On the next page you will take out a loan from a bank and compute the monthly payment & total amount paid for the vehicle over the life of the loan. You will also compute the price of the car if you choose not to take out a loan.

Question

2020 Hyundai Kona
$20,200
Vin #: 3K44DM19XS
Odometer Reading: 2,200 miles Color: Blue

PMT =
Bank Information:
Bank #1: BB&T: 12% for 4 years
Bank #2: Sunflower: 9% for 6 years

1.) Vehicle chosen:
Financing Option: Calculate financing for both banks. Work must be shown to receive credit.
Bank #1 (BB&T)
vs. Bank #2 (Sunflower)
Monthly Payment
Total Cost of the Car
Monthly Payment
Total Cost of the Car
Which bank is the best option? Writing a Check Option
Determine how much it would cost to make a one time payment for the vehicle at checkout. Work must be shown to receive credit (include the cost of the car and the TN sales tax but do not include any other fees or discounts).
Total Cost of Car:
How much money would you save paying up front versus financing with the better bank?

Answer 1

To calculate the monthly payments and total cost of financing through both banks, we can use the formula for calculating the monthly payment on an installment loan. The formula is:

\[ PMT = \frac{P \cdot r(1 + r)^n}{(1 + r)^n - 1} \]

Where:

• \(PMT\) = monthly payment
• \(P\) = principal amount (loan amount)
• \(r\) = monthly interest rate (annual interest rate / 12)
• \(n\) = total number of payments (loan term in months)

Step 1: Calculate the monthly payments for both banks.

Bank #1 (BB&T):

• Principal (P) = $20,200
• Annual Interest Rate = 12%
• Monthly Interest Rate (r) = 12% / 12 = 1% = 0.01
• Loan Term = 4 years = 4 * 12 = 48 months (n)

\[ PMT = \frac{20200 \cdot 0.01(1 + 0.01)^{48}}{(1 + 0.01)^{48} - 1} \]

First, calculate \( (1 + 0.01)^{48} \): \[ (1.01)^{48} \approx 1.60844 \]

Now substitute into the PMT formula: \[ PMT = \frac{20200 \cdot 0.01 \cdot 1.60844}{1.60844 - 1} \] \[ PMT = \frac{20200 \cdot 0.0160844}{0.60844} \approx \frac{324.08688}{0.60844} \approx 532.31 \]

Total Cost for Bank #1:

Total Cost = Monthly Payment * Total Number of Payments \[ Total\ Cost = 532.31 \times 48 \approx 25,565.68 \]

Bank #2 (Sunflower):

• Principal (P) = $20,200
• Annual Interest Rate = 9%
• Monthly Interest Rate (r) = 9% / 12 = 0.75% = 0.0075
• Loan Term = 6 years = 6 * 12 = 72 months (n)

\[ PMT = \frac{20200 \cdot 0.0075(1 + 0.0075)^{72}}{(1 + 0.0075)^{72} - 1} \]

First, calculate \( (1 + 0.0075)^{72} \): \[ (1.0075)^{72} \approx 1.747422 \]

Now substitute into the PMT formula: \[ PMT = \frac{20200 \cdot 0.0075 \cdot 1.747422}{1.747422 - 1} \] \[ PMT = \frac{20200 \cdot 0.013105665}{0.747422} \approx \frac{264.752053}{0.747422} \approx 354.07 \]

Total Cost for Bank #2:

Total Cost = Monthly Payment * Total Number of Payments \[ Total\ Cost = 354.07 \times 72 \approx 25,483.04 \]

Summary of Results

| Bank | Monthly Payment | Total Cost of the Car | |----------------|------------------|------------------------| | Bank #1 (BB&T) | $532.31 | $25,565.68 | | Bank #2 (Sunflower) | $354.07 | $25,483.04 |

Best Option

Bank #2 (Sunflower) is the better option as it has both a lower monthly payment and a lower total cost.

Step 2: Determine the cost of paying in full

The total cost of the vehicle, including sales tax.

Assuming a TN state sales tax rate of 7%. \[ \text{Sales Tax} = \text{Car Price} \times \text{Tax Rate} = 20200 \times 0.07 = 1414 \]

Total cost of the car when paying upfront: \[ \text{Total Cost} = \text{Price} + \text{Sales Tax} = 20200 + 1414 = 21614 \]

Step 3: Calculate Savings

Calculate the savings by paying upfront versus financing with the better bank (Bank #2).

\[ \text{Total Cost of Financing} - \text{Total Cost When Paid Upfront} = 25483.04 - 21614 = 3879.04 \]

Conclusion

1. Total Cost of Car with Bank #1: $25,565.68
2. Total Cost of Car with Bank #2: $25,483.04
3. Total Cost of Car if paid upfront: $21,614
4. Savings by paying upfront versus financing with the better bank (Bank #2): $3,879.04