To determine the net present value (NPV) of a project with given cash flows and a specified discount rate, follow these steps:
1. Identify the cash flows by year:
[tex]\[
\begin{aligned}
& \text{Year 0: } -27,500 \\
& \text{Year 1: } 14,800 \\
& \text{Year 2: } 19,400 \\
& \text{Year 3: } 5,200 \\
\end{aligned}
\][/tex]
2. Determine the discount rate:
[tex]\[
r = 12\% \text{ or } 0.12
\][/tex]
3. Calculate the present value (PV) of each cash flow:
[tex]\[
\text{PV of a cash flow in year t} = \frac{\text{Cash Flow}_t}{(1 + r)^t}
\][/tex]
4. Compute the present value for each year:
- Year 0 (Initial Investment, t = 0):
[tex]\[
\text{PV}_0 = \frac{-27,500}{(1 + 0.12)^0} = -27,500
\][/tex]
The PV for year 0 remains [tex]\(-27,500\)[/tex] because any number to the power of zero is 1.
- Year 1 (t = 1):
[tex]\[
\text{PV}_1 = \frac{14,800}{(1 + 0.12)^1} = \frac{14,800}{1.12} \approx 13,214.29
\][/tex]
- Year 2 (t = 2):
[tex]\[
\text{PV}_2 = \frac{19,400}{(1 + 0.12)^2} = \frac{19,400}{1.2544} \approx 15,465.56
\][/tex]
- Year 3 (t = 3):
[tex]\[
\text{PV}_3 = \frac{5,200}{(1 + 0.12)^3} = \frac{5,200}{1.404928} \approx 3,701.26
\][/tex]
5. Sum the present values to compute the NPV:
[tex]\[
\begin{aligned}
\text{NPV} &= \text{PV}_0 + \text{PV}_1 + \text{PV}_2 + \text{PV}_3 \\
&= -27,500 + 13,214.29 + 15,465.56 + 3,701.26 \\
&= 4,881.10
\end{aligned}
\][/tex]
6. Interpret the NPV result:
Thus, the net present value (NPV) of the project, given the cash flows [tex]\( -27,500 \)[/tex], [tex]\( 14,800 \)[/tex], [tex]\( 19,400 \)[/tex], and [tex]\( 5,200 \)[/tex] at a discount rate of 12%, is approximately \$4,881.10. This positive NPV suggests that the project is expected to add value and therefore may be considered a profitable investment.
