To determine the maximum cash back offer for Derek, we first need to calculate the present value of the 50-month cash flow from the dealership with zero percent financing.
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Monthly Payment Calculation: Since the dealership offers zero percent financing for 50 months on a $30,138 car, Derek will pay the total price in 50 equal monthly payments.
\[ \text{Monthly Payment} = \frac{30,138.00}{50} = 602.76 \]
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Present Value Calculation: Next, we find the present value of these monthly payments. The present value \( PV \) of an annuity can be calculated using the following formula:
\[ PV = PMT \times \left(1 - (1 + r)^{-n}\right) / r \]
where:
- \( PMT \) = monthly payment ($602.76)
- \( r \) = monthly interest rate (annual rate divided by 12 months)
- \( n \) = number of months (50)
The bank’s annual interest rate is 4.63%, so the monthly interest rate \( r \) is:
\[ r = \frac{4.63%}{12} = \frac{0.0463}{12} \approx 0.0038583 \]
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Calculating the Present Value:
Now we can substitute the values into the formula:
\[ PV = 602.76 \times \left(1 - (1 + 0.0038583)^{-50}\right) / 0.0038583 \]
First, calculate \( (1 + 0.0038583)^{-50} \):
\[ (1 + 0.0038583)^{-50} \approx (1.0038583)^{-50} \approx 0.826435 \]
Now, substitute back into the present value formula:
\[ PV = 602.76 \times \left(1 - 0.826435\right) / 0.0038583 \] \[ PV = 602.76 \times \left(0.173565\right) / 0.0038583 \] \[ PV = 602.76 \times 45.057528 \] \[ PV \approx 27,116.53 \]
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Cash Back Calculation:
The maximum cash back Derek would be willing to accept is the difference between the total price of the car and the present value of the financing offer:
\[ \text{Cash Back} = \text{Price of Car} - PV \] \[ \text{Cash Back} = 30,138.00 - 27,116.53 \approx 3,021.47 \]
Thus, the dealership would need to offer Derek approximately $3,021.47 cash back for him to consider paying for the car in full today.