Accepting Independent Projects by NPV

Why Evaluate Independent Projects Separately?

When a firm considers several projects, the decision is not always a contest in which one project must win and the others must be rejected. If projects are independent, accepting one does not prevent the firm from accepting another. When there is no capital constraint, the firm can therefore consider each project on its own merits: accept an investment that adds value and reject one that reduces value.

This decision rule relies on the project NPVs being estimated from relevant incremental cash flows and appropriate required returns. It also relies on the projects not changing one another’s cash flows, risks, or feasibility. A project that shares resources with another, displaces its sales, or must be undertaken as part of a bundle is not independent in this decision-making sense.

Definition: Projects are independent for an NPV decision when undertaking or rejecting one project does not change the cash flows, required return, availability, or feasibility of another. With independent projects and no binding capital constraint, each project can be accepted or rejected separately.

For project $j$, let $CF_{jt}$ be its incremental cash flow at the end of year $t$, including its initial investment at time 0, and let $k_j$ be its effective annual required return. Assuming annual periods and a rate appropriate to the project’s risk, its NPV is

\operatorname{NPV}_j=\sum_{t=0}^{T_j}\frac{CF_{jt}}{(1+k_j)^t}.

The required return and cash-flow timing must use consistent periods. The cash flows should be incremental project cash flows, not accounting earnings. A positive NPV indicates that the present value of a project’s benefits exceeds the investment and other costs represented in those cash flows; the required return is the benchmark used to discount them.

Decision rule: For an independent project that can be undertaken without displacing another project or breaching a capital limit, accept it if its NPV is positive, reject it if its NPV is negative, and treat it as value-neutral if its NPV is zero. When all positive-NPV projects are feasible, total value created is the sum of their NPVs.

How the Decision Rule Follows from NPV

The NPV rule compares a project’s value today with the resources it requires today. If the present value of its incremental benefits exceeds its present-value costs, the difference is positive and accepting the project adds that amount of value. If the difference is negative, undertaking the project uses resources worth more than the benefits received. Independence means that this comparison for one project does not alter the corresponding comparison for another.

The additivity of NPV explains how to calculate the value created by a group of accepted projects. If the projects’ values are measured using the appropriate discounting for each project, then their separate NPVs add. This remains true if different projects have different required returns: each project must first be valued at its own appropriate rate.

\begin{aligned} \text{Total NPV of selected projects} &=\sum_{j\in \mathcal{A}}\operatorname{NPV}_j\\ &=\sum_{j\in \mathcal{A}}\left(\sum_{t=0}^{T_j} \frac{CF_{jt}}{(1+k_j)^t}\right), \end{aligned}

Here, $\mathcal{A}$ is the set of projects the firm undertakes. The equality follows because adding project NPVs is ordinary addition: each project’s cash flows are discounted using its own required return, and the resulting present values are summed. One must not instead combine cash flows from projects with different required returns and discount the combined cash flows at one arbitrary rate.

Value each project separately.
Use its own incremental cash flows, timing, and risk-appropriate required return to calculate NPV.

Check feasibility and independence.
Confirm that accepting a project does not prevent another project from proceeding or change its economics, and that no capital limit binds.

Apply the sign rule.
Accept positive-NPV projects, reject negative-NPV projects, and recognize that a zero-NPV project adds no value.

Sum the accepted NPVs.
The sum is the total value created by the selected projects under the stated assumptions.

Worked Decisions for Independent Projects

Worked Example: Projects with Different Required Returns

Two feasible projects are independent, and there is no capital constraint. Project X requires $30,000 now and pays $16,000 at the end of year 1 and $18,000 at the end of year 2. Its required return is 10% effective annually. Project Y requires $24,000 now and pays $14,000 at the end of each of years 1 and 2. Its required return is 8% effective annually. Which projects should be accepted, and what is the total value created?

Method. Calculate each NPV separately using its stated annual required return. Since the projects are independent and there is no capital constraint, accept positive-NPV projects and reject negative-NPV projects.

For X, the initial outlay is the time-0 cash flow:

\begin{aligned} \operatorname{NPV}_X &=-30{,}000+\frac{16{,}000}{1.10} +\frac{18{,}000}{(1.10)^2}\\ &=-30{,}000+14{,}545.454545\ldots +14{,}876.033058\ldots\\ &=-578.512396\ldots \end{aligned}

X has a negative NPV, so reject it. For Y, discount both payments at 8%:

\begin{aligned} \operatorname{NPV}_Y &=-24{,}000+\frac{14{,}000}{1.08} +\frac{14{,}000}{(1.08)^2}\\ &=-24{,}000+12{,}962.962963\ldots +12{,}002.743484\ldots\\ &=965.706447\ldots \end{aligned}

Y has a positive NPV, so accept it. Only Y is undertaken; therefore total value created is its NPV, $965.71, rounded to the nearest cent after retaining full precision in the calculation.

Common mistake avoided: Do not round the present values separately before subtracting the initial cost. The exact calculation gives an NPV of $965.706447…; rounding the final result gives $965.71, not $965.70. Also do not use X’s 10% rate to value Y or Y’s 8% rate to value X.

Worked Example: Summing the Value of All Positive-NPV Projects

A firm can undertake either, both, or neither of two independent projects, with no capital constraint. Project Q costs $8,000 at time 0 and pays $4,800 at the end of year 1 and $4,840 at the end of year 2. Its required return is 10% effective annually. Project T costs $10,000 at time 0 and pays $6,500 at the end of each of years 1 and 2. Its required return is also 10% effective annually. Determine the optimal decisions and total value created.

Method. Find both NPVs using the common 10% annual rate, then accept each project with positive NPV. Because the projects are independent, accepting one does not affect the other’s NPV.

For Q:

\begin{aligned} \operatorname{NPV}_Q &=-8{,}000+\frac{4{,}800}{1.10} +\frac{4{,}840}{(1.10)^2}\\ &=-8{,}000+4{,}363.636364\ldots+4{,}000\\ &=363.636364\ldots \end{aligned}

Q is positive-NPV, so accept it. For T:

\begin{aligned} \operatorname{NPV}_T &=-10{,}000+\frac{6{,}500}{1.10} +\frac{6{,}500}{(1.10)^2}\\ &=-10{,}000+5{,}909.090909\ldots +5{,}371.900826\ldots\\ &=1{,}280.991736\ldots \end{aligned}

T is also positive-NPV, so accept it. Total value created is the sum of the unrounded NPVs:

363.636364\ldots+1{,}280.991736\ldots =1{,}644.628099\ldots

Thus, accept both projects and report total value created as $1,644.63, rounded to the nearest cent. Adding their separately rounded NPVs would happen to produce the same cent result here, but the reliable procedure is to retain full precision until the total is calculated.

Common mistake avoided: Do not choose only the project with the larger NPV. There is no requirement to choose between Q and T: both can be undertaken, and each adds value.

Checking Additivity with Combined Cash Flows

When independent projects use the same required return and have cash flows at the same dates, a useful check is to add their cash flows at each date and calculate the combined NPV. The result should equal the sum of the individual NPVs. This works because discounting at a common rate is linear: discounting a sum of cash flows gives the sum of their discounted values. The calculation is a check on additivity, not a reason to combine projects that interact.

Worked Example: Verifying the Sum by Combining Cash Flows

Projects A and B are independent and have an 8% effective annual required return. A costs $15,000 now and pays $9,000 at the end of each of years 1 and 2. B costs $8,000 now, pays $4,500 at the end of year 1, and $5,000 at the end of year 2. There is no capital constraint. Verify the total value created both by valuing projects separately and by combining their cash flows.

Method. First calculate each project’s NPV at 8%. Then sum cash flows by date and recalculate the NPV at the same rate.

For A:

\begin{aligned} \operatorname{NPV}_A &=-15{,}000+\frac{9{,}000}{1.08} +\frac{9{,}000}{(1.08)^2}\\ &=-15{,}000+8{,}333.333333\ldots +7{,}716.049383\ldots\\ &=1{,}049.382716\ldots \end{aligned}

For B:

\begin{aligned} \operatorname{NPV}_B &=-8{,}000+\frac{4{,}500}{1.08} +\frac{5{,}000}{(1.08)^2}\\ &=-8{,}000+4{,}166.666667\ldots +4{,}286.694101\ldots\\ &=453.360768\ldots \end{aligned}

Both NPVs are positive, so accept both. Their sum is

1{,}049.382716\ldots+453.360768\ldots =1{,}502.743484\ldots

As a check, the combined cash flows are $-$23,000$ at time 0, $13,500 at year 1, and $14,000 at year 2. Therefore:

\begin{aligned} \operatorname{NPV}_{A+B} &=-23{,}000+\frac{13{,}500}{1.08} +\frac{14{,}000}{(1.08)^2}\\ &=-23{,}000+12{,}500+12{,}002.743484\ldots\\ &=1{,}502.743484\ldots \end{aligned}

The combined-cash-flow result agrees with the sum of the individual NPVs. Total value created is $1,502.74, rounded to the nearest cent.

Common mistake avoided: The combined-cash-flow check uses one discount rate because both projects have the same required return. If projects have different required returns, calculate their NPVs separately and add those NPVs; do not discount combined cash flows at a single rate chosen without justification.

Conditions, Zero NPV, and Common Traps

The separate accept-or-reject rule is appropriate only when the assumptions behind it hold. If a capital limit prevents undertaking every positive-NPV project, the firm faces a selection problem rather than a set of independent accept-or-reject decisions. If accepting one project changes another’s cash flows—for example, because the projects share customers or equipment—the projects’ effects must be evaluated together. Those cases do not satisfy the setup used here.

A zero NPV is a boundary case. Under the stated assumptions, accepting a zero-NPV project neither adds nor destroys value relative to rejecting it. The firm is indifferent on the basis of NPV alone; other considerations may determine the choice. A negative-NPV project should not be accepted merely because another project has a larger positive NPV. With no interaction or constraint, the negative project remains value-reducing.

Rejecting a positive project because another has a higher NPV: When both can be undertaken, accept both; the decision is not a contest.
Accepting a negative project to “balance” a positive one: Under independence, the NPVs do not offset as a reason to undertake the negative project. Accept the positive project and reject the negative one.
Adding all candidate NPVs to report value created: If negative-NPV projects are rejected, total value created is the sum of accepted NPVs, not the sum of every candidate’s NPV.
Using one discount rate for every project: Each project should be valued using its own appropriate required return. A common rate is suitable for the combined-cash-flow check only when it is appropriate for all included cash flows.

Key takeaway: With independent, feasible projects and no binding capital constraint, evaluate each project using its own NPV, accept every positive-NPV project, reject every negative-NPV project, and sum the accepted NPVs to measure total value created. A zero-NPV project is value-neutral.

Check Your Understanding

For each problem, use the project-specific required returns given. Round dollar answers to the nearest cent only after completing the calculations.

A standalone project costs $18,000 now and pays $20,000 at the end of year 1. Its required return is 10% effective annually. Should it be accepted, and what value does it add? (Answer: Accept; NPV = $181.82.)
An independent project costs $12,000 now and pays $6,800 at the end of each of years 1 and 2. Its required return is 10% effective annually. Should it be accepted? (Answer: Reject; NPV = -$198.35.)
Two independent projects have no capital constraint. X costs $9,000 and pays $10,000 at the end of year 1; its required return is 10%. Y costs $7,000 and pays $7,500 at the end of year 1; its required return is 8%. Which projects should be accepted, and what is total value created? (Answer: Accept X and reject Y; total value created = $90.91.)
Two independent projects each have a 12% effective annual required return. A costs $20,000 and pays $12,000 at the end of each of years 1 and 2. B costs $15,000 and pays $8,000 at the end of each of years 1 and 2. With no capital constraint, find the optimal decisions and total value created. (Answer: Accept A and reject B; total value created = $280.61.)
Three independent feasible projects have NPVs of $2,400, -$900, and $0, respectively. There is no capital constraint or interaction. State the NPV-based decision for each and the total value created. (Answer: Accept the $2,400 project, reject the -$900 project, and treat the zero-NPV project as value-neutral; total value created = $2,400.)

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