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