Measuring risk and return - Covariance, correlation, and div...

Learning Outcomes

After studying this article, you will be able to explain the relationship between risk and return for investments, calculate and interpret covariance and correlation, and apply the principle of diversification to reduce portfolio risk. You will also understand how these statistical measures underpin modern portfolio theory and inform practical investment decisions.

ACCA Financial Management (FM) Syllabus

For ACCA Financial Management (FM), you are required to understand how risk and return are measured and managed. In particular, focus your revision on:

• The nature and types of investment risk
• The calculation and interpretation of covariance and correlation between asset returns
• The role of diversification in risk management
• Portfolio risk reduction through combining assets with different risk profiles
• The theoretical basis and practical implications of portfolio theory for exam scenarios

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. Which statement best describes diversification in investment?
   1. It eliminates all risk.
   2. It reduces portfolio risk by combining assets whose returns are not perfectly correlated.
   3. It guarantees higher returns.
   4. It requires investing only in government bonds.
2. How is correlation between two assets’ returns typically measured?
   1. As a percentage of variance explained.
   2. As a value between -1 and +1.
   3. As a ratio between mean and variance.
   4. As a probability value between 0 and 1.

3. True or false? If two assets have negative covariance, when one asset's return increases, the other's is likely to decrease.

4. Define covariance in the context of portfolio returns.

5. Explain why a portfolio of more than one asset can be less risky than holding a single asset.

Introduction

Investment decisions are shaped by the trade-off between risk and return. Understanding how different assets interact in a portfolio is key to making informed choices that align with shareholder wealth maximisation. This article examines how risk is measured, the significance of covariance and correlation, and how investors can exploit diversification to manage uncertainty.

Key Term: risk
The possibility that the actual return on an investment will differ from its expected return, often quantified as variability (variance or standard deviation) of returns.

MEASURING RISK IN INVESTMENTS

Investors assess the uncertainty of future returns using statistical measures. Two important measures are variance and standard deviation, both of which gauge the extent to which asset returns fluctuate around their expected values. However, when assets are combined in a portfolio, understanding how their returns move in relation to each other becomes critical.

Key Term: variance
A measure of how widely asset returns are dispersed from their mean, calculated as the average squared deviation from the mean.

Key Term: standard deviation
The square root of variance, representing the typical distance of returns from the average; commonly used as a measure of total risk.

COVARIANCE AND CORRELATION: UNDERSTANDING RETURN RELATIONSHIPS

Covariance

Covariance measures how two assets' returns move together. If the returns on two assets tend to rise or fall at the same time, their covariance is positive. If one rises while the other falls, covariance is negative.

Key Term: covariance
A statistical value indicating the degree to which two assets' returns move together; positive values indicate returns move in the same direction, negative values suggest inverse movement.

Correlation

While covariance indicates direction, correlation standardises this relationship, showing both strength and direction on a scale from -1 to +1.

Key Term: correlation
A statistical measure reflecting both the strength and direction of relationship between two assets' returns, ranging from -1 (perfectly negative) to +1 (perfectly positive).

Calculation and Interpretation

• Covariance requires knowledge of paired returns over time. It is not easily compared across asset pairs due to dependence on units.
• Correlation adjusts covariance by dividing by both assets' standard deviations, creating a dimensionless measure.

A correlation of:

• +1: perfect positive relationship (returns always move together)
• -1: perfect negative relationship (when one asset rises, the other falls)
• 0: no linear relationship (returns move independently)

Most real-world asset correlations lie between these extremes.

Worked Example 1.1

An investor is considering adding Asset A and Asset B to a portfolio. Over three years, their annual returns are as follows:

Calculate the correlation between Asset A and Asset B returns.

Answer:

1. Compute means: A = 9%, B = 10%.
2. Find each year's deviations from mean, multiply deviations for each year, sum results.
3. Divide sum by number of years for covariance.
4. Calculate standard deviations for A and B.
5. Correlation = covariance ÷ (std. dev. of A × std. dev. of B).

A positive correlation indicates returns move broadly together, though not perfectly.

DIVERSIFICATION AND PORTFOLIO RISK

Diversification means investing in different assets to reduce total risk. The effectiveness depends on how asset returns interact.

• If two assets are perfectly positively correlated (correlation = +1), combining them does not reduce risk.
• If two assets are negatively correlated (correlation = -1), a portfolio can, in theory, eliminate risk entirely.
• Most assets have correlations between these extremes; diversification still reduces risk, but cannot remove it.

Key Term: diversification
The strategy of spreading investments across assets with differing return patterns to reduce total portfolio risk.

Key Term: portfolio risk
The combined risk of all assets in a portfolio, dependent on individual asset risks and the correlations between them.

The Impact of Diversification

Adding assets with low or negative correlations to a portfolio results in lower risk compared to the average risk of individual assets. This is because negative or low correlations mean that poor performance in one asset may be offset by better performance in another.

Worked Example 1.2

Suppose you hold two shares: X and Y. Each alone has a standard deviation (risk) of 20%, but their correlation is –0.2. Will the combined portfolio be less risky than holding only one?

Answer:
Yes. The lower (or negative) the correlation, the more effective diversification becomes. In this case, combining X and Y results in portfolio risk lower than 20%. The greater the negative correlation, the more risk is reduced.

Portfolio Size and Remaining Risk

While adding more assets reduces risk, not all risks can be eliminated. Risks unique to a business (unsystematic risk) can be nearly removed, but risks shared by all companies (systematic risk) remain.

Key Term: systematic risk
Risk affecting the entire market and cannot be diversified away (e.g., recession, interest rate changes).

Key Term: unsystematic risk
Risk unique to a single company or industry, which can be reduced through diversification.

Summary

Measuring risk and return in investments requires more than considering individual assets. Covariance and correlation reveal how asset returns behave together, guiding effective diversification. By combining assets with less than perfect positive correlation, overall portfolio risk is reduced. This highlights the practical importance of portfolio construction for managing investment risk.

Key Point Checklist

This article has covered the following key knowledge points:

• Define risk, variance, standard deviation, covariance, and correlation in the context of investments
• Calculate and interpret covariance and correlation between asset returns
• Explain the effect of correlation on portfolio risk
• Describe systematic and unsystematic risk and the rationale for diversification
• Apply the concepts of diversification to construct less risky portfolios

Key Terms and Concepts

• risk
• variance
• standard deviation
• covariance
• correlation
• diversification
• portfolio risk
• systematic risk
• unsystematic risk