When can IRR go wrong?
Lessons from a 17th century Frenchman
In a perfect world, investment decisions would be made based on net present value (NPV). Suppose you are faced with a decision to make an investment now in exchange for a series of cash flows in the future. To decide if the investment is worthwhile, you compute the present value of the stream of cash flows based on some discount rate. If that present value exceeds the initial investment, then you proceed. If it doesn't, you don't.
The problem with this paradigm is that NPV is highly sensitive to the discount rate. And how the heck do you pick the discount rate? To get around this problem, one solution is to treat it like an inverse problem: instead of finding the NPV based on the discount rate, find the discount rate at which you "break even," meaning you have an NPV of exactly 0. This discount rate is called the internal rate of return (IRR). To make investment decisions with IRR, you could compare the IRR to your "hurdle rate," which is a targeted return on investment. If the IRR of a series of cash flows exceeds your hurdle rate, then you invest. If it doesn't, you don't.
What is IRR?
NPV and IRR are based on the same setup. Suppose you have an initial investment C_0, and a sequence of cash flows C_i for i = 1 … n occurring at the end of the i-th year. For an unknown discount rate r, the NPV is given by:
The IRR is the value of r that makes the above expression equal to 0. To solve this, you can define x = 1/(1+r). The first step to finding the IRR is then to find values of x that solve
If you have a solution x to this equation, then IRR = 1/x - 1.
Potential pitfall of IRR
Given a discount rate, we can always compute a unique NPV based on the first formula. Unfortunately, the same cannot be said for IRR. If you have cash flows going out to the n-th year, then the IRR equation above is an n-th degree polynomial, which can have up to n solutions. In other words, there might be n different discount rates that yield an NPV of 0 for that sequence of cash flows. If there is more than one, how do you pick the right IRR?
One thing you can do is just focus on positive solutions x to the IRR equation. After all, if x < 0, then r < -1, which means a potential IRR is less than -100%. If that's the case, you probably haven't shortlisted this investment opportunity. The question then becomes: how do you know how many positive solutions exist to the IRR equation? For that, we turn to a 17th century philosopher.
Descartes’ rule of signs
If you took Algebra 2 in high school, you might remember something called Descartes' rule of signs. The Descartes in question is René Descartes--the guy who said, "I think, therefore I am." (He probably said it with a French accent.) His rule of signs goes as follows. For the IRR polynomial above, make a sequence out of the coefficients in ascending (or descending) order in powers of x:
In general, some of those numbers will be positive, and others will be negative. In a realistic scenario, -C_0 will be negative because you're deciding whether to make an investment. What Descartes' rule of signs says is that the number of positive solutions x to the IRR equation is at most equal to the number of sign changes in the sequence of coefficients--call it S. Moreover, if there are fewer solutions than S, then the number of positive solutions must be S minus an even number.
Let's look at an example. Suppose you can purchase the following stream of cash flows for $1,000 today: +$400 at the end of year 1, -$300 at the end of year 2, +$1,000 at the end of year 3. For any discount rate r, the NPV of this opportunity is given by
The corresponding sequence of coefficients is (-1000, 400, -300, 1000), which has 3 sign changes. Descartes tells us that there are either one or three solutions to the IRR equation. In this case, there happens to be one: ~3.8%.
What about this opportunity? Invest $675 now to receive $975 in one year and $650 in three years, but with additional payments of $25 and $1,000 after two and four years, respectively. The NPV of this opportunity is
This time, the sequence of coefficients is (-675, 975, -25, 650, -1000), which has four sign changes. Applying Descartes, there are either zero, two, or four solutions to the IRR equation. In other words, we know right off the bat that IRR can't be unique! There happen to be two IRRs in this case: ~11% and ~33%. I don't have any advice on which one is right.
Why does IRR persist?
Given that having multiple IRRs for the same stream of cash flows is a pretty serious defect of IRR, why do many managers prefer to think about IRR instead NPV? I can only speculate. I mentioned one possible reason above: it can be hard to figure out what your discount rate should be. Another reason is that IRR works fine in probably the most common scenario: namely, when your initial investment is the only cash outflow, and all future payments are inflows. In that case, the sequence
has exactly one sign change. Per Descartes, that means that there is a unique IRR.
Beast mode: Sturm’s theorem
If you look on the internet, a lot of people have made the connection between Descartes and IRR. If you want to take your recreational business math game to the next level, I recommend taking a look at Sturm's theorem. In contrast to Descartes, which only gives you the possible number of IRR solutions, Sturm's theorem gives you an exact count. Moreover, you can determine the interval over which you want to look for IRRs. For instance, Descartes tells you how many positive solutions x exist for the IRR equation, but only x values between 0 and 1 give you a positive IRR. With Sturm, you can count exactly how many positive IRR solutions there are for a given stream of cash flows. The catch is that you have to do polynomial long division. (There’s always a catch.)
To close, I want to stress that this post is not intended as a dig at IRR or its fans. It was all about giving Descartes' rule of signs its moment in the sun. I hope you enjoyed reading.