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Chapter 9: Problem 19

If we define the NPV index as the ratio of NPV to cost, what is the relationship between this index and the profitability index?

Short Answer

Expert verified
The relationship between the NPV index and the profitability index is: \[\text{NPV Index} = \text{Profitability Index} - 1\] This means that the NPV index is equivalent to the profitability index minus 1, as it incorporates the initial cost of an investment project.

Step by step solution

01

Define NPV Index and Profitability Index

The NPV (Net Present Value) index is defined as the ratio of NPV (Net Present Value) to the initial cost of an investment project. NPV Index = \(\frac{NPV}{Initial\, Cost}\) The profitability index, also known as the benefit-cost ratio, is the ratio of present value of cash inflows to the present value of cash outflows. Profitability index = \(\frac{PV\, of\, Cash\, Inflows}{Initial\, Cost}\)
02

Establish Relationship between NPV Index and Profitability Index

Recall the NPV formula: NPV = \(PV\, of\, Cash\, Inflows - Initial\, Cost\) Let's rewrite the formula for NPV Index and Profitability Index in terms of NPV. We have: NPV Index = \(\frac{NPV}{Initial\, Cost}\) And the formula for Profitability Index is: Profitability Index = \(\frac{PV\, of\, Cash\, Inflows}{Initial\, Cost}\) Now, let's substitute the NPV formula into the NPV Index formula: NPV Index = \(\frac{PV\, of\, Cash\, Inflows - Initial\, Cost}{Initial\, Cost}\) By looking closer, we realize the relationship between the NPV index and Profitability Index: NPV Index = Profitability Index - 1
03

Conclusion

The relationship between the NPV index and the profitability index is: \[\text{NPV Index} = \text{Profitability Index} - 1\] This means that the NPV index is equivalent to the profitability index minus 1, as it incorporates the initial cost of an investment project.

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Most popular questions from this chapter

An investment has an installed cost of \(\$ 412,670\) The cash flows over the four-year life of the investment are projected to be \(\$ 212,817, \$ 153,408, \$ 102,389,\) and \(\$ 72,308 .\) If the discount rate is zero, what is the NPV? If the discount rate is infinite, what is the NPV? At what discount rate is the NPV just equal to zero? Sketch the NPV profile for this investment based on these three points.

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