Advanced Financial Management · Discounted cash flow techniques

Capital Rationing: Single and Multi-Period Techniques for AFM

Updated 11 October 2026 · Fact-checked

Capital rationing means you cannot fund every positive-NPV project because capital is limited. For one period, rank divisible projects by profitability index (NPV ÷ capital outlay) and fund them in order. For indivisible projects, test combinations. For several periods, use linear programming to maximise total NPV within each period's funding limit.

Understand Capital Rationing: Single and Multi-Period

Normally you accept every project with a positive NPV. That rule assumes the firm can raise all the money it needs at its cost of capital. Capital rationing is the situation where it cannot. Funds are limited, so you must choose the set of projects that gives the highest total NPV, not just the projects that look good one by one.

Hard rationing is imposed from outside. Examples are lenders refusing to lend, or a weak equity market that blocks a share issue. Soft rationing is imposed from inside. Examples are a board budget cap, a policy of not raising new equity, or a rule that each division gets a fixed sum. Soft rationing can be changed by management. Hard rationing cannot. In an exam, you should say that soft rationing may be a sign of poor planning, or may be a deliberate control tool for a business that is growing too fast to manage.

When capital is the scarce resource, the best measure is NPV per unit of capital used. This is the profitability index (PI). If projects are divisible (you can do part of a project and earn the same proportion of NPV), rank by PI and fund in order until the money runs out. The last project may be funded only in part.

If projects are indivisible (all or nothing), ranking by PI may not give the best answer, because unused money is left over. You must test feasible combinations and pick the one with the highest total NPV. Projects that are mutually exclusive cannot be combined.

With multi-period rationing, funds are limited in more than one year, for example this year and next year. One PI cannot capture two scarce resources. You use linear programming: an objective (maximise total NPV) and one constraint for each period. Solving gives the best mix, and the shadow price of a binding constraint tells you how much extra NPV one more unit of that period's funds would add.

Key rules to remember

Profitability index (single-period): PI = NPV ÷ capital outlay in the rationed period; Some texts use (PV of inflows ÷ outlay) = PI + 1. Rankings are the same. State which one you use.
Divisible projects: Rank by PI, fund in order, fund the last project in part: fraction = remaining funds ÷ its outlay; Total NPV = sum of NPVs of full projects + fraction × NPV of the last project.
Indivisible projects: Choose the feasible combination with the highest total NPV, with total outlay ≤ funds available; Check mutually exclusive projects. Leftover funds are assumed to earn only the cost of capital, so they add zero NPV.
Linear programming (multi-period): Maximise Σ(NPV of project j × xj) subject to Σ(outlay of j in year t × xj) ≤ funds in year t, and 0 ≤ xj ≤ 1; For indivisible projects, xj must be 0 or 1. Divisible projects let xj be any fraction from 0 to 1.
Shadow price: Shadow price = extra NPV from one more unit of a binding constraint's funds; A constraint with spare funds has a shadow price of zero.

How to solve Capital Rationing: Single and Multi-Period questions

Use this order for any capital rationing question. It works for single and multi-period cases.

1Identify the type of rationing (hard or soft), the period or periods that are limited, and whether projects are divisible, indivisible or mutually exclusive. Check the question carefully because this decides the method.
2Calculate the NPV of each project at the correct discount rate. Discard any with a negative NPV, unless they are needed for strategic reasons stated in the question.
3If only one period is limited, calculate PI = NPV ÷ outlay in that period for each project, then rank.
4For divisible projects, fund in rank order until the money runs out. Take a fraction of the last project and compute total NPV.
5For indivisible projects, list the feasible combinations within the budget, add up the NPVs, and choose the highest. Start from the PI ranking, but test alternatives that use the money better.
6If two or more periods are limited, define variables, write the objective and a constraint for each period, add 0 ≤ x ≤ 1 limits, and solve. State which constraints are binding.
7Interpret the result. State the plan, the total NPV, any unused funds, the shadow price if relevant, and what management could do (for example, relax a soft limit or find new finance).
8Add brief comments on assumptions: projects cannot be repeated, divisibility may be unrealistic, outlays and NPVs are estimates, and unused funds earn only the cost of capital.

Quickest way: PI ranking with a fast indivisible check

When to use it: Single-period rationing with three to six projects. This is the usual Section A or B situation.

Compute NPV ÷ outlay for each project and rank them in a small table.
If divisible, fill the budget down the table and fund the last project by fraction. You are done.
If indivisible, fill the budget in rank order first and note the unused funds.
Test two or three swaps that use the unused money, such as dropping a small project for a larger one. Compare total NPVs.
For multi-period, write the LP in two lines before solving. Then test whether only one constraint binds. If x = 1 for the high-value projects and only one period is short, fund the last project by the fraction allowed by that period.

Common mistakes in Capital Rationing: Single and Multi-Period

Ranking by NPV or IRR instead of PI when funds are limited.
Students are used to choosing the highest NPV or IRR when capital is not limited.
Fix: When capital is the scarce resource, rank by NPV per unit of that capital. Use IRR only for comment, not for the ranking.
Using PI ranking alone for indivisible projects.
The method is correct for divisible projects, so students apply it everywhere.
Fix: After the PI ranking, test other combinations. Use PI only as a starting point.
Dividing NPV by total outlay across all years in a multi-period question.
Students want a single ranking measure.
Fix: A single PI is not valid when more than one period is limited. Use linear programming, or state clearly why a PI is only approximate.
Forgetting to fund the last divisible project in part, or funding too much.
Students round up or skip the fraction calculation.
Fix: Fraction = remaining funds ÷ outlay. Check that total spend equals the budget exactly.
Treating hard and soft rationing as the same, or giving no comment at all.
The calculation takes all the time and the discussion is left out.
Fix: Add two or three sentences: who imposed the limit, whether management can relax it, and what it means for shareholder wealth. These are easy marks.
Including negative-NPV projects or counting unused funds as earning a return.
Students want to spend the full budget.
Fix: Leave unused funds unspent or invested at the cost of capital, which adds zero NPV. Never add a negative-NPV project just to use up funds.

Worked examples

Example 1

A company has $500,000 available for investment now. Funds will be limited to this amount. Four projects are available, none can be repeated. Figures are in $000. Project A: outlay 200, NPV 50. Project B: outlay 150, NPV 48. Project C: outlay 250, NPV 40. Project D: outlay 100, NPV 15. (a) Find the best plan if projects are divisible. (b) Find the best plan if projects are indivisible.

Show the solution

Calculate PI = NPV ÷ outlay. A = 50 ÷ 200 = 0.25. B = 48 ÷ 150 = 0.32. C = 40 ÷ 250 = 0.16. D = 15 ÷ 100 = 0.15.
Ranking: B (0.32), A (0.25), C (0.16), D (0.15).
(a) Divisible: fund B in full (150), leaving 350. Fund A in full (200), leaving 150. Fund C in part: 150 ÷ 250 = 0.6 of C, which uses all funds.
Total NPV = 48 + 50 + (0.6 × 40) = 48 + 50 + 24 = 122.
(b) Indivisible: rank order gives B + A = 350 outlay, NPV 98. Then C does not fit (250 > 150 left), but D does (100). B + A + D = 450 outlay, NPV 113, leaving 50 unused.
Test alternatives within 500: B + C + D = 500 outlay, NPV 48 + 40 + 15 = 103. A + C = 450 outlay, NPV 90. A + B + C = 600, not feasible. A + C + D = 550, not feasible.
The best feasible combination is A + B + D with NPV 113.

Answer: (a) Divisible: accept B and A in full and 60% of C. Total NPV is $122,000. (b) Indivisible: accept A, B and D. Total NPV is $113,000, with $50,000 of funds unused. The indivisibility costs $9,000 of NPV compared with (a).

Example 2

A company has funds of $400,000 now (year 0) and $200,000 in year 1. Three divisible projects, none repeatable, have the following outlays and NPVs in $000. Project X: year 0 outlay 150, year 1 outlay 100, NPV 60. Project Y: year 0 outlay 200, year 1 outlay 50, NPV 70. Project Z: year 0 outlay 100, year 1 outlay 120, NPV 45. Formulate the linear programme, find the optimal plan and the shadow price of year 1 funds.

Show the solution

Let x, y and z be the proportions of X, Y and Z undertaken, each between 0 and 1.
Objective: maximise NPV = 60x + 70y + 45z.
Year 0 constraint: 150x + 200y + 100z ≤ 400.
Year 1 constraint: 100x + 50y + 120z ≤ 200.
Try x = 1 and y = 1. Year 0 use is 350, leaving 50. Year 1 use is 150, leaving 50.
Z is limited by year 0 to 50 ÷ 100 = 0.5, and by year 1 to 50 ÷ 120 = 0.4167. Year 1 is tighter, so z = 5/12 (0.4167).
Check year 0: 150 + 200 + (100 × 5/12) = 391.67 ≤ 400, so year 0 has spare funds of 8.33. Year 1: 100 + 50 + 50 = 200, exactly used.
NPV = 60 + 70 + (45 × 5/12) = 60 + 70 + 18.75 = 148.75.
Test optimality: Z earns 45 ÷ 120 = 0.375 NPV per $ of year 1 funds. Cutting X frees 100 of year 1 funds but loses 60, which is 0.6 per $. Cutting Y frees 50 but loses 70, which is 1.4 per $. Both are worse than using the funds on Z, so the plan is optimal.
Shadow price of year 1 funds = 0.375 per $1, so $1,000 more year 1 funds adds $375 of NPV. Year 0 has spare funds, so its shadow price is zero.

Answer: Undertake X and Y in full and 5/12 (about 41.7%) of Z. Total NPV is $148,750. Year 1 funds are fully used and have a shadow price of $0.375 per $1. Year 0 funds are not binding, so their shadow price is zero.

Exam tips

Read the wording for divisible or indivisible, and for which years are limited. Write it at the top of your answer. The method depends on it.
Always show the PI table or the LP formulation. Marks go for method even if a number slips.
Add the discussion: hard versus soft rationing, why soft limits arise, and the assumptions behind the method. These marks are often missed.
In multi-period questions, define variables clearly and state which constraints bind. Give the shadow price meaning in words, for example what extra funds in that year would be worth.
Use the professional skills marks: recommend a plan, say what management could do (seek new finance, delay a project, relax an internal cap), and keep your tone suited to the board.

Practice questions from Discounted cash flow techniques

Capital Rationing: Single and Multi-Period in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Capital Rationing: Single and Multi-Period: frequently asked questions

What is the difference between hard and soft capital rationing?

Hard rationing comes from outside the firm, such as lenders refusing credit or a closed equity market. Soft rationing is set internally, such as a board budget cap or a no-new-equity policy. Management can change soft limits but not hard ones.

When do I use the profitability index in AFM?

Use it when only one period is rationed and projects are divisible. Rank projects by NPV ÷ outlay and fund them in order. For indivisible projects it gives a starting point, but you must test combinations.

Why can't I use PI for multi-period capital rationing?

PI measures return per unit of one scarce resource. With limits in two or more years, there are several scarce resources, and no single ratio ranks projects correctly. Linear programming handles all constraints at the same time.

What does a shadow price tell me in capital rationing?

It is the extra NPV you would gain from one more unit of funds in a period where the limit is binding. It helps management decide whether it is worth raising more finance or relaxing a soft limit. It is zero for a period with unused funds.