Projects With No Nonnegative IRR

Why a Project May Have No Nonnegative IRR

The internal rate of return is a rate that makes a project’s net present value equal to zero. Some projects do not have such a rate at all; others have real IRRs, but every one is negative. These are different outcomes. The distinction matters whenever an IRR calculation appears to fail or a project’s cash flows change direction more than once.

Assume cash flows occur at equally spaced year-ends, with $CF_t$ denoting the cash flow at time $t$, including the initial cash flow $CF_0$. For a finite project running through year $n$, the IRR equation uses the annual effective rate $r$. The standard domain is $r>-1$, so the discount factor is positive and finite. The nonnegative-rate domain is $r\geq 0$.

Definition: An IRR in the standard rate domain is a real rate $r>-1$ for which the project’s NPV is zero. A nonnegative IRR is an IRR with $r\geq 0$. A project has no nonnegative IRR if its NPV is never zero for $r\geq 0$; it may still have one or more negative IRRs.

To distinguish these cases, convert the rate equation into a polynomial and check its roots. A root is relevant only if it corresponds to a positive discount factor. A real root that is zero or negative does not produce an IRR in the standard rate domain.

From the IRR Equation to the Root Test

At an IRR, discounted cash flows sum to zero. Set $x=1/(1+r)$, the discount factor per year. Then $r=1/x-1$, and the IRR equation becomes a polynomial in $x$.

\operatorname{NPV}(r)=\sum_{t=0}^{n}\frac{CF_t}{(1+r)^t} =\sum_{t=0}^{n}CF_t x^t=0, \qquad x=\frac{1}{1+r}.

This substitution gives the key domain test. Since $r>-1$, a valid rate requires $x>0$. A nonnegative rate requires $0<x\leq 1$. If $x>1$, then $1/x<1$, so $r=1/x-1<0$. The value $x=0$ is not valid: it would require an infinite rate, not a finite IRR.

Theorem: Find the real roots of the IRR polynomial, but retain only its positive roots as valid discount factors. A positive root $x$ gives $r=1/x-1$. Roots in $0<x\leq 1$ give nonnegative IRRs; roots $x>1$ give negative IRRs. If the polynomial has no positive real roots, there is no IRR in the standard rate domain—even if the polynomial has real roots that are zero or negative.

The reason this test works is that the substitution is reversible on the standard domain. For every $r>-1$, $x=1/(1+r)>0$; conversely, every $x>0$ gives exactly one $r=1/x-1>-1$. Thus a positive polynomial root corresponds to one valid rate, and a nonpositive root does not. This changes the root-finding task into a domain check as well as an algebra problem.

Worked Examples

Worked Example: No Real IRR in the Standard Domain

A hypothetical project has annual cash flows, in thousands of dollars, of $-3$ at time 0, $+2$ at the end of year 1, and $-1$ at the end of year 2. Assume these amounts are certain, occur at the stated dates, and are discounted at an annual effective rate $r>-1$. Determine whether it has a nonnegative IRR or any IRR in the standard domain.

Method: form the IRR polynomial and check for positive roots. With $x=1/(1+r)$, the zero-NPV equation is

$$ -3+2x-x^2=0. $$

Complete the square:

-3+2x-x^2 =-(x^2-2x+3) =-\bigl((x-1)^2+2\bigr).

Because $(x-1)^2\geq 0$, the expression is at most $-2$, so it is strictly negative for every real $x$. In particular, it has no positive real root and therefore no IRR in the standard domain. At $r=0$, $x=1$ and NPV is $-3+2-1=-2$ thousand dollars, consistent with the result. The project has no nonnegative IRR and, more strongly, no real IRR in the standard rate domain.

Common mistake avoided: A changing sign in the cash flows does not guarantee an IRR. The completed-square result establishes that the NPV polynomial never reaches zero.

Worked Example: A Real IRR That Is Negative

A hypothetical equipment project has cash flows of $-4$, $+4$, and $-1$ thousand dollars at times 0, 1, and 2. Cash flows are certain and occur at year-end; use the annual effective rate $r>-1$. Find its IRR and determine whether it is nonnegative.

Method: factor the polynomial, then convert each valid root. The IRR equation is

-4+4x-x^2=0 \quad\Longleftrightarrow\quad -(x-2)^2=0.

The only root is $x=2$, which is positive and therefore valid. Converting back to the rate gives

r=\frac{1}{2}-1=-0.5=-50\%.

This is a real IRR, but it is negative. To check directly, at $r=-50\%$, $1+r=0.5$, so the NPV is

-4+\frac{4}{0.5}+\frac{-1}{(0.5)^2} =-4+8-4=0 \quad\text{(thousand dollars).}

For a further domain check, every nonnegative rate has $0<x\leq 1$, whereas the root $x=2$ lies above 1. Hence the project has no nonnegative IRR, though it does have a negative IRR.

Common mistake avoided: Reporting “no IRR” because there is no root between 0 and 1 discards a valid positive discount-factor root. The corresponding rate is negative, not absent.

Worked Example: Two Negative IRRs and No Nonnegative IRR

A hypothetical redevelopment project has cash flows of $-6$, $+5$, and $-1$ thousand dollars at times 0, 1, and 2. Assume certain year-end cash flows and an annual effective rate $r>-1$. Find all standard-domain IRRs and check whether any is nonnegative.

Method: factor the polynomial and test each root against the rate domain. Substituting $x=1/(1+r)$ gives

-6+5x-x^2=0 \quad\Longleftrightarrow\quad -(x-2)(x-3)=0.

The roots are $x=2$ and $x=3$. Both are positive, so both correspond to standard-domain IRRs. Convert them separately:

x=2:\quad r=\frac{1}{2}-1=-0.5=-50\%, \qquad x=3:\quad r=\frac{1}{3}-1=-\frac{2}{3}\approx-66.67\%.

Both rates are negative. Check the sign over the nonnegative-rate domain: if $r\geq 0$, then $0<x\leq 1$, so both $x-2$ and $x-3$ are negative. Their product is positive, and its negative is strictly negative. Thus NPV cannot equal zero for any $r\geq 0$. At $r=0$, NPV is $-6+5-1=-2$ thousand dollars, which confirms the sign at that rate.

Common mistake avoided: Counting roots without converting them can obscure the decision-relevant result. Both roots represent real rates, but neither is nonnegative.

Interpreting “No IRR” Carefully

The phrase “no IRR” is often used loosely. It may mean that there is no nonnegative IRR, or it may mean that there is no real IRR in the standard domain at all. State which meaning applies. The polynomial can have real roots that are not valid discount factors. For example, the polynomial $x+1=0$ has the real root $x=-1$, but that root corresponds to no rate $r>-1$. It is not a negative IRR: a negative IRR still requires $x>1$.

A project with no nonnegative IRR may have negative IRRs, as in the second and third examples, or no standard-domain IRR, as in the first. When the IRR polynomial has several roots, classify each positive root rather than deciding from the number of real roots alone.

The absence of a nonnegative IRR is not, by itself, an accept-or-reject rule. To make a decision, calculate NPV at the required return. If the required return is nonnegative and the project has no nonnegative IRR, NPV cannot equal zero at that required return; its sign still needs to be established. Since NPV is continuous for $r\geq 0$, if it has no zero on that interval its sign is constant there. Its value at one point, such as the required return or $r=0$, determines that sign. This is why a root diagnosis and an NPV decision are related but not interchangeable.

A Practical Root-Diagnosis Sequence

Use the following sequence when an IRR calculation produces no acceptable nonnegative rate or gives an unexpected answer.

Write the cash-flow equation.
Include the initial cash flow and each later cash flow at its correct time. Confirm the periods are equally spaced and that the rate is expressed per period.

Substitute the discount factor.
Set $x=1/(1+r)$, and write the resulting polynomial $\sum_{t=0}^{n}CF_t x^t=0$.

Find and classify real roots.
Keep only roots with $x>0$. A root with $0<x\leq 1$ gives a nonnegative rate; a root with $x>1$ gives a negative rate. A root $x\leq 0$ gives no standard-domain IRR.

Convert and verify.
For each valid root, compute $r=1/x-1$, substitute the rate into the original NPV equation, and report whether the rate is economically usable for the decision being considered.

Common Traps and Decision Relevance

Calling every missing nonnegative root “no real IRR.” A root $x>1$ is positive and valid; it corresponds to a negative rate.
Accepting any real polynomial root. The standard rate domain requires $x>0$. A zero or negative root does not correspond to an IRR in that domain.
Assuming a sign change guarantees an IRR. The first worked example has alternating cash-flow signs but a polynomial that is strictly negative for every real $x$.
Using “no nonnegative IRR” as the decision. Evaluate NPV at the relevant required return. IRR availability and NPV acceptability are not identical statements.
Ignoring the rate domain. This tutorial uses rates $r>-1$, the domain in which the annual discount factor is positive. A mathematical solution outside that domain is not treated as a standard IRR.

Key takeaway: A project has no nonnegative IRR when its IRR polynomial has no root in $0<x\leq 1$. It may still have valid negative IRRs from roots $x>1$, or it may have no standard-domain IRR if there are no positive real roots. Classify roots by the discount-factor domain before drawing a conclusion.

Check Your Understanding

For each hypothetical project, assume certain cash flows occur at year-ends, with $r>-1$. Use $x=1/(1+r)$ to classify the IRRs and, where requested, evaluate NPV.

A project has cash flows $-8,+3,-1$ thousand dollars at times 0, 1, and 2. Determine whether it has a real IRR in the standard domain. (Answer: No standard-domain IRR; the polynomial has no positive real roots.)
A project has cash flows $-6,-11,-6,-1$ dollars at times 0, 1, 2, and 3. Its polynomial is $-x^3-6x^2-11x-6$. Classify its real roots and state whether any gives an IRR. (Answer: The roots are $-1,-2,-3$; there is no standard-domain IRR.)
A project has cash flows $-10,+7,-1$ thousand dollars at times 0, 1, and 2. Find its IRRs and state whether it has a nonnegative IRR. (Answer: The discount-factor roots are $2$ and $5$, giving IRRs $-50\%$ and $-80\%$; no nonnegative IRR.)
A project has cash flows $-1,+2.5,-1$ thousand dollars at times 0, 1, and 2. Find all IRRs and identify any nonnegative one. (Answer: The discount-factor roots are $0.5$ and $2$, giving IRRs $100\%$ and $-50\%$; the nonnegative IRR is $100\%$.)
A project has cash flows $-4,+3,-1$ thousand dollars at times 0, 1, and 2. Confirm that it has no standard-domain IRR, then calculate NPV at a 10% annual effective required return. (Answer: No standard-domain IRR; NPV is $-254/121\approx-\$2.10$ thousand.)

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