Finding the Crossover Rate

Why Crossover Rates Matter

Two projects can have different NPVs at one discount rate and equal NPVs at another. The rate at which their NPVs are equal is the crossover rate. Finding it helps locate where their NPV rankings change as the discount rate varies. The calculation is a root-finding problem: form the difference between the projects’ cash flows and find the rate at which that difference has zero NPV.

This tutorial assumes that both projects are evaluated at the same valuation date, use annual cash flows paid at each year-end, and have cash flows specified over the same periods. Use a common timeline: if one project has no cash flow in a period, enter zero for that project in that period. Unless a problem says otherwise, consider nonnegative effective annual discount rates.

Definition: For projects A and B with cash flows $CF_{A,t}$ and $CF_{B,t}$ in period $t$, define the incremental cash flow as $D_t=CF_{A,t}-CF_{B,t}$. A crossover rate $r$ is a permitted discount rate for which $NPV_A(r)=NPV_B(r)$, or, equivalently, the NPV of the incremental cash flows is zero.

Constructing the Incremental Cash Flows

Subtract the cash flows period by period, keeping the order consistent. If $D_t=CF_{A,t}-CF_{B,t}$, a positive $D_t$ means A has a larger cash inflow, or a smaller cash outflow, than B in period $t$. A negative $D_t$ means the reverse. Reversing the subtraction changes the signs of all the differences, but it does not change the rates at which the NPV difference is zero.

With end-of-year cash flows and an effective annual discount rate $r$, each project’s NPV is the present value of its dated cash flows. The crossover condition follows by subtracting the two NPV expressions:

\begin{aligned} NPV_A(r)&=\sum_{t=0}^{n}\frac{CF_{A,t}}{(1+r)^t},\\ NPV_B(r)&=\sum_{t=0}^{n}\frac{CF_{B,t}}{(1+r)^t},\\ NPV_A(r)-NPV_B(r) &=\sum_{t=0}^{n}\frac{CF_{A,t}-CF_{B,t}}{(1+r)^t} =\sum_{t=0}^{n}\frac{D_t}{(1+r)^t}. \end{aligned}

Therefore $NPV_A(r)=NPV_B(r)$ precisely when the final sum is zero. That is the IRR equation for the incremental cash flows. This is a derivation of the crossover calculation, not a claim that either project’s standalone IRR is the crossover rate.

Key result: To find a crossover rate, subtract one project’s cash flows from the other’s in every period, set the NPV of those differences equal to zero, and solve for $r$. Then check each root in the original incremental NPV equation and retain only roots in the rate range allowed by the problem.

A Reliable Calculation Process

Align the periods.
Write both projects’ cash flows on the same timeline, including any zero cash flows.

Subtract consistently.
Calculate $D_t=CF_{A,t}-CF_{B,t}$ for each period, including time zero.

Set the incremental NPV to zero.
Use the effective annual rate and each cash flow’s correct time exponent.

Solve and check roots.
Check the roots against the original equation and the permitted discount-rate range. If the cash-flow differences change sign more than once, do not assume there can be only one root.

Worked Calculations

Worked Example: Two Projects with a 10% Crossover Rate

Two projects have year-end cash flows, in dollars, as follows. Find the nonnegative crossover rate. Project A has cash flows $(-500,250,350)$ at times 0, 1, and 2. Project B has cash flows $(-400,190,295)$ at those same times.

Method. Form the differences A minus B, then solve the incremental NPV equation. This method avoids comparing the projects’ standalone IRRs, which do not in general give the rate at which their NPVs are equal.

The incremental cash flows are $D_0=-500-(-400)=-100$, $D_1=250-190=60$, and $D_2=350-295=55$. Thus:

-100+\frac{60}{1+r}+\frac{55}{(1+r)^2}=0.

Set $x=1+r$, where $x>0$. Multiplying by $x^2$ gives $ -100x^2+60x+55=0$, or $100x^2-60x-55=0$. The quadratic formula gives $x=(60\pm\sqrt{60^2+4(100)(55)})/200=(60\pm\sqrt{25{,}600})/200$. Since $\sqrt{25{,}600}=160$, the roots are $x=1.10$ and $x=-0.50$. The negative $x$ root is inadmissible because $x=1+r>0$. The nonnegative crossover rate is $r=1.10-1=0.10$, or 10.00%.

Check. At 10%, A’s NPV is $-500+250/1.1+350/1.21=-500+227.2727+289.2562=16.5289$, or $16.53 to the nearest cent. B’s NPV is $-400+190/1.1+295/1.21=-400+172.7273+243.8017=16.5289$, also $16.53 to the nearest cent. The NPVs agree.

Worked Example: A 35.77% Crossover Rate

Project A has cash flows $(-250,160,117)$, while project B has cash flows $(-200,112,90)$, at times 0, 1, and 2. Find the nonnegative crossover rate and explain why the initial NPV gap is not the answer.

Method. Calculate A minus B by period. This avoids treating a dollar-valued NPV difference as if it were a percentage rate.

The incremental cash flows are $-250-(-200)=-50$ at time 0, $160-112=48$ at time 1, and $117-90=27$ at time 2. Set their NPV to zero:

-50+\frac{48}{1+r}+\frac{27}{(1+r)^2}=0.

Let $x=1+r$. Multiplying by $x^2$ gives $-50x^2+48x+27=0$, or $50x^2-48x-27=0$. Substituting $x=1+r$ and expanding gives $50r^2+52r-25=0$. The quadratic formula gives $r=(-52\pm\sqrt{52^2+4(50)(25)})/(2(50))=(-52\pm\sqrt{7704})/100$. The positive root is approximately $0.357724$; the other root is negative. The nonnegative crossover rate is therefore **35.77%**, rounded to two decimal places.

Check. The root satisfies $50r^2+52r-25=0$, which is the incremental NPV equation after multiplication by $(1+r)^2$. At a zero discount rate, A’s NPV is $-250+160+117=27$, while B’s is $-200+112+90=2$. Their initial NPV gap is $25, not 25%. Treating that $25 gap as a rate would give the wrong answer; the crossover rate comes from solving the equation.

Worked Example: Multiple Algebraic Roots and the Permitted Rate

Project A has cash flows $(-300,400,100)$, and project B has cash flows $(-208,171,149.5)$, at times 0, 1, and 2. Find all nonnegative crossover rates. Assume rates below zero are not permitted for this comparison.

Method. Solve the incremental NPV equation and inspect every root. This avoids assuming that a cash-flow difference with multiple sign changes must have just one algebraic root.

Subtract B from A to obtain incremental cash flows $-92$, $229$, and $-49.5$. The equation is $-92+229/(1+r)-49.5/(1+r)^2=0$. Put $x=1+r$ and multiply by $x^2$:

-92x^2+229x-49.5=0, \qquad 92x^2-229x+49.5=0.

The quadratic formula gives $x=(229\pm\sqrt{229^2-4(92)(49.5)})/184$. The discriminant is $52{,}441-18{,}216=34{,}225$, and its square root is 185. Thus $x=(229+185)/184=2.25$ or $x=(229-185)/184=44/184\approx0.23913$. These correspond to $r=1.25$, or 125%, and $r\approx-0.76087$, or $-76.09\%$. Under the stated nonnegative-rate condition, only **125.00%** is permitted.

Check. At 125%, $1+r=2.25$, and the incremental NPV is $-92+229/2.25-49.5/(2.25)^2=-92+101.7778-9.7778=0$. The other algebraic root is a valid root of the equation but is excluded by the problem’s rate condition.

Reading and Checking the Result

A crossover rate is a property of the difference between the projects’ cash flows. It is not generally the average of their standalone IRRs, nor is it necessarily the IRR of either project taken by itself. The subtraction order determines whether the incremental NPV is $NPV_A-NPV_B$ or $NPV_B-NPV_A$, but either order has the same zero-NPV rates.

A root is meaningful only in the domain being considered. In the usual effective annual-rate model, $r>-1$ is needed for $1+r$ to be positive. A problem may further restrict attention to $r\geq0$, as the third example did. If incremental cash flows change signs more than once, the equation can have multiple roots; check all roots rather than selecting one solely because it is positive or appears first in a calculation.

A practical verification is to substitute each candidate rate into the original incremental NPV equation. Alternatively, compute both project NPVs at the candidate rate and confirm that they agree, allowing for rounding. Use unrounded values during the calculation and round the reported rate only at the end.

Common exam trap: The difference between projects’ NPVs at a particular rate is a dollar amount; it is not a rate. For example, a $25 NPV gap at a zero discount rate does not imply a 25% crossover rate. Form the incremental cash flows and solve the zero-NPV equation instead. Another frequent error is to omit or reverse the time-zero difference; the initial cash flow is part of the equation and is not discounted.

Scope of the calculation: The crossover rate identifies a rate where two projects have equal NPVs. Finding that rate alone does not determine which project should be selected under a particular decision rule. That requires the relevant project-selection criterion and assumptions.

Check Your Understanding

For each question, form the incremental cash flows before solving. Unless stated otherwise, use annual end-of-year cash flows and report rates to two decimal places.

Project A’s cash flows at times 0, 1, and 2 are $(-300,150,180)$; project B’s are $(-240,117,143.7)$. Find the nonnegative crossover rate. (Answer: 10.00%)
The incremental cash flows A minus B are $(-80,100)$ at times 0 and 1. Find the nonnegative crossover rate. (Answer: 25.00%)
The incremental cash flows A minus B are $(-120,72,86.4)$ at times 0, 1, and 2. Verify that a 20% rate is a crossover rate. (Answer: 20.00%)
The incremental cash flows A minus B are $(-100,230,-132)$ at times 0, 1, and 2. Find all nonnegative crossover rates. (Answer: 10.00% and 20.00%)
The incremental cash flows A minus B are $(10,-20,5)$ at times 0, 1, and 2. Find all crossover rates greater than or equal to zero. (Answer: 70.71%)

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