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.
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