Why Project Scale Changes the Ranking
Two projects can both add value and still rank differently depending on what is being compared. A large project may create more total value even though each dollar invested produces less NPV. A smaller project may have the stronger NPV relative to its investment but add fewer dollars overall. This is a scale effect: the projects differ in the size of their initial outlays, so an absolute measure and a per-dollar measure need not give the same ranking.
The distinction matters when comparing projects. If the alternatives are mutually exclusive and there is no stated constraint that makes investment dollars scarce, total NPV measures the value added by choosing each project. NPV per initial outlay is a useful measure of relative performance, but it answers a different question. It should not automatically replace total NPV as the decision rule.
These expressions assume that the initial outlay occurs at time 0, the later cash flows are dated at the ends of annual periods, and the same annual effective rate \(i\) is appropriate for the cash flows being compared. If cash flows are not annual, the discount periods must match the cash-flow dates and the rate convention. Here, \(C_t\) denotes a net cash flow at time \(t\); if a later cash flow is a cost, it is entered as a negative amount.
What the Two Measures Tell You
Total NPV is an amount of value at the valuation date. It tells how many dollars the project adds, relative to the required rate of return used to discount its cash flows. NPV per initial outlay divides that dollar amount by the amount invested initially. It tells how much NPV is produced per dollar of initial investment. Neither measure is a rate of return on the invested funds, and neither should be confused with an internal rate of return.
When the only initial cash flow is \(-I_0\) and all later cash flows are benefits, the present value of those benefits is $$ \operatorname{PV}(\text{benefits})=\sum_{t=1}^{n}C_t(1+i)^{-t}. $$ Since \(\operatorname{NPV}=\operatorname{PV}(\text{benefits})-I_0\), it follows that $$ \frac{\operatorname{PV}(\text{benefits})}{I_0} =1+\frac{\operatorname{NPV}}{I_0}. $$ The left side is the present-value benefit ratio. It is not the same quantity as NPV per initial outlay: it includes recovery of the initial investment. The two ratios have the same ordering under these assumptions because one is the other plus 1, but their numerical meanings differ. If later cash flows include costs, the benefit-ratio interpretation may not apply without defining which cash flows count as benefits and which count as investment.
Why Scaling Changes Absolute NPV but Not the Ratio
Suppose every cash flow of a project, including its initial outlay, is multiplied by the same positive number \(k\). This represents scaling the project without changing its per-unit cash flows, and assumes that costs and benefits both scale proportionally with no fixed costs, capacity limits, or interaction effects. The scaled NPV is
The ratio for the scaled project is therefore $$ \frac{\operatorname{NPV}_{\text{scaled}}}{kI_0} =\frac{k\operatorname{NPV}}{kI_0} =\frac{\operatorname{NPV}}{I_0}. $$ Thus, exact proportional scaling increases or decreases total NPV in proportion to project size while leaving NPV per dollar unchanged. In practice, two projects are seldom perfect scaled copies, so their per-dollar results can differ. Fixed costs, different cash-flow patterns, or other project features can change the comparison.
Worked Example: A Higher NPV but a Lower NPV per Dollar
Two mutually exclusive projects each require one outlay at time 0. Project A costs $20,000 and pays $24,200 at the end of year 1. Project B costs $100,000 and pays $118,800 at the end of year 1. Use an effective annual interest rate of 10%. Calculate both projects’ NPV and NPV per initial outlay, then compare their rankings.
Method. Discount each year-1 receipt by \(1.10\), subtract the initial outlay to obtain NPV, and divide that NPV by the outlay to obtain the per-dollar measure.
For Project A, the discount factor is \(1/1.10\), so the present value of the receipt is $$ \frac{\$24{,}200}{1.10}=\$22{,}000. $$ Therefore, $$ \operatorname{NPV}_A=\$22{,}000-\$20{,}000=\$2{,}000, \qquad \frac{\operatorname{NPV}_A}{I_{0,A}} =\frac{\$2{,}000}{\$20{,}000}=0.10=10\%. $$
For Project B, the present value of the receipt is $$ \frac{\$118{,}800}{1.10}=\$108{,}000. $$ Therefore, $$ \operatorname{NPV}_B=\$108{,}000-\$100{,}000=\$8{,}000, \qquad \frac{\operatorname{NPV}_B}{I_{0,B}} =\frac{\$8{,}000}{\$100{,}000}=0.08=8\%. $$
The calculations check because, for each project, present value of the receipt equals initial outlay plus NPV: for A, \(\$22{,}000=\$20{,}000+\$2{,}000\); for B, \(\$108{,}000=\$100{,}000+\$8{,}000\). Project B has the larger total NPV, while Project A has the larger NPV per dollar invested. If the projects are mutually exclusive and the objective is to maximize total value, B is preferred on these figures. The per-dollar ranking alone does not reverse that conclusion.
Common mistake avoided: B's present-value benefit ratio is \(\$108{,}000/\$100{,}000=108\%\), but its NPV per initial outlay is \(\$8{,}000/\$100{,}000=8\%\). Reporting 108% as NPV per dollar incorrectly counts return of the original investment as value added.
Comparing Projects With Several Cash Flows
With multiple dated receipts, calculate the present value of each cash flow separately before finding NPV. Only after total NPV is known should it be divided by the initial outlay. This keeps the absolute and per-dollar measures tied to the same valuation date and discount rate.
Worked Example: Multi-Period Cash Flows and Opposite Rankings
At an effective annual interest rate of 10%, Project M requires $25,000 at time 0 and pays $16,500 at the end of year 1 and $18,150 at the end of year 2. Project N requires $120,000 at time 0 and pays $66,000 at the end of year 1 and $87,120 at the end of year 2. Find each NPV and NPV per initial outlay. If only one project can be selected and the goal is to maximize total value, which is preferred?
Method. Discount the year-1 cash flow by \(1.10\) and the year-2 cash flow by \(1.10^2=1.21\). Sum those present values, subtract the initial outlay, then divide by that outlay.
For Project M, the individual present values are $$ \frac{\$16{,}500}{1.10}=\$15{,}000, \qquad \frac{\$18{,}150}{1.21}=\$15{,}000. $$ Thus, its total present value of later cash flows is \(\$15{,}000+\$15{,}000=\$30{,}000\), and $$ \operatorname{NPV}_M=\$30{,}000-\$25{,}000=\$5{,}000, \qquad \frac{\operatorname{NPV}_M}{I_{0,M}} =\frac{\$5{,}000}{\$25{,}000}=0.20=20\%. $$
For Project N, the present values are $$ \frac{\$66{,}000}{1.10}=\$60{,}000, \qquad \frac{\$87{,}120}{1.21}=\$72{,}000. $$ The total is \(\$132{,}000\), so $$ \operatorname{NPV}_N=\$132{,}000-\$120{,}000=\$12{,}000, \qquad \frac{\operatorname{NPV}_N}{I_{0,N}} =\frac{\$12{,}000}{\$120{,}000}=0.10=10\%. $$
The present-value checks are \(\$30{,}000=\$25{,}000+\$5{,}000\) for M and \(\$132{,}000=\$120{,}000+\$12{,}000\) for N. M ranks higher per dollar, but N adds more total value. On the stated mutually exclusive choice, N is preferred when the objective is the larger total NPV. The error to avoid is dividing a present value by the wrong project’s outlay, or treating the larger percentage as automatically decisive.
When a Per-Dollar Comparison Helps
NPV per initial outlay is useful when the question explicitly asks for relative value added, or when comparing how much NPV is associated with each dollar committed. It can also clarify what a scale difference is doing: a project can be more efficient by this measure without creating the most total value. A ratio is especially informative when project sizes differ substantially, but it does not by itself establish which project is feasible or best under every decision constraint.
For example, if a project can be repeated in identical units and all cash flows scale exactly in proportion, the NPV per dollar is the same for one unit or several. The total NPV then grows with the number of units. That conclusion depends on the proportional-scaling assumption. If a larger version requires a fixed setup cost, has capacity limits, or changes its receipts per unit, the scaled cash flows must be recalculated rather than inferred from the smaller project.
For mutually exclusive projects, the absolute comparison is direct: choose the alternative with the greater NPV when the objective is to maximize value and the cash flows are evaluated consistently. The earlier material on incremental NPV gives another way to express that comparison: subtract one alternative’s cash flows from the other’s and evaluate the difference. A positive incremental NPV for the higher-outlay alternative means that the additional investment adds value at the stated rate. The ratio comparison is supplemental, not a substitute for this value comparison.
Check Your Understanding
Use the stated annual effective rate and assume each initial outlay occurs at time 0. Answers are given so you can check both the calculation and the ranking.
- At 5%, a project costs $40,000 and pays $45,360 at the end of year 1. Find its NPV and NPV per initial outlay. (Answer: NPV = $3,200; NPV per initial outlay = 8%.)
- At 10%, Project R costs $60,000 and pays $22,000 at the end of year 1 and $48,400 at the end of year 2. Find its NPV and NPV per initial outlay. (Answer: NPV = $0; NPV per initial outlay = 0%.)
- At 10%, Project S costs $30,000 and pays $16,500 at the end of year 1 and $22,506 at the end of year 2. Project T costs $90,000 and pays $55,000 at the end of year 1 and $57,112 at the end of year 2. Find both NPVs and ratios, then state which project has the higher total NPV. (Answer: S has NPV $3,600 and ratio 12%; T has NPV $7,200 and ratio 8%; T has the higher total NPV.)
- A project has an initial outlay of $50,000 and a present value of later benefits of $58,000. Find its NPV per initial outlay and its present-value benefit ratio. Explain the difference. (Answer: NPV per initial outlay = 16%; present-value benefit ratio = 116%. The latter includes recovery of the initial outlay.)
- A repeatable unit requires $20,000 initially and has later benefits with present value $22,400. Assuming all cash flows scale exactly and there are no fixed costs, how many units are needed for total NPV to exceed $7,500, and what are their combined outlay and NPV? (Answer: Each unit has NPV $2,400; four units are needed, with combined outlay $80,000 and NPV $9,600.)