Now we have a solid understanding of Net Present Value (NPV) and Internal rate of return (IRR) concepts, both can be effectively put into use for analyzing investment decisions, now an interesting question arises,

**will NPV and IRR always agree?**For answering this question, we will look into a sample problem (Problem 2.11) from Investment Science by Luenberger.

Problem 2.11 Luenberger:

Conflicting recommendations: Consider the two projects whose cash flows are shown in the table below. Find the IRRs of the two projects and the NPVs at 5%. Show that the IRR and NPV figures yield different recommendations. Can you explain this?

From the formulas for calculating Net Present Value (NPV) and Internal rate of return (IRR), and the spreadsheet way of solving them, we get

IRR(Project 1) = 15.24%, IRR(Project 2) = 12.38%

Therefore based on the Net Present Value Analysis, one would recommend Project 1, but based on Internal rate of return calculation, one would recommend Project 2, therefore the IRR and NPV figures yield different recommendations.

The Net Present Value (NPV) and IRR (Internal rate of return) may yield consistent results if cash flows are reinvested (at IRR), IRR calculation assumes that the cash flows are reinvested at IRR which is not the case in the above scenario, hence the conflict in recommendation, in general it's advisable to analyze an investment based on NPV (Net Present Value analysis).

Don't worry if the above statement is not clear to you, we should understand one thing from this article, Net Present Value analysis and IRR computations would yield consistent recommendations if the cash flows are reinvested at IRR

**But why?**The Net Present Value (NPV) and IRR (Internal rate of return) may yield consistent results if cash flows are reinvested (at IRR), IRR calculation assumes that the cash flows are reinvested at IRR which is not the case in the above scenario, hence the conflict in recommendation, in general it's advisable to analyze an investment based on NPV (Net Present Value analysis).

Don't worry if the above statement is not clear to you, we should understand one thing from this article, Net Present Value analysis and IRR computations would yield consistent recommendations if the cash flows are reinvested at IRR

**, else they may differ.**
## 1 comment:

did you ever get a chance to prove the concept in the problem 2.13 part 'a' ?

Question:

In general, we say that two projects with cash flows xi and yi, i=0,1,2,.. n, cross if x0 < y0 and summation i=0 to n (xi) > summation i=0 to n (yi). Let Px(d) and Py(d) denote the present values of these two projects when the discount factor is d.

Show that there is a crossover value c >0 such that Px(c) = Py(c)

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