Capital Rationing With Indivisible Projects

Why Capital Rationing Changes the Decision

A project can have a positive NPV and still be impossible to undertake if the organization does not have enough investment funds to accept every attractive opportunity. This is capital rationing: investment is limited by a budget, so management must choose a portfolio rather than assess each project in isolation. When projects are indivisible, each must be accepted in full or rejected; partial investment is not an available choice.

The decision has two parts. First, determine each project’s NPV using the appropriate required return. Then compare portfolios that fit within the budget. The best portfolio is the feasible one with the greatest total NPV, not necessarily the portfolio with the most projects or the greatest total investment.

Definition: Suppose project $j$ requires an initial investment $I_j$ and has future cash flows $CF_{jt}$ at times $t=1,\ldots,n$. At a required return $k$ per period, its NPV is $$ NPV_j=-I_j+\sum_{t=1}^{n}\frac{CF_{jt}}{(1+k)^t}. $$ For indivisible projects, let $x_j=1$ if project $j$ is accepted and $x_j=0$ otherwise. With a single investment budget $B$, the decision is to maximize $\sum_j x_jNPV_j$, subject to $\sum_j x_jI_j\leq B$ and $x_j\in\{0,1\}$ for every project.

This model assumes that each project’s cash flows and investment are known for the decision, that the NPVs are measured at the same valuation date using appropriate required returns, and that project NPVs add without interaction. It also assumes that the relevant restriction is one budget covering the initial investments. If accepting one project changes another project’s cash flows, projects are mutually exclusive, or separate budgets apply in different periods, this simple model needs additional constraints or a different formulation.

Why Portfolio NPV Is Additive

The objective follows from the linearity of discounting. If the accepted projects are independent in the sense that one does not change another’s cash flows, the portfolio cash flow at each date is the sum of its project cash flows. Discounting that sum gives the sum of the discounted cash flows:

\sum_{t=1}^{n}\frac{\sum_j x_jCF_{jt}}{(1+k)^t} = \sum_j x_j\left(\sum_{t=1}^{n}\frac{CF_{jt}}{(1+k)^t}\right).

The portfolio also incurs the sum of the accepted initial investments, $\sum_j x_jI_j$. Subtracting that amount from the discounted future cash flows gives $\sum_j x_jNPV_j$. Therefore, maximizing total portfolio NPV is equivalent to maximizing the value created by the selected projects under these assumptions.

The budget constraint is a separate test: it checks the initial investments, not the projects’ NPVs. A portfolio with high total NPV is not a valid choice if its required investment exceeds the available budget.

Worked Example: Comparing Feasible Portfolios

A firm has a $75,000 budget for four independent, indivisible projects. The required return is 10% per year, and each project pays one cash inflow at the end of year 1. Project A costs $40,000 and pays $62,700; B costs $35,000 and pays $60,500; C costs $25,000 and pays $44,000; and D costs $20,000 and pays $35,200. Select the portfolio with the highest NPV.

Method. Calculate each NPV at 10%, then check the total investment and total NPV of the feasible combinations. Because each inflow occurs after one year, its present value is the inflow divided by $1.10$.

For A, the present value of the inflow is $62{,}700/1.10=57{,}000$, so $NPV_A=57{,}000-40{,}000=\$17{,}000$. For B, $60{,}500/1.10=55{,}000$, so $NPV_B=55{,}000-35{,}000=\$20{,}000$. For C, $44{,}000/1.10=40{,}000$, so $NPV_C=40{,}000-25{,}000=\$15{,}000$. For D, $35{,}200/1.10=32{,}000$, so $NPV_D=32{,}000-20{,}000=\$12{,}000$.

Check the pairs. A+B costs $40{,}000+35{,}000=\$75{,}000$ and has NPV $17{,}000+20{,}000=\$37{,}000$. A+C costs $65,000 and has NPV $32,000; A+D costs $60,000 and has NPV $29,000; B+C costs $60,000 and has NPV $35,000; B+D costs $55,000 and has NPV $32,000; and C+D costs $45,000 and has NPV $27,000. Each pair fits the budget.

No three-project combination fits: the least expensive triple is B+C+D, costing $35{,}000+25{,}000+20{,}000=\$80{,}000$, which exceeds the budget. Any triple containing A costs at least $40{,}000+25{,}000+20{,}000=\$85{,}000$. Thus A+B is the feasible portfolio with the greatest NPV, $37,000.

Check and common mistake avoided. The selected investment is exactly $75,000, so the portfolio is feasible. Comparing A+B with B+C confirms that the largest total investment is not itself the objective: A+B has the larger NPV. Do not choose a combination until its total cost has been checked against the budget.

A Practical Selection Procedure

For a small set of projects, a disciplined comparison is usually clearer than an informal ranking. The candidate must be a complete portfolio, and its cost and NPV must both be calculated. The following process keeps those requirements explicit.

Calculate individual NPVs.
Use the stated required return and the timing of each project’s cash flows. Keep each project’s initial investment visible for the budget test.

Specify whole-project choices.
For every project, record either accepted or rejected. Do not treat an indivisible project as if a fraction of its cost or NPV could be selected.

Test feasibility.
Add the initial investments in the proposed portfolio. Reject the portfolio if its total exceeds the budget, even if its NPV is high.

Compare total NPVs among feasible choices.
Add the NPVs of the accepted projects and select the feasible portfolio with the greatest sum. Do not use unspent budget as a reason to accept a value-destroying project.

When the project set is small, a table of candidate combinations makes the checks easy to audit. For larger sets, the same objective and constraints can be handled with an integer optimization method. In either case, the method must preserve the all-or-nothing decision for every project.

Worked Example: A Budget That Creates a Three-Project Alternative

A company has an $80,000 budget and considers five independent, indivisible projects. The required return is 5% for one year. Their costs and year-end cash inflows are: A, $50,000 and $81,900; B, $40,000 and $67,200; C, $35,000 and $58,800; D, $25,000 and $39,900; and E, $20,000 and $31,500. Find the value-maximizing portfolio.

Method. Discount each one-year inflow by 1.05. This gives NPVs of A: $81{,}900/1.05-50{,}000=78{,}000-50{,}000=\$28{,}000$; B: $67{,}200/1.05-40{,}000=64{,}000-40{,}000=\$24{,}000$; C: $58{,}800/1.05-35{,}000=56{,}000-35{,}000=\$21{,}000$; D: $39{,}900/1.05-25{,}000=38{,}000-25{,}000=\$13{,}000$; and E: $31{,}500/1.05-20{,}000=30{,}000-20{,}000=\$10{,}000$.

Project A cannot be combined with B or C because those costs are $90,000 and $85,000. A+D costs $75,000 and has NPV $41,000; A+E costs $70,000 and has NPV $38,000. Among combinations without A, B+C costs $75,000 and has NPV $24{,}000+21{,}000=\$45{,}000$. C+D+E costs exactly $80,000 and has NPV $21{,}000+13{,}000+10{,}000=\$44{,}000$. B+D+E costs $85,000, so it is infeasible; all other triples without A also cost more than $80,000.

The leading feasible choices therefore include B+C at $45,000 NPV, C+D+E at $44,000, and A+D at $41,000. The best is B+C, which costs $75,000. The unused $5,000 cannot fund another whole project, and there is no requirement to spend the entire budget.

Check and common mistake avoided. B+C is feasible because $40{,}000+35{,}000=\$75{,}000\leq \$80{,}000$, and its NPV exceeds the other leading feasible alternatives. Do not reject it merely because C+D+E uses the entire budget: spending more does not imply creating more value.

Negative NPVs and the Feasibility Trap

If a project has negative NPV and can be rejected without affecting the other projects, accepting it reduces total portfolio NPV. A positive-NPV project may also be left out if the budget is insufficient to combine it with a more valuable feasible set. Capital rationing is therefore not a rule to accept every positive-NPV project, nor a rule to use every available dollar.

A frequent exam error is to identify an attractive set by adding its NPVs but forget to add its costs. For instance, suppose the budget is $70,000, project 1 costs $30,000 and has NPV $8,000, and project 2 costs $45,000 and has NPV $13,000. The tempting calculation $8{,}000+13{,}000=\$21{,}000$ does not make accepting both a valid answer: their combined cost is $30{,}000+45{,}000=\$75{,}000$, which exceeds the budget. The feasible single-project choices have NPVs of $8,000 and $13,000, so project 2 is better. The correction is to discard the infeasible pair before comparing portfolio values.

Worked Example: Rejecting a Negative-NPV Project

A firm has a $45,000 budget. The required return is 10% per year. Independent, indivisible project A costs $30,000 and pays $22,000 at the end of year 1 and $16,500 at the end of year 2. Project B costs $25,000 and pays $16,500 at the end of year 1 and $14,520 at the end of year 2. Project C costs $20,000 and pays $9,900 at the end of year 1 and $10,890 at the end of year 2. Which portfolio maximizes NPV?

Method. Discount year 1 by $1.10$ and year 2 by $1.10^2=1.21$. For A, the present values are $22{,}000/1.10=\$20{,}000$ and $16{,}500/1.21=\$13{,}636.36$ when rounded to cents. Using the unrounded sum, $NPV_A=20{,}000+16{,}500/1.21-30{,}000=\$3{,}636.36$ to the nearest cent.

For B, the present values are $16{,}500/1.10=\$15{,}000$ and $14{,}520/1.21=\$12{,}000$, so $NPV_B=15{,}000+12{,}000-25{,}000=\$2{,}000$. For C, the present values are $9{,}900/1.10=\$9{,}000$ and $10{,}890/1.21=\$9{,}000$, so $NPV_C=9{,}000+9{,}000-20{,}000=-\$2{,}000$.

A+B costs $55,000 and is infeasible. A+C costs $50,000 and is infeasible. B+C costs exactly $45,000 and has NPV $2{,}000-2{,}000=\$0$. The single-project alternatives have NPVs of $3,636.36 for A, $2,000 for B, and negative $2,000 for C; rejecting all projects gives $0. Therefore A alone is optimal, with NPV $3,636.36.

Check and common mistake avoided. A fits within the budget, while the pair with the highest cost of $45,000, B+C, produces no net NPV. Do not accept C merely because it fits the budget or because combining projects can seem preferable to leaving funds unused.

What the Model Does Not Decide

The result depends on the stated assumptions. If projects are mutually exclusive, the choices need an explicit restriction preventing both from being selected. If one project is required before another can proceed, a precedence restriction is needed. If a project changes another project’s revenues or costs, adding standalone NPVs may double-count value or omit a loss. And if investment limits apply in several different periods, one total-budget inequality is not enough.

Within the single-period, independent-project setting, the reliable decision rule is direct: calculate NPVs, preserve whole-project choices, test the cost of every candidate portfolio, and select the feasible one with the largest total NPV. This is aligned with the SOA VEE Accounting and Finance syllabus.

Check Your Understanding

For each problem, compare total NPVs only after checking whether the proposed portfolio fits the budget.

A firm has a $60,000 budget. Project A costs $35,000 and has NPV $18,000; B costs $30,000 and has NPV $16,000; C costs $25,000 and has NPV $13,000. Projects are independent and indivisible. Which portfolio maximizes NPV? (Answer: A+C; it costs $60,000 and has NPV $31,000.)
A $50,000 budget is available for three independent projects: A costs $24,000 with NPV $14,000, B costs $31,000 with NPV $17,000, and C costs $19,000 with NPV $9,000. Which portfolio is best? (Answer: B+C; it costs $50,000 and has NPV $26,000.)
The required return is 10% for one year. Project X costs $40,000 and pays $61,600 after one year; Y costs $30,000 and pays $46,200; Z costs $25,000 and pays $44,000. With a $70,000 budget and indivisible independent projects, which portfolio maximizes NPV? (Answer: X+Z; their NPVs are $16,000 and $15,000, respectively, and their $65,000 cost is feasible, giving total NPV $31,000.)
A company has an $80,000 budget. Independent projects A, B, C, and D have costs of $45,000, $40,000, $35,000, and $20,000, and NPVs of $30,000, $25,000, $22,000, and $11,000, respectively. Find the best portfolio. (Answer: A+C; it costs $80,000 and has NPV $52,000. B+C costs $75,000 and has NPV $47,000; A+B is infeasible.)
A $50,000 budget is available for independent, indivisible projects P, Q, and R. P costs $28,000 and has NPV $19,000; Q costs $22,000 and has NPV $11,000; R costs $30,000 and has NPV $-\$4,000$. Which portfolio maximizes NPV, allowing the firm to reject every project? (Answer: P+Q; it costs $50,000 and has NPV $30,000. R should be rejected.)

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