Net present value says to accept every project that earns more than it costs. That rule assumes money is always available at the cost of capital. Sometimes it is not: the bank will lend no more, the shareholders will not subscribe for new shares, or the board has set a ceiling on the year’s capital spending.
Then there are more good projects than money, and the company must choose, aiming to get the largest total net present value (NPV) from the money it has.
This topic explains capital rationing and its types (hard and soft; single period and multi-period), and then shows how to select projects: by the profitability index when projects can be done in part, by testing combinations when they cannot, and with the complications of mutually exclusive projects and a project that must be undertaken, with worked examples and the limits of the method.
It closes Area C. The next topic starts Area D with short-term sources of business finance.
The syllabus asks for single-period capital rationing, divisible and indivisible projects. Read asset replacement: equivalent annual cost and lowest common multiple first.
In 30 seconds
Capital rationing exists when funds are too few to invest in all projects with a positive NPV. Hard rationing is a real shortage of capital; soft rationing is a limit set by management policy. The objective is to maximise the total NPV from the funds available.
Divisible projects (can be done in part): rank by the profitability index = NPV ÷ capital invested in the limited period, and invest in order until the funds run out, taking the last project in part. Indivisible projects (all or nothing): test the combinations that fit within the funds and choose the one with the highest total NPV.
For mutually exclusive projects, evaluate each allowed group separately; for a project that must be undertaken, fund it first. Never rank by NPV alone or by internal rate of return (IRR).
The nature of capital rationing
Capital rationing occurs when there is not enough capital to invest in all the projects with a positive NPV, so a choice must be made. Two kinds are distinguished:
- Hard capital rationing: a real shortage, for example the company cannot raise new capital in the market or borrow more, so it can invest only the cash flow it retains from profits.
- Soft capital rationing: funds could be found, but management has set a budget limit on capital spending as a matter of policy, perhaps to limit risk, to avoid dilution or because of limited management capacity.
The methods assume that the aim is to maximise total NPV and that the projects are comparable in risk. Single-period rationing restricts funds in one period only, usually now (time 0), with funds freely available later; multi-period rationing restricts several years. Only the single-period case is covered in detail here.
Divisible projects: the profitability index
If projects are divisible, part of a project can be undertaken for a proportionate part of the NPV (for example half the investment gives half the NPV). The best use of limited funds is the same idea as the limiting factor in decision-making: rank projects by the profitability index, the NPV per naira invested in the year of rationing:
Profitability index = NPV ÷ capital invested at time 0
and invest in order of the highest index until the funds are used up, taking the last project in part. Projects with a negative NPV are rejected outright.
Example: Delta has ₦22m to invest at time 0 and four divisible projects.
| Project | Investment ₦m | NPV ₦m | Index | Rank |
|---|---|---|---|---|
| Q | 8 | 5.2 | 0.65 | 1st |
| R | 6 | 2.7 | 0.45 | 2nd |
| P | 12 | 5.0 | 0.42 | 3rd |
| S | 10 | 4.0 | 0.40 | 4th |
Allocate: Q takes ₦8m (NPV 5.2), leaving ₦14m; R takes ₦6m (NPV 2.7), leaving ₦8m; P takes the remaining ₦8m, two-thirds of the project (8 ÷ 12), for an NPV of 8 ÷ 12 × 5.0 = 3.33. The total NPV is 5.2 + 2.7 + 3.33 = ₦11.23m, from Q and R in full and two-thirds of P. Project S is not undertaken. Ranking by NPV alone would have put Q, P, S first and given a worse result.
Indivisible projects: testing combinations
If projects are indivisible (all or nothing), the profitability index may not give the best answer because the last project cannot be taken in part and some funds may be left over. The method is to list every combination that fits within the funds and choose the one with the highest total NPV. For Delta with ₦22m:
| Combination | Investment ₦m | Total NPV ₦m |
|---|---|---|
| Q and P | 20 | 10.2 |
| P and S | 22 | 9.0 |
| Q and S | 18 | 9.2 |
| Q and R | 14 | 7.9 |
| P and R | 18 | 7.7 |
| R and S | 16 | 6.7 |
(Combinations of three, such as Q, R and S at ₦24m, exceed the funds.) The best is Q and P, a total NPV of ₦10.2m, leaving ₦2m unused. This is lower than the ₦11.23m possible with divisible projects, as it must be: the all-or-nothing restriction can only reduce the total.
Complications
- Mutually exclusive projects. If R and S cannot both be undertaken, evaluate the groups without both: Q, P and R (the ranking above: ₦11.23m) and Q, P and S (Q 8, then P 12 for NPV 5.0, then 2 of S’s 10 for NPV 0.8: ₦11.0m). Choose the better group, here Q, R and two-thirds of P.
- A project that must be undertaken. for strategic or legal reasons is funded first, whatever its index.
If S must be done (₦10m, NPV 4.0), the remaining ₦12m goes to Q (8; NPV 5.2) and then R in part (4 of 6; NPV 1.8): total NPV 4.0 + 5.2 + 1.8 = ₦11.0m. - Multi-period rationing or several constraints (funds, labour, machine time) cannot be solved by ranking: mathematical programming is needed.
- Project timing and follow-on projects: a project may open the way to later ones, which the NPV of its own cash flows does not show.
The ideas in more depth
Why rank by NPV per naira of the scarce resource
The scarce resource is money at time 0, so the criterion is the NPV produced per naira of that money. A large project may have the highest NPV and the lowest index, and choosing it uses up funds that would have earned more elsewhere.
The last project’s index (here 0.42 for P) is the shadow price of the funds: each extra naira of capital would add about 42 kobo to the NPV, so a lender or shareholder offering funds at a cost below that return would add value.
Why capital rationing is a puzzle for theory
If capital markets worked perfectly the company could raise funds for every positive-NPV project, so rationing would not arise. When it does, it signals a market imperfection or an internal constraint. A company under soft rationing should ask whether the policy limit costs it value; under hard rationing it should look for new sources of finance, partners, joint ventures, leasing or the sale of assets.
Sources and references
- No accounting standard governs capital rationing. The linear programming used for multi-period problems is taught in the course on quantitative methods.
These points are stated as at 5 November 2026.
Worked examples
Example 1: the index. Project P costs ₦12m and has an NPV of ₦5.0m. Required: the profitability index.
- Index.
5.0 ÷ 12 = 0.42.
Example 2: the part-project. Only ₦8m remains for P (₦12m, NPV ₦5.0m). Required: the NPV earned.
- Share. 8 ÷ 12 = two-thirds.
- NPV.
8 ÷ 12 × 5.0 = ₦3.33m.
Example 3: a combination. Q (₦8m, NPV ₦5.2m) and P (₦12m, NPV ₦5.0m) are indivisible; funds ₦22m. Required: the total NPV.
- Investment.
8 + 12 = ₦20m, within ₦22m. - NPV.
5.2 + 5.0 = ₦10.2m.
How to answer in the exam
- State the type of rationing and whether the projects are divisible, and say that the aim is to maximise total NPV.
- Divisible: set out a table of the index and rank, allocate funds in order and compute the part-project.
- Indivisible: list every feasible combination with its investment and NPV.
- Deal with the complications (exclusive or compulsory projects) before ranking, and comment on the limits.
Common mistakes in the exam
- Ranking by NPV, or by IRR, instead of the profitability index.
- Taking a project in part when it is indivisible.
- Forgetting to fund a compulsory project first.
- Including a project with a negative NPV to use up the funds.
- Applying the index method when funds are limited in more than one period.
- Not comparing the allowed groups when projects are mutually exclusive.
Quick revision
- Rationing: not enough funds for all positive-NPV projects; hard or soft.
- Divisible: rank by NPV ÷ investment; last project in part.
- Indivisible: test combinations; best total NPV.
- Exclusive projects: separate groups; compulsory projects: fund first.
Practice questions
These are original questions written in the style of the Institute’s exams. Try each one before you open the answer. Section A is a recall check, because the paper itself is written.
Recall check: multiple choice
Question 1. Capital rationing exists when:
- Projects have negative NPVs
- Interest rates fall
- Funds are insufficient to undertake all positive-NPV projects
- There is no tax
Show the answer
C. So a choice must be made.
Question 2. Soft capital rationing arises from:
- A shortage in the market
- A management policy limit on spending
- A bank refusal
- A tax rule
Show the answer
B. Hard rationing is a real shortage.
Question 3. The objective under capital rationing is to:
- Maximise total NPV from the funds available
- Maximise the number of projects
- Minimise investment
- Maximise payback
Show the answer
A. Subject to the funds constraint.
Question 4. The profitability index is:
- NPV × outlay
- Outlay ÷ NPV
- IRR − cost of capital
- NPV ÷ capital invested in the rationed period
Show the answer
D. NPV per naira of scarce capital.
Question 5. For divisible projects you should:
- Rank by NPV
- Rank by the profitability index
- Rank by IRR
- Rank by size
Show the answer
B. And take the last project in part.
Question 6. For indivisible projects you should:
- Test the combinations that fit and choose the highest total NPV
- Take part of a project
- Rank by index only
- Choose the largest
Show the answer
A. Funds may be left over.
Question 7. Compared with divisible projects, the total NPV with indivisible projects is:
- Always higher
- Unrelated
- The same or lower
- Negative
Show the answer
C. The restriction cannot raise the NPV.
Question 8. A project that must be undertaken is:
- Ignored
- Funded last
- Ranked by index
- Funded first, then the balance is allocated
Show the answer
D. Whatever its index.
Question 9. For mutually exclusive projects, you should:
- Take both
- Take neither
- Evaluate each allowed group and choose the best
- Choose the larger
Show the answer
C. Only one of the pair can be in a group.
Question 10. The index method is NOT suitable when:
- Funds are limited in several periods
- Projects are divisible
- Only one period is limited
- NPVs are positive
Show the answer
A. Mathematical programming is then needed.
Section B style: written questions
Question 11 (10 marks). (a) Distinguish between hard and soft capital rationing. (4 marks) (b) State the decision rule for divisible projects. (3 marks) (c) State how indivisible projects are evaluated. (3 marks)
Question 12 (15 marks). A company has ₦10m at time 0 and four divisible projects: A (₦4m, NPV ₦3.2m), B (₦5m, NPV ₦3.0m), C (₦3m, NPV ₦1.5m) and D (₦6m, NPV ₦2.4m). (a) Calculate the profitability index of each. (4 marks) (b) Rank and allocate the funds. (6 marks) (c) Calculate the total NPV. (3 marks) (d) State why NPV alone is not the right ranking. (2 marks)
A further written question
Question 13 (20 marks). A company has ₦15m at time 0 and four indivisible projects: W (₦9m, NPV ₦4.5m), X (₦6m, NPV ₦3.3m), Y (₦5m, NPV ₦2.2m) and Z (₦4m, NPV ₦1.7m).
(a) List the combinations that fit within ₦15m, with their NPVs. (8 marks) (b) State the best combination. (3 marks) (c) Repeat the choice if W and X are mutually exclusive. (6 marks) (d) State TWO limits of the method. (3 marks)
Model answers
Show the model answer to Question 11
(a) Hard: a real shortage of capital, for example the company cannot raise or borrow more (2). Soft: funds are available, but management limits spending by policy (2).
(b) Rank the projects by the profitability index (NPV per naira invested in the rationed period) and invest in order until the funds are used, taking the last in part (3).
(c) List the combinations that fit within the funds and choose the one with the highest total NPV (3).
Show the model answer to Question 12
(a)
- A 3.2 ÷ 4 = 0.80
- B 3.0 ÷ 5 = 0.60
- C 1.5 ÷ 3 = 0.50
- D 2.4 ÷ 6 = 0.40 (4).
(b) Rank A, B, C, D. A takes ₦4m, B takes ₦5m, leaving ₦1m for C (one-third of the project); D is not undertaken (6).
(c) 3.2 + 3.0 + 1 ÷ 3 × 1.5 (0.5) = ₦6.7m (3).
(d) NPV ignores the amount of the scarce capital used by each project (2).
Show the model answer to Question 13
(a) W 4.5; X 3.3; Y 2.2; Z 1.7; W and X (₦15m) 7.8; W and Y (₦14m) 6.7; W and Z (₦13m) 6.2; X and Y (₦11m) 5.5; X and Z (₦10m) 5.0; Y and Z (₦9m) 3.9; X, Y and Z (₦15m) 7.2 (any combination of three including W exceeds ₦15m) (8).
(b) W and X, using all ₦15m, total NPV ₦7.8m (3).
(c) W and X cannot be taken together, so the options are W and Y (6.7), W and Z (6.2), and X, Y and Z (7.2): choose X, Y and Z, NPV ₦7.2m (6).
(d) (1.5 each, any two) It covers one period only; it assumes the NPVs and risks are comparable; the NPVs are forecasts; follow-on projects are ignored.
Spotted a mistake? Tell us and we will check it.
