You just bought equipment from Blue Valley Research. You will receive a cash rebate of $1,000 from Blue Valley Research in 5 years, pay $26,200 to Blue Valley Research in 7 years, receive a cash rebate of $4,500 from Blue Valley Research in 12 years, and pay $1,400 to Blue Valley Research in 18 years. The discount rate is 16.78 percent per year. What is the present value of the cash flows associated with this transaction? Note: the correct answer is less than zero.

To find the present value (PV) of the cash flows, we need to discount each cash flow back to the present using the formula for the present value of a future cash flow:

\[
PV = \frac{CF}{(1 + r)^t}
\]

Where:
- $ CF $ is the future cash flow,
- $ r $ is the discount rate (16.78% or 0.1678 as a decimal),
- $ t $ is the time in years when the cash flow occurs.

The given cash flows are:
- Receive $1,000 in 5 years,
- Pay $26,200 in 7 years,
- Receive $4,500 in 12 years,
- Pay $1,400 in 18 years.

We'll calculate the present value of each cash flow individually:

1. **Cash flow in 5 years (Receive $1,000):**
\[
PV_1 = \frac{1000}{(1 + 0.1678)^5}
\]

2. **Cash flow in 7 years (Pay $26,200):**
\[
PV_2 = \frac{-26200}{(1 + 0.1678)^7}
\]

3. **Cash flow in 12 years (Receive $4,500):**
\[
PV_3 = \frac{4500}{(1 + 0.1678)^12}
\]

4. **Cash flow in 18 years (Pay $1,400):**
\[
PV_4 = \frac{-1400}{(1 + 0.1678)^18}
\]

The total present value will be the sum of all these present values:

\[
PV_{\text{total}} = PV_1 + PV_2 + PV_3 + PV_4
\]

Let's calculate this.The present value of the cash flows associated with this transaction is approximately $-7,771.36$. This means the present value is less than zero, which indicates an overall net cost when accounting for the time value of money.

Would you like to explore more details or have any questions about the process? Here are five related questions for further understanding:

1. How does a discount rate affect the present value of future cash flows?
2. What does it mean if the present value is negative?
3. How would a higher or lower discount rate change the present value?
4. Can you explain how future payments and receipts are valued differently at present?
5. What are some real-life applications of discounting cash flows?

**Tip:** A higher discount rate decreases the present value of future cash flows, reflecting a higher opportunity cost of capital.

Learn how to calculate the present value of multiple cash flows with a discount rate of 16.78% over different time periods (5, 7, 12, and 18 years). This finance problem illustrates the time value of money concept and the formula PV = CF / (1 + r)^t, applied to real-world cash flows.

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