Problem 49 - Calculus: Early Transcendental Functions

Question

Relate to money investments. A house mortgage is a loan that is to be paid over a fixed period of time. Suppose \(\$ 150,000\) is borrowed at \(8 \%\) interest. If the monthly payment is \(\$ P\), then explain why the equation \(A^{\prime}(t)=0.08 A(t)-12 P, A(0)=150,000\) is a model of the amount owed after \(t\) years. For a 30 -year mortgage, the payment \(P\) is set so that \(A(30)=0 .\) Find \(P .\) Then, compute the total amount paid and the amount of interest paid.

Step-by-Step Solution

Answer

The monthly payment \(P\) for a 30-year mortgage needed for the balance to be 0 after 30 years is approximately $1,100.65. The total amount paid over 30 years would be approximately $396,234, while the amount of interest paid is approximately $246,234.

Step 1: Understand the provided equation

The differential equation is given as \(A'(t) = 0.08A(t) - 12P\) where \(A(t)\) represents the amount owed at time \(t\), \(A'(t)\) represents the rate of change of \(A(t)\), \(P\) is the monthly payment and 0.08 is the interest rate. It can be observed from the equation that the rate of change of amount owed is equal to the interest minus the monthly payments.

Step 2: Find the monthly payment (P)

We must find the value for \(P\) that would mean the loan is fully paid after 30 years or \(A(30)=0\). This is essentially finding for what payment the rate of change \(A'(t)\) goes to 0. However, we can solve this by integrating \(A'(t)\) from 0 to 30 years with the boundary condition of A(0) = $150,000, then equate it to 0 and solve for \(P\). You should find: \(P \approx 1100.65\)

Step 3: Compute the total amount paid

The total amount paid is simply the product of the monthly payment \(P\) and the number of payments which is (30years * 12 months/year) or 360. So, total amount paid = \(P * 360 = $396,234\)

Step 4: Calculate the total interest paid

The total interest paid is the difference between the total amount paid and the initial amount borrowed. So, total interest paid = total amount paid - initial amount borrowed = $396,234 - $150,000 = $246,234

Key Concepts

Mortgage Loans

A mortgage loan is a type of loan specifically used to purchase real estate, such as a home. It is a secured loan, meaning that you agree to let the lender take your home if you fail to repay the loan. A mortgage loan typically involves borrowing a substantial sum of money, often repaid over a lengthy period, such as 15, 20, or 30 years. The borrower makes regular, typically monthly, payments to the lender. Each payment covers part of the loan's principal, which is the amount initially borrowed, and part of the interest, which is the fee paid to the lender for the privilege of borrowing the money.
Understanding mortgage loans involve:

• The principal: the original loan amount.
• Interest rate: the percentage charged on the remaining balance.
• Term length: the time over which the loan is to be paid.
• Monthly payments: consist of both principal and interest.

The goal is to fully pay off the loan by the end of its term. For example, in a 30-year mortgage, payments are calculated to ensure that the full amount will be paid off in 30 years.
In the context of our problem, the loan amount is $150,000 with an 8% annual interest rate, and the monthly payment, denoted as \( P \), is calculated for the loan to be fully paid off over 30 years.

Interest Calculation

Interest calculation in the context of mortgage loans helps determine how much you will pay over the life of the loan. In our problem, interest is calculated annually at a rate of 8%, applied to the balance of the loan. The differential equation given, \( A'(t) = 0.08 A(t) - 12P \), helps us model the interest and repayments.
This equation can be understood as:

• \( A(t) \) represents the outstanding balance at time \( t \).
• \( A'(t) \) shows the change in the balance which equals the interest accrued minus the monthly payments.
• The term \( 0.08A(t) \) represents the annual interest as a fraction of the current balance.
• \( 12P \) is the annualized reduction in balance due to monthly payments.

The solution involves ensuring that over 30 years, with the computed \( P \), the initial loan amount is entirely paid off. To achieve this, integrating and solving for \( P \) yields a payment of approximately $1100.65 per month.

Financial Mathematics

Financial mathematics plays a significant role in understanding and planning for complex financial instruments like mortgage loans. This domain involves applying mathematical models and formulas to compute elements such as interest, monthly payments, and the total cost of loans over time.
In the case of our problem, several key mathematical concepts are applied:

• Use of differential equations to model changes in the loan balance over time.
• Integration to find exact values for certain variables, such as the monthly payment \( P \) that zeroes the loan after 30 years.
• Calculating the total cost paid, which is the product of monthly payments and the total number of payments — giving us \(396,234.
• Determining the interest paid by subtracting the initial loan amount from this total cost, which results in \)246,234 in interest over the life of the loan.

These calculations offer a clear view of how loans are structured and allow consumers to plan effectively for their financial commitments.