To find the Net Present Value (NPV) of the J-Mix 2000 machine, we need to calculate the present value of the future cash flows and then subtract the initial investment.
Given:
- Cost of the machine = $1,950,000
- Incremental cash flows per year = $657,598
- Number of years = 5
- Cost of capital (discount rate) = 10.95%
Step 1: Calculate the Present Value (PV) of the cash flows.
The formula for the present value of an annuity is:
\[ PV = C \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \]
Where:
- \( C \) is the cash flow per period ($657,598),
- \( r \) is the discount rate (10.95% or 0.1095),
- \( n \) is the number of periods (5 years).
Now, we can plug in the values:
\[ PV = 657,598 \times \left( \frac{1 - (1 + 0.1095)^{-5}}{0.1095} \right) \]
Calculating the term \((1 + 0.1095)^{-5}\):
\[ 1 + 0.1095 = 1.1095 \] \[ (1.1095)^{-5} \approx 0.5907 \]
Now substituting back into the formula:
\[ PV = 657,598 \times \left( \frac{1 - 0.5907}{0.1095} \right) \] \[ PV = 657,598 \times \left( \frac{0.4093}{0.1095} \right) \] \[ PV = 657,598 \times 3.7374 \approx 2,457,330.80 \]
Step 2: Calculate the NPV.
Now that we have the present value of cash flows, we can calculate the NPV:
\[ NPV = PV - \text{Initial Investment} \] \[ NPV = 2,457,330.80 - 1,950,000 = 507,330.80 \]
Conclusion: The net present value (NPV) of the J-Mix 2000 is approximately $507,330.80.
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