To calculate the monthly payments and total costs of financing each vehicle through the two banks, we'll use the following formula for a fixed-rate loan:
\[ PMT = P \times \frac{r(1 + r)^n}{(1 + r)^n - 1} \]
Where:
- \( PMT \) = monthly payment
- \( P \) = principal (loan amount)
- \( r \) = monthly interest rate (annual interest rate / 12)
- \( n \) = total number of payments (loan term in months)
We'll also add the Tennessee sales tax (7%) to the total cost of the vehicle for the up-front payment calculations.
1. 2020 Hyundai Kona
- Price: $20,200
- Sales Tax: 7% of $20,200 = $1,414
- Total Cost: $20,200 + $1,414 = $21,614
Bank #1: BB&T (12% for 4 years)
- Principal (P): $20,200
- Annual Interest Rate: 12%
- Monthly Interest Rate (r): \( \frac{12}{100} \div 12 = 0.01 \)
- Loan Term (n): 4 years = 48 months
Using the formula: \[ PMT_{BB&T} = 20200 \times \frac{0.01(1 + 0.01)^{48}}{(1 + 0.01)^{48} - 1} \approx 20200 \times 0.027778 \approx 560.00 \]
- Total Payment: \[ Total_{BB&T} = PMT_{BB&T} \times n = 560.00 \times 48 = 26,880.00 \]
Bank #2: Sunflower (9% for 6 years)
- Principal (P): $20,200
- Annual Interest Rate: 9%
- Monthly Interest Rate (r): \( \frac{9}{100} \div 12 = 0.0075 \)
- Loan Term (n): 6 years = 72 months
Using the formula: \[ PMT_{Sunflower} = 20200 \times \frac{0.0075(1 + 0.0075)^{72}}{(1 + 0.0075)^{72} - 1} \approx 20200 \times 0.0171 \approx 345.96 \]
- Total Payment: \[ Total_{Sunflower} = PMT_{Sunflower} \times n = 345.96 \times 72 \approx 24,157.00 \]
Comparison for 2020 Hyundai Kona
- Monthly Payment BB&T: $560.00
- Total Cost BB&T: $26,880.00
- Monthly Payment Sunflower: $345.96
- Total Cost Sunflower: $24,157.00
- Best Option: Bank #2 Sunflower
Upfront Savings Calculation: Savings = Total Cost of the better bank - Total Cost of upfront payment \[ Savings = 24157.00 - 21614.00 = 2,543.00 \]
2. 2021 Dodge Charger
- Price: $30,250
- Sales Tax: 7% of $30,250 = $2,118
- Total Cost: $30,250 + $2,118 = $32,368
Bank #1: BB&T (12% for 4 years)
- Principal (P): $30,250
- Monthly Interest Rate (r): 0.01
- Loan Term (n): 48 months
Using the formula: \[ PMT_{BB&T} = 30250 \times \frac{0.01(1 + 0.01)^{48}}{(1 + 0.01)^{48} - 1} \approx 30250 \times 0.027778 \approx 840.00 \]
- Total Payment: \[ Total_{BB&T} = 840.00 \times 48 = 40,320.00 \]
Bank #2: Sunflower (9% for 6 years)
- Principal (P): $30,250
- Monthly Interest Rate (r): 0.0075
- Loan Term (n): 72 months
Using the formula: \[ PMT_{Sunflower} = 30250 \times \frac{0.0075(1 + 0.0075)^{72}}{(1 + 0.0075)^{72} - 1} \approx 30250 \times 0.0171 \approx 517.00 \]
- Total Payment: \[ Total_{Sunflower} = 517.00 \times 72 \approx 37,224.00 \]
Comparison for 2021 Dodge Charger
- Monthly Payment BB&T: $840.00
- Total Cost BB&T: $40,320.00
- Monthly Payment Sunflower: $517.00
- Total Cost Sunflower: $37,224.00
- Best Option: Bank #2 Sunflower
Upfront Savings Calculation: Savings = Total Cost of the better bank - Total Cost of upfront payment \[ Savings = 37224 - 32368 = 4,856.00 \]
3. 2021 Ford Focus
- Price: $24,535
- Sales Tax: 7% of $24,535 = $1,717.45
- Total Cost: $24,535 + $1,717.45 = $26,252.45
Bank #1: BB&T (12% for 4 years)
- Principal (P): $24,535
- Monthly Interest Rate (r): 0.01
- Loan Term (n): 48 months
Using the formula: \[ PMT_{BB&T} = 24535 \times \frac{0.01(1 + 0.01)^{48}}{(1 + 0.01)^{48} - 1} \approx 24535 \times 0.027778 \approx 681.00 \]
- Total Payment: \[ Total_{BB&T} = 681.00 \times 48 = 32,688.00 \]
Bank #2: Sunflower (9% for 6 years)
- Principal (P): $24,535
- Monthly Interest Rate (r): 0.0075
- Loan Term (n): 72 months
Using the formula: \[ PMT_{Sunflower} = 24535 \times \frac{0.0075(1 + 0.0075)^{72}}{(1 + 0.0075)^{72} - 1} \approx 24535 \times 0.0171 \approx 419.00 \]
- Total Payment: \[ Total_{Sunflower} = 419.00 \times 72 \approx 30,168.00 \]
Comparison for 2021 Ford Focus
- Monthly Payment BB&T: $681.00
- Total Cost BB&T: $32,688.00
- Monthly Payment Sunflower: $419.00
- Total Cost Sunflower: $30,168.00
- Best Option: Bank #2 Sunflower
Upfront Savings Calculation: Savings = Total Cost of the better bank - Total Cost of upfront payment \[ Savings = 30168 - 26252.45 = 3,915.55 \]
Summary of Results
- 2020 Hyundai Kona:
- Best Bank: Sunflower
- Upfront Savings: $2,543.00
- 2021 Dodge Charger:
- Best Bank: Sunflower
- Upfront Savings: $4,856.00
- 2021 Ford Focus:
- Best Bank: Sunflower
- Upfront Savings: $3,915.55
In every case, Bank #2 (Sunflower) offers more favorable terms than Bank #1 (BB&T).
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