To calculate the net present value (NPV) of the RGM-7000 water filtration system, we can follow these steps:
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Calculate Annual After-Tax Cash Flows:
- Revenues: $3.99 million per year
- Operating Costs: $667,395 per year
- Depreciation: The initial cost of $15 million fully depreciated over 20 years results in: \[ \text{Depreciation} = \frac{15,000,000}{20} = 750,000 \text{ per year} \]
- Taxable Income: \[ \text{Taxable Income} = \text{Revenues} - \text{Operating Costs} - \text{Depreciation} \] \[ = 3,990,000 - 667,395 - 750,000 = 2,572,605 \]
- Tax: \[ \text{Tax} = \text{Taxable Income} \times \text{Tax Rate} \] \[ = 2,572,605 \times 0.31 = 798,706.55 \]
- After-Tax Income: \[ \text{After-Tax Income} = \text{Taxable Income} - \text{Tax} \] \[ = 2,572,605 - 798,706.55 = 1,773,898.45 \]
- Add Back Depreciation: \[ \text{Total After-Tax Cash Flow} = \text{After-Tax Income} + \text{Depreciation} \] \[ = 1,773,898.45 + 750,000 = 2,523,898.45 \]
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Calculate NPV: We will subtract the initial investment of $15 million from the present value of the cash flows: \[ NPV = \sum_{t=1}^{20} \frac{CF}{(1 + r)^t} - \text{Initial Investment} \] Where:
- \( CF \) = total after-tax cash flow = $2,523,898.45
- \( r \) = discount rate = 8% or 0.08
- \( t \) = years from 1 to 20
The present value of an annuity can be calculated using the formula: \[ PV = CF \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \] Where \( n = 20 \) years. Plugging in the numbers: \[ PV = 2,523,898.45 \times \left( \frac{1 - (1 + 0.08)^{-20}}{0.08} \right) \] First we calculate the annuity factor: \[ = \frac{1 - (1.08)^{-20}}{0.08} \approx 9.6463 \] Now, calculate the present value: \[ PV \approx 2,523,898.45 \times 9.6463 \approx 24,878,180.73 \]
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Calculate NPV: \[ NPV = 24,878,180.73 - 15,000,000 \approx 9,878,180.73 \]
Thus, the net present value of the RGM-7000 is approximately $9,878,180.73.