Practice Question
You’re considering a project with an initial cost of $70,189 and annual cash inflows of $17,304 per year forever. What is the discount rate that will produce a profitability index of 4.0?
Step 1: Define the Variables
- Initial Cost (\( C_0 \)) = $70,189
- Annual Cash Inflows = $17,304
- Desired Profitability Index (PI) = 4.0
Step 2: Use the Profitability Index Formula
\[ PI = \frac{PV\text{ of Cash Inflows}}{\text{Initial Cost}} \] Rearranged: \[ PV = PI \times \text{Initial Cost} = 4.0 \times 70,189 = 280,756 \]
Step 3: Use the Perpetuity Formula to Solve for Discount Rate
Since the inflows are perpetual: \[ PV = \frac{CF}{r} \Rightarrow 280,756 = \frac{17,304}{r} \] Solving for \( r \): \[ r = \frac{17,304}{280,756} \approx 0.0616 \text{ or } 6.16\% \]
Conclusion: A discount rate of 6.16% results in a profitability index of 4.0 for this project.