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How Cash-Flow Timing Reverses IRR Rankings

Learn to compare IRR and NPV rankings when projects receive cash at different times, and why the preferred ranking depends on the required rate.

Intermediate 10 min read

What You'll Learn

  • Compare project IRRs and NPVs when receipt schedules differ.
  • Calculate an IRR from a single receipt and from multiple receipts.
  • Explain why a higher IRR does not guarantee a higher NPV at a specified rate.
  • Use equal total receipts to isolate the effect of cash-flow timing.
  • Check rankings at the required rate and avoid treating IRR as an NPV substitute.

Why Timing Can Reverse an IRR Ranking

Two projects can have the same initial cost and even the same total future receipts, yet differ in when those receipts arrive. IRR summarizes each project with one rate, while NPV discounts its specific cash flows at a chosen required rate. Because these summaries respond differently to the timing pattern, the project with the higher IRR need not have the higher NPV at the required rate.

The distinction matters when comparing projects: an IRR ranking is not automatically an NPV ranking. The examples below show how to calculate both measures, interpret the difference, and check whether a conclusion depends on the required rate. Each example assumes annual cash flows at the stated year-ends.

Definition: For cash flows \(C_0,C_1,\ldots,C_n\) at times \(0,1,\ldots,n\), with \(C_0<0\), the net present value at effective annual rate \(i\) is $$ \operatorname{NPV}(i)=\sum_{t=0}^{n} C_t(1+i)^{-t}. $$ An internal rate of return \(j\) is a rate that makes \(\operatorname{NPV}(j)=0\). In the examples, each project has one initial outflow followed only by nonnegative receipts, so its NPV decreases with the rate and its IRR is unique.

Worked Example: Equal Total Receipts, Different Timing

Worked Example: Equal Total Receipts, Different Timing

Two hypothetical projects each require \$100 today. Project A pays \$100 at the end of year 1, \$600 at the end of year 2, and \$200 at the end of year 3. Project B pays \$900 at the end of year 2. Compare their IRRs and their NPVs at a required effective annual rate of 10%.

Method. Set each project's NPV to zero to find its IRR, then discount the original cash flows at 10% to compare NPVs. Both projects have total receipts of \$900, so their different timing patterns are central to this comparison.

Project B's IRR. Its equation of value at time 0 is

$$ -100+\frac{900}{(1+j_B)^2}=0. $$

Thus \((1+j_B)^2=9\). Taking the positive rate root gives \(1+j_B=3\), so \(j_B=2\), or 200.00%.

Project A's IRR. Let \(x=1+j_A\). Its equation is

$$ -100+\frac{100}{x}+\frac{600}{x^2}+\frac{200}{x^3}=0, \qquad x^3-x^2-6x-2=0. $$

The positive solution is \(x\approx3.1249\), so \(j_A\approx2.1249\), or 212.49%. As a check, at \(x\approx3.1249\), the three receipts have present values of about \$32.00, \$61.44, and \$6.55. Their sum is about \$100.00, as required.

NPV at 10%. For Project A, the discount factors for years 1, 2, and 3 are \(1/1.1\), \(1/1.1^2\), and \(1/1.1^3\). Therefore

$$ \begin{aligned} \operatorname{NPV}_A(0.10) &=-100+\frac{100}{1.1}+\frac{600}{1.1^2}+\frac{200}{1.1^3}\\ &=-100+90.9091+495.8678+150.2630\\ &=\$637.04. \end{aligned} $$

For Project B,

$$ \operatorname{NPV}_B(0.10) =-100+\frac{900}{1.1^2} =-100+743.8017 =\$643.80. $$

At the required rate, B's NPV exceeds A's by about \$6.76, even though A's IRR is higher. The comparison is checked directly from the discounted cash flows: the IRR ranking is A above B, but the NPV ranking at 10% is B above A. Money amounts are rounded to the nearest cent.

Common mistake avoided: Do not choose A at a 10% required rate just because its IRR is larger. The IRR describes the rate at which that project's own NPV is zero; it does not give the project's NPV relative to another project at 10%.

Why the Rankings Can Differ

The NPV comparison depends on the present values of the cash flows at the particular rate being used. In the first example, subtracting B's NPV from A's cancels the equal initial costs. If \(x=1+i\), the difference is

$$ \operatorname{NPV}_A(i)-\operatorname{NPV}_B(i) =\frac{100}{x}-\frac{300}{x^2}+\frac{200}{x^3}. $$

At 10%, this difference is negative: B has the larger NPV. But at B's IRR, \(x=3\), the difference is

$$ \frac{100}{3}-\frac{300}{9}+\frac{200}{27} =\$7.4074>0. $$

Since B's NPV is zero at its own IRR, this shows that A's NPV is positive at B's IRR. That is consistent with A having the higher IRR.

Justification: For a conventional project with an initial outflow and nonnegative future receipts, its NPV decreases as the discount rate increases. If \(C_t\geq0\) for every \(t\geq1\), then for \(i>-1\), $$ \frac{d}{di}\operatorname{NPV}(i) =-\sum_{t=1}^{n}tC_t(1+i)^{-t-1}\leq0. $$ When at least one future receipt is positive, the derivative is strictly negative. Therefore, if A has positive NPV at B's unique IRR, A's own zero-NPV rate must be higher than B's. This explains the IRR ordering, but it does not say which project has greater NPV at every other rate.

The first example also isolates a useful point: equal total nominal receipts do not force equal NPVs, because a dollar received at one date and a dollar received at another date have different present values when \(i>0\). Nor is one project simply earlier throughout: A has a year-1 receipt, while B concentrates all its receipts in year 2. Comparing the complete schedules is more reliable than describing either project as merely “earlier.”

Worked Example: A Larger Delayed Receipt

Worked Example: A Larger Delayed Receipt

A hypothetical project Fast costs \$100 today and pays \$130 at the end of year 1. Project Delayed also costs \$100 today, but pays \$160 at the end of year 2. Find both IRRs and compare NPVs at a required effective annual rate of 10%.

Fast's IRR. The equation \(-100+130/(1+j_F)=0\) gives

$$ 1+j_F=\frac{130}{100}=1.30, \qquad j_F=30.00\%. $$

Delayed's IRR. Its equation is \(-100+160/(1+j_D)^2=0\). Hence

$$ 1+j_D=\sqrt{\frac{160}{100}}=\sqrt{1.6}\approx1.2649, \qquad j_D\approx26.49\%. $$

Fast has the higher IRR. Now discount each receipt at 10%. Fast's NPV is

$$ \operatorname{NPV}_F(0.10) =-100+\frac{130}{1.1} =-100+118.1818 =\$18.18. $$

Delayed's NPV is

$$ \operatorname{NPV}_D(0.10) =-100+\frac{160}{1.1^2} =-100+132.2314 =\$32.23. $$

Thus Delayed has the greater NPV at 10%, despite its lower IRR. The larger receipt compensates for arriving one year later at this required rate. This example changes both receipt amount and timing; unlike the equal-total-receipts example, it does not isolate timing alone. Money amounts are rounded to the nearest cent.

Common mistake avoided: Do not infer from the word “delayed” that the project's NPV must be lower. Discount the stated amount at the stated date; a sufficiently larger later receipt can produce the greater NPV.

Worked Example: Multiple Receipts and a Different Ranking

Worked Example: Multiple Receipts and a Different Ranking

Project M costs \$100 today and pays \$50 at the end of year 1 and \$100 at the end of year 2. Project N costs \$100 today and pays \$160 at the end of year 2. Compare the IRRs and the NPVs at a required effective annual rate of 5%.

Project M's IRR. Put \(x=1+j_M\). Its equation of value is

$$ -100+\frac{50}{x}+\frac{100}{x^2}=0, \qquad 2x^2-x-2=0. $$

The positive root is \(x=(1+\sqrt{17})/4\approx1.2808\). Thus \(j_M\approx0.2808\), or 28.08%. Checking at that root, the receipt present values total approximately \(50/1.2808+100/1.2808^2=\$100.00\).

Project N's IRR. Its equation gives \(100=160/(1+j_N)^2\), so

$$ 1+j_N=\sqrt{1.6}\approx1.2649, \qquad j_N\approx26.49\%. $$

M therefore has the higher IRR. At 5%, however, M's NPV is

$$ \begin{aligned} \operatorname{NPV}_M(0.05) &=-100+\frac{50}{1.05}+\frac{100}{1.05^2}\\ &=-100+47.6190+90.7029\\ &=\$38.32. \end{aligned} $$

For N,

$$ \operatorname{NPV}_N(0.05) =-100+\frac{160}{1.05^2} =-100+145.1247 =\$45.12. $$

At the specified required rate, N has the greater NPV. The calculations use the actual year-1 and year-2 dates, rather than treating both receipts as if they arrived at the same time. Money amounts are rounded to the nearest cent.

Common mistake avoided: Do not add the \$50 and \$100 receipts and discount the resulting \$150 as though it all arrived in year 2. The year-1 payment uses one year of discounting; the year-2 payment uses two.

A Reliable Comparison Process

For an exam problem that asks for both measures, keep the IRR calculation separate from the NPV comparison. The IRR equation uses the project cash flows and sets their NPV to zero. The NPV comparison uses the required rate given in the problem, whether or not that rate is close to either project's IRR.

1
Write each timeline.
Record the amount and date of every outflow and receipt. Do not combine receipts at different dates.
2
Find each IRR separately.
Set that project's NPV equal to zero and solve for its rate. Check that the resulting rate makes its discounted receipts equal its initial cost.
3
Use the required rate for NPV.
Discount every cash flow at the stated required rate, using its own time period, then compare the resulting NPVs.
4
State the two rankings separately.
For example: “A has the higher IRR; B has the higher NPV at 10%.” This avoids implying the measures give the same ordering.

A positive NPV means the discounted value of receipts exceeds the initial cost at the specified rate; it does not mean the project has the higher IRR than another project. Also distinguish a ranking question from a decision about whether to accept a project. The required rate and the relevant decision setting matter, and the NPV calculation should not be replaced by an IRR ranking.

Common Mistakes and Exam Traps

  • Ranking by IRR when asked for NPV. Calculate NPV at the stated required rate; a higher IRR does not guarantee a higher NPV there.
  • Using one discount period for all receipts. A year-1 payment is discounted once, a year-2 payment twice, and so on.
  • Assuming the timing effect has one direction for every rate. The NPV comparison can depend on the required rate because the projects have different cash-flow patterns.
  • Comparing total receipts instead of their present values. Equal nominal totals can have unequal NPVs, and a larger nominal total does not by itself settle the ranking.
  • Mixing up a project's IRR and the required rate. Use the IRR to describe the rate at which that project's NPV is zero; use the required rate to evaluate NPV.
Key takeaway: IRR and NPV answer different questions. When cash flows arrive on different schedules, calculate both rankings from the relevant cash-flow dates. A project can have the higher IRR while another has the higher NPV at the required rate.

Check Your Understanding

For each problem, compare the requested measures using the stated annual cash-flow dates. Money answers are rounded to the nearest cent.

  1. Project A costs \$500 and pays \$650 at the end of year 1. Project B costs \$500 and pays \$800 at the end of year 2. Find both IRRs and compare NPVs at 8%. (Answer: A IRR = 30.00%; B IRR \(\approx26.49\%\); NPV\(_A=\$101.85\), NPV\(_B=\$185.87\), so B has the greater NPV.)
  2. For the equal-total-receipts projects in the first worked example, compare their NPVs at 50%. (Answer: NPV\(_A\approx\$292.59\), NPV\(_B\approx\$300.00\); B has the greater NPV.)
  3. Project P costs \$100 and pays \$300 at the end of year 1 and \$200 at the end of year 3. Project Q costs \$100 and pays \$500 at the end of year 2. Compare their NPVs at 10%. (Answer: NPV\(_P\approx\$322.99\), NPV\(_Q\approx\$313.22\); P has the greater NPV.)
  4. Project R costs \$100 and pays \$130 at the end of year 1. Project S costs \$100 and pays \$160 at the end of year 2. Find both IRRs and compare NPVs at 10%. (Answer: R IRR = 30.00%, S IRR \(\approx26.49\%\); NPV\(_R=\$18.18\), NPV\(_S=\$32.23\), so S has the greater NPV.)
  5. For the projects in the first worked example, evaluate Project A's NPV at Project B's IRR. What does the sign imply about their IRRs? (Answer: NPV\(_A(200\%)\approx\$7.41>0\); since B's NPV is zero there and both projects have unique IRRs, A's IRR is higher.)

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