Timing Conflicts Between NPV and IRR

Why Timing Can Create Conflicting Rankings

Two projects can each have a positive net present value and still compete for the same investment funds. When their cash flows arrive at different times, the project with the higher internal rate of return (IRR) need not have the higher net present value (NPV). The difference matters because IRR is a project-specific break-even rate, whereas NPV measures value at the required return for that project’s risk.

The discussion below assumes annual periods, end-of-year cash flows, and a constant effective annual required return $k$. Project cash flows are stated on a consistent basis, such as after-tax incremental cash flows. The required return must be appropriate for the risk of the cash flows being evaluated. These measures are used to compare mutually exclusive projects: selecting one means forgoing the other.

Definition: If $C_t$ is a project’s cash flow at the end of year $t$, with $C_0$ the initial cash flow, its NPV at required return $k$ is $NPV(k)=\sum_{t=0}^{n} C_t/(1+k)^t$. The project’s IRR is a rate $r$ that makes $NPV(r)=0$. For a conventional project, $C_0<0$ and later cash flows are nonnegative, with at least one positive inflow.

For a conventional project with an initial outflow and later positive cash flows, NPV decreases as the discount rate rises: each positive future cash flow has a smaller present value at a higher rate. If the project has one IRR, NPV is positive below that rate, zero at it, and negative above it. This is a useful accept-or-reject test, but it does not by itself rank mutually exclusive projects.

A Timing Conflict in a Two-Project Choice

Worked Example: Earlier Proceeds Versus a Larger Later Receipt

A firm can undertake Project R, which costs $90 now and pays $126 at the end of year 1, or Project S, which costs $90 now and pays $153 at the end of year 2. Both projects have the same risk, and the required return is 10% per year. Calculate each IRR and NPV, then decide which project is preferred by each measure. The projects are mutually exclusive.

Method. For each project, set its NPV to zero to find the IRR, and discount its cash inflow at the given 10% required return to find NPV. This separates a project’s break-even rate from the value it creates at the firm’s required return.

For Project R, the IRR solves $-90+126/(1+r_R)=0$. Therefore $1+r_R=126/90=1.4$, so $r_R=0.40$, or 40.00%.

For Project S, $-90+153/(1+r_S)^2=0$. Thus $(1+r_S)^2=153/90=1.7$, so $r_S=\sqrt{1.7}-1=0.303840\ldots$, or 30.38%.

At the 10% required return, Project R has $NPV_R=-90+126/1.10=-90+114.54545\ldots=$ $24.55. Project S has $NPV_S=-90+153/(1.10)^2=-90+153/1.21=-90+126.44628\ldots=$ $36.45. Currency amounts are rounded to the nearest cent only at the final step.

Project R has the higher IRR, but Project S has the higher NPV at the required return. For this mutually exclusive choice, the NPV criterion favors S: at 10%, it adds $36.45 of value rather than $24.55. Both NPVs are positive, so each would pass a standalone NPV accept-or-reject test.

Check. The present value of S’s $153 receipt is $126.44628…, greater than its $90 cost, giving a positive NPV. The present value of R’s $126 receipt is $114.54545…, also greater than its cost. The calculated IRRs make each project’s discounted receipt equal its $90 initial cost.

Common mistake avoided. A higher IRR does not mean higher dollar value at the firm’s required return. Here, choosing R solely because 40.00% exceeds 30.38% would overlook S’s larger NPV at 10%.

The Crossover Rate and the Required Return

The crossover rate is a discount rate at which the NPVs of two projects are equal. It can be found by setting their NPV difference to zero. Equivalently, form incremental cash flows by subtracting one project’s cash flows from the other’s, and find the rate at which the incremental NPV is zero. When the cash-flow difference changes sign over time, the relative present value of the projects can change as the discount rate changes.

NPV_A(k)-NPV_B(k)=\sum_{t=0}^{n}\frac{C_{A,t}-C_{B,t}}{(1+k)^t}

The expression follows directly from subtracting the NPV formula for B from the NPV formula for A, period by period. At a crossover rate, this difference is zero. On either side of that rate, its sign identifies which project has the higher NPV. A crossover rate is a ranking boundary, not a required return and not automatically the rate the firm should use.

Worked Example: Finding Where the NPV Ranking Reverses

Project A costs $100 now and pays $180 at the end of year 1. Project B costs $100 now and pays $260 at the end of year 3. They have the same risk. Find their IRRs, compare their NPVs at required returns of 10% and 30%, and find the crossover rate.

IRRs. For A, $r_A=180/100-1=0.80$, or 80.00%. For B, $(1+r_B)^3=260/100=2.6$, so $r_B=\sqrt[3]{2.6}-1=0.37507\ldots$, or 37.51%.

NPVs at 10%. $NPV_A=-100+180/1.10=-100+163.63636\ldots=$ $63.64. For B, $1.10^3=1.331$, so $NPV_B=-100+260/1.331=-100+195.34185\ldots=$ $95.34. B has the higher NPV at 10%.

NPVs at 30%. $NPV_A=-100+180/1.30=-100+138.46154\ldots=$ $38.46. Since $1.30^3=2.197$, $NPV_B=-100+260/2.197=-100+118.34320\ldots=$ $18.34. A has the higher NPV at 30%.

Crossover rate. The initial costs cancel when the NPVs are set equal: $180/(1+k)=260/(1+k)^3$. Multiplying by $(1+k)^3$ gives $180(1+k)^2=260$. Therefore $1+k=\sqrt{260/180}=\sqrt{13/9}$, and $k=\sqrt{13}/3-1=0.20185\ldots$, or 20.19%.

Below the crossover rate, B has the higher NPV; above it, A has the higher NPV. The calculations at 10% and 30% confirm the reversal. The firm should use its risk-appropriate required return, not choose whichever comparison rate produces a preferred ranking.

Check. At the crossover rate, the two present values of inflows are equal because the equation $180(1+k)^2=260$ holds. The initial cost of $100 is the same for both projects, so their NPVs are equal there as well.

Why the Required Return Governs the NPV Choice

For mutually exclusive projects with comparable risk, calculate each NPV using the same risk-appropriate required return and choose the project with the larger NPV, provided the choice is consistent with the firm’s constraints and objectives. IRR remains informative: it indicates the rate at which each individual project breaks even in present-value terms. But the IRRs answer separate break-even questions; they do not directly compare the dollar value created at the firm’s required return.

The crossover rate helps explain why a ranking changes. With earlier cash flows, less of the project’s value depends on discounting distant receipts. A project with later receipts can rank better when the required return is low enough, but lose that advantage as the rate rises. The direction and size of the effect depend on the actual cash-flow differences, so the ranking should be calculated rather than inferred from a verbal rule about “early” or “late” money.

Key rule: For mutually exclusive projects of comparable risk, rank them by NPV at the appropriate required return. Use IRR as supporting information, not as a substitute for the NPV comparison when the rankings conflict.

What Reinvestment Assumptions Explain

An algebraic identity gives a useful interpretation of NPV. For a project ending at year $n$, multiplying its NPV at $k$ by $(1+k)^n$ gives:

NPV(k)(1+k)^n=\sum_{t=0}^{n} C_t(1+k)^{n-t}

This follows by multiplying each term $C_t/(1+k)^t$ by $(1+k)^n$, leaving $C_t(1+k)^{n-t}$. If interim cash flows are reinvested through year $n$ at rate $k$, the right-hand side is their combined terminal value, with the initial outflow carried forward as a negative amount. This identity helps interpret the value measured by NPV. It does not mean that calculating NPV requires the firm actually to reinvest each receipt at exactly $k$; NPV itself is the present value comparison using the required return.

A similar terminal-value interpretation using IRR carries interim receipts forward at the project’s own IRR. Thus, an IRR-based comparison may implicitly treat different projects as if they can reinvest interim cash flows at different rates. Those rates may not be achievable in practice, especially when one project has an unusually high IRR. The required return gives the common benchmark for valuing the projects; actual reinvestment outcomes can be assessed separately.

Worked Example: Reinvestment Interpretation of Conflicting Rankings

Project X costs $100 now, pays $50 at the end of year 1, and pays $75 at the end of year 2. Project Y costs $100 now and pays $132.25 at the end of year 2. They have the same risk, and the required return is 10%. Compare IRRs and NPVs, then verify the terminal-value interpretation at 10%.

IRRs. For X, let $x=1+r_X$. Its break-even equation is $-100+50/x+75/x^2=0$, or $100x^2-50x-75=0$. The positive root is $x=(50+\sqrt{32{,}500})/200=1.1513878\ldots$, so $r_X=15.14\%$. For Y, $(1+r_Y)^2=132.25/100=1.3225$, so $1+r_Y=1.15$ and $r_Y=15.00\%$. X has the higher IRR.

NPVs at 10%. $NPV_X=-100+50/1.10+75/1.21=-100+45.45455\ldots+61.98347\ldots=$ $7.44. $NPV_Y=-100+132.25/1.21=-100+109.29752\ldots=$ $9.30. Y has the higher NPV.

Terminal values at the required return. If X’s year-1 receipt is carried forward one year at 10%, its year-2 terminal value is $50(1.10)+75=130$. Y’s terminal value is $132.25. The $100 initial investment carried forward two years at 10% is $100(1.10)^2=121$. Thus, the projects’ terminal surpluses over that benchmark are $130-121=$9.00$ for X and $132.25-121=$11.25$ for Y.

Check. Multiplying the unrounded NPVs by $1.21$ gives X’s surplus, $7.4380165\ldots(1.21)=$9.00$, and Y’s surplus, $9.2975207\ldots(1.21)=$11.25$. The NPV ranking agrees with the terminal-value comparison at the required return. By contrast, carrying X’s year-1 receipt forward at X’s own IRR makes its terminal value equal $100(1+r_X)^2$, by the IRR equation. That is a different reinvestment rate from the one used for Y.

Common mistake avoided. Do not conclude that X creates more value merely because its IRR is higher. At the stated 10% required return, Y has the higher NPV, and its terminal surplus is also larger when the interim cash flow is carried forward at 10%.

Common Traps in Timing Comparisons

Ranking by IRR alone. IRR is a break-even rate, not a measure of NPV at the firm’s required return. If IRR and NPV rankings disagree for mutually exclusive projects, use the NPV ranking at the appropriate required return.
Using the wrong discount period. A year-3 receipt is discounted by $(1+k)^3$, not $(1+k)^2$. Write the date of every cash flow before discounting.
Treating the crossover rate as the firm’s hurdle rate. It only identifies where project NPVs are equal. The risk-appropriate required return determines which side of that boundary applies.
Assuming every cash-flow pattern has a single IRR. Nonconventional cash flows, with more than one change in sign, can have multiple IRRs or no IRR. In that case IRR may not provide a clear ranking; NPV at the required return remains the direct value comparison.

The IRR ranking and NPV ranking can also agree. A crossover rate need not lie within the range of required returns relevant to a decision, and cash-flow patterns may not produce a ranking reversal at all. Compute the measures for the stated cash flows and rate rather than assuming a conflict just because project timing differs.

Check Your Understanding

For each problem, apply the required return to both projects when comparing NPVs. Round final currency amounts to the nearest cent and rates to two decimal places.

Project P costs $80 now and pays $112 in one year. Project Q costs $80 now and pays $145.20 in two years. At a 10% required return, find both IRRs and NPVs, and state which project each measure ranks higher. (Answer: $IRR_P=40.00\%$, $IRR_Q=34.72\%$; $NPV_P=$21.82$, $NPV_Q=$40.00$; IRR ranks P higher and NPV ranks Q higher.)
Project A costs $120 now and pays $180 in one year. Project B costs $120 now and pays $250 in two years. Find the crossover rate and compare NPVs at 20%. (Answer: Crossover rate $=38.89\%$; at 20%, $NPV_A=$30.00$ and $NPV_B=$53.61$, so B has the higher NPV.)
Project X costs $100 now, pays $70 in one year, and pays $50 in two years. Project Y costs $100 now and pays $130 in two years. At a 10% required return, calculate both NPVs and IRRs and identify the preferred project by NPV. (Answer: $NPV_X=$4.96$, $NPV_Y=$7.44$; $IRR_X=13.90\%$, $IRR_Y=14.02\%$; Y has the higher NPV and IRR.)
Project J costs $200 now, pays $100 at the end of year 1, and pays $140 at the end of year 2. At a 10% required return, find its NPV and its year-2 terminal surplus if the year-1 receipt is reinvested at 10%. Verify the relation between the two amounts. (Answer: $NPV_J=$6.61$; terminal surplus $=$8.00$, equal to the unrounded NPV multiplied by $1.10^2$.)
Project U costs $150 now, pays $90 at the end of each of years 1 and 2. Project V costs $150 now and pays $190 at the end of year 2. Find both IRRs and the crossover rate; then compare NPVs at required returns of 8% and 15%. (Answer: $IRR_U=13.07\%$, $IRR_V=12.55\%$; crossover rate $=11.11\%$. At 8%, $NPV_U=$10.49$ and $NPV_V=$12.89$, so V ranks higher; at 15%, $NPV_U=-$3.69$ and $NPV_V=-$6.33$, so U ranks higher, although both projects have negative NPV.)

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