- Identify the cash flows and the hurdle rate: Investment B has an initial cost of \(\$12,600\) and future inflows of \(\$6,080\), \(\$4,960\), \(\$4,320\), and \(\$2,440\) over four years.
- Calculate the present value (PV) for each inflow using the discount rate of \(10\%\): \(PV_n = \frac{CF_n}{(1.10)^n}\).
- Sum the present values to find the total present value of inflows, which is approximately \(\$14,538.68\).
- Divide the total PV by the initial investment to obtain the present value index (PVI): \(PVI = \frac{14,538.68}{12,600} \approx 1.15\).
- The final answer is \(\boxed{1.15}\).
Explanation
- Problem Analysis and Data Collection
To determine the present value index (PVI) of Investment B, we must calculate the present value of all future cash inflows and divide that total by the initial investment cost. The initial investment (cash outflow at Year 0) for Investment B is \(\$12,600\). The future cash inflows are as follows:
Year 1: \(\$6,080\)
Year 2: \(\$4,960\)
Year 3: \(\$4,320\)
Year 4: \(\$2,440\)
The hurdle rate, or discount rate, is \(10\%\) (\(i = 0.10\)). The formula for the present value (PV) of a future cash flow is \(PV = \frac{CF_n}{(1 + i)^n}\).
- Calculating Present Value of Inflows
We calculate the present value (PV) for each future year's cash inflow using the formula \(PV = \frac{CF_n}{(1 + 0.10)^n}\):
- Year 1: \(PV_1 = \frac{6,080}{(1.10)^1} = \frac{6,080}{1.10} \approx 5,527.2727\)
- Year 2: \(PV_2 = \frac{4,960}{(1.10)^2} = \frac{4,960}{1.21} \approx 4,099.1736\)
- Year 3: \(PV_3 = \frac{4,320}{(1.10)^3} = \frac{4,320}{1.331} \approx 3,245.6800\)
- Year 4: \(PV_4 = \frac{2,440}{(1.10)^4} = \frac{2,440}{1.4641} \approx 1,666.5528\)
Summing these individual present values gives the total present value (PV) of the inflows:
\(Total\ PV = 5,527.2727 + 4,099.1736 + 3,245.6800 + 1,666.5528 = 14,538.6791\).
- Calculating the Present Value Index (PVI)
The present value index (PVI) is calculated as the ratio of the total present value of future cash inflows to the initial investment:
\(PVI = \frac{Total\ PV}{|CF_0|} = \frac{14,538.6791}{12,600} \approx 1.15386\).
Rounding the result to two decimal places, we get \(1.15\). Comparing this result with the multiple-choice options, it corresponds to the value \(1.15\). The final answer is \(\boxed{1.15}\).
Examples
The Present Value Index is a vital tool for companies like Shenandoah Springs when deciding between multiple projects. It helps management determine which investments provide the most value relative to their initial cost. For instance, if a company has limited funds, they can use the PVI to rank projects and select those that yield the highest return for every dollar invested today, ensuring that capital is allocated as efficiently as possible.