Quantifying uncertainty - Decision trees and expected moneta...

Learning Outcomes

After studying this article, you will be able to explain the purpose and construction of decision trees, calculate expected monetary value (EMV) in uncertain situations, and apply these methods to multi-stage decision problems typical of ACCA Performance Management (PM) exam questions. You will recognise when these techniques are appropriate, interpret their results in context, and identify the limitations and assumptions that underpin their use.

ACCA Performance Management (PM) Syllabus

For ACCA Performance Management (PM), you are required to understand and apply decision-making techniques under conditions of risk and uncertainty. Ensure your revision covers:

• The relevance of uncertainty and the distinction between risk and uncertainty in decision making
• Construction and probabilistic calculation of expected values for a range of outcomes
• Application and interpretation of decision trees in multi-stage decision scenarios
• Calculation and practical interpretation of the value of perfect and imperfect information
• The strengths and limitations of the expected value approach and decision tree analysis

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. What is an expected monetary value (EMV)? How is it used in decision-making under risk?
2. When is it appropriate to use a decision tree?
3. A project has a 60% chance of earning $10,000 profit and a 40% chance of losing $5,000. What is the expected value?
4. Name two advantages and two disadvantages of using the expected value approach.
5. Briefly define the value of perfect information and explain how it is calculated.

Introduction

Management accountants are routinely asked to advise on choices where future outcomes are uncertain. Rather than relying on intuition, quantitative techniques allow you to structure choices and compare the financial consequences if different scenarios unfold. Decision trees and expected values are the main tools for formally quantifying risk and providing sound recommendations under uncertainty.

Key Term: risk
The situation where several possible outcomes exist and the probability of each outcome can be assigned based on past experience, data, or objective estimation.

Key Term: uncertainty
The situation where there are several possible outcomes, but the probabilities associated with each outcome are not known or cannot be reliably estimated.

Key Term: expected monetary value (EMV)
The weighted average of all possible monetary outcomes, calculated by multiplying each possible outcome by its probability and summing the results.

Key Term: decision tree
A diagrammatic technique used to show each possible course of action and subsequent events, together with probabilities, outcomes and expected values, for multi-stage or sequential decisions.

Quantifying Risk with Expected Monetary Value (EMV)

In decision making under risk, where probabilities can be estimated, the expected monetary value (EMV) represents the long-run average outcome if the decision were repeated many times. The EMV is used to compare alternative courses of action.

For each option:

1. List all possible outcomes and their associated probabilities.
2. Multiply each outcome by its probability.
3. Add the products to obtain the EMV for that option.

EMV = Σ (outcome × probability)

Worked Example 1.1

A company faces two investment options:

• Investment A: 50% chance of $8,000 profit, 50% chance of $2,000 loss.
• Investment B: 80% chance of $5,000 profit, 20% chance of $1,500 loss.

What is the EMV of each investment?

Answer:
Investment A: (0.5 × $8,000) + (0.5 × –$2,000) = $4,000 – $1,000 = $3,000 Investment B: (0.8 × $5,000) + (0.2 × –$1,500) = $4,000 – $300 = $3,700 On EMV grounds, Investment B is preferred. Exam Warning The EMV approach is suitable only for repeated or regular decisions. For one-off, high-stakes, or non-repetitive choices, EMV may mislead. Always state this limitation if relevant in your answer.

Using Decision Trees in Multi-Stage Problems

Complex decisions often contain a sequence of choices or events, where each choice leads to further uncertainty. Decision trees provide a clear, logical layout for such cases.

Constructing a Decision Tree:

• Decision points (where a choice is made) are represented by squares.
• Chance points (uncertain outcomes) are shown as circles.
• Each branch from a chance point is labelled with the outcome value and its probability.
• Probabilities on branches emanating from a chance point must sum to 1.
• At each final outcome, record the monetary value.

Evaluating a Decision Tree:

• Work right to left ("rolling back" the tree).
• At each chance point, calculate the EMV of that node.
• At each decision point, select the branch with the highest EMV.

Worked Example 1.2

A product has two launch strategies:

• Strategy X: Invest $20,000 in advertising. If the market response is strong (probability 0.3), profit is $70,000. If weak (probability 0.7), profit is $8,000.
• Strategy Y: Invest nothing. If strong (0.3), profit is $35,000, if weak (0.7), profit is $7,000.

Which strategy has the highest EMV?

Answer:
Strategy X EMV: (0.3 × $70,000) + (0.7 × $8,000) – $20,000 = $21,000 + $5,600 – $20,000 = $6,600 Strategy Y EMV: (0.3 × $35,000) + (0.7 × $7,000) = $10,500 + $4,900 = $15,400 Strategy Y has a higher EMV. The company should not invest in advertising based purely on EMV (disregarding qualitative factors).

Key Term: rolling back
The process of evaluating a decision tree starting from the end outcomes and moving backwards to the initial decision, calculating EMVs at each chance point.

The Value of Perfect and Imperfect Information

Key Term: value of perfect information (VOPI)
The increase in expected profit that would result from having 100% accurate predictions of uncertain future events before making a decision.

Key Term: value of imperfect information
The increase in expected profit that could be obtained using additional, but not totally reliable, information (e.g., market research) about uncertain future events.

Perfect information lets the decision-maker always pick the best outcome for each event. The VOPI is calculated as follows:

1. Calculate the EMV for each possible state if that state were known and the optimal action chosen every time ("with perfect information").
2. Subtract the EMV for the original decision (without extra information).
3. VOPI = EMV with perfect information – EMV without perfect information.

Worked Example 1.3

A business can launch Product Z or Product Y, but not both. Their profits, depending on future market conditions, are:

What is the value of perfect information?

Answer:
EMV for each product:

• Product Z: (0.3 × $60,000) + (0.5 × $35,000) + (0.2 × $0) = $18,000 + $17,500 + $0 = $35,500
• Product Y: (0.3 × $40,000) + (0.5 × $30,000) + (0.2 × $10,000) = $12,000 + $15,000 + $2,000 = $29,000 Optimal choice without information: choose Product Z (EMV = $35,500) If perfect information is available, always choose the better outcome for each state:
• Good: choose Z ($60,000)
• Moderate: choose Z ($35,000)
• Poor: choose Y ($10,000) EMV with perfect information = (0.3 × $60,000) + (0.5 × $35,000) + (0.2 × $10,000) = $18,000 + $17,500 + $2,000 = $37,500 VOPI = $37,500 – $35,500 = $2,000

Revision Tip

If the value of perfect information is less than the cost of acquiring it (e.g. market research), the investment is not justified.

Limitations and Assumptions of EMV and Decision Trees

While EMV and decision trees support objective analysis, they rely on accurate probability estimation and repetition of similar decisions. In one-off, high-risk scenarios, EMV may be a poor guide. Also, EMV ignores the range and variability of possible outcomes, which may matter more to risk-averse managers than the average.

Other practical considerations:

• Probabilities are often subjective or based on limited data.
• Decision trees may become unwieldy if there are many stages or options.
• EMV does not incorporate risk preferences (risk-averse or risk-prone decisions).
• These methods assume only monetary outcomes matter; qualitative aspects (reputation, staff morale, customer loyalty) may need qualitative discussion.

Key Term: risk-neutral
A decision-making attitude where the decision-maker bases choices only on expected values without regard to risk or variability.

Summary

Decision trees and expected value calculations allow management to quantify financial consequences where outcomes are uncertain but probabilities are available. The expected value ranks alternatives by long-run average gain. Decision trees help structure sequential or multi-stage problems, enabling systematic analysis. These tools support, but do not replace, professional judgement.

Key Point Checklist

This article has covered the following key knowledge points:

• Distinguish between risk and uncertainty for decision purposes
• Define and use expected monetary value (EMV) in risk analysis
• Construct and evaluate decision trees for multi-stage choices
• Calculate the value of perfect and imperfect information and interpret results
• Recognise assumptions and limitations of the EMV and decision tree approach

Key Terms and Concepts

• risk
• uncertainty
• expected monetary value (EMV)
• decision tree
• rolling back
• value of perfect information (VOPI)
• value of imperfect information
• risk-neutral