To calculate the profitability index (PI) for the J-Mix 2000, we first need to determine the present value (PV) of the future cash flows and then use that to obtain the PI.
Formula for the Profitability Index (PI):
\[ \text{Profitability Index} = \frac{\text{PV of Future Cash Flows}}{\text{Initial Investment}} \]
- Calculate the present value of the incremental cash flows:
The incremental cash flow is $568,891 per year for 5 years. We will use the formula for the present value of an annuity to compute this. The present value (PV) of an annuity can be calculated as follows:
\[ \text{PV} = C \times \left(1 - (1 + r)^{-n}\right) / r \]
where:
- \(C\) = annual cash flow ($568,891)
- \(r\) = discount rate (10.82% or 0.1082)
- \(n\) = number of years (5)
Plugging in the numbers:
\[ \text{PV} = 568,891 \times \left(1 - (1 + 0.1082)^{-5}\right) / 0.1082 \]
Now we compute this step by step:
- Calculate \((1 + 0.1082)^{-5}\):
\[ (1 + 0.1082)^{-5} \approx (1.1082)^{-5} \approx 0.586 \]
- Calculate \(1 - (1 + 0.1082)^{-5}\):
\[ 1 - 0.586 \approx 0.414 \]
- Divide by \(r\):
\[ \frac{0.414}{0.1082} \approx 3.828 \]
- Finally, multiply by \(C\):
\[ \text{PV} \approx 568,891 \times 3.828 \approx 2,178,748.38 \]
- Calculate the Profitability Index:
\[ \text{Profitability Index} = \frac{PV}{\text{Initial Investment}} = \frac{2,178,748.38}{1,890,000} \]
Calculating this gives:
\[ \text{Profitability Index} \approx 1.152 \]
Thus, the profitability index for the J-Mix 2000 is approximately 1.15.
This indicates that for every dollar invested, there is a return of $1.15, suggesting that the investment is worthwhile.
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