Crossover Rates from Competing NPV Profiles

Why NPV Profiles Can Cross

Two projects can have different investment amounts and different patterns of future cash flows. As the interest rate changes, those patterns receive different present-value weights, so the project with the higher net present value (NPV) can change. A crossover rate is an interest rate at which the projects have equal NPVs. Finding it identifies a boundary between rate ranges in which one project or the other has the higher NPV.

The comparison is relative: a project can have a higher NPV than its competitor while still having a negative NPV. If the decision also permits rejecting both projects, that separate choice must be considered. Here the focus is on when the NPV ranking changes.

Definition: For two projects with cash flows measured at the same valuation date, a crossover rate is an interest rate at which their NPVs are equal. The difference between the two projects’ cash flows is called the incremental cash flow. Its NPV is the difference between the projects’ NPVs.

Suppose the effective annual interest rate is \(i\), with \(i>-1\), and let \(v=1/(1+i)\). If project A has cash flow \(A_t\) and project B has cash flow \(B_t\) at time \(t\), their NPVs are the sums of their discounted cash flows. For cash flows at integer times \(0,1,\ldots,n\), the crossover equation is

\operatorname{NPV}_A(i)-\operatorname{NPV}_B(i) =\sum_{t=0}^{n}(A_t-B_t)v^t=0.

This equation is a polynomial in \(v\). A positive root \(v\) corresponds to the rate \(i=1/v-1\). For nonnegative interest rates, \(0<v\leq 1\), so a positive root greater than 1 corresponds to a negative interest rate, not a nonnegative crossover. If cash flows occur at irregular times, the same equality applies using the appropriate discount factors for those times; the equation need not be a polynomial in \(v\).

Why the Incremental Cash-Flow Equation Works

The crossover definition is simply equality of the projects’ present values. Subtracting one project’s NPV from the other gives the NPV of their cash-flow differences, because discounting and addition are linear operations.

Write each NPV.
Discount every project cash flow to the same valuation date at the same effective annual rate.

Subtract the NPVs.
Term by term, the difference is the discounted value of \(A_t-B_t\).

Set the difference to zero.
At a crossover, neither project has a higher NPV, so their NPV difference is zero.

\begin{aligned} \operatorname{NPV}_A(i)&=\sum_{t=0}^{n}A_tv^t,\\ \operatorname{NPV}_B(i)&=\sum_{t=0}^{n}B_tv^t,\\ \operatorname{NPV}_A(i)-\operatorname{NPV}_B(i) &=\sum_{t=0}^{n}(A_t-B_t)v^t. \end{aligned}

This justifies using the incremental cash flows to find crossovers. The method requires that both projects’ cash flows be expressed consistently: same valuation date, interest-rate basis, and time units. If project lives differ, the comparison must also use an appropriate common basis before their NPVs are compared.

Finding a Crossover and Reading the Ranking

Solving the equality gives candidate crossover rates, but the roots alone do not state which project ranks higher between them. After solving, check which roots lie in the rate range of interest and evaluate the NPV difference in each interval. A sign change in the difference indicates a ranking reversal; a repeated root may touch zero without changing sign.

Worked Example: One Crossover and a Ranking Reversal

At times 0, 1, and 2, project A has cash flows \((-100,70,70)\), and project B has cash flows \((-80,50,50)\), in dollars. Find their nonnegative crossover rate and state which project has the higher NPV below and above it. Assume an effective annual interest rate \(i\geq 0\), with annual periods.

Method. Subtract B’s cash flows from A’s, set the incremental NPV to zero, and check the sign on either side of the root. This avoids trying to infer the ranking from the projects’ IRRs or from the initial investments alone.

Step 1: Form the incremental cash flows. At time 0 the difference is \(-100-(-80)=-20\). At times 1 and 2 the differences are \(70-50=20\) at each time. Thus A minus B is \((-20,20,20)\).

Step 2: Write and solve the crossover equation. With \(v=1/(1+i)\),

-20+20v+20v^2=0 \quad\Longleftrightarrow\quad v^2+v-1=0.

The quadratic formula gives \(v=(-1\pm\sqrt{5})/2\). Only the positive root is an admissible discount factor:

v=\frac{\sqrt{5}-1}{2}\approx 0.618034.

The corresponding annual effective rate is

i=\frac{1}{v}-1 =\frac{\sqrt{5}+1}{2}-1 =\frac{\sqrt{5}-1}{2} \approx 0.618034.

Rounded to two decimal places as a percentage, the crossover rate is 61.80%.

Step 3: Determine the ranking. At \(i=0\), \(v=1\), and the difference is \(-20+20+20=20>0\), so A has the higher NPV. At \(i=100\%\), \(v=1/2\), and the difference is \(-20+10+5=-5<0\), so B has the higher NPV. The projects tie at 61.80%, and the ranking reverses there.

Check. Substitution of the root gives \(v^2+v=1\), so \(-20+20v+20v^2=0\), as required.

Roots Outside the Rate Range

A positive root in \(v\) is not automatically a nonnegative crossover rate. Since \(i=1/v-1\), the nonnegative-rate range \(i\geq0\) corresponds exactly to \(0<v\leq1\). A root with \(v>1\) represents a negative rate. Also, a polynomial may have no positive roots, or may have positive roots that lie outside the range being considered.

Worked Example: A Root That Is Not a Nonnegative Crossover

At times 0, 1, and 2, project A has cash flows \((-60,42,33)\), and project B has cash flows \((-50,40,30)\), in dollars. Determine whether there is a crossover at a nonnegative effective annual rate, and identify the higher-NPV project for all such rates.

Method. Use the incremental NPV and examine its roots on \(0<v\leq1\). This avoids treating every positive discount-factor root as a nonnegative-rate crossover.

Step 1: Calculate the incremental cash flows. A minus B is \((-60-(-50),42-40,33-30)=(-10,2,3)\). Therefore,

\operatorname{NPV}_A-\operatorname{NPV}_B=-10+2v+3v^2.

Step 2: Solve the equality. Setting the expression to zero gives \(3v^2+2v-10=0\). Its roots are

v=\frac{-2\pm\sqrt{124}}{6} =\frac{-1\pm\sqrt{31}}{3}.

The positive root is \((\sqrt{31}-1)/3\approx1.5226\), which is greater than 1. It corresponds to a negative rate, approximately \(-34.32\%\), and is not in the nonnegative-rate range. The other root is negative and cannot be a discount factor.

Step 3: Establish the ranking for all \(i\geq0\). For \(0<v\leq1\), the expression is increasing in \(v\), since its derivative is \(2+6v>0\). Its largest value on this range is at \(v=1\), where it equals \(-10+2+3=-5<0\). Thus A’s NPV is always lower than B’s for nonnegative rates.

Check. At \(i=0\), the difference is \(-5\); at \(i=100\%\), \(v=1/2\) and it is \(-10+1+0.75=-8.25\). Both checks agree with the conclusion.

More Than One Crossover

With cash flows at three or more times, the incremental NPV can be a polynomial of degree two or higher. It can have multiple positive roots in \(0<v\leq1\), and therefore multiple nonnegative crossover rates. In that case, evaluate signs across the rate intervals. Remember that increasing \(i\) decreases \(v\), so the order of roots in \(v\) is the reverse of their order as rates.

Worked Example: Two Crossovers and Three Ranking Ranges

At times 0, 1, and 2, project A has cash flows \((-96,37,70)\), and project B has cash flows \((-100,50,60)\), in dollars. Find all nonnegative crossover rates and compare the projects at 0%, 50%, and 200%. Assume an effective annual rate and annual cash-flow periods.

Method. Factor the incremental NPV, convert each admissible discount-factor root to a rate, and test the sign in each interval. This prevents overlooking a second crossing or reversing the ranking between roots.

Step 1: Form the equation. The incremental cash flows A minus B are \((4,-13,10)\), so

\operatorname{NPV}_A-\operatorname{NPV}_B =4-13v+10v^2 =10(v-0.5)(v-0.8).

The roots \(v=0.5\) and \(v=0.8\) are both in \(0<v\leq1\). They correspond respectively to \(i=1/0.5-1=1\), or 100%, and \(i=1/0.8-1=0.25\), or 25%. Thus there are two nonnegative crossovers.

Step 2: Check the requested rates. At 0%, \(v=1\), and the difference is \(4-13+10=1>0\), so A has the higher NPV. At 50%, \(v=2/3\), and

4-\frac{26}{3}+\frac{40}{9} =\frac{36-78+40}{9} =-\frac{2}{9}<0,

so B has the higher NPV. At 200%, \(v=1/3\), and

4-\frac{13}{3}+\frac{10}{9} =\frac{36-39+10}{9} =\frac{7}{9}>0,

so A has the higher NPV.

Conclusion and check. In increasing rate order, the crossovers are 25% and 100%. A ranks higher below 25%, B between 25% and 100%, and A above 100%. The test rates 0%, 50%, and 200% fall in those three ranges, respectively, matching the calculations.

Common Mistakes and Exam Checks

Crossover analysis is aligned with the SOA Exam FM (Financial Mathematics) syllabus because it links an NPV profile to an incremental cash-flow equation. The central task is not merely solving for roots; it is interpreting the roots on the correct rate domain and checking how the rankings behave.

Using the wrong difference. State whether the expression is A minus B or B minus A. Reversing it changes the signs used to describe which project leads, although it does not change the crossover rates.
Accepting an out-of-range root. For nonnegative rates, require \(0<v\leq1\). A root \(v>1\) represents a negative rate.
Assuming one root means one ranking reversal. Check signs on either side. A repeated root can produce equality without a change in ranking.
Reading rate order from discount-factor order. Since \(v=1/(1+i)\), higher rates have smaller discount factors.
Comparing inconsistent cash flows. Use the same valuation date and period units. An annual rate cannot be applied as though a monthly cash-flow period were one full year.

A reliable solution therefore states the incremental cash flows, solves the equality, filters roots by the stated rate range, converts valid roots into rates, and tests the NPV difference to establish rankings. When there are several valid roots, test each interval rather than assuming the ranking alternates.

Key takeaway: A crossover rate is a root of the incremental NPV equation. A root identifies equality; the sign of the incremental NPV on either side determines whether the projects’ ranking actually changes.

Check Your Understanding

For each problem, use A minus B as the NPV difference and assume annual cash-flow periods. Unless stated otherwise, consider nonnegative effective annual rates.

At times 0 and 1, project A has cash flows \((-30,24)\), and project B has cash flows \((-24,12)\). Find the crossover rate and state which project has the higher NPV at 0% and 200%. (Answer: Crossover at 100%; A leads at 0%, and B leads at 200%.)
At times 0, 1, and 2, the incremental cash flows A minus B are \((-4,1,2)\). Determine whether a nonnegative crossover exists and which project has the higher NPV for every nonnegative rate. (Answer: No nonnegative crossover; B has the higher NPV for all \(i\geq0\).)
At times 0, 1, and 2, the incremental cash flows A minus B are \((3,-8,4)\). Find the nonnegative crossover rate and compare the projects at 0% and 200%. (Answer: Crossover at 100%; B leads at 0%, and A leads at 200%.)
At times 0, 1, and 2, the incremental cash flows A minus B are \((8,-18,10)\). Find all nonnegative crossover rates and identify the higher-NPV project at 10%, 50%, and 100%. (Answer: Crossovers at 0% and 25%; B leads at 10%, and A leads at 50% and 100%.)

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