Discounted Cash Flow – Further Aspects
1 Introduction
This chapter deals with four further aspects of investment appraisal: capital rationing, replacement decisions, the qualitative factors that no discounted cash flow can carry, and real options.
The first two are calculations, and both are worth marks. The third is the reason a positive net present value is a recommendation rather than an instruction. The fourth changes the shape of the appraisal itself: it drops the assumption, made everywhere in chapters 10 to 12, that once a project starts nothing about it can be changed.
2 Capital rationing
This recording works Example 1 end to end. It is correct throughout and it adds one useful point: the reason a manager might expect the answer to be "project B" and why that is wrong — "if you have four projects available, you'll do every project that gives a positive NPV".
Notice one. The tutor says the hard/soft distinction "isn't mentioned in the notes, I don't think". It is — it opens section 2.1. Nothing turns on it; the explanation he gives agrees with the notes.
Notice two. He works in dollars for the projects and in pounds for the personal car-loan analogy. The paper is in dollars.
Capital rationing describes the situation in which a company has only a limited amount of funds available for investment, so that it cannot undertake every project with a positive net present value.
2.1 Hard and soft capital rationing
Who imposes the limit | What it means in practice | |
Hard capital rationing | External providers of finance — lenders will not lend more and shareholders will not subscribe more. | A constraint to be worked within. The causes are usually outside the company: a lack of a track record, restrictive covenants on existing debt, or conditions in the capital markets. |
Soft capital rationing | The company itself, even though more finance could be raised. | A management decision, and therefore one that can be argued with. Reasons include a policy on gearing, unwillingness to dilute existing shareholders, a shortage of management capacity to run more projects at once, or a deliberate wish to force project sponsors to compete. |
The distinction does not change any of the arithmetic below — the selection method is the same either way. It changes the advice: under soft rationing, the recommendation may be to relax the limit.
2.2 Divisible projects and the profitability index
A project is divisible if the company can invest in part of it rather than all of it. We assume throughout that a 10% investment gives 10% of the outlay, 10% of the returns and therefore 10% of the net present value, and that the most that can be taken of any one project is 100%.
Where projects are divisible the selection is made by ranking on the profitability index — the net present value generated per $1 of the scarce resource:
Profitability index = Net present value ÷ Initial investment
Then invest as much as possible in the project with the highest index, then the next highest, and so on until the capital available is exhausted.
You will meet a second definition of the profitability index — present value of the future cash inflows divided by the initial investment. It gives a figure exactly 1.0 higher than the version above, because the numerator includes the outlay instead of netting it off, so the two rank projects identically. Use whichever the question gives you, and say which you are using.
2.3 Indivisible projects
If projects are indivisible the company must take all of a project or none of it. There is no shortcut: list every combination that the available capital will fund, total the net present values of each, and choose the highest.
Ranking on the profitability index does not work for indivisible projects, and it is the standard trap. Its answer will often be right by accident, but it is not the method: the index tells you the best use of the marginal dollar, and with indivisible projects the choice is between whole bundles. List the combinations.
Capital rationing
A company has the following four projects available. All figures are in $000.
Time | A | B | C | D |
0 | (500) | (600) | (300) | (400) |
1 | 221 | 207 | 194 | 181 |
2 | 221 | 207 | 194 | 181 |
3 | 221 | 207 | – | 181 |
4 | – | 207 | – | – |
NPV @ 10% | 50 | 57 | 36 | 50 |
Required
What should the company's investment decision be if:
there is no capital rationing;
capital is restricted to $1,600 at time 0 and the projects are divisible;
capital is restricted to $1,600 at time 0 and the projects are indivisible?
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Rationing across more than one period, where the constraint binds in several years at once, is solved by linear programming and is not examinable at P2.
3 Replacement decisions
This recording covers section 3 and works Example 2 — the one-, two- and three-year replacement cycles — and closes with section 4's qualitative factors.
Notice one. Check one figure against the answer below: the one-year equivalent annual cost is $65,959 (57,384 ÷ 0.870). The two- and three-year figures are $44,878 and $38,152, so the machine is replaced every three years.
Notice two. Fix two things as you watch: the one-year cycle starts with the same $72,000 machine as the others, and the cheapest cycle is three years. Where you hear pounds, read dollars.
Many assets have to be replaced periodically rather than used until they stop working. The longer an asset is kept, two things happen: running and maintenance costs rise, and the proceeds on eventual sale fall. Replacing more often avoids both — but means paying for a new asset more frequently.
The question is therefore not whether to buy but how often to replace, and the objective is the optimal replacement cycle: the cycle with the lowest cost, given that the asset will be needed indefinitely.
Because the asset is needed indefinitely, the cost of a two-year cycle cannot be compared directly with the cost of a three-year cycle — one repeats every two years and the other every three. They are made comparable by converting each into an equivalent annual cost:
Equivalent annual cost = PV of the costs of one replacement cycle ÷ Annuity factor for the length of that cycle
Two steps, every time
Step 1. Calculate the present value of one complete cycle — the purchase cost, the running costs, and the sale proceeds at the end of the cycle.
Step 2. Divide that present value by the annuity factor for the length of the cycle. The result is the annual cost of running the asset forever on that cycle, and the cycles can now be compared directly.
Operating revenues are ignored in a replacement calculation. The asset earns the same money whichever cycle is chosen, so only the costs differ — which is why the answer is the lowest equivalent annual cost rather than the highest net present value.
Optimal replacement cycle
A machine costs $72,000 and has a maximum life of 3 years.
The running costs each year are as follows:
Year | Running cost $ |
1 | 7,200 |
2 | 9,600 |
3 | 12,000 |
The estimated scrap values are as follows:
Sold after | Scrap value $ |
1 year | 24,000 |
2 years | 16,600 |
3 years | 9,600 |
The cost of capital is 15%.
Required
How often should the machine be replaced?
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4 Qualitative factors in project selection
Every appraisal in chapters 10 to 13 reaches its conclusion on financial grounds alone: given the projected cash flows and the required rate of return, the project is or is not viable. That is a statement about the numbers, not about the decision.
Decision-makers must also weigh factors that cannot be expressed in financial terms, and which can be as important as the numbers in determining whether a project actually succeeds:
compatibility with existing operations and systems;
the reaction and preferences of the workforce, and the effect on morale;
the quality and reliability of the product or service, and of the equipment itself;
supplier warranties, service levels and the risk of dependence on a single supplier;
environmental and social consequences, and the reputational effect of them;
the effect on customers, and on the organisation's ability to serve them during the change;
whether the organisation has the management capacity to run the project alongside everything else it is doing.
The syllabus lists qualitative issues and non-financial information among the data required for decision-making. Where a question asks you to advise rather than to calculate, the qualitative half of the answer is where the marks that separate candidates are.
5 Real options
This recording is a short introduction to sections 5.1 and 5.2: what a real option is, and the types. The 2027 outcome is a decision — “decision to make follow-on investment, abandon or wait” — and section 5.4 works that decision with numbers.
Notice one. Where the recording says real options involve no calculations at this level, P2 does expect the decision with numbers: Example 3 values the option to abandon and a follow-on investment by comparing the appraisal with the option against the appraisal without it (section 5.4). Option-pricing models are beyond P2.
Notice two. It describes four types: delay, abandon, switch, and expand or contract. The three the syllabus names are follow-on investment, abandon and wait; the follow-on option is not on the tape, and section 5.2 adds it. Where it groups the option to switch with the puts, section 5.2 treats it as an exchange of one use for another rather than a simple put.
5.1 What a real option is
Every appraisal so far has assumed that a project is taken or refused as a whole: pay the cost, receive the forecast returns, run it for the stated life. Real life is rarely like that. A project that turns out badly can usually be stopped early; one that turns out well can usually be extended. Those choices have value.
Real option
The right, but not the obligation, to take some future action in relation to a real (non-financial) asset or project — to delay it, abandon it, switch it, expand it or follow it with a further investment.
Because it is a right and not an obligation, its holder uses it only when it is beneficial. That asymmetry — the downside can be walked away from, the upside cannot — is why an option always has a value greater than or equal to zero, and why it may be worth paying for.
5.2 Types of real option
Type | What it is | Illustration | Option type |
Option to delay (wait) | The right to postpone the start of a project until more is known. | A project's returns depend on the outcome of an election in six months. Waiting removes the risk of committing to the wrong answer. The supplier may charge for holding the price — and that fee is what the option costs. | Call |
Option to abandon | The right to stop a project part-way through and recover whatever the assets will realise. | A five-year project is running badly after two years. Rather than complete it, sell the equipment and stop. | Put |
Option to switch or redeploy | The right to move the assets to a different use if market conditions change. | A college equips itself to teach one qualification and, when demand moves, uses the same rooms, staff and systems to teach another. | An exchange of one use for another — not a simple put |
Option to expand or contract | The right to scale the project up or down after it has started. | Production was planned at 20,000 units a year; demand proves stronger, so a further line is added. | Expand = call; contract = put |
Option to make a follow-on investment | The right — created by undertaking a first project — to make a larger second investment that would not otherwise have been available. | A pilot in one region establishes the platform, the data and the credibility for a national roll-out. The pilot may not be worth doing on its own; the roll-out it makes possible may make it worth doing several times over. | Call |
The syllabus names three of these explicitly — "decision to make follow-on investment, abandon or wait" — and the follow-on option is the one that most often changes an answer, because it is the one whose value is not visible anywhere in the first project's own cash flows.
5.3 What makes a real option valuable
Uncertainty.
An option's value comes from being able to respond to information that has not arrived yet. Holding the other inputs to the valuation constant, the wider the range of possible outcomes, the more the right to choose later is worth. Two cautions go with that. If the outcome is already certain, waiting adds nothing — but the right itself can still be valuable: a certain right to buy for $100 an asset certain to be worth $120 is worth $20, not nothing. And greater uncertainty raises the value of the flexibility, not the value of the underlying project; it is not a rule that any increase in uncertainty makes a project better.
Time.
A longer exercise window — one that keeps the earlier opportunities to choose, rather than merely pushing a single fixed date back — means more information has arrived before the decision has to be taken. Moving a single fixed exercise date later is not automatically worth more: a fixed pay-off is then discounted for longer, and that can outweigh the extra information.
The size of the asymmetry.
The option's value comes from the difference between what happens if it is exercised and what happens if it is not — the abandonment proceeds against the value of continuing, or the follow-on NPV against zero.
The cost of keeping it open.
Options are rarely free. A supplier may charge to hold a price; a modular design may cost more than a purpose-built one; a pilot costs money that a full roll-out would not have spent twice. The option is worth having only if it is worth more than it costs to keep.
5.4 Valuing the decision
A full valuation of a real option uses option-pricing models and is beyond P2. What is examinable is the decision: recognising that the option exists, and showing what it is worth by comparing the appraisal with the option against the appraisal without it.
Real options and the decision
Part A — the option to abandon
Kaunas Co is considering a project requiring an immediate outlay of $450,000 and expected to run for four years. The outcome depends on how the market develops, and this will be known at the end of year 1. At that point:
if the market is favourable (probability 0.6), the present value at time 1 of the remaining cash flows will be $700,000;
if the market is unfavourable (probability 0.4), the present value at time 1 of the remaining cash flows will be $150,000.
The equipment could be sold at the end of year 1 for $300,000.
The cost of capital is 10%. The one-year discount factor at 10% is 0.909.
Required
Calculate the expected net present value on the assumption that, once started, the project must be run for the full four years.
Calculate the expected net present value on the assumption that the project can be abandoned at the end of year 1, and advise whether the project should be undertaken.
Calculate the value of the option to abandon.
Part B — the follow-on investment
Klaipeda Co is considering a pilot scheme costing $120,000 now, which on its own has a net present value of $(30,000).
If the pilot succeeds — the directors assess the probability at 0.45 — the company will be able to make a much larger follow-on investment at the end of year 2, whose net present value measured at that date would be $400,000. If the pilot fails, the follow-on investment will not be made and nothing further is spent.
The cost of capital is 10%. The two-year discount factor at 10% is 0.826.
Required
Advise whether the pilot scheme should go ahead.
Comment on the limitations of the analysis in Part A and Part B.
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5.5 Why this matters to the appraisal
Two conclusions follow, and both are examinable as discussion:
A conventional NPV understates a flexible project.
The standard calculation assumes the project runs to plan for its full life. Where genuine flexibility exists, the calculation is a floor, not an estimate — and a project rejected on a conventional NPV may be worth undertaking once the options are recognised.
Flexibility is worth designing in.
If the option to abandon is worth $54,540, it is worth up to $54,540 to build the project so that abandonment is possible — modular equipment, shorter contracts, assets with a resale market. That is a design decision taken before the appraisal, not after it.
Chapter 9 applies both conclusions to the case where they bite hardest: digital transformation projects, which are routinely delivered as a pilot followed by a staged roll-out precisely so that the abandon and follow-on options exist.
Under capital rationing, rank divisible projects by NPV per $ of the scarce resource and take them in order; for indivisible projects, list the affordable combinations and pick the highest total NPV. For replacement, find the present value of one cycle and divide by the annuity factor for that cycle's length — the lowest equivalent annual cost wins. Qualitative factors decide whether a viable project is a sensible one. Real options — delay, abandon, switch, expand or contract, and follow-on — are rights and not obligations, so the right itself is never worth less than zero — though keeping it open can cost money — and the examinable skill is showing the decision with and without the option.
6 Test your knowledge
Two quick checks before you move on: work through the flashcards to fix this chapter’s key terms and definitions, then sit the objective questions for exam-style practice. Both mark themselves and explain the answers as you go.
Discounted Cash Flow – Further Aspects
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