Answer :
To determine the cash flows for each year of the project, we need to follow a series of well-defined steps. This involves calculating initial investments, annual free cash flows (FCFs), depreciation, taxes, and incorporating the machine's salvage value. Here's the detailed, step-by-step solution:
### Initial Investment (Year 0)
- Machine Cost: \[tex]$ 210,000 - Shipping and Installation Costs: \$[/tex] 51,000
- Increase in Net Working Capital: \[tex]$ 105,000 Initial Investment Calculation: \[ \text{Initial Investment} = \text{Machine Cost} + \text{Shipping and Installation} + \text{Increase in Working Capital} \] \[ \text{Initial Investment} = 210,000 + 51,000 + 105,000 = 366,000 \] So, the Free Cash Flow in Year 0 (FCF Year 0) is: \[ \text{FCF}_0 = -366,000 \] ### Depreciation (MACRS 7-year class life for the first 3 years) We use the MACRS depreciation rates for the 7-year class life: - Year 1 Rate: 0.1429 - Year 2 Rate: 0.2449 - Year 3 Rate: 0.1749 Machine cost excluding shipping and installation costs: \[ \text{Depreciable Cost} = 210,000 \] Depreciation for each year: - Year 1 Depreciation: \( 210,000 \times 0.1429 = 30,009 \) - Year 2 Depreciation: \( 210,000 \times 0.2449 = 51,429 \) - Year 3 Depreciation: \( 210,000 \times 0.1749 = 36,729 \) ### Calculating EBIT and Taxes Revenue and costs for each year: - Revenue: \$[/tex] 610,000 per year
- Costs: \$ 260,000 per year
Year 1:
[tex]\[ \text{EBIT}_1 = \text{Revenue} - \text{Costs} - \text{Depreciation}_1 \][/tex]
[tex]\[ \text{EBIT}_1 = 610,000 - 260,000 - 30,009 = 319,991 \][/tex]
[tex]\[ \text{Tax}_1 = \text{EBIT}_1 \times \text{Tax Rate} \][/tex]
[tex]\[ \text{Tax}_1 = 319,991 \times 0.21 = 67,198.11 \][/tex]
Year 2:
[tex]\[ \text{EBIT}_2 = \text{Revenue} - \text{Costs} - \text{Depreciation}_2 \][/tex]
[tex]\[ \text{EBIT}_2 = 610,000 - 260,000 - 51,429 = 298,571 \][/tex]
[tex]\[ \text{Tax}_2 = \text{EBIT}_2 \times \text{Tax Rate} \][/tex]
[tex]\[ \text{Tax}_2 = 298,571 \times 0.21 = 62,700.91 \][/tex]
Year 3:
[tex]\[ \text{EBIT}_3 = \text{Revenue} - \text{Costs} - \text{Depreciation}_3 \][/tex]
[tex]\[ \text{EBIT}_3 = 610,000 - 260,000 - 36,729 = 313,271 \][/tex]
[tex]\[ \text{Tax}_3 = \text{EBIT}_3 \times \text{Tax Rate} \][/tex]
[tex]\[ \text{Tax}_3 = 313,271 \times 0.21 = 65,786.91 \][/tex]
### Calculating Free Cash Flows (FCFs)
[tex]\[ \text{FCF} = (\text{Revenue} - \text{Costs} - \text{Taxes} + \text{Depreciation}) - \text{Increase in NWC} \][/tex]
Year 1:
[tex]\[ \text{FCF}_1 = (610,000 - 260,000 - 67,198.11 + 30,009) - 105,000 = 207,810.89 \][/tex]
Year 2:
[tex]\[ \text{FCF}_2 = (610,000 - 260,000 - 62,700.91 + 51,429) - 105,000 = 233,729.09 \][/tex]
Year 3:
[tex]\[ \text{FCF}_3 = (610,000 - 260,000 - 65,786.91 + 36,729) - 105,000 = 216,942.09 \][/tex]
### Including After-Tax Salvage Value in Year 3
[tex]\[ \text{After-Tax Salvage Value} = \text{Resale Value} \times (1 - \text{Tax Rate}) \][/tex]
[tex]\[ \text{After-Tax Salvage Value} = 76,000 \times (1 - 0.21) = 60,040 \][/tex]
Adding this to the FCF for Year 3:
[tex]\[ \text{Final FCF}_3 = 216,942.09 + 60,040 = 276,982.09 \][/tex]
### Summary of Free Cash Flows
[tex]\[ \begin{tabular}{|c|c|c|c|c|} \hline \text{Year} & 0 & 1 & 2 & 3 \\ \hline \text{FCF} & -366,000 & 207,810.89 & 233,729.09 & 276,982.09 \\ \hline \end{tabular} \][/tex]
### Net Present Value (NPV) Calculation
[tex]\[ \text{NPV} = -\text{Initial Investment} + \sum_{t=1}^{3} \left( \frac{\text{FCF}_t}{(1 + \text{Discount Rate})^t} \right) \][/tex]
[tex]\[ \text{NPV} = -366,000 + \left( \frac{207,810.89}{(1 + 0.11)^1} \right) + \left( \frac{233,729.09}{(1 + 0.11)^2} \right) + \left( \frac{276,982.09}{(1 + 0.11)^3} \right) \][/tex]
[tex]\[ \text{NPV} = -366,000 + 187,181.08 + 189,887.87 + 201,643.57 = 212,712.52 \][/tex]
### Final Cash Flows and NPV
[tex]\[ \text{FCFs and NPV}: \begin{tabular}{|c|c|c|c|c|c|} \hline \text{Year} & 0 & 1 & 2 & 3 & \text{NPV} \\ \hline \text{FCF} & -366,000 & 207,810.89 & 233,729.09 & 276,982.09 & 212,712.52 \\ \hline \end{tabular} \][/tex]
### Initial Investment (Year 0)
- Machine Cost: \[tex]$ 210,000 - Shipping and Installation Costs: \$[/tex] 51,000
- Increase in Net Working Capital: \[tex]$ 105,000 Initial Investment Calculation: \[ \text{Initial Investment} = \text{Machine Cost} + \text{Shipping and Installation} + \text{Increase in Working Capital} \] \[ \text{Initial Investment} = 210,000 + 51,000 + 105,000 = 366,000 \] So, the Free Cash Flow in Year 0 (FCF Year 0) is: \[ \text{FCF}_0 = -366,000 \] ### Depreciation (MACRS 7-year class life for the first 3 years) We use the MACRS depreciation rates for the 7-year class life: - Year 1 Rate: 0.1429 - Year 2 Rate: 0.2449 - Year 3 Rate: 0.1749 Machine cost excluding shipping and installation costs: \[ \text{Depreciable Cost} = 210,000 \] Depreciation for each year: - Year 1 Depreciation: \( 210,000 \times 0.1429 = 30,009 \) - Year 2 Depreciation: \( 210,000 \times 0.2449 = 51,429 \) - Year 3 Depreciation: \( 210,000 \times 0.1749 = 36,729 \) ### Calculating EBIT and Taxes Revenue and costs for each year: - Revenue: \$[/tex] 610,000 per year
- Costs: \$ 260,000 per year
Year 1:
[tex]\[ \text{EBIT}_1 = \text{Revenue} - \text{Costs} - \text{Depreciation}_1 \][/tex]
[tex]\[ \text{EBIT}_1 = 610,000 - 260,000 - 30,009 = 319,991 \][/tex]
[tex]\[ \text{Tax}_1 = \text{EBIT}_1 \times \text{Tax Rate} \][/tex]
[tex]\[ \text{Tax}_1 = 319,991 \times 0.21 = 67,198.11 \][/tex]
Year 2:
[tex]\[ \text{EBIT}_2 = \text{Revenue} - \text{Costs} - \text{Depreciation}_2 \][/tex]
[tex]\[ \text{EBIT}_2 = 610,000 - 260,000 - 51,429 = 298,571 \][/tex]
[tex]\[ \text{Tax}_2 = \text{EBIT}_2 \times \text{Tax Rate} \][/tex]
[tex]\[ \text{Tax}_2 = 298,571 \times 0.21 = 62,700.91 \][/tex]
Year 3:
[tex]\[ \text{EBIT}_3 = \text{Revenue} - \text{Costs} - \text{Depreciation}_3 \][/tex]
[tex]\[ \text{EBIT}_3 = 610,000 - 260,000 - 36,729 = 313,271 \][/tex]
[tex]\[ \text{Tax}_3 = \text{EBIT}_3 \times \text{Tax Rate} \][/tex]
[tex]\[ \text{Tax}_3 = 313,271 \times 0.21 = 65,786.91 \][/tex]
### Calculating Free Cash Flows (FCFs)
[tex]\[ \text{FCF} = (\text{Revenue} - \text{Costs} - \text{Taxes} + \text{Depreciation}) - \text{Increase in NWC} \][/tex]
Year 1:
[tex]\[ \text{FCF}_1 = (610,000 - 260,000 - 67,198.11 + 30,009) - 105,000 = 207,810.89 \][/tex]
Year 2:
[tex]\[ \text{FCF}_2 = (610,000 - 260,000 - 62,700.91 + 51,429) - 105,000 = 233,729.09 \][/tex]
Year 3:
[tex]\[ \text{FCF}_3 = (610,000 - 260,000 - 65,786.91 + 36,729) - 105,000 = 216,942.09 \][/tex]
### Including After-Tax Salvage Value in Year 3
[tex]\[ \text{After-Tax Salvage Value} = \text{Resale Value} \times (1 - \text{Tax Rate}) \][/tex]
[tex]\[ \text{After-Tax Salvage Value} = 76,000 \times (1 - 0.21) = 60,040 \][/tex]
Adding this to the FCF for Year 3:
[tex]\[ \text{Final FCF}_3 = 216,942.09 + 60,040 = 276,982.09 \][/tex]
### Summary of Free Cash Flows
[tex]\[ \begin{tabular}{|c|c|c|c|c|} \hline \text{Year} & 0 & 1 & 2 & 3 \\ \hline \text{FCF} & -366,000 & 207,810.89 & 233,729.09 & 276,982.09 \\ \hline \end{tabular} \][/tex]
### Net Present Value (NPV) Calculation
[tex]\[ \text{NPV} = -\text{Initial Investment} + \sum_{t=1}^{3} \left( \frac{\text{FCF}_t}{(1 + \text{Discount Rate})^t} \right) \][/tex]
[tex]\[ \text{NPV} = -366,000 + \left( \frac{207,810.89}{(1 + 0.11)^1} \right) + \left( \frac{233,729.09}{(1 + 0.11)^2} \right) + \left( \frac{276,982.09}{(1 + 0.11)^3} \right) \][/tex]
[tex]\[ \text{NPV} = -366,000 + 187,181.08 + 189,887.87 + 201,643.57 = 212,712.52 \][/tex]
### Final Cash Flows and NPV
[tex]\[ \text{FCFs and NPV}: \begin{tabular}{|c|c|c|c|c|c|} \hline \text{Year} & 0 & 1 & 2 & 3 & \text{NPV} \\ \hline \text{FCF} & -366,000 & 207,810.89 & 233,729.09 & 276,982.09 & 212,712.52 \\ \hline \end{tabular} \][/tex]
