Problem 27 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

You take out a loan of \(\$ 3000\) at an interest rate of \(6 \%\) compounded monthly. You start paying back the loan exactly one year later. How much should each payment be if the loan is paid off after 24 equal monthly payments? Give an exact answer and an approximation correct to the nearest penny.

Step-by-Step Solution

Answer

After calculating and rounding up, the monthly payment should be approximately \$.

Step 1: Calculate the Monthly Interest Rate

Divide the annual interest rate of \(6\%\) by \(12\) months to find the monthly interest rate. Therefore, the monthly interest rate (\(i\)) is \(0.06 / 12 = 0.005 \).

Step 2: Determine the Number of Payments

After one year, the payments start and are made for another 2 years, which is 24 months. So, calculate the total number of payments (\(n\)) to be \(1 year = 12 months + 24 months = 36 months\).

Step 3: Calculate the Future Value

The future value \(FV\) of the loan after the first year (12 months) is obtained by compounding the initial principal using the formula: \(FV = P * ((1 + i) ^ n)\), where \(P = \$3000\), \(i = 0.005\) and \(n = 12\). So, \(FV = 3000 * ((1+0.005) ^ {12})\).

Step 4: Calculate the Monthly Payments

Using the future value obtained from step 3, calculate the monthly payments \(PMT\) by using the annuity formula: \(PMT = FV * i / ((1 + i) ^ n - 1)\), where \(FV\) is the future value calculated in Step 3, \(i = 0.005\) and \(n=24\). Calculate \(PMT = FV * 0.005 / ((1+0.005) ^ {24} - 1)\).

Step 5: Round to the Nearest Penny

The last step is to round the annuity payments to the nearest penny. Use standard rules for rounding, where values 5 or above are rounded up and values 4 or below are rounded down.

Key Concepts

Monthly Payments

When thinking about loans, monthly payments become a key element in understanding how much money you need to regularly pay back. In our exercise, a loan of \(\$3000\) has an interest rate compounded every month. This simply means that interest is added to the principal (original loan amount) every month, and the total amount grows slightly each time.

The monthly interest rate in the exercise is given as \(0.5\%\), which was found by dividing the annual interest rate of \(6\%\) by 12 (since interest is compounded monthly). This interest rate makes it crucial for determining the exact payment needed each month to ultimately pay off the loan.

• Monthly Interest Rate: Divide the annual rate by 12.
• Total Payments: Consider the total period over which repayments will be made.
• Consistent Payments: Must be regular and counted as part of a schedule for paying off the loan.

By calculating your monthly payments accurately, you ensure that you have a clear understanding of your financial obligations and can avoid unexpected financial strain.

Future Value Calculation

Future Value (FV) calculations are an essential part of determining how much the loan will grow as it is subject to interest over time. The future value represents the amount of money that a principal balance will grow to after interest is applied.

In this context, you begin by calculating the FV of the loan after a year, because payments start one year after taking it. For this, you use the formula:

\[ FV = P \times (1 + i) ^ n \]

• \(P\) represents the principal, or original amount of the loan, \(\$3000\).
• \(i\) is the monthly interest rate, \(0.005\) in our exercise.
• \(n\) is the number of compounding periods, which is 12 months for one year.

The result gives you the future value after one year, showing the total amount owed before starting your monthly payments. Understanding future value helps plan how to manage loan repayment, by offering a clear picture of financial requirements.

Annuity Formula

The annuity formula is crucial in figuring out your monthly payments for loans like the one in the exercise. Essentially, an annuity is a series of equal payments made at regular intervals, such as the 24 monthly payments needed in our scenario.

The formula to calculate these payments (PMT) based on the future value is:

\[ PMT = \frac{FV \times i}{(1 + i)^n - 1} \]

• \(FV\) is the future value of the loan after initial growth (from the Future Value Calculation).
• \(i\) remains the monthly interest rate, \(0.005\).
• \(n\) is now 24, as it considers the number of months over which the payments are to be spread.

The formula helps determine the exact amount that must be paid each month to ensure that the total loan is covered by the end of the payment period. It's an efficient way to balance loan repayment, interest, and payment stability.