Why Cash-Flow Timing Can Change a Project Ranking
Two mutually exclusive projects can require the same initial investment and still deliver their cash flows at different times. A project that pays sooner may have a higher internal rate of return (IRR), while a project with larger or later receipts may have a higher net present value (NPV) at a low discount rate. As the discount rate rises, the later receipts lose more present value, and the preferred project can change.
The important distinction is that IRR describes a rate for one project's cash flows, whereas NPV measures the project's dollar contribution at a specified required return. When choosing between mutually exclusive projects, the economically preferred option is the one with the larger NPV at the appropriate required return, provided the comparison uses consistent assumptions.
For mutually exclusive projects with the same relevant comparison horizon and risk, use the same required return \(i\) to evaluate each project's NPV. Select the project with the greatest NPV if the choice is mandatory. If the decision is whether to undertake a project rather than which one to choose, a positive NPV indicates value creation at that required return; a negative NPV indicates that the project does not meet it.
An IRR ranking can conflict with this decision because a higher percentage return does not necessarily create more dollars of value. In particular, a high IRR may be associated with cash received early, while a lower-IRR project may produce more total cash later. Whether those later amounts are valuable depends on the required return used to discount them.
How the Discount Rate Affects Timing
At a positive discount rate, a dollar received later is worth less today than a dollar received earlier. Moreover, a change in the discount rate has a larger proportional effect on a cash flow that is farther in the future. This is the key mechanism behind timing conflicts.
For a conventional project with an initial cost \(I_0>0\) and nonnegative future inflows \(C_t\), its NPV can be written as \(-I_0+\sum_{t=1}^{n}C_t(1+i)^{-t}\). Differentiating with respect to the required return gives:
Each term in the sum is nonnegative when \(i>-1\) and \(C_t\geq 0\), so the derivative is nonpositive: the NPV of a conventional project does not increase as its required return rises. The factor \(t\) shows why a later inflow contributes more to the rate sensitivity than an equal earlier inflow.
For two projects, compare their NPVs by subtracting one from the other. The difference is the present value of the differences in their cash flows. If one project pays more early and the other pays more late, the signs of those differences can oppose each other. Changing \(i\) changes the relative weights on the early and late differences, so the NPV ranking can reverse.
Worked Example: A Higher IRR Does Not Always Mean a Higher NPV
A hypothetical firm must choose one of two same-risk projects. Project S costs $100 immediately and pays $130 at the end of year 1. Project L costs $100 immediately and pays $150 at the end of year 2. Assume annual end-of-year cash flows, no cash flows after the stated payment, and the same annual effective required return for both projects. Compare their IRRs and NPVs at 10% and 20%.
Method and IRRs. For each one-payment conventional project, set NPV to zero and solve for its IRR. Project S satisfies \(-100+130/(1+i_{\mathrm{IRR}})=0\), so \(1+i_{\mathrm{IRR}}=130/100=1.30\), giving \(i_{\mathrm{IRR},S}=30.00\%\). Project L satisfies \(-100+150/(1+i_{\mathrm{IRR}})^2=0\), so \((1+i_{\mathrm{IRR}})^2=1.50\). Thus \(i_{\mathrm{IRR},L}=\sqrt{1.50}-1\approx22.47\%\). The IRR ranking favors S.
NPVs at 10%. For S, \(\operatorname{NPV}_S(0.10)=-100+130/1.10=-100+118.181818\ldots=\$18.18\). For L, \(\operatorname{NPV}_L(0.10)=-100+150/(1.10)^2=-100+123.966942\ldots=\$23.97\). The unrounded difference is \(23.966942\ldots-18.181818\ldots=\$5.785124\ldots\), so L has the higher NPV by $5.79. At this required return, choose L.
NPVs at 20%. For S, \(\operatorname{NPV}_S(0.20)=-100+130/1.20=-100+108.333333\ldots=\$8.33\). For L, \(\operatorname{NPV}_L(0.20)=-100+150/(1.20)^2=-100+104.166666\ldots=\$4.17\). Before rounding, S's NPV advantage is \(130/1.20-150/(1.20)^2=8.333333\ldots-4.166666\ldots=\$4.166666\ldots\), which rounds to $4.17. At this required return, choose S.
Check and common mistake. The IRR ranking stays S over L, but the NPV ranking changes: L is preferred at 10%, while S is preferred at 20%. A common mistake is to treat the higher IRR as the automatic choice between mutually exclusive projects. The correction is to compare NPVs at the relevant required return. Amounts here are rounded to the nearest cent only at the final step.
Reading the Ranking at the Relevant Required Return
The project ranking is not a universal property independent of the decision context. A project can have a higher NPV at one required return and a lower NPV at another. The company's required return represents the return required for the relevant level of risk; it is the rate used to make the NPV choice, not a rate selected merely because it favors one option.
The comparison assumes that the cash-flow estimates are stated on a consistent basis, the projects have comparable risk so the same required return is appropriate, and their cash flows cover a suitable common comparison horizon. If projects differ materially in risk, they should not automatically be discounted at the same rate. If their lives or consequences beyond the listed cash flows differ, those differences must be handled before interpreting the comparison.
Worked Example: A Delayed Receipt Loses More Value at a High Required Return
A hypothetical utility must choose between two mutually exclusive equipment plans with the same risk and comparison horizon. Plan E costs $400 now and pays $500 at the end of year 1. Plan D costs $400 now and pays $650 at the end of year 3. Assume annual end-of-year cash flows, no other cash flows, and a common annual effective required return. Compare the IRR ranking with the NPV ranking at 0% and 25%.
Method and IRRs. Plan E has \(-400+500/(1+i_{\mathrm{IRR}})=0\), so \(i_{\mathrm{IRR},E}=500/400-1=25.00\%\). Plan D has \(-400+650/(1+i_{\mathrm{IRR}})^3=0\), so \(i_{\mathrm{IRR},D}=(650/400)^{1/3}-1=1.625^{1/3}-1\approx17.57\%\). The IRR ranking favors E.
NPVs at 0%. Plan E has \(-400+500=\$100.00\). Plan D has \(-400+650=\$250.00\). At a zero required return, D has the higher NPV by $150.00 because its larger receipt is not discounted.
NPVs at 25%. Plan E has \(-400+500/1.25=-400+400=\$0.00\). For Plan D, \(1.25^3=1.953125\), so its NPV is \(-400+650/1.953125=-400+332.80=-\$67.20\). E is higher by $67.20. If the choice is mandatory, E is preferred at this rate; if the firm can reject both, E is only at break-even and D has negative NPV.
Check and common mistake. The earlier payment supports E's higher IRR, but D's larger receipt gives it the higher NPV when the discount rate is zero. At 25%, the delayed payment is discounted much more, and the ranking reverses. Do not compare the nominal amounts of the receipts without accounting for their dates. Amounts are rounded to the nearest cent; the stated values are exact at the displayed rates.
Comparing Cash-Flow Patterns, Not Just Final Totals
Timing conflicts are not limited to projects with a single future payment. With multiple inflows, compare each dated cash flow at the required return. A project with more early cash may be preferred at a high rate even when another project has a larger undiscounted total. Conversely, at a low rate, the larger total may dominate because later cash is discounted less heavily.
The following example isolates this effect. Both projects have the same initial cost, so the ranking comes from the timing and amount of their future receipts rather than from a difference in the initial investment.
Worked Example: Ranking Two Projects With Two Inflows Each
A hypothetical company has a fixed budget and must select one of two same-risk projects. Project A costs $1,000 now, pays $600 at the end of year 1, and pays $500 at the end of year 3. Project B costs $1,000 now, pays $300 at the end of year 1, and pays $900 at the end of year 3. Assume annual end-of-year cash flows, no other cash flows, and a common annual effective required return. Compare the NPVs at 0% and 20%.
Method. Discount each inflow separately to time 0, then subtract the initial cost. At 0%, Project A has \(-1{,}000+600+500=\$100.00\), while Project B has \(-1{,}000+300+900=\$200.00\). B is higher by $100.00.
At 20%, the year-1 discount factor is \(1/1.20\), and the year-3 discount factor is \(1/(1.20)^3=1/1.728\). Project A has \[ -1{,}000+\frac{600}{1.20}+\frac{500}{1.728} =-1{,}000+500+289.351851\ldots =-\$210.65. \] Project B has \[ -1{,}000+\frac{300}{1.20}+\frac{900}{1.728} =-1{,}000+250+520.833333\ldots =-\$229.17. \] A is higher by \(18.518518\ldots\), or $18.52, so A is preferred at 20% if a choice is mandatory. Both projects have negative NPV at that rate, so rejecting both is preferable if that is allowed.
Check and common mistake. At 0%, B's total inflows exceed A's by $100. At 20%, B's larger year-3 payment is discounted more heavily, and A's advantage in the year-1 payment produces the higher NPV. Do not discount a year-3 payment as if it arrived in year 1. Amounts are rounded to the nearest cent only at the final step.
Common Traps in Timing Comparisons
- Choosing the larger IRR automatically. IRR ranks each project's percentage return; NPV ranks its dollar value at the required return. For mutually exclusive projects, use the NPV ranking at the relevant rate.
- Ignoring the date of a cash flow. A receipt in year 3 is divided by \((1+i)^3\), not by \(1+i\). Match every cash flow to its actual period.
- Changing rates between projects. If the projects have comparable risk, discount both at the same required return. Using a different rate for each can make the comparison inconsistent.
- Assuming a ranking cannot change. Early and late cash-flow differences can receive different relative weights at different rates. Calculate both NPVs at the specified decision rate rather than extrapolating from one rate.
- Rounding before comparing. Keep full precision while calculating NPVs and their difference, then round the final dollar amounts. Subtracting rounded NPVs can create a one-cent error.
Check Your Understanding
For each problem, use the stated annual effective required return and assume all cash flows occur at year-end unless the problem says otherwise.
- Project R costs $250 now and pays $325 in one year. Project T costs $250 now and pays $390 in two years. Calculate both NPVs at 10% and identify the higher-NPV project. (Answer: R has NPV $45.45; T has NPV $72.31; choose T.)
- Project U costs $600 now and pays $720 in one year. Project V costs $600 now and pays $850 in two years. Find each IRR and compare their NPVs at 5%. (Answer: U's IRR is 20.00%, V's is approximately 19.02%; U's NPV is $85.71, V's is $170.98, so choose V.)
- Project F costs $800 now and pays $1,000 in one year. Project G costs $800 now and pays $1,350 in three years. At 20%, calculate both NPVs and identify the higher-NPV project. (Answer: F has NPV $33.33; G has NPV −$18.75; choose F if selection is mandatory, though rejecting both is better than accepting either.)
- Project A costs $500 now and pays $400 in year 1 and $200 in year 2. Project B costs $500 now and pays $200 in year 1 and $450 in year 2. Compare NPVs at 0% and 50%, and state whether the ranking reverses. (Answer: At 0%, A has NPV $100 and B has NPV $150, so B is higher; at 50%, A has NPV −$144.44 and B has NPV −$166.67, so A is higher. Yes, the ranking reverses.)