To calculate the internal rate of return (IRR) for the RGM-7000 device, we need to determine the annual cash flows generated by the system over its 15-year life and then find the discount rate that makes the net present value (NPV) of these cash flows equal to zero.
Step 1: Calculate Annual Cash Flows
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Revenues: The additional revenues generated by the RGM-7000 are $2.81 million per year.
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Operating Costs: The increased operating costs are $671,538 per year.
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Depreciation: The RGM-7000 costs $13 million and is fully depreciated over 15 years. Therefore, the annual depreciation expense is: \[ \text{Depreciation} = \frac{13,000,000}{15} = 866,666.67 \text{ (per year)} \]
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Tax Impact: Since the company is subject to a 25% marginal tax rate, the taxable income and cash flows must incorporate taxes: \[ \text{Taxable Income} = \text{Revenues} - \text{Operating Costs} - \text{Depreciation} \] \[ \text{Taxable Income} = 2,810,000 - 671,538 - 866,666.67 = 1,271,795.33 \]
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Taxes: Calculate the tax on the taxable income: \[ \text{Taxes} = 1,271,795.33 \times 0.25 = 317,948.83 \]
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Net Income: After tax, the net income is: \[ \text{Net Income} = \text{Taxable Income} - \text{Taxes} = 1,271,795.33 - 317,948.83 = 953,846.50 \]
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Cash Flow: The cash flow to be considered is net income plus depreciation (as depreciation is a non-cash expense): \[ \text{Annual Cash Flow} = \text{Net Income} + \text{Depreciation} \] \[ \text{Annual Cash Flow} = 953,846.50 + 866,666.67 = 1,820,513.17 \text{ (per year)} \]
Step 2: NPV and IRR Calculation
The initial investment is $13 million. The annual cash inflow is $1,820,513.17 for 15 years.
The formula for IRR is that it is the rate \( r \) that solves: \[ NPV = 0 = -C_0 + \sum_{t=1}^{n} \frac{C_t}{(1 + r)^t} \] Where:
- \( C_0 = \) initial investment
- \( C_t = \) cash inflow in year \( t \)
- \( n = \) lifetime of the asset
For our case:
- \( C_0 = 13,000,000 \)
- \( C_t = 1,820,513.17 \) for \( t = 1 \) to \( t = 15 \)
Using a financial calculator or Excel/financial software, input these cash flows to find the IRR.
In Excel, you would enter:
- Year 0: -13,000,000
- Year 1-15: 1,820,513.17
Using the IRR function, Excel would return approximately \( 3.87% \) as the IRR for the RGM-7000 water filtration system.
Conclusion
The internal rate of return (IRR) for the RGM-7000 is approximately 3.87%. This indicates the expected annual rate of return on the investment over its 15-year life based on the estimated cash flows.