Why Portfolio NPV Matters
When a firm has more positive-NPV projects than its capital budget can support, it cannot simply accept every attractive project. It must choose a portfolio: a set of projects to undertake together. If projects are indivisible, each must be accepted in full or rejected; if they are independent, choosing one does not change the cash flows or value of another. Under those conditions, the decision can be made by comparing the total NPV of every feasible combination.
The required return and project cash flows determine each project's NPV. Here, project NPVs are treated as already calculated at the stated common required return. The task is to combine those project NPVs without violating the spending limit. A portfolio with the highest individual project NPV is not necessarily the best portfolio: the choice depends on the full set of feasible combinations.
Set Up the Choice Before Comparing Portfolios
The word indivisible is the key to the decision structure. For every project, \(x_j\) can only be 0 or 1. A project cannot be chosen at, for example, 60% of its stated investment to use the remaining budget. The budget constraint applies to the sum of the initial investments of the projects selected, not to the sum of their NPVs.
The total-NPV formula also depends on a substantive assumption: the projects' cash flows do not interact. If accepting two projects together changes their costs, revenues, timing, or risks, their combined NPV cannot automatically be found by adding their separate NPVs. In that case, the cash flows for the combined choice must be assessed appropriately. For independent projects, however, adding NPVs follows directly from the definition of NPV.
A reliable manual method is to list each possible subset of projects, calculate its total investment, mark whether that investment is within budget, and then compare the total NPVs of the feasible subsets. With \(n\) projects there are \(2^n\) subsets, including the choice to accept none. For a small number of projects, a table makes it easier to check that no combination has been overlooked.
List each project's initial investment and NPV, all measured on a consistent basis.
Include the empty portfolio and each possible combination; do not treat a project as divisible.
Add the selected projects' initial investments. Exclude any portfolio whose total exceeds the budget.
Add the selected projects' NPVs and identify every feasible portfolio attaining the maximum.
Why the NPVs Add
For a project with dated cash flows \(CF_{j,t}\), valued at a common required return \(r\), its NPV at time 0 is the present value of its future cash flows minus its initial investment. This expression uses the same valuation date and discounting basis for every project in the portfolio.
If projects are independent, the portfolio's time-\(t\) cash flow is the sum of the cash flows for its selected projects. Substituting that sum into the NPV calculation and distributing the sum across the terms gives the sum of the separate project NPVs. Thus the additivity is a consequence of linearity in the NPV calculation, not a separate assumption that every project's value is identical.
This derivation requires that each project's cash flows remain the same whether or not other projects are accepted. It also requires NPVs measured at the same valuation date and on a consistent required-return basis. A cost saving available only if two projects are undertaken together, for example, is an interaction; it would have to be included in the combined cash flows rather than ignored.
Worked Example: Choosing Between Three Independent Projects
A hypothetical firm has a capital budget of $90,000. It can choose among three indivisible, independent projects. Their initial investments and NPVs, in thousands of dollars, are A: investment 50, NPV 24; B: investment 40, NPV 20; and C: investment 30, NPV 17. Each project's NPV is given at a common 10% effective annual required return. Enumerate the combinations and find the maximizing portfolio.
Method. Enumerate all \(2^3=8\) portfolios. For each, add investment amounts to test the $90,000 budget, then add the project NPVs. Because amounts are stated in thousands, the budget is 90 in the table.
| Portfolio | Investment ($000) | Total NPV ($000) | Feasible? |
|---|---|---|---|
| None | 0 | 0 | Yes |
| A | 50 | 24 | Yes |
| B | 40 | 20 | Yes |
| C | 30 | 17 | Yes |
| A and B | 50 + 40 = 90 | 24 + 20 = 44 | Yes |
| A and C | 50 + 30 = 80 | 24 + 17 = 41 | Yes |
| B and C | 40 + 30 = 70 | 20 + 17 = 37 | Yes |
| A, B, and C | 50 + 40 + 30 = 120 | 24 + 20 + 17 = 61 | No |
Among feasible portfolios, A and B have the greatest total NPV: \(24+20=44\) thousand dollars, or $44,000. Their total investment is exactly the $90,000 budget. The three-project combination has a higher arithmetic total NPV of $61,000, but it is infeasible because its investment of $120,000 exceeds the budget.
Common mistake avoided: Selecting all three projects because their total NPV is largest ignores the spending constraint. The decision is to maximize NPV among feasible portfolios, so the infeasible combination cannot be selected.
Ties Are Valid Optimal Results
An optimization problem can have more than one maximizing portfolio. If two feasible combinations have the same largest total NPV, both are optimal under the stated objective and constraints. A conclusion that reports only one of them is incomplete unless additional information supplies a tie-breaker. The example below illustrates why each combination should be evaluated directly, even when two projects share an NPV.
Worked Example: Identifying Two Equally Valuable Portfolios
A hypothetical company has $11,000 to invest in three independent, indivisible projects. In thousands of dollars, A requires 6 and has NPV 4; B requires 5 and has NPV 3; C requires 4 and has NPV 3. The NPVs are stipulated at a common 8% effective annual required return. Find every portfolio with the maximum feasible NPV.
Step 1: Enumerate the eight portfolios. The empty portfolio and the three single-project portfolios are feasible because each costs no more than 11. The pair investments are A+B: \(6+5=11\), A+C: \(6+4=10\), and B+C: \(5+4=9\). Each pair is feasible. The three-project investment is \(6+5+4=15\), so it is infeasible.
Step 2: Add NPVs for each feasible portfolio. None has NPV 0; A has 4; B has 3; C has 3; A+B has \(4+3=7\); A+C has \(4+3=7\); and B+C has \(3+3=6\), all in thousands of dollars. The infeasible combination A+B+C has a mathematical total of \(4+3+3=10\), but it cannot be chosen under the budget.
The highest feasible total NPV is 7 thousand dollars. Both A+B and A+C are maximizing portfolios, each with total NPV $7,000. Neither the shared maximum nor the fact that A+C leaves $1,000 of the budget unspent makes it inferior under the stated objective: its total NPV ties A+B.
Method and common mistake avoided. The method is exhaustive subset comparison. A common error is to assume that the portfolio using the full budget must be uniquely best. Here A+B uses the full budget, but A+C is also feasible and has the same total NPV. Report both.
Exhaustive Enumeration with Four Projects
As the number of projects grows, it becomes more important to organize the enumeration. A table can show all \(2^n\) possible portfolios, including infeasible ones, so that the budget test is explicit. The following example has four projects. The table is longer than the one for three projects, but each row follows the same two calculations: total investment and total NPV.
Worked Example: Comparing All Portfolios of Four Projects
A hypothetical insurer is considering four independent, indivisible technology projects under a $13,000 capital budget. Their initial investments and NPVs, in thousands, are A: 8 and 6; B: 7 and 5; C: 6 and 4; and D: 5 and 2. The supplied NPVs use a common 12% effective annual required return. Determine the maximizing portfolio.
Step 1: List all 16 subsets. For example, portfolio A+D costs \(8+5=13\) and has NPV \(6+2=8\). Portfolio B+C costs \(7+6=13\) and has NPV \(5+4=9\). Any triple costs more than 13: the least expensive triple is B+C+D, costing \(7+6+5=18\). The table verifies every subset.
| Portfolio | Investment ($000) | Total NPV ($000) | Feasible? |
|---|---|---|---|
| None | 0 | 0 | Yes |
| A | 8 | 6 | Yes |
| B | 7 | 5 | Yes |
| C | 6 | 4 | Yes |
| D | 5 | 2 | Yes |
| A+B | 8 + 7 = 15 | 6 + 5 = 11 | No |
| A+C | 8 + 6 = 14 | 6 + 4 = 10 | No |
| A+D | 8 + 5 = 13 | 6 + 2 = 8 | Yes |
| B+C | 7 + 6 = 13 | 5 + 4 = 9 | Yes |
| B+D | 7 + 5 = 12 | 5 + 2 = 7 | Yes |
| C+D | 6 + 5 = 11 | 4 + 2 = 6 | Yes |
| A+B+C | 8 + 7 + 6 = 21 | 6 + 5 + 4 = 15 | No |
| A+B+D | 8 + 7 + 5 = 20 | 6 + 5 + 2 = 13 | No |
| A+C+D | 8 + 6 + 5 = 19 | 6 + 4 + 2 = 12 | No |
| B+C+D | 7 + 6 + 5 = 18 | 5 + 4 + 2 = 11 | No |
| A+B+C+D | 8 + 7 + 6 + 5 = 26 | 6 + 5 + 4 + 2 = 17 | No |
Step 2: Find the largest feasible total NPV. The feasible pair NPVs are 8, 9, 7, and 6 thousand dollars. Each feasible single-project NPV is at most 6, and the empty portfolio has NPV 0. Therefore B+C is the unique maximizing portfolio, with total NPV \(5+4=9\) thousand dollars, or $9,000, at a total investment of \(7+6=13\) thousand dollars.
Check. B+C meets the budget exactly. The largest feasible alternative, A+D, has NPV 8 thousand dollars, which is less than 9. The higher totals in the table belong only to portfolios that exceed the budget.
Common Errors and Practical Checks
The arithmetic is straightforward, but a correct answer depends on applying the model and constraints consistently. These checks help distinguish a valid portfolio decision from a calculation that looks plausible but answers a different question.
- Do not optimize NPV without first checking feasibility. A portfolio whose investment exceeds \(B\) is excluded even if its total NPV is higher than every feasible alternative.
- Do not add investments to NPVs. Investment determines whether the budget constraint is met; the project's NPV is the value used in the objective. If NPVs are supplied, they already reflect the initial investment within their calculation.
- Do not choose a fraction of an indivisible project. A choice such as \(x_j=0.5\) is not allowed. The permitted values are \(x_j=0\) and \(x_j=1\).
- Do not assume a unique winner. Compare all feasible totals and report every portfolio tied at the maximum.
- Check the independence assumption. If joint acceptance changes project cash flows, separately calculated NPVs may not add to the correct portfolio NPV.
- Use consistent units and valuation assumptions. Costs and the budget must be in the same units; project NPVs must use the same valuation date and a consistent required-return basis.
Check Your Understanding
For each problem, treat the projects as independent and indivisible. The stated NPVs have already been calculated at the common required return given for that problem. Compare feasible combinations; amounts are in thousands unless otherwise stated.
- A hypothetical firm has a budget of $70,000. Project A costs $45,000 and has NPV $18,000; project B costs $30,000 and has NPV $14,000. Their NPVs use a 9% effective annual required return. Which portfolio maximizes NPV? (Answer: A alone, with NPV $18,000.)
- A hypothetical utility has a budget of $60,000. Projects X, Y, and Z cost 40, 35, and 25, and have NPVs 22, 18, and 12, respectively. Their NPVs use a common 10% effective annual required return. Find the maximizing portfolio and its NPV. (Answer: Y and Z, costing $60,000 and producing NPV $30,000.)
- A hypothetical company has a budget of $6,000. A costs 4 and has NPV 5; B costs 3 and has NPV 3; C costs 2 and has NPV 2. The supplied NPVs use an 8% effective annual required return. Identify the maximizing portfolio and check whether it is tied. (Answer: A and C, with NPV $7,000; this is the unique maximum.)
- A hypothetical firm has a $9,000 budget. Four projects have these investment and NPV pairs: A (6, 8), B (5, 7), C (4, 5), and D (3, 3). Their NPVs use a common 11% effective annual required return. Which feasible portfolio has the highest NPV? (Answer: B and C, costing $9,000 and producing NPV $12,000.)
- A hypothetical organization has an $8,000 budget. Independent projects A, B, and C have costs 5, 3, and 2, and NPVs 7, 2, and \(-1\), respectively. Their NPVs use a common 10% effective annual required return. Compare feasible portfolios and state the maximizing choice. (Answer: A and B, costing $8,000 and producing NPV $9,000.)