Why Compare Portfolios?
When a firm has more positive-NPV projects than it can fund, the decision is not simply which project looks best on its own. The firm must choose a portfolio: a combination of projects whose total initial investment fits the available budget. A project with an attractive individual ranking may use capital that could have produced more value in a different combination.
For the single-period problem in this tutorial, each project is indivisible: it is either accepted in full or rejected. Projects are independent in the relevant financial sense: accepting one does not change another project's investment or cash flows. The budget applies to initial investments at time 0, and there are no additional constraints such as project dependencies or mutually exclusive choices. Each project's NPV is measured at the same valuation date and using a consistent discount rate.
Representing and Comparing Portfolios
Let \(x_j=1\) if project \(j\) is accepted and \(x_j=0\) if it is rejected. Let \(I_j\) be its initial investment, \(N_j\) its NPV, and \(B\) the available budget. Under the assumptions above, the portfolio's total investment and NPV are:
A portfolio is feasible when its total investment is at most \(B\). The decision is to maximize total NPV subject to that budget condition and the binary choices \(x_j\in\{0,1\}\). For a small number of projects, direct enumeration is a reliable way to solve the problem: list the possible portfolios, calculate both totals, remove the infeasible choices, and compare the remaining NPVs.
With \(n\) indivisible projects, there are \(2^n\) possible accept-or-reject combinations, including the portfolio that accepts none. As \(n\) grows, listing every combination becomes less practical, but the underlying test is unchanged. The examples below show the method for small portfolios and how to check that the chosen combination is truly best.
A Systematic Enumeration Method
Confirm the budget, the projects’ initial investments and NPVs, and whether projects are indivisible and independent.
Represent each project as accepted or rejected. With a few projects, list every combination.
Add the initial investments and NPVs separately for each portfolio.
Discard any portfolio whose total initial investment exceeds the available budget.
Select the feasible portfolio with the highest total NPV, and verify that no feasible alternative has a greater value.
Worked Example: Enumerating Every Portfolio
A firm has a budget of $15 thousand. Four independent projects have the following initial investments and NPVs, also in thousands: A costs 9 and has NPV 12; B costs 7 and has NPV 10; C costs 6 and has NPV 8; D costs 4 and has NPV 5. Identify the portfolio with the highest NPV that fits the budget.
Method. Enumerate all \(2^4=16\) portfolios, including the empty portfolio. Add investment and NPV for each combination, then test investment against the $15 thousand budget.
| Portfolio | Investment ($000) | NPV ($000) | Feasible? |
|---|---|---|---|
| None | 0 | 0 | Yes |
| A | 9 | 12 | Yes |
| B | 7 | 10 | Yes |
| C | 6 | 8 | Yes |
| D | 4 | 5 | Yes |
| A + B | 16 | 22 | No |
| A + C | 15 | 20 | Yes |
| A + D | 13 | 17 | Yes |
| B + C | 13 | 18 | Yes |
| B + D | 11 | 15 | Yes |
| C + D | 10 | 13 | Yes |
| A + B + C | 22 | 30 | No |
| A + B + D | 20 | 27 | No |
| A + C + D | 19 | 25 | No |
| B + C + D | 17 | 23 | No |
| A + B + C + D | 26 | 35 | No |
Among the feasible portfolios, A + C has the highest NPV, $20 thousand, and uses the entire $15 thousand budget. The next-best feasible combination is B + C, with NPV $18 thousand. Thus A + C is optimal.
Common mistake avoided. Selecting projects in order of an individual ratio or ranking is not a proof of the best combination. For instance, ranking by NPV divided by investment puts B first: \(10/7\approx1.429\), compared with \(12/9\approx1.333\) for A and \(8/6\approx1.333\) for C. Taking B and then C gives NPV $18 thousand, but the feasible combination A + C gives $20 thousand. The correction is to compare feasible portfolios rather than assume that a project ranking settles the combination decision.
Calculating Project NPVs Before Comparing Portfolios
If the project's NPV is not supplied, calculate it from its cash flows before enumerating combinations. Use a consistent discount rate and valuation date. The following example gives cash flows in thousands and an effective annual discount rate of 10%; each project's initial investment occurs at time 0.
Worked Example: Discount Cash Flows, Then Choose a Portfolio
A firm has a $55 thousand budget and considers three independent projects. Project A costs $30 thousand and pays $11 thousand at the end of year 1 and $29.04 thousand at the end of year 2. Project B costs $25 thousand and pays $16.50 thousand and $18.15 thousand at those dates. Project C costs $18 thousand and pays $5.50 thousand and $18.15 thousand at those dates. Calculate each NPV at 10%, enumerate the portfolios, and find the maximum NPV.
Step 1: Calculate NPVs. The discount factors are \(1/1.10\) for year 1 and \(1/1.10^2=1/1.21\) for year 2.
For A, the present values of the two receipts are \(11/1.10=10\) and \(29.04/1.21=24\), so \(N_A=-30+10+24=4\) thousand.
For B, the present values are \(16.50/1.10=15\) and \(18.15/1.21=15\), so \(N_B=-25+15+15=5\) thousand.
For C, the present values are \(5.50/1.10=5\) and \(18.15/1.21=15\), so \(N_C=-18+5+15=2\) thousand.
Step 2: Enumerate the portfolios. There are \(2^3=8\) combinations.
| Portfolio | Investment ($000) | NPV ($000) | Feasible? |
|---|---|---|---|
| None | 0 | 0 | Yes |
| A | 30 | 4 | Yes |
| B | 25 | 5 | Yes |
| C | 18 | 2 | Yes |
| A + B | 55 | 9 | Yes |
| A + C | 48 | 6 | Yes |
| B + C | 43 | 7 | Yes |
| A + B + C | 73 | 11 | No |
A + B is the best feasible portfolio, with NPV $9 thousand. The portfolio containing all three projects has a larger NPV of $11 thousand, but it is not feasible because its $73 thousand investment exceeds the $55 thousand budget.
Common mistake avoided. Do not compare the NPVs of portfolios without checking their investment totals. A higher NPV cannot make an over-budget portfolio feasible.
When a Project Should Be Left Out
A project can have positive NPV and still be excluded from the optimal portfolio. Its NPV is attractive in isolation, but accepting it may use budget that could be allocated to a better combination. Conversely, when a project has negative NPV and is optional, independent, and has no other benefit or constraint attached to it, including it reduces total NPV and consumes additional budget. Removing it preserves feasibility and improves the objective.
Worked Example: Use Dominance to Simplify the Comparison
A company has a $22 thousand budget. Independent projects have these investment and NPV pairs, in thousands: A (12, 15), B (10, 13), C (7, 9), and D (5, -2). Find the optimal portfolio.
Method. First check whether any project can be ruled out. D has negative NPV and positive investment. Any portfolio containing D has lower NPV and higher investment than the otherwise identical portfolio without D. Since D is optional and independent, it cannot improve the solution. Compare the portfolios formed by A, B, and C.
| Portfolio without D | Investment ($000) | NPV ($000) | Feasible? |
|---|---|---|---|
| None | 0 | 0 | Yes |
| A | 12 | 15 | Yes |
| B | 10 | 13 | Yes |
| C | 7 | 9 | Yes |
| A + B | 22 | 28 | Yes |
| A + C | 19 | 24 | Yes |
| B + C | 17 | 22 | Yes |
| A + B + C | 29 | 37 | No |
The highest feasible NPV is $28 thousand from A + B, exactly using the budget. Adding C would raise total NPV to $37 thousand, but the $29 thousand investment would exceed the budget. Including D cannot improve A + B: it would cost \(22+5=27\) thousand and reduce NPV to \(28-2=26\) thousand.
Common mistake avoided. Do not treat every positive-NPV project as automatically acceptable under capital rationing, or every negative-NPV project as an unexamined bookkeeping detail. Test the budget and the stated decision assumptions. Here C is positive-NPV but excluded because of the constraint; D is excluded because it is optional, independent, and value-reducing.
Conditions and Common Exam Traps
Enumeration gives a defensible answer only when the quantities being added are comparable and the constraints are correctly stated. Check the following before finalizing a portfolio:
- Use one valuation basis. Project NPVs must use the same valuation date and a consistent discount-rate assumption.
- Apply the budget to investment, not NPV. Feasibility depends on initial investment; NPV is the quantity being maximized.
- Respect indivisibility. A project is either selected in full or not selected. Treating a fraction of a project as available changes the problem.
- Include every stated constraint. If acceptance of one project changes another project's cash flows, or if projects depend on one another, simple additive enumeration needs to reflect that interaction.
- Verify the maximum. The first feasible portfolio found is not necessarily optimal. Compare its NPV with every other feasible alternative, or show that excluded choices are dominated or infeasible.
The method is aligned with the SOA VEE Accounting and Finance syllabus: it combines project cash-flow valuation with the practical decision of allocating scarce capital among indivisible choices. The key discipline is to keep the two tests separate: first determine whether a portfolio fits the budget, then compare its NPV with those of the other feasible portfolios.
Check Your Understanding
For each problem, compare total investments and total NPVs, and state the best feasible choice.
- A firm has a budget of $8 thousand. Project A costs $5 thousand and has NPV $8 thousand; B costs $4 thousand and has NPV $6 thousand; C costs $3 thousand and has NPV $4 thousand. Which portfolio maximizes NPV? (Answer: A + C, investment $8 thousand and NPV $12 thousand.)
- A budget is $14 thousand. Three independent projects have investment and NPV pairs, in thousands: A (9, 12), B (7, 9), and C (5, 6). Which feasible portfolio has the greatest NPV? (Answer: A + C, investment $14 thousand and NPV $18 thousand.)
- At an effective annual rate of 10%, project X costs $20 thousand and pays $11 thousand at year 1 and $12.10 thousand at year 2. Project Y costs $15 thousand and pays $11 thousand and $12.10 thousand at those dates. With a $35 thousand budget, what is the maximum portfolio NPV? (Answer: $5 thousand; Y alone attains it, and X + Y also attains it with investment $35 thousand.)
- A firm has a $19 thousand budget. Its independent projects have investment and NPV pairs, in thousands: A (11, 16), B (8, 11), C (6, 8), and D (3, -1). What is the optimal portfolio? (Answer: A + B, investment $19 thousand and NPV $27 thousand.)
- A budget of $13 thousand is available for four independent projects, with investment and NPV pairs, in thousands: A (8, 11), B (6, 8), C (5, 7), and D (4, 5). Which feasible portfolio has the greatest NPV? (Answer: A + C, investment $13 thousand and NPV $18 thousand.)