ACCA Applied Skills · Performance Management

Dealing with risk and uncertainty in decision-making: formula sheet

Full chapter guide

Key formulas

Expected value (EV): EV = Σ (probability × outcome); Used when probabilities are known (risk). It is a long-run average, not an outcome that will actually occur in a single decision.
Risk: Risk = outcomes possible AND probabilities known or estimable; Link to expected values, decision trees and value of information.
Uncertainty: Uncertainty = outcomes possible AND probabilities not known; Link to maximin, maximax, minimax regret and sensitivity analysis.
Risk-neutral rule: Choose the option with the highest EV of profit (or lowest EV of cost); Ignores variability of outcomes.
Risk-averse rule of thumb: Prefers lower variability, often maximin; Maximin picks the best of the worst outcomes. It is a rule linked to caution, not a definition.
Risk-seeking rule of thumb: Prefers highest possible payoff, often maximax; Maximax picks the best of the best outcomes.
Expected value: EV = Σ (probability × outcome) = Σ px; Probabilities must add to 1. Use the same measure (profit, cost, contribution) for every outcome.
EV decision rule: Choose the option with the highest EV of profit, or the lowest EV of cost; Assumes the decision maker is risk neutral.
Probability check: Σ p = 1; If probabilities add to less than or more than 1, recheck the data before calculating.
Joint probability (independent events): P(A and B) = P(A) × P(B); Used when a question has two uncertain factors, such as price and volume.
Maximin (profits): Choose the option with the highest minimum pay-off; Pessimistic, risk-averse. Find the lowest figure in each row, then pick the largest of those.
Maximax (profits): Choose the option with the highest maximum pay-off; Optimistic, risk-seeking. Find the highest figure in each row, then pick the largest of those.
Regret for a cell: Regret = best pay-off in that state − pay-off of the option in that state; Work down each column. The best option in a state has regret of zero. For costs, regret = option's cost − lowest cost in that state.
Minimax regret: Choose the option with the lowest maximum regret; Find the largest regret in each row of the regret table, then pick the smallest of those.
Rules for costs: Minimax: choose the lowest of the maximum costs. Minimin: choose the lowest of the minimum costs; When the table shows costs, the logic flips. Pessimist looks at the highest cost for each option and wants the smallest of those.
Sensitivity (NPV approach): Sensitivity % = NPV of project ÷ PV of the cash flows affected by the variable × 100%; Use the present value of the specific cash flow, after tax if tax applies. This is the percentage fall in that variable that makes NPV zero.
Sensitivity (profit or contribution approach): Sensitivity % = Profit ÷ the total amount of the item flexed × 100%; Use when there is no discounting, for example a one-period decision. The item flexed is the total contribution for volume, total revenue for price, total variable costs for variable costs and total fixed costs for fixed costs.
Discount rate sensitivity: Break-even discount rate = IRR; The IRR is the discount rate at which NPV is zero. Compare it with the cost of capital; the gap shows how much the rate can rise.
Reading the result: Smaller percentage = more sensitive = more critical variable; Rank the variables by percentage to see which matter most.
Expected value: EV = Σ (probability × payoff); Calculate it for every option. Payoffs can be profit, contribution or cost, so check whether you want the highest or the lowest.
Expected value of perfect information: EVPI = EV with perfect information − EV without perfect information; EV with perfect information = Σ (probability of each outcome × best payoff for that outcome). EV without = the best EV from the original table.
Value of imperfect information: Value = EV with imperfect information − EV without information; EV with research = Σ (probability of each report × EV of the best option after that report).
Revised (posterior) probability: P(outcome | report) = P(outcome and report) ÷ P(report); P(outcome and report) = prior probability × probability of that report given the outcome.
Decision on buying information: Buy if value of information > cost of information; The most you would pay is the value of the information. Its cost must be below this.
Expected value: EV = Σ (probability × outcome); Used at every chance node. Probabilities leaving a node must total 1.
Rollback at a decision node: Value of decision node = best of the values of its branches; Best means highest for profit or NPV, lowest for cost. Use the value after deducting any cost on that branch.
Net value of a branch: Net EV = EV of later outcomes − cost incurred on the branch; Deduct costs like a survey fee or investment at the point they are paid.
Random number allocation: Allocate random numbers in proportion to probability, e.g. P = 0.30 → numbers 00–29 from a 00–99 set; Each outcome gets a range of numbers matching its probability. Ranges must not overlap or leave gaps.

Quick revision

Risk: outcomes and probabilities are known or estimable. Uncertainty: probabilities cannot be assigned.
Expected value = Σ (probability × outcome).
Expected value suits repeated decisions best; for a one-off decision the average outcome may never actually occur.
Expected value ignores the spread of outcomes and the decision maker's attitude to risk.
Maximin picks the option with the best of the worst outcomes (pessimist).
Maximax picks the option with the best of the best outcomes (optimist).
Minimax regret picks the option with the smallest of the maximum regrets. Regret = best outcome for that state minus the outcome you got.
Sensitivity = change needed for the decision to switch ÷ value of the variable, as a percentage. The smaller the percentage, the more sensitive the variable.
Sensitivity analysis changes one variable at a time and gives no probabilities.
Value of perfect information = EV with perfect information − EV without it. It is the most you should pay for information.
Imperfect information is worth less than perfect information, so its value cannot exceed that of perfect information.
Decision trees: squares are decisions, circles are chance events. Work from right to left, calculating expected values at each circle.
Simulation uses random numbers to model many variables and produce a range of outcomes. It can be costly and is only as good as its inputs.

Common mistakes

Treating risk and uncertainty as the same thing. Fix: Test for probabilities. Known or estimable probabilities mean risk. No probabilities means uncertainty.
Saying the expected value is the result that will happen. Fix: Describe it as a weighted average over many repeats. In a one-off decision the actual result will differ from the EV.
Choosing the most likely outcome instead of calculating the EV. Fix: Always weight every outcome by its probability and add them. The most likely outcome ignores the rest of the distribution.
Probabilities that do not add to 1, or using percentages inconsistently. Fix: Convert everything to decimals and total them before you multiply.
Calculating regret across the row instead of down the column. Fix: Regret is always measured within one state. Take the best figure in the column and compare each option against it.
Choosing the largest regret instead of the smallest of the maximum regrets. Fix: Find the maximum regret for each option, then choose the option where that maximum is lowest.
Dividing NPV by the undiscounted total of the variable. Fix: Always use the present value of the variable's cash flows, using the same discount factors as the NPV.
Using contribution instead of sales revenue when flexing selling price. Fix: A price change flows straight to profit with no change in variable cost, so use the full PV of revenue.
Subtracting the cost of the information when finding its value. Fix: Find the value first, without the cost. Then compare the value with the cost. The maximum price is the value.
Using the EV of the best option for each outcome in the without-information figure. Fix: Without information you must pick one option and stick with it. Calculate EV for each option and take the best single one.

Exam tips

Start every answer by deciding risk or uncertainty. Examiners often reward a clear distinction before any calculation.
In objective questions, wrong options often swap maximin and maximax. Link averse to maximin and seeker to maximax.
When asked to discuss expected values, always mention that probabilities may be subjective and the EV may not be a possible outcome.
In written parts, finish with a recommendation tied to the decision maker's attitude to risk, not just the numbers.
Check whether figures are profit or cost before picking the best option.
Objective questions give no partial marks, so check that probabilities add to 1 and that you chose highest profit or lowest cost before you submit.
In a Section C answer, lay out the calculation in a clear table-like list of outcome, probability and weighted value, so marks are given even if you slip on arithmetic.
When asked to discuss limitations, give three or four points and link at least one to the scenario. Name one-off decisions, risk attitude and subjective probabilities.