Why a Capital Budget Changes the NPV Decision
When a company can fund every independent project with positive NPV, the NPV accept-or-reject rule gives a straightforward decision: accept each positive-NPV project. A fixed investment budget changes the problem. If the total cost of all positive-NPV proposals exceeds the available funds, the company must choose which projects to fund. The goal is to obtain the greatest total NPV from a set of projects whose initial investments fit within the budget.
The restriction is on the investment required, not on the projects’ NPVs. A project’s NPV measures its estimated contribution to value after discounting its future cash flows at the required return. Its initial investment is the amount of budget it uses. These are different quantities and should not be substituted for one another.
For this setup, define \(x_j=1\) if project \(j\) is selected and \(x_j=0\) otherwise. The selection rule can be written as:
The objective is total selected NPV; the constraint is total selected initial investment. A project with positive NPV is desirable on its own, but it is not automatically part of the best feasible set. The limited budget means accepting one project can prevent accepting another.
Why Project NPVs Add
The addition in the objective follows from the definition of NPV. Let \(CF_{jt}\) be project \(j\)’s net cash flow at time \(t\), and let \(i\) be the effective annual required return when the cash flows are annual. If the project’s initial investment occurs at time 0, then:
For independent projects, selecting several means combining their cash flows; it does not change any selected project’s cash flows or required return. Discounting the combined cash flows therefore gives the sum of the project NPVs:
The equality holds because addition and discounting are linear: discounting a sum of cash flows is the same as summing the discounted cash flows. This justification depends on the projects being independent in the relevant sense. If choosing one project changes another’s costs, revenues, or risk, the projects cannot simply be assessed as unchanged standalone proposals.
Worked Example: Three Positive-NPV Projects and a Tight Budget
A firm has a budget of $100 at time 0 and three independent proposals. Each proposal pays one cash inflow at the end of year 1. The annual effective required return is 10%. Project A costs $60 and pays $92.40; B costs $50 and pays $78.10; C costs $40 and pays $61.60. Find the selection with the highest total NPV without exceeding the budget.
Method. First calculate each NPV using the same required return, then compare the feasible sets. At 10%, the one-year discount factor is \(1/1.10\).
Project A. Its present value of inflow is \(92.40/1.10=\$84.00\). Thus \(NPV_A=84.00-60.00=\$24.00\).
Project B. Its present value of inflow is \(78.10/1.10=\$71.00\). Thus \(NPV_B=71.00-50.00=\$21.00\).
Project C. Its present value of inflow is \(61.60/1.10=\$56.00\). Thus \(NPV_C=56.00-40.00=\$16.00\).
A and B together require \(60+50=\$110\), so they are infeasible. A and C require \(60+40=\$100\) and produce total NPV \(24+16=\$40\). B and C require \(50+40=\$90\) and produce total NPV \(21+16=\$37\). The single-project choices have NPVs of $24, $21, and $16; selecting no project has NPV zero. Therefore, select A and C, for total investment of $100 and total NPV of $40. The selected costs meet the constraint exactly, and no other feasible set has a higher NPV.
Common mistake avoided: Selecting only A because it has the highest individual NPV would yield $24. A and C together are feasible and yield $40, so choosing the best single proposal is not the same as choosing the best set.
Compare Feasible Sets, Not Just Individual Projects
A reliable approach is to list the project combinations that fit the budget and compare their summed NPVs. For a small number of proposals, direct enumeration is often the clearest method. For a larger set, the same objective and budget constraint still define the problem, even if a systematic optimization method is needed to check the choices.
Calculate or verify each project’s NPV using the cash flows and required return appropriate to that project. Keep the initial investment separate as the amount charged to the budget.
For each candidate set, add the initial investments. Discard any set whose total exceeds the stated budget.
Add the NPVs of the projects in each remaining set. Select a feasible set with the largest total.
Worked Example: The Highest-Value Feasible Pair
A company can invest up to $120 at time 0 in four independent projects. Each pays one inflow at the end of year 1. The annual effective required return is 10%. Project A costs $70 and pays $112.20; B costs $60 and pays $97.90; C costs $50 and pays $82.50; D costs $40 and pays $63.80. Determine the optimal project set.
The discount factor is \(1/1.10\). The discounted inflows and NPVs are:
A: \(112.20/1.10=\$102.00\), so \(NPV_A=102-70=\$32\).
B: \(97.90/1.10=\$89.00\), so \(NPV_B=89-60=\$29\).
C: \(82.50/1.10=\$75.00\), so \(NPV_C=75-50=\$25\).
D: \(63.80/1.10=\$58.00\), so \(NPV_D=58-40=\$18\).
The A-and-B pair costs \(70+60=\$130\) and is infeasible. The other pairs are feasible: A and C cost $120 and have NPV \(32+25=\$57\); A and D cost $110 and have NPV $50; B and C cost $110 and have NPV $54; B and D cost $100 and have NPV $47; C and D cost $90 and have NPV $43. No three-project combination fits: even the least costly triple, B, C, and D, costs \(60+50+40=\$150\). Among the feasible pairs and single projects, A and C have the largest total NPV, $57. Thus the optimal choice is A and C, with investment of $120.
Common mistake avoided: A and B are the two projects with the largest individual NPVs, but their combined cost exceeds the budget. They cannot be selected together, so their NPVs do not determine the best feasible choice.
When Cash Flows Occur at Different Times
The selection method does not change when projects have multiple cash flows, but their NPVs must reflect their timing. Discount every cash flow to time 0 at the required return before comparing sets. The budget comparison, in the situation considered here, uses each project’s initial investment at time 0. A constraint on funds available at several different dates would require additional timing-specific budget constraints; a single total-budget comparison would not fully represent that problem.
Worked Example: Discounting Before Comparing Project Sets
A firm has $65,000 available at time 0 and considers four independent projects, with all amounts stated in thousands of dollars. The annual effective required return is 10%, and each project’s cash flows occur at the ends of years 1 and 2. Project A costs 40 and pays 25 at year 1 and 30 at year 2; B costs 35 and pays 20 and 25; C costs 30 and pays 18 and 20; D costs 25 and pays 18 and 15. Identify the feasible set with the greatest NPV.
Use \(NPV=-I+CF_1/1.10+CF_2/(1.10)^2\), with \((1.10)^2=1.21\). Keeping the calculation in thousands:
A: \(NPV_A=-40+25/1.10+30/1.21=-40+22.727273+24.793388=7.520661\), or 7.521 to three decimal places.
B: \(NPV_B=-35+20/1.10+25/1.21=-35+18.181818+20.661157=3.842975\), or 3.843.
C: \(NPV_C=-30+18/1.10+20/1.21=-30+16.363636+16.528926=2.892562\), or 2.893.
D: \(NPV_D=-25+18/1.10+15/1.21=-25+16.363636+12.396694=3.760331\), or 3.760.
A and B cost 75, and A and C cost 70, so both sets exceed the budget. A and D cost 65 and have total NPV \(7.520661+3.760331=11.280992\), or 11.281. B and C cost 65 and have total NPV \(3.842975+2.892562=6.735537\), or 6.736. B and D cost 60 and have total NPV 7.603306, or 7.603; C and D cost 55 and have total NPV 6.652893, or 6.653. The least costly triple, B, C, and D, costs 90, so no triple is feasible. Each single-project NPV is below 11.281. Therefore, select A and D. Their initial investment is $65,000 and their combined NPV is approximately $11,281. Comparisons use the unrounded NPVs; displayed NPVs are rounded to three decimal places in thousands.
Common mistake avoided: Adding undiscounted future receipts and comparing those totals would ignore their dates and the required return. The NPVs must be calculated first; only then should the feasible sets be compared.
Interpretation and Common Traps
This decision rule applies when the proposals are independent, the required return is suitable for each project, and the stated budget is the binding restriction on their initial investments. A different situation may need a different model. For example, if projects share facilities, one project changes another’s cash flows, or cash is limited separately in several years, the simple single-budget formulation may not capture all relevant effects.
- Confusing NPV with investment cost. NPV is a value measure; the initial investment is what uses the stated budget.
- Assuming every positive-NPV proposal can be accepted. A positive NPV supports acceptance when funding is available, but a fixed budget can make the proposals compete for funds.
- Choosing projects one at a time without checking combinations. A locally attractive first choice can rule out a better feasible set, as the worked example with a $100 budget shows.
- Ignoring cash-flow timing. Project NPVs must be computed at the required return before their values are added.
- Applying the rule to dependent projects as if they were independent. If one project changes another’s cash flows, evaluate the relevant combined cash flows rather than simply adding standalone NPVs.
The core test is therefore two-part: does the proposed set fit the investment budget, and does it have the highest total NPV among the feasible alternatives? Answering both questions is what turns the ordinary NPV accept-or-reject rule into a capital-budget selection decision.
Check Your Understanding
For each problem, compare total initial investment with the stated budget and use the given effective annual required return to calculate or compare NPVs.
- A firm has a $70 budget. At an annual effective required return of 8%, project X costs $40 and pays $54 at year 1; project Y costs $30 and pays $43.20 at year 1. Which projects should be selected, and what is their total NPV? (Answer: Select X and Y; their cost is $70 and total NPV is $20.)
- A company has a $80 budget. The annual effective required return is 10%, and each project pays at year 1. P costs $50 and pays $66; Q costs $40 and pays $52.80; R costs $30 and pays $46.20. Which feasible set maximizes total NPV? (Answer: Select P and R; their cost is $80 and total NPV is $22.)
- A firm has a $75 budget. At a 10% annual effective required return, three one-year projects have these cost and inflow pairs: A, $45 and $66; B, $40 and $61.60; C, $35 and $52.80. Find the optimal set and total NPV. (Answer: Select B and C; their cost is $75 and total NPV is $29.)
- All amounts are in thousands. A company has a budget of 70 and an annual effective required return of 10%. Project K costs 45 and pays 28 at year 1 and 30 at year 2; L costs 40 and pays 25 at each year-end for two years; M costs 30 and pays 18 at year 1 and 20 at year 2. Which set maximizes NPV without exceeding the budget? (Answer: Select L and M; their cost is 70 and total NPV is approximately 6.281 thousand.)
- A company has a $60 budget and a 10% annual effective required return. Each of three independent projects pays at year 1: A costs $40 and pays $55; B costs $35 and pays $49.50; C costs $20 and pays $19.80. Which feasible set has the highest NPV? (Answer: Select A alone or B alone; each costs less than $60 and has NPV $10, which is the maximum.)