Question
Caspian Sea Drinks is considering buying the J-Mix 2000. It will allow them to make and sell more product. The machine cost $1.89 million and create incremental cash flows of $568,891.00 each year for the next five years. The cost of capital is 10.82%. What is the profitability index for the J-Mix 2000?
Answers
GPT-4o mini
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):**
\[
\text{Profitability Index} = \frac{\text{PV of Future Cash Flows}}{\text{Initial Investment}}
\]
1. **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:
1. Calculate \((1 + 0.1082)^{-5}\):
\[
(1 + 0.1082)^{-5} \approx (1.1082)^{-5} \approx 0.586
\]
2. Calculate \(1 - (1 + 0.1082)^{-5}\):
\[
1 - 0.586 \approx 0.414
\]
3. Divide by \(r\):
\[
\frac{0.414}{0.1082} \approx 3.828
\]
4. Finally, multiply by \(C\):
\[
\text{PV} \approx 568,891 \times 3.828 \approx 2,178,748.38
\]
2. **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.
**Formula for the Profitability Index (PI):**
\[
\text{Profitability Index} = \frac{\text{PV of Future Cash Flows}}{\text{Initial Investment}}
\]
1. **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:
1. Calculate \((1 + 0.1082)^{-5}\):
\[
(1 + 0.1082)^{-5} \approx (1.1082)^{-5} \approx 0.586
\]
2. Calculate \(1 - (1 + 0.1082)^{-5}\):
\[
1 - 0.586 \approx 0.414
\]
3. Divide by \(r\):
\[
\frac{0.414}{0.1082} \approx 3.828
\]
4. Finally, multiply by \(C\):
\[
\text{PV} \approx 568,891 \times 3.828 \approx 2,178,748.38
\]
2. **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.