NPV Ranking Without a Capital Constraint

Why Rank Projects When Funding Is Unlimited?

A firm may compare several independent investments even when it has enough funding to undertake every worthwhile one. Ranking the projects by net present value (NPV) shows which contributes the most value in dollars. But a ranking is an ordering, not a rule that says to accept only the project at the top. When funding is genuinely unlimited and projects are independent, the decision is made separately for each project: accept every positive-NPV project, reject every negative-NPV project, and regard a zero-NPV project as value-neutral.

That distinction matters because a project can rank below another and still increase value. If one project adds $15.45 and another adds $9.09, taking both adds $24.54, provided the projects are independent and the firm can fund both. Rejecting the second project merely because it ranks lower would leave value uncreated.

Definition: Projects are independent when accepting or rejecting one does not change the cash flows, feasibility, or acceptance decision of another. With no capital constraint, the firm can fund every project it chooses to accept. Under these conditions, rank projects from highest to lowest NPV to compare their value contributions, but accept each project with positive NPV rather than only the highest-ranked one.

Calculating and Interpreting a Ranking

For project $j$, let $CF_{j,t}$ be its cash flow at time $t$, with the initial investment usually recorded as a negative cash flow at time 0. If cash flows occur at annual dates and the effective annual required return is $i$, the project's NPV at time 0 is

\operatorname{NPV}_j=\sum_{t=0}^{n}\frac{CF_{j,t}}{(1+i)^t} =\sum_{t=0}^{n}CF_{j,t}v^t, \qquad v=\frac{1}{1+i}.

This model assumes that the cash flows and required return are stated consistently, that the cash flows are measured from the same valuation date, and that $i$ is appropriate for the project’s risk. For the examples below, all cash flows occur at year-end, all amounts are in dollars, and the stated effective annual rate applies to each project. For cash flows at nonannual dates, use discount factors that match those dates rather than treating each payment as if it occurred at a year-end.

Calculate each NPV before ordering the projects. A higher NPV means a larger dollar contribution to value at the valuation date; it does not mean that the project is the only one worth accepting. A positive NPV exceeds the value required by the project's required return. A negative NPV falls short of that requirement. A zero NPV meets the requirement exactly and, under the model, neither adds nor subtracts value.

Worked Example: Ranking Four Independent Investments

A firm can fund any number of four independent projects. Each requires a time-0 investment and produces the listed year-end cash flows. The effective annual required return is $i=10\%$. Calculate each NPV, rank the projects, and state the decision for each.

Method and discount factors. Use $v=1/1.10=10/11$ and $v^2=1/1.21=100/121$. Discount each future receipt to time 0, include the initial outlay at time 0, and then order the resulting NPVs from largest to smallest.

Project A. Its cash flows are $-\$100$ at time 0 and $+\$120$ at year 1: $120(10/11)=\$109.0909\ldots$, so $\operatorname{NPV}_A=-100+109.0909\ldots=\$9.09$, rounded to cents.

Project B. Its cash flows are $-\$150$ at time 0 and $+\$180$ at year 1: $180(10/11)=\$163.6363\ldots$, so $\operatorname{NPV}_B=-150+163.6363\ldots=\$13.64$, rounded to cents.

Project C. Its cash flows are $-\$80$ at time 0, $+\$95$ at year 1, and $+\$11$ at year 2. The year-1 present value is $95(10/11)=\$86.3636\ldots$. The year-2 present value is $11(100/121)=\$9.0909\ldots$. Therefore, $\operatorname{NPV}_C=-80+86.3636\ldots+9.0909\ldots=\$15.45$, rounded to cents.

Project D. Its cash flows are $-\$50$ at time 0 and $+\$55$ at year 1: $55(10/11)=\$50$, so $\operatorname{NPV}_D=-50+50=\$0.00$.

The descending ranking is C (\$15.45), B (\$13.64), A (\$9.09), and D (\$0.00). Accept A, B, and C because each has positive NPV; D is value-neutral, so the firm is indifferent about accepting it under the stated assumptions. As a check, the sum of the three positive NPVs is $\$15.45+\$13.64+\$9.09=\$38.18$, rounded to cents. The common mistake avoided here is treating the top-ranked project as the only acceptable one: B and A also add value.

Why Independent Project NPVs Add

The reason a lower-ranked positive-NPV project should not be discarded is that independent projects contribute separate cash flows. With a common required return and valuation date, the NPV of the set of projects is the sum of their NPVs. To see this, write the combined cash flow at time $t$ as the sum of the cash flows from the projects undertaken. Discounting that combined cash flow and rearranging the finite sums gives:

\operatorname{NPV}_{\text{combined}} =\sum_{t=0}^{n}v^t\left(\sum_{j=1}^{m}CF_{j,t}\right) =\sum_{j=1}^{m}\left(\sum_{t=0}^{n}v^tCF_{j,t}\right) =\sum_{j=1}^{m}\operatorname{NPV}_j.

The equality follows from the distributive property of addition: multiplying a sum of cash flows by a discount factor gives the sum of the individually discounted cash flows. It relies on evaluating the projects on a consistent basis and on there being no interactions that change their cash flows or feasibility. If one project changes another’s costs, revenues, or ability to proceed, the projects are not independent as defined here; evaluate the relevant combined cash flows instead.

When every positive-NPV project can be undertaken, adding one such project increases the combined NPV by that project's positive amount. Adding a negative-NPV project decreases the combined NPV. Thus, choosing all and only positive-NPV projects gives the highest total NPV among the available independent accept-or-reject decisions. Ranking identifies the size order of the contributions; the sign identifies whether each contribution raises or lowers value.

Worked Example: Using NPV Signs to Form the Project Set

Three independent projects have the following cash flows, with an effective annual required return of $i=10\%$. The firm has no funding limit and may accept or reject each project separately. Calculate the NPVs, rank the projects, and compare the value of accepting all three with the value of accepting only positive-NPV projects.

Project X. Cash flows are $-\$90$ at time 0, $+\$55$ at year 1, and $+\$55$ at year 2. Using $v=10/11$ and $v^2=100/121$, the present values are $55(10/11)=\$50.00$ and $55(100/121)=\$45.4545\ldots$. Thus, $\operatorname{NPV}_X=-90+50+45.4545\ldots=\$5.45$, rounded to cents.

Project Y. Cash flows are $-\$90$ at time 0 and $+\$110$ at year 2. The receipt has present value $110(100/121)=\$90.9090\ldots$. Therefore, $\operatorname{NPV}_Y=-90+90.9090\ldots=\$0.91$, rounded to cents.

Project Z. Cash flows are $-\$40$ at time 0 and $+\$43.20$ at year 1. The receipt has present value $43.20(10/11)=\$39.2727\ldots$. Therefore, $\operatorname{NPV}_Z=-40+39.2727\ldots=-\$0.73$, rounded to cents.

The descending ranking is X (\$5.45), Y (\$0.91), and Z ($-\$0.73$). Accept X and Y; reject Z. The combined NPV of the accepted projects, calculated before rounding, is $5.4545\ldots+0.9090\ldots=\$6.3636\ldots$, or $\$6.36$ to cents. Accepting all three would instead give $6.3636\ldots-0.7272\ldots=\$5.6363\ldots$, or $\$5.64$. The check confirms that taking the negative-NPV project reduces value. The common mistake avoided is accepting every project in a set just because the set’s total NPV remains positive; with independent, separately selectable projects, assess each project’s own NPV.

A Portfolio Check by Addition

The additivity result also gives a useful check on calculations. Calculate and round only after the project NPVs have been summed if a total is required, because adding individually rounded values can create a small rounding difference. The following example uses unrounded values in the portfolio calculation and reports final monetary answers to the nearest cent.

Worked Example: Checking the Value of the Accepted Portfolio

A firm evaluates three independent projects at an effective annual required return of $i=10\%$. Project E costs \$60 now and pays \$35 at each of years 1 and 2. Project F costs \$40 now and pays \$48 at year 1. Project G costs \$25 now and pays \$26 at year 1. Determine their ranking, the value-maximizing accept-or-reject decisions with unlimited funding, and the NPV of the accepted portfolio.

Project E. The receipts have present values $35(10/11)=\$31.8181\ldots$ and $35(100/121)=\$28.9256\ldots$. Hence, $\operatorname{NPV}_E=-60+31.8181\ldots+28.9256\ldots=\$0.7438\ldots$, or $\$0.74$ to cents.

Project F. The year-1 receipt has present value $48(10/11)=\$43.6363\ldots$. Hence, $\operatorname{NPV}_F=-40+43.6363\ldots=\$3.6363\ldots$, or $\$3.64$ to cents.

Project G. The year-1 receipt has present value $26(10/11)=\$23.6363\ldots$. Hence, $\operatorname{NPV}_G=-25+23.6363\ldots=-\$1.3636\ldots$, or $-\$1.36$ to cents.

The ranking is F (\$3.64), E (\$0.74), and G ($-\$1.36$). Accept F and E and reject G. Using unrounded NPVs, the accepted portfolio has NPV $3.6363\ldots+0.7438\ldots=\$4.3801\ldots$, or $\$4.38$ to cents. Including G would reduce that total by \$1.3636… and produce a portfolio NPV of about \$3.02. The check is consistent with the separate decisions: F and E each add value, while G subtracts value. The common mistake avoided is confusing a project’s rank with its decision; E ranks second but should still be accepted.

Common Ranking Traps

Accepting only the highest-ranked project. That is not the rule when projects are independent and funding is unlimited. Every positive-NPV project adds value.
Ranking before calculating NPV. Initial investment, the timing of receipts, and all later cash flows affect NPV. Compare the completed present values, not just one cash-flow amount.
Ignoring the sign of a lower-ranked project. A lower positive NPV is still value-adding; a negative NPV should not be accepted merely because it ranks above another negative project.
Comparing figures from inconsistent valuations. Use the same valuation date and appropriate required returns. Discount each cash flow using a rate and period that match its timing and risk assumptions.
Assuming independence when projects interact. If accepting one project changes another’s cash flows or feasibility, separate NPVs do not represent the actual combined decision without adjusting for that interaction.

Key takeaway: For independent projects with unlimited funding, rank NPV to compare dollar value contributions, but decide project by project: accept positive NPV, reject negative NPV, and treat zero NPV as indifferent. The highest-ranked project is not the only one worth undertaking.

Check Your Understanding

For each problem, use the stated effective annual required return and year-end cash-flow dates. Round final dollar NPVs to the nearest cent.

A project costs \$100 at time 0 and pays \$118 at year 1. At $i=8\%$, calculate its NPV and state the decision. (Answer: NPV = \$9.26; accept.)
Two independent projects are evaluated at $i=10\%$. Project A costs \$75 and pays \$82.50 at year 1; Project B costs \$75 and pays \$90 at year 1. Give the ranking and decision for each. (Answer: B ranks above A; accept B, and be indifferent about A, whose NPV is \$0.00.)
A project costs \$50 and pays \$30 at each of years 1 and 2. At $i=10\%$, calculate its NPV and state the decision. (Answer: NPV = \$2.07; accept.)
Three independent projects are evaluated at $i=10\%$: P costs \$100 and pays \$115 at year 1; Q costs \$200 and pays \$220 at year 1; R costs \$40 and pays \$42 at year 1. Rank them and state each decision. (Answer: P ranks first with NPV \$4.55, Q second with NPV \$0.00, and R third with NPV $-\$1.82$; accept P, be indifferent about Q, and reject R.)
At $i=10\%$, independent projects A, B, and C have cash flows as follows: A costs \$90 and pays \$55 at each of years 1 and 2; B costs \$50 and pays \$60 at year 1; C costs \$30 and pays \$31 at year 1. Rank them and find the NPV of the value-maximizing accepted set. (Answer: A ranks first with NPV \$5.45, B second with NPV \$4.55, and C third with NPV $-\$1.82$; accept A and B, reject C, and the accepted set has NPV \$10.00.)

What You'll Learn