Risk and Uncertainty
1 Introduction
Budgets and decisions are about the future, and the future is not known. Section D of the P1 syllabus — the whole of which this chapter carries — asks you to apply basic risk-management tools in the short term: to explain what risk and uncertainty are, and to apply sensitivity analysis both to budgeting and to short-term decision-making.
Key decisions — a new product launch, a special contract, a change of capacity — have to be made now, on estimates, and the results are not known until later. Risk exists where the actual outcome of a decision may differ from the forecast one, so that results turn out different from those planned or hoped for.
The techniques in this chapter are quantitative and they lend themselves to objective test questions. They are also the foundation of the more advanced risk work in P2 and P3.
Two of this chapter's techniques belong to other chapters as well, and are deliberately not re-taught there:
Stress testing is a named P1D topic, but it is taught in Chapter 11 §8.4 in its budgeting setting, where it belongs. Section 8.4 below links the two and does not repeat it.
What-if analysis of a budget is worked in Chapter 11 §8.3, because preparing budgets is where the syllabus asks for it. Section 8.4 below gives the method from this chapter's side: the same arithmetic, aimed at a different target.
2 Risk and uncertainty
Fifty-five minutes and the spine of the chapter: attitudes to risk, risk against uncertainty, expected values and the decision rules, worked on parts (a) and (b) of Example 2 in section 6.1. Its payoff table and regret table agree with Answer 2: contract for 700 units on expected value ($4,500), 800 on maximin, 300 on maximax and 700 on minimax regret. Three points before you play it.
Keep one classification straight: maximax is the risk seeker’s rule and maximin the risk-averse decision-maker’s. Minimax regret is neither — it is the rule of the “sore loser”, who minimises the largest opportunity loss, and it is applied to the regret table (the pitfall in section 3).
Example numbers: it calls Example 2 “exercise two” and passes over Example 1, which is answered in section 4.
Part (c): the value of perfect information is the next recording, at section 6.
The two words are used interchangeably in ordinary speech, but there is a technical difference and the syllabus expects you to make it.
Risk and uncertainty
Risk exists where the possible outcomes of a decision are known and a probability can be attached to each of them. The exposure can therefore be quantified. We do not know what a single roll of a dice will give, but we know it must be one of six outcomes and we know the probability of each.
Uncertainty exists where the possible outcomes are not well defined and no reliable probabilities can be attached to them. Probabilities cannot then be used as a basis for prediction.
In practice most short-term business problems are somewhere between the two. Estimates of demand, selling price, material prices and machine downtime are rarely as clean as a dice, and where probabilities are used they are themselves estimates. That is worth remembering every time a technique in this chapter produces a confident-looking number.
The risks and uncertainties that matter in the short term are the ones that can move inside a single budget period: the level of demand, the price a customer will pay, the price of a bought-in material, the availability of a scarce resource, the reliability of a supplier, and the accuracy of the estimates in the budget itself.
3 Attitudes to risk
There is no single correct way to make a decision under risk. Which criterion is appropriate depends on the amounts of money involved and on the decision-maker's attitude to risk. In an examination you are never asked which criterion is best; you are told which one to apply, or asked what each one would choose.
Risk profile | What the decision-maker looks at | Matching decision rule (section 5) |
Risk seeker | The best possible outcome, however small its chance. An optimistic attitude, which is reckless if the likelihood of outcomes is ignored altogether. | Maximax |
Risk neutral | The most likely, or average, outcome. | Expected value |
Risk averse | The worst possible outcome. A pessimistic attitude: options are chosen on the basis of what happens if things go badly. | Maximin |
Minimax regret is not the risk seeker's rule.
Maximax is the risk seeker's rule and maximin is the risk-averse decision-maker's rule. Minimax regret belongs to neither: it is the rule of a decision-maker who is chiefly concerned not to look foolish afterwards — the “sore loser” — and who therefore minimises the largest opportunity loss they could suffer. It is worked out from a regret table, not from the profit table, which is what makes it different from the other three.
4 Expected values
An expected value can be calculated where a decision has several possible outcomes and the probability of each is known. It is the weighted average of the outcomes, weighted by their probabilities.
EV = Σpx
where p is the probability of an outcome and x is the value of that outcome. The expected value represents a long-run average result: the figure the decision-maker would expect if the same decision were repeated many times.
The outcome of a new venture has been forecast below.
Probability of $50,000 profit = 0.3
Probability of ($20,000) loss = 0.7
What is the expected value of this project?
Should the decision maker go ahead with the venture?
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4.1 Limitations of the expected value method
Expected value is normally the approach of a risk-neutral decision-maker. It has serious limitations, and a question asking you to comment on an expected value is asking for these:
It is a long-run average applied to a one-off event. The expected value is what you would get on average if the decision were repeated. Most business decisions are taken once.
The expected value may not be a possible outcome at all. In Example 1 no outcome of $1,000 exists.
It depends entirely on the probabilities, which are themselves estimates. A small change in a probability can change the decision, and in practice nobody can be certain the probabilities are right.
It ignores the spread of the outcomes. Two projects with the same expected value can carry very different risk — see section 9 on standard deviation.
It ignores the decision-maker's attitude to risk and the absolute size of the sums involved. A possible loss that would be an inconvenience to one organisation would close another.
5 Decision rules
Given a range of possible outcomes — usually profits or payoffs — an examination question may ask you to identify the decision that would be taken under each of the criteria below. Each is applied to a payoff table: a table whose rows are the courses of action open to the decision-maker and whose columns are the uncertain states of the world.
Maximax — the choice of an optimist, who prefers the option with the best possible return regardless of how likely it is. Identify the best outcome of each option; choose the option with the best of those.
Maximin — the choice of a pessimist, who considers the worst result of every option and seeks to make that worst result as good as possible. Identify the worst outcome of each option; choose the option with the best of those.
Minimax regret — the choice of a “sore loser”, who seeks to minimise the maximum regret. Regret is the profit given up by not having chosen the option that turned out best for the state of the world that actually occurred.
5.1 Building a regret table
Minimax regret needs a second table, built from the payoff table one column at a time:
Take one column of the payoff table — that is, assume one state of the world actually happens.
Find the highest payoff in that column. That is what the decision-maker could have earned with the benefit of hindsight.
The regret in every cell of the column is that highest payoff less the payoff actually earned. The best cell in each column therefore has a regret of nil.
Repeat for every column, then take the largest regret in each row and choose the row whose largest regret is the smallest.
6 The value of perfect information
Fourteen minutes on part (c) of Example 2 alone, and every step of it is right: the four best-decision payoffs, an expected value with perfect information of $4,800, and a limit of $300 on what the market research can be worth. It is worth the time for two passages in particular — why the calculation is hard to get right even though the arithmetic is trivial, and why $300 is a limit rather than a forecast of what you will gain. One cosmetic point: the money drifts between £ and $ while the question, like all of P1, is in dollars. The arithmetic is unaffected.
An organisation facing an uncertain outcome can sometimes pay to remove the uncertainty — market research, a survey, a test market. The question is how much such information is worth.
Value of perfect information
The maximum worth paying for information that would remove the uncertainty entirely is the difference between the expected value of the decision made with that information and the expected value of the best decision made without it.
The method has three steps, and the arithmetic is easy provided the logic is straight:
For each possible state of the world, ask what the best decision would be if the decision-maker knew in advance that this state would occur, and record the payoff that decision would give.
The probability of receiving each piece of news is the probability of the state itself, because the information is perfect. Multiply and add to get the expected value with perfect information.
Deduct the expected value of the best decision that would be taken without the information. The difference is the most that should be paid.
Notice that the answer is a limit, not a forecast of what will happen. The organisation will actually end up with one of the individual payoffs, not with the expected value; and the fee for the research is paid whatever the news turns out to be.
Perfect and imperfect information. Information is described as perfect when it is assumed to be right — if the research says demand will be 400 units, demand will be 400 units. In reality research is rarely perfect: it shifts the probabilities without removing the uncertainty. That case is dealt with in section 7 on decision trees, and it is worth less than perfect information.
6.1 Worked example — all four decision rules and perfect information
John has a factory capacity of 1,200 units per month.
Units cost him $6 each to make and his normal selling price is $11 each. However, the demand per month is uncertain and is as follows:
Demand (units) | Probability |
400 | 0.2 |
500 | 0.3 |
700 | 0.4 |
900 | 0.1 |
He has been approached by a customer who is prepared to contract to a fixed quantity per month at a price of $9 per unit. The customer is prepared to sign a contract to purchase 300, 500, 700 or 800 units per month.
The company can vary production levels during the month up to the maximum capacity, but cannot carry forward any unsold units in inventory.
(a) Calculate all possible profits that could result from the various demand levels.
(b) Determine for what quantity John should sign the contract, under each of the following criteria:
(i) expected value (ii) maximin (iii) maximax (iv) minimax regret
(c) What is the most that John would be prepared to pay in order to obtain perfect knowledge as to the level of demand?
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7 Decision trees
Forty minutes on the decision tree in Example 3, and all eleven of its figures are correct, finishing at 7.43 for commissioning the market research against 7.17 for refurbishing immediately. Watch how it is built as much as what it contains: the tree is drawn before a single number is written on it, which is the draw-forward, evaluate-backwards method of §7.1, and the closing point is that the answer is the decision now together with the decisions that follow it. Two points on presentation. The recording letters the decision points and numbers the chance points; Answer 3 below follows the recording, so the video and this chapter agree with each other. And the money moves between £ and $ where the question is in $m.
A decision tree is a diagram of the alternatives open to a decision-maker and the outcomes that may follow each of them. It is used with the expected value approach, and it earns its place when there are several decisions to be made in sequence — where the later decision depends on how an earlier uncertainty resolves. In that situation it is very easy to lose track of what is being compared with what, and the diagram makes the options understandable.
7.1 Convention and method
A square marks a decision point: a branching where we choose.
A circle marks a chance point: a branching where the outcome is uncertain and each branch carries a probability. The probabilities out of any one circle must sum to 1.
The branches carry the cash flows, shown as they arise; the payoffs sit at the ends.
The tree is drawn forward — from today's decision out to the final outcomes — and then evaluated backwards, which is called rolling back:
At every circle, replace the uncertainty with its expected value.
At every square, choose the branch with the highest net value and carry that value back. The branches not chosen are struck out.
Work leftwards until the first decision point is reached. The answer is the decision to be taken now, together with the decisions that would be taken later depending on what happens.
Combi plc are having problems with one of their offices and have decided that there are three courses of action available to them:
(a) shut down the office, raising proceeds of $5 million
(b) have an expensive refurbishment of the office costing $4,000,000
(c) have a cheaper refurbishment of the office at a cost of $2,000,000
If they do the expensive refurbishment, then a good result will yield a return of $13,500,000 whereas a poor result will yield a return of only $6,500,000.
If they alternatively decide to do the cheaper refurbishment, then a good result will yield a return of $8,500,000 whereas a poor result will yield $4,000,000.
In either case, the probability of the refurbishment achieving a good result has been estimated to be 2/3.
An independent company has offered to undertake market research for them in order to identify in advance whether the result of refurbishment is likely to be good or poor. The research will cost $200,000 and there is a 68% probability that it will indicate a good result.
Unfortunately, the research cannot be guaranteed to be accurate. However, if the research indicates a good result, then the probability of the actual result being good is 91%.
If the survey indicates a poor result, then the probability of the actual result being good is 13%.
Combi have already decided that if they do have market research, and if the research indicates a poor result, then they will only be prepared to consider the cheaper refurbishment.
Use a decision tree to recommend what actions should be taken.
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Imperfect knowledge. The market research in Example 3 is not guaranteed to be accurate — it is an example of imperfect knowledge, which is what is normally available in real life. Compare it with section 6: perfect knowledge in Example 2 was worth $300 because it removed the uncertainty; the research here is worth only $0.46m before its fee because it merely shifts the probabilities from 2/3 towards 0.91 or 0.13.
8 Sensitivity analysis
The best recording in the chapter, and thirteen minutes that add three things worth knowing: it gives each sensitivity a sign — −16.7% on revenue, −25% on volume, +50% on variable costs, +33% on fixed costs — it explains why sales volume is measured against contribution and not revenue, which is the single most-missed point in the topic, and it is honest about the one-variable-at-a-time limitation. It also carries the standard deviation and coefficient of variation working, and all of that is correct too. Two small things. The revenue sensitivity is 16.7% to one decimal place, which is the same number as the 16.67% stated on tape, more roughly rounded. And if you hear 20 × 0.3 called 0.6 in the coefficient of variation working, it is 6, and the recording puts it right in the same breath.
Expected values, decision rules and decision trees all need probabilities. Sensitivity analysis needs none. It is used where an outcome depends on a number of uncertain input variables — a new product's demand, selling price, variable cost and fixed cost, for instance — and it asks a different question: by how much would this estimate have to be wrong before the decision changed?
A decision to go ahead is worthwhile as long as the result is a profit. So the sensitivity of any one variable is the percentage change in that variable which would reduce the profit to zero, with every other variable held at its estimate.
Sensitivity (%) = Profit ÷ Value of the variable under test × 100
The denominator is the figure through which a change in that variable feeds into profit. Get it right and the rest is arithmetic:
Variable under test | Denominator | Because |
Selling price (sales revenue) | Total sales revenue | A price change alters revenue only; variable costs are unaffected. |
Sales volume | Total contribution | A volume change alters revenue and variable cost together, so it is contribution that moves. |
Variable cost per unit (total variable costs) | Total variable costs | A cost change alters variable cost only. |
Fixed costs | Total fixed costs | A change in fixed costs feeds straight into profit. |
The smallest percentage identifies the most critical variable — the estimate that most needs to be right, and therefore the one on which further research is worth commissioning. It is good practice to give each sensitivity a sign, because a variable is only a threat in one direction.
Harry is considering a new business opportunity. Based on his current estimates the opportunity looks profitable.
His forecast sales revenue is $30,000 per year, based on 1,000 units sold.
His accountant has helped him to estimate fixed costs of $15,000 per annum.
The variable costs are likely to be $10 per unit.
(a) Confirm, on the basis of the above figures, that the new opportunity is worthwhile.
(b) Calculate the sensitivity to change of:
(i) sales revenue (selling price) (ii) sales volume (iii) total variable costs (iv) fixed costs
(c) Comment on the results.
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8.1 Reading a sensitivity
A small percentage means the estimate is critical and the project is risky in that respect; a large percentage means the decision is robust to error in that variable. A sensitivity of 3% and a sensitivity of 60% call for very different management responses, even though the calculation behind them is identical.
8.2 What-if analysis
What-if analysis is the same idea run the other way round. Instead of asking how far a variable could move before the answer changed, it fixes a change — “what if demand is 10% below budget?”, “what if the material price rises by $2 a kilo?” — and recalculates the outcome. A spreadsheet or a planning model makes this cheap, so a set of what-if runs can be produced for several variables at once, which single-variable sensitivity analysis cannot do.
8.3 Limitations
One variable at a time. Each sensitivity assumes every other estimate is correct. In reality several will be wrong together, and a much smaller change in each would then be enough to remove the profit.
It ignores interdependence. The variables are not independent: a lower selling price would normally raise sales volume, and higher volume may raise variable cost per unit or step up fixed costs.
It says nothing about likelihood. Sensitivity analysis tells you how far a variable could move, not how likely it is to move that far. A 10% sensitivity on a stable variable may be less worrying than a 40% sensitivity on a volatile one.
It gives no decision rule. There is no percentage above which a project is “safe”. The figures inform judgement; they do not replace it.
8.4 Sensitivity analysis applied to a budget
The syllabus requires basic sensitivity analysis to be applied to budgeting as well as to short-term decisions, and the arithmetic does not change — only what is being flexed and what is being protected.
In a decision the target is the point at which profit becomes nil. In a budget the target is usually a different one: the budgeted profit, a covenant, a cash balance that must not go negative, or a bonus threshold. The question becomes “by how much can the budgeted sales volume fall before the year-end cash balance is exhausted?” rather than “before profit is nil”.
Flexing a budget assumption. Take one budget input — sales volume, selling price, a material price, a wage rate, the collection period for receivables — and compute the change that would consume the whole of the budgeted margin. That percentage is the budget's sensitivity to that assumption.
What-if analysis of a budget, and scenario analysis. Re-run the budget model on a stated change, or on several changes that belong to one coherent story, and read off the effect on profit and on cash. Chapter 11 §8.3 works both in their budgeting setting — including why changing assumptions one at a time understates risk — and they are not repeated here.
Stress testing. Push the assumptions to deliberately severe but plausible levels and ask whether the organisation would survive: whether the facility would still be available, the covenant still met, the payroll still payable. Stress testing budgets is taught in Chapter 11 §8.4 and is not repeated here either; it is the same technique as sensitivity analysis, used to test survival rather than to rank estimates.
The output of all three is the same in kind: a list of the assumptions the budget cannot afford to have wrong, which is where the effort in preparing and monitoring the budget should be concentrated.
9 Probability distributions, standard deviation and the coefficient of variation
A list of the possible outcomes of a decision with a probability attached to each is a probability distribution. Everything in sections 4 to 7 uses one. The expected value is its mean — its centre. What the expected value does not tell you is how widely the outcomes are spread around that centre, and spread is risk.
For example, project A and project B both have an expected value of $40,000.
Project A | Project B | |
Expected value | $40,000 | $40,000 |
Standard deviation | $2,000 | $6,500 |
Project A's returns lie mostly within ±$2,000 of $40,000, while project B's lie within ±$6,500. The two projects have the same expected value but project B is markedly the riskier. A risk-averse decision-maker would choose A; a decision made on expected value alone could not tell them apart.
9.1 Standard deviation
The standard deviation measures dispersion: the extent to which individual values in a distribution differ from their mean. The higher the standard deviation, the more spread out the outcomes and the higher the risk.
where x is each outcome, x̄ the expected value (the mean), and p the probability of that outcome. The deviations are squared because positive and negative deviations would otherwise cancel to nil; the square root then returns the answer to the units of the original data. If the returns are percentages the standard deviation is a percentage; if they are dollars it is dollars.
9.2 Coefficient of variation
A standard deviation of 3.5% is a small spread around a mean of 90% and a very large one around a mean of 5%. The coefficient of variation puts the spread in proportion to the mean, so that distributions of different sizes can be compared:
Coefficient of variation = σ ÷ x̄
The higher the coefficient, the more risk is being carried per unit of return.
The following are likely returns from project Z.
Return | Probability |
10% | 0.2 |
15% | 0.5 |
20% | 0.3 |
Calculate the expected value and the standard deviation for project Z.
What is the coefficient of variation in this case?
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10 Choosing between the techniques
Technique | Needs probabilities? | Answers the question |
Expected value | Yes | What is the average outcome of each option in the long run? |
Maximax / maximin | No | What is the best, or the worst, that each option could do? |
Minimax regret | No | Which option minimises the largest opportunity loss I could suffer? |
Value of perfect information | Yes | How much is it worth paying to remove the uncertainty? |
Decision tree | Yes | What should I decide now, and what should I decide later, when decisions come in sequence? |
Sensitivity analysis | No | How wrong can each estimate be before the decision changes? |
Standard deviation / coefficient of variation | Yes | How much spread — how much risk — sits behind the expected value? |
11 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.
Risk and Uncertainty
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