## **Learning Outcomes**

After reading this article, you will be able to distinguish between risk and uncertainty, apply sensitivity analysis to investment appraisal, explain and calculate expected values using scenario (probability) analysis, and describe the use and interpretation of simulation techniques. You will understand how these methods support project selection and risk management in investment decisions for the ACCA FM exam.

## **ACCA Financial Management (FM) Syllabus**

For ACCA Financial Management (FM), you are required to understand how to identify, evaluate, and manage risk in investment appraisal under conditions of uncertainty. In particular, you should be able to:

- Explain the difference between risk and uncertainty in investment project evaluation
- Apply sensitivity analysis to assess project risk and identify key variables
- Use probability analysis to estimate expected values and decision-making
- Discuss and interpret the use of simulation for dealing with multiple risk variables
- Evaluate the usefulness and limitations of these risk analysis techniques for 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. How does sensitivity analysis help a financial manager assess investment project risk?
2. What is the formula for expected value (EV), and what does it represent in project appraisal?
3. True or false? Simulation enables the assessment of the combined effect of changes in multiple variables at once.
4. Briefly distinguish between risk and uncertainty in the context of investment appraisal.

## **Introduction**

Investment decisions are rarely certain. Project cash flows, costs, and even project lives are estimated, meaning that actual outcomes can deviate from forecasts. Risk analysis provides tools for financial managers to appraise projects considering these uncertainties. This article covers the main techniques you need to know for the ACCA FM exam: sensitivity analysis, scenario (probability) analysis, and simulation.

> **Key Term: risk**
> The chance that future outcomes will differ from predictions due to known or unknown factors, typically with quantifiable probabilities.
>
> **Key Term: uncertainty**
> Any situation where future outcomes are not known and cannot be reliably assigned probabilities.

## **Risk, Uncertainty, and Key Variables**

Accurate investment appraisal requires more than calculating an NPV or IRR. You must consider:

- **Risk:** When it is possible to assign probabilities to possible outcomes (the impact of unknown events on project cash flows can be quantified).
- **Uncertainty:** When probabilities cannot be attached to all possible outcomes. Here, lack of information makes outcomes unpredictable.

Most project appraisals face a mixture of risk (quantified) and uncertainty (not quantifiable).

Projects can fail due to over-optimistic estimates of sales volume, selling price, costs, or due to adverse changes in economic conditions. Identifying the most critical estimates is central to risk analysis.

## **Sensitivity Analysis**

Sensitivity analysis tests how sensitive a project's NPV (or other measure) is to changes in key variables—such as sales price, volume, costs, or discount rate. In other words, it answers: "by how much can a variable change before the project becomes financially unacceptable?"

The results help managers focus attention and monitoring on the variables that matter most.

> **Key Term: sensitivity analysis**
> A technique that measures how the result of a project appraisal—such as NPV—will change when a key input variable is altered, keeping all others constant.

### Sensitivity Calculation

The sensitivity margin for a variable is typically calculated as:

$$
\text{Sensitivity (%)}
= \frac{\text{NPV}}{\text{Present Value of cash flow(s) affected by variable}} \times 100
$$

A low percentage means the project is highly sensitive—small changes may make it unviable.

### **Worked Example 1.1**

**A project has a four-year NPV of \$8,000. The present value of total expected sales revenue over the same period is \$400,000. By what percentage could sales revenue fall before the project would have a zero NPV?**

> **Answer:**
> The sensitivity to sales revenue is:
>
> $$
> > \frac{8,000}{400,000} \times 100 = 2\%
> $$
>
> So, if sales revenue fell by more than 2%, the project’s NPV would be zero.

### Strengths and Weaknesses

Sensitivity analysis:

- Is simple to apply and interpret
- Highlights which assumptions are most critical

However, it:

- Considers one variable at a time, ignoring the likelihood of simultaneous changes
- Does not assign probabilities or tell you the likelihood of a variable changing enough to cause project failure

### **Revision Tip**

Focus your calculations on the variables that drive major cash flows (e.g., revenue, major costs) — they typically have the largest impact on NPV.

## **Scenario and Probability Analysis**

While sensitivity analysis varies one input at a time, scenario (probability) analysis assesses outcomes under different combined assumptions and quantifies the expected result.

For example, a company might estimate “optimistic”, “most likely”, and “pessimistic” cash flows, each with an associated probability.

> **Key Term: expected value (EV)**
> The weighted average of possible outcomes, found by multiplying each outcome by its probability and summing the results.

### Expected Value Formula

$$ \text{EV} = \sum ( \text{Outcome} \times \text{Probability} ) $$

### **Worked Example 1.2**

**A firm is considering a project with the following NPVs and probabilities:**

- Probability 0.3: NPV = \$10,000
- Probability 0.5: NPV = \$5,000
- Probability 0.2: NPV = -\$2,000

**Calculate the expected value (EV) of the project.**

> **Answer:**
>
> $$
> > EV = (0.3 \times 10,000) + (0.5 \times 5,000) + (0.2 \times -2,000) \\
> EV = 3,000 + 2,500 - 400 \\
> EV = $5,100
> $$
>
> The weighted average NPV is \$5,100.

#### How to use EV results

A positive expected value suggests financial acceptability. However, decision-makers must consider the spread of possible outcomes (risk) and the probability of financial loss.

#### Standard Deviation

Standard deviation can be calculated to measure the "risk" (variability) of possible outcomes. A higher standard deviation means greater potential deviation from the EV.

#### Strengths and Weaknesses (Probability Analysis)

- EV analysis brings probabilities into project appraisal
- Provides a basis for long-term or frequent decision-making

However:

- Uses subjective probabilities—accuracy depends on forecasting quality
- The EV may not actually occur (especially for one-off projects)
- Ignores the dispersion or likelihood of extreme (bad) outcomes

## **Simulation**

Simulation goes further than sensitivity and scenario analysis by allowing multiple variables (e.g., both sales and costs) to change at once, each according to a specified probability distribution. This gives a probability distribution for project outcomes, not just a single expected value.

> **Key Term: simulation**
> A quantitative technique that uses mathematical models and random sampling to predict a range of possible outcomes based on varying multiple uncertain inputs simultaneously.

Simulation produces:

- A range of possible NPVs (not one outcome)
- The probability of achieving various results, helping decision-makers assess not just the average outcome, but the chance of failure or large losses

Simulation often requires special software and is more common for large, complex, or high-value projects.

### **Worked Example 1.3**

**A company uses computer simulation to test a project's NPV when both sales price and variable costs are uncertain. After 10,000 iterations, the simulation shows:**

- 70% probability of a positive NPV
- 30% probability of a negative NPV

**How can this help in project selection?**

> **Answer:**
> The simulation shows there is a significant (30%) risk of a financial loss. Decision-makers might set a minimum acceptable probability for success (e.g., at least 90%), or further analyze whether accepting this risk level matches company risk policy.

### Advantages and Disadvantages

Simulation:

- Offers a realistic picture when many variables are uncertain
- Quantifies risks across a range of possible outcomes
- Helps management see the probability of extreme losses

But:

- Relies on accurate estimates of variable distributions and relationships
- Can be costly and complex
- Interpretation requires understanding of probability and statistics

## **Summary**

Risk analysis is essential in project selection. Sensitivity analysis identifies critical assumptions, probability analysis quantifies expected outcomes and their variability, while simulation models multiple uncertainties at once. In practice, these tools complement — not replace — management’s judgment.

## **Key Point Checklist**

_This article has covered the following key knowledge points:_

- Define and distinguish risk and uncertainty in investment appraisal
- Explain and apply sensitivity analysis to investment projects
- Apply and interpret expected values in scenario/probability analysis
- Describe the use and value of simulation in modelling multiple uncertainties
- Evaluate the strengths and limitations of each risk analysis technique

## **Key Terms and Concepts**

- risk
- uncertainty
- sensitivity analysis
- expected value (EV)
- simulation

Risk analysis and project selection - Sensitivity, scenario, and simulation