Incremental IRR for Mutually Exclusive Projects

Why Compare Incremental Cash Flows?

When projects are mutually exclusive, choosing the one with the largest standalone IRR can be misleading. A smaller project may have a higher percentage return, while a larger project may produce more value after accounting for the additional capital required. Incremental IRR addresses that question directly: what return is earned on the extra investment needed to move from one project to the other?

The comparison assumes the projects provide alternatives for the same decision, have cash flows measured at the same dates, and are evaluated over a common horizon. Use the same sign convention for both projects: amounts paid by the investor are negative, and amounts received are positive. If their horizons or timing do not match, the cash flows must first be put on a comparable basis; a direct period-by-period subtraction would not be valid.

Definition: For two mutually exclusive projects $A$ and $B$, the incremental cash flow at time $t$ for choosing $B$ instead of $A$ is $\Delta C_t=C_{B,t}-C_{A,t}$. The incremental IRR, written $i_{\Delta}$, is a rate that makes the present value of these incremental cash flows equal to zero. If $B$ requires the larger initial outlay, then $\Delta C_0<0$: the incremental cash flows represent the additional investment and the additional receipts from choosing $B$.

For annual cash flows $C_t$ at times $t=0,1,\ldots,n$, the net present value at effective annual rate $i$ is the sum of each cash flow discounted to time 0. The incremental IRR is found by setting the NPV of the difference cash flows to zero.

\operatorname{NPV}_{\Delta}(i) =\sum_{t=0}^{n}\frac{\Delta C_t}{(1+i)^t}, \qquad \operatorname{NPV}_{\Delta}(i_{\Delta})=0.

This tutorial’s decision rule applies most directly when the incremental cash flows are conventional: the initial incremental cash flow is negative, followed only by nonnegative cash flows, with at least one positive receipt. In that setting there is at most one economically relevant IRR above $-1$, and, if a root exists, the incremental NPV is positive below that root and negative above it. If the incremental cash flows change sign more than once, there may be multiple IRRs or no IRR; use the incremental NPV at the required rate rather than relying on a simple comparison with one reported IRR.

The Incremental Decision Rule

Let project $B$ require more money at time 0 than project $A$. At the required effective annual rate $i$, compare the present values of their cash flows by subtracting one NPV from the other. The difference is exactly the NPV of choosing $B$ instead of $A$.

\operatorname{NPV}_{B}(i)-\operatorname{NPV}_{A}(i) =\sum_{t=0}^{n}\frac{C_{B,t}-C_{A,t}}{(1+i)^t} =\operatorname{NPV}_{\Delta}(i).

For conventional incremental cash flows, the derivative of the incremental NPV with respect to the discount rate is negative: each positive future incremental receipt is discounted more heavily as the rate rises. Therefore, the incremental NPV falls as $i$ rises and crosses zero at $i_{\Delta}$, if a root exists. This gives the rule below. It is not a rule to compare each project’s standalone IRR.

Theorem: If $B$ requires the larger initial investment and its incremental cash flows are conventional, choose $B$ when $i_{\Delta}>i$, choose $A$ when $i_{\Delta}<i$, and regard the projects as tied on NPV when $i_{\Delta}=i$. If the incremental pattern is nonconventional, compare $\operatorname{NPV}_{\Delta}(i)$ directly: choose $B$ if it is positive and $A$ if it is negative.

Set the order.
Identify which project requires the larger initial investment. Subtract the smaller-investment project’s cash flows from the larger-investment project’s cash flows, consistently at every date.

Inspect the incremental signs.
A negative initial amount followed by nonnegative amounts is the conventional pattern for applying the simple incremental IRR decision rule.

Find the incremental IRR or evaluate incremental NPV.
For conventional cash flows, solve for $i_{\Delta}$ and compare it with the required rate. For nonconventional cash flows, evaluate $\operatorname{NPV}_{\Delta}(i)$ at the required rate.

State the choice.
A positive incremental NPV means the larger-investment project adds value at the required rate; a negative incremental NPV means it does not.

Worked Comparisons

The examples report cash flows in thousands of dollars, with annual receipts at the end of each year. The required rate $i$ is an effective annual rate. Each comparison uses the same timeline for both projects.

Worked Example: Extra Investment Earns More Than the Required Rate

A firm must choose between two equipment plans. Plan $A$ costs $10 thousand initially and returns $6 thousand at the end of each of the next two years. Plan $B$ costs $15 thousand initially and returns $8 thousand after year 1 and $10 thousand after year 2. The required effective annual rate is $i=10\%$. Which plan should it select?

Method: form the incremental cash flows for choosing $B$ instead of $A$. Subtract $A$ from $B$ at each date:

\begin{array}{c|rrr} t&0&1&2\\ A&-10&6&6\\ B&-15&8&10\\ \Delta C_t=B-A&-5&2&4 \end{array}

These incremental cash flows are conventional. Let $x=1+i_{\Delta}$. The incremental IRR equation is

-5+\frac{2}{x}+\frac{4}{x^2}=0, \qquad 5x^2-2x-4=0.

The positive quadratic root is

x=\frac{2+\sqrt{84}}{10} =\frac{1+\sqrt{21}}{5} \approx 1.116515, \qquad i_{\Delta}\approx 0.116515=11.6515\%.

Since $11.6515\%>10\%$, the additional investment in $B$ earns more than the required rate. Select plan $B$. As a check, its incremental NPV at 10% is

-5+\frac{2}{1.10}+\frac{4}{1.10^2} =-5+1.818182+3.305785 =0.123967\text{ thousand dollars}.

This is about $123.97, a positive amount, confirming the choice. Common mistake avoided: comparing standalone IRRs instead of the return on the extra investment. Plan $A$’s standalone IRR is about 13.07%, while $B$’s is about 12.56%; ranking those rates alone would incorrectly favor $A$ at the 10% required rate.

Worked Example: Extra Investment Does Not Meet the Required Rate

A warehouse can install system $A$, with a $20 thousand initial cost and receipts of $12 thousand at the end of each of the next two years, or system $B$, with a $26 thousand initial cost and receipts of $16 thousand after year 1 and $15 thousand after year 2. The required effective annual rate is $i=12\%$. Choose a system.

Method: evaluate the return on the additional $6 thousand required by $B$. The incremental cash flows are

\Delta C_0=-26-(-20)=-6,\qquad \Delta C_1=16-12=4,\qquad \Delta C_2=15-12=3.

The pattern is conventional. With $x=1+i_{\Delta}$, solve

-6+\frac{4}{x}+\frac{3}{x^2}=0, \qquad 6x^2-4x-3=0.

Thus

x=\frac{4+\sqrt{88}}{12} \approx 1.115069, \qquad i_{\Delta}\approx 11.5069\%.

The incremental IRR is below 12%, so the extra investment does not meet the required rate. Select $A$. Check by discounting the incremental cash flows at 12%:

\operatorname{NPV}_{\Delta}(0.12) =-6+\frac{4}{1.12}+\frac{3}{1.12^2} =-6+3.571429+2.391582 =-0.036990\text{ thousand dollars}.

The result is approximately $-\$36.99$, confirming that $B$ has lower NPV at the required rate. Common mistake avoided: treating a positive incremental IRR as sufficient reason to choose $B$. The incremental return must exceed the required rate; here it does not.

When the Incremental Cash Flows Change Sign More Than Once

Subtracting two project cash-flow streams can create a nonconventional pattern, even if each project seems straightforward on its own. A later incremental cost, such as a removal or restoration expense, can introduce another sign change. In that case, multiple incremental IRRs may occur, and comparing a single IRR with the required rate can give the wrong decision. The reliable test is the sign of incremental NPV at the required rate.

Worked Example: Two Incremental IRRs Require an NPV Check

Consider two hypothetical facility options over two years. Option $A$ has a $100 thousand initial cost and no later cash flows. Option $B$ has a $200 thousand initial cost, a $230 thousand receipt after year 1, and a $132 thousand removal cost after year 2. The required effective annual rate is $i=15\%$. Which option has the greater NPV?

Method: subtract $A$ from $B$, then compare incremental NPV at the required rate. In thousands of dollars, the incremental cash flows are $-100,\ 230,\ -132$. Their signs change twice, so the simple one-IRR decision rule is not appropriate. To identify the issue, set $x=1+i_{\Delta}$:

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

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

x=\frac{230\pm 10}{200}=1.10\ \text{or}\ 1.20, \qquad i_{\Delta}=10\%\ \text{or}\ 20\%.

There are two incremental IRRs, one below and one above the required rate. Neither a single “incremental IRR” comparison nor the instruction “choose $B$ if an IRR exceeds 15%” resolves the choice. Evaluate incremental NPV directly:

\operatorname{NPV}_{\Delta}(0.15) =-100+\frac{230}{1.15}-\frac{132}{1.15^2} =-100+200-99.810965 =0.189035\text{ thousand dollars}.

The incremental NPV is about $189.04, so $B$ has the greater NPV at 15%. As a check, option $A$ has NPV $-\$100{,}000$, while option $B$ has NPV $-\$99{,}810.97$; their difference is approximately $189.04. Common mistake avoided: applying the conventional incremental IRR rule despite multiple sign changes. The NPV comparison gives the decision at the specified rate.

Common Mistakes and Exam Checks

Subtracting in inconsistent directions. If the intended comparison is $B$ instead of $A$, calculate $B-A$ at every date. Reversing the order reverses the signs and changes the interpretation.
Choosing the larger standalone IRR. Standalone rates measure returns on different total investments. For mutually exclusive projects, the incremental comparison answers whether the extra investment is worthwhile at the required rate.
Ignoring the cash-flow sign pattern. A conventional incremental pattern supports the direct IRR decision rule. Multiple sign changes require an incremental NPV check at the required rate.
Forgetting the equality case. If the incremental IRR equals the required rate, incremental NPV is zero: the projects tie on NPV at that rate.
Subtracting cash flows from unmatched dates. Each difference must compare amounts at the same time. Different project horizons need a justified common comparison before incremental IRR is used.

Key takeaway: Incremental IRR is the return on the extra investment required to move from one mutually exclusive project to another. When the incremental cash flows are conventional, choose the higher-investment project only if that return exceeds the required rate. If the incremental signs are nonconventional, compare incremental NPV at the required rate.

Check Your Understanding

For each question, assume annual end-of-year cash flows, equal project horizons, and the stated required effective annual rate. Report money in thousands of dollars unless stated otherwise.

Project $A$ has cash flows $(-12,7,7)$, and project $B$ has cash flows $(-16,9,10)$. Find the incremental IRR for $B-A$ and choose at a required rate of 12%. (Answer: $i_{\Delta}\approx 15.14\%$; choose $B$.)
Project $A$ has cash flows $(-20,12,12)$, and project $B$ has cash flows $(-28,17,17)$. Find the incremental IRR and choose at a required rate of 18%. (Answer: $i_{\Delta}\approx 16.26\%$; choose $A$.)
Project $A$ has cash flows $(-30,18,18)$, and project $B$ has cash flows $(-40,24,24)$. Use the incremental IRR rule to choose at a required rate of 8%. (Answer: $i_{\Delta}\approx 13.07\%$; choose $B$.)
The incremental cash flows for choosing $B$ are $(-50,120,-72)$. Find the incremental IRRs, then decide between $A$ and $B$ at a required rate of 15%. (Answer: incremental IRRs are 20% and 20%; incremental NPV is negative at 15%, so choose $A$.)
Two projects have the same initial outlay, but their later cash flows differ. Is a conventional incremental IRR decision rule automatically available? State the appropriate comparison at a specified required rate. (Answer: No; the time-0 difference is zero, so compare the projects’ NPVs directly at the required rate, or equivalently evaluate the NPV of their later cash-flow differences.)

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