Capital rationing

Capital rationing exists when a company has more positive-NPV projects than it has money to fund. The appraisal question changes shape: it is no longer whether each project is worth doing, but which combination of them squeezes the most value out of a fixed amount of capital.

Everything up to this point has said accept every project with a positive NPV. That rule quietly assumed the company could raise whatever the projects needed. Most cannot, and a Zambian firm working with a bank facility capped at ZMW5 million for the year is in exactly this position — three good projects and enough money for two of them.

So the decision becomes a ranking problem, and the trap is ranking on the wrong thing. The largest NPV is not automatically the best use of a constrained kwacha, because a project that adds ZMW22,000 while consuming the whole budget can be beaten by two smaller ones that add ZMW27,000 between them.

The shortage comes from one of two places, and where it comes from decides whether anyone can do anything about it.

Soft rationing is self-imposed, and often for reasons that are perfectly sensible: a family-owned business that will not issue shares to outsiders is choosing control over capital. It is still a choice, and it can be revisited if a project is good enough.

Hard rationing is imposed by the market. A small Zambian company that has already borrowed against everything it owns, in a year when the banks have tightened, faces a limit it cannot argue with.

The arithmetic that follows is the same in both cases. The distinction matters for what you write around it — under soft rationing the honest recommendation may be that the constraint itself should be reconsidered.

The profitability index measures how much present value each kwacha of outlay buys. Rank the projects by it, work down the list until the money runs out, and you have the combination with the highest total NPV — provided the projects are divisible.

An index above 1.00 says the same thing a positive NPV says. What it adds is comparability: a project scoring 1.30 returns ZMW1.30 of value for every kwacha it ties up, against ZMW1.22 from one scoring 1.22, whatever their sizes. The same ranking comes from dividing NPV by the outlay directly, which is the form most students find quicker.

Take four projects and ZMW200,000 to spend. Each can be part-funded, and a project taken at 40% returns 40% of its NPV.

Now spend the budget in rank order.

Rank instead by NPV alone and you would take C first, then A, then a third of B — for ZMW48,000. The same budget, ZMW3,400 less value, because the biggest project was not the most efficient one.

Most real projects are indivisible. You cannot build 20% of a depot or install a fifth of a mill and collect a fifth of the returns. Once fractions are off the table the profitability index stops being reliable, and you have to test the combinations that fit.

Take the same four projects and the same ZMW200,000, but now each is all or nothing.

B + C + D wins with ZMW49,000. The index ranking would have chosen B, then A, then D, leaving ZMW20,000 that could not buy anything and ZMW2,000 of value on the table. It went wrong because it took A — efficient per kwacha, but the wrong size to fit alongside the others.

So with indivisible projects the index is a starting point, not the answer. Use it to see roughly where the value is, then list the feasible combinations and total them. Note also that using the whole budget is not the objective in itself — leftover cash is only a problem when it could have bought more NPV.

When capital is limited in more than one period, listing combinations stops being practical. Each project draws on several budgets at once, and a combination that fits in year 1 can breach year 2. This is what linear programming is for.

The board of Bazza Inc has approved ZMW16,000 of investment in year 1, ZMW14,000 in year 2 and ZMW17,000 in year 3. Four projects are available.

Write the problem down as an objective and a set of constraints. Let Y₁ to Y₄ stand for the four projects, each taking the value 1 if the project is undertaken and 0 if it is not. The objective is the total NPV, and it is to be made as large as possible.

Then one constraint per rationed year, each saying that what the chosen projects consume in that year cannot exceed what the board has approved.

Solved, the answer is Y₁ = 1, Y₂ = 1, Y₃ = 0, Y₄ = 0: take projects 1 and 2 in full, leave 3 and 4. Total NPV ZMW19,000. Check it against the constraints — year 1 uses the full ZMW16,000, year 2 uses ZMW10,000 of the ZMW14,000, year 3 uses ZMW16,000 of the ZMW17,000.

Projects 3 and 4 together also fit comfortably, and would be a perfectly valid answer to the question “what can we afford?”. They are worth ZMW10,000. That is the difference between feasible and optimal, and it is the reason the objective function is written down at all.

It is worth seeing why the two rejected projects cannot simply be squeezed into the money left over. Project 4 needs ZMW5,000 in year 1, and year 1 is fully spent. Project 3 needs nothing in year 1, but wants ZMW6,000 in year 2 against ZMW4,000 spare and ZMW8,000 in year 3 against ZMW1,000. A different year blocks each of them, which is why all three constraints have to be written down rather than just the tightest-looking one.