Problem 18 - Algebra and Trigonometry

Question

Interest Rate Janet's payments on her \(\$ 12,500\) car are \(\$ 420\) a month for 3 years. Assuming that interest is compounded monthly, what interest rate is she paying on the car loan?

Step-by-Step Solution

Answer

The interest rate is approximately 16.77% annually, compounded monthly.

Step 1: Understand the Loan Parameters

The principal of the loan is \( P = \\(12,500 \). Janet makes monthly payments of \( M = \\)420 \) for \( n = 3 \times 12 = 36 \) months. We aim to find the monthly interest rate \( r \), such that the future value of the loan equals zero after all payments have been made.

Step 2: Identify the Loan Formula

In this context, we use the formula for the monthly payment of an installment loan, which is \( M = \frac{P \cdot r \cdot (1+r)^n}{(1+r)^n - 1} \), where \( M \) is the monthly payment, \( P \) is the principal, \( r \) is the monthly interest rate, and \( n \) is the number of payments.

Step 3: Rearrange for Interest Rate

To find the interest rate \( r \), we rearrange the loan formula to isolate \( r \). This requires iterative methods such as trial and error or numerical techniques like Newton's method, as there is no algebraic way to solve directly for \( r \).

Step 4: Solve for Monthly Interest Rate

Apply an iterative approach to solve for \( r \). Assume a starting point for \( r \), evaluate if the equality holds, and adjust \( r \) as required. This eventually converges to \( r \approx 0.013 \) or 1.3% monthly.

Step 5: Convert to Annual Interest Rate

The monthly interest rate \( r \approx 0.013 \) corresponds to an annual interest rate compounded monthly of \( R = (1 + r)^{12} - 1 \). Calculate this to find \( R \approx 16.77\% \).

Key Concepts

Compound Interest

Compound interest plays a critical role when it comes to calculating loan repayments. Unlike simple interest, where interest is calculated only on the principal amount, compound interest is accumulated on both the initial principal and the interest that has been added to it over time. In the context of loans, this means that every month the remaining balance includes both the principal and the interest from previous months. Thus, making understanding compound interest crucial as it impacts the total amount owed over time.
With monthly compounding, the interest rate is applied every month, and the formula used is:

• Future Value = Principal \( \times (1 + \text{monthly interest rate})^n \)

where \( n \) is the number of compounding periods. This formula shows how each month, the interest builds upon itself, affecting the amount Janet owes on her car loan.

Monthly Payments

Monthly payments are a fixed amount you pay each month to cover both the principal and the interest. In Janet's case, the monthly payment of \( \$420 \) includes both these components. The calculation of monthly payments in the context of a loan is vital because it helps borrowers understand how much they need to budget regularly.
The formula used for calculating monthly payments is:

• \( M = \frac{P \cdot r \cdot (1+r)^n}{(1+r)^n - 1} \)

where \( M \) is the monthly payment, \( P \) is the principal, \( r \) is the monthly interest rate, and \( n \) is the total number of payments. By knowing the monthly payment, borrowers can effectively plan their finances and ensure they meet their loan obligations.

Iterative Methods

Iterative methods are essential numerical techniques used when a problem cannot be solved analytically. In the case of calculating the interest rate for Janet's car loan, an iterative approach is necessary because direct algebraic methods do not work for isolating the interest rate \( r \) in the monthly payment formula.
Iterative methods like trial and error involve:

• Assuming a starting point for \( r \)
• Checking if the assumed rate satisfies the payment formula
• Adjusting \( r \) and recalculating until convergence is achieved

This gradual approach leads us to the exact interest rate needed for Janet’s payment plan. Iterative methods are practical and often used in real-world scenarios where precision is crucial.

Newton's Method

Newton's Method, also known as the Newton-Raphson method, is a more advanced iterative technique used to find successively better approximations to the roots (or zeroes) of a real-valued function. In our scenario, it helps to solve for the root of the equation derived from the loan payment formula when isolating \( r \).
For Newton’s Method, you would:

• Select an initial guess for the interest rate \( r \)
• Calculate the function \( f(r) \) and its derivative \( f'(r) \)
• Update \( r \) using the formula: \( r_{ ext{new}} = r - \frac{f(r)}{f'(r)} \)
• Repeat the process until \( r \) converges on a stable value

This method provides a systematic way of refining the value. It's notably faster than simple trial and error, especially when a good initial estimate is available. Using Newton's Method, we determined Janet's monthly interest rate to approximately 1.3%, which when compounded monthly yields the annual rate she’s paying.