You want to buy a new sports coupe for $76,500, and the finance office at the dealership has quoted you an APR of 5.8 percent for a 72-month loan to buy the car.

What will your monthly payments be?

**Note: Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.**

What is the effective annual rate on this loan?

**Note: Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.**


To solve this problem, we'll calculate the following:

1. **Monthly Payment:** Using the formula for a fixed-rate loan:
   \[
   M = \frac{P \times r \times (1 + r)^n}{(1 + r)^n - 1}
   \]
   where:
   - $ M $ is the monthly payment.
   - $ P $ is the principal loan amount ($76,500).
   - $ r $ is the monthly interest rate (APR divided by 12).
   - $ n $ is the total number of payments (loan term in months).

2. **Effective Annual Rate (EAR):** Using the formula:
   \[
   \text{EAR} = \left(1 + \frac{r}{n}\right)^n - 1
   \]
   where:
   - $ r $ is the nominal monthly interest rate.
   - $ n $ is the number of compounding periods per year (12 for monthly).

Let's calculate each of these step by step.

### 1. Monthly Payment Calculation

Given:
- Principal $ P = \$76,500 $
- APR = 5.8%
- Loan term = 72 months

Convert APR to a monthly rate:
\[
r = \frac{5.8\%}{12}
\]

Calculate the monthly payment using the loan payment formula.

### 2. Effective Annual Rate (EAR) Calculation

Using the formula:
\[
\text{EAR} = \left(1 + \frac{r}{12}\right)^{12} - 1
\]

I'll calculate these now.### Results:
1. **Monthly Payment:** $ \$1,260.62 $
2. **Effective Annual Rate (EAR):** $ 5.96\% $

These values represent the monthly payment required for the loan and the effective annual rate, which accounts for the compounding effect of the monthly interest rate.

Do you need further details or have any other questions?

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Here are 5 related questions you might consider:
1. How would the monthly payment change if the loan term was 60 months instead of 72 months?
2. What would be the total interest paid over the life of the loan?
3. How would an increase in the APR affect the monthly payment and EAR?
4. What is the impact on the EAR if the loan is compounded daily instead of monthly?
5. How does the principal amount affect the monthly payment for different APR values?

**Tip:** Always calculate the EAR when comparing loans, as it gives a better understanding of the actual cost of borrowing over time.

Learn how to calculate monthly payments and the effective annual rate (EAR) for a $76,500 car loan with a 5.8% APR over 72 months. Understand the financial implications and compare loan options. Suitable for adults looking to understand car financing.

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