Question
Derek has the opportunity to buy a money machine today. The money machine will pay Derek $20,249.00 exactly 9.00 years from today. Assuming that Derek believes the appropriate discount rate is 10.00%, how much is he willing to pay for this money machine?
Answers
GPT-4o mini
To find out 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 $20,249.00 received in 9 years, discounted at a rate of 10.00%.
The formula for calculating the present value of a future cash flow is:
\[
PV = \frac{FV}{(1 + r)^n}
\]
Where:
- \( PV \) = Present Value
- \( FV \) = Future Value ($20,249.00)
- \( r \) = discount rate (10% or 0.10)
- \( n \) = number of years until the cash flow is received (9 years)
Now, we can plug in the values:
\[
PV = \frac{20249}{(1 + 0.10)^9}
\]
\[
PV = \frac{20249}{(1.10)^9}
\]
First, calculate \( (1.10)^9 \):
\[
(1.10)^9 \approx 2.3579477
\]
Now, substitute this value back into the present value formula:
\[
PV = \frac{20249}{2.3579477} \approx 8585.70
\]
Thus, the present value of the money machine is approximately **$8,585.70**.
Therefore, Derek should be willing to pay about **$8,585.70** for the money machine today.
The formula for calculating the present value of a future cash flow is:
\[
PV = \frac{FV}{(1 + r)^n}
\]
Where:
- \( PV \) = Present Value
- \( FV \) = Future Value ($20,249.00)
- \( r \) = discount rate (10% or 0.10)
- \( n \) = number of years until the cash flow is received (9 years)
Now, we can plug in the values:
\[
PV = \frac{20249}{(1 + 0.10)^9}
\]
\[
PV = \frac{20249}{(1.10)^9}
\]
First, calculate \( (1.10)^9 \):
\[
(1.10)^9 \approx 2.3579477
\]
Now, substitute this value back into the present value formula:
\[
PV = \frac{20249}{2.3579477} \approx 8585.70
\]
Thus, the present value of the money machine is approximately **$8,585.70**.
Therefore, Derek should be willing to pay about **$8,585.70** for the money machine today.