To calculate the profitability index (PI) for the J-Mix 2000, we must first determine the present value (PV) of the incremental cash flows generated by the machine over the next five years, using the cost of capital as the discount rate.
Steps to calculate the profitability index:
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Calculate the Present Value (PV) of the Cash Flows: The present value of an annuity formula is used when cash flows are the same each year: \[ PV = C \times \frac{(1 - (1 + r)^{-n})}{r} \] where:
- \(C\) = cash flow per period ($511,457)
- \(r\) = discount rate (8.88% or 0.0888)
- \(n\) = number of periods (5 years)
Plug in the values: \[ PV = 511,457 \times \frac{(1 - (1 + 0.0888)^{-5})}{0.0888} \]
Calculate \((1 + r)^{-n}\): \[ (1 + 0.0888)^{-5} \approx 0.634 \]
Then: \[ PV = 511,457 \times \frac{(1 - 0.634)}{0.0888} \approx 511,457 \times \frac{0.366}{0.0888} \approx 511457 \times 4.128 \approx 2,110,622.84 \]
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Calculate the Net Present Value (NPV): The NPV is calculated as follows: \[ NPV = PV - Initial Investment \] Initial Investment = $1,670,000 \[ NPV = 2,110,622.84 - 1,670,000 = 440,622.84 \]
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Calculate the Profitability Index (PI): The profitability index is calculated by the formula: \[ PI = \frac{PV}{Initial Investment} \] \[ PI = \frac{2,110,622.84}{1,670,000} \approx 1.26 \]
Conclusion:
The profitability index for the J-Mix 2000 is approximately 1.26. This indicates that for every dollar invested, there is an expected return of $1.26, making this investment a potentially profitable decision for Caspian Sea Drinks.