A company has ₦10m to spend and five projects, all with positive NPVs, costing ₦30m in total. Every project passes the NPV test, yet the company can fund only a third of them. Choosing by the largest NPV is not enough: the right choice depends on how much capital each project uses. This topic sets out the method.
This topic covers hard and soft capital rationing, single-period rationing with divisible projects (profitability index), with indivisible projects (trial of combinations), mutually exclusive projects, a project that must be undertaken, multi-period rationing and its linear programming formulation, and post-completion audits with their benefits and problems. The previous topic covered asset replacement. The next covers the cost of capital and the weighted average cost of capital.
Neither the May 2014 nor the November 2014 Pathfinder sets a stand-alone question on this topic.
In 30 seconds
Capital rationing means there is not enough capital for all positive-NPV projects. Hard rationing is a real shortage; soft rationing is a management limit. The aim is to maximise total NPV from the capital available. Divisible projects: rank by profitability index = NPV ÷ capital invested in the rationed year, and fund in that order, taking a fraction of the last one.
Indivisible projects: test combinations and pick the highest total NPV, because ranking can fail. Mutually exclusive projects: never take both. A must-do project is funded first. Multi-period rationing needs linear programming. A post-completion audit reviews actual against forecast results to improve control and forecasting, but it has costs and problems.
Key terms in plain English
| Term | What it means |
|---|---|
| Capital rationing | A limit on capital that prevents funding every positive-NPV project |
| Hard rationing | A real external shortage of capital |
| Soft rationing | A limit that management sets itself |
| Divisible project | A project that can be done in part, with a proportionate NPV |
| Profitability index | NPV per ₦1 of capital invested in the rationed period |
| Mutually exclusive | Projects where taking one rules out the other |
| Linear programming | A method that maximises an objective subject to constraints |
| Post-completion audit | A review of a finished project against its forecast |
Syllabus check
This topic is in syllabus area B (advanced investment appraisal, 30% of the marks). It covers:
- the optimum investment plan under single-period rationing with divisible and indivisible projects;
- the formulation of a linear programme for multi-period rationing;
- post-completion auditing.
CAMA 2020 and capital allocation
The Companies and Allied Matters Act 2020 (CAMA 2020) is the company law of Nigeria.
| Section | What it says | Link to this topic |
|---|---|---|
| s. 305 | A director must act in good faith in the best interests of the company | Scarce capital should go where it adds most value for the company |
| s. 308 | A director must use the care, diligence and skill of a reasonably prudent director | Directors should test the choices made when capital is limited |
| s. 377(1) and s. 378 | Directors prepare statements that follow IFRS and give a true and fair view | Decisions later affect reported assets, profit and cash |
| s. 385 | The directors’ report covers performance and risk | Major plans and risks are reported to shareholders |
The nature of capital rationing
Capital rationing arises when funds are too few for every positive-NPV project. Hard rationing is a real shortage, for example when the company cannot raise new capital or borrow much, so its limit is the cash from retained profits. Soft rationing is a policy limit on capital spending. The methods assume that the aim is to maximise total NPV and that the projects have similar risk.
There are three problem types: single period with divisible projects, single period with indivisible projects, and multi-period with divisible projects.
Single period, divisible projects
Capital is short at time 0 only. A fraction of a project can be done for the same fraction of its NPV. Rank the projects by profitability index (PI) = NPV ÷ capital invested at time 0, fund them in rank order and take only the fraction of the last one that the remaining capital allows. This is like key-factor analysis in management accounting.
ICAN’s example. Capital is ₦5,000k. Project I costs ₦5,000k for an NPV of ₦6,250k (PI 1.25); II costs ₦2,100k for ₦4,200k (PI 2.00); III costs ₦1,400k for ₦1,540k (PI 1.10); IV costs ₦1,500k for ₦1,950k (PI 1.30).
- Ranking: II (2.00), IV (1.30), I (1.25), III (1.10).
- Invest in II (₦2,100k) and IV (₦1,500k), then ₦1,400k of I (28% of it) for ₦1,750k.
- Total NPV.
4,200 + 1,950 + 1,750 = ₦7,900k.
Single period, indivisible projects
Each project is all or nothing. Ranking by PI may not give the best answer, so list every affordable combination and choose the highest total NPV. Unspent cash earns a zero NPV. With ICAN’s data, I alone gives ₦6,250k, but II + III + IV costs ₦5,000k and gives ₦7,690k, so the plan is II, III and IV. Total NPV with divisible projects is never lower than with indivisible ones.
Complications
- Mutually exclusive projects. If II and IV exclude each other, test combinations without both. ICAN’s result for indivisible projects: I alone (₦6,250k) beats II + III (₦5,740k) and III + IV (₦3,490k), so invest in I only.
- A project that must be done. Fund it first, then allocate what is left. If III (₦1,400k, NPV ₦1,540k) is a must, ₦3,600k remains, which funds II and IV in full, for a total NPV of ₦7,690k.
Multi-period rationing
When capital is limited in more than one period, ranking by PI no longer works, because a project may use capital in several years. The method is linear programming. Let xi be the fraction of project i undertaken (between 0 and 1). The objective is to maximise Σ NPVi × xi, subject to a constraint for each period: Σ capital required in period t × xi ≤ capital available in period t.
The syllabus asks you to formulate the programme, not to solve it.
Post-completion audit
A post-completion audit compares the actual cash flows of a project, after it ends or some years in, with the original appraisal, and asks the responsible manager to explain big variances. It cannot reverse the decision, but it has control value:
- managers who know a project will be audited take more care over costs and benefits;
- if done mid-life, it can lead to better efficiency or to closing a failing project;
- it shows who forecasts well and who does not, and where forecasting methods are weak;
- it helps improve how capital projects are chosen and run in general.
Problems:
- it is hard to separate a project’s costs and benefits from the rest of the business (the disentanglement problem);
- projects may be unique, so the lessons may not apply again;
- it is costly, so firms audit only some projects;
- only accepted projects are audited, often only the poor ones, which biases the picture;
- an inquisitorial review reduces co-operation, so the review team should be balanced and impartial;
- analysts may put forward only safe projects, which encourages risk aversion;
- changes in the environment can move the goalposts.
The rules in more depth
Why rank by PI and not by NPV
The scarce resource is capital. The best plan gets the most NPV from each naira of that capital.
Why PI can fail for indivisible projects
Leftover cash cannot be used for part of a project, so a lower-ranked combination may use capital more completely.
Why unspent capital has a zero NPV
It is simply kept or not borrowed. It adds nothing to the plan’s value.
Why multi-period problems need LP
Each project draws on several periods’ limits at once, so the choices are linked and a single ranking cannot capture them.
Effect on analysis
State the type of rationing, the method, the plan and the total NPV, and comment on the assumptions of divisibility and similar risk.
Worked examples
Example 1: divisible projects. Capital ₦10m at time 0. A costs ₦4m (NPV ₦1.7m); B costs ₦3m (₦1.5m); C costs ₦5m (₦2.0m); D costs ₦2m (₦0.9m).
- PIs: B 0.50, D 0.45, A 0.425, C 0.40.
- Fund B (3), D (2), A (4) = ₦9m, then ₦1m of C (one-fifth).
- NPV = 1.5 + 0.9 + 1.7 + 0.4 = ₦4.5m.
Example 2: indivisible projects. The same projects cannot be split.
- B + C + D.
₦10m, NPV = 1.5 + 2.0 + 0.9 = ₦4.4m. - A + B + D.
₦9m, NPV = 4.1
A + C = ₦9m, NPV = 3.7. - Choose B, C and D. The divisible plan (₦4.5m) is higher, as it should be.
Example 3: a must-do project. In Example 1, D must be undertaken for strategic reasons.
- Fund D first: ₦2m, NPV ₦0.9m. ₦8m remains.
- Rank the rest: B (3), A (4), then ₦1m of C.
- NPV = 0.9 + 1.5 + 1.7 + 0.4 = ₦4.5m (it matches because D was already in the best plan).
Example 4: a linear programme. Capital is ₦20m at time 0 and ₦3m at time 1. Projects A to D need ₦8m, ₦5m, ₦6m and ₦10m at time 0 and ₦0, ₦1m, ₦0 and ₦2m at time 1, with NPVs ₦3.0m, ₦2.0m, ₦2.5m and ₦4.2m.
- Maximise 3.0A + 2.0B + 2.5C + 4.2D.
- Subject to 8A + 5B + 6C + 10D ≤ 20 and B + 2D ≤ 3.
- With 0 ≤ A, B, C, D ≤ 1.
How to score full marks
- Identify the type of rationing and the periods involved.
- Compute the PI for each project in the rationed period.
- Rank and allocate, showing the fraction of the last project.
- For indivisible projects, show every combination you tested.
- Handle mutually exclusive and must-do projects first.
- State the total NPV and the assumptions.
Common mistakes in the exam
- Ranking by NPV, not by NPV per naira of capital.
- Using PI for indivisible projects without testing combinations.
- Dividing NPV by total capital over all years, not the rationed period.
- Forgetting that unspent cash earns nothing.
- Funding both of two mutually exclusive projects.
- Leaving out a must-do project.
Quick revision
- Hard rationing is external; soft is a management limit.
- Divisible: PI = NPV ÷ capital; fund in rank order.
- Indivisible: test combinations.
- Mutually exclusive: not both. Must-do: fund first.
- Multi-period: linear programming, objective and constraints.
- Post-completion audit: control and forecasting, with cost and bias problems.
- CAMA 2020: ss. 305, 308, 377(1), 378, 385.
Practice questions
These are original questions written in the style of the Institute of Chartered Accountants of Nigeria (ICAN) exam. Try each one before you open the answer. Full model answers follow the written questions.
Recall check: multiple choice
Recent ICAN papers are written, although older papers such as May 2014 included multiple-choice questions. These questions are for quick recall. The written questions that follow are closer to the exam.
Question 1. Capital rationing means:
- Capital is unlimited
- Funds are too few for all positive-NPV projects
- NPV is negative
- Only debt is used
Show the answer
B. A choice among projects is needed.
Question 2. Hard capital rationing is caused by:
- A management policy
- A real external shortage of capital
- High tax
- Inflation
Show the answer
B. Soft rationing is a policy limit.
Question 3. The profitability index is:
- NPV ÷ capital invested
- Capital ÷ NPV
- NPV × capital
- IRR ÷ cost of capital
Show the answer
A. In the rationed period.
Question 4. For divisible projects, fund projects in order of:
- Highest NPV
- Highest PI
- Lowest cost
- Longest life
Show the answer
B. Until capital runs out.
Question 5. For indivisible projects you should:
- Rank by PI only
- Test combinations of projects
- Fund all
- Take the cheapest
Show the answer
B. Ranking may fail.
Question 6. Total NPV with divisible projects, compared with indivisible, is:
- Never higher
- At least as high
- Always lower
- Unrelated
Show the answer
B. Fractions allow fuller use of capital.
Question 7. Multi-period rationing is solved by:
- Linear programming
- The payback rule
- The accounting rate of return
- Ranking by NPV
Show the answer
A. Objective plus constraints.
Question 8. A post-completion audit compares:
- Actual results with the appraisal forecast
- Two projects
- Tax with profit
- Debt with equity
Show the answer
A. It cannot reverse the decision.
Question 9. A problem of post-audits is:
- They are free
- Disentangling project costs from the business
- They always improve forecasts
- They remove risk
Show the answer
B. Among others.
Question 10. Which section of CAMA 2020 sets the director’s duty of care and skill?
- Section 308
- Section 400
- Section 405
- Section 409
Show the answer
A. Directors should test the assumptions behind a decision.
Written questions
Question 11 (10 marks). (a) Distinguish hard and soft capital rationing. (3 marks) (b) Explain how to choose projects when they are divisible and capital is limited in one period. (4 marks) (c) State why the answer may differ if the projects are indivisible. (3 marks)
Question 12 (20 marks). Ukpe Ltd has ₦10m at time 0. A costs ₦4m (NPV ₦1.7m), B ₦3m (₦1.5m), C ₦5m (₦2.0m), D ₦2m (₦0.9m). (a) Calculate the profitability indices and the best plan and NPV if projects are divisible. (7 marks) (b) Find the best plan and NPV if projects are indivisible. (6 marks) (c) Find the plan if D must be undertaken and the projects are divisible. (3 marks) (d) State two assumptions of the method. (4 marks)
Question 13 (10 marks). (a) Formulate, without solving, a linear programme for a company with ₦20m at time 0 and ₦3m at time 1 and projects A to D needing ₦8m, ₦5m, ₦6m and ₦10m at time 0 and ₦0, ₦1m, ₦0 and ₦2m at time 1, with NPVs ₦3.0m, ₦2.0m, ₦2.5m and ₦4.2m. (5 marks) (b) State three benefits and two problems of post-completion audits. (5 marks)
Model answers
Show the model answer to Question 11
(a) Hard and soft (3 marks). Hard rationing is a real shortage of capital, for example when new capital cannot be raised. (1.5) Soft rationing is a limit set by management policy. (1.5)
(b) Divisible projects (4 marks). Calculate the profitability index (NPV per naira of capital in the rationed year). (1.5) Rank the projects by PI and fund them in order. (1.5) Take the fraction of the last project that remaining capital allows. (1)
(c) Indivisible projects (3 marks). A project must be done fully or not at all, so ranking may leave capital unused. (1.5) Combinations must be tested to find the highest total NPV. (1.5)
Show the model answer to Question 12
(a) Divisible (7 marks).
- PIs.
B 1.5 ÷ 3 = 0.50 - D 0.9 ÷ 2 = 0.45
- A 1.7 ÷ 4 = 0.425
- C 2.0 ÷ 5 = 0.40. (2)
- Fund B, D and A (₦9m), then one-fifth of C with the last ₦1m. (2)
- NPV = 1.5 + 0.9 + 1.7 + 0.4 = ₦4.5m. (3)
(b) Indivisible (6 marks).
- Combinations.
B + C + D = ₦10m, NPV 4.4 - A + B + D = ₦9m, NPV 4.1
- A + C = ₦9m, NPV 3.7. (4)
- Choose B, C and D for NPV ₦4.4m. (2)
(c) D must be done (3 marks).
- Fund D (₦2m, NPV 0.9), then rank the rest with ₦8m: B (3), A (4) and ₦1m of C.
- NPV = 0.9 + 1.5 + 1.7 + 0.4 = ₦4.5m. (3)
(d) Assumptions (4 marks). The aim is to maximise total NPV. (1) The projects have similar risk. (1) Capital is limited in one period only, at time 0. (1) For (a), projects can be split with proportionate NPV. (1)
Show the model answer to Question 13
(a) Formulation (5 marks). Let A, B, C and D be the fractions undertaken. Maximise 3.0A + 2.0B + 2.5C + 4.2D. (2) Subject to 8A + 5B + 6C + 10D ≤ 20 at time 0 and B + 2D ≤ 3 at time 1. (2) With 0 ≤ A, B, C, D ≤ 1. (1)
(b) Post-audits (5 marks). Benefits: managers take more care knowing the project will be reviewed; weak forecasting is found; poor projects can be improved or stopped. Problems: costs and benefits are hard to separate; the audit is costly and biased towards poor projects. Any three benefits and two problems. (1 each)
Pathfinder on Capital rationing, profitability index and linear programming
These are the ICAN Pathfinder questions that test this topic. ICAN owns the questions and the suggested solutions, so we point you to them and do not copy them. Open your Pathfinder, attempt each question under exam conditions, then check your answer against the notes above.
Neither the May 2014 nor the November 2014 Pathfinder sets a stand-alone question on this topic. Capital rationing appears in investment appraisal questions. Worked examples 1 to 4 show how to apply the method.
Revise first: Worked examples
Questions ask for the best plan and total NPV when capital is limited. See the section on divisible projects.
Revise first: Single period, divisible projects
Source: ICAN Pathfinder, Strategic Financial Management. Diets covered: No stand-alone question in the May 2014 or November 2014 Pathfinder.
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