P1: Uncertainty

Answer: B. Expected value = Σ(probability × outcome) = (0.4 × £80,000) + (0.4 × £40,000) + (0.2 × −£10,000) = £32,000 + £16,000 − £2,000 = £46,000. £48,000 leaves out the loss outcome altogether (£32,000 + £16,000), and £44,000 deducts the expected loss twice. The £36,667 figure is the simple (unweighted) average of the three outcomes, ignoring the probabilities. Expected value is a long-run average, not a guaranteed outcome, and can mask significant downside risk.

Answer: C. Risk describes situations where the possible outcomes are known and probabilities can be assigned to them (enabling tools such as expected value and decision trees). Uncertainty describes situations where the range of outcomes or their probabilities cannot meaningfully be determined — often in novel or unprecedented situations. Both risk and uncertainty can lead to positive or negative outcomes. Insurance transfers financial risk but cannot eliminate uncertainty. The distinction determines which analytical tools are appropriate.

Answer: D. Maximin is a conservative, pessimistic decision rule: the decision maker assumes the worst will happen for each option and then chooses the option whose worst case (minimum payoff) is the highest. It is used under conditions of uncertainty where probabilities cannot be assigned. Maximising the best possible outcome is the maximax rule (optimistic). Minimising maximum regret is the minimax regret rule. Maximising expected value requires probabilities to be assigned and is used under risk, not pure uncertainty.

Answer: A. Sensitivity analysis asks how robust a decision is by calculating how much a critical variable — such as selling price, volume, variable cost or fixed cost — would need to change (relative to its expected value) before the preferred option would no longer be optimal. The variable with the smallest percentage change required is the most sensitive and deserves the most management attention. It does not assign probabilities, handle cost drivers, or replace budgets — it is a 'what-if' technique used alongside the main analysis.

Answer: D. Maximax is an optimistic decision rule: the decision maker focuses only on the best possible outcome of each option and chooses the one with the highest maximum payoff, ignoring the risk of poor outcomes. It is the opposite of maximin (which focuses on the worst case). A maximax decision maker is effectively risk-seeking. Choosing the highest minimum payoff is maximin; minimising the maximum loss is minimax; lowest variance is associated with risk-averse behaviour under uncertainty.

Answer: A. Minimax regret calculates the regret (the difference between the best possible payoff in each state of the world and the payoff actually achieved under each option) and selects the option whose maximum regret is smallest. It is a compromise between the pessimism of maximin and the optimism of maximax. Maximising expected payoff requires probabilities and is used under risk. Minimising overall cost may not be the same as minimising maximum regret. Maximising minimum guaranteed payoff is maximin.

Answer: B. In a decision tree, square nodes are decision points — the decision maker chooses between alternative courses of action. Circular nodes are chance (probability) nodes — the outcome is determined by chance, with each branch assigned a probability. The tree is drawn left to right and evaluated by working back from right to left (rolling back), replacing chance nodes with expected values and choosing the decision node branch with the highest expected value. This structure is the standard convention in management accounting and decision analysis.

Answer: D. Expected value of Project X = (0.6 × £80,000) + (0.4 × £20,000) = £48,000 + £8,000 = £56,000. Expected value of Project Y = £55,000 (certain). Since £56,000 > £55,000, Project X is preferred on an expected value basis. The rollback method values chance nodes as expected values, then selects the highest value at each decision node. The £50,000 figure misapplies probabilities; a risk-averse decision maker might prefer Project Y despite its lower expected value.

Answer: A. Expected value with perfect information (EVWPI) = (0.6 × £60,000) + (0.4 × £27,500) = £36,000 + £11,000 = £47,000. EVPI = EVWPI − expected value without information = £47,000 − £40,000 = £7,000. This represents the maximum amount the company should pay for information that perfectly predicts which state of the world will occur. The £47,000 figure is the expected payoff with perfect information, not the value of that information. EVPI is always non-negative; if it is small, acquiring information may not be worth the cost. £20,000 and £3,000 do not follow from the figures.

Answer: B. Sensitivity of variable costs = project NPV ÷ PV of the variable being tested = £20,000 ÷ £100,000 = 20%. If variable costs were 20% higher, the negative impact on NPV would be £100,000 × 20% = £20,000, exactly eliminating the positive NPV. This is the percentage change that renders the project marginal. A variable with a low sensitivity percentage is the most critical because a small adverse change makes the project unviable. The 5% and 10% figures understate the sensitivity. 25% divides the NPV by £80,000 instead of £100,000.

Answer: C. Simulation — particularly Monte Carlo simulation — models the combined effect of uncertainty in multiple variables simultaneously, generating thousands of random scenarios and producing a probability distribution of outcomes (such as profit or NPV). This gives decision makers a richer picture of risk than single-variable sensitivity analysis. Simulation does not guarantee the correct answer (it provides a probability distribution, not certainty), does not eliminate risk, and can be used with assumed or empirically estimated distributions for each variable.

Answer: B. Maximin identifies the minimum payoff for each option and selects the highest: Option 1 minimum = £10,000; Option 2 minimum = £30,000; Option 3 minimum = −£5,000 (a loss). The highest minimum payoff is Option 2 (£30,000), so maximin selects Option 2. This is a pessimistic, risk-averse rule that protects against the worst outcome. A risk-seeking maximax rule would choose Option 3 (highest maximum of £90,000), and an expected value approach would require probabilities.

Answer: C. Regret table: In a good state (best = £90): Option 1 regret = 90−80 = £10; Option 2 regret = 90−60 = £30; Option 3 regret = 0. In a poor state (best = £30): Option 1 regret = 30−10 = £20; Option 2 regret = 0; Option 3 regret = 30−(−5) = £35. Maximum regret: Option 1 = max(10,20) = £20; Option 2 = max(30,0) = £30; Option 3 = max(0,35) = £35. Minimax regret selects the option with the lowest maximum regret: Option 1 (£20,000). Option 2 has maximum regret £30,000 (in good state) and Option 3 has £35,000 (in poor state). Option 1 is preferred by minimax regret. Note: a risk-seeking maximax approach would choose Option 3; maximin would choose Option 2 (highest minimum of £30,000).

Answer: D. Standard deviation measures the spread of possible outcomes around the expected value; a higher standard deviation means greater variability and therefore greater risk. When expected returns are equal, a risk-averse decision maker would prefer the project with the lower standard deviation. Higher standard deviation does not imply higher expected return (they are compared here on the same expected return), and it implies greater, not lesser, unpredictability. In this case the coefficient of variation (standard deviation ÷ expected return) gives the same ranking because expected returns are equal.

Answer: B. Strategy B EV = (0.7 × £70,000) + (0.3 × −£20,000) = £49,000 − £6,000 = £43,000. Since Strategy A EV (£45,000) > Strategy B EV (£43,000), Strategy A should be chosen on an expected value basis. The £55,000 figures arise from adding probabilities instead of computing expected values (0.7 × £70,000 + 0.3 × £20,000 = £49,000 + £6,000 = £55,000 — treating the loss as a gain). Expected value calculation must correctly account for the sign of each payoff.

Answer: C. A risk-averse decision maker dislikes uncertainty and, all else equal, prefers the option with more predictable outcomes — lower variance, standard deviation or spread of possible results — even at the same expected return. Risk-seeking behaviour involves preferring more variance (maximax thinking). Risk-neutral decision makers base decisions purely on expected values and are genuinely indifferent when expected values are equal. Always choosing the highest payoff is maximax, which is risk-seeking, not risk-averse.

Answer: D. The sensitivity percentage for a variable is the percentage change required in that variable before the NPV falls to zero. The variable with the lowest sensitivity percentage requires the smallest adverse change to destroy the project's viability — it is therefore the most sensitive and most critical. A low sensitivity percentage means little room for error. The highest sensitivity percentage means the project can withstand a large change before failing. Neither absolute value nor management control determines sensitivity.

Answer: A. Each Monte Carlo run randomly samples a value for each uncertain variable from its assumed probability distribution — for example, picking a selling price from a uniform or normal distribution — then calculates the model output (such as NPV) for that combination of values. After thousands of runs, the results form a probability distribution of the output, showing the range of outcomes and their relative likelihoods. Using the most likely value, best-case values or expected values would all give a single deterministic result rather than the distribution that Monte Carlo generates.

Answer: C. The expected value is a probability-weighted average; the actual outcome will be one of the specific scenarios (not the weighted average), which may differ substantially from the EV. Two projects with the same EV may have very different risk profiles — one with a narrow range of outcomes, another with a wide range including large losses. EV also assumes the decision is repeated many times, which may not be true for a one-off decision. Probabilities can be estimated even without certainty, EV does not guarantee correctness, and it is not inherently conservative.

Answer: D. Single-variable sensitivity varies one input at a time while holding all others constant — useful but unrealistic, since in practice multiple variables change together. Two-way analysis shows, for example, how NPV changes as both selling price and volume vary, producing a matrix of results that reflects the combined effect. It does not calculate probabilities (that is simulation), does not eliminate uncertainty, and produces a range of results rather than a single answer. It gives a richer picture than single-variable analysis and helps identify critical combinations of conditions.

Answer: A. Imperfect information (such as market research) tells the decision maker something about the likely state of the world but does not guarantee accuracy — there is still a chance of incorrect signals. Perfect information would always indicate the correct state. Therefore, the improvement in decision making (and expected payoff) from imperfect information is less than from perfect information, making EVII ≤ EVPI. EVII is the maximum worth paying for imperfect information and is always non-negative (you cannot be worse off by having more information if you can choose to ignore it).

Answer: B. The coefficient of variation (CV) = standard deviation ÷ expected return expresses risk relative to return, allowing fair comparison when returns differ. Project A: CV = £50,000 ÷ £200,000 = 25%. Project B: CV = £60,000 ÷ £300,000 = 20%. Project A is relatively more risky (higher CV), despite having the lower absolute standard deviation. Comparing standard deviations alone (£50,000 vs £60,000) is misleading when expected returns differ significantly. A risk-averse investor comparing equal-return projects uses standard deviation; comparing different-return projects should use the coefficient of variation.