Problem 45 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Emilio is securing a 7 -yr Fannie Mae "balloon" mortgage for $$\$ 280,000$$ to finance the purchase of his first home. The monthly payments are based on a 30 -yr amortization. If the prevailing interest rate is \(7.5 \% /\) year compounded monthly, what will be Emilio's monthly payment? What will be his "balloon" payment at the end of 7 yr?

Step-by-Step Solution

Answer

Emilio's monthly payment will be $$\$ 1,935.57$$. The balloon payment at the end of 7 years will be $$\$ 268,706.01$$.

Step 1: Calculate the Monthly Interest Rate

First, we need to find the monthly interest rate. Since the interest rate is given yearly, we can calculate the monthly interest rate by dividing the annual interest rate by 12. In this case: \[i = \frac{7.5\%}{12} = 0.625 \%\]

Step 2: Calculate the Total Number of Payments

Now we need to find the total number of payments in the amortization schedule based on the 30-year period. Since there are 12 payments per year, the total number of payments is: \[n = 30 \times 12 = 360\]

Step 3: Calculate the Monthly Mortgage Payment

Now we can calculate the monthly mortgage payment using the following formula: \[P = \frac{Pr \times (1 + r)^n}{(1 + r)^n - 1}\] Where: - P is the monthly payment - Pr is the principal amount ($$\$ 280,000$$) - r is the monthly interest rate - n is the total number of payments Substituting the given values, we get: \[P = \frac{280,000 \times (1 + 0.00625)^{360}}{(1 + 0.00625)^{360} - 1} = \$ 1,935.57\] So, Emilio's monthly payment will be $$\$ 1,935.57$$.

Step 4: Calculate the Outstanding Balance after 7 years

Now, let's find the outstanding balance on the loan after 7 years, which will be the balloon payment. We can use the following formula: \[B = Pr \times ((1 + r)^n - (1 + r)^p) \div ((1 + r)^n - 1)\] Where: - B is the outstanding balance - p is the number of payments made (7 years * 12 payments per year) Calculating the outstanding balance: \[B = 280,000 \times ((1 + 0.00625)^{360} - (1 + 0.00625)^{84}) \div ((1 + 0.00625)^{360} - 1) = \$ 268,706.01\] So, Emilio's balloon payment at the end of 7 years will be $$\$ 268,706.01$$.

Key Concepts

Amortization Schedule

An amortization schedule offers a detailed breakdown of the repayment process for a mortgage over time. Each payment includes a portion that goes toward the interest as well as a portion that pays down the principal balance. For a balloon mortgage, like the one Emilio has secured, the schedule is based on a longer term (30 years in Emilio's case), but with a final balloon payment concluding the agreement much sooner (after 7 years here).
An important aspect of the amortization schedule is that it shows how, with each payment, a greater fraction goes toward the principal and a smaller fraction toward the interest. This helps borrowers to understand how much they owe at any point during the term and how their payments impact the total debt over time. For those seeking to calculate a custom amortization schedule, many online mortgage calculators are available or one can create a spreadsheet that delineates this payment plan.

Monthly Interest Rate Calculation

The monthly interest rate is crucial for understanding how much you'll pay in interest over each period of your mortgage. It is typically derived from the annual interest rate, which is the rate advertised by lenders. To calculate the monthly rate, you divide the annual rate by 12, because there are 12 months in a year.
For Emilio, with an annual rate of 7.5%, the monthly interest rate calculation is straightforward: divide 7.5 by 12 to get 0.625%. This conversion helps in determining the part of each monthly payment that will go toward interest, which is necessary when creating an amortization schedule.

Mortgage Payment Formula

The mortgage payment formula is used to calculate the consistent monthly payments that will be due over the course of the mortgage. Emilio's monthly payment, based on a classic formula, involves the loan principal, the monthly interest rate, and the total number of payments.
Within this formula, the monthly payment is computed by multiplying the principal amount by the monthly rate and then by the number of total payments, which is then divided by the number of payments raised to the power of one plus the monthly rate minus one. The formula accounts for both the repayment of the principal loan amount and the interest on the loan, ensuring each monthly payment is the same until the final balloon payment.

Outstanding Balance Calculation

Calculating the outstanding balance on a mortgage after a certain number of payments is crucial for understanding the size of a balloon payment in a balloon mortgage scenario. The formula includes the original principal, the monthly interest rate, the total number of payments over the full amortization period, and the number of payments made to date.
For Emilio's situation, to find the balloon payment after 7 years, we'd subtract the paid portion from the total amount payable over 30 years. This calculation quantifies the remaining principal amount, which is what constitutes the balloon payment. It is a lump sum that either must be refinanced or paid off at the end of the balloon mortgage term, and it can be significantly larger than the monthly installments paid thus far.