Multiple IRRs from Sign-Changing Cash Flows

Why a Project Can Have More Than One IRR

For a conventional investment, the initial cash flow is an outflow and every later cash flow is an inflow. Its net present value generally moves in one direction as the rate changes, which supports a unique IRR. A project with later outflows as well as inflows is different: discounting those amounts at different rates can make the NPV equation equal zero more than once. In that case, the project has multiple internal rates of return.

The essential task is to solve the entire IRR equation, not to stop at the first rate found by a calculator or numerical method. For equally spaced annual cash flows, a change of variable turns the equation into a polynomial. Its roots can then be checked and translated into rates.

Definition: Given cash flows $C_0,C_1,\ldots,C_n$ at times $0,1,\ldots,n$, an annual effective internal rate of return (IRR) $i$ is a rate satisfying $$ \sum_{t=0}^{n} C_t(1+i)^{-t}=0, $$ with $i>-1$. A positive IRR is a solution with $i>0$. Cash flows are signed from the investor’s perspective: an investment outflow is negative and a receipt is positive.

The timing assumption matters. The equation above applies when each cash flow occurs at the stated integer year and the same annual effective rate is used to discount all periods. For other payment spacings, the exponent must match the actual time in years. This tutorial uses annual periods.

Use the Discount Factor as the Unknown

Directly solving for $i$ can leave terms involving powers of $1+i$. Instead, define the discount factor per year by

x=\frac{1}{1+i}.

For any admissible rate $i>-1$, the factor $x$ is positive. If the rate is positive, then $1+i>1$, so $0<x<1$. The IRR equation becomes a polynomial:

C_0+C_1x+C_2x^2+\cdots+C_nx^n=0.

Every root $x$ that is positive can be translated back using

i=\frac{1}{x}-1.

A positive root $x$ below 1 gives a positive IRR; a root above 1 gives a negative IRR between $-1$ and 0. A zero or negative root cannot represent a finite rate satisfying $i>-1$. Thus, finding polynomial roots and identifying admissible IRRs are separate steps.

Why the Polynomial Method Works

The substitution does not approximate the IRR equation. It rewrites the same equation exactly. For an annual cash flow $C_t$, its present value at rate $i$ is $C_t/(1+i)^t$. Since $x=1/(1+i)$, this equals $C_tx^t$. Adding the present values of all cash flows gives the polynomial form.

\sum_{t=0}^{n}\frac{C_t}{(1+i)^t} = \sum_{t=0}^{n}C_t\left(\frac{1}{1+i}\right)^t = \sum_{t=0}^{n}C_tx^t.

Consequently, each admissible root of the polynomial corresponds to an IRR, and substituting the rate obtained from that root back into the original NPV equation must give zero. Factoring is especially useful when the polynomial has simple roots; otherwise a numerical method can be used to locate roots, followed by the same admissibility and verification checks.

Solving and Checking for Multiple IRRs

The following process keeps the cash-flow timing, algebra, and rate interpretation separate. That separation is important because a correct polynomial root can still be an invalid IRR if it is not a positive discount factor.

Write the dated cash flows.
Keep the sign convention explicit and identify the period associated with each amount.

Form the NPV equation.
Discount each cash flow using its actual period and set their sum equal to zero.

Substitute the discount factor.
For annual periods, set $x=1/(1+i)$ and collect the coefficients into a polynomial.

Solve and screen the roots.
Retain positive roots for rates above $-1$; retain only roots in $(0,1)$ when seeking positive IRRs.

Convert and verify.
Calculate $i=1/x-1$ for each retained root and substitute into the original NPV equation.

Worked Example: Two IRRs from Three Annual Cash Flows

A hypothetical project has cash flows of $-\$20$ at time 0, $+\$65$ at the end of year 1, and $-\$50$ at the end of year 2. Assume annual effective compounding and find all positive IRRs. Method: form and factor the discount-factor polynomial. The main mistake to avoid is finding one root and assuming it is the only IRR.

Let $x=1/(1+i)$. The NPV equation is

$$ -20+65x-50x^2=0. $$

Factor the polynomial, checking that the expansion reproduces each cash-flow coefficient:

-50\left(x-\frac12\right)\left(x-\frac45\right) =-50x^2+65x-20.

The roots are $x=1/2$ and $x=4/5$. Both are in $(0,1)$, so both correspond to positive IRRs. Convert each one:

x=\frac12:\quad i=\frac{1}{1/2}-1=1=100\%, \qquad x=\frac45:\quad i=\frac{1}{4/5}-1=\frac14=25\%.

Check both rates in the NPV equation. At $100\%$, $x=1/2$, so $-20+65(1/2)-50(1/2)^2=-20+32.5-12.5=0$. At $25\%$, $x=4/5$, so $-20+65(4/5)-50(4/5)^2=-20+52-32=0$. Thus both rates are verified. The two negative cash flows separated by an inflow make it possible for the NPV to be zero at two distinct rates.

Worked Example: A Cubic with Two Positive IRRs

A hypothetical equipment project has annual cash flows $-\$30,+\$85,+\$15,-\$100$ at times 0 through 3, respectively. Assume annual effective compounding. Find all positive IRRs. Method: factor the cubic after changing variables. The common mistake to avoid is using a guessed factorization without expanding it to check every coefficient.

With $x=1/(1+i)$, the NPV polynomial is

$$ -30+85x+15x^2-100x^3=0. $$

The polynomial factors as

-100\left(x-\frac25\right)\left(x-\frac34\right)(x+1) =-100x^3+15x^2+85x-30.

The expansion matches the cash-flow coefficients: the constant is $-30$, the coefficient of $x$ is $85$, the coefficient of $x^2$ is $15$, and the coefficient of $x^3$ is $-100$. The roots are $2/5$, $3/4$, and $-1$. The root $-1$ is not a possible discount factor, because $x=1/(1+i)$ is positive whenever $i>-1$. The other two roots are between 0 and 1.

Convert the admissible roots:

x=\frac25:\quad i=\frac52-1=\frac32=150\%, \qquad x=\frac34:\quad i=\frac43-1=\frac13=33.333\ldots\%.

Check directly in the polynomial. For $x=2/5$, $-30+85(2/5)+15(4/25)-100(8/125)=-30+34+2.4-6.4=0$. For $x=3/4$, $-30+85(3/4)+15(9/16)-100(27/64) =-30+63.75+8.4375-42.1875=0$. Both rates therefore satisfy the original NPV equation. Rounded to two decimal places, they are $150.00\%$ and $33.33\%$.

Worked Example: Distinguishing Positive and Negative IRRs

A hypothetical project has annual cash flows $-\$20,+\$130,-\$220,+\$80$ at times 0 through 3. Assume annual effective compounding. Find its positive IRRs, and identify any other admissible rate. Method: factor the cubic and classify each root by its location. The mistake to avoid is reporting every polynomial root as a positive IRR.

Set $x=1/(1+i)$. The NPV equation is

$$ -20+130x-220x^2+80x^3=0. $$

Factoring and expanding to check the coefficients gives

80\left(x-\frac14\right)\left(x-\frac12\right)(x-2) =80x^3-220x^2+130x-20.

The roots are $1/4,1/2,$ and $2$. The first two are in $(0,1)$, so they produce positive rates:

x=\frac14:\quad i=4-1=300\%, \qquad x=\frac12:\quad i=2-1=100\%.

The root $x=2$ is positive, so it is admissible for some rate above $-1$, but it is greater than 1 and therefore corresponds to a negative rate:

x=2:\quad i=\frac12-1=-50\%.

Verify the roots in the polynomial. At $x=1/4$, the cash-flow terms sum to $-20+32.5-13.75+1.25=0$. At $x=1/2$, they sum to $-20+65-55+10=0$. At $x=2$, they sum to $-20+260-880+640=0$. Thus the project has two positive IRRs and one negative IRR. Reporting the $x=2$ root as a positive rate would confuse a positive discount factor with a discount factor below 1.

Common Mistakes and Exam Traps

Stopping at the first root. A solver may return one IRR for an initial guess without revealing other roots. Factor the polynomial when possible, or investigate other roots systematically.
Using the wrong coefficient order. The coefficient of $x^t$ is the cash flow at time $t$. In particular, the time-0 cash flow is the constant term.
Mixing up the rate and discount factor. The conversion is $i=1/x-1$, not $i=x-1$. For example, $x=4/5$ means $i=25\%$, not $-20\%$.
Calling every root a positive IRR. Positive rates require $0<x<1$. A root greater than 1 gives a negative rate, while a zero or negative root is not a valid discount factor.
Trusting a factorization without checking it. Expand the factors and compare every coefficient with the dated cash flows. A small coefficient error can change all claimed roots.
Checking only the transformed equation. After converting each candidate $x$ into $i$, verify the rate in the original NPV equation. This confirms that the timing and conversion were applied consistently.

Multiple IRRs matter because an IRR is a root of an equation, not automatically a unique summary of a project. When cash flows alternate between outflows and inflows, different rates can discount those amounts so that the same project has zero NPV. For exam calculations, the reliable sequence is to preserve the timeline, solve the complete equation, screen roots using the discount-factor conditions, and verify each reported rate. This approach is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each problem, assume the cash flows occur at the stated integer years and are discounted at one annual effective rate. Find the requested positive IRR or IRRs and check that the reported roots are feasible.

A project has cash flows $-\$6,+\$26,-\$24$ at times 0, 1, and 2. Find all positive IRRs. (Answer: $200\%$ and $33.33\%$)
A project has cash flows $-\$10,+\$60,-\$30,-\$100$ at times 0 through 3. Find all positive IRRs. (Answer: $400\%$ and $100\%$)
A project has cash flows $-\$18,+\$87,-\$125,+\$50$ at times 0 through 3. Its discount-factor polynomial factors as $50(x-\frac25)(x-\frac35)(x-\frac32)$. Find the positive IRRs and identify the rate from the remaining root. (Answer: Positive IRRs $150\%$ and $66.67\%$; remaining rate $-33.33\%$)
A project has cash flows $-\$45,+\$140,-\$100$ at times 0, 1, and 2. Factor its NPV polynomial, calculate every positive IRR, and verify by substitution. (Answer: $100\%$ and $11.11\%$)

What You'll Learn