Why Capital Rationing Must Be Checked Period by Period
A project can be attractive on an NPV basis and still be impossible to undertake when the firm has a limited investment budget. The difficulty becomes more pronounced when the limits apply in several periods: a portfolio may fit the total amount available over the planning horizon but exceed the budget in one particular period. The decision therefore requires both a value calculation and a period-by-period feasibility check.
This tutorial considers projects that are indivisible: each is either accepted in full or rejected. Their investment commitments can occur at different dates, while their operating cash flows may occur later. All project NPVs must be measured at the same valuation date and using the applicable discount rate. The objective is to find the feasible portfolio with the greatest total NPV.
Formulating the Selection Problem
Let \(x_j\) indicate whether project \(j\) is selected: \(x_j=1\) if the project is accepted and \(x_j=0\) otherwise. With optional, independent projects, the selection problem is
There is a separate budget constraint for each period. A portfolio that satisfies the period-0 constraint but fails the period-1 constraint is infeasible, even if it leaves substantial funds unused in period 2. Unless the problem explicitly allows borrowing across periods or carrying unused funds forward, a budget in one period cannot be moved to another.
The commitments \(c_{jt}\) are the amounts counted against the stated capital budgets. NPV, by contrast, is calculated from the project’s full relevant cash-flow stream, including its investment outlays and later operating cash flows. Follow the problem’s definition of the budget: if it constrains investment commitments, do not insert operating receipts into the commitment constraint. If the problem explicitly defines a different funding constraint, model that constraint as stated.
Why Portfolio NPV Is the Sum of Project NPVs
When projects are independent, selecting a portfolio adds their cash flows date by date. Present value is a linear operation: the present value of a sum is the sum of the present values. Subtracting each project’s investment outlays from its discounted cash inflows therefore gives the same total whether NPV is calculated project by project and then added, or from the combined portfolio cash flow.
Here \(CF_{jt}\) is project \(j\)’s net cash flow at time \(t\), and \(i\) is the effective annual discount rate when periods are annual. This additivity justifies the objective in the selection model. It does not eliminate the period-specific constraints: the commitments still have to be checked separately at every budget date.
Make a row for each project and a column for each budget period. Enter the amount charged to the budget in that period.
Use the same valuation date and the stated discount rate or rates to calculate each project’s NPV from its full cash-flow stream.
For each candidate portfolio, sum selected commitments separately in each period and compare each sum with that period’s budget.
Add the NPVs of the selected projects for feasible portfolios and choose the greatest total.
Worked Examples
Worked Example: Comparing Feasible Portfolios Over Three Periods
A firm is considering four independent, indivisible projects. Amounts are in thousands of dollars. The effective annual discount rate is \(10\%\); all cash flows occur at year-end. Budgets for periods 0, 1, and 2 are respectively \(100\), \(50\), and \(40\). The commitment schedule and project cash flows are:
| Project | Commitments in periods 0, 1, 2 | Net cash flows at times 0, 1, 2, 3 |
|---|---|---|
| A | (60, 20, 0) | (−60, −20, 105, 0) |
| B | (40, 0, 30) | (−40, 0, −30, 100) |
| C | (30, 30, 0) | (−30, −30, 75, 0) |
| D | (50, 0, 10) | (−50, 0, −10, 85) |
Method. First discount each project’s entire cash-flow stream to time 0. Then compare portfolio commitments with each period’s budget. Avoid the mistake of testing only the sum of all three budgets: the amount available in one period cannot be assumed available in another.
The discount factors are \(v=1/1.10\), \(v^2=1/1.21\), and \(v^3=1/1.331\). The NPVs, calculated from unrounded values and rounded only at the end to four decimal places in thousands, are:
For A, \(NPV_A=-60-20/1.10+105/1.21=-60-18.181818\ldots+86.776859\ldots=8.5950\).
For B, \(NPV_B=-40-30/1.21+100/1.331=-40-24.793388\ldots+75.131481\ldots=10.3381\).
For C, \(NPV_C=-30-30/1.10+75/1.21=-30-27.272727\ldots+61.983471\ldots=4.7107\).
For D, \(NPV_D=-50-10/1.21+85/1.331=-50-8.264463\ldots+63.861758\ldots=5.5973\).
Now compare the feasible pairs. A and B require \((60+40,\ 20+0,\ 0+30)=(100,20,30)\), so they meet all three budgets; their combined NPV is \(8.595041\ldots+10.338092\ldots=18.9331\). A and C require \((90,50,0)\), so they are feasible, with combined NPV \(13.3058\). B and C require \((70,30,30)\), so they are feasible, with combined NPV \(15.0488\). B and D require \((90,0,40)\), so they are feasible, with combined NPV \(15.9354\). C and D require \((80,30,10)\), so they are feasible, with combined NPV \(10.3080\).
A and D are infeasible because their period-0 commitments total \(60+50=110\), above the period-0 budget of \(100\). Every three-project portfolio exceeds the period-0 budget: even the least expensive triple, B+C+D, requires \(40+30+50=120\). Thus A+B has the greatest NPV among feasible portfolios. Check: its commitments are exactly \(100\) in period 0, \(20\le50\) in period 1, and \(30\le40\) in period 2.
Worked Example: Rounding NPV Before Comparing Portfolios
A company has budgets of \(50\) thousand dollars in period 0 and \(25\) thousand dollars in period 1. It considers three independent projects. The effective annual discount rate is \(10\%\), cash flows occur at year-end, and amounts are in thousands. Project A has commitments \((25,0)\) and cash flows \((-25,36.3)\); project B has commitments \((20,10)\) and cash flows \((-20,-10,44)\); project C has commitments \((25,15)\) and cash flows \((-25,-15,50)\), with the final cash flow at time 2. Which feasible portfolio has the greatest NPV?
Method. Calculate each NPV without rounding intermediate terms, then round the final amounts to four decimal places in thousands. The key trap is rounding the discounted cash flows before combining them, which can change the last reported digit.
For A, \(NPV_A=-25+36.3/1.10=-25+33=8.0000\).
For B, \(NPV_B=-20-10/1.10+44/1.21=-20-9.090909\ldots+36.363636\ldots=7.272727\ldots\), or \(7.2727\).
For C, \(NPV_C=-25-15/1.10+50/1.10^2=-25-13.636363\ldots+41.322314\ldots=2.685950\ldots\), or \(2.6860\). The final result is rounded from the unrounded NPV, not obtained by combining prematurely rounded terms.
A+B commits \((25+20,\ 0+10)=(45,10)\) and has NPV \(8+7.272727\ldots=15.2727\). A+C commits \((50,15)\) and has NPV \(10.6860\). B+C commits \((45,25)\) and has NPV \(9.9587\). Each pair is feasible, but all three projects together require \(70\) in period 0, exceeding the budget of \(50\). Therefore A+B is optimal. Check: its commitments \(45\le50\) and \(10\le25\), and its total NPV exceeds that of every other feasible pair.
Worked Example: A Strong Project That Cannot Fit Beside Another
A firm has a period-0 budget of \(80\) thousand dollars and a period-1 budget of \(50\) thousand dollars. The effective annual discount rate is \(10\%\), with year-end cash flows. In thousands, project P has commitments \((50,20)\) and cash flows \((-50,-20,90)\); project Q has commitments \((40,30)\) and cash flows \((-40,-30,95)\); project R has commitments \((35,0)\) and cash flows \((-35,50)\). Identify the highest-NPV feasible portfolio.
Method. Compare each portfolio’s commitment vector with the two budgets, rather than ranking projects by NPV alone. This avoids treating a high-NPV project as automatically selectable alongside another attractive project.
The NPVs are \(NPV_P=-50-20/1.10+90/1.21=-50-18.181818\ldots+74.380165\ldots=6.1983\); \(NPV_Q=-40-30/1.10+95/1.21=-40-27.272727\ldots+78.512397\ldots=11.2397\); and \(NPV_R=-35+50/1.10=-35+45.454545\ldots=10.4545\). Results are rounded to four decimal places in thousands.
P+Q is infeasible because its period-0 commitments are \(50+40=90>80\). P+R is infeasible because its period-0 commitments are \(50+35=85>80\). Q+R requires \((40+35,\ 30+0)=(75,30)\), which meets both budgets and has combined NPV \(11.239669\ldots+10.454545\ldots=21.6942\). All three projects together require \(125\) in period 0, so they are infeasible as well. The best feasible portfolio is Q+R. Check: no portfolio containing P can add another project within the period-0 limit, while Q+R is feasible and has greater NPV than any single project.
Common Traps and Practical Checks
A single-period ranking does not solve a multi-period capital-rationing problem. A project with a favorable NPV or profitability index can consume funds in a scarce period and prevent the firm from selecting a better combination. The multi-period constraints also mean that unused funds in one period do not automatically make an otherwise infeasible portfolio acceptable.
- Checking only total investment. A portfolio may fit the sum of all budgets but exceed one period’s limit. Check every period separately.
- Comparing NPVs measured at different dates. Discount cash flows to the same valuation date before adding project NPVs.
- Using investment commitments as the NPV. Commitments determine budget use; NPV uses the full discounted project cash-flow stream.
- Rounding intermediate values. Keep full precision through the calculations and round only the reported result. This matters when portfolio NPVs are close.
- Assuming projects are independent without checking. If one project requires another, or if projects are mutually exclusive, include the corresponding condition. The basic model does not represent such restrictions automatically.
For a small number of projects, enumerating feasible portfolios is transparent and makes it easy to verify each constraint. With many projects, the same formulation is an integer optimization problem. In either case, the decision rule is unchanged: among the portfolios satisfying every period’s constraints, select the one with the greatest total NPV.
Check Your Understanding
For each problem, test feasibility period by period before comparing total NPV. Amounts are in thousands unless stated otherwise.
- A firm has budgets of \(60\) in periods 0 and 1 of \(20\). Projects A, B, and C have commitment vectors \((30,20)\), \((40,0)\), and \((20,20)\), and NPVs of \(5\), \(7\), and \(6\), respectively. Which feasible portfolio has the greatest NPV? (Answer: B+C, with NPV \(13\) and commitments \((60,20)\).)
- Budgets for periods 0, 1, and 2 are \((90,50,30)\). Four indivisible projects have the following commitment vectors and NPVs: P \((50,10,0)\), \(12\); Q \((40,0,30)\), \(11\); R \((30,40,0)\), \(10\); S \((20,20,20)\), \(8\). What is the best feasible portfolio? (Answer: P+Q, with total NPV \(23\) and commitments \((90,10,30)\).)
- At an effective annual rate of \(10\%\), project X requires \(18\) at time 0 and \(12\) at time 1, and pays \(40\) at time 2. Project Y requires \(25\) at time 0 and has NPV \(7\). Budgets are \(43\) at time 0 and \(12\) at time 1. What is the optimal selection? (Answer: Select both; \(NPV_X=-18-12/1.10+40/1.21=4.1488\), so the total NPV is \(11.1488\), and commitments are \((43,12)\).)
- At a \(10\%\) effective annual discount rate, project M has cash flows \((-30,-20,66)\) at times 0, 1, and 2, and commitments \((30,20)\). Project N has cash flows \((-35,0,50)\) at times 0 and 1, and commitments \((35,0)\). Budgets are \(65\) at time 0 and \(20\) at time 1. Which portfolio maximizes NPV? (Answer: M+N is feasible with commitments \((65,20)\); \(NPV_M=-30-20/1.10+66/1.21=6.3636\), \(NPV_N=-35+50/1.10=10.4545\), and total NPV is \(16.8182\).)
- A company has budgets of \(70\) in period 0 and \(25\) in period 1. Project U has commitments \((45,25)\) and NPV \(14\); project V has commitments \((30,0)\) and NPV \(9\); project W has commitments \((25,10)\) and NPV \(8\). Each project is optional and indivisible. What portfolio is optimal, and why is the other two-project combination not feasible? (Answer: U+V, with NPV \(23\) and commitments \((75,25)\), is infeasible because period-0 commitments exceed \(70\); V+W is feasible with commitments \((55,10)\) and NPV \(17\), while U+W requires \(70\) in period 0 and \(35\) in period 1, so it is infeasible. Thus V+W is optimal.)