucial for borrowers to comprehend how their payments and the total interest paid over the life of a loan are affected by different interest rates.

Financial Mathematics

Financial Mathematics is the application of mathematical methods to financial problems. It involves formulas and calculations that help in making informed financial decisions. Central to this field is the time value of money, which explains why receiving money today is more valuable than the same amount in the future due to its potential earning capacity.

Applying financial mathematics, individuals can calculate present and future values of money, analyze investments, and optimize loan payments. The exercise provided is a perfect example, where financial mathematics is used to determine how much less the Sandersons will pay in interest by refinancing their loan. The formula utilized incorporates the principles of financial mathematics to provide a clear comparison between two different financial scenarios. Through accurate mathematical models, borrowers can anticipate their financial commitments and savings more reliably.