Problem 13

Question

The monthly payments, $P$ dollars, on a mortgage in which $A$ dollars were borrowed at an annual interest rate of $r \%$ for $t$ years is given by $P=f(A, r, t)$. Is $f$ an increasing or decreasing function of $A$ ? Of $r$ ? Of $t$ ?

Step-by-Step Solution

Verified

Answer

$ f $ is increasing in $ A $ and $ r $, but decreasing in $ t $.

1Step 1: Understanding the Mortgage Formula

The monthly payment, $ P $, on a mortgage is determined by the formula $ P = f(A, r, t) $. This function depends on three variables: the amount borrowed ($ A $), the annual interest rate ($ r $), and the number of years ($ t $) over which the loan is to be repaid.

2Step 2: Analyzing $ P $ as a Function of $ A $

When the amount borrowed $ A $ increases, the total amount to be repaid also increases. Since the monthly payment $ P $ is directly related to $ A $, $ P $ is an increasing function of $ A $. This means as $ A $ increases, $ P $ also increases.

3Step 3: Analyzing $ P $ as a Function of $ r $

As the interest rate $ r $ increases, the cost of borrowing the same amount $ A $ increases. Hence, the monthly payment $ P $ will also increase. Therefore, $ P $ is an increasing function of $ r $.

4Step 4: Analyzing $ P $ as a Function of $ t $

When the loan term $ t $ increases, the amount borrowed is spread out over a longer period, generally resulting in smaller monthly payments. However, the interest paid over a longer term increases. From the borrower’s perspective, $ P $ generally decreases as $ t $ increases, making $ P $ a decreasing function of $ t $ for fixed $ A $ and $ r $.

Key Concepts

Mortgage Payment FormulaIncreasing and Decreasing FunctionsInterest Rate ImpactLoan Term

Mortgage Payment Formula

The mortgage payment formula is a crucial tool for anyone entering the world of home loans and financing. It serves to calculate the monthly payment, denoted by $ P $, required to pay off a mortgage. This payment depends on several factors, namely the amount borrowed ($ A $), the annual interest rate ($ r $), and the loan term ($ t $), which is the duration over which the loan is to be repaid. The general form of this formula can be written as $ P = f(A, r, t) $.
When applying this formula, it's important to keep these variables in mind:

Amount Borrowed ($ A $): This is the principal amount of the loan. The larger $ A $ is, the higher the monthly payment will generally be.
Interest Rate ($ r $): Expressed as a percentage, this rate determines how much additional money you'll pay back on top of the principal.
Loan Term ($ t $): Measured in years, it's the period over which you agree to repay the loan.

Increasing and Decreasing Functions

In calculus, determining whether a function is increasing or decreasing is vital to understanding how the variables affect outcomes. For the mortgage payment function $ P = f(A, r, t) $, we must understand how changes in $ A $, $ r $, and $ t $ impact $ P $.

Function of $ A $: P is an increasing function of $ A $. This means that as more money is borrowed, the monthly payment increases. This correlation is direct because the principal amount is being amortized over the loan term.
Function of $ r $: Similarly, $ P $ is an increasing function of $ r $. Higher interest rates increase the cost of borrowing and, consequently, the monthly payment.
Function of $ t $: In contrast, $ P $ is a decreasing function of $ t $. Extending the repayment period generally lowers monthly payments, as they are spread out over a longer time. However, this can lead to more total interest paid over time.

Interest Rate Impact

Interest rates play a significant role in calculating mortgage payments. They determine how much you pay on top of your loan's principal. Here's why interest rates are crucial:

Cost of Borrowing: Higher rates mean higher payments because the lender charges more for providing the loan. This effect makes $ P $ an increasing function of $ r $.
Variable Rates: Be aware that some mortgages have variable rates, which can change. This potential variation can lead to increased costs over time.
Long-Term Impact: Even small changes in interest rates can have a significant impact due to the compounding effect over the loan term.

Managing and understanding interest rates can save borrowers a lot of money in the long run. It pays to shop around for the best rate.

Loan Term

The loan term is the period over which the mortgage is repaid. It has a direct impact on the size of the monthly payment and the total interest paid.

Longer Loan Terms: These often come with lower monthly payments because the principal is spread over more months. However, they usually result in higher total interest payments over the life of the loan.
Shorter Loan Terms: These generally lead to higher monthly payments but less interest paid in total. If you can afford higher payments, this option can save money in the long run.

Balancing between a comfortable monthly payment and minimizing total interest paid is key to choosing the right loan term. Calculating different scenarios using the mortgage payment formula can help in making the best decision for your financial situation.

Problem 13

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