## **Learning Outcomes**

After reading this article, you will be able to explain the concept of quantifying uncertainty in decision making, distinguish between perfect and imperfect information, calculate the value of perfect and imperfect information, and apply these concepts directly to exam-style problems using decision trees and expected values.

## **ACCA Performance Management (PM) Syllabus**

For ACCA Performance Management (PM), you are required to understand how uncertainty impacts decision making and how quantitative techniques can help support management decisions. In particular, you should focus on the following syllabus points:

- Understand and apply the use of expected values in decision making
- Use decision trees to analyse multi-stage decision problems
- Calculate and interpret the value of perfect information (VPI) and imperfect information (VII)
- Discuss how information (and its accuracy) can affect management's ability to make better decisions 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. What is the expected value of perfect information (VPI), and how is it calculated in a decision scenario?
2. A company faces two possible outcomes: a gain of \$10,000 (probability 0.4) or a loss of \$5,000 (probability 0.6). What is the expected value of the project?
3. What is the main difference between perfect and imperfect information in the context of decision making?
4. True or false? The value of perfect information can never be negative.
5. Briefly describe the purpose of using decision trees in uncertainty analysis.

## **Introduction**

Management decisions often involve uncertainty. The outcomes of different actions are not always known in advance, so managers must work with probabilities and estimates to make informed choices. Quantifying this uncertainty can help managers improve decisions, especially when considering whether to purchase extra information—such as market research or technical analysis—before committing to a course of action.

This article covers the key techniques for quantifying uncertainty, specifically focusing on how to calculate the value of perfect and imperfect information and integrate these calculations into decision-making frameworks like decision trees and expected value analysis.

> **Key Term: expected value**
> The average outcome, weighted by probability, that would result if a decision were repeated many times.
>
> **Key Term: perfect information**
> Information that predicts future events with complete accuracy, allowing the manager to always select the best possible action for each state of nature.
>
> **Key Term: imperfect information**
> Information that increases the likelihood of making the right decision, but is not always accurate—forecasts may be correct or incorrect.
>
> **Key Term: value of perfect information (VPI)**
> The additional expected profit that could be earned if perfect information was available before making a decision, compared to making the decision without such information.
>
> **Key Term: value of imperfect information (VII)**
> The extra expected profit from making a decision with imperfect (fallible) forecasts, minus the expected profit from making the decision without that information.

## UNCERTAINTY AND MANAGEMENT DECISION MAKING

Uncertainty is a feature of most business decisions. Risk is present when the possible outcomes are known, along with their probabilities. Uncertainty exists when the possible outcomes are known, but probabilities are not. In practice, most exam questions provide probabilities, so we can apply expected value analysis.

Decision trees are commonly used to model multi-stage problems, such as entering new markets, developing products, or considering large investments.

## CALCULATING EXPECTED VALUES

Expected value (EV) provides a single average outcome for a decision, calculated by multiplying each possible outcome by its probability and summing the results. EV is most relevant when a decision will be repeated many times, as actual results in a single instance can vary significantly from the mean.

### **Worked Example 1.1**

A business can launch a product, earning a profit of \$80,000 if the market is strong (probability 0.3), or \$10,000 if the market is weak (probability 0.7). What is the expected value?

> **Answer:**
>
> Expected value = (0.3 × \$80,000) + (0.7 × \$10,000) = \$24,000 + \$7,000 = **\$31,000**
>
> **Revision Tip**
> Underline that expected value averages out outcomes and is most useful for repeated decisions, not for one-off situations.

## THE VALUE OF PERFECT AND IMPERFECT INFORMATION

Information can improve decision making. However, acquiring additional information has a cost. Managers must decide if the benefit from improved decisions (the value of information) is greater than the cost of obtaining it.

### Perfect Information

If management could get guaranteed, error-free forecasts—perfect information—they would always choose the best action for the actual outcome.

> **Key Term: value of perfect information (VPI)**
> The maximum amount a business should be willing to pay for information that predicts the true outcome with total accuracy.

The value of perfect information is calculated as:

> VPI = Expected value with perfect information – Expected value without information

### **Worked Example 1.2**

A company can drill for oil at a cost of \$10,000. If oil is found, profit will be \$190,000. Probability of finding oil is 0.10. If not found, there is no profit and the \$10,000 is lost. Alternatively, the company can decide not to drill, earning nothing.

**a) What is the expected value if the company decides to drill without any additional information?**

> **Answer:**
>
> EV = (0.10 × \$190,000) + (0.90 × –\$10,000) = \$19,000 + (–\$9,000) = **\$10,000**

**b) What is the value of perfect information?**

- With perfect information, the company drills only if oil is found (probability 0.10), earning \$190,000, and does not drill otherwise (probability 0.90), earning \$0.

> EV with perfect information = (0.10 × \$190,000) + (0.90 × \$0) = \$19,000
> Value of perfect information = \$19,000 – \$10,000 = **\$9,000**

### Imperfect Information

Most information sources (such as market research or technical forecasts) are not always accurate. This is **imperfect information**—useful, but not guaranteed. The value of imperfect information will always be less than or equal to the value of perfect information.

Calculation steps for the value of imperfect information:

1. **Calculate the revised probabilities** that the information source signals each possible outcome, using Bayes’ Theorem if required.
2. **Find the expected value with imperfect information,** taking into account the accuracy of the information.
3. **Subtract the expected value without information** from the expected value with imperfect information.

### **Worked Example 1.3**

Suppose a company can pay for a market report predicting strong or weak market demand for a product. The report is 80% reliable—it correctly identifies the true state 8 times out of 10.

- If demand is strong, profit = \$100,000; probability = 0.5.
- If demand is weak, profit = \$20,000; probability = 0.5.
- The report costs \$5,000.

**What is the value of imperfect information?**

> **Answer:**
>
> Step 1: Without any information, choose the option with the highest EV:
> EV = (0.5 × \$100,000) + (0.5 × \$20,000) = \$50,000 + \$10,000 = \$60,000
> Step 2: With the imperfect report, calculate the revised EV:
>
> - Probability that the report predicts strong = (0.8 × 0.5) + (0.2 × 0.5) = 0.5
> - Probability actual demand is strong given a “strong” report = (0.8 × 0.5) / 0.5 = 0.8
>   If the report says strong: 80% chance of strong demand (\$100,000), 20% weak (\$20,000).
>   EV if “strong” report: (0.8 × \$100,000) + (0.2 × \$20,000) = \$80,000 + \$4,000 = \$84,000
>   If the report says weak: 20% chance strong (\$100,000), 80% weak (\$20,000).
>   EV if “weak” report: (0.2 × \$100,000) + (0.8 × \$20,000) = \$20,000 + \$16,000 = \$36,000
>   Since the report says “strong” half the time and “weak” half the time:
>   EV with report: (0.5 × \$84,000) + (0.5 × \$36,000) = \$42,000 + \$18,000 = \$60,000
>   Value of imperfect information = \$60,000 – \$60,000 = \$0
>   (In this scenario, the imperfect report does not add value, but in many questions, with close probabilities or outcomes, imperfect information adds positive value.)
>
> **Exam Warning**
> Remember: The value of imperfect information can never exceed the value of perfect information. Do not forget to _deduct the cost_ of purchasing information when deciding if it is worthwhile.

## DECISION TREES AND INFORMATION VALUES

Decision trees are used to structure multi-stage decisions, showing possible actions, events, probabilities, and payoffs. Rolling back the tree allows management to compare expected values at each stage and select the best option.

> **Key Term: decision tree**
> A visual representation of decision options and their possible consequences, including chance events, probabilities, and payoffs.

Each branch of a decision tree ends in a payoff and is weighted by the probability of the event occurring. The expected value at a decision point is calculated by multiplying each possible payoff by its probability and summing the results.

### **Worked Example 1.4**

A business can market a new service or not. If marketed, there is a 60% chance the service is successful (\$50,000 profit) and a 40% chance it fails (\$10,000 loss). Market research (cost \$6,000) is available and 90% accurate. Should the business buy the research?

> **Answer:**
>
> - Calculate the EV without research: (0.6 × \$50,000) + (0.4 × (–\$10,000)) = \$30,000 – \$4,000 = \$26,000
> - Calculate the EV with perfect information:
>   - If the market is successful, always market: \$50,000
>   - If not, do not market: \$0
>   - EV = (0.6 × \$50,000) + (0.4 × \$0) = \$30,000
>   - Value of perfect information: \$30,000 – \$26,000 = \$4,000
> - Calculate the EV with imperfect (90%-accurate) information:
>   - If research predicts success: 0.9 × 0.6 + 0.1 × 0.4 = 0.54 + 0.04 = 0.58
>   - Probability research predicts success = 0.58
>   - EV if “success” predicted: (0.9 × \$50,000) + (0.1 × (–\$10,000)) = \$45,000 – \$1,000 = \$44,000
>   - If research predicts failure: 0.1 × 0.6 + 0.9 × 0.4 = 0.06 + 0.36 = 0.42
>   - EV if “failure” predicted: (0.1 × \$50,000) + (0.9 × (–\$10,000)) = \$5,000 – \$9,000 = –\$4,000 (do not market, so choose \$0 profit)
>   - Total EV with information: (0.58 × \$44,000) + (0.42 × \$0) = \$25,520 + \$0 = \$25,520
>   - Value of imperfect information: \$25,520 – \$26,000 = –\$480 (negative, so do not buy the research, do not proceed)
>     In practice, recalculate with different costs or probabilities as needed.

## **Summary**

The value of perfect or imperfect information is the maximum gain that can be achieved from improved decision making, not including the cost of acquiring that information. Calculating this value helps managers decide if additional information is worth purchasing. Always compare expected value with and without information, and subtract information costs when making the final judgment.

## **Key Point Checklist**

_This article has covered the following key knowledge points:_

- Define expected value, perfect information, and imperfect information in decision making
- Calculate expected values for uncertain events
- Calculate and interpret the value of perfect and imperfect information
- Apply decision tree analysis to multi-stage decisions under uncertainty
- Explain why the value of imperfect information is always less than or equal to that of perfect information

## **Key Terms and Concepts**

- expected value
- perfect information
- imperfect information
- value of perfect information (VPI)
- value of imperfect information (VII)
- decision tree


Quantifying uncertainty - Value of perfect and imperfect information