Interpreting Multiple or Missing IRRs

Why IRR Needs Interpretation

An internal rate of return is a rate that makes a cash flow’s net present value equal to zero. For a conventional cash-flow pattern, the IRR is often a single useful summary. When cash flows change sign more than once, however, the equation can have multiple roots, a repeated root, or no positive root. In those cases, reporting one rate without explaining the other possibilities can give a misleading impression.

The key is to treat the IRR equation as an equation to solve, not as a rule that automatically selects one answer. Report the relevant roots, state whether they are positive, and use the cash-flow pattern and NPV at any specified rate to describe the result. A candidate working with these cases in material aligned with the SOA Exam FM (Financial Mathematics) syllabus should not pick the most convenient root merely because it was found first.

Definition: For cash flows $C_0,C_1,\ldots,C_n$ at times $0,1,\ldots,n$, with $C_t$ positive for a receipt and negative for an outflow, the NPV at an effective rate per period $r$ is $$ \operatorname{NPV}(r)=\sum_{t=0}^{n}\frac{C_t}{(1+r)^t}. $$ An IRR is a rate $r>-1$ satisfying $\operatorname{NPV}(r)=0$. A positive IRR is an admissible root with $r>0$. The restriction $r>-1$ ensures the discount factor $1/(1+r)$ is positive and defined.

The period in the rate must match the spacing of the cash flows. If the flows occur annually, $r$ is an effective annual rate; if they occur quarterly at equally spaced dates, $r$ is an effective quarterly rate. A root of the equation is not automatically positive, and “no positive IRR” does not mean “no admissible IRR.”

How to Interpret the Root Set

For equally spaced cash flows, set $x=1+r$. Since $r>-1$, the relevant domain is $x>0$. Multiplying the NPV equation by $x^n$ gives a polynomial equation:

\operatorname{NPV}(r)=0 \quad\Longleftrightarrow\quad C_0x^n+C_1x^{n-1}+\cdots+C_n=0, \qquad x=1+r>0.

This transformation helps reveal all roots, but it also requires a domain check: a polynomial solution with $x\leq 0$ does not correspond to an admissible rate. For each admissible solution, convert back using $r=x-1$, then determine whether that rate is positive, zero, or negative.

There is a second interpretive check. A root need not be accompanied by a change in the sign of NPV. If the root has even multiplicity, NPV can touch zero and turn back without crossing it. A method that searches only for opposite NPV signs at two trial rates can therefore miss such a root.

Key takeaway: The correct IRR result may be a list of rates, a repeated rate, or “no positive IRR.” State which case applies. Do not report a single rate unless the cash-flow equation and the question justify doing so.

Worked Examples

Worked Example: Two Positive Roots and a Specified Hurdle Rate

A hypothetical three-year project has cash flows of $-\$100$ now, $+\$230$ at the end of year 1, and $-\$132$ at the end of year 2. Find all positive IRRs. Then evaluate the project’s NPV at 15% per year and interpret the result. Assume the cash flows occur annually and the rate is effective annual.

Method. Write the NPV equation, transform it to a polynomial in $x=1+r$, and solve for every admissible root.

The NPV equation is

-100+\frac{230}{1+r}-\frac{132}{(1+r)^2}=0.

Let $x=1+r$, with $x>0$. Multiplying by $x^2$ gives

-100x^2+230x-132=0, \qquad 100x^2-230x+132=0.

The discriminant is $230^2-4(100)(132)=52{,}900-52{,}800=100$. Thus

x=\frac{230\pm 10}{200}=1.10\ \text{or}\ 1.20.

Both values are positive, so both are admissible. Converting back gives $r=1.10-1=0.10$, or 10%, and $r=1.20-1=0.20$, or 20%. The project has two positive IRRs.

At the stated hurdle rate of 15%, $1+r=1.15$. The NPV is

\operatorname{NPV}(0.15) =-100+\frac{230}{1.15}-\frac{132}{1.15^2} =-100+200-\frac{52{,}800}{529} \approx \$0.19.

The NPV is positive by about 19 cents, rounded to the nearest cent. This calculation evaluates the cash flows at the specified rate; it does not turn either IRR into the unique answer. For a check, at 10% the discounted receipts are $230/1.10=209.0909$ and $132/1.10^2=109.0909$, so NPV is $-100+209.0909-109.0909=0$. At 20%, NPV is $-100+230/1.20-132/1.20^2=0$ as well.

Common mistake avoided: Reporting only 10% because it is the smaller root, or only 20% because it is the larger root, arbitrarily discards a valid solution. Report both roots and separately state the NPV result at the given rate.

Worked Example: A Repeated IRR Without an NPV Sign Change

A hypothetical project has annual cash flows $-\$100$, $+\$240$, and $-\$144$ at times 0, 1, and 2. Find its positive IRR and determine whether NPV changes sign as the rate passes through that root. Assume an effective annual rate.

Method. Factor the transformed polynomial and inspect the sign on both sides of the root.

With $x=1+r>0$, multiplying NPV by $x^2$ gives

$$ -100x^2+240x-144 =-4(5x-6)^2=0. $$

The only polynomial root is $x=6/5=1.20$, and it has multiplicity two. Therefore $r=x-1=0.20$, or 20%. To check the NPV shape, divide the factored expression by $x^2$:

\operatorname{NPV}(r) =-4\left(5-\frac{6}{x}\right)^2 =-4\left(5-\frac{6}{1+r}\right)^2.

A negative number multiplied by a square is never positive. NPV equals zero only when $5-6/x=0$, which gives $x=1.20$. Thus NPV is negative immediately below and immediately above the 20% root; it touches zero but does not cross it. For instance, at 19%, $x=1.19$ and NPV is approximately $-\$0.0071$; at 21%, $x=1.21$ and NPV is approximately $-\$0.0068$.

The positive IRR is 20%, with multiplicity two. A bracketing procedure that requires one trial NPV to be positive and the other negative will not detect this root across a bracket surrounding 20%.

Common mistake avoided: Concluding that there is no root because NPV has the same sign on both sides is incorrect. The equation itself shows a repeated root; a sign change is sufficient evidence of a crossing root, but it is not necessary for every root.

Worked Example: No Positive IRR but One Negative Admissible IRR

A hypothetical project has annual cash flows $-\$100$, $+\$50$, and $+\$40$ at times 0, 1, and 2. Determine whether it has a positive IRR. If possible, identify its other admissible IRR. Assume an effective annual rate.

Method. First bound NPV over all nonnegative rates. Then solve the transformed equation and check the domain.

For $r\geq 0$, $x=1+r\geq1$, so $50/x\leq50$ and $40/x^2\leq40$. Therefore

\operatorname{NPV}(r) =-100+\frac{50}{x}+\frac{40}{x^2} \leq -100+50+40=-10<0.

NPV cannot equal zero at any nonnegative rate, so the project has no positive IRR. To check whether it has any other admissible roots, multiply the IRR equation by $x^2$:

-100x^2+50x+40=0, \qquad 100x^2-50x-40=0.

The quadratic formula gives

x=\frac{50\pm\sqrt{50^2+4(100)(40)}}{200} =\frac{50\pm\sqrt{18{,}500}}{200} \approx 0.9300735,\ -0.4300735.

The negative $x$ value is outside the admissible domain $x>0$. The positive value gives $r=0.9300735-1=-0.0699265$, approximately $-6.993\%$, rounded to three decimal places. It is an admissible negative IRR, not a positive IRR.

Common mistake avoided: “No positive IRR” is not the same as “no root.” Here an admissible negative root exists, while the NPV bound proves that no root can occur for $r\geq0$.

A Reliable Reporting Process

The algebra and the interpretation are separate tasks. Solving for roots identifies rates that make NPV zero; it does not establish that one root should be preferred. A disciplined report makes the distinction explicit.

State the timing and rate unit.
Specify the cash-flow periods and whether the result is an effective rate per year, quarter, or other period.

Solve for all relevant roots.
Use the NPV equation or its polynomial form, and retain only solutions satisfying $x=1+r>0$.

Classify the rates.
Separate positive roots from zero or negative roots. If a root is repeated, report that fact when it explains the NPV profile.

Check the interpretation.
Substitute each reported root into NPV, and evaluate NPV at any rate specified in the problem. Do not infer a sign change merely from the existence of a root.

This process matters because an IRR is a property of the cash-flow equation, while choosing how to assess an investment requires a stated comparison framework. With multiple roots, there is no general mathematical reason to select the highest, lowest, or first root found. With no positive root, a candidate should report that fact rather than invent a positive rate or mistake a negative root for a positive one.

A sound answer therefore gives the root set and its interpretation in context. For example: “The annual cash flows have IRRs of 10% and 20%; therefore the cash flows do not identify a unique IRR. At the stated 15% rate, NPV is positive by approximately 19 cents.” This reports the mathematical result and the specified-rate calculation without arbitrarily choosing one root.

Check Your Understanding

For each problem, identify the positive IRRs or state that none exist. Assume equally spaced annual cash flows and effective annual rates.

A project has cash flows $-\$50$, $+\$115$, and $-\$66$ at times 0, 1, and 2. Find all positive IRRs. (Answer: 10% and 20%.)
A project has cash flows $-\$100$, $+\$240$, and $-\$144$. Find the IRR and state whether NPV changes sign at it. (Answer: 20%, a repeated root; NPV touches zero without changing sign.)
A project has cash flows $-\$120$, $+\$60$, and $+\$48$. Determine whether a positive IRR exists. (Answer: No positive IRR; for $r\geq0$, NPV $\leq-12<0$.)
A project has cash flows $-\$100$, $+\$260$, and $-\$165$. Find all positive IRRs. (Answer: 10% and 50%.)
A project has cash flows $-\$100$, $+\$50$, and $+\$40$. State whether it has a positive IRR and, if not, give its admissible negative IRR to three decimal places. (Answer: No positive IRR; the admissible negative IRR is approximately $-6.993\%$.)

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