PastPaperHero | Measuring risk and return - Expected return, variance, and standard deviation

Learning Outcomes

After completing this article, you will be able to define and calculate expected return, variance, and standard deviation for individual assets and portfolios. You will understand how these measures are applied in evaluating investment risk and return, interpret what they mean for decision-making, and explain their importance in assessing investments in line with the ACCA Financial Management exam.

ACCA Financial Management (FM) Syllabus

For ACCA Financial Management (FM), you are required to understand how to analyse and quantify risk and return in investment decisions. In this article, focus your revision on:

Defining and calculating expected return for investments
Computing variance and standard deviation as measures of investment risk
Interpreting what these measures indicate about risk and return
Applying these calculations to single assets and basic portfolios
Explaining the relevance of these statistics for financial management decisions

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.

What does a higher standard deviation tell you about the returns from an investment?
Which formula is used to compute the expected return for a set of possible investment outcomes and their probabilities?
True or false? Variance can be negative.
Explain in your own words why a financial manager is interested in both expected return and standard deviation when comparing two projects.
If an asset's possible returns are 5% (p=0.4), 10% (p=0.5), and 15% (p=0.1), what is the expected return?

Introduction

Assessing investments means looking beyond just potential returns—evaluating risk is just as important. The key statistical measures to quantify risk and return are expected return, variance, and standard deviation. These provide a basis for judging how much reward to expect, and how much uncertainty is involved. Being able to interpret and calculate these measures is essential for the ACCA exam and real-world financial decision-making.

Key Term: expected return
The average, probability-weighted value of all possible future returns for an investment, representing what an investor can generally expect to earn.

Key Term: variance
A measure of the spread of possible investment returns around the expected return. It quantifies how much actual results are likely to differ from the average.

Key Term: standard deviation
The square root of variance. It expresses the average amount returns differ from the expected value, using the same units as the returns themselves.

Measuring Return: Expected Return

The expected return is a statistical forecast of average outcome, based on all possible scenarios weighted by their likelihood. For ACCA purposes, expected return is calculated as:

$E(R) = \sum\_{i=1}^n p_i \times r_i$

Where:

$E(R)$ = expected return
$p_i$ = probability of outcome $i$
$r_i$ = return in outcome $i$
$n$ = total number of outcomes

The expected return is not a guarantee—it represents the mean outcome over many trials.

Worked Example 1.1

A project has three possible returns: 6% (probability 0.2), 10% (probability 0.5), and 15% (probability 0.3). Calculate the expected return.

Answer:
$E(R) = (0.2 \times 6\%) + (0.5 \times 10\%) + (0.3 \times 15\%) = 1.2\% + 5\% + 4.5\% = 10.7\%$ The expected return is 10.7%.

Measuring Risk: Variance and Standard Deviation

Variance and standard deviation evaluate how much returns fluctuate around the expected value—this is the basis of risk in financial management.

Variance

Variance is the average of squared differences between each possible return and the expected return, weighted by probability:

$\text{Variance} = \sum\_{i=1}^n p_i \times (r_i - E(R))^2$

A higher variance means returns are more unpredictable.

Standard Deviation

Standard deviation is the square root of variance:

$\text{Standard deviation} = \sqrt{\text{Variance}}$

This restores the units back to percentage (or monetary units), making the result more interpretable.

Worked Example 1.2

Using the previous example (possible returns: 6%, 10%, 15%, probabilities: 0.2, 0.5, 0.3, expected return 10.7%), calculate the variance and standard deviation.

Answer:

Compute squared deviations:

(6% - 10.7%)² = (–4.7%)² = 22.09

(10% - 10.7%)² = (–0.7%)² = 0.49

(15% - 10.7%)² = 4.3%² = 18.49

Weight by probability and sum:

0.2 × 22.09 = 4.418

0.5 × 0.49 = 0.245

0.3 × 18.49 = 5.547

Total variance = 4.418 + 0.245 + 5.547 = 10.21

Standard deviation = √10.21 = 3.2% The variance is 10.21 (squared percent), and the standard deviation is 3.2%.

Why Use Standard Deviation?

Standard deviation tells you, on average, how far the actual return could differ from the expected return. An investment with a higher standard deviation is riskier—even if it has a high expected return.

Exam Warning Never report variance as negative—it is always zero or positive, since it is based on squared differences. If you get a negative variance, check your calculations for errors.

Risk and Return for Portfolios (Introduction)

In practice, investors often hold more than one investment at a time. The total expected return and risk of a portfolio depend not just on the statistics for each asset, but also on how returns move in relation to each other (correlation). Portfolio risk may be less than the weighted average of individual risks due to diversification.

Worked Example 1.3

Suppose you have a portfolio made up of 50% Asset X (expected return 8%, standard deviation 4%) and 50% Asset Y (expected return 12%, standard deviation 8%). If the assets are uncorrelated (for basic exam purposes), what is the portfolio’s expected return?

Answer:
Portfolio expected return = (0.5 × 8%) + (0.5 × 12%) = 4% + 6% = 10%. Portfolio risk (standard deviation) requires further calculations involving correlation, but expected return is the weighted average.

Interpreting Risk and Return Measures

Expected return tells you the average outcome.
Standard deviation quantifies risk: how spread out returns could be.
Investments with high standard deviation are riskier, even if they promise higher returns.
For two projects with the same expected return, the one with lower standard deviation is less risky.

Revision Tip

Focus on understanding the formulas and practicing with small probability tables—being comfortable with expected return and standard deviation will save time under exam pressure.

Summary

Expected return shows the average profit you expect, while variance and standard deviation show how unpredictable the results are likely to be. These measures are essential for analysing investments and making decisions about risk versus reward.

Key Point Checklist

This article has covered the following key knowledge points:

Define expected return, variance, and standard deviation
Calculate expected return given possible outcomes and probabilities
Compute variance and standard deviation for risky assets
Interpret standard deviation as a measure of risk
Apply these concepts to both individual assets and simple portfolios
Understand why both risk and return matter for investment decisions

Key Terms and Concepts

expected return
variance
standard deviation

Measuring risk and return - Expected return, variance, and s...