Let's break down the financing options for both banks and then calculate the one-time payment option.
Bank #1 (BB&T)
- Loan Amount: $30,250
- Interest Rate: 12%
- Loan Term: 4 years (48 months)
Monthly Payment Calculation: The formula to calculate monthly payments on an installment loan is:
\[ M = P \frac{r(1+r)^n}{(1+r)^n - 1} \]
Where:
- \( M \) = total monthly payment
- \( P \) = loan principal (amount borrowed) = $30,250
- \( r \) = monthly interest rate (annual interest rate / 12) = 12% / 12 = 1% = 0.01
- \( n \) = number of payments (loan term in months) = 4 years * 12 months/year = 48 months
Substituting the values into the formula:
\[ M = 30250 \frac{0.01(1+0.01)^{48}}{(1+0.01)^{48} - 1} \]
Calculating \( (1 + 0.01)^{48} \):
\[ (1.01)^{48} \approx 1.608 \]
Now substituting back into the formula:
\[ M = 30250 \frac{0.01 \times 1.608}{1.608 - 1} = 30250 \frac{0.01608}{0.608} = 30250 \times 0.0264 \approx 798.60 \]
Total Cost of the Car with BB&T: Total Cost = Monthly Payment × Number of Payments
\[ Total\ Cost = 798.60 \times 48 \approx 38,188.80 \]
Bank #2 (Sunflower)
- Loan Amount: $30,250
- Interest Rate: 9%
- Loan Term: 6 years (72 months)
Monthly Payment Calculation:
Using the same formula above, with:
- \( r = 9% / 12 = 0.0075 \)
- \( n = 6 \times 12 = 72 \)
\[ M = 30250 \frac{0.0075(1+0.0075)^{72}}{(1+0.0075)^{72} - 1} \]
Calculating \( (1 + 0.0075)^{72} \):
\[ (1.0075)^{72} \approx 1.747 \]
Substituting back into the formula:
\[ M = 30250 \frac{0.0075 \times 1.747}{1.747 - 1} = 30250 \frac{0.0131}{0.747} \approx 30250 \times 0.0175 \approx 529.38 \]
Total Cost of the Car with Sunflower: \[ Total\ Cost = 529.38 \times 72 \approx 38,116.36 \]
Comparison
-
Monthly Payment for BB&T: $798.60
-
Total Cost for BB&T: $38,188.80
-
Monthly Payment for Sunflower: $529.38
-
Total Cost for Sunflower: $38,116.36
The bank with the better financing option is Sunflower as it has a lower total cost.
One-Time Payment Calculation
- Cost of the Car: $30,250
- TN Sales Tax Rate: Let's assume a sales tax of 7% (this may vary, check local regulations).
\[ Sales\ Tax = 30,250 \times 0.07 = 2,117.50 \]
Total one-time payment:
\[ Total\ Cost\ at\ Checkout = $30,250 + Sales\ Tax = 30,250 + 2,117.50 = 32,367.50 \]
Savings Calculation
Cost of financing with the better bank (Sunflower):
\[ Total\ Cost = 38,116.36 \]
Cost of paying up front:
\[ Total\ Cost\ upfront = 32,367.50 \]
Amount Saved by Paying Up Front:
\[ Savings = Total Cost of Financing - Total Cost upfront \] \[ Savings = 38,116.36 - 32,367.50 \approx 5,748.86 \]
Summary:
- Best Option: Bank #2 (Sunflower)
- Upfront Total Cost: $32,367.50
- Savings by paying upfront: $5,748.86
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