Question

Derek decides to buy a new car. The dealership offers him a choice of paying $544.00 per month for 5 years (with the first payment due next month) or paying some amount today. He can borrow money from his bank to buy the car. The bank requires a 5.00% interest rate. What is the most that he would be willing to pay today rather than making the payments?

Answer 1

To determine how much Derek should be willing to pay today rather than make the monthly payments of $544.00 for 5 years at an interest rate of 5.00%, we can calculate the present value of an annuity.

The formula for the present value of an annuity is:

\[
PV = P \times \left(1 - (1 + r)^{-n}\right) / r
\]

Where:
- \( PV \) = present value
- \( P \) = payment amount per period
- \( r \) = interest rate per period
- \( n \) = total number of payments

**Step 1: Identify Variables**
- \( P = 544 \)
- Annual interest rate = 5.00%, so monthly interest rate \( r = \frac{5.00\%}{12} = \frac{0.05}{12} \approx 0.0041667 \)
- Number of payments \( n = 5 \times 12 = 60 \) (since the payments are monthly for 5 years)

**Step 2: Calculate Present Value (PV)**

\[
PV = 544 \times \left(1 - (1 + 0.0041667)^{-60}\right) / 0.0041667
\]

**Step 3: Calculate \((1 + r)^{-n}\)**

Calculating \((1 + r)^{-n}\):

\[
(1 + 0.0041667)^{-60} \approx 0.7792
\]

**Step 4: Plug Into the Formula**

Now substituting into the formula:

\[
PV = 544 \times \left(1 - 0.7792\right) / 0.0041667
\]

Calculating:

\[
PV = 544 \times \left(0.2208\right) / 0.0041667
\]

\[
PV \approx 544 \times 52.416 = 28534.784
\]

**Step 5: Conclusion**

Thus, the most Derek would be willing to pay today instead of making the payments is approximately **$28,534.78**.