From Project NPVs to a Budgeted Portfolio
When a firm can fund every positive-NPV opportunity, accepting each independent project is straightforward. A fixed capital budget changes the decision: projects may compete for scarce funds, and the best portfolio need not contain the project with the largest individual NPV. The task is to compare feasible combinations, not just individual proposals.
This workshop treats projects as indivisible: a project is either undertaken in full or not undertaken. Each project requires a stated time-zero investment, and its NPV is evaluated using the stated required return. The examples use hypothetical proposals; monetary amounts are in thousands of dollars when stated.
A Selection Model and Why NPVs Add
Suppose there are \(n\) projects. For project \(j\), let \(C_j\) be its time-zero investment and \(N_j\) its NPV at the required return. Let \(x_j=1\) if project \(j\) is selected and \(x_j=0\) otherwise. With a budget \(B\), the decision can be written as:
The binary condition matters: it rules out funding a fraction of a project. The constraint allows some budget to remain unused. If a project has a negative NPV, selecting it reduces the objective, so it will not be chosen in this model unless some additional requirement or interaction changes the decision.
Before comparing combinations, calculate each project NPV consistently. With a required effective annual return \(i\), time-zero investment \(C_j\), and cash flow \(CF_{jt}\) at the end of year \(t\), the standard discrete-compounding model is:
This formula assumes the investment is paid at time zero, the cash flows occur at the stated year-ends, and all amounts are measured on a consistent nominal or real basis. It does not describe a project with cash flows at other dates unless those dates are reflected in the discount exponents.
The additivity of portfolio NPV follows directly from the definition. If each selected project has its own cash flows and taking one project does not change another's cash flows, the portfolio cash flow at each date is the sum of the selected projects' cash flows. Discounting that sum gives:
Thus, adding project NPVs is valid under the stated independence assumption. If projects share costs, compete for the same operating capacity, or change each other's cash flows, the simple sum can fail; the interaction must then be included in the portfolio cash flows.
Worked Example: Selecting Among Four Independent Proposals
A hypothetical firm has a budget of $100,000 and four indivisible, independent projects. Each requires an investment at time zero and pays one cash flow at the end of year 1. All amounts below are in thousands of dollars. The required effective annual return is 10%. Find the combination with the greatest total NPV.
Method. First calculate each NPV from its time-zero cost and year-one cash flow. Then list feasible combinations and compare their summed NPVs.
| Project | Time-zero cost | Year-one cash flow | Present value of cash flow | NPV |
|---|---|---|---|---|
| A | 40 | 63.8 | 63.8 / 1.10 = 58 | 58 − 40 = 18 |
| B | 35 | 56.1 | 56.1 / 1.10 = 51 | 51 − 35 = 16 |
| C | 30 | 47.3 | 47.3 / 1.10 = 43 | 43 − 30 = 13 |
| D | 25 | 38.5 | 38.5 / 1.10 = 35 | 35 − 25 = 10 |
For example, project A has \(NPV_A=-40+63.8/1.10=-40+58=18\). Each entry follows the same calculation. The costs and NPVs are in thousands, so an NPV of 18 means $18,000.
The total cost of all four projects is \(40+35+30+25=130\), so that combination is infeasible. Among the three-project combinations, A+B+C costs \(40+35+30=105\), A+B+D costs 100, A+C+D costs 95, and B+C+D costs 90. The feasible combinations of two or fewer projects and the relevant three-project combinations are:
| Combination | Cost | Total NPV | Feasible? |
|---|---|---|---|
| A+B | 75 | 34 | Yes |
| A+C | 70 | 31 | Yes |
| A+D | 65 | 28 | Yes |
| B+C | 65 | 29 | Yes |
| B+D | 60 | 26 | Yes |
| C+D | 55 | 23 | Yes |
| A+B+D | 100 | 44 | Yes |
| A+C+D | 95 | 41 | Yes |
| B+C+D | 90 | 39 | Yes |
The best single project has NPV 18, while the best feasible pair has NPV 34. Of the feasible three-project combinations, A+B+D has the greatest NPV: \(18+16+10=44\). It is therefore the best overall combination. Its cost is exactly \(40+35+25=100\), and its total NPV is \(44\) thousand dollars.
Common mistake and correction. A time-zero cost is already expressed at the valuation date. Incorrectly discounting project A's cost again gives \(-40/1.10+63.8/1.10=21.64\), which overstates its NPV. The correct calculation is \(-40+63.8/1.10=18\). The selected combination is checked against the budget and the summed NPVs above.
Systematic Combination Testing
For a small number of projects, a table is a reliable way to avoid overlooking a feasible portfolio. Start with the empty combination if useful, then test single projects, pairs, and larger subsets. For each subset, add the costs and NPVs separately. Reject a subset as soon as its total cost exceeds the budget; do not compare its NPV as if it were attainable.
Use the common required return and the dates of the project cash flows. Record each project's cost separately from its NPV.
For every subset, add the selected project costs. Because the projects are indivisible, do not scale a cost or NPV by a fraction.
Keep a subset only if its total cost is at most the budget. A combination costing more is infeasible even if its NPV is high.
Add the NPVs within each feasible subset and select the largest total, including the possibility that not all budget is used.
Confirm the chosen cost satisfies the constraint, recompute the selected NPVs' sum, and check that no feasible alternative has a greater total.
When the number of proposals grows, exhaustive listing becomes tedious, but the decision principle does not change. At this level, the important discipline is to distinguish the feasible set from the objective: the budget determines which combinations can be chosen, and total NPV determines which of those is best.
Worked Example: A Budget That Cannot Fund Every Positive-NPV Project
A hypothetical firm has a budget of $80,000. Four independent, indivisible projects each require a time-zero investment and pay one cash flow at the end of year 1. The effective annual required return is 10%. Amounts are in thousands of dollars.
| Project | Cost | Year-one cash flow | Present value | NPV |
|---|---|---|---|---|
| P | 50 | 81.4 | 81.4 / 1.10 = 74 | 74 − 50 = 24 |
| Q | 40 | 64.9 | 64.9 / 1.10 = 59 | 59 − 40 = 19 |
| R | 30 | 49.5 | 49.5 / 1.10 = 45 | 45 − 30 = 15 |
| S | 25 | 39.6 | 39.6 / 1.10 = 36 | 36 − 25 = 11 |
All individual NPVs are positive, but taking all four would cost \(50+40+30+25=145\), which is greater than the budget of 80. No three-project combination is feasible: the cheapest possible triple, Q+R+S, costs \(40+30+25=95\). It remains to compare feasible single projects and pairs.
| Feasible pair | Cost | Total NPV |
|---|---|---|
| P+R | 50+30 = 80 | 24+15 = 39 |
| P+S | 50+25 = 75 | 24+11 = 35 |
| Q+R | 40+30 = 70 | 19+15 = 34 |
| Q+S | 40+25 = 65 | 19+11 = 30 |
| R+S | 30+25 = 55 | 15+11 = 26 |
P+Q costs 90 and is infeasible. The best single project has NPV 24, less than the best feasible pair's NPV of 39. Therefore, select P+R: its cost is exactly \(50+30=80\), and its total NPV is \(24+15=39\), or $39,000.
Common mistake and correction. Accepting every positive-NPV project without checking the capital constraint would call for spending 145 against a budget of 80. Positive NPV indicates value creation in isolation; it does not make an over-budget combination feasible. The correction is to compare only combinations within the budget.
Several Cash-Flow Dates, One Portfolio Decision
A project can have multiple future cash flows without changing the selection method. First discount each dated amount to time zero, subtract the time-zero cost, and record the resulting NPV. Then use the cost and NPV columns to test combinations. The following example illustrates why the cash-flow calculation and the portfolio calculation are separate steps.
Worked Example: Combining Projects With Two-Year Cash Flows
A hypothetical firm has a $100,000 budget and four independent, indivisible projects. Each project has a time-zero cost and cash flows at the ends of years 1 and 2. The effective annual required return is 10%. Amounts are in thousands of dollars. Find the best feasible combination.
The discount factors are \(1/1.10=0.909091\) for year 1 and \(1/1.10^2=1/1.21=0.826446\) for year 2. Calculate each NPV:
- A: \(PV=55/1.10+60.5/1.21=50+50=100\). Thus \(NPV_A=100-50=50\).
- B: \(PV=44/1.10+36.3/1.21=40+30=70\). Thus \(NPV_B=70-40=30\).
- C: \(PV=0/1.10+84.7/1.21=0+70=70\). Thus \(NPV_C=70-35=35\).
- D: \(PV=33/1.10+24.2/1.21=30+20=50\). Thus \(NPV_D=50-30=20\).
The total cost of all four is \(50+40+35+30=155\), so it is infeasible. Every three-project combination also exceeds the budget: A+B+C costs 125, A+B+D costs 120, A+C+D costs 115, and B+C+D costs 105. The pairs are:
| Pair | Cost | Total NPV | Feasible? |
|---|---|---|---|
| A+B | 90 | 80 | Yes |
| A+C | 85 | 85 | Yes |
| A+D | 80 | 70 | Yes |
| B+C | 75 | 65 | Yes |
| B+D | 70 | 50 | Yes |
| C+D | 65 | 55 | Yes |
The largest feasible pair NPV is \(50+35=85\) for A+C. The best single project has NPV 50, so no single project beats that pair. A+C costs \(50+35=85\), within the budget, and produces total NPV 85, or $85,000. Since every triple and the four-project portfolio exceed the budget, this completes the comparison.
Common mistake and correction. Cash flows received in year 2 must be discounted for two years, not one. For project A, using \(60.5/1.10=55\) would incorrectly produce a present value of \(50+55=105\) and an NPV of 55. The correct year-2 present value is \(60.5/1.21=50\), giving NPV 50. After that correction, the portfolio comparison uses the corrected project NPVs.
Decision Checks and Common Traps
A good selection is supported by both a feasibility check and an optimality check. The feasibility check confirms the chosen costs sum to no more than the budget. The optimality check confirms that no other feasible combination has a higher sum of NPVs. For a small case, a table makes both checks visible.
- Do not confuse cost with NPV. The cost is used in the budget constraint; NPV is the value measure being maximized.
- Do not treat indivisible projects as divisible. If a project is specified as all-or-nothing, a fractional investment is not a permitted candidate.
- Do not assume the budget must be fully spent. A feasible project may not fit in the remaining funds, and spending more does not automatically add value.
- Do not add NPVs when projects interact. Shared costs or changed cash flows require an explicit portfolio-level calculation.
- Keep the discounting date consistent. A time-zero investment is not discounted, while a future cash flow is discounted according to its stated date.
The selection framework is aligned with the SOA VEE Accounting and Finance syllabus. Its central result is concise: under the stated assumptions, choose the feasible subset with the greatest sum of project NPVs. The work lies in accurately calculating each NPV, respecting the budget, and checking the competing combinations.
Check Your Understanding
For each original problem, identify the feasible combination with the greatest total NPV. Unless a question states otherwise, projects are independent and indivisible, and amounts are in thousands of dollars.
- A firm has a budget of 60. Projects A, B, and C cost 35, 30, and 25, with NPVs 20, 17, and 13, respectively. Which combination is best, and what is its cost and total NPV? (Answer: A+C; cost 60; total NPV 33.)
- A budget is 90. Projects A, B, C, and D have respective costs 55, 45, 35, and 30, and NPVs 32, 27, 20, and 16. Find the best feasible combination and verify that no triple can be funded. (Answer: A+C; cost 90; total NPV 52; the cheapest triple costs 30+35+45=110.)
- A firm has a budget of 75 and a 10% effective annual required return. Project A costs 45 and pays 33 at year 1 and 60.5 at year 2. Project B costs 40 and pays 44 at year 1 and 24.2 at year 2. Project C costs 30 and pays 60.5 at year 2 only. Project D costs 25 and pays 33 at year 1 only. Calculate the project NPVs and choose the best feasible combination. (Answer: \(NPV_A=35\), \(NPV_B=20\), \(NPV_C=20\), \(NPV_D=5\); choose A+C, cost 75 and total NPV 55.)
- A budget is 100. Five independent projects have costs and NPVs as follows: A (60, 40), B (45, 30), C (35, 22), D (25, 14), and E (20, 10). Find every combination that attains the greatest total NPV and give the cost and NPV of each. (Answer: A+C costs 95 and has NPV 62; B+C+E costs 100 and also has NPV 62.)