Incremental IRR and Mutually Exclusive Projects

Why Compare Projects Incrementally?

When projects are mutually exclusive, accepting one rules out the other. Comparing each project's IRR separately can therefore give a misleading ranking: IRR is a rate of return, while the decision also depends on how much additional cash must be invested and what additional cash flows that investment produces. The incremental method focuses on that difference.

Suppose project B requires more money at the outset than project A. The relevant question is not simply whether B has a higher IRR than A. It is whether the extra initial outlay for B earns an adequate return through B's extra future cash flows. The incremental IRR measures that return; incremental NPV at the specified discount rate provides a direct value comparison.

Definition: For two projects with cash flows on the same dates, the incremental cash flow for choosing B rather than A is $B-A$: subtract A's cash flow from B's cash flow at each date. The incremental IRR is a rate $i^*$ that makes the NPV of those difference cash flows equal to zero.

Assume annual cash flows occur at the ends of years $1,\ldots,n$, with an initial cash flow at time 0. At an effective annual discount rate $i>-1$, let $C_{A,t}$ and $C_{B,t}$ denote the cash flows of projects A and B at time $t$. The NPV difference is

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

This identity follows by subtracting the two NPV sums term by term: both projects use the same discount rate and the same dates, so each pair of discounted cash flows subtracts to the discounted difference. It is the central reason the incremental method works. If the incremental cash flows are conventional—an initial outflow followed by nonnegative inflows—and have a unique IRR, then the incremental NPV is positive below that rate and negative above it. Thus, for a higher-investment B, B is preferred to A when $i$ is below the incremental IRR; A is preferred when $i$ is above it. At equality, the projects have equal NPV.

A Reliable Comparison Procedure

Use the method only after confirming that the alternatives are being compared on a common basis: the same valuation date, consistent cash-flow dates, and a stated discount rate. If the projects have different horizons, include all relevant cash flows through a common comparison horizon or otherwise make the valuation basis consistent. Do not compare amounts that omit different fees, terminal values, or financing cash flows.

Choose the comparison direction.
For B versus A, calculate B minus A at every date. State clearly which alternative the resulting difference represents.

Inspect the incremental cash flows.
Check their signs and timing. A conventional pattern supports a single economically interpretable incremental IRR; a pattern with multiple sign changes may not.

Solve for the incremental IRR.
Set the NPV of the difference cash flows to zero and solve for the rate. Verify that the rate is admissible and satisfies the equation.

Check NPV at the required rate.
Calculate the incremental NPV at the specified discount rate. A positive result means B has higher NPV than A; a negative result means A has higher NPV.

The incremental IRR is a crossover rate: the rate at which the two NPVs are equal, provided the equation has an appropriate solution. It is not automatically the return earned by either project on its own. If the option to reject both projects is available, also check whether the preferred project has positive NPV relative to that outside option.

Worked Example: A Corrected Incremental IRR Calculation

Projects A and B have annual cash flows at times 0, 1, and 2. A has cash flows $-\$60,\ \$40,\ \$40$, and B has cash flows $-\$75,\ \$50,\ \$50$. Find the incremental IRR for choosing B rather than A. Then decide between the projects at a 10% effective annual discount rate.

Method and direction. Calculate B minus A. This makes a positive incremental NPV evidence in favor of B. The incremental cash flows are $-\$15,\ \$10,\ \$10$: B requires an extra \$15 initially and provides an extra \$10 at each later date.

Set the incremental NPV equal to zero. Write $x=1+i^*$, so $x$ is the one-year accumulation factor:

-15+\frac{10}{x}+\frac{10}{x^2}=0, \qquad 15x^2=10x+10, \qquad 3x^2-2x-2=0.

The quadratic formula gives $x=(2+\sqrt{28})/6=(1+\sqrt{7})/3$, using the positive root. Therefore

i^*=\frac{1+\sqrt{7}}{3}-1 =0.215250\ldots =21.53\%.

The incremental IRR is approximately 21.53%, rounded to two decimal places. Check the decision at 10% directly:

\operatorname{NPV}_{B-A}(0.10) =-15+\frac{10}{1.10}+\frac{10}{1.10^2} =-15+9.090909\ldots+8.264463\ldots =\$2.355372\ldots.

The incremental NPV is positive, so B has higher NPV than A at 10%. This agrees with the rate comparison: 10% is below the incremental IRR of 21.53%. As a check, the difference NPV is zero when evaluated at $i^*$, by construction. Common mistake avoided: rounding the root too early or reporting 21.52%; the value rounds to 21.53% to two decimal places.

Incremental IRR and the Specified Discount Rate

The incremental IRR can summarize where the alternatives switch in NPV ranking, but the decision is always tied to the rate used to value the cash flows. The next example illustrates why a positive incremental result does not necessarily mean that either project should be undertaken when rejecting both is an option.

Worked Example: Choosing Between Projects When Both NPVs Are Negative

Project A has cash flows $-\$100,\ \$80,\ \$40$, and project B has cash flows $-\$150,\ \$110,\ \$80$, at times 0, 1, and 2. The alternatives are mutually exclusive. Calculate the incremental IRR for B over A, compare the projects at a 20% effective annual discount rate, and state what to do if doing neither is allowed.

Method. Subtract A from B to value the extra investment in B. The difference cash flows are $-\$50,\ \$30,\ \$40$, a conventional pattern.

-50+\frac{30}{1+i^*}+\frac{40}{(1+i^*)^2}=0.

Let $x=1+i^*$. Multiplication by $x^2$ gives $50x^2-30x-40=0$, or $5x^2-3x-4=0$. The positive root is $x=(3+\sqrt{89})/10=1.243398\ldots$, so

i^*=0.243398\ldots=24.34\%.

At 20%, calculate the incremental NPV:

\operatorname{NPV}_{B-A}(0.20) =-50+\frac{30}{1.20}+\frac{40}{1.20^2} =-50+25+27.777778\ldots =\$2.777778\ldots.

Thus B has higher NPV than A at 20%. The individual NPVs show why this relative conclusion is not the same as an accept decision:

\begin{aligned} \operatorname{NPV}_A(0.20) &=-100+\frac{80}{1.20}+\frac{40}{1.20^2} =-100+66.666667\ldots+27.777778\ldots =-\$5.555556\ldots,\\ \operatorname{NPV}_B(0.20) &=-150+\frac{110}{1.20}+\frac{80}{1.20^2} =-150+91.666667\ldots+55.555556\ldots =-\$2.777778\ldots. \end{aligned}

The difference between these NPVs is $ -2.777778\ldots-(-5.555556\ldots)=\$2.777778\ldots$, matching the incremental NPV. If the choice is strictly A or B, choose B. If doing neither is available and has NPV zero, neither project is preferred to that option at 20%, because both project NPVs are negative. Common mistake avoided: interpreting “B is better than A” as “B is worthwhile” without checking NPV against zero.

The Crossover Rate and Direction of the Difference

A crossover rate is the discount rate at which two projects have equal NPV. The incremental IRR gives that rate when it solves the difference-NPV equation. The sign of the incremental cash flows matters for interpreting the result: if A is subtracted from B, positive incremental NPV favors B; reverse the subtraction and every incremental cash flow and the NPV difference change sign.

Worked Example: Finding Which Project to Prefer Above and Below the Crossover

Project B has cash flows $-\$80,\ \$50,\ \$50$, and project A has cash flows $-\$120,\ \$74,\ \$74$, at times 0, 1, and 2. Find the incremental IRR for A over B. Compare the projects at 10% and 15%.

Method. A requires \$40 more initially and produces \$24 more at each later date. The incremental cash flows A minus B are $-\$40,\ \$24,\ \$24$. Set their NPV to zero and let $x=1+i^*$:

-40+\frac{24}{x}+\frac{24}{x^2}=0, \qquad 5x^2-3x-3=0, \qquad x=\frac{3+\sqrt{69}}{10}=1.130662\ldots.

Therefore $i^*=0.130662\ldots=13.07\%$, rounded to two decimal places. At 10%, the incremental NPV is

\operatorname{NPV}_{A-B}(0.10) =-40+\frac{24}{1.10}+\frac{24}{1.10^2} =-40+21.818182\ldots+19.834711\ldots =\$1.652893\ldots.

A is preferred at 10%, which is below the crossover rate. At 15%,

\operatorname{NPV}_{A-B}(0.15) =-40+\frac{24}{1.15}+\frac{24}{1.15^2} =-40+20.869565\ldots+18.147448\ldots =-\$0.982987\ldots.

B is preferred at 15%, which is above the crossover rate. The sign change in incremental NPV confirms the switch in ranking. Common mistake avoided: treating the incremental IRR as a recommendation on its own. It identifies the crossover; the specified discount rate determines which side of the crossover applies.

Conditions, Limits, and Common Exam Traps

For conventional incremental cash flows, the pattern is an initial negative difference followed by nonnegative differences, with at least one positive future difference. In this setting, the incremental IRR has a clear interpretation as the rate earned on the extra investment, and its position relative to the discount rate determines the NPV ranking. The rate should still be checked by substitution, especially when solving a polynomial or using a numerical method.

If the incremental cash flows change sign more than once, the equation may have multiple IRRs or no economically useful real IRR. A single “incremental IRR” comparison may then be ambiguous. Use incremental NPV at the stated discount rate to compare the projects directly; the NPV difference identity remains valid. This is particularly important if one project's early cash flows exceed the other's but later cash flows reverse that relationship.

Subtract in a stated direction. Write “B minus A” or “A minus B” before constructing the row. Reversing direction reverses which project a positive difference favors.
Keep timing consistent. Discount each difference cash flow at its actual date; do not treat a year-2 cash flow as though it occurred at year 1.
Do not compare only individual IRRs. A larger standalone IRR does not by itself establish which mutually exclusive project has the higher NPV.
Distinguish relative and absolute decisions. The better of two projects can still have negative NPV compared with doing neither.
Do not rely on an incremental IRR with an irregular sign pattern. Check the incremental NPV directly at the specified rate when uniqueness or interpretation is uncertain.

Key takeaway: Construct the difference cash flows, solve their zero-NPV equation when an incremental IRR is meaningful, and verify the ranking using incremental NPV at the required discount rate. The rate comparison is a shortcut for a conventional incremental pattern, not a replacement for checking the project's value and the available alternatives.

Check Your Understanding

For each problem, state the incremental cash flows in the direction specified, then use the stated discount rate to make the project comparison. Assume annual cash flows occur at times 0, 1, and 2, and all rates are effective annual rates.

Project A has cash flows $-\$40,\ \$25,\ \$25$, and B has $-\$50,\ \$31,\ \$31$. Find the incremental IRR for B over A and decide at 8%. (Answer: Incremental IRR $=13.07\%$; incremental NPV at 8% $=\$0.70$ approximately, so choose B.)
Project A has cash flows $-\$100,\ \$65,\ \$65$, and B has $-\$130,\ \$83,\ \$83$. Find the incremental IRR for B over A and decide at 15%. (Answer: Incremental IRR $=13.07\%$; incremental NPV at 15% $=-\$0.74$ approximately, so choose A.)
Project A has cash flows $-\$70,\ \$45,\ \$45$, and B has $-\$95,\ \$60,\ \$60$. Find the incremental IRR for B over A and decide at 10%. (Answer: Incremental IRR $=13.07\%$; incremental NPV at 10% $=\$1.03$ approximately, so choose B.)
Project A has cash flows $-\$50,\ \$80,\ -\$35$, and B has $-\$70,\ \$100,\ -\$40$. Calculate the B-minus-A incremental NPV at 10%. What limits the use of a single incremental IRR decision rule here? (Answer: Difference cash flows are $-\$20,\ \$20,\ -\$5$; incremental NPV at 10% is approximately $-\$5.95$, so A has higher NPV. The difference cash flows change sign twice, so a single incremental IRR rule may be ambiguous.)
Project A has cash flows $-\$80,\ \$60,\ \$40$, and B has $-\$110,\ \$80,\ \$60$. Find the incremental IRR for B over A and decide at 18%. (Answer: Incremental IRR $=21.53\%$; incremental NPV at 18% is approximately $\$1.31$, so choose B.)

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