$ F $	Income stream ($/time)
$ \text{NPV} $	Net present value
$ V_t $	Value of cash received at time $t$
$ r_r $	Hypothetical rate of return
$ g(r_r) $	Error function - root of $g$ is the internal rate of return

Internal rate of return - what it is and when it is undefined

Photo credit to Ibrahim Rifath

Goals of this article

Internal rate of return (IRR) has been criticized since at least 1955 [1]. Nonetheless it is still commonly used in business decision-making because better alternatives do not seem to be available. In this article we are going to rigorously define several versions of the IRR, discuss situations in which more than one IRR is possible for a given situation, and touch on some semantic reasons the IRR is criticized.

What is the IRR

The concept of internal rate of return is similar to net present value (NPV) and both concepts are formalized by the same equations. When we discuss NPVs, we are choosing a rate of return, $r_r$, and asking how much would we need today at that $r_r$ to generate those future cash flows. When we discuss IRRs, we are asking what $r_r$ would give us an NPV of 0. IRRs are commonly used in business to compare the profitability of different potential projects or by individuals to evaluate the profitability of potential investments.

It is critical to remember than IRRs are based on predictions about the future and the future is an uncertain place. You will never know the IRR of an investment to three significant figures until you are looking back on it in hindsight.

IRR written three ways

Here are NPV/IRR calculations for a continuous streams of cashflow on a continuous time scale.

An NPV calculation.

$$ \text{NPV} = \int^{t}_{0} F(\tau) e^{-r_r\tau} d\tau $$

An IRR calculation. The root of $g(r_r)$ is the internal rate of return.

$$ g(r_r) = \int^{t}_{0} F(\tau) e^{-r_r\tau} d\tau = 0 $$

NPV/IRR calculations for discrete cashflows on a continuous time scale.

$$ \text{NPV} = \sum_{i} V_i \int^{t}_{0} e^{-r_r\tau} d\tau $$

$$ g(r_r) = \sum_i V_i \int^{t}_{0} e^{-r_r\tau} d\tau = 0 $$

NPV/IRR calculations for discrete cashflows on a discrete time scale. This is the most common version in literature [1,2,3].

$$ \text{NPV} = \sum_{i} \frac{V_i}{(1+r_r)^{t_i} } $$

$$ g(r_r) = \sum_{i} \frac{V_i}{(1+r_r)^{t_i} } = 0 $$

The third case (discrete cashflow, discrete time) is the most commonly used and the least useful for developing understanding. For that reason, we will ignore it almost entirely despite its popularity. The first and second cases each have their place depending on the consistency of the cashflows predicted.

A demonstration of multiple roots

For illustration, let us consider the simple case of a few discrete cash flows. In this hypothetical, we are putting money into an investment every year for 4 years and at year 5 we withdraw 25% more than the total invested.

Time ($t_i$) [years]	Value ($V_t$)
0	$100
1	$100
2	$100
3	$100
4	-$500

Because these are discrete cash flows, we can use a discrete version of the IRR equation.

$$ g(r_r) = \sum_{i=0}^4 V_t e^{-r_r t_i} = 0 $$

Components and error function of hypothetical IRR

That looks reasonable, the one root of $g(r_r)$, 0.087, is the IRR. This rate of return aligns with our intuition ($100\times 1.09^4$+$100\times 1.09^3$+$100\times 1.09^2$+$100\times 1.09= 498$). However there is no reason the investment has to be linear or reasonable. Suppose we buy $10 of bitcoin, then sell a portion as it climbs. A year later we realize we made a mistake and buy $120 more. The following year we have a computer crash and lose all the bitcoin. What is our internal rate of return?

Time ($t_i$) [years]	Value ($V_t$)
0	$10
1	-$150
2	$120
3	$0

$$ \begin{align} g(r_r) &= 10e^{-0r_r}-150e^{-r_r}+120e^{-2r_r} = 0\\ r_r &= -0.165\quad \text{or} \quad 2.65 \end{align} $$

Neither answer here makes any sense. If you make 265% per year on $10 for 3 years, you would have just over $28,000. A negative rate of return makes no sense. In three years you put in $130 and pulled out $150; you have $20 in total profit. Obviously internal rates of return can give troublingly unreasonable results if the investment is highly volatile and actively managed (money regularly flowing in and out).

Descartes' sign rule and IRR

To make sense of the IRR calculation, to confirm that the rate of return function is sensible, we need to know how many roots $g(r_r)$ has. If we convert $g$ to a polynomial, we can apply Descartes' sign rule and find out how many real roots $g$ has without finding them all (which could be very difficult). For the set of $t_i$ and $V_i$ listed above and change of variable $x=e^{-r_r}$:

$$ \begin{align} g(r_r) &= \sum_i V_i e^{-r_r t_i} = 10e^{-0r_r}-150e^{-r_r}+120e^{-2r_r} \\ g(r_r) &= \sum_i V_i \left( e^{-r_r}\right)^{t_i} = 10\left(e^{-r_r}\right) ^{0}-150\left(e^{-r_r}\right)^1+120\left(e^{-r_r}\right)^2\\ g(x) &= \sum_i V_i x^{t_i} = 120x^2-150x^1+10x^0 \end{align} $$

Because $x=e^{-r_r}$, all real (positive or negative) rates of return, $r_r$, are mapped onto the $x>0$ space. Therefore, we are only interested in the positive real roots of $g(x)$.

Descartes' sign rule: When a polynomial is ordered by descending variable exponent, the number of positive real roots is less than or equal to the number of sign changes. If it is less, it is less by an increment of 2.

In the example case, we have two sign changes (positive 120 to negative 150 and negative 150 to positive 10) so we have either 2 or 0 positive roots. In the more general case, we have shown that any investment scheme may have multiple roots if there are a series of alternating investments and withdrawls over the course of the investment lifetime. If there are only investments and a single withdrawl at the end, there will always be exactly 1 real root, one IRR, because there will be only one sign change in the coefficients of $g(x)$.

References

[1] J. H. Lorie, and L. J. Savage, "Three problems in rationing capital," The Journal of Business, vol. 28, pp. 229—239, 1955.

[2] G. B. Hazen, "A new perspective on multiple internal rates of return," The Engineering Economist, vol. 48, pp. 31—51, 2003.

[3] J. R. Lohmann, "The IRR, NPV and the fallacy of the reinvestment rate assumptions," The Engineering Economist, vol. 33, pp. 303—330, 1988.

\( F \)	Income stream ($/time)
\( \text{NPV} \)	Net present value
\( V_t \)	Value of cash received at time \(t\)
\( r_r \)	Hypothetical rate of return
\( g(r_r) \)	Error function - root of \(g\) is the internal rate of return