Problem 10 - Applied Calculus

Question

The monthly mortgage payment in dollars, \(P\), for a house is a function of three variables: $$ P=f(\boldsymbol{A}, \boldsymbol{r}, \boldsymbol{N}) $$ where \(A\) is the amount borrowed in dollars, \(r\) is the interest rate, and \(N\) is the number of years before the mortgage is paid off. (a) \(f(92000,14,30)=1090.08 .\) What does this tell you, in financial terms? (b) \(\left.\frac{\partial P}{\partial r}\right|_{(92000,14,30)}=72.82 .\) What is the financial significance of the number \(72.82 ?\) (c) Would you expect \(\partial P / \partial A\) to be positive or negative? Why? (d) Would you expect \(\partial P / \partial N\) to be positive or negative? Why?

Step-by-Step Solution

Answer

(a) $1090.08/month mortgage; (b) $72.82 increase per 1% interest; (c) Positive; (d) Negative.

Step 1: Understanding Part (a)

Part (a) asks for the interpretation of the function value \( f(92000,14,30) = 1090.08 \). This function value represents the monthly mortgage payment, \( P \), when \(92,000 is borrowed, the interest rate is 14% per annum, and the loan term is 30 years. Financially, this means that under these conditions, the homeowner would pay \)1,090.08 each month to fully repay the mortgage over 30 years.

Step 2: Analyzing Part (b)

Part (b) provides \( \left.\frac{\partial P}{\partial r}\right|_{(92000,14,30)} = 72.82 \). This partial derivative represents the rate at which the monthly payment \( P \) changes with respect to the interest rate \( r \) while keeping the other variables constant. Financially, it means that for every 1% increase in the interest rate, the monthly payment will increase by approximately $72.82.

Step 3: Predicting Part (c)

Part (c) asks about the expected sign of \( \frac{\partial P}{\partial A} \). Since the mortgage payment \( P \) generally increases as the amount borrowed \( A \) increases, we expect \( \frac{\partial P}{\partial A} \) to be positive. This is because a larger loan amount typically results in higher monthly payments.

Step 4: Predicting Part (d)

Part (d) inquires about the expected sign of \( \frac{\partial P}{\partial N} \). Normally, as the number of years \( N \) increases, the monthly payment \( P \) decreases since the total amount is spread over more months. Thus, we expect \( \frac{\partial P}{\partial N} \) to be negative.

Key Concepts

Partial Derivative Interpretation

In the context of a mortgage, a partial derivative helps us understand how changes in one of the variables affecting the mortgage payment alter the payment amount, while holding the other variables constant. When we see \( \left.\frac{\partial P}{\partial r}\right|_{(92000,14,30)} = 72.82 \), it indicates how sensitive the monthly payment is concerning the interest rate. In simpler terms, it provides us with the change in the monthly mortgage payment for a tiny change in the interest rate, assuming the loan amount and term remain fixed.

When you look at this derivative, think of it as checking the effect of a very slight increase (or decrease) in the interest rate on the amount you need to pay monthly toward your mortgage. In this specific case, for every additional 1% in the interest rate, your monthly payment would increase by approximately $72.82.

Partial derivatives are essential in financial calculations because they help predict changes in costs or payments when conditions fluctuate. This understanding enables you to make informed decisions about whether to lock in a current rate or how changes might affect future financial commitments.

Interest Rate Impact

Interest rates play a pivotal role in determining the cost of a mortgage. They represent the percentage of the loan charged as interest by the lender. As the interest rate increases, the total interest paid over the loan term also rises.

A higher interest rate implies higher monthly payments since the borrower is not only repaying the principal (amount borrowed) but also a larger amount in interest. This is because interest gets calculated as a percentage of the outstanding mortgage balance and compounds over time.

• An increase in the interest rate increases the cost of borrowing.
• More of each monthly payment goes toward paying interest rather than reducing the principal balance.

Knowing how interest rates impact your mortgage allows you to decide when to buy or refinance a home. Lower rates make borrowing cheaper, potentially saving you thousands over the life of the loan.

Loan Term Effect

The loan term, defined as the period over which the mortgage is to be repaid, significantly affects the monthly payment amount. Typically, a longer loan term implies lower monthly payments, as the principal and interest disperse over more installments.

However, a more extended loan term can lead to paying more interest over the period of the loan, even though each payment might seem more manageable.

• Longer terms lower monthly payments but increase total interest paid.
• Shorter terms increase monthly payments, reducing overall interest but increasing monthly financial pressure.

Deciding on a loan term requires balancing monthly affordability with total loan cost. Generally, if you can afford higher payments, a shorter term can save you money in interest.

Amount Borrowed Influence

The amount borrowed, or the principal, is the foundation of any mortgage. It dictates not only the starting point for calculations of interest but also strongly influences the monthly payment.

As logic dictates, the more you borrow, the higher your monthly payments will be, assuming interest rates and loan terms remain constant. This is because there’s a larger amount to pay off within the same timeframe, and interest is calculated on a larger sum.

• Higher loan amounts lead to higher monthly payments.
• The increase in the amount borrowed translates to increased total interest paid if other factors like interest rate and term remain unchanged.

Understanding how the borrowing amount affects your mortgage helps you evaluate how much home you can realistically afford and informs decisions on potential savings for a larger down payment. This choice can have long-term impacts on financial health and flexibility.