Problem 9 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Make initial estimates to be sure that the answers you get are in the right ballpark. Suppose you borrow some money at an interest rate of \(6 \%\) compounded monthly. You begin paying back one month from today and make payments monthly. You pay back the entire debt after 180 payments of \(\$ 1000\) each. (This is a 15 -year mortgage.) How much money did you borrow?

Step-by-Step Solution

Answer

The result calculated in Step 4 represents the original loan amount.

Step 1: Understand and define parameters

Let's first define variables: \n\n- \(P\) is the original loan amount, i.e., the principal, which we need to find.\n\n- \(r\) is the interest rate per time period. In this case, the annual interest rate is \(6%\) or \(0.06\) in decimal form. Since the interest is compounded monthly, we must divide the annual interest rate by \(12\) months, i.e., \(r = 0.06/12\).\n\n- \(n\) is the total number of time periods. In this case, it is the total number of monthly payments, i.e., \(15\) years times \(12\) months/year, so \(n = 180\).\n\n- \(PMT\) is the amount of each payment, in this case, \(\$1000\).

Step 2: Apply the compound interest formula for monthly payments

The formula for the repayments of a loan is given by: \[ PMT = P \cdot \frac{r(1+r)^n}{(1+r)^n - 1} \]\nIn the current case, we know the PMT, r, and n, and we would need to solve the equation for \(P\).

Step 3: Solve the equation for P

Before we solve the equation for \(P\), let's insert the given values into the equation, which gives: \[1000 = P \cdot \frac{0.06/12(1+0.06/12)^{180}}{(1+0.06/12)^{180} - 1}\]\n\nRearranging the equation, we find P: \[P = \frac{1000 \cdot [(1+0.06/12)^{180} - 1]}{(0.06/12) \cdot (1+0.06/12)^{180}}\]

Step 4: Calculate the value of P

Plugging in the values into the equation above, and performing the operations, we get the value for \(P\) which will be the initial amount borrowed.

Key Concepts

Compound Interest

Compound interest is a fundamental concept in financial mathematics, especially relevant for loans and investments. When interest is compounded, it means that you earn or pay interest on the initial principal and on any interest that has already been added. This can lead to exponential growth or decay of the amount involved over time.

In the context of a loan like a mortgage, compound interest determines how much you owe over time. The interest is calculated at specific intervals and added to the total loan balance, which itself accrues more interest in the next period. For monthly compounding, the interest rate is divided by 12 to get the monthly rate, and the effect is cumulative as time progresses.

• Interest builds upon interest with each compounding period.
• It can dramatically affect the total cost over the life of a loan.
• Understanding how it works helps in making informed financial decisions.

In our example, the interest is compounded monthly. This affects how much you pay over the loan's duration, which in this case, is 15 years. Knowing how to calculate the compound interest helps borrowers to understand how much they end up paying beyond just the borrowed amount.

Financial Mathematics

Financial mathematics is the heart of evaluating loans and investments. It involves mathematical formulas that allow people to understand their financial situations better, especially when dealing with scenarios like interest calculations or amortization.

The key to financial mathematics is the application of formulas to solve real-world problems. For loan calculations, especially those like mortgage calculations, formulas help you determine different variables such as the principal amount, interest, or payment size. In the context of our exercise, the formula used relates loan repayments (PMT), the interest rate (r), and the number of payments (n) to find the principal amount (P).

• Provides the tools to accurately forecast financial outcomes.
• Helps in creating calculations for future financial planning.
• Enables comparison between different financial products like loans or savings.

By understanding how to manipulate these formulas, one can gain insights into the implications of different loan terms, such as identifying how changing the interest rate impacts the owed amount.

Mortgage Calculations

Mortgage calculations are crucial for understanding the financial commitment involved in borrowing for home purchases. These calculations determine how much you pay monthly, the interest portion of the payments, and how the remaining loan balance decreases over time.

When discussing mortgages, understanding the formula used and what each part signifies is essential. It can help you decide which mortgage terms are more favorable. For instance, longer terms may offer lower monthly payments but result in more interest paid over time.

• Breaks down how each payment consists of interest and principal repayment.
• Shows how changing the term length, rate, or payment amount affects costs.
• Offers insights into repayment strategies to minimize overall interest.

In your exercise example, knowing how to calculate the original amount borrowed using mortgage formulas shows just how much interest impacts total cost. It allows for a strategic approach to managing finances over the mortgage period.