Problem 24 - Calculus and its applications

Question

Maximum loan amount. Curtis plans to purchase a new car. He qualifies for a loan at an annual interest rate of \(7 \%,\) compounded monthly for 5 yr. He is willing to pay up to \(\$ 200\) per month. What is the largest loan he can afford?

Step-by-Step Solution

Answer

Curtis can afford a loan of approximately \$10,102.75.

Step 1: Understand the Formula

The maximum loan amount he can afford relates to a fixed monthly payment on a loan. We use the formula for the present value of an annuity to find the loan amount, which is: \[ P = \frac{PMT \times (1 - (1 + r)^{-n})}{r} \]where \(P\) is the loan principal (amount borrowed), \(PMT\) is the monthly payment, \(r\) is the monthly interest rate, and \(n\) is the total number of payments.

Step 2: Convert Annual Interest Rate to Monthly

First, convert the annual interest rate to a monthly rate by dividing by 12. Given the annual rate is 7%, the monthly interest rate \(r\) is:\[ r = \frac{7\%}{12} \approx 0.5833\% = 0.005833 \]

Step 3: Calculate Total Number of Payments

The loan duration is 5 years, with monthly payments. Calculate the total number of monthly payments \(n\) as:\[ n = 5 \text{ years} \times 12 \text{ months/year} = 60 \text{ payments} \]

Step 4: Set Up the Present Value Formula

Now, substitute the monthly payment \(PMT = \$200\), monthly interest rate \(r = 0.005833\), and total payments \(n = 60\) into the annuity formula to solve for \(P\):\[ P = \frac{200 \times (1 - (1 + 0.005833)^{-60})}{0.005833} \]

Step 5: Solve for the Loan Amount

Calculate the term \((1 + 0.005833)^{-60}\):\[ (1.005833)^{-60} \approx 0.7054 \]Substitute back into the formula to find \(P\):\[ P \approx \frac{200 \times (1 - 0.7054)}{0.005833} \approx \frac{200 \times 0.2946}{0.005833} \approx \$10,102.75 \]

Step 6: Conclusion

The largest loan Curtis can afford, with a monthly payment of \(200, at a 7% annual interest rate compounded monthly for 5 years, is approximately \\)10,102.75.

Key Concepts

Monthly Interest Rate

When dealing with loans, especially those like Curtis's car loan that are compounded monthly, understanding how to calculate the monthly interest rate is essential. The monthly interest rate connects the annual interest rate to the payments that occur each month.

• To convert an annual interest rate to a monthly one, you simply divide the annual rate by 12, since there are twelve months in a year.
• For example, with an annual rate of 7%, the calculation looks like this: \[ r = \frac{7\%}{12} = 0.5833\% \]

This result is then converted to a decimal, which is necessary for mathematical calculations:\[ r = 0.005833 \]
This seemingly small number has a significant impact on the overall loan calculation since it is used in determining other factors, such as the loan principal and the monthly payments.

Loan Principal Calculation

The loan principal is the core amount of money that Curtis will borrow for his car. Calculating the available loan principal relies on the concept of the present value of an annuity.
An annuity in this scenario refers to the regular monthly payments made over the term of the loan. The formula used is: \[ P = \frac{PMT \times (1 - (1 + r)^{-n})}{r} \]
In this equation:

• \( P \) is the loan principal.
• \( PMT \) is the fixed monthly payment, which is $200 in Curtis's case.
• \( r \) represents the monthly interest rate.
• \( n \) denotes the total number of monthly payments over the loan's duration. Since the loan term is 5 years, you calculate \( n \) as:\[ n = 5 \times 12 = 60 \text{ payments} \]

By substituting the values into the formula, Curtis can determine how much he can afford to borrow._This approach ensures that he knows exactly what the present value of his regular future payments is in terms of today's money."},{