After deciding to acquire a new car, you realize you can either lease it or purchase it with a four-year loan. The car you want costs $40,000. The dealer has a leasing arrangement where you pay $109 today and $509 per month for the next four years. If you purchase the car, you will pay it off in monthly payments over the next four years. If you believe that you will be able to sell the car for $29,177 in four years, What is the cost (present value) of purchasing the car if the APR is 7%? When finding the present value of the resale price, discount the resale price monthly. 

To find the present value (PV) of purchasing the car, we need to consider both the monthly loan payments and the present value of the car's resale value at the end of four years.

### Step 1: Calculate the monthly loan payment
- **Principal (loan amount):** $40,000
- **APR (Annual Percentage Rate):** 7%
- **Monthly interest rate (i):** $ \frac{7\%}{12} = \frac{0.07}{12} $
- **Loan term (n):** 4 years or $ 4 \times 12 = 48 $ months

The monthly payment for a loan can be calculated using the formula for the annuity:
\[
PMT = \frac{P \times i}{1 - (1 + i)^{-n}}
\]

Where:
- $ PMT $ = monthly payment
- $ P $ = principal amount ($40,000)
- $ i $ = monthly interest rate
- $ n $ = total number of payments

### Step 2: Calculate the present value of the resale price
- **Future value (FV):** $29,177
- We need to discount this amount back to the present value over 48 months using the monthly interest rate.

The formula for the present value is:
\[
PV = \frac{FV}{(1 + i)^n}
\]

### Step 3: Find the total present value of purchasing the car
- The total present value of purchasing the car is the sum of the present value of all monthly payments and the present value of the resale price.

Let's calculate these values now.It seems like I can’t do more advanced data analysis right now. Please try again later. However, I can explain the process of solving this problem so you can follow the steps manually if needed.

Here’s a summary of how to solve the problem:
1. **Calculate the monthly loan payment** using the annuity formula:
   \[
   PMT = \frac{P \times i}{1 - (1 + i)^{-n}}
   \]
   - $ P $ = $40,000 (principal)
   - $ i $ = monthly interest rate ($ \frac{0.07}{12} $)
   - $ n $ = 48 months

2. **Calculate the present value of the resale price**:
   \[
   PV = \frac{FV}{(1 + i)^n}
   \]
   - $ FV $ = $29,177
   - $ i $ = monthly interest rate
   - $ n $ = 48 months

3. **Find the total present value of purchasing the car**:
   - Sum the present value of all monthly payments and subtract the present value of the resale price.

If you have any questions or want to delve into any part of this problem in more detail, please let me know!

Here are 5 related questions you might be interested in:
1. How does the APR impact the total cost of a loan?
2. What is the difference between leasing and purchasing a car in terms of total cost?
3. How does changing the loan term affect the monthly payments and total cost?
4. What is the impact of a higher or lower resale value on the present value of the purchase?
5. How do monthly payments change with varying interest rates?

**Tip:** When calculating present value, always ensure to match the compounding period of the interest rate (monthly, in this case) with the period of the cash flows being discounted.

After deciding to acquire a new car, you realize you can either lease it or purchase it with a four-year loan. The car you want costs $40,000. The dealer has a leasing arrangement where you pay $109 today and $509 per month for the next four years. If you purchase the car, you will pay it off in monthly payments over the next four years. If you believe that you will be able to sell the car for $29,177 in four years, what is the cost (present value) of purchasing the car if the APR is 7%? When finding the present value of the resale price, discount the resale price monthly.

Learn how to calculate the present value of purchasing a car, considering loan payments and resale value, using a 7% APR. This problem demonstrates the time value of money, loan payment formulas, and discounting future resale value. Suitable for advanced high school or undergraduate finance students.

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