You want to buy a new sports coupe for $74,800, and the finance office at the dealership has...

Question:

You want to buy a new sports coupe for $74,800, and the finance office at the dealership has quoted you a loan with an APR of 7.2% for 48 months to buy the car. What will your monthly payments be? What is the effective annual rate on this loan?

Monthly Loan Payment:

Many loans are structured to be repaid with monthly payments. That means interest is calculated monthly on both the outstanding principal and the accrued interest. Therefore, the effective annual rate on the loan is higher than the stated annual percentage rate.

Answer and Explanation:

The monthly payment is $1798.13. The effective annual rate is 7.44%.

We can use the following formula to compute the monthly payment for a loan with principal {eq}P {/eq}, monthly interest rate {eq}r{/eq} and number of monthly payments {eq}T{/eq}:

  • {eq}\displaystyle \frac{Pr}{1 - (1 + r)^{-T}} {/eq}

In this question, the loan amount is 74,800, the APR is 7.2%, which implies an effective monthly rate of 7.2% / 12 = 0.6%. The term on the loan is 48 months. Applying the formula, the monthly payment is:

  • {eq}\displaystyle \frac{74,800*0.6\%}{1 - (1 + 0.6\%)^{-48}} = 1,798.13 {/eq}

We can use the following formula to compute the effective annual rate:

  • {eq}(1 + \text{APR} / T)^T - 1 {/eq}

where {eq}T{/eq} is the number of times interest compounds in a year.

Applying the formula, the effective annual rate is:

  • {eq}(1 + 0.6\%)^{12} - 1 = 7.44\% {/eq}

Learn more about this topic:

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Calculating Monthly Loan Payments

from Remedial Algebra I

Chapter 25 / Lesson 8
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