## **Learning Outcomes**

After reading this article, you will be able to explain and apply decision trees to investment appraisal under uncertainty, outline the process of sequential decision-making, and interpret probability-adjusted project values. You will practice structuring and evaluating risk using graphical methods and learn how to incorporate probabilities and potential outcomes for ACCA AFM exam scenarios.

## **ACCA Advanced Financial Management (AFM) Syllabus**

For ACCA Advanced Financial Management (AFM), you are required to understand techniques for incorporating risk and uncertainty into investment appraisal decisions. This article focuses on the following examinable areas:

- Use of decision trees to appraise projects under risk and uncertainty
- Construction and interpretation of probability-based decision models
- Evaluation of sequential decisions within investment appraisal
- Calculation and analysis of expected monetary values and related risk metrics
- Application of risk analysis techniques in project selection

## **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 the primary purpose of using a decision tree in investment appraisal?
2. In a sequential decision involving several stages of uncertainty, why is it important to evaluate decisions from the end of the tree backwards?
3. How does a company use expected monetary value (EMV) to choose between alternatives in a risky project?
4. Briefly explain how probabilities assigned to branches in a decision tree affect the final project appraisal.

## **Introduction**

Risk and uncertainty are present in almost every investment decision. Standard techniques such as Net Present Value (NPV) assume all inputs can be forecast with confidence, but in real projects many future events are uncertain. Decision trees are graphical tools designed to structure complex, multi-stage decisions where outcomes depend on future events, each with assigned probabilities and values. Sequential decisions allow financial managers to incorporate future choices made in response to new information, supporting more flexible and realistic investment appraisals.

## Decision Trees: The Structure

A decision tree is a diagram that maps out possible decisions and their consequences over time. It consists of decision nodes, chance nodes, branches showing possible outcomes, and end nodes with resulting cash flows.

> **Key Term: decision tree**  
> A graphical technique for mapping out, calculating, and visually analyzing a sequence of decisions and uncertain outcomes, using nodes, branches, and probabilities.
>
> **Key Term: decision node**  
> A point in a decision tree where the decision maker must select between alternative actions, typically illustrated by a square.
>
> **Key Term: chance node**  
> A point in a decision tree where more than one possible outcome may occur due to uncertainty, each associated with a probability; shown as a circle.

### Building a Decision Tree

To construct a decision tree for investment appraisal:

1. Start with the initial investment decision (square).
2. From each decision, draw branches showing every possible action.
3. At each chance node (circle), show possible outcomes with an estimated probability assigned to each branch.
4. Continue until all terminal nodes (ends) are reached, noting monetary outcomes (such as project NPVs).

### Calculating Expected Monetary Value (EMV)

For each chance node, calculate the expected value by multiplying the monetary values by their respective probabilities and summing these products.

> **Key Term: expected monetary value (EMV)**  
> The weighted average of all possible outcomes in a risky or uncertain scenario, calculated by multiplying each outcome by its probability and adding the results.

By comparing the EMVs of different decision paths, you can select the course of action with the highest expected value.

## Sequential Decisions and Project Flexibility

Many investment decisions are made in stages. At each stage, new information may become available, allowing the company to revise its strategy. Sequential decision-making uses decision trees to model this process, enabling the incorporation of real-world flexibility into the financial appraisal.

- The tree allows for options such as abandoning, delaying, or expanding projects at certain points, depending on earlier outcomes.

- Sequential analysis always proceeds by "rolling back" the tree: working from the farthest (right-side) branches back towards the initial decision, updating the expected values at each node.

### **Worked Example 1.1**

**A company can invest £1 million in a project. After one year, market research (at a cost of £100,000) will inform the company whether to invest a further £1.5 million for full-scale launch or abandon the project.**

- Probability that research is positive: 60%
- Probability research is negative: 40%
- If positive, expected NPV of the launch: £3 million (probability 70%), £0 (probability 30%)
- If negative, project is abandoned (no further investment or returns).

What is the expected monetary value (EMV) of the initial investment?

> **Answer:**
>
> 1. First, calculate the EMV of proceeding after a positive research outcome:
>    - £3 million × 70% = £2.1 million; £0 × 30% = £0; Total EMV = £2.1 million
> 2. Subtract the £1.5 million additional investment: £2.1 million – £1.5 million = £0.6 million
> 3. Adjust for the probability of positive research: £0.6 million × 60% = £360,000
> 4. Subtract the cost of research: £100,000 × 100% = £100,000
> 5. Final EMV: £360,000 – £100,000 = £260,000
> 6. Deduct the initial investment: £260,000 – £1 million = –£740,000
>
> The negative EMV suggests the initial investment is not financially justified given estimated outcomes and probabilities.

## Valuing Real Options in Trees

Decision trees can incorporate managerial flexibility:

- Delay starting or expanding a project if uncertainty is high
- Abandon a project following adverse early outcomes

This approach values options to respond to new information by mapping them explicitly in the tree, improving the accuracy of the project’s risk-adjusted value.

### **Worked Example 1.2**

**A firm can enter a new market today, or conduct further research (for £50,000) that may indicate a better entry strategy. If the research outcome is favorable (probability 0.8), expected profit is £150,000; if unfavorable (probability 0.2), company decides not to proceed (no profit or loss). Should the firm conduct research or proceed directly?**

> **Answer:**  
> EMV with research: (0.8 × £150,000) + (0.2 × £0) = £120,000.  
> Subtract research cost: £120,000 – £50,000 = £70,000.  
> If the firm proceeds without research, assume best estimate profit is £100,000.  
> Comparison: £70,000 (with research) vs £100,000 (without research).  
> The firm should proceed without research if it wants to maximize expected monetary value.

## Probability Assessment and Objective Decision-Making

Probabilities in decision trees should be estimated using objective information wherever possible—such as historical data, simulation, or expert judgment. Care must be taken to ensure that all relevant outcomes and branches are considered, and that probabilities for branches from each chance node sum to one.

### **Worked Example 1.3**

**A company faces a maintenance decision: repair equipment now (£50,000 cost, 80% chance of no further expense this year, 20% chance a replacement will still be necessary, costing £200,000 later), or wait and take a 50% risk the equipment will fail, requiring the same £200,000 replacement. Which is the better decision by EMV?**

> **Answer:**  
> If repair now:
>
> - 80%: cost = £50,000
> - 20%: cost = £50,000 + £200,000 = £250,000  
>   EMV = (0.8 × £50,000) + (0.2 × £250,000) = £40,000 + £50,000 = £90,000  
>   If wait: 50% chance of paying £200,000, 50% chance of no cost.  
>   EMV = (0.5 × £200,000) + (0.5 × £0) = £100,000  
>   Cheaper to repair now, with an expected cost of £90,000 vs £100,000 if waiting.

### **Exam Warning**

> When building decision trees, ensure every possible outcome and decision point is mapped and each set of probabilities at a chance node adds to 1. Otherwise, you risk incorrect EMV calculations and flawed recommendations.

### **Revision Tip**

> In ACCA AFM, always "roll back" through the tree: calculate expected values from the end nodes to the start, updating at each chance and decision node.

## **Summary**

Decision trees help visualize and analyze multi-stage investment decisions with risk and uncertainty. Probabilities are assigned at each chance node, and the expected monetary value is calculated by rolling back from outcomes to the original decision. Sequential decision analysis allows incorporation of managerial flexibility, making appraisal more realistic and supporting sound financial choices under uncertainty.

## **Key Point Checklist**

_This article has covered the following key knowledge points:_

- Explain the structure and use of decision trees for risk analysis in investment appraisal
- Define and calculate expected monetary value (EMV) at each chance node
- Apply sequential decision-making techniques using decision trees
- Incorporate probabilities and real options in project evaluation
- Recognize common errors in decision tree analysis for ACCA AFM

## **Key Terms and Concepts**

- decision tree
- decision node
- chance node
- expected monetary value (EMV)

Risk analysis in appraisal - Decision trees and sequential decisions