Quantifying uncertainty - Expected values, payoff tables, an...

Learning Outcomes

After reading this article, you will be able to define and calculate expected values, construct and analyse payoff tables, and apply probability concepts to decision making under uncertainty. You will learn the differences between risk and uncertainty, use expected value to aid decisions, and interpret results in an ACCA Performance Management exam context.

ACCA Performance Management (PM) Syllabus

For ACCA Performance Management (PM), you are required to understand quantitative approaches for dealing with risk and uncertainty in decision making. This article covers the following syllabus areas:

• Distinguish between risk and uncertainty in business decision making
• Calculate and interpret expected values for decision problems with known probabilities
• Construct and analyse payoff tables for alternative actions and outcomes
• Apply probability rules relevant to business scenarios
• Identify limitations of expected value and related quantitative tools
• Use these techniques to support recommendations under uncertainty

Test Your Knowledge

Attempt these questions before reading this article. If you find some difficult or cannot remember the answers, remember to look more closely at that area during your revision.

1. A project can result in a profit of $10,000 (probability 0.6) or a loss of $4,000 (probability 0.4). What is its expected value?
2. Which decision rule is used by a manager who chooses the alternative with the best possible pay-off, regardless of risk?
3. What is a payoff table used for in decision making under uncertainty?
4. In which situations is probability described as objective, and when is it subjective?

Introduction

All business decisions involve uncertainty. Some outcomes can be predicted with reasonable confidence, while others depend on unknown or random events. To make structured decisions in risky situations, accountants use quantitative methods such as expected values and payoff tables. These methods are essential for evaluating project choices, one-off contracts, or operational plans where several outcomes are possible.

Key Term: Risk
A situation in which the possible outcomes of a decision are known and can be assigned numerical probabilities based on data or experience.

Key Term: Uncertainty
A situation where several outcomes are possible, but the probabilities of these outcomes are unknown or cannot be reliably quantified.

Key Term: Expected value
The weighted average of all possible outcomes, each multiplied by its respective probability, representing the long-run average result.

Expected Values in Decision Making

The expected value (EV) provides an average result if a decision could be repeated many times. To calculate it, multiply each potential outcome by its probability and sum the results.

$\text{Expected value} = \sum (\text{Probability} \times \text{Pay-off})$

Worked Example 1.1

A company is considering launching a new product. There is a 40% chance of earning a profit of $25,000, a 35% chance of breaking even ($0), and a 25% chance of incurring a $10,000 loss. What is the expected value?

Answer:

\text{EV} = (0.4 \times \25,000) + (0.35 \times $0) + (0.25 \times -$10,000) = $10,000 + $0 - $2,500 = $7,500$.

A positive expected value suggests the project is worthwhile if the decision is repeated over time.

Key Term: Payoff table
A table summarising all possible actions, states of nature, and resulting outcomes (payoffs), used to compare and choose between alternatives under risk or uncertainty.

Payoff Tables for Business Decisions

Payoff tables allow you to systematically analyse decisions with multiple possible outcomes. Create rows for each state of nature (e.g., market conditions) and columns for each action, listing the payoffs at each intersection.

Worked Example 1.2

A distributor can stock either 100, 150, or 200 units of a seasonal item. Demand can be low (100 units), medium (150), or high (200). The profit per unit sold is $15; excess stock has no value.

If the probability of each demand level is 1/3, what is the expected profit for each stocking option?

Answer:

• Stock 100: EV = (1/3 x $1,500) + (1/3 x $1,500) + (1/3 x $1,500) = $1,500
• Stock 150: EV = (1/3 x $1,500) + (1/3 x $2,250) + (1/3 x $2,250) = $2,000
• Stock 200: EV = (1/3 x $1,500) + (1/3 x $2,250) + (1/3 x $3,000) = $2,250

Stocking 200 units gives the highest expected value.

Applying Probability Concepts

Probabilities can be objective (based on past data) or subjective (based on opinion or judgement). Probabilities for exclusive outcomes always sum to 1.

Key Term: Probability
The likelihood of an event occurring, expressed as a number between 0 (impossible) and 1 (certain), or as a percentage between 0% and 100%.

For two independent events, the combined probability is found by multiplying their individual probabilities. For mutually exclusive events, add the probabilities.

Worked Example 1.3

A company faces machine failure risk next quarter. Probability of a minor fault is 0.10, a major fault is 0.05, and no fault is 0.85. What is the probability that there will be any fault next quarter?

Answer:

Probability of any fault = Probability of minor + Probability of major = 0.10 + 0.05 = 0.15.

Decision Rules and the Role of Expected Value

Although expected value is a key metric, it is not always the sole basis for choices. Other approaches include:

• Maximax: Choose the alternative with the highest possible payoff (optimistic, risk-seeking).
• Maximin: Choose the alternative with the best among the worst possible payoffs (pessimistic, risk-averse).
• Minimax regret: Minimise the maximum potential regret from not choosing the best alternative.

These rules help managers make decisions reflecting their risk preferences, especially if an activity will only happen once and the expected value is not realised.

Revision Tip

Always check that the probabilities used in your analysis sum to 1. If not, review your scenario for missing or overlapping outcomes.

Limitations of Expected Value and Payoff Tables

Expected values assume a decision can be repeated many times. In unique, one-off decisions, actual outcomes may differ greatly from the calculated EV. EV considers average results but ignores the variability (risk) of those results—important when deciding between a safe, lower-return and a risky, higher-return option.

Exam Warning Do not use expected value as the sole decision criterion for one-off, high-risk projects. Always comment on the spread of possible outcomes and the decision makers' risk appetite.

Summary

Expected value is an important quantitative tool for summarising the long-run outcome of uncertain events. Payoff tables provide structure in comparing alternative actions under different scenarios. Probabilities are central to these calculations. While these methods bring discipline to decisions under risk, always recognise their limitations and supplement results with other decision rules when required in the exam.

Key Point Checklist

This article has covered the following key knowledge points:

• Define and distinguish between risk and uncertainty
• Calculate expected values by weighting outcomes by probability
• Construct and interpret payoff tables for alternative actions and states of nature
• Apply probability rules to calculate joint and mutually exclusive event probabilities
• Recognise the use and limitations of expected value in ACCA exam scenarios
• Explain decision rules: maximax, maximin, and minimax regret

Key Terms and Concepts

• Risk
• Uncertainty
• Expected value
• Payoff table
• Probability