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NPV Rules for Independent Projects

Apply the positive-NPV rule project by project, and determine when the value of accepted independent projects can be added.

Intermediate 10 min read

What You'll Learn

  • Define independent projects and identify the assumptions needed for the positive-NPV rule.
  • Calculate each project’s NPV from its own initial outlay and later cash flows.
  • Decide what to do with projects having positive, negative, or zero NPV.
  • Add project NPVs to find the value of a portfolio of independent projects.
  • Recognize why a binding budget or project interaction changes the decision problem.

Why Independent Projects Are Evaluated Separately

When projects are independent, undertaking one does not change the cash flows or feasibility of another. If there is no binding limit on the funds available, each project can be screened on its own: undertake a project that adds value and do not undertake one that reduces value. This differs from a choice between mutually exclusive projects, where selecting one rules out the others.

The rule depends on a consistent valuation model. In this tutorial, cash flows occur at the ends of annual periods, each project has an initial outlay at time 0, and one effective annual discount rate is used to evaluate the projects. Each project’s NPV is measured at time 0.

Definition: For a project with signed cash flow \(C_t\) at the end of year \(t\), initial cash flow \(C_0\), and effective annual discount rate \(i\), its net present value at time 0 is
$$ \operatorname{NPV}=\sum_{t=0}^{n}\frac{C_t}{(1+i)^t}, \qquad v=\frac{1}{1+i}. $$
Here \(C_0\) is usually negative for an initial outlay. A positive NPV means the discounted value of the project’s net cash flows is positive under the stated assumptions.

For an independent project with no binding resource constraint, the decision rule is to accept a positive-NPV project, reject a negative-NPV project, and be financially indifferent to accepting or rejecting a zero-NPV project. The zero-NPV result means that the discounted value of the project’s benefits exactly offsets its costs; it does not mean that the project has no cash flows.

The Positive-NPV Rule and Its Assumptions

Applying the rule project by project requires more than calculating a number. Check that the projects can in fact be considered independently and that the decision maker is able to undertake every positive-NPV project. In particular:

  • One project’s cash flows do not change because another project is undertaken.
  • Choosing one project does not make another project unavailable.
  • There is no binding budget, staffing limit, or other constraint that prevents the undertaking of all positive-NPV projects.
  • The cash flows and discount rate are stated on compatible timing and valuation-date assumptions.

If a budget is binding, projects may remain independent in their cash flows while the choices are no longer independent: funds spent on one project cannot also be spent on another. The positive-NPV test still identifies whether a project has positive value when considered on its own, but it does not by itself determine which set of projects is feasible and best under the constraint. Similarly, if projects interact, their joint cash flows must be assessed rather than treating each project’s standalone NPV as decisive.

Key result: When projects are independent, each can be undertaken, and the same valuation date and discounting model are used, accept every project with positive NPV and reject every project with negative NPV. A zero-NPV project does not change total value under the model.

Why Portfolio NPV Is the Sum of Project NPVs

Additivity explains why separate NPVs give a consistent decision for an unconstrained set of independent projects. Suppose project \(j\) has cash flow \(C_{j,t}\) at time \(t\), and let \(x_j\) be 1 if it is undertaken and 0 otherwise. The portfolio’s cash flow at time \(t\) is the sum of the cash flows of the projects selected. Discounting that sum gives the same result as summing the individual discounted cash flows.

$$ \begin{aligned} \operatorname{NPV}_{\text{portfolio}} &=\sum_{t=0}^{n}v^t\left(\sum_j x_j C_{j,t}\right)\\ &=\sum_j x_j\left(\sum_{t=0}^{n}v^t C_{j,t}\right)\\ &=\sum_j x_j\operatorname{NPV}_j. \end{aligned} $$

The second line follows by rearranging finite sums: each project’s cash flow at each date is discounted by the same factor before the project values are added. To maximize this total without a constraint, consider each term \(x_j\operatorname{NPV}_j\) separately. If \(\operatorname{NPV}_j>0\), choosing \(x_j=1\) raises the total; if \(\operatorname{NPV}_j<0\), choosing \(x_j=1\) lowers it; and if \(\operatorname{NPV}_j=0\), either choice leaves it unchanged. This gives the positive-NPV rule.

Applying the Rule to Several Projects

The initial outlay belongs in the NPV of the project that requires it. Keep each project’s timeline intact when evaluating it; do not treat an outlay for one project as if it were the outlay for another. Once each NPV has been calculated at the same valuation date, the portfolio NPV can be found by adding the NPVs of the projects selected.

Worked Example: Screening Four Projects

A firm considers four independent projects. Each requires a separate initial outlay, and there is no binding capital limit. All cash flows occur at year-end. The effective annual discount rate is 10%. The cash flows, in dollars, are:

  • Project A: \(-450\) at time 0; \(220\) at year 1, \(242\) at year 2, and \(133.10\) at year 3.
  • Project B: \(-400\) at time 0; \(220\) at year 1 and \(242\) at year 2.
  • Project C: \(-360\) at time 0; \(220\) at year 1 and \(242\) at year 2.
  • Project D: \(-500\) at time 0; \(330\) at year 1 and \(363\) at year 2.

Method. Calculate each NPV separately using \(v=1/1.10\). Keep the unrounded discounting in the calculation; these cash flows are chosen so their present values are exact dollars.

For A, the discounted receipts are \(220/1.10=200\), \(242/1.10^2=242/1.21=200\), and \(133.10/1.10^3=133.10/1.331=100\). Therefore,

$$ \operatorname{NPV}_A=-450+200+200+100=50. $$

For B, the discounted receipts are \(220/1.10=200\) and \(242/1.21=200\), so

$$ \operatorname{NPV}_B=-400+200+200=0. $$

For C, the same two discounted receipts total \(400\), giving

$$ \operatorname{NPV}_C=-360+200+200=40. $$

For D, the discounted receipts are \(330/1.10=300\) and \(363/1.21=300\), so

$$ \operatorname{NPV}_D=-500+300+300=100. $$

A and C and D have positive NPV, so undertake them. B has zero NPV, so it is financially optional under the model. The selected positive-NPV projects add \(50+40+100=190\) dollars of value at time 0. Including B would leave that total unchanged. As a check, the portfolio NPV for all four projects is \(50+0+40+100=190\).

Common mistake avoided: Do not reject B merely because it has no positive NPV, or describe it as a loss-making project. Its NPV is exactly zero; under these assumptions it neither adds nor subtracts value.

Worked Example: Undertaking Two Projects Together

Two independent projects are available, and the firm has enough funds to undertake both. Each has its own initial outlay. All cash flows occur at year-end, and the effective annual discount rate is 10%.

  • Project P: \(-650\) at time 0, followed by \(440\) at year 1 and \(363\) at year 2.
  • Project Q: \(-550\) at time 0, followed by \(330\) at year 1 and \(363\) at year 2.

Method. First evaluate the projects separately, then verify the combined result using the portfolio cash flows.

For P, the present values of the two receipts are \(440/1.10=400\) and \(363/1.21=300\). Thus,

$$ \operatorname{NPV}_P=-650+400+300=50. $$

For Q, the present values are \(330/1.10=300\) and \(363/1.21=300\). Thus,

$$ \operatorname{NPV}_Q=-550+300+300=50. $$

Both NPVs are positive, so undertake both. Their combined NPV is \(50+50=100\) dollars. The direct check combines the cash flows by date: the time-0 outlay is \(-650-550=-1{,}200\), the year-1 receipt is \(440+330=770\), and the year-2 receipt is \(363+363=726\). Therefore,

$$ \operatorname{NPV}_{P+Q} =-1{,}200+\frac{770}{1.10}+\frac{726}{1.21} =-1{,}200+700+600 =100. $$

Common mistake avoided: Do not calculate a single NPV from only one project’s initial outlay while including both projects’ receipts. Combining cash flows is valid only when the outlays and receipts of every selected project are included at their correct dates. The direct portfolio calculation agrees with the sum of the separate NPVs.

Independent Projects Are Not a Reason to Bundle Decisions

A project with negative NPV should not be accepted simply because another independent project has a positive NPV. Additivity means their values can be combined to describe a portfolio; it does not erase the separate accept-or-reject information. A positive project may make a combined portfolio positive even when the portfolio includes a value-reducing project. If the projects are truly independently selectable and there is no binding constraint, the negative-NPV project can be left out.

Worked Example: Do Not Let a Positive Project Hide a Negative One

A decision maker can undertake any combination of three independent projects. There is no binding resource constraint. All cash flows occur at year-end, and the effective annual discount rate is 10%.

  • Project M: \(-500\) at time 0, \(330\) at year 1, and \(242\) at year 2.
  • Project N: \(-250\) at time 0 and \(330\) at year 1.
  • Project O: \(-400\) at time 0, \(220\) at year 1, and \(121\) at year 2.

Method. Evaluate each project on its own before considering any combined portfolio. For M, the discounted receipts are \(330/1.10=300\) and \(242/1.21=200\), so

$$ \operatorname{NPV}_M=-500+300+200=0. $$

For N, the discounted receipt is \(330/1.10=300\), giving

$$ \operatorname{NPV}_N=-250+300=50. $$

For O, the discounted receipts are \(220/1.10=200\) and \(121/1.21=100\), giving

$$ \operatorname{NPV}_O=-400+200+100=-100. $$

Undertake N, reject O, and treat M as financially optional. If all three were bundled, their combined NPV would be \(0+50-100=-50\). That negative total does not make N unattractive: N can be undertaken without O, giving an NPV of \(50\).

Common mistake and correction. A mistaken bundled decision might calculate the value of N and O together as \(50-100=-50\), then conclude that N should also be rejected. That conclusion is wrong because N and O are independently selectable. Evaluate each project separately; accept N for its \(+50\) NPV and reject O for its \(-100\) NPV.

Common Exam Traps

  • Confusing independent with mutually exclusive. Independent projects can all be undertaken if resources allow. A mutually exclusive choice permits only one alternative and needs a different comparison.
  • Ranking instead of screening. A project does not need to have the largest NPV among a set to be worth accepting. If projects are independent and there is no binding constraint, every positive-NPV project adds value.
  • Forgetting the project’s own outlay. Each NPV includes that project’s initial cost. Omitting an outlay can turn a negative project into an apparent positive one.
  • Treating zero NPV as negative NPV. Zero means no change in value under the stated assumptions, not a loss.
  • Ignoring constraints or interactions. If the firm cannot fund all positive-NPV projects, or if one project changes another’s cash flows, the simple independent-project rule is not a complete portfolio decision method.
Key takeaway: Under a consistent discounting model, add the NPV of each selected independent project to obtain portfolio NPV. With no binding resource limit, accept positive-NPV projects, reject negative-NPV projects, and recognize that zero-NPV projects leave value unchanged.

Check Your Understanding

For each problem, assume the stated annual cash flows occur at year-end and the initial outlay occurs at time 0. Apply the positive-NPV rule using the stated effective annual discount rate.

  1. A project requires \(200\) dollars initially and returns \(110\) dollars at the end of year 1 and \(121\) dollars at the end of year 2. The effective annual discount rate is 10%. Should it be accepted, rejected, or treated as optional? (Answer: NPV \(=0\); financially optional.)
  2. An independent project requires \(250\) dollars initially and returns \(220\) dollars at the end of year 1 and \(121\) dollars at the end of year 2. The effective annual discount rate is 10%. What is its NPV and decision? (Answer: NPV \(=50\) dollars; accept.)
  3. A project requires \(500\) dollars initially and returns \(315\) dollars at the end of year 1 and \(220.50\) dollars at the end of year 2. The effective annual discount rate is 5%. Find its NPV and decision. (Answer: NPV \(=0\) dollars; financially optional.)
  4. Two independent projects can both be undertaken. At a 10% effective annual rate, X has cash flows \(-300\) at time 0, \(220\) at year 1, and \(121\) at year 2; Y has cash flows \(-350\) at time 0, \(330\) at year 1, and \(121\) at year 2. Find each NPV and the combined NPV if both are selected. (Answer: \(\operatorname{NPV}_X=0\), \(\operatorname{NPV}_Y=50\), combined NPV \(=50\) dollars; Y should be accepted and X is optional.)
  5. At a 10% effective annual rate, C requires \(600\) dollars initially and returns \(550\) at year 1 and \(242\) at year 2. D requires \(400\) dollars initially and returns \(220\) at year 1 and \(121\) at year 2. The projects are independent, and there is no binding resource limit. What is the correct decision for each, and what is the NPV if both are undertaken? (Answer: \(\operatorname{NPV}_C=100\) dollars, so accept C; \(\operatorname{NPV}_D=-100\) dollars, so reject D; undertaking both would have NPV \(0\) dollars.)

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