Sure, let's compute the Net Present Value (NPV) of Project X step-by-step and determine whether to accept or reject the project given a 12 percent cost of capital.
### Step-by-Step Calculation:
1. Given Data:
- Cash Flows: [tex]\([-120, -120, 0, 190, 165, 140]\)[/tex]
- Cost of Capital ([tex]\(r\)[/tex]) = 12% or 0.12
2. NPV Formula:
[tex]\[
NPV = \sum_{t=0}^{n} \frac{CF_t}{(1 + r)^t}
\][/tex]
where [tex]\(CF_t\)[/tex] represents the cash flow at time [tex]\(t\)[/tex], [tex]\(r\)[/tex] is the discount rate, and [tex]\(t\)[/tex] is the time period.
3. Calculate the Present Value of Each Cash Flow:
- For t = 0:
[tex]\[
\frac{-120}{(1 + 0.12)^0} = -120
\][/tex]
- For t = 1:
[tex]\[
\frac{-120}{(1 + 0.12)^1} = \frac{-120}{1.12} \approx -107.14
\][/tex]
- For t = 2:
[tex]\[
\frac{0}{(1 + 0.12)^2} = 0
\][/tex]
- For t = 3:
[tex]\[
\frac{190}{(1 + 0.12)^3} = \frac{190}{1.404928} \approx 135.30
\][/tex]
- For t = 4:
[tex]\[
\frac{165}{(1 + 0.12)^4} = \frac{165}{1.5748016} \approx 104.79
\][/tex]
- For t = 5:
[tex]\[
\frac{140}{(1 + 0.12)^5} = \frac{140}{1.7623411} \approx 79.44
\][/tex]
4. Sum these Present Values:
[tex]\[
NPV = -120 - 107.14 + 0 + 135.30 + 104.79 + 79.44 \approx 92.40
\][/tex]
5. Decision Rule:
- If NPV > 0: Accept the project.
- If NPV < 0: Reject the project.
6. Conclusion:
- NPV [tex]\( \approx 92.40 \)[/tex]
- Since the NPV is positive, we accept Project X.
Therefore, the NPV of Project X is approximately 92.40, and based on this calculation, we should accept the project.
