Scale Conflicts Between IRR and NPV

Why Project Size Can Change the Ranking

The internal rate of return (IRR) expresses a project’s return as a rate. Net present value (NPV) expresses its value contribution in dollars at a specified required return. Those are different measures, so they can rank projects differently when the projects require different amounts of investment. A smaller project may earn the higher percentage return while a larger alternative adds more value.

This matters when projects are mutually exclusive: accepting one rules out the other. If there is no capital constraint that changes the decision and the alternatives have comparable risk, the relevant question is which choice adds more value at the firm’s required return. A higher IRR alone does not answer that question.

Definition: A project’s net present value at required return $k$ is the present value of its expected cash inflows and outflows, discounted at $k$. Its internal rate of return is the discount rate that makes its NPV equal to zero. A scale conflict occurs when mutually exclusive projects of different sizes have different IRR and NPV rankings.

Model and Decision Rule

Assume each project has an initial investment at time 0 followed by conventional cash flows: the initial cash flow is negative and later cash flows are nonnegative. Cash flows occur at the ends of annual periods, and the required return $k$ is an effective annual rate appropriate to the project’s risk. For a project with cash flow $CF_t$ at time $t$, its NPV is

NPV(k)=\sum_{t=0}^{n}\frac{CF_t}{(1+k)^t}.

For a conventional project with a unique IRR, that rate is the value $r$ satisfying

0=\sum_{t=0}^{n}\frac{CF_t}{(1+r)^t}.

For an independent project, NPV greater than zero means its discounted inflows exceed its investment at the required return, so it adds value under this model. For mutually exclusive projects, compare their NPVs at the same required return and choose the project with the higher NPV, provided a project must be chosen and no other constraint changes the decision. If doing nothing is an available alternative, it has NPV zero and should also be considered.

Key result: For mutually exclusive projects evaluated at the same required return, IRR ranks percentage returns, while NPV ranks dollar value added. When their rankings conflict, the higher NPV identifies the project that contributes more value under the stated assumptions.

Worked Examples

Worked Example: A One-Year Scale Conflict

A firm must choose one of two unrelated equipment upgrades. Upgrade A costs $100 now and returns $130 at the end of one year. Upgrade B costs $1,000 now and returns $1,200 at the end of one year. Both have the same risk, and the annual required return is 10%. Which project should the firm choose?

Method. Compute each project’s IRR from its one-year break-even equation, then compute both NPVs at the common required return of 10%. The cash flows are conventional and each project has one IRR.

For A, the IRR solves $-100+130/(1+r)=0$. Thus $1+r=130/100=1.30$, so A’s IRR is 30%. Its NPV at 10% is

NPV_A(10\%)=-100+\frac{130}{1.10} =-100+118.1818 =\$18.18.

For B, $-1{,}000+1{,}200/(1+r)=0$, so $1+r=1{,}200/1{,}000=1.20$, and B’s IRR is 20%. Its NPV is

NPV_B(10\%)=-1{,}000+\frac{1{,}200}{1.10} =-1{,}000+1{,}090.9091 =\$90.91.

A has the higher IRR, but B has the higher NPV. Because the alternatives are mutually exclusive and the decision is based on value added at the stated required return, choose B. Common mistake avoided: choosing A just because its percentage return is higher would ignore that B creates $72.73 more value at 10% ($$90.91-\$18.18$).

Worked Example: Comparing Two-Year Projects

A company can fund only one of two conventional projects. Project A costs $200 at time 0 and pays $100 after one year and $150 after two years. Project B costs $1,000 and pays $600 after each of the next two years. The required annual return is 10%. Compare the IRR and NPV rankings.

Method. Discount both projects’ dated cash flows at the same required return. For the IRRs, set each NPV equal to zero and solve for its positive break-even rate.

For A, discounting at 10% gives $100/1.10=90.9091$ and $150/(1.10)^2=123.9669$. Therefore,

NPV_A(10\%)=-200+90.9091+123.9669=\$14.88.

Let $x=1+r$. A’s IRR equation is $-200+100/x+150/x^2=0$. Multiplying by $x^2$ and rearranging gives $4x^2-2x-3=0$. The positive root is $x=(1+\sqrt{13})/4=1.1514$, so A’s IRR is approximately 15.14%.

For B, the discounted inflows are $600/1.10=545.4545$ and $600/(1.10)^2=495.8678$. Thus,

NPV_B(10\%)=-1{,}000+545.4545+495.8678=\$41.32.

B’s IRR satisfies $-1{,}000+600/x+600/x^2=0$, or $5x^2-3x-3=0$. The positive root is $x=(3+\sqrt{69})/10=1.1307$, giving an IRR of approximately 13.07%.

A has the higher IRR, but B has the higher NPV: $41.32 compared with $14.88. Choose B if one project must be selected. Common mistake avoided: treating the IRR ranking as a dollar-value ranking would select A even though B adds $26.45 more present value at 10%.

Why NPV Measures Value Added

The scale conflict is not a defect in either calculation; the measures answer different questions. IRR asks, “What discount rate makes this project’s discounted net cash flow zero?” NPV asks, “After discounting cash flows at the required return, how many dollars of value remain?” The NPV comparison follows directly from subtracting the projects’ discounted cash flows.

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

This identity holds because subtraction distributes across the two NPV sums. It shows that the difference in NPVs is the present value of choosing B’s cash flows rather than A’s. If that difference is positive, B contributes more value at $k$; if negative, A contributes more. IRRs do not provide this dollar difference because each is calculated using that project’s own cash flows and initial investment.

This interpretation assumes that the cash-flow estimates are comparable, the required return reflects project risk, and the projects are being assessed on the same valuation basis. If projects differ materially in risk, using the same required return may not be appropriate. If funding is limited or other constraints apply, the choice may require additional analysis; the simple higher-NPV rule assumes those constraints do not alter the decision.

Worked Example: The Larger Project Adds More Value

A manufacturer is choosing between two three-year service systems. System A costs $6,849.68 and produces $3,000 at each year-end for three years. System B costs $19,214.65 and produces $8,000 at each year-end for three years. The projects have comparable risk, the required annual return is 10%, and the company can fund either system. Compare their returns and value added.

Method. First verify each IRR using the present value of a level annual cash-flow stream. Then calculate both NPVs at 10%. For $n$ equal end-of-year payments $C$, the present value at rate $j$ is $C\sum_{t=1}^{n}(1+j)^{-t}$.

For three years, the annuity factors are $2.283225$ at 15%, $2.401831$ at 12%, and $2.486852$ at 10%. These follow by adding the three annual discount factors at each rate. System A’s cost is approximately $3{,}000(2.283225)=\$6{,}849.68$, so its IRR is approximately 15%. System B’s cost is approximately $8{,}000(2.401831)=\$19{,}214.65$, so its IRR is approximately 12%. Displayed costs are rounded to cents.

At 10%, System A’s inflows have present value $3{,}000(2.486852)=\$7{,}460.56$. Its NPV is therefore $\$7{,}460.56-\$6{,}849.68=\$610.88$. System B’s inflows have present value $8{,}000(2.486852)=\$19{,}894.82$. Its NPV is $\$19{,}894.82-\$19{,}214.65=\$680.17$.

A has the higher IRR, while B has the higher NPV by $69.29. Choose B if the objective is to add the most value at the stated required return. Common mistake avoided: comparing only the rates, without accounting for the larger dollar scale of B’s cash flows, would point to A.

Common Traps and Practical Interpretation

A scale conflict is easiest to recognize when one project needs a much larger initial outlay than another. The IRR is a percentage, so a small project can have a large percentage return without adding as many dollars as a larger project. Conversely, a larger project does not automatically have the higher NPV: its discounted inflows must exceed its cost by more than the competing project’s discounted inflows exceed its cost.

For example, a tempting but incorrect calculation is to compare only the IRRs in the first worked example and choose A because $30\%>20\%$. That calculation answers which project has the higher break-even rate; it does not answer which project adds more value at 10%. The corrected comparison is $NPV_A(10\%)=\$18.18$ versus $NPV_B(10\%)=\$90.91$, so B is preferred under the stated assumptions.

Keep the comparison disciplined:

Use the same required return when comparing alternatives with comparable risk.
Use each project’s own cash flows and investment when calculating its IRR and NPV.
Do not confuse an IRR hurdle-rate test for an independent project with a ranking rule for mutually exclusive projects.
Compare NPVs in the same currency and at the same valuation date.
Check whether doing nothing or a binding capital constraint changes the available choices.

The central conclusion is precise: an IRR ranking describes relative percentage returns, whereas an NPV ranking describes value added at the chosen required return. When mutually exclusive projects of different scale conflict on those rankings, NPV is the appropriate value-creation comparison under the stated model.

Check Your Understanding

For each problem, use the stated required return for the NPV comparison. Assume annual end-of-year cash flows, comparable project risk, conventional cash flows, and no binding funding constraint unless the problem says otherwise.

A firm must select one of two one-year projects. Project A costs $800 and pays $1,000 after one year; Project B costs $6,000 and pays $7,200 after one year. At a 10% required return, calculate both IRRs and NPVs and identify the preferred project. (Answer: A IRR = 25%, NPV = $109.09; B IRR = 20%, NPV = $545.45; choose B.)
Two two-year projects are mutually exclusive. Project C costs $500 and pays $300 at each year-end. Project D costs $2,500 and pays $1,480 at each year-end. At a 10% required return, compare their IRRs and NPVs. (Answer: C IRR ≈ 13.07%, NPV = $20.66; D IRR ≈ 12.04%, NPV = $68.60; choose D.)
At a 10% required return, Project E costs $1,000 and pays $550 after one year and $600 after two years. Project F costs $5,000 and pays $2,800 after one year and $2,900 after two years. Calculate each NPV and decide whether either should be accepted if doing nothing is allowed. (Answer: E NPV = −$4.13; F NPV = −$57.85; choose neither, because both have negative NPV.)
A firm must choose one of two three-year systems. System G costs $6,849.68 and pays $3,000 at each year-end. System H costs $19,214.65 and pays $8,000 at each year-end. At a 10% required return, use the three-year annuity factors $2.283225$ at 15%, $2.401831$ at 12%, and $2.486852$ at 10% to compare the approximate IRRs and NPVs. (Answer: G IRR ≈ 15%, NPV = $610.88; H IRR ≈ 12%, NPV = $680.17; choose H.)

What You'll Learn