To determine how much cash back Derek would need in order to justify paying for the car in full today instead of taking the zero percent financing option, we first need to calculate the present value of the future payments he would make under the financing plan.
Step 1: Calculate the monthly payment
Since the car costs $31,231.00 and the financing term is 56 months with zero percent interest, the monthly payment would be simply the cost of the car divided by the number of months:
\[ \text{Monthly Payment} = \frac{\text{Cost of the Car}}{\text{Number of Months}} = \frac{31,231.00}{56} \approx 557.18 \]
Step 2: Calculate the total amount paid under financing
Now, we calculate the total amount he would pay if he took the financing option:
\[ \text{Total Payment over 56 Months} = \text{Monthly Payment} \times \text{Number of Months} = 557.18 \times 56 = 31,231.00 \]
Step 3: Calculate the present value of these payments
Next, we need to find the present value of these monthly payments at the bank's interest rate of 5.00%. The present value of an annuity formula is used since Derek will be making regular monthly payments:
\[ PV = PMT \times \left(1 - (1 + r)^{-n}\right) / r \]
Where:
- \(PMT\) = monthly payment = 557.18
- \(r\) = monthly interest rate = \(5% / 12 = 0.0041667\)
- \(n\) = total number of payments = 56
Substituting in the values gives:
Calculating \(PV\):
-
Calculate \( (1 + r)^{-n} \): \[ (1 + 0.0041667)^{-56} \approx 0.7867 \]
-
Calculate \( PV \): \[ PV = 557.18 \times \left(1 - 0.7867\right) / 0.0041667 \] \[ = 557.18 \times \frac{0.2133}{0.0041667} \approx 557.18 \times 51.996 \approx 28,914.59 \]
Step 4: Determine the cash back amount
To find out how much cash back Derek would need, we compare the present value of the payments to the total price of the car:
\[ \text{Cash Back Needed} = \text{Cost of the Car} - PV \] \[ \text{Cash Back Needed} = 31,231.00 - 28,914.59 \approx 2,316.41 \]
Conclusion
Derek would need to receive approximately $2,316.41 cash back from the dealership to be indifferent between paying for the car today and taking the zero percent financing option.