2.3 Solution to Problem NPV vs. IRR
We can calculate the NPV Profiles for each project as found below. The NPV Profile depicts how the NPV for a project will change when assuming a set of different discount rates.
| NPV Profiles for | ||
| NPVA | NPVB | |
| 0.0 | $890 | $399 |
| 10.0 | 283 | 179 |
| 12.0 | 200 | 146 |
| 18.1 | 000 | 62 |
| 20.0 | (49) | 42 |
| 24.0 | (138) | 00 |
| 30.0 | (238) | (51) |
The “NPV Profiles” are the sets of NPVs at varying discount rates.
First, take note that the IRRs for the two projects are as follows:
IRRA = .181
IRRB = .240
We recognize that, in truth, there will be multiple IRRs – due to the presence of negative free cash flow projections. For present purposes, we will use only those IRRs that “fit” in the range denoted in the table.
You may also have observed that the “crossover rate” will be between 12% – 18.1% (note that the IRR numbers have been marked above in bold). The crossover rate is the discount rate at which the preference for one project over another, on an NPV basis, changes from one project to the other – as you go from lower to higher rates.
This rate is both interesting and relevant because, as may be readily seen, at lower discount rates, project “A” will produce relatively higher NPVs than project “B”, while at higher discount rates project “B” will produce higher NPVs. Since the discount rate can change over time, the moment at which one calculates the NPV, may yield an outcome that may be arbitrary. The exact crossover rate is therefore noteworthy. (At this point, you should diagram this table. A few pages hence, you will find a diagram to fill in.)
Let’s get back to the crossover rate. To discover the crossover rate, we wish to find that one rate at which the NPVs for both projects are the same. That is to say mechanically, we will calculate the NPV of the differences between the two projects’ respective cash flows to accomplish this mathematical task. Thus, we create a third, fictitious project data set – let’s call it “Project C,” whose cash flows are as follows:
| Project A | Project B | “Project C” CFA – CFB | |
| 0 | ($300) | ($405) | 105 |
| 1 | (387) | 134 | (521) |
| 2 | (193) | 134 | (327) |
| 3 | (100) | 134 | (234) |
| 4 | 600 | 134 | 466 |
| 5 | 600 | 134 | 466 |
| 6 | 850 | 134 | 716 |
| 7 | (180) | 0 | (180) |
In other words, we calculate the differences between the cash flows of Projects “A” and “B,” and treat those differences as though it were another, third project, which we are calling “Project C” or “CFA – CFB.” We must work iteratively in order to arrive at the correct crossover rate. (We will use just the one rate that falls within a useful range for this illustration, i.e., between 12-18%. We are really not concerned with the problem of multiple IRRs because we are more focused on the NPV.…)
If Project C is the difference in the cash flows of Projects “A” and “B,” then the IRR for “C” must produce a difference in the NPVs for the two projects equal to Zero. This is the very crux of the concept being discussed here.
The precise crossover rate for both projects A and B = .1453. (And IRRC = 0.1453.) Once again, while there will, of course, be numerous solutions for the IRR, given that there are some negative cash flows; the one solution (of many) that falls within our 12-18.1% percent range is 14.53%.
The crossover rate is the discount rate where the NPV for both projects will be the same, i.e., NPVA = NPVB. The table below proves that out – with an immaterial rounding error.
| $ | CF Project A | Discount Rate | PV of CFA | CF Project B | PV of CFB |
| 0 | (300) | ÷ 1.14530 | (300) | (405) | (405) |
| 1 | (387) | ÷ 1.14531 | (337.90) | 134 | 117.00 |
| 2 | (193) | ÷ 1.14532 | (147.14) | 134 | 102.16 |
| 3 | (100) | ÷ 1.14533 | (66.56) | 134 | 89.20 |
| 4 | 600 | ÷ 1.14534 | 348.72 | 134 | 77.88 |
| 5 | 600 | ÷ 1.14535 | 304.48 | 134 | 68.00 |
| 6 | 850 | ÷ 1.14536 | 376.62 | 134 | 59.37 |
| 7 | (180) | ÷ 1.14537 | (69.64) | 0 | 0 |
| NPV=108.58 | NPV=108.61 |
(The numbers in the NPV table were rounded to the nearest cent, so in the end, there is a rounding error of about $0.03; this should not be of any conceptual concern.)
This also says that, at the 0.1453 discount rate, the difference in the two respective projects’ NPVs will be zero, i.e., NPVA – NPVB = 0. Project C represents the differences in the projects’ cash flows and thus must also reflect the difference in the two NPVs; in other words, the NPV for Project C should be zero, i.e., NPVC = 0. The rate that causes the NPV of Project C to be zero is its IRR: 0.1453. The following table bears this truth out.
| “Project C” CFA – CFB | ÷ 1.1453N | = | |
| 0 | 105 | ÷ 1.14530 | 105.00 |
| 1 | (521) | ÷ 1.14531 | (455.78) |
| 2 | (327) | ÷ 1.14532 | (249.29) |
| 3 | (234) | ÷ 1.14533 | (155.76) |
| 4 | 466 | ÷ 1.14534 | 270.84 |
| 5 | 466 | ÷ 1.14535 | 236.48 |
| 6 | 716 | ÷ 1.14536 | 317.25 |
| 7 | (180) | ÷ 1.14537 | (65.53) |
| Sum/NPV= | 0 |
To repeat, the “crossover point” depicts the point at which the NPVs for both projects will be the same; that also says that the difference between the two projects’ NPVs will be zero. Because “Project C” itself represents the difference in the two projects, the NPV of Project C will be zero, and the IRR of Project C will be the rate that caused the difference in the NPVs to be zero. That rate was calculated to be 0.1453.
So, while the IRR for Project C is 0.1453 and its NPV = 0, it does not mean that the respective NPVs for either projects “A” or “B” will be equal to zero at the IRR for project “C”; the individual NPVs will be positive, as you can see in the graph (below). However, at the IRR for project “C,” the NPVs for both projects “A” and “B” will be the same. That is the IRR C will be the point at which NPVA – NPVB = 0. Remember: Project “C” represents the difference between the two projects’ cash flows.
As a practical matter, an analyst who studies two (or more) competing projects whose cash flow profiles are radically dissimilar, should also calculate the crossover point. Why? If it so happens that the company’s cost of capital is very close to the crossover point, say within a percentage point, he may reconsider whether to recommend one project over the other solely based on the decision rule of thumb of choosing the one with the higher NPV. In the case just evaluated, suppose the firm’s k = 14% or 15%. Based on this, the analyst, using the rule of thumb, will “flip” his decision from one to the other project. Isn’t that just arbitrary? Instead, he should rely on some other rationale, maybe even a qualitative one, heaven forbid!
Since a picture is worth a thousand words, a diagram is presented below. See if you can fill it in (below).