To determine how much Derek is willing to pay for the money machine today, we need to calculate the present value (PV) of the future cash flow of $47,368.00 that he will receive in 13 years. The present value can be calculated using the formula:
\[ PV = \frac{FV}{(1 + r)^n} \]
Where:
- \( PV \) = Present Value
- \( FV \) = Future Value (the amount he will receive in the future)
- \( r \) = Discount rate (as a decimal)
- \( n \) = Number of years until the payment is received
In Derek's case:
- \( FV = 47,368.00 \)
- \( r = 0.10 \) (10% as a decimal)
- \( n = 13 \)
Now, substituting the values into the formula:
\[ PV = \frac{47,368}{(1 + 0.10)^{13}} \] \[ PV = \frac{47,368}{(1.10)^{13}} \] First, calculate \( (1.10)^{13} \):
\[ (1.10)^{13} \approx 4.439 \]
Now, substituting that back into the present value formula:
\[ PV = \frac{47,368}{4.439} \approx 10,654.69 \]
Therefore, Derek is willing to pay approximately $10,654.69 today for the money machine.