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Exam FM · Internal Rate of Return · Tutorial 498 of 998

How Project Scale Reverses IRR Rankings

Learn why a smaller project can have a higher IRR but a lower NPV, and how to make a consistent choice using the required rate of return.

Intermediate 10 min read

What You'll Learn

  • Calculate and compare IRRs and NPVs for projects with different initial investments.
  • Explain why a higher percentage return does not necessarily create more dollar value.
  • Use the required rate to evaluate NPV when comparing project values.
  • Distinguish proportional scaling, which preserves IRR, from comparing projects with different returns.
  • Avoid ranking mutually exclusive projects by IRR alone.

Why Project Scale Can Reverse an IRR Ranking

The internal rate of return (IRR) is a percentage, while net present value (NPV) is an amount of money. That difference matters when projects require different initial investments. A small project can earn a high percentage return on a modest amount of capital, while a larger project earns a lower percentage return but creates more dollar value.

For independent projects, the earlier material on IRR as a decision rule explains how to compare each project with a required rate of return. When projects are alternatives and only one can be selected, however, choosing the project with the highest IRR can be misleading. This tutorial focuses on the scale effect: compare each project’s IRR with its NPV at the same required rate, and use the NPV criterion to identify which alternative adds more value.

Definition: For dated cash flows \(C_0,C_1,\ldots,C_n\), with \(C_0\) usually the initial investment, the net present value at an annual effective required rate \(i\) is $$ \operatorname{NPV}(i)=\sum_{t=0}^{n}\frac{C_t}{(1+i)^t}. $$ The internal rate of return is a rate \(r\) that solves $$ \sum_{t=0}^{n}\frac{C_t}{(1+r)^t}=0. $$

Unless stated otherwise, the examples use deterministic cash flows at the ends of whole years, a constant annual effective required rate, and no taxes, transaction costs, or additional financing constraints. Each has one initial outflow followed by positive inflows. This conventional cash-flow pattern gives a unique IRR when a positive IRR exists. Cash flows and rates must be measured consistently: an annual effective rate is used with year-end annual periods.

Worked Example: A Smaller Investment Has the Higher IRR

Worked Example: A Smaller Investment Has the Higher IRR

Two mutually exclusive projects have year-end cash flows in dollars. Project A requires 100 now and pays 60 after one year and 72 after two years. Project B requires 1,000 now and pays 600 after one year and 632.50 after two years. Find both IRRs, then compare their NPVs at a 10% annual effective required rate.

Method. First verify each IRR by discounting its inflows at the candidate rate. Then calculate both NPVs at the common required rate of 10%. This avoids treating the largest percentage return as automatically the best alternative.

For Project A, test \(r=20\%\):

$$ \operatorname{NPV}_A(20\%)=-100+\frac{60}{1.20}+\frac{72}{1.20^2} =-100+50+50=0. $$

Thus Project A’s IRR is 20%. For Project B, test \(r=15\%\):

$$ \operatorname{NPV}_B(15\%)=-1{,}000+\frac{600}{1.15}+\frac{632.50}{1.15^2} =-1{,}000+521.7391+478.2609=0. $$

Thus Project B’s IRR is 15%. Now use the required rate, not either project’s own IRR, to evaluate both NPVs:

$$ \begin{aligned} \operatorname{NPV}_A(10\%) &=-100+\frac{60}{1.10}+\frac{72}{1.10^2}\\ &=-100+54.5455+59.5041\\ &=14.0496\approx \$14.05, \\[4pt] \operatorname{NPV}_B(10\%) &=-1{,}000+\frac{600}{1.10}+\frac{632.50}{1.10^2}\\ &=-1{,}000+545.4545+522.7273\\ &=68.1818\approx \$68.18. \end{aligned} $$

Rounded to the nearest cent, Project A has a 20% IRR and an NPV of $14.05; Project B has a 15% IRR and an NPV of $68.18. Project A ranks higher by IRR, but Project B creates more value at the 10% required rate. The common mistake is to select A simply because 20% exceeds 15%; the correction is to compare the NPVs when choosing between these alternatives.

Check. At 10%, both NPVs are positive, so each individually exceeds the required return. Because the projects are mutually exclusive, the higher NPV makes B the preferred choice under the stated assumptions.

Why the Rankings Can Differ

IRR measures return relative to the cash invested; NPV measures the discounted dollar surplus after accounting for that investment. A higher percentage does not guarantee a higher dollar surplus. For example, earning a larger percentage on 100 may add less value than earning a smaller percentage on 1,000.

For conventional cash flows, NPV decreases as the discount rate increases. Each positive future cash flow has a present value that falls as the rate rises, while the initial outflow at time zero does not change. Therefore, if the IRR is unique, comparing the IRR with the required rate gives the same accept-or-reject result as checking whether NPV at that rate is positive. That equivalence for a single project does not mean IRR alone ranks competing projects by value.

1
Find the IRR for each project.
Use the rate that makes that project’s own NPV equal to zero. The IRR is a project-specific percentage.
2
Set one required rate for the comparison.
Use the same rate for every project so that all NPVs are evaluated on the same basis.
3
Compare the NPVs.
For mutually exclusive alternatives, the highest NPV identifies the greatest dollar value added at the required rate, subject to the model assumptions.

A Justification: What Scaling Does to IRR and NPV

It is useful to distinguish two ideas: scaling every cash flow in a project by the same positive factor, and comparing two projects that happen to have different investment amounts. Pure proportional scaling does not change the IRR. It does, however, scale the NPV by the same factor.

Derivation: Suppose every cash flow \(C_t\) is multiplied by a positive constant \(k\). At any rate \(r\), the scaled project has $$ \sum_{t=0}^{n}\frac{kC_t}{(1+r)^t} =k\sum_{t=0}^{n}\frac{C_t}{(1+r)^t}. $$ Since \(k>0\), the scaled project’s NPV equals zero at exactly the same rates as the original project’s NPV. Its IRR is therefore unchanged. At the required rate \(i\), its NPV is \(k\operatorname{NPV}(i)\). Thus exact scaling preserves the percentage return and multiplies the dollar value.

Consequently, “larger scale” by itself does not cause an IRR reversal. The reversal occurs when projects of different sizes also have different returns: a smaller project may have a higher IRR, while the larger project’s lower rate applies to a much larger investment and produces a higher NPV. The next example makes that comparison explicit.

Worked Example: Comparing Projects at an 8% Required Rate

Project S requires $200 and pays $100 after one year and $149.50 after two years. Project L requires $1,200 and pays $600 after one year and $833.28 after two years. Find their IRRs and NPVs at an 8% annual effective required rate. All payments are certain year-end amounts, and only one project may be chosen.

Method. Verify the proposed IRRs by discounting the inflows, then compute NPV at 8% for both projects. Do not infer dollar value from the IRR percentage.

For S, test \(r=15\%\):

$$ \operatorname{NPV}_S(15\%)=-200+\frac{100}{1.15}+\frac{149.50}{1.15^2} =-200+86.9565+113.0435=0. $$

So S’s IRR is 15%. For L, test \(r=12\%\):

$$ \operatorname{NPV}_L(12\%)=-1{,}200+\frac{600}{1.12}+\frac{833.28}{1.12^2} =-1{,}200+535.7143+664.2857=0. $$

So L’s IRR is 12%. At the common 8% required rate:

$$ \begin{aligned} \operatorname{NPV}_S(8\%) &=-200+\frac{100}{1.08}+\frac{149.50}{1.08^2}\\ &=-200+92.5926+128.1722\\ &=20.7648\approx \$20.76, \\[4pt] \operatorname{NPV}_L(8\%) &=-1{,}200+\frac{600}{1.08}+\frac{833.28}{1.08^2}\\ &=-1{,}200+555.5556+714.4033\\ &=69.9589\approx \$69.96. \end{aligned} $$

Project S has the higher IRR, but Project L has the higher NPV at 8%. With only one project selectable, L is preferred by the NPV criterion. The trap is to assume that a 15% return must be more valuable than a 12% return. The figures show why that is incorrect: the dollar surplus depends on the size and pattern of all the cash flows as well as the rate.

Check. Discounting each project at its own verified IRR gives an NPV of zero. Discounting both at 8% gives positive NPVs, and L’s is larger.

Worked Example: Exact Scaling Preserves the IRR

Worked Example: Exact Scaling Preserves the IRR

Project P requires $100 and pays $60 after one year and $72 after two years. Project Q has every cash flow of P multiplied by 3: it requires $300 and pays $180 and $216 at the same respective times. Find both IRRs and compare their NPVs at a 10% annual effective required rate.

Method. Verify the rate for one project, apply the scaling argument to the other, and calculate each NPV. This example avoids the mistake of assuming that a bigger initial investment necessarily changes the IRR.

For P, at 20%:

$$ \operatorname{NPV}_P(20\%)=-100+\frac{60}{1.20}+\frac{72}{1.20^2} =-100+50+50=0. $$

P’s IRR is 20%. Since Q multiplies every cash flow by 3, its NPV at 20% is also zero:

$$ \operatorname{NPV}_Q(20\%)=-300+\frac{180}{1.20}+\frac{216}{1.20^2} =-300+150+150=0. $$

Thus Q’s IRR is also 20%. At 10%:

$$ \begin{aligned} \operatorname{NPV}_P(10\%) &=-100+\frac{60}{1.10}+\frac{72}{1.10^2}\\ &=14.0496\approx \$14.05, \\ \operatorname{NPV}_Q(10\%) &=-300+\frac{180}{1.10}+\frac{216}{1.10^2}\\ &=42.1488\approx \$42.15. \end{aligned} $$

Q has three times P’s NPV, just as it has three times each cash flow, but the two projects have the same IRR. This is not an IRR ranking reversal: exact scaling preserves the rate ranking because there is no rate difference to rank. A reversal requires comparing projects with different cash-flow returns and different dollar scales.

Check. The computed NPV of Q is three times P’s before rounding: \(3(14.0495868)=42.1487604\). The rounded values, $14.05 and $42.15, are consistent.

Choosing and Reporting a Comparison

When a question asks which of two mutually exclusive projects is preferable at a stated required rate, calculate both NPVs at that rate. A higher NPV means a greater present-value surplus under the given assumptions. The IRRs remain useful summaries of each project’s percentage return, but do not substitute for the dollar-value comparison.

For independent projects, the decision differs: each project can be considered on its own, so a project with positive NPV at the required rate meets the value criterion without requiring rejection of another project. For mutually exclusive projects, positive NPV for both does not settle the choice; compare their NPVs. A project with a higher IRR can still be a reasonable project, but the IRR percentage alone does not show that it is the better exclusive alternative.

A concise exam-style conclusion should name both rankings and the basis for the decision. For example: “Project A has the higher IRR, but Project B has the higher NPV at the 10% required rate; therefore, choose B if the projects are mutually exclusive and the stated assumptions apply.” This prevents a common error: calculating both measures correctly but using the wrong one to answer the selection question.

Key takeaway: IRR is a percentage return; NPV at the required rate is a dollar-value measure. A smaller project can have a higher IRR and a lower NPV than a larger project. For mutually exclusive projects, use the NPV comparison at the common required rate rather than ranking by IRR alone.

Check Your Understanding

For each problem, assume the stated cash flows occur at year-end, rates are annual effective, and there are no taxes or transaction costs.

  1. Project A requires $400 and pays $520 after one year. Project B requires $2,000 and pays $2,400 after one year. Calculate each IRR and NPV at a 10% required rate. If the projects are mutually exclusive, which has the higher NPV? (Answer: A has IRR 30% and NPV $72.73; B has IRR 20% and NPV $181.82, so B has the higher NPV.)
  2. Project P has cash flows \(-\$300,\ \$150,\ \$252\) at times 0, 1, and 2. Project Q has cash flows \(-\$750,\ \$375,\ \$630\) at those times. Find their IRRs and NPVs at 10%. (Answer: Both IRRs are 20%; P has NPV $44.63 and Q has NPV $111.57.)
  3. Project S requires $500 and pays $300 after one year and $360 after two years. Project L requires $1,500 and pays $900 after one year and $923.40 after two years. Verify their IRRs and compare NPVs at 10%. (Answer: S has IRR 20% and NPV $70.25; L has IRR 14% and NPV $81.32, so L has the higher NPV.)
  4. Project A requires $1,000 and pays $1,250 after one year. Project B requires $5,000 and pays $5,900 after one year. Calculate their IRRs and NPVs at a 12% required rate, then state which is preferred if mutually exclusive. (Answer: A has IRR 25% and NPV $116.07; B has IRR 18% and NPV $267.86, so B is preferred by NPV.)

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